Hybrid and Relaxed Mann Iterations for General Systems of ...

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Aug 10, 2013 - Variational Inequalities and Nonexpansive Mappings. L. C. Ceng,1 .... The norm of is said to be the Frechet differential if for each ∈  ...
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 102820, 44 pages http://dx.doi.org/10.1155/2013/102820

Research Article Hybrid and Relaxed Mann Iterations for General Systems of Variational Inequalities and Nonexpansive Mappings L. C. Ceng,1 A. E. Al-Mazrooei,2 A. A. N. Abdou,2 and A. Latif2 1

Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China 2 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Correspondence should be addressed to A. Latif; [email protected] Received 4 August 2013; Accepted 10 August 2013 Academic Editor: Jen-Chih Yao Copyright Β© 2013 L. C. Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce hybrid and relaxed Mann iteration methods for a general system of variational inequalities with solutions being also common solutions of a countable family of variational inequalities and common fixed points of a countable family of nonexpansive mappings in real smooth and uniformly convex Banach spaces. Here, the hybrid and relaxed Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method, and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for hybrid and relaxed Mann iteration algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gateaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

1. Introduction Let 𝑋 be a real Banach space whose dual space is denoted by βˆ— π‘‹βˆ— . The normalized duality mapping 𝐽 : 𝑋 β†’ 2𝑋 is defined by σ΅„© σ΅„©2 𝐽 (π‘₯) = {π‘₯βˆ— ∈ π‘‹βˆ— : ⟨π‘₯, π‘₯βˆ— ⟩ = β€–π‘₯β€–2 = σ΅„©σ΅„©σ΅„©π‘₯βˆ— σ΅„©σ΅„©σ΅„© } , βˆ€π‘₯ ∈ 𝑋, (1) where βŸ¨β‹…, β‹…βŸ© denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach theorem that 𝐽(π‘₯) is nonempty for each π‘₯ ∈ 𝑋. Let 𝐢 be a nonempty closed convex subset of 𝑋. A mapping 𝑇 : 𝐢 β†’ 𝐢 is called nonexpansive if ‖𝑇π‘₯ βˆ’ 𝑇𝑦‖ ≀ β€–π‘₯ βˆ’ 𝑦‖ for every π‘₯, 𝑦 ∈ 𝐢. The set of fixed points of 𝑇 is denoted by Fix(𝑇). We use the notation ⇀ to indicate the weak convergence and the one β†’ to indicate the strong convergence. A mapping 𝐴 : 𝐢 β†’ 𝑋 is said to be (i) accretive if for each π‘₯, 𝑦 ∈ 𝐢 there exists 𝑗(π‘₯ βˆ’ 𝑦) ∈ 𝐽(π‘₯ βˆ’ 𝑦) such that ⟨𝐴π‘₯ βˆ’ 𝐴𝑦, 𝑗 (π‘₯ βˆ’ 𝑦)⟩ β‰₯ 0;

(2)

(ii) 𝛼-strongly accretive if for each π‘₯, 𝑦 ∈ 𝐢 there exists 𝑗(π‘₯ βˆ’ 𝑦) ∈ 𝐽(π‘₯ βˆ’ 𝑦) such that σ΅„©2 σ΅„© ⟨𝐴π‘₯ βˆ’ 𝐴𝑦, 𝑗 (π‘₯ βˆ’ 𝑦)⟩ β‰₯ 𝛼󡄩󡄩󡄩π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 ,

(3)

for some 𝛼 ∈ (0, 1); (iii) 𝛽-inverse strongly accretive if, for each π‘₯, 𝑦 ∈ 𝐢, there exists 𝑗(π‘₯ βˆ’ 𝑦) ∈ 𝐽(π‘₯ βˆ’ 𝑦) such that σ΅„©2 σ΅„© ⟨𝐴π‘₯ βˆ’ 𝐴𝑦, 𝑗 (π‘₯ βˆ’ 𝑦)⟩ β‰₯ 𝛽󡄩󡄩󡄩𝐴π‘₯ βˆ’ 𝐴𝑦󡄩󡄩󡄩 ,

(4)

for some 𝛽 > 0; (iv) πœ†-strictly pseudocontractive [1] (see also [2]) if for each π‘₯, 𝑦 ∈ 𝐢 there exists 𝑗(π‘₯ βˆ’ 𝑦) ∈ 𝐽(π‘₯ βˆ’ 𝑦) such that σ΅„©2 σ΅„©2 σ΅„© σ΅„© ⟨𝐴π‘₯ βˆ’ 𝐴𝑦, 𝑗 (π‘₯ βˆ’ 𝑦)⟩ ≀ σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 βˆ’ πœ†σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦 βˆ’ (𝐴π‘₯ βˆ’ 𝐴𝑦)σ΅„©σ΅„©σ΅„© (5) for some πœ† ∈ (0, 1).

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Abstract and Applied Analysis

It is worth emphasizing that the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, for example, [3–5]. Let π‘ˆ = {π‘₯ ∈ 𝑋 : β€–π‘₯β€– = 1} denote the unite sphere of 𝑋. A Banach space 𝑋 is said to be uniformly convex if, for each πœ– ∈ (0, 2], there exists 𝛿 > 0 such that, for all π‘₯, 𝑦 ∈ π‘ˆ, σ΅„© σ΅„©σ΅„© σ΅„©π‘₯ + 𝑦󡄩󡄩󡄩 σ΅„© σ΅„©σ΅„© ≀ 1 βˆ’ 𝛿. σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 β‰₯ πœ– 󳨐⇒ σ΅„© 2

(6)

It is known that a uniformly convex Banach space is reflexive and strict convex. A Banach space 𝑋 is said to be smooth if the limit σ΅„©σ΅„©σ΅„©π‘₯ + 𝑑𝑦󡄩󡄩󡄩 βˆ’ β€–π‘₯β€– σ΅„© (7) lim σ΅„© 𝑑→0 𝑑 exists for all π‘₯, 𝑦 ∈ π‘ˆ; in this case, 𝑋 is also said to have a Gateaux differentiable norm. 𝑋 is said to have a uniformly, Gateaux differentiable norm if, for each 𝑦 ∈ π‘ˆ, the limit is attained uniformly for π‘₯ ∈ π‘ˆ. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for π‘₯, 𝑦 ∈ π‘ˆ. The norm of 𝑋 is said to be the Frechet differential if for each π‘₯ ∈ π‘ˆ, this limit is attained uniformly for 𝑦 ∈ π‘ˆ. In the meantime, we define a function 𝜌 : [0, ∞) β†’ [0, ∞) called the modulus of smoothness of 𝑋 as follows: 1 σ΅„© σ΅„© σ΅„© σ΅„© 𝜌 (𝜏) = sup { (σ΅„©σ΅„©σ΅„©π‘₯ + 𝑦󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩) βˆ’ 1 : π‘₯, 𝑦 ∈ 𝑋, 2 σ΅„© σ΅„© β€–π‘₯β€– = 1, 󡄩󡄩󡄩𝑦󡄩󡄩󡄩 = 𝜏} .

(8)

(i) 1 ≀ 𝛽𝑛 ≀ 1 βˆ’ 𝜌, for all 𝑛 β‰₯ 𝑛0 for some integer 𝑛0 β‰₯ 1, (ii) βˆ‘βˆž 𝑛=0 𝛽𝑛 = ∞, (iii) 0 < lim inf 𝑛 β†’ ∞ 𝛼𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛼𝑛 < 1, (iv) lim𝑛 β†’ ∞ (𝛽𝑛+1 /(1 βˆ’ (1 βˆ’ 𝛽𝑛+1 )𝛼𝑛+1 ) βˆ’ 𝛽𝑛 /(1 βˆ’ (1 βˆ’ 𝛽𝑛 )𝛼𝑛 )) = 0. Ceng and Yao [8] proved that π‘₯𝑛 󳨀→ π‘ž ⇐⇒ 𝛽𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 ) 󳨀→ 0,

(9)

where π‘ž ∈ Fix(𝑇) solves the variational inequality problem (VIP): βŸ¨π‘ž βˆ’ 𝑓 (π‘ž) , 𝐽 (π‘ž βˆ’ 𝑝)⟩ ≀ 0,

βˆ€π‘ ∈ Fix (𝑇) .

(10)

Such a result includes [7, Theorem 1] as a special case. Let 𝐢 be a nonempty closed convex subset of a real Banach space 𝑋 and 𝑓 ∈ Ξ𝐢 with a contractive coefficient 𝜌 ∈ (0, 1), where Ξ𝐢 is the set of all contractive self-mappings on 𝐢. Let {𝑇𝑛 }∞ 𝑛=0 be a sequence of nonexpansive self-mappings on 𝐢 and {πœ† 𝑛 }∞ 𝑛=0 a sequence of nonnegative numbers in [0, 1]. For any 𝑛 β‰₯ 0, define a self-mapping π‘Šπ‘› on 𝐢 as follows: π‘ˆπ‘›,𝑛+1 = 𝐼, π‘ˆπ‘›,𝑛 = πœ† 𝑛 𝑇𝑛 π‘ˆπ‘›,𝑛+1 + (1 βˆ’ πœ† 𝑛 ) 𝐼,

It is known that 𝑋 is uniformly smooth if and only if lim𝜏 β†’ 0 𝜌(𝜏)/𝜏 = 0. Let π‘ž be a fixed real number with 1 < π‘ž ≀ 2. Then, a Banach space 𝑋 is said to be π‘ž-uniformly smooth if there exists a constant 𝑐 > 0 such that 𝜌(𝜏) ≀ π‘πœπ‘ž for all 𝜏 > 0. As pointed out in [6], no Banach space is π‘ž-uniformly smooth for π‘ž > 2. In addition, it is also known that 𝐽 is single valued if and only if 𝑋 is smooth, whereas if 𝑋 is uniformly smooth, then the mapping 𝐽 is norm-to-norm uniformly continuous on bounded subsets of 𝑋. If 𝑋 has a uniformly Gateaux differentiable norm, then the duality mapping 𝐽 is norm-toweakβˆ— uniformly continuous on bounded subsets of 𝑋. Recently, Yao et al. [7] combined the viscosity approximation method and Mann iteration method and gave the following hybrid viscosity approximation method. Let 𝐢 be a nonempty closed convex subset of a real uniformly smooth Banach space 𝑋, 𝑇 : 𝐢 β†’ 𝐢 a nonexpansive mapping with Fix(𝑇) =ΜΈ 0, and 𝑓 : 𝐢 β†’ 𝐢 a contraction with coefficient 𝜌 ∈ (0, 1). For an arbitrary π‘₯0 ∈ 𝐢, define {π‘₯𝑛 } in the following way: 𝑦𝑛 = 𝛼𝑛 π‘₯𝑛 + (1 βˆ’ 𝛼𝑛 ) 𝑇π‘₯𝑛 , π‘₯𝑛+1 = 𝛽𝑛 𝑓 (π‘₯𝑛 ) + (1 βˆ’ 𝛽𝑛 ) 𝑦𝑛 ,

a fixed point of 𝑇. Subsequently, under the following control conditions on {𝛼𝑛 } and {𝛽𝑛 }:

βˆ€π‘› β‰₯ 0,

(YCY)

where {𝛼𝑛 } and {𝛽𝑛 } are two sequences in (0, 1). They proved under certain control conditions on the sequences {𝛼𝑛 } and {𝛽𝑛 } that {π‘₯𝑛 } converges strongly to

π‘ˆπ‘›,π‘›βˆ’1 = πœ† π‘›βˆ’1 π‘‡π‘›βˆ’1 π‘ˆπ‘›,𝑛 + (1 βˆ’ πœ† π‘›βˆ’1 ) 𝐼, .. . π‘ˆπ‘›,π‘˜ = πœ† π‘˜ π‘‡π‘˜ π‘ˆπ‘›,π‘˜+1 + (1 βˆ’ πœ† π‘˜ ) 𝐼,

(CY)

π‘ˆπ‘›,π‘˜βˆ’1 = πœ† π‘˜βˆ’1 π‘‡π‘˜βˆ’1 π‘ˆπ‘›,π‘˜ + (1 βˆ’ πœ† π‘˜βˆ’1 ) 𝐼, .. . π‘ˆπ‘›,1 = πœ† 1 𝑇1 π‘ˆπ‘›,2 + (1 βˆ’ πœ† 1 ) 𝐼, π‘Šπ‘› = π‘ˆπ‘›,0 = πœ† 0 𝑇0 π‘ˆπ‘›,1 + (1 βˆ’ πœ† 0 ) 𝐼. Such a mapping π‘Šπ‘› is called the π‘Š-mapping generated by 𝑇𝑛 , π‘‡π‘›βˆ’1 , . . . , 𝑇0 , and πœ† 𝑛 , πœ† π‘›βˆ’1 , . . . , πœ† 0 ; see [9]. In 2008, Ceng and Yao [10] introduced and analyzed the following relaxed viscosity approximation method for finding a common fixed point of an infinite family of nonexpansive mappings in a strictly convex and reflexive Banach space with a uniformly Gateaux differentiable norm. Theorem 1 (see [10]). Let 𝑋 be a strictly convex and reflexive Banach space with a uniformly Gateaux differentiable norm, 𝐢 a nonempty closed convex subset of 𝑋, {𝑇𝑛 }∞ 𝑛=0 a sequence of nonexpansive self-mappings on 𝐢 such that the common fixed point set 𝐹 := β‹‚βˆž 𝑛=0 Fix(𝑇𝑛 ) =ΜΈ 0, and 𝑓 ∈ Ξ𝐢 with a contractive

Abstract and Applied Analysis

3

coefficient 𝜌 ∈ (1/2, 1). For any given π‘₯0 ∈ 𝐢, let {π‘₯𝑛 }∞ 𝑛=0 be the iterative sequence defined by 𝑦𝑛 = (1 βˆ’ 𝛾𝑛 ) π‘₯𝑛 + 𝛾𝑛 π‘Šπ‘› π‘₯𝑛 , π‘₯𝑛+1 = (1 βˆ’ 𝛼𝑛 βˆ’ 𝛽𝑛 ) π‘₯𝑛 + 𝛼𝑛 𝑓 (𝑦𝑛 ) + 𝛽𝑛 π‘Šπ‘› 𝑦𝑛 ,

βˆ€π‘› β‰₯ 0, (11)

∞ where {𝛼𝑛 }∞ 𝑛=0 and {𝛽𝑛 }𝑛=0 are two sequences in (0, 1) with 𝛼𝑛 + ∞ 𝛽𝑛 ≀ 1 (𝑛 β‰₯ 0), {𝛾𝑛 }𝑛=0 is a sequence in [0, 1], and π‘Šπ‘› is the π‘Š-mapping generated by (CY). Assume that

(i) lim𝑛 β†’ ∞ 𝛼𝑛 = 0, βˆ‘βˆž = ∞ and 0 𝑛=0 𝛼𝑛 lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1;


0 a given number, which is known as the extragradient method (see also [16]). The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [3, 11, 13, 17–33] and references therein, to name but a few. In particular, whenever 𝑋 is still a real smooth Banach space, 𝐡1 = 𝐡2 = 𝐴 and π‘₯βˆ— = π‘¦βˆ— , then GSVI (13) reduces to the variational inequality problem (VIP) of finding π‘₯βˆ— ∈ 𝐢 such that ⟨𝐴π‘₯βˆ— , 𝐽 (π‘₯ βˆ’ π‘₯βˆ— )⟩ β‰₯ 0,

βˆ€π‘₯ ∈ 𝐢,

(18)

which was considered by Aoyama et al. [34]. Note that VIP (18) is connected with the fixed point problem for nonlinear mapping (see, e.g., [35]), the problem of finding a zero point of a nonlinear operator (see, e.g., [36]), and so on. It is clear that VIP (18) extends VIP (15) from Hilbert spaces to Banach spaces. In order to find a solution of VIP (18), Aoyama et al. [34] introduced the following Mann-type iterative scheme for an accretive operator 𝐴: π‘₯𝑛+1 = 𝛼𝑛 π‘₯𝑛 + (1 βˆ’ 𝛼𝑛 ) Π𝐢 (π‘₯𝑛 βˆ’ πœ† 𝑛 𝐴π‘₯𝑛 ) ,

βˆ€π‘› β‰₯ 1, (19)

where Π𝐢 is a sunny nonexpansive retraction from 𝑋 onto 𝐢. Then, they proved a weak convergence theorem. For the related work, see [37] and the references therein. Let 𝐢 be a nonempty convex subset of a real Banach space 𝑋. Let {𝑇𝑖 }𝑁 𝑖=1 be a finite family of nonexpansive mappings of

4

Abstract and Applied Analysis

𝐢 into itself and let πœ† 1 , . . . , πœ† 𝑁 be real numbers such that 0 ≀ πœ† 𝑖 ≀ 1 for every 𝑖 = 1, . . . , 𝑁. Define a mapping 𝐾 : 𝐢 β†’ 𝐢 as follows: π‘ˆ1 = πœ† 1 𝑇1 + (1 βˆ’ πœ† 1 ) 𝐼, π‘ˆ2 = πœ† 2 𝑇2 π‘ˆ1 + (1 βˆ’ πœ† 2 ) π‘ˆ1 , π‘ˆ3 = πœ† 3 𝑇3 π‘ˆ2 + (1 βˆ’ πœ† 3 ) π‘ˆ2 , (20)

.. .

Beyond doubt, it is an interesting and valuable problem of constructing some algorithms with strong convergence for solving GSVI (13) which contains VIP (18) as a special case. Very recently, Cai and Bu [11] constructed an iterative algorithm for solving GSVI (13) and a common fixed point problem of a countable family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a 2uniformly smooth Banach space 𝑋. Lemma 4 (see [39]). Let 𝑋 be a 2-uniformly smooth Banach space. Then,

π‘ˆπ‘βˆ’1 = πœ† π‘βˆ’1 π‘‡π‘βˆ’1 π‘ˆπ‘βˆ’2 + (1 βˆ’ πœ† π‘βˆ’1 ) π‘ˆπ‘βˆ’2 , 𝐾 = π‘ˆπ‘ = πœ† π‘π‘‡π‘π‘ˆπ‘βˆ’1 + (1 βˆ’ πœ† 𝑁) π‘ˆπ‘βˆ’1 . Such a mapping 𝐾 is called the 𝐾-mapping generated by 𝑇1 , . . . , 𝑇𝑁 and πœ† 1 , . . . , πœ† 𝑁. Very recently, Kangtunyakarn [38] introduced and analyzed an iterative algorithm by the modification of Mann’s iteration process for finding a common element of the set of solutions of a finite family of variational inequalities and the set of fixed points of an πœ‚-strictly pseudocontractive mapping and a nonexpansive mapping in uniformly convex and 2uniformly smooth Banach spaces.

σ΅„© σ΅„©2 σ΅„©2 σ΅„©σ΅„© 2 σ΅„©σ΅„©π‘₯ + 𝑦󡄩󡄩󡄩 ≀ β€–π‘₯β€– + 2 βŸ¨π‘¦, 𝐽 (π‘₯)⟩ + 2σ΅„©σ΅„©σ΅„©πœ…π‘¦σ΅„©σ΅„©σ΅„© ,

βˆ€π‘₯, 𝑦 ∈ 𝑋, (23)

where πœ… is the 2-uniformly smooth constant of 𝑋 and 𝐽 is the normalized duality mapping from 𝑋 into π‘‹βˆ— . Define the mapping 𝐺 : 𝐢 β†’ 𝐢 as follows: 𝐺 (π‘₯) := Π𝐢 (𝐼 βˆ’ πœ‡1 𝐡1 ) Π𝐢 (𝐼 βˆ’ πœ‡2 𝐡2 ) π‘₯,

βˆ€π‘₯ ∈ 𝐢.

(24)

The fixed point set of 𝐺 is denoted by Ω. Then, their strong convergence theorem on the proposed method is stated as follows.

Theorem 3 (see [38]). Let 𝐢 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢. Let 𝐴 𝑖 : 𝐢 β†’ 𝑋 be an 𝛼𝑖 -inverse-strongly accretive mapping for each 𝑖 = 1, . . . , 𝑁. Define the mapping 𝐺𝑖 : 𝐢 β†’ 𝐢 by 𝐺𝑖 = Π𝐢(𝐼 βˆ’ πœ† 𝑖 𝐴 𝑖 ) for 𝑖 = 1, . . . , 𝑁, where πœ† 𝑖 ∈ (0, 𝛼𝑖 /πœ…2 ) and πœ… is the 2-uniformly smooth constant of 𝑋. Let 𝐡 : 𝐢 β†’ 𝐢 be the 𝐾-mapping generated by 𝐺1 , . . . , 𝐺𝑁 and 𝜌1 , . . . , πœŒπ‘, where πœŒπ‘– ∈ (0, 1), for all 𝑖 = 1, . . . , 𝑁 βˆ’ 1, and πœŒπ‘ ∈ (0, 1]. Let 𝑓 : 𝐢 β†’ 𝐢 a contraction with coefficient 𝜌 ∈ (0, 1). Let 𝑉 : 𝐢 β†’ 𝐢 be an πœ‚-strictly pseudocontractive mapping and 𝑆 : 𝐢 β†’ 𝐢 be a nonexpansive mapping such that 𝐹 = Fix(𝑆) ∩ Fix(𝑉) ∩ (⋂𝑁 𝑖=1 VI(𝐢, 𝐴 𝑖 )) =ΜΈ 0. For arbitrarily given π‘₯0 ∈ 𝐢, let {π‘₯𝑛 } be the sequence generated by

Theorem 5 (see [11]). Let 𝐢 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢. Let the mapping 𝐡𝑖 : 𝐢 β†’ 𝑋 be 𝛽𝑖 -inverse-strongly accretive with 0 < πœ‡π‘– < 𝛽𝑖 /πœ…2 for 𝑖 = 1, 2. Let 𝑓 be a contraction of 𝐢 into itself with coefficient 𝛿 ∈ (0, 1). Let {𝑇𝑛 }∞ 𝑛=1 be a countable family of nonexpansive mappings of 𝐢 into itself such that 𝐹 = β‹‚βˆž 𝑖=1 Fix(𝑇𝑖 ) ∩ Ξ© =ΜΈ 0, where Ξ© is the fixed point set of the mapping 𝐺 defined by (24). For arbitrarily given π‘₯1 ∈ 𝐢, let {π‘₯𝑛 } be the sequence generated by

π‘₯𝑛+1 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡π‘₯𝑛 + 𝛿𝑛 𝑆 ((1 βˆ’ 𝛼) 𝐼 + 𝛼𝑉) π‘₯𝑛 ,

𝑦𝑛 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + (1 βˆ’ 𝛼𝑛 ) 𝑧𝑛 ,

βˆ€π‘› β‰₯ 0, (21) 2

where 𝛼 ∈ (0, πœ‚/πœ… ). Suppose that {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, and {𝛿𝑛 } are the sequences in [0, 1], 𝛼𝑛 + 𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 and satisfy the following conditions:

𝑧𝑛 = Π𝐢 (𝑒𝑛 βˆ’ πœ‡1 𝐡1 𝑒𝑛 ) , 𝑒𝑛 = Π𝐢 (π‘₯𝑛 βˆ’ πœ‡2 𝐡2 π‘₯𝑛 ) ,

(25)

βˆ€π‘› β‰₯ 1.

Suppose that {𝛼𝑛 } and {𝛽𝑛 } are two sequences in (0, 1) satisfying the following conditions: (i) lim𝑛 β†’ ∞ 𝛼𝑛 = 0 and βˆ‘βˆž 𝑛=1 𝛼𝑛 = ∞;

(i) lim𝑛 β†’ ∞ 𝛼𝑛 = 0 and βˆ‘βˆž 𝑛=0 𝛼𝑛 = ∞;

(ii) 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1.

(ii) {𝛾𝑛 }, {𝛿𝑛 } βŠ‚ [𝑐, 𝑑] for some 𝑐, 𝑑 ∈ (0, 1);

(iii) βˆ‘βˆž 𝑛=1 (|𝛽𝑛 βˆ’ π›½π‘›βˆ’1 | + |𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 | + |𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 |) < ∞; (iv) 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1. Then, {π‘₯𝑛 } converges strongly to π‘ž ∈ 𝐹, which solves the following VIP: βŸ¨π‘ž βˆ’ 𝑓 (π‘ž) , 𝐽 (π‘ž βˆ’ 𝑝)⟩ ≀ 0,

π‘₯𝑛+1 = 𝛽𝑛 π‘₯𝑛 + (1 βˆ’ 𝛽𝑛 ) 𝑆𝑛 𝑦𝑛 ,

βˆ€π‘ ∈ 𝐹.

(22)

Assume that βˆ‘βˆž 𝑛=1 supπ‘₯∈𝐷 ‖𝑇𝑛+1 π‘₯βˆ’π‘‡π‘› π‘₯β€– < ∞ for any bounded subset 𝐷 of 𝐢 and let 𝑇 be a mapping of 𝐢 into 𝑋 defined by 𝑇π‘₯ = lim𝑛 β†’ ∞ 𝑇𝑛 π‘₯ for all π‘₯ ∈ 𝐢 and suppose that Fix(𝑇) = β‹‚βˆž 𝑛=1 Fix(𝑇𝑛 ). Then, {π‘₯𝑛 } converges strongly to π‘ž ∈ 𝐹, which solves the following VIP: βŸ¨π‘ž βˆ’ 𝑓 (π‘ž) , 𝐽 (π‘ž βˆ’ 𝑝)⟩ ≀ 0,

βˆ€π‘ ∈ 𝐹.

(26)

Abstract and Applied Analysis

5

It is easy to see that the iterative scheme in Theorem 5 is essentially equivalent to the following two-step iterative scheme: 𝑦𝑛 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + (1 βˆ’ 𝛼𝑛 ) 𝐺π‘₯𝑛 , π‘₯𝑛+1 = 𝛽𝑛 π‘₯𝑛 + (1 βˆ’ 𝛽𝑛 ) 𝑇𝑛 𝑦𝑛 ,

βˆ€π‘› β‰₯ 1.

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For the convenience of implementing the argument techniques in [14], the authors of [11] have used the following inequality in a real smooth and uniform convex Banach space 𝑋. Proposition 6 (see [40]). Let 𝑋 be a real smooth and uniform convex Banach space and let π‘Ÿ > 0. Then, there exists a strictly increasing, continuous, and convex function 𝑔 : [0, 2π‘Ÿ] β†’ R, 𝑔(0) = 0 such that σ΅„© σ΅„©2 σ΅„© σ΅„© 𝑔 (σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩) ≀ β€–π‘₯β€–2 βˆ’ 2 ⟨π‘₯, 𝐽 (𝑦)⟩ + 󡄩󡄩󡄩𝑦󡄩󡄩󡄩 ,

βˆ€π‘₯, 𝑦 ∈ π΅π‘Ÿ , (28)

where π΅π‘Ÿ = {π‘₯ ∈ 𝑋 : β€–π‘₯β€– ≀ π‘Ÿ}. Let 𝐢 be a nonempty closed convex subset of a real smooth Banach space 𝑋. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢 and 𝑓 : 𝐢 β†’ 𝐢 a contraction with coefficient 𝜌 ∈ (0, 1). Motivated and inspired by the research going on this area, we consider and introduce hybrid and relaxed Mann iteration methods for finding solutions of the GSVI (13) which are also common solutions of a countable family of variational inequalities and common fixed points of a countable family of nonexpansive mappings in 𝑋. Here, the hybrid and relaxed Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method, and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for hybrid and relaxed Mann iteration algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gateaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature; see, for example, [8, 10, 11, 14, 33, 38].

2. Preliminaries We list some lemmas that will be used in the sequel. Lemma 7 (see [41]). Let {𝑠𝑛 } be a sequence of nonnegative real numbers satisfying 𝑠𝑛+1 ≀ (1 βˆ’ 𝛼𝑛 ) 𝑠𝑛 + 𝛼𝑛 𝛽𝑛 + 𝛾𝑛 ,

βˆ€π‘› β‰₯ 0,

where {𝛼𝑛 }, {𝛽𝑛 }, and {𝛾𝑛 } satisfy the following conditions: (i) {𝛼𝑛 } βŠ‚ [0, 1] and

βˆ‘βˆž 𝑛=0

(ii) lim sup𝑛 β†’ ∞ 𝛽𝑛 ≀ 0;

𝛼𝑛 = ∞;

(iii) 𝛾𝑛 β‰₯ 0, for all 𝑛 β‰₯ 0, and βˆ‘βˆž 𝑛=0 𝛾𝑛 < ∞. Then, lim sup𝑛 β†’ ∞ 𝑠𝑛 = 0.

(29)

The following lemma is an immediate consequence of the subdifferential inequality of the function (1/2)β€– β‹… β€–2 . Lemma 8 (see [42]). Let 𝑋 be a real Banach space 𝑋. Then, for all π‘₯, 𝑦 ∈ 𝑋 (i) β€–π‘₯ + 𝑦‖2 ≀ β€–π‘₯β€–2 + 2βŸ¨π‘¦, 𝑗(π‘₯ + 𝑦)⟩ for all 𝑗(π‘₯ + 𝑦) ∈ 𝐽(π‘₯ + 𝑦); (ii) β€–π‘₯ + 𝑦‖2 β‰₯ β€–π‘₯β€–2 + 2βŸ¨π‘¦, 𝑗(π‘₯)⟩ for all 𝑗(π‘₯) ∈ 𝐽(π‘₯). Let 𝐷 be a subset of 𝐢 and let Ξ  be a mapping of 𝐢 into 𝐷. Then, Ξ  is said to be sunny if Ξ  [Ξ  (π‘₯) + 𝑑 (π‘₯ βˆ’ Ξ  (π‘₯))] = Ξ  (π‘₯) ,

(30)

whenever Ξ (π‘₯) + 𝑑(π‘₯ βˆ’ Ξ (π‘₯)) ∈ 𝐢 for π‘₯ ∈ 𝐢 and 𝑑 β‰₯ 0. A mapping Ξ  of 𝐢 into itself is called a retraction if Ξ 2 = Ξ . If a mapping Ξ  of 𝐢 into itself is a retraction, then Ξ (𝑧) = 𝑧 for every 𝑧 ∈ 𝑅(Ξ ) where 𝑅(Ξ ) is the range of Ξ . A subset 𝐷 of 𝐢 is called a sunny nonexpansive retract of 𝐢 if there exists a sunny nonexpansive retraction from 𝐢 onto 𝐷. The following lemma concerns the sunny nonexpansive retraction. Lemma 9 (see [43]). Let 𝐢 be a nonempty closed convex subset of a real smooth Banach space 𝑋. Let 𝐷 be a nonempty subset of 𝐢. Let Ξ  be a retraction of 𝐢 onto 𝐷. Then, the following are equivalent: (i) Ξ  is sunny and nonexpansive; (ii) β€–Ξ (π‘₯) βˆ’ Ξ (𝑦)β€–2 ≀ ⟨π‘₯ βˆ’ 𝑦, 𝐽(Ξ (π‘₯) βˆ’ Ξ (𝑦))⟩, for all π‘₯, 𝑦 ∈ 𝐢; (iii) ⟨π‘₯ βˆ’ Ξ (π‘₯), 𝐽(𝑦 βˆ’ Ξ (π‘₯))⟩ ≀ 0, for all π‘₯ ∈ 𝐢, 𝑦 ∈ 𝐷. It is well known that if 𝑋 = 𝐻 a Hilbert space, then a sunny nonexpansive retraction Π𝐢 is coincident with the metric projection from 𝑋 onto 𝐢; that is, Π𝐢 = 𝑃𝐢. If 𝐢 is a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space 𝑋 and if 𝑇 : 𝐢 β†’ 𝐢 is a nonexpansive mapping with the fixed point set Fix(𝑇) =ΜΈ 0, then the set Fix(𝑇) is a sunny nonexpansive retract of 𝐢. Lemma 10. Let 𝐢 be a nonempty closed convex subset of a smooth Banach space 𝑋. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢 and let 𝐡1 , 𝐡2 : 𝐢 β†’ 𝑋 be nonlinear mappings. For given π‘₯βˆ— , π‘¦βˆ— ∈ 𝐢, (π‘₯βˆ— , π‘¦βˆ— ) is a solution of GSVI (13) if and only if π‘₯βˆ— = Π𝐢(π‘¦βˆ— βˆ’ πœ‡1 𝐡1 π‘¦βˆ— ), where π‘¦βˆ— = Π𝐢(π‘₯βˆ— βˆ’ πœ‡2 𝐡2 π‘₯βˆ— ). Proof. We can rewrite GSVI (13) as ⟨π‘₯βˆ— βˆ’ (π‘¦βˆ— βˆ’ πœ‡1 𝐡1 π‘¦βˆ— ) , 𝐽 (π‘₯ βˆ’ π‘₯βˆ— )⟩ β‰₯ 0,

βˆ€π‘₯ ∈ 𝐢,

βŸ¨π‘¦βˆ— βˆ’ (π‘₯βˆ— βˆ’ πœ‡2 𝐡2 π‘₯βˆ— ) , 𝐽 (π‘₯ βˆ’ π‘¦βˆ— )⟩ β‰₯ 0,

βˆ€π‘₯ ∈ 𝐢,

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which is obviously equivalent to π‘₯βˆ— = Π𝐢 (π‘¦βˆ— βˆ’ πœ‡1 𝐡1 π‘¦βˆ— ) , π‘¦βˆ— = Π𝐢 (π‘₯βˆ— βˆ’ πœ‡2 𝐡2 π‘₯βˆ— ) , because of Lemma 9. This completes the proof.

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6

Abstract and Applied Analysis In terms of Lemma 10, we observe that π‘₯βˆ— = Π𝐢 [Π𝐢 (π‘₯βˆ— βˆ’ πœ‡2 𝐡2 π‘₯βˆ— ) βˆ’ πœ‡1 𝐡1 Π𝐢 (π‘₯βˆ— βˆ’ πœ‡2 𝐡2 π‘₯βˆ— )] , (33)

which implies that π‘₯βˆ— is a fixed point of the mapping 𝐺. Throughout this paper, the set of fixed points of the mapping 𝐺 is denoted by Ξ©. Lemma 11 (see [44]). Let 𝑋 be a uniformly convex Banach space and π΅π‘Ÿ = {π‘₯ ∈ 𝑋 : β€–π‘₯β€– ≀ π‘Ÿ}, π‘Ÿ > 0. Then, there exists a continuous, strictly increasing, and convex function 𝑔 : [0, ∞] β†’ [0, ∞], 𝑔(0) = 0 such that σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„©σ΅„© 2 2 󡄩󡄩𝛼π‘₯ + 𝛽𝑦 + 𝛾𝑧󡄩󡄩󡄩 ≀ 𝛼‖π‘₯β€– + 𝛽󡄩󡄩󡄩𝑦󡄩󡄩󡄩 + 𝛾‖𝑧‖ βˆ’ 𝛼𝛽𝑔 (σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩) (34)

a continuous strictly increasing function 𝑔 : [0, ∞) β†’ [0, ∞), 𝑔(0) = 0, such that σ΅„© σ΅„©2 σ΅„©2 σ΅„©σ΅„© 2 σ΅„©σ΅„©πœ†π‘₯ + (1 βˆ’ πœ†) 𝑦󡄩󡄩󡄩 ≀ πœ†β€–π‘₯β€– + (1 βˆ’ πœ†) 󡄩󡄩󡄩𝑦󡄩󡄩󡄩 σ΅„© σ΅„© βˆ’ πœ† (1 βˆ’ πœ†) 𝑔 (σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩)

(37)

for all πœ† ∈ [0, 1] and π‘₯, 𝑦 ∈ 𝑋 such that β€–π‘₯β€– ≀ π‘Ÿ and ‖𝑦‖ ≀ π‘Ÿ. Lemma 16 (see [48, Lemma 3.2]). Let 𝐢 be a nonempty closed convex subset of a strictly convex Banach space 𝑋. Let {𝑇𝑛 }∞ 𝑛=0 be a sequence of nonexpansive self-mappings on 𝐢 such that ∞ ⋂𝑛=0 Fix(𝑇𝑛 ) =ΜΈ 0 and let {πœ† 𝑛 }∞ 𝑛=0 be a sequence of positive numbers in (0, 𝑏] for some 𝑏 ∈ (0, 1). Then, for every π‘₯ ∈ 𝐢 and π‘˜ β‰₯ 0, the limit lim𝑛 β†’ ∞ π‘ˆπ‘›,π‘˜ π‘₯ exists. Using Lemma 16, one can define a mapping π‘Š : 𝐢 β†’ 𝐢 as follows:

for all π‘₯, 𝑦, 𝑧 ∈ π΅π‘Ÿ , and all 𝛼, 𝛽, 𝛾 ∈ [0, 1] with 𝛼 + 𝛽 + 𝛾 = 1.

π‘Šπ‘₯ = lim π‘Šπ‘› π‘₯ = lim π‘ˆπ‘›,0 π‘₯

Lemma 12 (see [45]). Let 𝐢 be a nonempty closed convex subset of a Banach space 𝑋. Let 𝑆0 , 𝑆1 , . . . be a sequence of mappings of 𝐢 into itself. Suppose that βˆ‘βˆž 𝑛=1 sup{‖𝑆𝑛 π‘₯βˆ’π‘†π‘›βˆ’1 π‘₯β€– : π‘₯ ∈ 𝐢} < ∞. Then for each 𝑦 ∈ 𝐢, {𝑆𝑛 𝑦} converges strongly to some point of 𝐢. Moreover, let 𝑆 be a mapping of 𝐢 into itself defined by 𝑆𝑦 = lim𝑛 β†’ ∞ 𝑆𝑛 𝑦 for all 𝑦 ∈ 𝐢. Then lim𝑛 β†’ ∞ sup{‖𝑆π‘₯ βˆ’ 𝑆𝑛 π‘₯β€– : π‘₯ ∈ 𝐢} = 0.

for every π‘₯ ∈ 𝐢. Such a π‘Š is called the π‘Š-mapping generated ∞ by the sequences {𝑇𝑛 }∞ 𝑛=0 and {πœ† 𝑛 }𝑛=0 . Throughout this paper, ∞ we always assume that {πœ† 𝑛 }𝑛=0 is a sequence of positive numbers in (0, 𝑏] for some 𝑏 ∈ (0, 1).

Let 𝐢 be a nonempty closed convex subset of a Banach space 𝑋 and 𝑇 : 𝐢 β†’ 𝐢 a nonexpansive mapping with Fix(𝑇) =ΜΈ 0. As previous, let Ξ𝐢 be the set of all contractions on 𝐢. For 𝑑 ∈ (0, 1) and 𝑓 ∈ Ξ𝐢, let π‘₯𝑑 ∈ 𝐢 be the unique fixed point of the contraction π‘₯ 󳨃→ 𝑑𝑓(π‘₯) + (1 βˆ’ 𝑑)𝑇π‘₯ on 𝐢; that is, π‘₯𝑑 = 𝑑𝑓 (π‘₯𝑑 ) + (1 βˆ’ 𝑑) 𝑇π‘₯𝑑 .

(35)

Lemma 13 (see [35, 46]). Let 𝑋 be a uniformly smooth Banach space, or a reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm. Let 𝐢 be a nonempty closed convex subset of 𝑋, 𝑇 : 𝐢 β†’ 𝐢 a nonexpansive mapping with Fix(𝑇) =ΜΈ 0, and 𝑓 ∈ Ξ𝐢. Then, the net {π‘₯𝑑 } defined by π‘₯𝑑 = 𝑑𝑓(π‘₯𝑑 ) + (1 βˆ’ 𝑑)𝑇π‘₯𝑑 converges strongly to a point in Fix(𝑇). If we define a mapping 𝑄 : Ξ𝐢 β†’ Fix(𝑇) by 𝑄(𝑓) := 𝑠 βˆ’ lim𝑑 β†’ 0 π‘₯𝑑 , for all 𝑓 ∈ Ξ𝐢, then 𝑄(𝑓) solves the VIP: ⟨(𝐼 βˆ’ 𝑓) 𝑄 (𝑓) , 𝐽 (𝑄 (𝑓) βˆ’ 𝑝)⟩ ≀ 0,

βˆ€π‘“ ∈ Ξ𝐢, 𝑝 ∈ Fix (𝑇) . (36)

Lemma 14 (see [47]). Let 𝐢 be a nonempty closed convex subset of a strictly convex Banach space 𝑋. Let {𝑇𝑛 }∞ 𝑛=0 be a sequence of nonexpansive mappings on 𝐢. Suppose that β‹‚βˆž 𝑛=0 Fix(𝑇𝑛 ) is nonempty. Let {πœ† 𝑛 } be a sequence of positive numbers with βˆ‘βˆž 𝑛=0 πœ† 𝑛 = 1. Then, a mapping 𝑆 on 𝐢 defined by 𝑆π‘₯ = βˆ‘βˆž 𝑛=0 πœ† 𝑛 𝑇𝑛 π‘₯ for π‘₯ ∈ 𝐢 is defined well; nonexpansive and Fix(𝑆) = β‹‚βˆž 𝑛=0 Fix(𝑇𝑛 ) holds. Lemma 15 (see [39]). Given a number π‘Ÿ > 0, A real Banach space 𝑋 is uniformly convex if and only if there exists

π‘›β†’βˆž

π‘›β†’βˆž

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Lemma 17 (see [48]). Let 𝐢 be a nonempty closed convex subset of a strictly convex Banach space 𝑋. Let {𝑇𝑛 }∞ 𝑛=0 be a sequence of nonexpansive self-mappings on 𝐢 such that ∞ β‹‚βˆž 𝑛=0 Fix(𝑇𝑛 ) =ΜΈ 0 and let {πœ† 𝑛 }𝑛=0 be a sequence of positive numbers in (0, 𝑏] for some 𝑏 ∈ (0, 1). Then, Fix(π‘Š) = β‹‚βˆž 𝑛=0 Fix(𝑇𝑛 ). Let πœ‡ be a continuous linear functional on π‘™βˆž and 𝑠 = (π‘Ž0 , π‘Ž1 , . . .) ∈ π‘™βˆž . One writes πœ‡π‘› (π‘Žπ‘› ) instead of πœ‡(𝑠). πœ‡ is called a Banach limit if πœ‡ satisfies β€–πœ‡β€– = πœ‡π‘› (1) = 1 and πœ‡π‘› (π‘Žπ‘›+1 ) = πœ‡π‘› (π‘Žπ‘› ) for all (π‘Ž0 , π‘Ž1 , . . .) ∈ π‘™βˆž . If πœ‡ is a Banach limit, then, there hold the following: (i) for all 𝑛 β‰₯ 0, π‘Žπ‘› ≀ 𝑐𝑛 implies πœ‡π‘› (π‘Žπ‘› ) ≀ πœ‡π‘› (𝑐𝑛 ); (ii) πœ‡π‘› (π‘Žπ‘›+π‘Ÿ ) = πœ‡π‘› (π‘Žπ‘› ) for any fixed positive integer π‘Ÿ; (iii) lim inf 𝑛 β†’ ∞ π‘Žπ‘› ≀ πœ‡π‘› (π‘Žπ‘› ) ≀ lim sup𝑛 β†’ ∞ π‘Žπ‘› for all (π‘Ž0 , π‘Ž1 , . . .) ∈ π‘™βˆž . Lemma 18 (see [49]). Let π‘Ž ∈ R be a real number and a sequence {π‘Žπ‘› } ∈ π‘™βˆž satisfy the condition πœ‡π‘› (π‘Žπ‘› ) ≀ π‘Ž for all Banach limit πœ‡. If lim sup𝑛 β†’ ∞ (π‘Žπ‘›+π‘Ÿ βˆ’ π‘Žπ‘› ) ≀ 0, then lim sup𝑛 β†’ ∞ π‘Žπ‘› ≀ π‘Ž. In particular, if π‘Ÿ = 1 in Lemma 18, then we immediately obtain the following corollary. Corollary 19 (see [50]). Let π‘Ž ∈ R be a real number and a sequence {π‘Žπ‘› } ∈ π‘™βˆž satisfy the condition πœ‡π‘› (π‘Žπ‘› ) ≀ π‘Ž for all Banach limit πœ‡. If lim sup𝑛 β†’ ∞ (π‘Žπ‘›+1 βˆ’ π‘Žπ‘› ) ≀ 0, then, lim sup𝑛 β†’ ∞ π‘Žπ‘› ≀ π‘Ž. Lemma 20 (see [51]). Let {π‘₯𝑛 } and {𝑧𝑛 } be bounded sequences in a Banach space 𝑋 and let {𝛽𝑛 } be a sequence of nonnegative numbers in [0, 1] with 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1. Suppose that π‘₯𝑛+1 = 𝛽𝑛 π‘₯𝑛 + (1 βˆ’ 𝛽𝑛 )𝑧𝑛 for all integers

Abstract and Applied Analysis

7

𝑛 β‰₯ 0 and lim sup𝑛 β†’ ∞ (‖𝑧𝑛+1 βˆ’ 𝑧𝑛 β€– βˆ’ β€–π‘₯𝑛+1 βˆ’ π‘₯𝑛 β€–) ≀ 0. Then, lim𝑛 β†’ ∞ β€–π‘₯𝑛 βˆ’ 𝑧𝑛 β€– = 0.

0 < πœ‡π‘– < 𝛽̂𝑖 /πœ…2 for 𝑖 = 1, 2. For arbitrarily given π‘₯0 ∈ 𝐢, let {π‘₯𝑛 } be the sequence generated by

Lemma 21 (see [34]). Let 𝐢 be a nonempty closed convex subset of a smooth Banach space 𝑋. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢 and 𝐴 : 𝐢 β†’ 𝑋 an accretive mapping. Then for all πœ† > 0,

𝑦𝑛 = 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 ,

VI (𝐢, 𝐴) = Fix (Π𝐢 (𝐼 βˆ’ πœ†π΄)) .

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Lemma 22 (see [11]). Let 𝐢 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝑋. Let the mapping 𝐡𝑖 : 𝐢 β†’ 𝑋 be 𝛽̂𝑖 -inverse-strongly accretive. Then, one has σ΅„©2 σ΅„©σ΅„© σ΅„©σ΅„©(𝐼 βˆ’ πœ‡π‘– 𝐡𝑖 ) π‘₯ βˆ’ (𝐼 βˆ’ πœ‡π‘– 𝐡𝑖 ) 𝑦󡄩󡄩󡄩 σ΅„©2 σ΅„© σ΅„© σ΅„©2 ≀ σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 + 2πœ‡π‘– (πœ‡π‘– πœ…2 βˆ’ 𝛽̂𝑖 ) 󡄩󡄩󡄩𝐡𝑖 π‘₯ βˆ’ 𝐡𝑖 𝑦󡄩󡄩󡄩 ,

for 𝑖 = 1, 2 where πœ‡π‘– > 0. In particular, if 0 < πœ‡π‘– ≀ 𝛽̂𝑖 /πœ…2 , then 𝐼 βˆ’ πœ‡π‘– 𝐡𝑖 is nonexpansive for 𝑖 = 1, 2. Lemma 23 (see [11]). Let 𝐢 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝑋. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢. Let the mapping 𝐡𝑖 : 𝐢 β†’ 𝑋 be 𝛽̂𝑖 -inverse-strongly accretive for 𝑖 = 1, 2. Let 𝐺 : 𝐢 β†’ 𝐢 be the mapping defined by

βˆ€π‘₯ ∈ 𝐢.

βˆ€π‘› β‰₯ 0,

(42)

where {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, and {𝛿𝑛 } are the sequences in [0, 1] such that 𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 for all 𝑛 β‰₯ 0. Suppose that the following conditions hold: (i) βˆ‘βˆž 𝑛=0 𝛼𝑛 = ∞ and 0 ≀ 𝛼𝑛 ≀ 1 βˆ’ 𝜌, for all 𝑛 β‰₯ 𝑛0 for some integer 𝑛0 β‰₯ 0; (ii) lim inf 𝑛 β†’ ∞ 𝛾𝑛 > 0 and lim inf 𝑛 β†’ ∞ 𝛿𝑛 > 0; (iii) lim𝑛 β†’ ∞ (|𝛼𝑛+1 /(1 βˆ’ (1 βˆ’ 𝛼𝑛+1 )𝛽𝑛+1 ) βˆ’ 𝛼𝑛 /(1 βˆ’ (1 βˆ’ 𝛼𝑛 )𝛽𝑛 )| + |𝛿𝑛+1 /(1 βˆ’ 𝛽𝑛+1 ) βˆ’ 𝛿𝑛 /(1 βˆ’ 𝛽𝑛 )|) = 0; (iv) 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1.

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βˆ€π‘₯, 𝑦 ∈ 𝐢,

𝐺π‘₯ = Π𝐢 [Π𝐢 (π‘₯ βˆ’ πœ‡2 𝐡2 π‘₯) βˆ’ πœ‡1 𝐡1 Π𝐢 (π‘₯ βˆ’ πœ‡2 𝐡2 π‘₯)] ,

π‘₯𝑛+1 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + (1 βˆ’ 𝛼𝑛 ) 𝑦𝑛 ,

(41)

If 0 < πœ‡π‘– ≀ 𝛽̂𝑖 /πœ…2 for 𝑖 = 1, 2, then 𝐺 : 𝐢 β†’ 𝐢 is nonexpansive.

3. Hybrid Mann Iterations and Their Convergence Criteria In this section, we introduce our hybrid Mann iteration algorithms in real smooth and uniformly convex Banach spaces and present their convergence criteria. Theorem 24. Let 𝐢 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢. Let {πœŒπ‘› }∞ 𝑛=0 be a sequence of positive numbers in (0, 𝑏] for some 𝑏 ∈ (0, 1) and 𝐴 𝑖 : 𝐢 β†’ 𝐸 an 𝛼̂𝑖 -inverse strongly accretive mapping for each 𝑖 = 0, 1, . . .. Define a mapping 𝐺𝑖 : 𝐢 β†’ 𝐢 by Π𝐢(𝐼 βˆ’ πœ† 𝑖 𝐴 𝑖 )π‘₯ = 𝐺𝑖 π‘₯ for all π‘₯ ∈ 𝐢 and 𝑖 = 0, 1, . . ., where πœ† 𝑖 ∈ (0, 𝛼̂𝑖 /πœ…2 ], πœ… is the 2-uniformly smooth constant of 𝑋. Let 𝐡𝑛 : 𝐢 β†’ 𝐢 be the π‘Š-mapping generated by 𝐺𝑛 , πΊπ‘›βˆ’1 , . . . , 𝐺0 and πœŒπ‘› , πœŒπ‘›βˆ’1 , . . . , 𝜌0 . Let the mapping 𝐡𝑖 : 𝐢 β†’ 𝑋 be 𝛽̂𝑖 inverse strongly accretive for 𝑖 = 1, 2. Let 𝑓 : 𝐢 β†’ 𝐢 be a contraction with coefficient 𝜌 ∈ (0, 1). Let {𝑆𝑖 }∞ 𝑖=0 be a countable family of nonexpansive mappings of 𝐢 into itself such that 𝐹 = ∞ (β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 )) ∩ Ξ© ∩ (⋂𝑖=0 VI(𝐢, 𝐴 𝑖 )) =ΜΈ 0, where Ξ© is the fixed point set of the mapping 𝐺 = Π𝐢(𝐼 βˆ’ πœ‡1 𝐡1 )Π𝐢(𝐼 βˆ’ πœ‡2 𝐡2 ) with

Assume that βˆ‘βˆž 𝑛=0 supπ‘₯∈𝐷 ‖𝑆𝑛+1 π‘₯ βˆ’ 𝑆𝑛 π‘₯β€– < ∞ for any bounded subset 𝐷 of 𝐢 and let 𝑆 be a mapping of 𝐢 into itself defined by 𝑆π‘₯ = lim𝑛 β†’ ∞ 𝑆𝑛 π‘₯ for all π‘₯ ∈ 𝐢 and suppose that Fix(𝑆) = β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 ). Then, there hold the following: (I) lim𝑛 β†’ ∞ β€–π‘₯𝑛+1 βˆ’ π‘₯𝑛 β€– = 0; (II) π‘₯𝑛 β†’ π‘ž ⇔ 𝛼𝑛 (𝑓(π‘₯𝑛 ) βˆ’ π‘₯𝑛 ) β†’ 0 provided 𝛽𝑛 ≑ 𝛽 for some fixed 𝛽 ∈ (0, 1), where π‘ž ∈ 𝐹 solves the following VIP: βŸ¨π‘ž βˆ’ 𝑓 (π‘ž) , 𝐽 (π‘ž βˆ’ 𝑝)⟩ ≀ 0,

βˆ€π‘ ∈ 𝐹.

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Proof. First of all, since 0 < πœ† 𝑖 < 𝛼̂𝑖 /πœ…2 for 𝑖 = 0, 1, . . ., it is easy to see that 𝐺𝑖 is a nonexpansive mapping for each 𝑖 = 0, 1, . . .. Since 𝐡𝑛 : 𝐢 β†’ 𝐢 is the π‘Š-mapping generated by 𝐺𝑛 , πΊπ‘›βˆ’1 , . . . , 𝐺0 and πœŒπ‘› , πœŒπ‘›βˆ’1 , . . . , 𝜌0 , by Lemma 16 we know that, for each π‘₯ ∈ 𝐢 and π‘˜ β‰₯ 0, the limit lim𝑛 β†’ ∞ π‘ˆπ‘›,π‘˜ π‘₯ exists. Moreover, one can define a mapping 𝐡 : 𝐢 β†’ 𝐢 as follows: 𝐡π‘₯ = lim 𝐡𝑛 π‘₯ = lim π‘ˆπ‘›,0 π‘₯ π‘›β†’βˆž

π‘›β†’βˆž

(44)

for every π‘₯ ∈ 𝐢. That is, such a 𝐡 is the π‘Š-mapping generated ∞ by the sequences {𝐺𝑛 }∞ 𝑛=0 and {πœŒπ‘› }𝑛=0 . According to Lemma 17, ∞ we know that Fix(𝐡) = ⋂𝑖=0 Fix(𝐺𝑖 ). From Lemma 15 and the definition of 𝐺𝑖 , we have Fix(𝐺𝑖 ) = VI(𝐢, 𝐴 𝑖 ) for each 𝑖 = 0, 1, . . .. Hence, we have ∞

∞

𝑖=0

𝑖=0

Fix (𝐡) = β‹‚ Fix (𝐺𝑖 ) = β‹‚ VI (𝐢, 𝐴 𝑖 ) .

(45)

Next, let us show that the sequence {π‘₯𝑛 } is bounded. Indeed, take a fixed 𝑝 ∈ 𝐹 arbitrarily. Then, we get 𝑝 = 𝐺𝑝, 𝑝 = 𝐡𝑛 𝑝, and 𝑝 = 𝑆𝑛 𝑝 for all 𝑛 β‰₯ 0. By Lemma 23 we know that 𝐺 is nonexpansive. Then, from (42), we have σ΅„© σ΅„© σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© 󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 , (46)

8

Abstract and Applied Analysis

and hence

=(

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„©σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ 𝛼𝑛 ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩

σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 (󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (𝑝)σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩)

βˆ’

σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩

+ 𝛿𝑛+1 𝑆𝑛+1 𝐺π‘₯𝑛+1 ]

σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩

βˆ’ πœŽπ‘›+1 π‘₯𝑛+1 Γ— (1 βˆ’ πœŽπ‘›+1 )

(47)

σ΅„© σ΅„© = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩

=(

σ΅„© σ΅„©σ΅„© 󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛼𝑛 (1 βˆ’ 𝜌) σ΅„© 1βˆ’πœŒ σ΅„© σ΅„© σ΅„© 󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© ≀ max {σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 , σ΅„© }. 1βˆ’πœŒ

βˆ’ =(

βˆ€π‘› β‰₯ 0. (48)

+

Thus, {π‘₯𝑛 } is bounded, and so are the sequences {𝑦𝑛 }, {𝐺π‘₯𝑛 } and {𝑓(π‘₯𝑛 )}. Let us show that σ΅„© σ΅„© lim σ΅„©σ΅„©π‘₯ βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0. 𝑛 β†’ ∞ σ΅„© 𝑛+1

(1 βˆ’ 𝛼𝑛+1 ) (1 βˆ’ 𝛽𝑛+1 ) 1 βˆ’ πœŽπ‘›+1

+[ Γ—

βˆ€π‘› β‰₯ 𝑛0 , (50)

(51)

(52)

𝛾𝑛+1 𝐡𝑛+1 π‘₯𝑛+1 + 𝛿𝑛+1 𝑆𝑛+1 𝐺π‘₯𝑛+1 1 βˆ’ 𝛽𝑛+1 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 ] 1 βˆ’ 𝛽𝑛

(1 βˆ’ 𝛼𝑛+1 ) (1 βˆ’ 𝛽𝑛+1 ) (1 βˆ’ 𝛼𝑛 ) (1 βˆ’ 𝛽𝑛 ) βˆ’ ] 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘›

𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 1 βˆ’ 𝛽𝑛

𝛼𝑛+1 (𝑓 (π‘₯𝑛+1 ) βˆ’ 𝑓 (π‘₯𝑛 )) 1 βˆ’ πœŽπ‘›+1 +( +

Define π‘₯𝑛+1 = πœŽπ‘› π‘₯𝑛 + (1 βˆ’ πœŽπ‘› ) 𝑧𝑛 .

𝛼𝑛+1 𝑓 (π‘₯𝑛+1 ) 𝛼𝑛 𝑓 (π‘₯𝑛 ) βˆ’ ) 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘›

(49)

and hence π‘›β†’βˆž

1 βˆ’ 𝛼𝑛 (𝛾 𝐡 π‘₯ + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 ) 1 βˆ’ πœŽπ‘› 𝑛 𝑛 𝑛

βˆ’

= 0 < lim inf πœŽπ‘› ≀ lim sup πœŽπ‘› < 1.

𝛼𝑛+1 𝑓 (π‘₯𝑛+1 ) 𝛼𝑛 𝑓 (π‘₯𝑛 ) 1 βˆ’ 𝛼𝑛+1 βˆ’ )+ 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘› 1 βˆ’ πœŽπ‘›+1

Γ—[

As a matter of fact, put πœŽπ‘› = (1βˆ’π›Όπ‘› )𝛽𝑛 , for all 𝑛 β‰₯ 0. Then, it follows from (i) and (iv) that 𝛽𝑛 β‰₯ πœŽπ‘› = (1 βˆ’ 𝛼𝑛 ) 𝛽𝑛 β‰₯ (1 βˆ’ (1 βˆ’ 𝜌)) 𝛽𝑛 = πœŒπ›½π‘› ,

𝛼𝑛+1 𝛼𝑛 βˆ’ ) 𝑓 (π‘₯𝑛 ) 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘›

(1 βˆ’ 𝛼𝑛+1 ) (1 βˆ’ 𝛽𝑛+1 ) 1 βˆ’ πœŽπ‘›+1

Γ—[

Observe that

𝛾𝑛+1 (𝐡 π‘₯ βˆ’ 𝐡𝑛 π‘₯𝑛 ) 𝛾𝑛+1 + 𝛿𝑛+1 𝑛+1 𝑛+1 +(

𝑧𝑛+1 βˆ’ 𝑧𝑛 π‘₯𝑛+2 βˆ’ πœŽπ‘›+1 π‘₯𝑛+1 π‘₯𝑛+1 βˆ’ πœŽπ‘› π‘₯𝑛 βˆ’ 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘›

+

=

𝛼𝑛+1 𝑓 (π‘₯𝑛+1 ) + (1 βˆ’ 𝛼𝑛+1 ) 𝑦𝑛+1 βˆ’ πœŽπ‘›+1 π‘₯𝑛+1 1 βˆ’ πœŽπ‘›+1

+(

𝛼𝑛 𝑓 (π‘₯𝑛 ) + (1 βˆ’ 𝛼𝑛 ) 𝑦𝑛 βˆ’ πœŽπ‘› π‘₯𝑛 1 βˆ’ πœŽπ‘›

βˆ’(

𝛾𝑛 𝛾𝑛+1 βˆ’ ) 𝐡𝑛 π‘₯𝑛 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛

𝛿𝑛+1 (𝑆 𝐺π‘₯ βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 ) 𝛾𝑛+1 + 𝛿𝑛+1 𝑛+1 𝑛+1

=

βˆ’

βˆ’1

Γ— (𝛾𝑛+1 𝐡𝑛+1 π‘₯𝑛+1 + 𝛿𝑛+1 𝑆𝑛+1 𝐺π‘₯𝑛+1 )

By induction, we obtain

π‘›β†’βˆž

(1 βˆ’ 𝛼𝑛 ) [𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 ] βˆ’ πœŽπ‘› π‘₯𝑛 1 βˆ’ πœŽπ‘›

+ (1 βˆ’ 𝛼𝑛+1 ) [𝛽𝑛+1 π‘₯𝑛+1 + 𝛾𝑛+1 𝐡𝑛+1 π‘₯𝑛+1

σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 (𝜌 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩)

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© 󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© }, σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ≀ max {σ΅„©σ΅„©σ΅„©π‘₯0 βˆ’ 𝑝󡄩󡄩󡄩 , σ΅„© 1βˆ’πœŒ

𝛼𝑛+1 𝑓 (π‘₯𝑛+1 ) 𝛼𝑛 𝑓 (π‘₯𝑛 ) βˆ’ ) 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘›

𝛿𝑛+1 𝛿𝑛 βˆ’ ) 𝑆 𝐺π‘₯ ] 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 𝑛 𝑛

𝛾 𝐡 π‘₯ + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 𝛼𝑛+1 𝛼𝑛 βˆ’ ) 𝑛 𝑛 𝑛 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘› 𝛾𝑛 + 𝛿𝑛

Abstract and Applied Analysis =

9

𝛼𝑛+1 (𝑓 (π‘₯𝑛+1 ) βˆ’ 𝑓 (π‘₯𝑛 )) 1 βˆ’ πœŽπ‘›+1 +(

𝛼𝑛+1 𝛼𝑛 βˆ’ ) 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘›

Γ— (𝑓 (π‘₯𝑛 ) βˆ’ +

𝛿𝑛+1 σ΅„©σ΅„© σ΅„© 󡄩𝑆 𝐺π‘₯ βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© 𝛾𝑛+1 + 𝛿𝑛+1 σ΅„© 𝑛+1 𝑛+1 󡄨󡄨 𝛿 𝛿𝑛 󡄨󡄨󡄨󡄨 σ΅„©σ΅„© 󡄨 σ΅„© 𝑛+1 + 󡄨󡄨󡄨 βˆ’ 󡄨 󡄩𝑆 𝐺π‘₯ σ΅„©σ΅„©] . 󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨󡄨󡄨 σ΅„© 𝑛 𝑛 σ΅„©

+

𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 ) 𝛾𝑛 + 𝛿𝑛

(54)

1 βˆ’ πœŽπ‘›+1 βˆ’ 𝛼𝑛+1 1 βˆ’ πœŽπ‘›+1

Γ—[

On the other hand, we note that, for all 𝑛 β‰₯ 0,

𝛾𝑛+1 (𝐡 π‘₯ βˆ’ 𝐡𝑛 π‘₯𝑛 ) 𝛾𝑛+1 + 𝛿𝑛+1 𝑛+1 𝑛+1

σ΅„©σ΅„© σ΅„© 󡄩󡄩𝑆𝑛+1 𝐺π‘₯𝑛+1 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© σ΅„© σ΅„© ≀ 󡄩󡄩󡄩𝑆𝑛+1 𝐺π‘₯𝑛+1 βˆ’ 𝑆𝑛+1 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛+1 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©

𝛾𝑛 𝛾𝑛+1 βˆ’ ) 𝐡𝑛 π‘₯𝑛 +( 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 +

σ΅„© σ΅„© σ΅„© σ΅„© ≀ 󡄩󡄩󡄩𝐺π‘₯𝑛+1 βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛+1 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛+1 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© .

𝛿𝑛+1 (𝑆 𝐺π‘₯ βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 ) 𝛾𝑛+1 + 𝛿𝑛+1 𝑛+1 𝑛+1

+(

𝛿𝑛+1 𝛿𝑛 βˆ’ ) 𝑆 𝐺π‘₯ ] , 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 𝑛 𝑛 (53)

and hence σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑧𝑛+1 βˆ’ 𝑧𝑛 σ΅„©σ΅„©σ΅„© ≀

𝛼𝑛+1 σ΅„©σ΅„© σ΅„© 󡄩𝑓 (π‘₯𝑛+1 ) βˆ’ 𝑓 (π‘₯𝑛 )σ΅„©σ΅„©σ΅„© 1 βˆ’ πœŽπ‘›+1 σ΅„© 󡄨󡄨 𝛼 𝛾 𝐡 π‘₯ + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©σ΅„© 𝛼𝑛 󡄨󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©σ΅„© 󡄨 + 󡄨󡄨󡄨 𝑛+1 βˆ’ 󡄨󡄨 󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑛 𝑛 𝑛 σ΅„©σ΅„© 󡄨󡄨 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘› 󡄨󡄨 σ΅„©σ΅„© σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛 1 βˆ’ πœŽπ‘›+1 βˆ’ 𝛼𝑛+1 1 βˆ’ πœŽπ‘›+1 σ΅„©σ΅„© 𝛾 σ΅„© 𝑛+1 Γ— σ΅„©σ΅„©σ΅„© (𝐡 π‘₯ βˆ’ 𝐡𝑛 π‘₯𝑛 ) σ΅„©σ΅„© 𝛾𝑛+1 + 𝛿𝑛+1 𝑛+1 𝑛+1 +

𝛾𝑛 𝛾𝑛+1 βˆ’ ) 𝐡𝑛 π‘₯𝑛 +( 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 𝛿𝑛+1 (𝑆 𝐺π‘₯ βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 ) 𝛾𝑛+1 + 𝛿𝑛+1 𝑛+1 𝑛+1 σ΅„©σ΅„© 𝛿𝑛+1 𝛿𝑛 σ΅„© βˆ’ ) 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© +( σ΅„©σ΅„© 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨 πœŒπ›Όπ‘›+1 σ΅„©σ΅„© 𝛼𝑛 󡄨󡄨󡄨󡄨 σ΅„© 󡄨󡄨 𝛼 ≀ 󡄨󡄨 σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄨󡄨󡄨 𝑛+1 βˆ’ 1 βˆ’ πœŽπ‘›+1 󡄨󡄨 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘› 󡄨󡄨 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— (󡄩󡄩󡄩𝑓 (π‘₯𝑛 )σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©) +

1 βˆ’ πœŽπ‘›+1 βˆ’ 𝛼𝑛+1 + 1 βˆ’ πœŽπ‘›+1 Γ—[

𝛾𝑛+1 σ΅„©σ΅„© σ΅„© 󡄩𝐡 π‘₯ βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 𝛾𝑛+1 + 𝛿𝑛+1 σ΅„© 𝑛+1 𝑛+1 󡄨󡄨󡄨 𝛾𝑛+1 𝛾𝑛 󡄨󡄨󡄨󡄨 σ΅„©σ΅„© σ΅„© + 󡄨󡄨󡄨 βˆ’ 󡄨󡄨 󡄩󡄩𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨󡄨

(55)

Furthermore, by (CY), since 𝐺𝑖 and π‘ˆπ‘›,𝑖 are nonexpansive, we deduce that for each 𝑛 β‰₯ 0 σ΅„©σ΅„© σ΅„© 󡄩󡄩𝐡𝑛+1 π‘₯𝑛+1 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 󡄩󡄩󡄩𝐡𝑛+1 π‘₯𝑛+1 βˆ’ 𝐡𝑛+1 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛+1 π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛+1 π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©πœ† 0 𝐺0 π‘ˆπ‘›+1,1 π‘₯𝑛 βˆ’ πœ† 0 𝐺0 π‘ˆπ‘›,1 π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + πœ† 0 σ΅„©σ΅„©σ΅„©π‘ˆπ‘›+1,1 π‘₯𝑛 βˆ’ π‘ˆπ‘›,1 π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + πœ† 0 σ΅„©σ΅„©σ΅„©πœ† 1 𝐺1 π‘ˆπ‘›+1,2 π‘₯𝑛 βˆ’ πœ† 1 𝐺1 π‘ˆπ‘›,2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© (56) σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + πœ† 0 πœ† 1 σ΅„©σ΅„©σ΅„©π‘ˆπ‘›+1,2 π‘₯𝑛 βˆ’ π‘ˆπ‘›,2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© .. . 𝑛

σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + (βˆπœ† 𝑖 ) σ΅„©σ΅„©σ΅„©π‘ˆπ‘›+1,𝑛+1 π‘₯𝑛 βˆ’ π‘ˆπ‘›,𝑛+1 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 𝑖=0

𝑛

σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 𝑀0 βˆπœ† 𝑖 , 𝑖=0

for some constant 𝑀0 > 0. Utilizing (54)–(56), we have σ΅„©σ΅„© σ΅„© 󡄩󡄩𝑧𝑛+1 βˆ’ 𝑧𝑛 σ΅„©σ΅„©σ΅„© ≀

󡄨 πœŒπ›Όπ‘›+1 σ΅„©σ΅„© 𝛼𝑛 󡄨󡄨󡄨󡄨 σ΅„© 󡄨󡄨 𝛼 󡄨 σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄨󡄨󡄨 𝑛+1 βˆ’ 󡄨󡄨 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘› 󡄨󡄨󡄨 1 βˆ’ πœŽπ‘›+1 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— (󡄩󡄩󡄩𝑓 (π‘₯𝑛 )σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©)

+

1 βˆ’ πœŽπ‘›+1 βˆ’ 𝛼𝑛+1 1 βˆ’ πœŽπ‘›+1

10

Abstract and Applied Analysis Γ—[

𝑛 𝛾𝑛+1 σ΅„© σ΅„© (σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 𝑀0 βˆπœ† 𝑖 ) 𝛾𝑛+1 + 𝛿𝑛+1 𝑖=0

where sup𝑛β‰₯0 {‖𝑓(π‘₯𝑛 )β€–+‖𝐡𝑛 π‘₯𝑛 β€–+‖𝑆𝑛 𝐺π‘₯𝑛 β€–+𝑀0 } ≀ 𝑀 for some 𝑀 > 0. So, from (58), condition (iii), and the assumption on {𝑆𝑛 }, it follows that (noting that 0 < πœ† 𝑖 ≀ 𝑏 < 1, for all 𝑖 β‰₯ 0)

󡄨󡄨 𝛾 𝛾𝑛 󡄨󡄨󡄨󡄨 σ΅„©σ΅„© 󡄨 σ΅„© 𝑛+1 + 󡄨󡄨󡄨 βˆ’ 󡄨 󡄩𝐡 π‘₯ σ΅„©σ΅„© 󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨󡄨󡄨 σ΅„© 𝑛 𝑛 σ΅„© +

σ΅„© σ΅„© σ΅„© σ΅„© lim sup (󡄩󡄩󡄩𝑧𝑛+1 βˆ’ 𝑧𝑛 σ΅„©σ΅„©σ΅„© βˆ’ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) ≀ 0.

𝛿𝑛+1 σ΅„© σ΅„© σ΅„© σ΅„© (σ΅„©σ΅„©π‘₯ βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛+1 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©) 𝛾𝑛+1 + 𝛿𝑛+1 σ΅„© 𝑛+1

lim 󡄩𝑧 π‘›β†’βˆž σ΅„© 𝑛

1 βˆ’ πœŽπ‘›+1 βˆ’ 𝛼𝑛+1 (1 βˆ’ 𝜌) σ΅„©σ΅„© σ΅„© = σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© 1 βˆ’ πœŽπ‘›+1 󡄨󡄨 𝛼 𝛼𝑛 󡄨󡄨󡄨󡄨 σ΅„©σ΅„© 󡄨 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + 󡄨󡄨󡄨 𝑛+1 βˆ’ 󡄨 (󡄩𝑓 (π‘₯𝑛 )σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©) 󡄨󡄨 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘› 󡄨󡄨󡄨 σ΅„©

σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(60)

It follows from (51) and (52) that σ΅„©σ΅„©

lim σ΅„©π‘₯ 𝑛 β†’ ∞ σ΅„© 𝑛+1

σ΅„© σ΅„© σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = lim (1 βˆ’ πœŽπ‘› ) 󡄩󡄩󡄩𝑧𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0. π‘›β†’βˆž

(61)

From (42), we have π‘₯𝑛+1 βˆ’ π‘₯𝑛 = 𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 ) + (1 βˆ’ 𝛼𝑛 ) (𝑦𝑛 βˆ’ π‘₯𝑛 ) ,

1 βˆ’ πœŽπ‘›+1 βˆ’ 𝛼𝑛+1 1 βˆ’ πœŽπ‘›+1

(59)

Consequently, by Lemma 20, we have σ΅„©σ΅„©

󡄨󡄨 𝛿 𝛿𝑛 󡄨󡄨󡄨󡄨 σ΅„©σ΅„© 󡄨 σ΅„© 𝑛+1 + 󡄨󡄨󡄨 βˆ’ 󡄨 󡄩𝑆 𝐺π‘₯ σ΅„©σ΅„© ] 󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨󡄨󡄨 σ΅„© 𝑛 𝑛 σ΅„©

+

π‘›β†’βˆž

(62)

which hence implies that

𝑛 𝛾𝑛+1 Γ—[ 𝑀0 βˆπœ† 𝑖 𝛾𝑛+1 + 𝛿𝑛+1 𝑖=0

σ΅„© σ΅„© σ΅„© σ΅„© 𝜌 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = (1 βˆ’ (1 βˆ’ 𝜌)) 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© ≀ (1 βˆ’ 𝛼𝑛 ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©

𝛿𝑛+1 σ΅„©σ΅„© σ΅„© + 󡄩𝑆 𝐺π‘₯ βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© 𝛾𝑛+1 + 𝛿𝑛+1 σ΅„© 𝑛+1 𝑛 󡄨󡄨 𝛿 𝛿𝑛 󡄨󡄨󡄨󡄨 σ΅„©σ΅„© 󡄨 σ΅„© σ΅„© σ΅„© 𝑛+1 + 󡄨󡄨󡄨 βˆ’ 󡄨 (󡄩𝐡 π‘₯ σ΅„©σ΅„© + 󡄩󡄩𝑆 𝐺π‘₯ σ΅„©σ΅„©) ] 󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨󡄨󡄨 σ΅„© 𝑛 𝑛 σ΅„© σ΅„© 𝑛 𝑛 σ΅„© 󡄨 𝛼𝑛 󡄨󡄨󡄨󡄨 σ΅„© σ΅„© 󡄨󡄨 𝛼 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄨󡄨󡄨 𝑛+1 βˆ’ 󡄨𝑀 󡄨󡄨 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘› 󡄨󡄨󡄨 𝑛

σ΅„© σ΅„© + π‘€βˆπœ† 𝑖 + 󡄩󡄩󡄩𝑆𝑛+1 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 βˆ’ 𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©

(63)

σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„© . Since π‘₯𝑛+1 βˆ’ π‘₯𝑛 β†’ 0 and 𝛼𝑛 (𝑓(π‘₯𝑛 ) βˆ’ π‘₯𝑛 ) β†’ 0, we get σ΅„©σ΅„©

lim 󡄩𝑦 π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(64)

Next, we show that β€–π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 β€– β†’ 0 as 𝑛 β†’ ∞. Indeed, for simplicity, put π‘ž = Π𝐢(𝑝 βˆ’ πœ‡2 𝐡2 𝑝), 𝑒𝑛 = Π𝐢(π‘₯𝑛 βˆ’ πœ‡2 𝐡2 π‘₯𝑛 ) and V𝑛 = Π𝐢(𝑒𝑛 βˆ’ πœ‡1 𝐡1 𝑒𝑛 ). Then, V𝑛 = 𝐺π‘₯𝑛 for all 𝑛 β‰₯ 0. From Lemma 22, we have

𝑖=0

󡄨󡄨 𝛿 𝛿𝑛 󡄨󡄨󡄨󡄨 󡄨 𝑛+1 + 󡄨󡄨󡄨 βˆ’ 󡄨𝑀 󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨󡄨󡄨 σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄨󡄨 𝛼 𝛼𝑛 󡄨󡄨󡄨󡄨 󡄨 + 𝑀 ( 󡄨󡄨󡄨 𝑛+1 βˆ’ 󡄨󡄨 󡄨󡄨 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘› 󡄨󡄨

σ΅„©2 σ΅„© σ΅„©2 σ΅„©σ΅„© 󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© = 󡄩󡄩󡄩Π𝐢 (π‘₯𝑛 βˆ’ πœ‡2 𝐡2 π‘₯𝑛 ) βˆ’ Π𝐢 (𝑝 βˆ’ πœ‡2 𝐡2 𝑝)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝 βˆ’ πœ‡2 (𝐡2 π‘₯𝑛 βˆ’ 𝐡2 𝑝)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 2πœ‡2 (𝛽̂2 βˆ’ πœ…2 πœ‡2 ) 󡄩󡄩󡄩𝐡2 π‘₯𝑛 βˆ’ 𝐡2 𝑝󡄩󡄩󡄩 , (65)

󡄨󡄨 𝛿 𝛿𝑛 󡄨󡄨󡄨󡄨 𝑛 󡄨 𝑛+1 + 󡄨󡄨󡄨 βˆ’ 󡄨 + βˆπœ† ) 󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨󡄨󡄨 𝑖=0 𝑖 σ΅„© σ΅„© + 󡄩󡄩󡄩𝑆𝑛+1 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© , (57) which hence yields 󡄩󡄩󡄩𝑧𝑛+1 βˆ’ 𝑧𝑛 σ΅„©σ΅„©σ΅„© βˆ’ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© 󡄨󡄨 𝛼 𝛼𝑛 󡄨󡄨󡄨󡄨 󡄨󡄨󡄨󡄨 𝛿𝑛+1 𝛿𝑛 󡄨󡄨󡄨󡄨 󡄨 ≀ 𝑀 ( 󡄨󡄨󡄨 𝑛+1 βˆ’ βˆ’ 󡄨󡄨 + 󡄨󡄨 󡄨 󡄨󡄨 1 βˆ’ πœŽπ‘›+1 1 βˆ’ πœŽπ‘› 󡄨󡄨 󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨󡄨󡄨 𝑛

σ΅„© σ΅„© +βˆπœ† 𝑖 ) + 󡄩󡄩󡄩𝑆𝑛+1 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© , 𝑖=0

(58)

σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 = 󡄩󡄩󡄩Π𝐢 (𝑒𝑛 βˆ’ πœ‡1 𝐡1 𝑒𝑛 ) βˆ’ Π𝐢 (π‘ž βˆ’ πœ‡1 𝐡1 π‘ž)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 ≀ 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘ž βˆ’ πœ‡1 (𝐡1 𝑒𝑛 βˆ’ 𝐡1 π‘ž)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© βˆ’ 2πœ‡1 (𝛽̂1 βˆ’ πœ…2 πœ‡1 ) 󡄩󡄩󡄩𝐡1 𝑒𝑛 βˆ’ 𝐡1 π‘žσ΅„©σ΅„©σ΅„© . (66) Substituting (65) for (66), we obtain σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„©2 2 σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 2πœ‡2 (𝛽̂2 βˆ’ πœ… πœ‡2 ) 󡄩󡄩󡄩𝐡2 π‘₯𝑛 βˆ’ 𝐡2 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 βˆ’ 2πœ‡1 (𝛽̂1 βˆ’ πœ…2 πœ‡1 ) 󡄩󡄩󡄩𝐡1 𝑒𝑛 βˆ’ 𝐡1 π‘žσ΅„©σ΅„©σ΅„© .

(67)

Abstract and Applied Analysis

11

From (42) and (67), we have

In the same way, we derive

σ΅„©σ΅„© σ΅„©2 󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 + 𝛿𝑛 [σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 2πœ‡2 (𝛽̂2 βˆ’ πœ…2 πœ‡2 ) σ΅„©2 σ΅„© Γ— 󡄩󡄩󡄩𝐡2 π‘₯𝑛 βˆ’ 𝐡2 𝑝󡄩󡄩󡄩

σ΅„©2 σ΅„©σ΅„© σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 = 󡄩󡄩󡄩Π𝐢 (𝑒𝑛 βˆ’ πœ‡1 𝐡1 𝑒𝑛 ) βˆ’ Π𝐢 (π‘ž βˆ’ πœ‡1 𝐡1 π‘ž)σ΅„©σ΅„©σ΅„© ≀ βŸ¨π‘’π‘› βˆ’ πœ‡1 𝐡1 𝑒𝑛 βˆ’ (π‘ž βˆ’ πœ‡1 𝐡1 π‘ž) , 𝐽 (V𝑛 βˆ’ 𝑝)⟩ = βŸ¨π‘’π‘› βˆ’ π‘ž, 𝐽 (V𝑛 βˆ’ 𝑝)⟩ + πœ‡1 ⟨𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 , 𝐽 (V𝑛 βˆ’ 𝑝)⟩ ≀ (68)

σ΅„© σ΅„©2 βˆ’ 2πœ‡1 (𝛽̂1 βˆ’ πœ…2 πœ‡1 ) 󡄩󡄩󡄩𝐡1 𝑒𝑛 βˆ’ 𝐡1 π‘žσ΅„©σ΅„©σ΅„© ]

1 σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 [󡄩𝑒 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 2 σ΅„© 𝑛 σ΅„© σ΅„© βˆ’ 𝑔2 (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) ] σ΅„© σ΅„©σ΅„© σ΅„© + πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ,

σ΅„©2 σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 which implies that

σ΅„© σ΅„©2 βˆ’ 2𝛿𝑛 [πœ‡2 (𝛽̂2 βˆ’ πœ…2 πœ‡2 ) 󡄩󡄩󡄩𝐡2 π‘₯𝑛 βˆ’ 𝐡2 𝑝󡄩󡄩󡄩

σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„© σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ≀ 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© βˆ’ 𝑔2 (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 .

σ΅„© σ΅„©2 +2πœ‡1 (𝛽̂1 βˆ’ πœ…2 πœ‡1 ) 󡄩󡄩󡄩𝐡1 𝑒𝑛 βˆ’ 𝐡1 π‘žσ΅„©σ΅„©σ΅„© ] , which hence implies that

σ΅„©2 σ΅„© σ΅„©2 σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© .

(69)

σ΅„©σ΅„©

σ΅„© βˆ’ 𝐡2 𝑝󡄩󡄩󡄩 = 0,

σ΅„©σ΅„©

lim 󡄩𝐡 𝑒 π‘›β†’βˆž σ΅„© 1 𝑛

σ΅„© βˆ’ 𝐡1 π‘žσ΅„©σ΅„©σ΅„© = 0. (70)

Utilizing Proposition 6 and Lemma 9, we have σ΅„©σ΅„© σ΅„©2 󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 = 󡄩󡄩󡄩Π𝐢 (π‘₯𝑛 βˆ’ πœ‡2 𝐡2 π‘₯𝑛 ) βˆ’ Π𝐢 (𝑝 βˆ’ πœ‡2 𝐡2 𝑝)σ΅„©σ΅„©σ΅„© ≀ ⟨π‘₯𝑛 βˆ’ πœ‡2 𝐡2 π‘₯𝑛 βˆ’ (𝑝 βˆ’ πœ‡2 𝐡2 𝑝) , 𝐽 (𝑒𝑛 βˆ’ π‘ž)⟩ = ⟨π‘₯𝑛 βˆ’ 𝑝, 𝐽 (𝑒𝑛 βˆ’ π‘ž)⟩ + πœ‡2 ⟨𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 , 𝐽 (𝑒𝑛 βˆ’ π‘ž)⟩ ≀

1 σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„© [σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© βˆ’ 𝑔1 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©)] 2 σ΅„© σ΅„©σ΅„© σ΅„© + πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© , (71)

(75)

σ΅„© σ΅„© σ΅„©σ΅„© + 2πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 .

By Lemma 8(i), we have from (68) and (75) σ΅„©σ΅„© σ΅„©2 󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 σ΅„© σ΅„© + 𝛿𝑛 [σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝑔1 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„© σ΅„© βˆ’ 𝑔2 (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) + 2πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© Γ— 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ] σ΅„© σ΅„©2 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩

σ΅„© σ΅„© βˆ’ 𝛿𝑛 [𝑔1 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©)

σ΅„© σ΅„© +𝑔2 (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©)]

which implies that σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„© 󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝑔1 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© .

σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝑔1 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© βˆ’ 𝑔2 (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©)

Since 0 < πœ‡π‘– < 𝛽̂𝑖 /πœ…2 for 𝑖 = 1, 2, and {π‘₯𝑛 }, {𝑦𝑛 } are bounded, we obtain from (64), (69), and condition (ii) that lim 󡄩𝐡 π‘₯ π‘›β†’βˆž σ΅„© 2 𝑛

(74)

Substituting (72) for (74), we get

σ΅„© σ΅„©2 2𝛿𝑛 [πœ‡2 (𝛽̂2 βˆ’ πœ…2 πœ‡2 ) 󡄩󡄩󡄩𝐡2 π‘₯𝑛 βˆ’ 𝐡2 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 +πœ‡1 (𝛽̂1 βˆ’ πœ…2 πœ‡1 ) 󡄩󡄩󡄩𝐡1 𝑒𝑛 βˆ’ 𝐡1 π‘žσ΅„©σ΅„©σ΅„© ]

(73)

σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© (72)

σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ,

(76)

12

Abstract and Applied Analysis σ΅„© σ΅„©2 = 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + (𝛽𝑛 + 𝛾𝑛 )

which hence leads to σ΅„© σ΅„© σ΅„© σ΅„© 𝛿𝑛 [𝑔1 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) + 𝑔2 (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©)] σ΅„© σ΅„© σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 2πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©

σ΅„©σ΅„© 𝛽 σ΅„©σ΅„©2 𝛾𝑛 σ΅„© σ΅„© 𝑛 Γ— σ΅„©σ΅„©σ΅„© (π‘₯𝑛 βˆ’ 𝑝) + (𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© 𝛽𝑛 + 𝛾𝑛 σ΅„©σ΅„© 𝛽𝑛 + 𝛾𝑛 σ΅„© σ΅„©2 ≀ 𝛿𝑛 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + (𝛽𝑛 + 𝛾𝑛 ) Γ—[

σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 .

βˆ’ (77)

From (70), (77), condition (ii), and the boundedness of {π‘₯𝑛 }, {𝑦𝑛 }, {𝑒𝑛 }, and {V𝑛 }, we deduce that lim 𝑔 π‘›β†’βˆž 1

σ΅„© σ΅„© (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) = 0,

σ΅„© σ΅„© lim 𝑔 (󡄩󡄩𝑒 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) = 0. π‘›β†’βˆž 2 σ΅„© 𝑛

(78)

σ΅„© σ΅„© lim 󡄩󡄩𝑒 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„© = 0. π‘›β†’βˆž σ΅„© 𝑛

𝛽𝑛 𝛾𝑛

2

(𝛽𝑛 + 𝛾𝑛 )

σ΅„© σ΅„© 𝑔3 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©)]

σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 σ΅„©2 ≀ 𝛿𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’

𝛽𝑛 𝛾𝑛 σ΅„© σ΅„© 𝑔 (σ΅„©σ΅„©π‘₯ βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) 𝛽𝑛 + 𝛾𝑛 3 σ΅„© 𝑛

𝛽𝛾 σ΅„© σ΅„© σ΅„©2 σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝑛 𝑛 𝑔3 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) , 𝛽𝑛 + 𝛾𝑛 (83) which immediately implies that

Utilizing the properties of 𝑔1 and 𝑔2 , we deduce that σ΅„© σ΅„© lim σ΅„©σ΅„©π‘₯ βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„© = 0, π‘›β†’βˆž σ΅„© 𝑛

𝛽𝑛 σ΅„©σ΅„© 𝛾𝑛 σ΅„©σ΅„© σ΅„©2 σ΅„©2 σ΅„©π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩𝐡 π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 𝛽𝑛 + 𝛾𝑛 σ΅„© 𝑛 𝛽𝑛 + 𝛾𝑛 σ΅„© 𝑛 𝑛

σ΅„© σ΅„© 𝛽𝑛 𝛾𝑛 𝑔3 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) (79)

≀

σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© .

From (79), we get σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛 βˆ’ V𝑛 σ΅„©σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„© 󳨀→ 0 as 𝑛 󳨀→ ∞. (80) That is,

𝛽𝑛 𝛾𝑛 σ΅„© σ΅„© 𝑔 (σ΅„©σ΅„©π‘₯ βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) 𝛽𝑛 + 𝛾𝑛 3 σ΅„© 𝑛

(84)

So, from (64), the boundedness of {π‘₯𝑛 }, {𝑦𝑛 }, and conditions (ii), (iv), it follows that lim 𝑔 π‘›β†’βˆž 3

σ΅„© σ΅„© (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) = 0.

(85)

From the properties of 𝑔3 , we have σ΅„©σ΅„©

lim σ΅„©π‘₯ π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

σ΅„©σ΅„©

σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0,

σ΅„© βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(86)

Taking into account that

Next, let us show that lim 󡄩𝐡 π‘₯ π‘›β†’βˆž σ΅„© 𝑛 𝑛

σ΅„©σ΅„©

lim σ΅„©π‘₯ π‘›β†’βˆž σ΅„© 𝑛

(81)

σ΅„© σ΅„© lim 󡄩󡄩𝑆π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

π‘›β†’βˆž σ΅„©

(82)

𝑦𝑛 βˆ’ π‘₯𝑛 = 𝛾𝑛 (𝐡𝑛 π‘₯𝑛 βˆ’ π‘₯𝑛 ) + 𝛿𝑛 (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 ) , we have σ΅„© σ΅„© 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© = 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘₯𝑛 βˆ’ 𝛾𝑛 (𝐡𝑛 π‘₯𝑛 βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© .

Indeed, utilizing Lemma 15 and (42), we have σ΅„©σ΅„© σ΅„©2 󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© σ΅„©σ΅„©2 𝛽 π‘₯ + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 σ΅„© σ΅„© = 󡄩󡄩󡄩𝛿𝑛 (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝) + (𝛽𝑛 + 𝛾𝑛 ) ( 𝑛 𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© 𝛽𝑛 + 𝛾𝑛 σ΅„© σ΅„©2 ≀ 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + (𝛽𝑛 + 𝛾𝑛 ) σ΅„©σ΅„©σ΅„©2 σ΅„©σ΅„©σ΅„© 𝛽 π‘₯ + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 Γ— σ΅„©σ΅„©σ΅„© 𝑛 𝑛 𝛽𝑛 + 𝛾𝑛 σ΅„©σ΅„© σ΅„©σ΅„©

(87)

(88)

From (64), (86), and condition (ii), it follows that σ΅„©σ΅„©

lim 󡄩𝑆 𝐺π‘₯𝑛 π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0,

σ΅„©σ΅„©

lim 󡄩𝑆 𝐺π‘₯𝑛 π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0. (89)

Abstract and Applied Analysis

13 σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 𝛼𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛼𝑛 )

Note that

σ΅„© σ΅„© Γ— [𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©

σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 󡄩󡄩󡄩𝑆𝑛 π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„©

(90)

σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 󡄩󡄩󡄩𝑆𝑛 π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© .

So, in terms of (81), (89), and Lemma 12, we have σ΅„©σ΅„©

lim σ΅„©π‘₯ π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(91)

Suppose that 𝛽𝑛 ≑ 𝛽 for some fixed 𝛽 ∈ (0, 1) such that 𝛽 + 𝛾𝑛 + 𝛿𝑛 = 1 for all 𝑛 β‰₯ 0. Define a mapping 𝑉π‘₯ = (1 βˆ’ πœƒ1 βˆ’ πœƒ2 )𝑆π‘₯ + πœƒ1 𝐡π‘₯ + πœƒ2 𝐺π‘₯, where πœƒ1 , πœƒ2 ∈ (0, 1) are two constants with πœƒ1 + πœƒ2 < 1. Then, by Lemmas 14 and 17, we have that Fix(𝑉) = Fix(𝑆) ∩ Fix(𝐡) ∩ Fix(𝐺) = 𝐹. For each π‘˜ β‰₯ 1, let {π‘π‘˜ } be a unique element of 𝐢 such that π‘π‘˜ =

1 1 𝑓 (π‘π‘˜ ) + (1 βˆ’ ) π‘‰π‘π‘˜ . π‘˜ π‘˜

(92)

From Lemma 13, we conclude that π‘π‘˜ β†’ π‘ž ∈ Fix(𝑉) = 𝐹 as π‘˜ β†’ ∞. Observe that for every 𝑛, π‘˜ σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑦𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© = 󡄩󡄩󡄩𝛽 (π‘₯𝑛 βˆ’ π΅π‘π‘˜ ) + 𝛾𝑛 (𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ ) + 𝛿𝑛 (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π΅π‘π‘˜ )σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© σ΅„© σ΅„© + 𝛿𝑛 (󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„© σ΅„© = 𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© ,

σ΅„© σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛽) (󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝐡𝑛 π‘π‘˜ σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 𝛼𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 ) [𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛽) (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„© = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + (𝛽 + 𝛼𝑛 (1 βˆ’ 𝛽)) σ΅„© σ΅„© σ΅„© σ΅„© Γ— σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛼𝑛 ) (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 ) [(1 βˆ’ 𝛽) 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + (𝛽 + 𝛼𝑛 (1 βˆ’ 𝛽)) σ΅„© σ΅„© σ΅„© σ΅„© Γ— (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 ) (1 βˆ’ 𝛽) (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 ) [(1 βˆ’ 𝛽) 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© ] σ΅„© σ΅„© = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + (𝛽 + 𝛼𝑛 (1 βˆ’ 𝛽)) σ΅„© σ΅„© σ΅„© σ΅„© Γ— σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛼𝑛 ) (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 ) [(1 βˆ’ 𝛽) 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©] + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© . (94) So, it immediately follows from 0 ≀ 𝛼𝑛 ≀ 1 βˆ’ 𝜌, for all 𝑛 β‰₯ 𝑛0 , that

(93)

and hence σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛼𝑛 ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 (󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 𝛼𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛼𝑛 ) σ΅„© σ΅„© Γ— [𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛽) 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©]

σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© 1 (1 βˆ’ 𝛼𝑛 ) (1 βˆ’ 𝛽) σ΅„© σ΅„© σ΅„© σ΅„© Γ— (󡄩󡄩󡄩𝛼𝑛 (π‘₯𝑛 βˆ’ 𝑓 (π‘₯𝑛 ))σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„©)

σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© +

𝛿𝑛 σ΅„©σ΅„© σ΅„© 󡄩𝑆 𝐺π‘₯ βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 1βˆ’π›½σ΅„© 𝑛 𝑛 σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© +

1 𝜌 (1 βˆ’ 𝛽) σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© Γ— (󡄩󡄩𝛼𝑛 (π‘₯𝑛 βˆ’ 𝑓 (π‘₯𝑛 ))σ΅„©σ΅„© + σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœƒπ‘› , βˆ€π‘› β‰₯ 𝑛0 , σ΅„© σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© +

(95)

14

Abstract and Applied Analysis

where πœƒπ‘› = ‖𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ β€– + ‖𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 β€– + (1/𝜌(1 βˆ’ 𝛽))(‖𝛼𝑛 (π‘₯𝑛 βˆ’ 𝑓(π‘₯𝑛 ))β€– + β€–π‘₯𝑛 βˆ’ π‘₯𝑛+1 β€–). Since lim𝑛 β†’ ∞ ‖𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ β€– = lim𝑛 β†’ ∞ ‖𝑆𝑛 𝐺π‘₯𝑛 βˆ’π΅π‘› π‘₯𝑛 β€– = lim𝑛 β†’ ∞ ‖𝛼𝑛 (π‘₯𝑛 βˆ’π‘“(π‘₯𝑛 ))β€– = lim𝑛 β†’ ∞ β€–π‘₯𝑛 βˆ’ π‘₯𝑛+1 β€– = 0, we know that πœƒπ‘› β†’ 0 as 𝑛 β†’ ∞. From (95), we obtain

1 1 (1 βˆ’ ) (π‘₯𝑛 βˆ’ π‘‰π‘π‘˜ ) = π‘₯𝑛 βˆ’ π‘π‘˜ βˆ’ (π‘₯𝑛 βˆ’ 𝑓 (π‘π‘˜ )) . π‘˜ π‘˜

(102)

It follows from Lemma 8 (ii) and (102) that

σ΅„©2 σ΅„© σ΅„©2 σ΅„©σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© + πœƒπ‘› (2 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœƒπ‘› ) ,

that is,

βˆ€π‘› β‰₯ 𝑛0 .

(96)

For any Banach limit πœ‡, from (96), we derive σ΅„© σ΅„©2 σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© = πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© = πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© .

1 2σ΅„© σ΅„©2 (1 βˆ’ ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘‰π‘π‘˜ σ΅„©σ΅„©σ΅„© π‘˜ σ΅„©2 2 σ΅„© β‰₯ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© βˆ’ ⟨π‘₯𝑛 βˆ’ π‘π‘˜ + π‘π‘˜ βˆ’ 𝑓 (π‘π‘˜ ) , 𝐽 (π‘₯𝑛 βˆ’ π‘π‘˜ )⟩ π‘˜

(97)

2 σ΅„© σ΅„©2 2 = (1 βˆ’ ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + βŸ¨π‘“ (π‘π‘˜ ) βˆ’ π‘π‘˜ , 𝐽 (π‘₯𝑛 βˆ’ π‘π‘˜ )⟩ . π‘˜ π‘˜ (103)

In addition, note that σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„©

So by (100) and (103), we have

σ΅„©2 σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 + 𝐺π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„©

1 2 σ΅„© σ΅„©2 (1 βˆ’ ) πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© π‘˜

σ΅„© σ΅„© σ΅„©2 σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„©2 σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— (2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©) , σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 + 𝑆π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„©2 σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— (2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„©) .

2 σ΅„© σ΅„©2 β‰₯ (1 βˆ’ ) πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© π‘˜

σ΅„© σ΅„©2 σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„© ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© .

and hence 1 σ΅„©σ΅„© σ΅„©2 2 πœ‡ σ΅„©π‘₯ βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© β‰₯ πœ‡π‘› βŸ¨π‘“ (π‘π‘˜ ) βˆ’ π‘π‘˜ , 𝐽 (π‘₯𝑛 βˆ’ π‘π‘˜ )⟩ . π‘˜2 𝑛 σ΅„© 𝑛 π‘˜

1 σ΅„©σ΅„© σ΅„©2 πœ‡ σ΅„©π‘₯ βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© β‰₯ πœ‡π‘› βŸ¨π‘“ (π‘π‘˜ ) βˆ’ π‘π‘˜ , 𝐽 (π‘₯𝑛 βˆ’ π‘π‘˜ )⟩ . 2π‘˜ 𝑛 σ΅„© 𝑛

(99) πœ‡π‘› βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛 βˆ’ π‘ž)⟩ ≀ 0.

󡄨 󡄨 lim σ΅„¨σ΅„¨βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ βˆ’ βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛 βˆ’ π‘ž)βŸ©σ΅„¨σ΅„¨σ΅„¨ = 0.

π‘›β†’βˆž 󡄨

(108) (100)

So, utilizing Lemma 18 we deduce from (107) and (108) that lim sup βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛 βˆ’ π‘ž)⟩ ≀ 0, π‘›β†’βˆž

σ΅„© σ΅„©2 ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© .

(109)

which together with (49) and the norm-to-norm uniform continuity of 𝐽 on bounded subsets of 𝑋, implies that

Also, observe that 1 1 (π‘₯𝑛 βˆ’ 𝑓 (π‘π‘˜ )) + (1 βˆ’ ) (π‘₯𝑛 βˆ’ π‘‰π‘π‘˜ ) ; π‘˜ π‘˜

(107)

On the other hand, from (49) and the norm-to-norm uniform continuity of 𝐽 on bounded subsets of 𝑋, it follows that

σ΅„© σ΅„©2 σ΅„© σ΅„©2 + πœƒ1 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœƒ2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„©

π‘₯𝑛 βˆ’ π‘π‘˜ =

(106)

Since π‘π‘˜ β†’ π‘ž ∈ Fix(𝑉) = 𝐹 as π‘˜ β†’ ∞, by the uniform Frechet differentiability of the norm of 𝑋 we have

σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘‰π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© = πœ‡π‘› σ΅„©σ΅„©σ΅„©(1 βˆ’ πœƒ1 βˆ’ πœƒ2 ) (π‘₯𝑛 βˆ’ π‘†π‘π‘˜ ) σ΅„© σ΅„©2 ≀ (1 βˆ’ πœƒ1 βˆ’ πœƒ2 ) πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„©

(105)

This implies that

Utilizing (97) and (99), we deduce that

σ΅„©2 +πœƒ1 (π‘₯𝑛 βˆ’ π΅π‘π‘˜ ) + πœƒ2 (π‘₯𝑛 βˆ’ πΊπ‘π‘˜ )σ΅„©σ΅„©σ΅„©

2 + πœ‡π‘› βŸ¨π‘“ (π‘π‘˜ ) βˆ’ π‘π‘˜ , 𝐽 (π‘₯𝑛 βˆ’ π‘π‘˜ )⟩ , π‘˜

(98)

It is easy to see from (81) and (91) that σ΅„© σ΅„©2 σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„© ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© ,

(104)

(101)

lim sup βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ ≀ 0. π‘›β†’βˆž

(110)

Abstract and Applied Analysis

15

Finally, let us show that π‘₯𝑛 β†’ π‘ž as 𝑛 β†’ ∞. Utilizing Lemma 8 (i), from (42) and the convexity of β€– β‹… β€–2 , we get σ΅„©σ΅„© σ΅„©2 󡄩󡄩𝑦𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© (111) σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 + 𝛿𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© , σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© = 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘ž)) + (1 βˆ’ 𝛼𝑛 ) (𝑦𝑛 βˆ’ π‘ž) σ΅„©2 + 𝛼𝑛 (𝑓 (π‘ž) βˆ’ π‘ž)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 ≀ 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘ž)) + (1 βˆ’ 𝛼𝑛 ) (𝑦𝑛 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„© + 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘ž)σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛼𝑛 ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ 𝛼𝑛 πœŒσ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛼𝑛 ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

(112)

+ 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ σ΅„© σ΅„©2 = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ σ΅„© σ΅„©2 = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© 2 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ + 𝛼𝑛 (1 βˆ’ 𝜌) . 1βˆ’πœŒ Applying Lemma 7 to (112), we obtain that π‘₯𝑛 β†’ π‘ž as 𝑛 β†’ ∞. Conversely, if π‘₯𝑛 β†’ π‘ž ∈ 𝐹 as 𝑛 β†’ ∞, then from (42) it follows that σ΅„©σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑦𝑛 βˆ’ π‘žσ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© (113) σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© ≀ 𝛽𝑛 σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„© + 𝛾𝑛 σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„© + 𝛿𝑛 σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© 󳨀→ 0 as 𝑛 󳨀→ ∞, that is, 𝑦𝑛 β†’ π‘ž. Again from (42) we obtain that σ΅„©σ΅„© σ΅„©σ΅„© 󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 βˆ’ (1 βˆ’ 𝛼𝑛 ) (𝑦𝑛 βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛼𝑛 ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 ) (󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©)

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© . (114) Since π‘₯𝑛 β†’ π‘ž and 𝑦𝑛 β†’ π‘ž, we get 𝛼𝑛 (𝑓(π‘₯𝑛 ) βˆ’ π‘₯𝑛 ) β†’ 0. This completes the proof. Corollary 25. Let 𝐢 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢. Let {πœŒπ‘› }∞ 𝑛=0 be a sequence of positive numbers in (0, 𝑏] for some 𝑏 ∈ (0, 1) and 𝐴 𝑖 : 𝐢 β†’ 𝐸 an 𝛼̂𝑖 -inverse strongly accretive mapping for each 𝑖 = 0, 1, . . .. Define a mapping 𝐺𝑖 : 𝐢 β†’ 𝐢 by Π𝐢(𝐼 βˆ’ πœ† 𝑖 𝐴 𝑖 )π‘₯ = 𝐺𝑖 π‘₯ for all π‘₯ ∈ 𝐢 and 𝑖 = 0, 1, . . ., where πœ† 𝑖 ∈ (0, 𝛼̂𝑖 /πœ…2 ] and πœ… is the 2-uniformly smooth constant of 𝑋. Let 𝐡𝑛 : 𝐢 β†’ 𝐢 be the π‘Š-mapping generated by 𝐺𝑛 , πΊπ‘›βˆ’1 , . . . , 𝐺0 and πœŒπ‘› , πœŒπ‘›βˆ’1 , . . . , 𝜌0 . Let 𝑉 : 𝐢 β†’ 𝐢 be an 𝛼-strictly pseudocontractive mapping. Let 𝑓 : 𝐢 β†’ 𝐢 be a contraction with coefficient 𝜌 ∈ (0, 1). Let {𝑆𝑖 }∞ 𝑖=0 be a countable family of nonexpansive mappings of 𝐢 into itself such ∞ that 𝐹 = (β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 )) ∩ Fix(𝑉) ∩ (⋂𝑖=0 VI(𝐢, 𝐴 𝑖 )) =ΜΈ 0. For arbitrarily given π‘₯0 ∈ 𝐢, let {π‘₯𝑛 } be the sequence generated by 𝑦𝑛 = 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 ((1 βˆ’ 𝑙) 𝐼 + 𝑙𝑉) π‘₯𝑛 , π‘₯𝑛+1 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + (1 βˆ’ 𝛼𝑛 ) 𝑦𝑛 ,

βˆ€π‘› β‰₯ 0,

(115)

where 0 < 𝑙 < 𝛼/πœ…2 , {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, and {𝛿𝑛 } are the sequences in [0, 1] such that 𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 for all 𝑛 β‰₯ 0. Suppose that the following conditions hold: (i) βˆ‘βˆž 𝑛=0 𝛼𝑛 = ∞ and 0 ≀ 𝛼𝑛 ≀ 1 βˆ’ 𝜌, for all 𝑛 β‰₯ 𝑛0 for some integer 𝑛0 β‰₯ 0; (ii) lim inf 𝑛 β†’ ∞ 𝛾𝑛 > 0 and lim inf 𝑛 β†’ ∞ 𝛿𝑛 > 0; (iii) lim𝑛 β†’ ∞ (|𝛼𝑛+1 /(1 βˆ’ (1 βˆ’ 𝛼𝑛+1 )𝛽𝑛+1 ) βˆ’ 𝛼𝑛 /(1 βˆ’ (1 βˆ’ 𝛼𝑛 )𝛽𝑛 )| + |𝛿𝑛+1 /(1 βˆ’ 𝛽𝑛+1 ) βˆ’ 𝛿𝑛 /(1 βˆ’ 𝛽𝑛 )|) = 0; (iv) 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1. Assume that βˆ‘βˆž 𝑛=0 supπ‘₯∈𝐷 ‖𝑆𝑛+1 π‘₯ βˆ’ 𝑆𝑛 π‘₯β€– < ∞ for any bounded subset 𝐷 of 𝐢 and let 𝑆 be a mapping of 𝐢 into itself defined by 𝑆π‘₯ = lim𝑛 β†’ ∞ 𝑆𝑛 π‘₯ for all π‘₯ ∈ 𝐢 and suppose that Fix(𝑆) = β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 ). Then, there hold the following: (I) lim𝑛 β†’ ∞ β€–π‘₯𝑛+1 βˆ’ π‘₯𝑛 β€– = 0; (II) π‘₯𝑛 β†’ π‘ž ⇔ 𝛼𝑛 (𝑓(π‘₯𝑛 ) βˆ’ π‘₯𝑛 ) β†’ 0 provided 𝛽𝑛 ≑ 𝛽 for some fixed 𝛽 ∈ (0, 1), where π‘ž ∈ 𝐹 solves the following VIP βŸ¨π‘ž βˆ’ 𝑓 (π‘ž) , 𝐽 (π‘ž βˆ’ 𝑝)⟩ ≀ 0,

βˆ€π‘ ∈ 𝐹.

(116)

Proof. In Theorem 24, we put 𝐡1 = 𝐼 βˆ’ 𝑉, 𝐡2 = 0, and πœ‡1 = 𝑙, where 0 < 𝑙 < 𝛼/πœ…2 . Then, GSVI (13) is equivalent to the VIP of finding π‘₯βˆ— ∈ 𝐢 such that ⟨𝐡1 π‘₯βˆ— , 𝐽 (π‘₯ βˆ’ π‘₯βˆ— )⟩ β‰₯ 0,

βˆ€π‘₯ ∈ 𝐢.

(117)

16

Abstract and Applied Analysis

In this case, 𝐡1 : 𝐢 β†’ 𝑋 is 𝛼-inverse strongly accretive. It is not hard to see that Fix(𝑉) = VI(𝐢, 𝐡1 ). As a matter of fact, we have, for 𝑙 > 0, 𝑒 ∈ VI (𝐢, 𝐡1 )

σ΅„© σ΅„©σ΅„© σ΅„© 1 σ΅„©σ΅„© σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 . σ΅„©σ΅„©(𝐼 βˆ’ 𝐡𝑖 ) π‘₯ βˆ’ (𝐼 βˆ’ 𝐡𝑖 ) 𝑦󡄩󡄩󡄩 ≀ πœ†π‘– σ΅„©

βˆ€π‘¦ ∈ 𝐢

≀ (1 +

⇐⇒ 𝑒 = Π𝐢 (𝑒 βˆ’ 𝑙𝐡1 𝑒) ⇐⇒ 𝑒 = Π𝐢 (𝑒 βˆ’ 𝑙𝑒 + 𝑙𝑉𝑒)

(118)

⇐⇒ βŸ¨π‘’ βˆ’ 𝑙𝑒 + 𝑙𝑉𝑒 βˆ’ 𝑒, 𝐽 (𝑒 βˆ’ 𝑦)⟩ β‰₯ 0

βˆ€π‘¦ ∈ 𝐢

(123)

1 σ΅„©σ΅„© σ΅„© ) σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 . πœ†π‘– σ΅„©

Utilizing the 𝛼𝑖 -strong accretivity and πœ† 𝑖 -strict pseudocontractivity of 𝐡𝑖 , we get σ΅„© σ΅„©2 πœ† 𝑖 σ΅„©σ΅„©σ΅„©(𝐼 βˆ’ 𝐡𝑖 ) π‘₯ βˆ’ (𝐼 βˆ’ 𝐡𝑖 ) 𝑦󡄩󡄩󡄩

⇐⇒ βŸ¨π‘’ βˆ’ 𝑉𝑒, 𝐽 (𝑒 βˆ’ 𝑦)⟩ ≀ 0 βˆ€π‘¦ ∈ 𝐢

σ΅„©2 σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 βˆ’ βŸ¨π΅π‘– π‘₯ βˆ’ 𝐡𝑖 𝑦, 𝐽 (π‘₯ βˆ’ 𝑦)⟩

⇐⇒ 𝑒 = 𝑉𝑒

(124)

σ΅„©2 σ΅„© ≀ (1 βˆ’ 𝛼𝑖 ) σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 .

⇐⇒ 𝑒 ∈ Fix (𝑉) . Accordingly, we know that 𝐹 = (β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 )) ∩ Ξ© ∩ ∞ VI(𝐢, 𝐴 )) = (β‹‚ Fix(𝑆 )) ∩ Fix(𝑉) ∩ (β‹‚βˆž (β‹‚βˆž 𝑖 𝑖 𝑖=0 𝑖=0 𝑖=0 VI(𝐢, 𝐴 𝑖 )), and Π𝐢 (𝐼 βˆ’ πœ‡1 𝐡1 ) Π𝐢 (𝐼 βˆ’ πœ‡2 𝐡2 ) π‘₯𝑛 = Π𝐢 (𝐼 βˆ’ πœ‡1 𝐡1 ) π‘₯𝑛 = Π𝐢 ((1 βˆ’ 𝑙) π‘₯𝑛 + 𝑙𝑉π‘₯𝑛 )

So, we have 1 βˆ’ 𝛼𝑖 σ΅„©σ΅„© σ΅„© σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©(𝐼 βˆ’ 𝐡𝑖 ) π‘₯ βˆ’ (𝐼 βˆ’ 𝐡𝑖 ) 𝑦󡄩󡄩󡄩 ≀ √ σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 . πœ† σ΅„©

(119)

So, the scheme (42) reduces to (115). Therefore, the desired result follows from Theorem 24. Here, we prove the following important lemmas which will be used in the sequel. Lemma 26. Let 𝐢 be a nonempty closed convex subset of a smooth Banach space 𝑋 and let the mapping 𝐡𝑖 : 𝐢 β†’ 𝑋 be πœ† 𝑖 -strictly pseudocontractive and 𝛼𝑖 -strongly accretive with 𝛼𝑖 + πœ† 𝑖 β‰₯ 1 for 𝑖 = 1, 2. Then, for πœ‡π‘– ∈ (0, 1] one has σ΅„© σ΅„©σ΅„© σ΅„©σ΅„©(𝐼 βˆ’ πœ‡π‘– 𝐡𝑖 ) π‘₯ βˆ’ (𝐼 βˆ’ πœ‡π‘– 𝐡𝑖 ) 𝑦󡄩󡄩󡄩

σ΅„© σ΅„©σ΅„© σ΅„©σ΅„©(𝐼 βˆ’ πœ‡π‘– 𝐡𝑖 ) π‘₯ βˆ’ (𝐼 βˆ’ πœ‡π‘– 𝐡𝑖 ) 𝑦󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©(𝐼 βˆ’ 𝐡𝑖 ) π‘₯ βˆ’ (𝐼 βˆ’ 𝐡𝑖 ) 𝑦󡄩󡄩󡄩 + (1 βˆ’ πœ‡π‘– ) 󡄩󡄩󡄩𝐡𝑖 π‘₯ βˆ’ 𝐡𝑖 𝑦󡄩󡄩󡄩 β‰€βˆš

1 βˆ’ 𝛼𝑖 σ΅„©σ΅„© 1 σ΅„© σ΅„© σ΅„© σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 + (1 βˆ’ πœ‡π‘– ) (1 + ) σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 πœ†π‘– σ΅„© πœ†π‘–

= {√

1 βˆ’ 𝛼𝑖 1 σ΅„© σ΅„© + (1 βˆ’ πœ‡π‘– ) (1 + )} σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 . πœ†π‘– πœ†π‘– (126)

Since 1 βˆ’ (πœ† 𝑖 /(1 + πœ† 𝑖 ))(1 βˆ’ √(1 βˆ’ 𝛼𝑖 )/πœ† 𝑖 ) ≀ πœ‡π‘– ≀ 1, it follows immediately that √

1 βˆ’ 𝛼𝑖 1 σ΅„© σ΅„© + (1 βˆ’ πœ‡π‘– ) (1 + )} σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 , πœ†π‘– πœ†π‘–

(120)

βˆ€π‘₯, 𝑦 ∈ 𝐢, for 𝑖 = 1, 2. In particular, if 1βˆ’(πœ† 𝑖 /(1+πœ† 𝑖 ))(1βˆ’βˆš(1 βˆ’ 𝛼𝑖 )/πœ† 𝑖 ) ≀ πœ‡π‘– ≀ 1, then 𝐼 βˆ’ πœ‡π‘– 𝐡𝑖 is nonexpansive for 𝑖 = 1, 2. Proof. Taking into account the πœ† 𝑖 -strict pseudocontractivity of 𝐡𝑖 , we derive for every π‘₯, 𝑦 ∈ 𝐢

σ΅„© σ΅„©σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©(𝐼 βˆ’ 𝐡𝑖 ) π‘₯ βˆ’ (𝐼 βˆ’ 𝐡𝑖 ) 𝑦󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 ,

1 βˆ’ 𝛼𝑖 1 + (1 βˆ’ πœ‡π‘– ) (1 + ) ≀ 1. πœ†π‘– πœ†π‘–

(127)

This implies that 𝐼 βˆ’ πœ‡π‘– 𝐡𝑖 is nonexpansive for 𝑖 = 1, 2. Lemma 27. Let 𝐢 be a nonempty closed convex subset of a smooth Banach space 𝑋. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢 and let the mapping 𝐡𝑖 : 𝐢 β†’ 𝑋 be πœ† 𝑖 -strictly pseudocontractive and 𝛼𝑖 -strongly accretive with 𝛼𝑖 + πœ† 𝑖 β‰₯ 1 for 𝑖 = 1, 2. Let 𝐺 : 𝐢 β†’ 𝐢 be the mapping defined by 𝐺 (π‘₯) = Π𝐢 [Π𝐢 (π‘₯ βˆ’ πœ‡2 𝐡2 π‘₯) βˆ’ πœ‡1 𝐡1 Π𝐢 (π‘₯ βˆ’ πœ‡2 𝐡2 π‘₯)] , βˆ€π‘₯ ∈ 𝐢. (128)

σ΅„© σ΅„©2 πœ† 𝑖 σ΅„©σ΅„©σ΅„©(𝐼 βˆ’ 𝐡𝑖 ) π‘₯ βˆ’ (𝐼 βˆ’ 𝐡𝑖 ) 𝑦󡄩󡄩󡄩 ≀ ⟨(𝐼 βˆ’ 𝐡𝑖 ) π‘₯ βˆ’ (𝐼 βˆ’ 𝐡𝑖 ) 𝑦, 𝐽 (π‘₯ βˆ’ 𝑦)⟩

(125)

𝑖

Therefore, for πœ‡π‘– ∈ (0, 1] we have

= ((1 βˆ’ 𝑙) 𝐼 + 𝑙𝑉) π‘₯𝑛 .

≀ {√

(122)

Hence, σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© 󡄩󡄩𝐡𝑖 π‘₯ βˆ’ 𝐡𝑖 𝑦󡄩󡄩󡄩 ≀ σ΅„©σ΅„©σ΅„©(𝐼 βˆ’ 𝐡𝑖 ) π‘₯ βˆ’ (𝐼 βˆ’ 𝐡𝑖 ) 𝑦󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩

⇐⇒ ⟨𝐡1 𝑒, 𝐽 (𝑦 βˆ’ 𝑒)⟩ β‰₯ 0 βˆ€π‘¦ ∈ 𝐢 ⇐⇒ βŸ¨π‘’ βˆ’ 𝑙𝐡1 𝑒 βˆ’ 𝑒, 𝐽 (𝑒 βˆ’ 𝑦)⟩ β‰₯ 0

which implies that

(121)

If 1βˆ’(πœ† 𝑖 /(1+πœ† 𝑖 ))(1βˆ’βˆš(1 βˆ’ 𝛼𝑖 )/πœ† 𝑖 ) ≀ πœ‡π‘– ≀ 1, then 𝐺 : 𝐢 β†’ 𝐢 is nonexpansive.

Abstract and Applied Analysis

17

Proof. According to Lemma 26, we know that 𝐼 βˆ’ πœ‡π‘– 𝐡𝑖 is nonexpansive for 𝑖 = 1, 2. Hence, for all π‘₯, 𝑦 ∈ 𝐢, we have σ΅„© σ΅„©σ΅„© 󡄩󡄩𝐺 (π‘₯) βˆ’ 𝐺 (𝑦)σ΅„©σ΅„©σ΅„© σ΅„© = 󡄩󡄩󡄩Π𝐢 [Π𝐢 (π‘₯ βˆ’ πœ‡2 𝐡2 π‘₯) βˆ’ πœ‡1 𝐡1 Π𝐢 (π‘₯ βˆ’ πœ‡2 𝐡2 π‘₯)]

(iv) 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1.

σ΅„© βˆ’ Π𝐢 [Π𝐢 (𝑦 βˆ’ πœ‡2 𝐡2 𝑦) βˆ’ πœ‡1 𝐡1 Π𝐢 (𝑦 βˆ’ πœ‡2 𝐡2 𝑦)]σ΅„©σ΅„©σ΅„©

σ΅„© = 󡄩󡄩󡄩Π𝐢 (𝐼 βˆ’ πœ‡1 𝐡1 ) Π𝐢 (𝐼 βˆ’ πœ‡2 𝐡2 ) π‘₯

(II) π‘₯𝑛 β†’ π‘ž ⇔ 𝛼𝑛 (𝑓(π‘₯𝑛 ) βˆ’ π‘₯𝑛 ) β†’ 0 provided 𝛽𝑛 ≑ 𝛽 for some fixed 𝛽 ∈ (0, 1), where π‘ž ∈ 𝐹 solves the following VIP

σ΅„© ≀ σ΅„©σ΅„©σ΅„©(𝐼 βˆ’ πœ‡1 𝐡1 ) Π𝐢 (𝐼 βˆ’ πœ‡2 𝐡2 ) π‘₯

σ΅„© βˆ’ (𝐼 βˆ’ πœ‡1 𝐡1 ) Π𝐢 (𝐼 βˆ’ πœ‡2 𝐡2 ) 𝑦󡄩󡄩󡄩

βŸ¨π‘ž βˆ’ 𝑓 (π‘ž) , 𝐽 (π‘ž βˆ’ 𝑝)⟩ ≀ 0,

σ΅„© σ΅„© ≀ 󡄩󡄩󡄩Π𝐢 (𝐼 βˆ’ πœ‡2 𝐡2 ) π‘₯ βˆ’ Π𝐢 (𝐼 βˆ’ πœ‡2 𝐡2 ) 𝑦󡄩󡄩󡄩 σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©(𝐼 βˆ’ πœ‡2 𝐡2 ) π‘₯ βˆ’ (𝐼 βˆ’ πœ‡2 𝐡2 ) 𝑦󡄩󡄩󡄩

(129)

This shows that 𝐺 : 𝐢 β†’ 𝐢 is nonexpansive. This completes the proof. Theorem 28. Let 𝐢 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢. Let {πœŒπ‘› }∞ 𝑛=0 be a sequence of positive numbers in (0, 𝑏] for some 𝑏 ∈ (0, 1) and 𝐴 𝑖 : 𝐢 β†’ 𝑋 be πœ‰π‘– -strictly pseudocontractive and 𝛼̂𝑖 -strongly accretive with πœ‰π‘– + 𝛼̂𝑖 β‰₯ 1 for each 𝑖 = 0, 1, . . .. Define a mapping 𝐺𝑖 : 𝐢 β†’ 𝐢 by Π𝐢(𝐼 βˆ’ πœ† 𝑖 𝐴 𝑖 )π‘₯ = 𝐺𝑖 π‘₯ for all π‘₯ ∈ 𝐢 and 𝑖 = 0, 1, . . ., where 1 βˆ’ (πœ‰π‘– /(1 + πœ‰π‘– ))(1 βˆ’ √(1 βˆ’ 𝛼̂𝑖 )/πœ‰π‘– ) ≀ πœ† 𝑖 ≀ 1 for all 𝑖 = 0, 1, . . .. Let 𝐡𝑛 : 𝐢 β†’ 𝐢 be the π‘Š-mapping generated by 𝐺𝑛 , πΊπ‘›βˆ’1 , . . . , 𝐺0 and πœŒπ‘› , πœŒπ‘›βˆ’1 , . . . , 𝜌0 . Let the mapping 𝐡𝑖 : 𝐢 β†’ 𝑋 πœπ‘– -strictly pseudocontractive and 𝛽̂𝑖 -strongly accretive with πœπ‘– + 𝛽̂𝑖 β‰₯ 1 for 𝑖 = 1, 2. Let 𝑓 : 𝐢 β†’ 𝐢 be a contraction with coefficient 𝜌 ∈ (0, 1). Let {𝑆𝑖 }∞ 𝑖=0 be a countable family of nonexpansive mappings of 𝐢 into itself such that 𝐹 = ∞ (β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 )) ∩ Ξ© ∩ (⋂𝑖=0 VI(𝐢, 𝐴 𝑖 )) =ΜΈ 0, where Ξ© is the fixed point set of the mapping 𝐺 = Π𝐢(𝐼 βˆ’ πœ‡1 𝐡1 )Π𝐢(𝐼 βˆ’ πœ‡2 𝐡2 ) with 1 βˆ’ (𝜁 /(1 + 𝜁 ))(1 βˆ’ √(1 βˆ’ 𝛽̂ )/𝜁 ) ≀ πœ‡ ≀ 1 for 𝑖 = 1, 2. For 𝑖

𝑖

𝑖

arbitrarily given π‘₯0 ∈ 𝐢, let {π‘₯𝑛 } be the sequence generated by 𝑦𝑛 = 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 , π‘₯𝑛+1 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + πœŽπ‘› 𝐺π‘₯𝑛 + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) 𝑦𝑛 ,

βˆ€π‘› β‰₯ 0, (130)

where {πœŽπ‘› }, {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, and {𝛿𝑛 } are the sequences in [0, 1] such that 𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 and 𝛼𝑛 + πœŽπ‘› ≀ 1 for all 𝑛 β‰₯ 0. Suppose that the following conditions hold: (i) βˆ‘βˆž 𝑛=0 𝛼𝑛 = ∞ and 0 ≀ 𝛼𝑛 + πœŽπ‘› ≀ 1 βˆ’ 𝜌, for all 𝑛 β‰₯ 𝑛0 for some integer 𝑛0 β‰₯ 0; (ii) lim inf 𝑛 β†’ ∞ πœŽπ‘› > lim inf 𝑛 β†’ ∞ 𝛿𝑛 > 0;

0, lim inf 𝑛 β†’ ∞ 𝛾𝑛

βˆ€π‘ ∈ 𝐹.

(131)

Proof. First of all, take a fixed 𝑝 ∈ 𝐹 arbitrarily. Then we obtain 𝑝 = 𝐺𝑝, 𝑝 = 𝐡𝑛 𝑝 and 𝑆𝑛 𝑝 = 𝑝 for all 𝑛 β‰₯ 0. By Lemma 27, we get from (130)

σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯ βˆ’ 𝑦󡄩󡄩󡄩 .

𝑖

Assume that βˆ‘βˆž 𝑛=0 supπ‘₯∈𝐷 ‖𝑆𝑛+1 π‘₯ βˆ’ 𝑆𝑛 π‘₯β€– < ∞ for any bounded subset 𝐷 of 𝐢 and let 𝑆 be a mapping of 𝐢 into itself defined by 𝑆π‘₯ = lim𝑛 β†’ ∞ 𝑆𝑛 π‘₯ for all π‘₯ ∈ 𝐢 and suppose that Fix(𝑆) = β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 ). Then there hold the following: (I) lim𝑛 β†’ ∞ β€–π‘₯𝑛+1 βˆ’ π‘₯𝑛 β€– = 0;

σ΅„© βˆ’ Π𝐢 (𝐼 βˆ’ πœ‡1 𝐡1 ) Π𝐢 (𝐼 βˆ’ πœ‡2 𝐡2 ) 𝑦󡄩󡄩󡄩

𝑖

(iii) lim𝑛 β†’ ∞ (|𝛼𝑛+1 /(1βˆ’(1βˆ’π›Όπ‘›+1 βˆ’πœŽπ‘›+1 )𝛽𝑛+1 )βˆ’(𝛼𝑛 /(1βˆ’(1βˆ’ 𝛼𝑛 βˆ’πœŽπ‘› )𝛽𝑛 ))|+|πœŽπ‘›+1 /(1βˆ’(1βˆ’π›Όπ‘›+1 βˆ’πœŽπ‘›+1 )𝛽𝑛+1 )βˆ’(πœŽπ‘› /(1βˆ’ (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› )𝛽𝑛 ))| + |𝛿𝑛+1 /(1 βˆ’ 𝛽𝑛+1 ) βˆ’ 𝛿𝑛 /(1 βˆ’ 𝛽𝑛 )|) = 0;

>

0 and

σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩

(132)

σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ,

and hence σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 (󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (𝑝)σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩) σ΅„© σ΅„© σ΅„© σ΅„© + πœŽπ‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 (𝜌 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩) σ΅„© σ΅„© σ΅„© σ΅„© + πœŽπ‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©σ΅„© 󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛼𝑛 (1 βˆ’ 𝜌) σ΅„© 1βˆ’πœŒ σ΅„© σ΅„© σ΅„© 󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© ≀ max {σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 , σ΅„© }. 1βˆ’πœŒ (133) By induction, we have σ΅„© σ΅„© σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© 󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 }, σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ≀ max {σ΅„©σ΅„©σ΅„©π‘₯0 βˆ’ 𝑝󡄩󡄩󡄩 , σ΅„© 1βˆ’πœŒ

βˆ€π‘› β‰₯ 0, (134)

which implies that {π‘₯𝑛 } is bounded and so are the sequences {𝑦𝑛 }, {𝐺π‘₯𝑛 }, and {𝑓(π‘₯𝑛 )}.

18

Abstract and Applied Analysis Let us show that σ΅„©σ΅„©

lim σ΅„©π‘₯ 𝑛 β†’ ∞ σ΅„© 𝑛+1

Γ—[

σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(135)

βˆ’

As a matter of fact, put πœƒπ‘› = (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› )𝛽𝑛 , for all 𝑛 β‰₯ 0. Then, it follows from (i) and (iv) that 𝛽𝑛 β‰₯ πœƒπ‘› = (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) 𝛽𝑛 β‰₯ (1 βˆ’ (1 βˆ’ 𝜌)) 𝛽𝑛 = πœŒπ›½π‘› , βˆ€π‘› β‰₯ 𝑛0 ,

+[ (136)

π‘›β†’βˆž

π‘›β†’βˆž

Γ—

(137)

Define

= π‘₯𝑛+1 = πœƒπ‘› π‘₯𝑛 + (1 βˆ’ πœƒπ‘› ) 𝑧𝑛 .

(138)

𝑧𝑛+1 βˆ’ 𝑧𝑛 =

π‘₯𝑛+2 βˆ’ πœƒπ‘›+1 π‘₯𝑛+1 π‘₯𝑛+1 βˆ’ πœƒπ‘› π‘₯𝑛 βˆ’ 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘›

Γ— (1 βˆ’ πœƒπ‘›+1 ) βˆ’

𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 1 βˆ’ 𝛽𝑛

πœŽπ‘›+1 (𝐺π‘₯𝑛+1 βˆ’ 𝐺π‘₯𝑛 ) 1 βˆ’ πœƒπ‘›+1

+(

𝛼𝑛+1 𝛼𝑛 βˆ’ ) 𝑓 (π‘₯𝑛 ) 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘›

+(

πœŽπ‘›+1 πœŽπ‘› βˆ’ ) 𝐺π‘₯𝑛 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘›

= (𝛼𝑛+1 𝑓 (π‘₯𝑛+1 ) + πœŽπ‘›+1 𝐺π‘₯𝑛+1 +

+ (1 βˆ’ 𝛼𝑛+1 βˆ’ πœŽπ‘›+1 ) 𝑦𝑛+1 βˆ’ πœƒπ‘›+1 π‘₯𝑛+1 ) βˆ’1

(1 βˆ’ 𝛼𝑛+1 βˆ’ πœŽπ‘›+1 ) (1 βˆ’ 𝛽𝑛+1 ) 1 βˆ’ πœƒπ‘›+1

Γ—[

𝛼𝑛 𝑓 (π‘₯𝑛 ) + πœŽπ‘› 𝐺π‘₯𝑛 + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) 𝑦𝑛 βˆ’ πœƒπ‘› π‘₯𝑛 1 βˆ’ πœƒπ‘›

(1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (1 βˆ’ 𝛽𝑛 ) ] 1 βˆ’ πœƒπ‘›

𝛼𝑛+1 (𝑓 (π‘₯𝑛+1 ) βˆ’ 𝑓 (π‘₯𝑛 )) 1 βˆ’ πœƒπ‘›+1 +

Observe that

𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 ] 1 βˆ’ 𝛽𝑛

(1 βˆ’ 𝛼𝑛+1 βˆ’ πœŽπ‘›+1 ) (1 βˆ’ 𝛽𝑛+1 ) 1 βˆ’ πœƒπ‘›+1 βˆ’

and hence 0 < lim inf πœƒπ‘› ≀ lim sup πœƒπ‘› < 1.

𝛾𝑛+1 𝐡𝑛+1 π‘₯𝑛+1 + 𝛿𝑛+1 𝑆𝑛+1 𝐺π‘₯𝑛+1 1 βˆ’ 𝛽𝑛+1

𝛾𝑛+1 (𝐡 π‘₯ βˆ’ 𝐡𝑛 π‘₯𝑛 ) 𝛾𝑛+1 + 𝛿𝑛+1 𝑛+1 𝑛+1 +(

𝛼 𝑓 (π‘₯𝑛+1 + πœŽπ‘›+1 𝐺π‘₯𝑛+1 ) 𝛼𝑛 𝑓 (π‘₯𝑛 ) + πœŽπ‘› 𝐺π‘₯𝑛 = ( 𝑛+1 βˆ’ ) 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘›

+

(1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) [𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 ] βˆ’ πœƒπ‘› π‘₯𝑛 βˆ’ 1 βˆ’ πœƒπ‘›

𝛿𝑛+1 (𝑆 𝐺π‘₯ βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 ) 𝛾𝑛+1 + 𝛿𝑛+1 𝑛+1 𝑛+1

+(

+ (1 βˆ’ 𝛼𝑛+1 βˆ’ πœŽπ‘›+1 ) Γ— [𝛽𝑛+1 π‘₯𝑛+1 + 𝛾𝑛+1 𝐡𝑛+1 π‘₯𝑛+1 + 𝛿𝑛+1 𝑆𝑛+1 𝐺π‘₯𝑛+1 ]

βˆ’(

βˆ’1

βˆ’ πœƒπ‘›+1 π‘₯𝑛+1 Γ— (1 βˆ’ πœƒπ‘›+1 )

𝛾𝑛 𝛾𝑛+1 βˆ’ ) 𝐡𝑛 π‘₯𝑛 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛

𝛿𝑛+1 𝛿𝑛 βˆ’ ) 𝑆 𝐺π‘₯ ] 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 𝑛 𝑛

𝛼𝑛+1 + πœŽπ‘›+1 𝛼𝑛 + πœŽπ‘› 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 βˆ’ ) 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘› 𝛾𝑛 + 𝛿𝑛

𝛼𝑛+1 (𝑓 (π‘₯𝑛+1 ) βˆ’ 𝑓 (π‘₯𝑛 )) 1 βˆ’ πœƒπ‘›+1

=(

𝛼𝑛+1 𝑓 (π‘₯𝑛+1 ) + πœŽπ‘›+1 𝐺π‘₯𝑛+1 𝛼𝑛 𝑓 (π‘₯𝑛 ) + πœŽπ‘› 𝐺π‘₯𝑛 βˆ’ ) 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘›

+

1 βˆ’ 𝛼𝑛+1 βˆ’ πœŽπ‘›+1 (𝛾𝑛+1 𝐡𝑛+1 π‘₯𝑛+1 + 𝛿𝑛+1 𝑆𝑛+1 𝐺π‘₯𝑛+1 ) 1 βˆ’ πœƒπ‘›+1

βˆ’

1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› (𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 ) 1 βˆ’ πœƒπ‘›

+(

𝛾 𝐡 π‘₯ + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 𝛼𝑛+1 𝛼𝑛 βˆ’ ) (𝑓 (π‘₯𝑛 ) βˆ’ 𝑛 𝑛 𝑛 ) 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘› 𝛾𝑛 + 𝛿𝑛

𝛼𝑛+1 𝑓 (π‘₯𝑛+1 ) + πœŽπ‘›+1 𝐺π‘₯𝑛+1 𝛼𝑛 𝑓 (π‘₯𝑛 ) + πœŽπ‘› 𝐺π‘₯𝑛 βˆ’ ) 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘›

+(

𝛾 𝐡 π‘₯ + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 πœŽπ‘›+1 πœŽπ‘› βˆ’ ) (𝐺π‘₯𝑛 βˆ’ 𝑛 𝑛 𝑛 ) 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘› 𝛾𝑛 + 𝛿𝑛

(1 βˆ’ 𝛼𝑛+1 βˆ’ πœŽπ‘›+1 ) (1 βˆ’ 𝛽𝑛+1 ) 1 βˆ’ πœƒπ‘›+1

+

=( +

=

+

πœŽπ‘›+1 (𝐺π‘₯𝑛+1 βˆ’ 𝐺π‘₯𝑛 ) 1 βˆ’ πœƒπ‘›+1

1 βˆ’ 𝛼𝑛+1 βˆ’ πœŽπ‘›+1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘›+1

Abstract and Applied Analysis Γ—[

19 󡄨󡄨 𝛼 𝛼𝑛 󡄨󡄨󡄨󡄨 󡄨 + 󡄨󡄨󡄨 𝑛+1 βˆ’ 󡄨 󡄨󡄨 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘› 󡄨󡄨󡄨 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— (󡄩󡄩󡄩𝑓 (π‘₯𝑛 )σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©) 󡄨󡄨 𝜎 πœŽπ‘› 󡄨󡄨󡄨󡄨 󡄨 + 󡄨󡄨󡄨 𝑛+1 βˆ’ 󡄨 󡄨󡄨 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘› 󡄨󡄨󡄨 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— (󡄩󡄩󡄩𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©)

𝛾𝑛+1 (𝐡 π‘₯ βˆ’ 𝐡𝑛 π‘₯𝑛 ) 𝛾𝑛+1 + 𝛿𝑛+1 𝑛+1 𝑛+1

+( +

𝛾𝑛 𝛾𝑛+1 βˆ’ ) 𝐡𝑛 π‘₯𝑛 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛

𝛿𝑛+1 (𝑆 𝐺π‘₯ βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 ) 𝛾𝑛+1 + 𝛿𝑛+1 𝑛+1 𝑛+1

+(

𝛿𝑛+1 𝛿𝑛 βˆ’ ) 𝑆 𝐺π‘₯ ] , 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 𝑛 𝑛

+ (139)

and hence 󡄩󡄩󡄩𝑧𝑛+1 βˆ’ 𝑧𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© 𝛼𝑛+1 σ΅„©σ΅„© σ΅„© ≀ 󡄩𝑓 (π‘₯𝑛+1 ) βˆ’ 𝑓 (π‘₯𝑛 )σ΅„©σ΅„©σ΅„© 1 βˆ’ πœƒπ‘›+1 σ΅„©

1 βˆ’ 𝛼𝑛+1 βˆ’ πœŽπ‘›+1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘›+1

Γ—[

󡄨󡄨 𝛾 𝛾𝑛 󡄨󡄨󡄨󡄨 σ΅„©σ΅„© 󡄨 σ΅„© 𝑛+1 + 󡄨󡄨󡄨 βˆ’ 󡄨 󡄩𝐡 π‘₯ σ΅„©σ΅„© 󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨󡄨󡄨 σ΅„© 𝑛 𝑛 σ΅„©

πœŽπ‘›+1 σ΅„©σ΅„© σ΅„© 󡄩𝐺π‘₯ βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© 1 βˆ’ πœƒπ‘›+1 σ΅„© 𝑛+1 󡄨󡄨 𝛼 𝛾 𝐡 π‘₯ + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©σ΅„© 𝛼𝑛 󡄨󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©σ΅„© 󡄨 + 󡄨󡄨󡄨 𝑛+1 βˆ’ 󡄨󡄨 󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑛 𝑛 𝑛 σ΅„©σ΅„© 󡄨󡄨 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘› 󡄨󡄨 σ΅„©σ΅„© σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛 󡄨󡄨 𝜎 𝛾 𝐡 π‘₯ + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©σ΅„© πœŽπ‘› 󡄨󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©σ΅„© 󡄨 + 󡄨󡄨󡄨 𝑛+1 βˆ’ 󡄨󡄨 󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑛 𝑛 𝑛 σ΅„©σ΅„© 󡄨󡄨 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘› 󡄨󡄨 σ΅„©σ΅„© σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛

+

+

+

1 βˆ’ 𝛼𝑛+1 βˆ’ πœŽπ‘›+1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘›+1 𝛾𝑛+1 σ΅„©σ΅„© σ΅„© 󡄩𝐡 π‘₯ βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 𝛾𝑛+1 + 𝛿𝑛+1 σ΅„© 𝑛+1 𝑛+1 󡄨󡄨 𝛾 𝛾𝑛 󡄨󡄨󡄨󡄨 σ΅„©σ΅„© 󡄨 σ΅„© 𝑛+1 + 󡄨󡄨󡄨 βˆ’ 󡄨 󡄩𝐡 π‘₯ σ΅„©σ΅„© 󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨󡄨󡄨 σ΅„© 𝑛 𝑛 σ΅„©

=

1 βˆ’ 𝛼𝑛+1 (1 βˆ’ 𝜌) βˆ’ πœƒπ‘›+1 σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© 1 βˆ’ πœƒπ‘›+1 1 βˆ’ 𝛼𝑛+1 βˆ’ πœŽπ‘›+1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘›+1

Γ—[

𝑛 󡄨󡄨 𝛿 𝛾𝑛+1 𝛿𝑛 󡄨󡄨󡄨󡄨 󡄨 𝑛+1 𝑀0 βˆπœ† 𝑖 + 󡄨󡄨󡄨 βˆ’ 󡄨 󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝑖=0

σ΅„© σ΅„© σ΅„© σ΅„© Γ— (󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©)

𝛿𝑛+1 σ΅„©σ΅„© σ΅„© 󡄩𝑆 𝐺π‘₯ βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© 𝛾𝑛+1 + 𝛿𝑛+1 σ΅„© 𝑛+1 𝑛+1 󡄨󡄨 𝛿 𝛿𝑛 󡄨󡄨󡄨󡄨 σ΅„©σ΅„© 󡄨 σ΅„© 𝑛+1 + 󡄨󡄨󡄨 βˆ’ 󡄨󡄨 󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©] . 󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨󡄨

+

+

(140) On the other hand, repeating the same arguments as those of (55) and (56) in the proof of Theorem 24, we can get 󡄩󡄩󡄩𝑆𝑛+1 𝐺π‘₯𝑛+1 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛+1 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© , σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© 𝑛

σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© 󡄩󡄩𝐡𝑛+1 π‘₯𝑛+1 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 𝑀0 βˆπœ† 𝑖 ,

𝛿𝑛+1 σ΅„©σ΅„© σ΅„© 󡄩𝑆 𝐺π‘₯ βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© ] 𝛾𝑛+1 + 𝛿𝑛+1 σ΅„© 𝑛+1 𝑛

󡄨󡄨 𝛼 𝛼𝑛 󡄨󡄨󡄨󡄨 󡄨 + 󡄨󡄨󡄨 𝑛+1 βˆ’ 󡄨 󡄨󡄨 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘› 󡄨󡄨󡄨 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— (󡄩󡄩󡄩𝑓 (π‘₯𝑛 )σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©) 󡄨󡄨 𝜎 πœŽπ‘› 󡄨󡄨󡄨󡄨 󡄨 + 󡄨󡄨󡄨 𝑛+1 βˆ’ 󡄨󡄨 󡄨󡄨 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘› 󡄨󡄨 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— (󡄩󡄩󡄩𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©)

σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©

𝑖=0

(141) for some constant 𝑀0 > 0. Utilizing (140)-(141), we have 󡄩󡄩󡄩𝑧𝑛+1 βˆ’ 𝑧𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© 𝛼𝑛+1 σ΅„© σ΅„© ≀ 𝜌 σ΅„©σ΅„©π‘₯ βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© 1 βˆ’ πœƒπ‘›+1 σ΅„© 𝑛+1 πœŽπ‘›+1 σ΅„©σ΅„© σ΅„© σ΅„©π‘₯ βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© 1 βˆ’ πœƒπ‘›+1 σ΅„© 𝑛+1

𝛿𝑛+1 σ΅„© σ΅„© σ΅„© σ΅„© (σ΅„©σ΅„©π‘₯ βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛+1 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©) 𝛾𝑛+1 + 𝛿𝑛+1 σ΅„© 𝑛+1

󡄨󡄨 𝛿 𝛿𝑛 󡄨󡄨󡄨󡄨 σ΅„©σ΅„© 󡄨 σ΅„© 𝑛+1 + 󡄨󡄨󡄨 βˆ’ 󡄨 󡄩𝑆 𝐺π‘₯ σ΅„©σ΅„© ] 󡄨󡄨 𝛾𝑛+1 + 𝛿𝑛+1 𝛾𝑛 + 𝛿𝑛 󡄨󡄨󡄨 σ΅„© 𝑛 𝑛 σ΅„©

+

Γ—[

+

𝑛 𝛾𝑛+1 σ΅„© σ΅„© (σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 𝑀0 βˆπœ† 𝑖 ) 𝛾𝑛+1 + 𝛿𝑛+1 𝑖=0

𝑛 󡄨󡄨 𝛿 𝛿𝑛 󡄨󡄨󡄨󡄨 󡄨 + 𝑀 (βˆπœ† 𝑖 + 󡄨󡄨󡄨 𝑛+1 βˆ’ 󡄨 󡄨󡄨 1 βˆ’ 𝛽𝑛+1 1 βˆ’ 𝛽𝑛 󡄨󡄨󡄨 𝑖=0

󡄨󡄨 𝛼 𝛼𝑛 󡄨󡄨󡄨󡄨 󡄨󡄨󡄨󡄨 πœŽπ‘›+1 πœŽπ‘› 󡄨󡄨󡄨󡄨 󡄨 + 󡄨󡄨󡄨 𝑛+1 βˆ’ βˆ’ 󡄨󡄨 + 󡄨󡄨 󡄨) 󡄨󡄨 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘› 󡄨󡄨 󡄨󡄨 1 βˆ’ πœƒπ‘›+1 1 βˆ’ πœƒπ‘› 󡄨󡄨󡄨 σ΅„© σ΅„© + 󡄩󡄩󡄩𝑆𝑛+1 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© , (142)

20

Abstract and Applied Analysis

where sup𝑛β‰₯0 {‖𝑓(π‘₯𝑛 )β€– + ‖𝐺π‘₯𝑛 β€– + ‖𝐡𝑛 π‘₯𝑛 β€– + ‖𝑆𝑛 𝐺π‘₯𝑛 β€– + 𝑀0 } ≀ 𝑀 for some 𝑀 > 0. So, from (142), condition (iii), and the assumption on {𝑆𝑛 } it follows that (noting that 0 < πœ† 𝑖 ≀ 𝑏 < 1, for all 𝑖 β‰₯ 0) σ΅„© σ΅„© σ΅„© σ΅„© lim sup (󡄩󡄩󡄩𝑧𝑛+1 βˆ’ 𝑧𝑛 σ΅„©σ΅„©σ΅„© βˆ’ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) ≀ 0. π‘›β†’βˆž

(143)

Consequently, by Lemma 20, we have σ΅„©σ΅„©

lim 󡄩𝑧 π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(144)

It follows from (137) and (138) that σ΅„©σ΅„©

lim σ΅„©π‘₯ 𝑛 β†’ ∞ σ΅„© 𝑛+1

σ΅„© σ΅„© σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = lim (1 βˆ’ πœƒπ‘› ) 󡄩󡄩󡄩𝑧𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0. π‘›β†’βˆž

σ΅„© Γ— [2 󡄩󡄩󡄩𝛼𝑛 (π‘₯𝑛 βˆ’ 𝑝) + πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ 𝑝) σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (𝑦𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©]

σ΅„© σ΅„©2 ≀ 𝛼𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© 𝜎 σ΅„©σ΅„©2 1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 ) σ΅„©σ΅„©σ΅„© 𝑛 (𝐺π‘₯𝑛 βˆ’ 𝑝) + (𝑦𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© 1 βˆ’ 𝛼𝑛 σ΅„©σ΅„© 1 βˆ’ 𝛼𝑛 σ΅„© σ΅„© + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— [2 (𝛼𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩) + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„©2 ≀ 𝛼𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ 𝛼𝑛 )

(145) Γ—[

Next, we show that β€–π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 β€– β†’ 0 as 𝑛 β†’ ∞. Indeed, in terms of Lemma 11, from (130), we have σ΅„©2 σ΅„©σ΅„© 󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 , σ΅„©2 σ΅„©σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© = 󡄩󡄩󡄩𝛼𝑛 (π‘₯𝑛 βˆ’ 𝑝) + πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ 𝑝) + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (𝑦𝑛 βˆ’ 𝑝) σ΅„©2 + 𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© ≀ [󡄩󡄩󡄩𝛼𝑛 (π‘₯𝑛 βˆ’ 𝑝) + πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ 𝑝) + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (𝑦𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„©2 = 󡄩󡄩󡄩𝛼𝑛 (π‘₯𝑛 βˆ’ 𝑝) + πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ 𝑝) + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (𝑦𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©

σ΅„© Γ— [2 󡄩󡄩󡄩𝛼𝑛 (π‘₯𝑛 βˆ’ 𝑝) + πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ 𝑝) σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (𝑦𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©] σ΅„©σ΅„© σ΅„© = 󡄩󡄩󡄩𝛼𝑛 (π‘₯𝑛 βˆ’ 𝑝) + (1 βˆ’ 𝛼𝑛 ) σ΅„©σ΅„© σ΅„©σ΅„©2 1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› πœŽπ‘› σ΅„© (𝐺π‘₯𝑛 βˆ’ 𝑝) + (𝑦𝑛 βˆ’ 𝑝)]σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© 1 βˆ’ 𝛼𝑛 1 βˆ’ 𝛼𝑛 σ΅„© σ΅„© + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„© Γ—[

πœŽπ‘› σ΅„©σ΅„© σ΅„©2 1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› σ΅„©σ΅„© σ΅„©2 󡄩𝐺π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩𝑦 βˆ’ 𝑝󡄩󡄩󡄩 1 βˆ’ 𝛼𝑛 σ΅„© 𝑛 1 βˆ’ 𝛼𝑛 σ΅„© 𝑛 βˆ’

πœŽπ‘› (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) 2

(1 βˆ’ 𝛼𝑛 )

σ΅„© σ΅„© 𝑔 (󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©)]

σ΅„© σ΅„© + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— [2 (𝛼𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + πœŽπ‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„© σ΅„© + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„©2 ≀ 𝛼𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ 𝛼𝑛 ) Γ—[

πœŽπ‘› σ΅„©σ΅„© σ΅„©2 1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 1 βˆ’ 𝛼𝑛 1 βˆ’ 𝛼𝑛 σ΅„© 𝑛 βˆ’

πœŽπ‘› (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) 2

(1 βˆ’ 𝛼𝑛 )

σ΅„© σ΅„© 𝑔 (󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©)]

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„© (2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„©2 𝜎 (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) σ΅„©σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝑛 𝑔 (󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©) 1 βˆ’ 𝛼𝑛 σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + 󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„© (2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©) . (146) Then, it immediately follows from 0 ≀ 𝛼𝑛 + πœŽπ‘› ≀ 1 βˆ’ 𝜌, for all 𝑛 β‰₯ 𝑛0 that σ΅„© σ΅„© πœŒπœŽπ‘› 𝑔 (󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©) ≀

πœŽπ‘› (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) σ΅„©σ΅„© σ΅„© 𝑔 (󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©) 1 βˆ’ 𝛼𝑛

σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„© (2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©)

Abstract and Applied Analysis

21

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© σ΅„© σ΅„© Γ— (2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„©) , (147) for all 𝑛 β‰₯ 𝑛0 . Since ‖𝛼𝑛 (𝑓(π‘₯𝑛 ) βˆ’ π‘₯𝑛 )β€– β†’ 0 and {π‘₯𝑛 } is bounded, we deduce from (145) and condition (ii) that σ΅„© σ΅„© lim 𝑔 (󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©) = 0. (148) π‘›β†’βˆž Utilizing the properties of 𝑔, we have σ΅„© σ΅„© lim 󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© = 0. π‘›β†’βˆž σ΅„©

(149)

Also, from (130) we have π‘₯𝑛+1 βˆ’ π‘₯𝑛 = 𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 ) + πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 ) + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (𝑦𝑛 βˆ’ π‘₯𝑛 ) = 𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 ) + πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 + 𝑦𝑛 βˆ’ π‘₯𝑛 )

(150)

+ (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (𝑦𝑛 βˆ’ π‘₯𝑛 ) = 𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 ) + πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 ) + (1 βˆ’ 𝛼𝑛 ) (𝑦𝑛 βˆ’ π‘₯𝑛 ) , which hence leads to σ΅„© σ΅„© 𝜌 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© ≀ (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© ≀ (1 βˆ’ 𝛼𝑛 ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 βˆ’ 𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 ) βˆ’ πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 )σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„© + πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© .

(151)

So, it is easy to see from (145), (149), and ‖𝛼𝑛 (𝑓(π‘₯𝑛 ) βˆ’ π‘₯𝑛 )β€– β†’ 0 that σ΅„© σ΅„© lim 󡄩󡄩𝑦 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0. (152) π‘›β†’βˆž σ΅„© 𝑛 We note that σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© 󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© ≀ 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© .

(153)

Therefore, from (149) and (152) it follows that σ΅„© σ΅„© lim 󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0. π‘›β†’βˆž σ΅„©

(154)

Repeating the same arguments as those of (86), (89), and (91) in the proof of Theorem 24, we can obtain σ΅„© σ΅„© σ΅„© σ΅„© lim σ΅„©σ΅„©π‘₯ βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© = lim 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© π‘›β†’βˆž σ΅„© 𝑛 π‘›β†’βˆž (155) σ΅„© σ΅„© = lim σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0. π‘›β†’βˆž

Suppose that 𝛽𝑛 ≑ 𝛽 for some fixed 𝛽 ∈ (0, 1) such that 𝛽 + 𝛾𝑛 + 𝛿𝑛 = 1 for all 𝑛 β‰₯ 0. Define a mapping 𝑉π‘₯ = (1 βˆ’ πœƒ1 βˆ’ πœƒ2 )𝑆π‘₯ + πœƒ1 𝐡π‘₯ + πœƒ2 𝐺π‘₯, where πœƒ1 , πœƒ2 ∈ (0, 1) are two constants with πœƒ1 + πœƒ2 < 1. Then, by Lemmas 14 and 17, we have that Fix(𝑉) = Fix(𝑆) ∩ Fix(𝐡) ∩ Fix(𝐺) = 𝐹. For each π‘˜ β‰₯ 1, let {π‘π‘˜ } be a unique element of 𝐢 such that π‘π‘˜ =

1 1 𝑓 (π‘π‘˜ ) + (1 βˆ’ ) π‘‰π‘π‘˜ . π‘˜ π‘˜

(156)

From Lemma 13, we conclude that π‘π‘˜ β†’ π‘ž ∈ Fix(𝑉) = 𝐹 as π‘˜ β†’ ∞. Observe that for every 𝑛, π‘˜ σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑦𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + 𝛿𝑛 (󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„© σ΅„© = 𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© ,

(157)

and hence σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 (󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„© σ΅„© + πœŽπ‘› (󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) σ΅„© σ΅„© σ΅„© σ΅„© Γ— [𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 (󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„© σ΅„© + πœŽπ‘› (󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) σ΅„© σ΅„© Γ— [𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛽) (󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝐡𝑛 π‘π‘˜ σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 (󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„© σ΅„© + πœŽπ‘› (󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— [𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„© σ΅„© σ΅„© = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + [𝛽 + (𝛼𝑛 + πœŽπ‘› ) (1 βˆ’ 𝛽)] σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) σ΅„© σ΅„© σ΅„© σ΅„© Γ— [(1 βˆ’ 𝛽) 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©]

22

Abstract and Applied Analysis σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©

Repeating the same arguments as those of (99), in the proof of Theorem 24, we can get

+ [𝛽 + (𝛼𝑛 + πœŽπ‘› ) (1 βˆ’ 𝛽)] σ΅„© σ΅„© σ΅„© σ΅„© Γ— (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (1 βˆ’ 𝛽) σ΅„© σ΅„© σ΅„© σ΅„© Γ— (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„©) + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) σ΅„© σ΅„© σ΅„© σ΅„© Γ— [(1 βˆ’ 𝛽) 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„© σ΅„© σ΅„© = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + [𝛽 + (𝛼𝑛 + πœŽπ‘› ) (1 βˆ’ 𝛽)] σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) σ΅„© σ΅„© σ΅„© σ΅„© Γ— [(1 βˆ’ 𝛽) 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© . (158) So, it immediately follows from 0 ≀ 𝛼𝑛 ≀ 1 βˆ’ 𝜌, for all 𝑛 β‰₯ 𝑛0 that σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© 1 (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (1 βˆ’ 𝛽) σ΅„© σ΅„© σ΅„© σ΅„© Γ— (󡄩󡄩󡄩𝛼𝑛 (π‘₯𝑛 βˆ’ 𝑓 (π‘₯𝑛 ))σ΅„©σ΅„©σ΅„© + πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©

+

σ΅„© σ΅„©2 σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„© ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© , σ΅„© σ΅„©2 σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„© ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© .

(162)

Utilizing (161) and (162), we deduce that σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘‰π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 ≀ (1 βˆ’ πœƒ1 βˆ’ πœƒ2 ) πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„©2 + πœƒ1 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœƒ2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„©

(163)

σ΅„© σ΅„©2 ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© . Also, observe that 1 1 (1 βˆ’ ) (π‘₯𝑛 βˆ’ π‘‰π‘π‘˜ ) = π‘₯𝑛 βˆ’ π‘π‘˜ βˆ’ (π‘₯𝑛 βˆ’ 𝑓 (π‘π‘˜ )) . π‘˜ π‘˜

(164)

Repeating the same arguments as those of (106) in the proof of Theorem 24, we can get

𝛿 σ΅„© σ΅„© σ΅„© σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„©) + 𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 1βˆ’π›½ σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©

(159)

1 σ΅„©σ΅„© σ΅„©2 πœ‡ σ΅„©π‘₯ βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© β‰₯ πœ‡π‘› βŸ¨π‘“ (π‘π‘˜ ) βˆ’ π‘π‘˜ , 𝐽 (π‘₯𝑛 βˆ’ π‘π‘˜ )⟩ . 2π‘˜ 𝑛 σ΅„© 𝑛

(165)

Since π‘π‘˜ β†’ π‘ž ∈ Fix(𝑉) = 𝐹 as π‘˜ β†’ ∞, by the uniform Gateaux differentiability of the norm of 𝑋, we have

1 𝜌 (1 βˆ’ 𝛽) σ΅„© σ΅„© Γ— (󡄩󡄩󡄩𝛼𝑛 (π‘₯𝑛 βˆ’ 𝑓 (π‘₯𝑛 ))σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© ) σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœπ‘› , βˆ€π‘› β‰₯ 𝑛0 , σ΅„© σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© +

πœ‡π‘› βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛 βˆ’ π‘ž)⟩ ≀ 0.

(166)

On the other hand, from (135) and the norm-to-weakβˆ— uniform continuity of 𝐽 on bounded subsets of 𝑋, it follows that

where πœƒπ‘› = ‖𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ β€– + ‖𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 β€– + 1/(𝜌(1 βˆ’ 𝛽))(‖𝛼𝑛 (π‘₯𝑛 βˆ’ 𝑓(π‘₯𝑛 ))β€– + ‖𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 β€– + β€–π‘₯𝑛 βˆ’ π‘₯𝑛+1 β€–). Since lim𝑛 β†’ ∞ ‖𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ β€– = lim𝑛 β†’ ∞ ‖𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 β€– = lim𝑛 β†’ ∞ ‖𝛼𝑛 (π‘₯𝑛 βˆ’ 𝑓(π‘₯𝑛 ))β€– = lim𝑛 β†’ ∞ ‖𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 β€– = lim𝑛 β†’ ∞ β€–π‘₯𝑛 βˆ’ π‘₯𝑛+1 β€– = 0, we know that πœπ‘› β†’ 0 as 𝑛 β†’ ∞. From (159), we obtain σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœπ‘› (2 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœπ‘› ) , βˆ€π‘› β‰₯ 𝑛0 . (160)

σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© = πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© .

So, utilizing Lemma 18, we deduce from (166) and (167) that lim sup βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛 βˆ’ π‘ž)⟩ ≀ 0, π‘›β†’βˆž

(168)

which, together with (135) and the norm-to-norm uniform continuity of 𝐽 on bounded subsets of 𝑋, implies that

For any Banach limit πœ‡, from (160) we derive σ΅„© σ΅„©2 σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© = πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©

󡄨 󡄨 lim σ΅„¨σ΅„¨βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ βˆ’ βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛 βˆ’ π‘ž)βŸ©σ΅„¨σ΅„¨σ΅„¨ = 0. (167)

π‘›β†’βˆž 󡄨

(161)

lim sup βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ ≀ 0. π‘›β†’βˆž

(169)

Abstract and Applied Analysis

23

Finally, let us show that π‘₯𝑛 β†’ π‘ž as 𝑛 β†’ ∞. Utilizing Lemma 8 (i), from (130) and the convexity of β€– β‹… β€–2 , we get σ΅„©2 σ΅„©2 σ΅„©2 σ΅„© σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑦𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„©2 + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© , σ΅„©2 σ΅„©σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© = 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘ž)) + πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ π‘ž)

(170)

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + πœŽπ‘› (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©)

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 3 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© .

(173) Since π‘₯𝑛 β†’ π‘ž and 𝑦𝑛 β†’ π‘ž, we get 𝛼𝑛 (𝑓(π‘₯𝑛 ) βˆ’ π‘₯𝑛 ) β†’ 0. This completes the proof.

σ΅„©2 + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (𝑦𝑛 βˆ’ π‘ž) + 𝛼𝑛 (𝑓 (π‘ž) βˆ’ π‘ž) σ΅„©σ΅„©σ΅„©

σ΅„© ≀ 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘ž)) + πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ π‘ž) σ΅„©2 + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (𝑦𝑛 βˆ’ π‘ž) σ΅„©σ΅„©σ΅„© + 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩

σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘ž)σ΅„©σ΅„©σ΅„© + πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 πœŒσ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + πœŽπ‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

Corollary 29. Let 𝐢 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢. Let {πœŒπ‘› }∞ 𝑛=0 be a sequence of positive numbers in (0, 𝑏] for some 𝑏 ∈ (0, 1) and 𝐴 𝑖 : 𝐢 β†’ π‘‹πœ‰π‘– strictly pseudocontractive and 𝛼̂𝑖 -strongly accretive with πœ‰π‘– + 𝛼̂𝑖 β‰₯ 1 for each 𝑖 = 0, 1, . . .. Define a mapping 𝐺𝑖 : 𝐢 β†’ 𝐢 by Π𝐢(𝐼 βˆ’ πœ† 𝑖 𝐴 𝑖 )π‘₯ = 𝐺𝑖 π‘₯ for all π‘₯ ∈ 𝐢 and 𝑖 = 0, 1, . . ., where 1βˆ’(πœ‰π‘– /(1+πœ‰π‘– ))(1βˆ’βˆš(1 βˆ’ 𝛼̂𝑖 )/πœ‰π‘– ) ≀ πœ† 𝑖 ≀ 1 for all 𝑖 = 0, 1, . . .. Let 𝐡𝑛 : 𝐢 β†’ 𝐢 be the π‘Š-mapping generated by 𝐺𝑛 , πΊπ‘›βˆ’1 , . . . , 𝐺0 and πœŒπ‘› , πœŒπ‘›βˆ’1 , . . . , 𝜌0 . Let 𝑉 : 𝐢 β†’ 𝐢 be a self-mapping such that 𝐼 βˆ’ 𝑉 : 𝐢 β†’ 𝑋 is πœ†-strictly pseudocontractive and 𝛼strongly accretive with 𝛼 + πœ† β‰₯ 1. Let 𝑓 : 𝐢 β†’ 𝐢 be a contraction with coefficient 𝜌 ∈ (0, 1). Let {𝑆𝑖 }∞ 𝑖=0 be a countable family of nonexpansive mappings of 𝐢 into itself such that 𝐹 = ∞ (β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 )) ∩ Fix(𝑉) ∩ (⋂𝑖=0 VI(𝐢, 𝐴 𝑖 )) =ΜΈ 0. For arbitrarily given π‘₯0 ∈ 𝐢, let {π‘₯𝑛 } be the sequence generated by 𝑦𝑛 = 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 ((1 βˆ’ 𝑙) 𝐼 + 𝑙𝑉) π‘₯𝑛 ,

+ 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩

π‘₯𝑛+1 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + πœŽπ‘› ((1 βˆ’ 𝑙) 𝐼 + 𝑙𝑉) π‘₯𝑛 + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) 𝑦𝑛 ,

σ΅„© σ΅„©2 = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

βˆ€π‘› β‰₯ 0, (174)

+ 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ σ΅„© σ΅„©2 = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛼𝑛 (1 βˆ’ 𝜌)

2 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ . 1βˆ’πœŒ (171)

Applying Lemma 7 to (171), we obtain that π‘₯𝑛 β†’ π‘ž as 𝑛 β†’ ∞. Conversely, if π‘₯𝑛 β†’ π‘ž ∈ 𝐹 as 𝑛 β†’ ∞, then from (130) it follows that σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑦𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© 󳨀→ 0

(172)

as 𝑛 β†’ ∞; that is, 𝑦𝑛 β†’ π‘ž. Again from (130) we obtain that σ΅„©σ΅„© σ΅„© 󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 βˆ’ πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 ) βˆ’ (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (𝑦𝑛 βˆ’ π‘₯𝑛 )σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + πœŽπ‘› (󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› ) (󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©)

where 1 βˆ’ (πœ†/(1 + πœ†))(1 βˆ’ √(1 βˆ’ 𝛼)/πœ†) ≀ 𝑙 ≀ 1 and {πœŽπ‘› }, {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, and {𝛿𝑛 } are the sequences in [0, 1] such that 𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 and 𝛼𝑛 + πœŽπ‘› ≀ 1 for all 𝑛 β‰₯ 0. Suppose that the following conditions hold: (i) βˆ‘βˆž 𝑛=0 𝛼𝑛 = ∞ and 0 ≀ 𝛼𝑛 + πœŽπ‘› ≀ 1 βˆ’ 𝜌, for all 𝑛 β‰₯ 𝑛0 for some integer 𝑛0 β‰₯ 0; (ii) lim inf 𝑛 β†’ ∞ πœŽπ‘› > lim inf 𝑛 β†’ ∞ 𝛿𝑛 > 0;

0, lim inf 𝑛 β†’ ∞ 𝛾𝑛

>

0 and

(iii) lim𝑛 β†’ ∞ (|𝛼𝑛+1 /(1βˆ’(1βˆ’π›Όπ‘›+1 βˆ’πœŽπ‘›+1 )𝛽𝑛+1 )βˆ’π›Όπ‘› /(1βˆ’(1βˆ’ 𝛼𝑛 βˆ’πœŽπ‘› )𝛽𝑛 )|+|πœŽπ‘›+1 /(1βˆ’(1βˆ’π›Όπ‘›+1 βˆ’πœŽπ‘›+1 )𝛽𝑛+1 )βˆ’πœŽπ‘› /(1βˆ’ (1 βˆ’ 𝛼𝑛 βˆ’ πœŽπ‘› )𝛽𝑛 )| + |𝛿𝑛+1 /(1 βˆ’ 𝛽𝑛+1 ) βˆ’ 𝛿𝑛 /(1 βˆ’ 𝛽𝑛 )|) = 0; (iv) 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1. Assume that βˆ‘βˆž 𝑛=0 supπ‘₯∈𝐷 ‖𝑆𝑛+1 π‘₯ βˆ’ 𝑆𝑛 π‘₯β€– < ∞ for any bounded subset 𝐷 of 𝐢 and let 𝑆 be a mapping of 𝐢 into itself defined by 𝑆π‘₯ = lim𝑛 β†’ ∞ 𝑆𝑛 π‘₯ for all π‘₯ ∈ 𝐢 and suppose that Fix(𝑆) = β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 ). Then there hold the following: (I) lim𝑛 β†’ ∞ β€–π‘₯𝑛+1 βˆ’ π‘₯𝑛 β€– = 0; (II) π‘₯𝑛 β†’ π‘ž ⇔ 𝛼𝑛 (𝑓(π‘₯𝑛 ) βˆ’ π‘₯𝑛 ) β†’ 0 provided 𝛽𝑛 ≑ 𝛽 for some fixed 𝛽 ∈ (0, 1), where π‘ž ∈ 𝐹 solves the following VIP βŸ¨π‘ž βˆ’ 𝑓 (π‘ž) , 𝐽 (π‘ž βˆ’ 𝑝)⟩ ≀ 0,

βˆ€π‘ ∈ 𝐹.

(175)

24

Abstract and Applied Analysis

Proof. In Theorem 28, we put 𝐡1 = 𝐼 βˆ’ 𝑉, 𝐡2 = 0, and πœ‡1 = 𝑙, where 1 βˆ’ (πœ†/(1 + πœ†))(1 βˆ’ √(1 βˆ’ 𝛼)/πœ†) ≀ 𝑙 ≀ 1. Then, GSVI (13) is equivalent to the VIP of finding π‘₯βˆ— ∈ 𝐢 such that ⟨𝐡1 π‘₯βˆ— , 𝐽 (π‘₯ βˆ’ π‘₯βˆ— )⟩ β‰₯ 0,

βˆ€π‘₯ ∈ 𝐢.

(176)

In this case, 𝐡1 : 𝐢 β†’ 𝑋 is πœ†-strictly pseudocontractive and 𝛼-strongly accretive. Repeating the same arguments as those in the proof of Corollary 25, we can infer that Fix(𝑉) = VI(𝐢, 𝐡1 ). Accordingly, 𝐹 = β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 ) ∩ Ξ© ∩ ∞ ∞ (β‹‚βˆž 𝑖=0 VI(𝐢, 𝐴 𝑖 )) = ⋂𝑖=0 Fix(𝑆𝑖 ) ∩ Fix(𝑉) ∩ (⋂𝑖=0 VI(𝐢, 𝐴 𝑖 )), and 𝐺π‘₯𝑛 = ((1 βˆ’ 𝑙) 𝐼 + 𝑙𝑉) π‘₯𝑛 ,

βˆ€π‘› β‰₯ 0.

(177)

So, scheme (130) reduces to (174). Therefore, the desired result follows from Theorem 31. Remark 30. Our Theorems 24 and 28 improve, extend, supplement and develop Ceng and Yao’s [10, Theorem 3.2], Cai and Bu’s [11, Theorem 3.1], Kangtunyakarn’s [38, Theorem 3.1], and Ceng and Yao’s [8, Theorem 3.1], in the following aspects. (i) The problem of finding a point π‘ž ∈ (β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 )) ∩ Ξ© ∩ (β‹‚βˆž 𝑖=0 VI(𝐢, 𝐴 𝑖 )) in our Theorems 24 and 28 is more general and more subtle than every one of the problem of finding a point π‘ž ∈ β‹‚βˆž 𝑖=0 Fix(𝑇𝑖 ) in [10, Theorem 3.2], the problem of finding a point π‘ž ∈ β‹‚βˆž 𝑖=1 Fix(𝑇𝑖 ) ∩ Ξ© in [11, Theorem 3.1], the problem of finding a point π‘ž ∈ Fix(𝑆) ∩ Fix(𝑉) ∩ (⋂𝑁 𝑖=1 VI(𝐢, 𝐴 𝑖 )) in [38, Theorem 3.1], and the problem of finding a point π‘ž ∈ Fix(𝑇) in [8, Theorem 3.1]. (ii) The iterative scheme in [8, Theorem 3.1] is extended to develop the iterative schemes (42) and (130) in our Theorems 24 and 28 by virtue of the iterative schemes of [11, Theorem 3.1] and [10, Theorems 3.2]. The iterative schemes (42) and (130) in our Theorems 24 and 28 are more advantageous and more flexible than the iterative scheme of [8, Theorem 3.1] because they can be applied to solving three problems (i.e., GSVI (13), fixed point problem and infinitely many VIPs), and involve several parameter sequences {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, {𝛿𝑛 }, (and {πœŽπ‘› }). (iii) Our Theorems 24 and 28 extend and generalize Ceng and Yao [8, Theorem 3.1] from a nonexpansive mapping to a countable family of nonexpansive mappings, and Ceng and Yao’s [10, Theorems 3.2], to the setting of the GSVI (13) and infinitely many VIPs, Kangtunyakarn [38, Theorem 3.1], from finitely many VIPs to infinitely many VIPs, from a nonexpansive mapping to a countable family of nonexpansive mappings and from a strict pseudocontraction to the GSVI (13). In the meantime, our Theorems 24 and 28 extend and generalize Cai and Bu’s [11, Theorem 3.1], to the setting of infinitely many VIPs. (iv) The iterative schemes (42) and (130) in our Theorems 24 and 28 are very different from every one in [10, Theorem 3.2], [11, Theorem 3.1], [38, Theorem 3.1],

and [8, Theorem 3.1] because the mappings 𝐺 and 𝑇𝑛 in [11, Theorem 3.1] and the mapping 𝑇 in [8, Theorem 3.1] are replaced with the same composite mapping 𝑆𝑛 𝐺 in the iterative schemes (42) and (130) and the mapping π‘Šπ‘› in [10, Theorem 3.2] is replaced with 𝐡𝑛 . (v) Cai and Bu’s proof in [11, Theorem 3.1] depends on the argument techniques in [14], the inequality in 2uniformly smooth Banach spaces (see Lemma 4), and the inequality in smooth and uniform convex Banach spaces (see Proposition 6). Because the composite mapping 𝑆𝑛 𝐺 appears in the iterative scheme (42) of our Theorem 24, the proof of our Theorem 24 depends on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), the inequality in smooth and uniform convex Banach spaces (see Proposition 6), the inequality in uniform convex Banach spaces (see Lemma 15 in Section 2 of this paper), and the properties of the π‘Š-mapping and the Banach limit (see Lemmas 16–18 in Section 2 of this paper). However, the proof of our Theorem 28 does not depend on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), and the inequality in smooth and uniform convex Banach spaces (see Proposition 6). It depends on only the inequality in uniform convex Banach spaces (see Lemma 15 in Section 2 of this paper) and the properties of the π‘Š-mapping and the Banach limit (see Lemmas 16–18 in Section 2 of this paper). (vi) The assumption of the uniformly convex and 2uniformly smooth Banach space 𝑋 in [11, Theorem 3.1] is weakened to the one of the uniformly convex Banach space 𝑋 having a uniformly Gateaux differentiable norm in our Theorem 28. Moreover, the assumption of the uniformly smooth Banach space 𝑋 in [8, Theorem 3.1] is replaced with the one of the uniformly convex Banach space 𝑋 having a uniformly Gateaux differentiable norm in our Theorem 28. It is worth emphasizing that there is no assumption on the convergence of parameter sequences {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, and {𝛿𝑛 } (and {πœŽπ‘› }) to zero in our Theorems 24 and 28.

4. Relaxed Mann Iterations and Their Convergence Criteria In this section, we introduce our relaxed Mann iteration algorithms in real smooth and uniformly convex Banach spaces and present their convergence criteria. Theorem 31. Let 𝐢 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢. Let {πœŒπ‘› }∞ 𝑛=0 be a sequence of positive numbers in (0, 𝑏] for some 𝑏 ∈ (0, 1) and 𝐴 𝑖 : 𝐢 β†’ 𝑋 an 𝛼̂𝑖 -inverse strongly accretive mapping for each 𝑖 = 0, 1, . . .. Define a mapping 𝐺𝑖 : 𝐢 β†’ 𝐢 by Π𝐢(𝐼 βˆ’ πœ† 𝑖 𝐴 𝑖 )π‘₯ = 𝐺𝑖 π‘₯ for all π‘₯ ∈ 𝐢 and 𝑖 = 0, 1, . . .,

Abstract and Applied Analysis

25

where πœ† 𝑖 ∈ (0, 𝛼̂𝑖 /πœ…2 ] and πœ… is the 2-uniformly smooth constant of 𝑋. Let 𝐡𝑛 : 𝐢 β†’ 𝐢 be the π‘Š-mapping generated by 𝐺𝑛 , πΊπ‘›βˆ’1 , . . . , 𝐺0 and πœŒπ‘› , πœŒπ‘›βˆ’1 , . . . , 𝜌0 . Let the mapping 𝐡𝑖 : 𝐢 β†’ 𝑋 be 𝛽̂𝑖 -inverse strongly accretive for 𝑖 = 1, 2. Let 𝑓 : 𝐢 β†’ 𝐢 be a contraction with coefficient 𝜌 ∈ (0, 1). Let {𝑆𝑖 }∞ 𝑖=0 be a countable family of nonexpansive mappings of 𝐢 into itself ∞ such that 𝐹 = (β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 )) ∩ Ξ© ∩ (⋂𝑖=0 VI(𝐢, 𝐴 𝑖 )) =ΜΈ 0, where Ξ© is the fixed point set of the mapping 𝐺 = Π𝐢(πΌβˆ’πœ‡1 𝐡1 )Π𝐢(πΌβˆ’ πœ‡2 𝐡2 ) with 0 < πœ‡π‘– < 𝛽̂𝑖 /πœ…2 for 𝑖 = 1, 2. For arbitrarily given π‘₯0 ∈ 𝐢, let {π‘₯𝑛 } be the sequence generated by π‘₯𝑛+1 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 ,

βˆ€π‘› β‰₯ 0, (178)

where {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, and {𝛿𝑛 } are the sequences in (0, 1) such that 𝛼𝑛 +𝛽𝑛 +𝛾𝑛 +𝛿𝑛 = 1 for all 𝑛 β‰₯ 0. Suppose that the following conditions hold: (i) lim𝑛 β†’ ∞ 𝛼𝑛 = 0 and βˆ‘βˆž 𝑛=0 𝛼𝑛 = ∞; (ii) {𝛾𝑛 }, {𝛿𝑛 } βŠ‚ [𝑐, 𝑑] for some 𝑐, 𝑑 ∈ (0, 1); (iii) lim𝑛 β†’ ∞ (|𝛽𝑛 βˆ’ π›½π‘›βˆ’1 | + |𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 | + |𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 |) = 0; (iv) 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1. Assume that βˆ‘βˆž 𝑛=1 supπ‘₯∈𝐷 ‖𝑆𝑛 π‘₯ βˆ’ π‘†π‘›βˆ’1 π‘₯β€– < ∞ for any bounded subset 𝐷 of 𝐢 and let 𝑆 be a mapping of 𝐢 into itself defined by 𝑆π‘₯ = lim𝑛 β†’ ∞ 𝑆𝑛 π‘₯ for all π‘₯ ∈ 𝐢 and suppose that Fix(𝑆) = β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 ). Then, there hold the following: (I) lim𝑛 β†’ ∞ β€–π‘₯𝑛+1 βˆ’ π‘₯𝑛 β€– = 0; (II) the sequence {π‘₯𝑛 }∞ 𝑛=0 converges strongly to some π‘ž ∈ 𝐹 which is the unique solution of the variational inequality problem (VIP) ⟨(𝐼 βˆ’ 𝑓) π‘ž, 𝐽 (π‘ž βˆ’ 𝑝)⟩ ≀ 0,

βˆ€π‘ ∈ 𝐹,

π‘›β†’βˆž

(180)

for every π‘₯ ∈ 𝐢. That is, such a 𝐡 is the π‘Š-mapping generated ∞ by the sequences {𝐺𝑛 }∞ 𝑛=0 and {πœŒπ‘› }𝑛=0 . According to Lemma 17, ∞ we know that Fix(𝐡) = ⋂𝑖=0 Fix(𝐺𝑖 ). From Lemma 21 and the definition of 𝐺𝑖 , we have Fix(𝐺𝑖 ) = VI(𝐢, 𝐴 𝑖 ) for each 𝑖 = 0, 1, . . .. Hence, we have ∞

∞

𝑖=0

𝑖=0

Fix (𝐡) = β‹‚ Fix (𝐺𝑖 ) = β‹‚ VI (𝐢, 𝐴 𝑖 ) .

σ΅„© σ΅„© σ΅„© 󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© ≀ max {σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 , σ΅„© }. 1βˆ’πœŒ (182) By induction, we obtain σ΅„© σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© { }, π‘₯ ≀ max , π‘₯ βˆ’ 𝑝 βˆ’ 𝑝 σ΅„©σ΅„© 0 σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© 𝑛 1βˆ’πœŒ

βˆ€π‘› β‰₯ 0. (183)

Hence, {π‘₯𝑛 } is bounded, and so are the sequences {𝐺π‘₯𝑛 } and {𝑓(π‘₯𝑛 )}. Let us show that σ΅„©σ΅„©

lim σ΅„©π‘₯ 𝑛 β†’ ∞ σ΅„© 𝑛+1

σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(184)

As a matter of fact, observe that π‘₯𝑛+1 can be rewritten as follows:

𝛼𝑖 /πœ…2 ] for 𝑖 = 0, 1, . . ., it Proof. First of all, since 0 < πœ† 𝑖 < [Μ‚ is easy to see that 𝐺𝑖 is a nonexpansive mapping for each 𝑖 = 0, 1, . . .. Since 𝐡𝑛 : 𝐢 β†’ 𝐢 is the π‘Š-mapping generated by 𝐺𝑛 , πΊπ‘›βˆ’1 , . . . , 𝐺0 , and πœŒπ‘› , πœŒπ‘›βˆ’1 , . . . , 𝜌0 , by Lemma 16 we know that, for each π‘₯ ∈ 𝐢 and π‘˜ β‰₯ 0, the limit lim𝑛 β†’ ∞ π‘ˆπ‘›,π‘˜ π‘₯ exists. Moreover, one can define a mapping 𝐡 : 𝐢 β†’ 𝐢 as follows: π‘›β†’βˆž

σ΅„© σ΅„©σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 (󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (𝑝)σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩) σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 (𝜌 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩) σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©σ΅„© 󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛼𝑛 (1 βˆ’ 𝜌) σ΅„© 1βˆ’πœŒ

(179)

provided 𝛽𝑛 ≑ 𝛽 for some fixed 𝛽 ∈ (0, 1).

𝐡π‘₯ = lim 𝐡𝑛 π‘₯ = lim π‘ˆπ‘›,0 π‘₯

𝑝 = 𝐡𝑛 𝑝, and 𝑝 = 𝑆𝑛 𝑝 for all 𝑛 β‰₯ 0. By Lemma 23, we know that 𝐺 is nonexpansive. Then, from (178), we have

(181)

Next, let us show that the sequence {π‘₯𝑛 } is bounded. Indeed, take a fixed 𝑝 ∈ 𝐹 arbitrarily. Then, we get 𝑝 = 𝐺𝑝,

π‘₯𝑛+1 = 𝛽𝑛 π‘₯𝑛 + (1 βˆ’ 𝛽𝑛 ) 𝑧𝑛 ,

(185)

where 𝑧𝑛 = (𝛼𝑛 𝑓(π‘₯𝑛 ) + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 )/(1 βˆ’ 𝛽𝑛 ). Observe that σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑧𝑛 βˆ’ π‘§π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© 𝛼 𝑓 (π‘₯ ) + 𝛾 𝐡 π‘₯ + 𝛿 𝑆 𝐺π‘₯ σ΅„© 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 = σ΅„©σ΅„©σ΅„©σ΅„© 𝑛 1 βˆ’ 𝛽𝑛 σ΅„©σ΅„© βˆ’

π›Όπ‘›βˆ’1 𝑓 (π‘₯π‘›βˆ’1 ) + π›Ύπ‘›βˆ’1 π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 + π›Ώπ‘›βˆ’1 π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© 1 βˆ’ π›½π‘›βˆ’1 σ΅„©

σ΅„©σ΅„© π‘₯ βˆ’ 𝛽 π‘₯ π‘₯ βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©σ΅„© σ΅„© 𝑛 𝑛 = σ΅„©σ΅„©σ΅„© 𝑛+1 βˆ’ 𝑛 σ΅„© σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 1 βˆ’ π›½π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© π‘₯ βˆ’ 𝛽 π‘₯ π‘₯ βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„© 𝑛 𝑛 = σ΅„©σ΅„©σ΅„© 𝑛+1 βˆ’ 𝑛 1 βˆ’ 𝛽𝑛 σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 +

π‘₯𝑛 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 π‘₯𝑛 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©σ΅„© βˆ’ σ΅„© 1 βˆ’ 𝛽𝑛 1 βˆ’ π›½π‘›βˆ’1 σ΅„©σ΅„©σ΅„©

26

Abstract and Applied Analysis σ΅„©σ΅„© π‘₯ βˆ’ 𝛽 π‘₯ π‘₯ βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©σ΅„© σ΅„© 𝑛 𝑛 βˆ’ 𝑛 ≀ σ΅„©σ΅„©σ΅„© 𝑛+1 σ΅„©σ΅„© σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 σ΅„©σ΅„© π‘₯ βˆ’ 𝛽 π‘₯ π‘₯ βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©σ΅„© σ΅„© π‘›βˆ’1 π‘›βˆ’1 + σ΅„©σ΅„©σ΅„© 𝑛 βˆ’ 𝑛 σ΅„© σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 1 βˆ’ π›½π‘›βˆ’1 σ΅„©σ΅„©σ΅„© =

=

=

σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + πœ† 0 σ΅„©σ΅„©σ΅„©π‘ˆπ‘›,1 π‘₯π‘›βˆ’1 βˆ’ π‘ˆπ‘›βˆ’1,1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + πœ† 0 σ΅„©σ΅„©σ΅„©πœ† 1 𝐺1 π‘ˆπ‘›,2 π‘₯π‘›βˆ’1 βˆ’ πœ† 1 𝐺1 π‘ˆπ‘›βˆ’1,2 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + πœ† 0 πœ† 1 σ΅„©σ΅„©σ΅„©π‘ˆπ‘›,2 π‘₯π‘›βˆ’1 βˆ’ π‘ˆπ‘›βˆ’1,2 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©

1 σ΅„©σ΅„© σ΅„© σ΅„©π‘₯ βˆ’ 𝛽𝑛 π‘₯𝑛 βˆ’ (π‘₯𝑛 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 σ΅„© 𝑛+1 󡄨󡄨 1 󡄨󡄨 1 󡄨 󡄨󡄨 σ΅„©σ΅„© σ΅„© + 󡄨󡄨󡄨 βˆ’ 󡄨 σ΅„©π‘₯ βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 󡄨󡄨 1 βˆ’ 𝛽𝑛 1 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 σ΅„© 𝑛

.. . π‘›βˆ’1

σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + (βˆπœ† 𝑖 ) σ΅„©σ΅„©σ΅„©π‘ˆπ‘›,𝑛 π‘₯π‘›βˆ’1 βˆ’ π‘ˆπ‘›βˆ’1,𝑛 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 𝑖=0

1 σ΅„©σ΅„© σ΅„© σ΅„©π‘₯ βˆ’ 𝛽𝑛 π‘₯𝑛 βˆ’ (π‘₯𝑛 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 σ΅„© 𝑛+1 󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 σ΅„© σ΅„©σ΅„© + σ΅„©π‘₯ βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© (1 βˆ’ π›½π‘›βˆ’1 ) (1 βˆ’ 𝛽𝑛 ) σ΅„© 𝑛

π‘›βˆ’1

σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + π‘€βˆπœ† 𝑖 , 𝑖=0

(188)

1 σ΅„©σ΅„© 󡄩𝛼 𝑓 (π‘₯𝑛 ) + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π›Όπ‘›βˆ’1 𝑓 (π‘₯π‘›βˆ’1 ) 1 βˆ’ 𝛽𝑛 σ΅„© 𝑛 σ΅„© βˆ’ π›Ύπ‘›βˆ’1 π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 βˆ’ π›Ώπ‘›βˆ’1 π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 σ΅„© σ΅„©σ΅„© + σ΅„©π‘₯ βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© (1 βˆ’ 𝛽 ) (1 βˆ’ 𝛽 ) σ΅„© 𝑛 π‘›βˆ’1

𝑛

1 σ΅„© σ΅„© ≀ [𝛼 󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 𝑛 σ΅„© σ΅„© σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© 󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©] 󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 σ΅„© σ΅„©σ΅„© + σ΅„©π‘₯ βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© . (1 βˆ’ π›½π‘›βˆ’1 ) (1 βˆ’ 𝛽𝑛 ) σ΅„© 𝑛 (186)

for some constant 𝑀 > 0. Taking into account 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1, we may assume, Μ‚ Utilizing (186)– without loss of generality, that {𝛽𝑛 } βŠ‚ [̂𝑐, 𝑑]. (188), we have σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑧𝑛 βˆ’ π‘§π‘›βˆ’1 σ΅„©σ΅„©σ΅„© ≀

≀

On the other hand, we note that, for all 𝑛 β‰₯ 1,

1 σ΅„© σ΅„© [𝛼 󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 𝑛 σ΅„© σ΅„© σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© 󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©] 󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 σ΅„© σ΅„©σ΅„© + σ΅„©π‘₯ βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© (1 βˆ’ π›½π‘›βˆ’1 ) (1 βˆ’ 𝛽𝑛 ) σ΅„© 𝑛 π‘›βˆ’1 1 σ΅„© σ΅„© σ΅„© σ΅„© {𝛼𝑛 𝜌 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 𝛾𝑛 [σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + π‘€βˆπœ† 𝑖 ] 1 βˆ’ 𝛽𝑛 𝑖=0

σ΅„© σ΅„© σ΅„© σ΅„© + 𝛿𝑛 [σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©]

σ΅„©σ΅„© σ΅„© 󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© σ΅„© σ΅„© ≀ 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© .

󡄨 󡄨󡄩 󡄨󡄩 σ΅„© σ΅„© 󡄨 + 󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© (187)

󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© } +

Furthermore, by (CY), since 𝐺𝑖 and π‘ˆπ‘›,𝑖 are nonexpansive, we deduce that for each 𝑛 β‰₯ 1 σ΅„© σ΅„©σ΅„© 󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯π‘›βˆ’1 βˆ’ π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯π‘›βˆ’1 βˆ’ π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©πœ† 0 𝐺0 π‘ˆπ‘›,1 π‘₯π‘›βˆ’1 βˆ’ πœ† 0 𝐺0 π‘ˆπ‘›βˆ’1,1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©

=

󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 σ΅„© σ΅„©σ΅„© σ΅„©π‘₯ βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© (1 βˆ’ π›½π‘›βˆ’1 ) (1 βˆ’ 𝛽𝑛 ) σ΅„© 𝑛

1 σ΅„© σ΅„© { (1 βˆ’ 𝛽𝑛 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 π‘›βˆ’1

σ΅„© σ΅„© + 𝛾𝑛 π‘€βˆπœ† 𝑖 + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 𝑖=0

󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„©

Abstract and Applied Analysis

27

󡄨󡄩 σ΅„© 󡄨 + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©

Next we show that β€–π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 β€– β†’ 0 as 𝑛 β†’ ∞. Indeed, for simplicity, put π‘ž = Π𝐢(𝑝 βˆ’ πœ‡2 𝐡2 𝑝), 𝑒𝑛 = Π𝐢(π‘₯𝑛 βˆ’ πœ‡2 𝐡2 π‘₯𝑛 ) and V𝑛 = Π𝐢(𝑒𝑛 βˆ’ πœ‡1 𝐡1 𝑒𝑛 ). Then, V𝑛 = 𝐺π‘₯𝑛 for all 𝑛 β‰₯ 0. From Lemma 26 we have

󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© } +

󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 σ΅„© σ΅„©σ΅„© σ΅„©π‘₯ βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© (1 βˆ’ π›½π‘›βˆ’1 ) (1 βˆ’ 𝛽𝑛 ) σ΅„© 𝑛

= (1 βˆ’

σ΅„©2 σ΅„©σ΅„© 󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 = 󡄩󡄩󡄩Π𝐢 (π‘₯𝑛 βˆ’ πœ‡2 𝐡2 π‘₯𝑛 ) βˆ’ Π𝐢 (𝑝 βˆ’ πœ‡2 𝐡2 𝑝)σ΅„©σ΅„©σ΅„© (193) σ΅„© σ΅„©2 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝 βˆ’ πœ‡2 (𝐡2 π‘₯𝑛 βˆ’ 𝐡2 𝑝)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 2πœ‡2 (𝛽̂2 βˆ’ πœ…2 πœ‡2 ) 󡄩󡄩󡄩𝐡2 π‘₯𝑛 βˆ’ 𝐡2 𝑝󡄩󡄩󡄩 ,

π‘›βˆ’1 𝛼𝑛 (1 βˆ’ 𝜌) σ΅„©σ΅„© σ΅„© 𝛾𝑀 ) σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 𝑛 βˆπœ† 𝑖 1 βˆ’ 𝛽𝑛 1 βˆ’ 𝛽𝑛 𝑖=0

𝛿𝑛 σ΅„©σ΅„© 1 σ΅„© 󡄩𝑆 𝐺π‘₯ βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 1 βˆ’ 𝛽𝑛 σ΅„© 𝑛 π‘›βˆ’1 1 βˆ’ 𝛽𝑛 󡄨󡄩 󡄨󡄩 σ΅„© 󡄨 σ΅„© 󡄨 Γ— [󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©] 󡄨󡄨 󡄨󡄨 σ΅„© σ΅„©σ΅„© 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨 + σ΅„©π‘₯ βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© (1 βˆ’ 𝛽 ) (1 βˆ’ 𝛽 ) σ΅„© 𝑛 +

π‘›βˆ’1

σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 = 󡄩󡄩󡄩Π𝐢 (𝑒𝑛 βˆ’ πœ‡1 𝐡1 𝑒𝑛 ) βˆ’ Π𝐢 (π‘ž βˆ’ πœ‡1 𝐡1 π‘ž)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 ≀ 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘ž βˆ’ πœ‡1 (𝐡1 𝑒𝑛 βˆ’ 𝐡1 π‘ž)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© βˆ’ 2πœ‡1 (𝛽̂1 βˆ’ πœ…2 πœ‡1 ) 󡄩󡄩󡄩𝐡1 𝑒𝑛 βˆ’ 𝐡1 π‘žσ΅„©σ΅„©σ΅„© .

𝑛

π‘›βˆ’1

σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + π‘€βˆπœ† 𝑖 + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 𝑖=0

Substituting (193) for (194), we obtain

1 󡄨 σ΅„© 󡄨 󡄨󡄩 󡄨 + [󡄨󡄨𝛼 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 1 βˆ’ 𝛽𝑛 󡄨 𝑛 σ΅„© σ΅„© 󡄨 󡄨󡄩 σ΅„© Γ— σ΅„©σ΅„©σ΅„©π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©] 󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 + (1 βˆ’ π›½π‘›βˆ’1 ) (1 βˆ’ 𝛽𝑛 ) σ΅„© σ΅„© Γ— σ΅„©σ΅„©σ΅„©π›Όπ‘›βˆ’1 𝑓 (π‘₯π‘›βˆ’1 ) + π›Ύπ‘›βˆ’1 π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 + π›Ώπ‘›βˆ’1 π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©

σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„©2 2 σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 2πœ‡2 (𝛽̂2 βˆ’ πœ… πœ‡2 ) 󡄩󡄩󡄩𝐡2 π‘₯𝑛 βˆ’ 𝐡2 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 βˆ’ 2πœ‡1 (𝛽̂1 βˆ’ πœ…2 πœ‡1 ) 󡄩󡄩󡄩𝐡1 𝑒𝑛 βˆ’ 𝐡1 π‘žσ΅„©σ΅„©σ΅„© .

By Lemma 8, we have from (178) and (195) σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© = 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (𝑝)) + 𝛽𝑛 (π‘₯𝑛 βˆ’ 𝑝)

π‘›βˆ’1

󡄨 󡄨 + 𝑀1 [βˆπœ† 𝑖 + 󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 𝑖=0

󡄨 󡄨 󡄨 󡄨 󡄨 󡄨 + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 + 󡄨󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 ]

+ 𝛾𝑛 (𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝) + 𝛿𝑛 (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝) σ΅„©2 + 𝛼𝑛 (𝑓 (𝑝) βˆ’ 𝑝)σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© , (189) Μ‚ 2 )(‖𝑓(π‘₯ )β€–+‖𝐡 π‘₯ β€–+‖𝑆 𝐺π‘₯ β€–+𝑀)} ≀ where sup𝑛β‰₯0 {(1/(1βˆ’ 𝑑) 𝑛 𝑛 𝑛 𝑛 𝑛 𝑀1 for some 𝑀1 > 0. Thus, from (189), conditions (i), (iii) and the assumption on {𝑆𝑛 }, it follows that (noting that 0 < πœ† 𝑖 ≀ 𝑏 < 1, for all 𝑖 β‰₯ 0) σ΅„© σ΅„© σ΅„© σ΅„© lim (󡄩󡄩𝑧 βˆ’ π‘§π‘›βˆ’1 σ΅„©σ΅„©σ΅„© βˆ’ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©) ≀ 0. (190) π‘›β†’βˆž σ΅„© 𝑛 Since 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1, by Lemma 20 we get σ΅„© σ΅„© lim σ΅„©σ΅„©π‘₯ βˆ’ 𝑧𝑛 σ΅„©σ΅„©σ΅„© = 0. (191) π‘›β†’βˆž σ΅„© 𝑛 Consequently, σ΅„© σ΅„© σ΅„© σ΅„© lim σ΅„©σ΅„©π‘₯ βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = lim (1 βˆ’ 𝛽𝑛 ) 󡄩󡄩󡄩𝑧𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0. 𝑛 β†’ ∞ σ΅„© 𝑛+1 π‘›β†’βˆž

(194)

σ΅„© ≀ 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (𝑝)) + 𝛽𝑛 (π‘₯𝑛 βˆ’ 𝑝) σ΅„©2 + 𝛾𝑛 (𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝) + 𝛿𝑛 (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© + 2𝛼𝑛 βŸ¨π‘“ (𝑝) βˆ’ 𝑝, 𝐽 (π‘₯𝑛+1 βˆ’ 𝑝)⟩ σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (𝑝)σ΅„©σ΅„©σ΅„© + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„©2 σ΅„©2 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 2𝛼𝑛 βŸ¨π‘“ (𝑝) βˆ’ 𝑝, 𝐽 (π‘₯𝑛+1 βˆ’ 𝑝)⟩ σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 𝜌2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„©2 σ΅„©2 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩

(192)

+ 2𝛼𝑛 βŸ¨π‘“ (𝑝) βˆ’ 𝑝, 𝐽 (π‘₯𝑛+1 βˆ’ 𝑝)⟩

(195)

28

Abstract and Applied Analysis σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 πœŒσ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩

which implies that

σ΅„© σ΅„©2 σ΅„© σ΅„©2 + 𝛿𝑛 [σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 2πœ‡2 (𝛽̂2 βˆ’ πœ…2 πœ‡2 ) 󡄩󡄩󡄩𝐡2 π‘₯𝑛 βˆ’ 𝐡2 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 βˆ’2πœ‡1 (𝛽̂1 βˆ’ πœ…2 πœ‡1 ) 󡄩󡄩󡄩𝐡1 𝑒𝑛 βˆ’ 𝐡1 π‘žσ΅„©σ΅„©σ΅„© ] + 2𝛼𝑛 βŸ¨π‘“ (𝑝) βˆ’ 𝑝, 𝐽 (π‘₯𝑛+1 βˆ’ 𝑝)⟩ σ΅„© σ΅„©2 = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩

σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩

σ΅„© σ΅„©2 +πœ‡1 (𝛽̂1 βˆ’ πœ…2 πœ‡1 ) 󡄩󡄩󡄩𝐡1 𝑒𝑛 βˆ’ 𝐡1 π‘žσ΅„©σ΅„©σ΅„© ]

σ΅„© σ΅„©2 = 󡄩󡄩󡄩Π𝐢 (𝑒𝑛 βˆ’ πœ‡1 𝐡1 𝑒𝑛 ) βˆ’ Π𝐢 (π‘ž βˆ’ πœ‡1 𝐡1 π‘ž)σ΅„©σ΅„©σ΅„©

+ 2𝛼𝑛 βŸ¨π‘“ (𝑝) βˆ’ 𝑝, 𝐽 (π‘₯𝑛+1 βˆ’ 𝑝)⟩ σ΅„©2 σ΅„© σ΅„©2 σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 2𝛿𝑛 [πœ‡2 (𝛽̂2 βˆ’ πœ…2 πœ‡2 ) 󡄩󡄩󡄩𝐡2 π‘₯𝑛 βˆ’ 𝐡2 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 + πœ‡1 (𝛽̂1 βˆ’ πœ…2 πœ‡1 ) 󡄩󡄩󡄩𝐡1 𝑒𝑛 βˆ’ 𝐡1 π‘žσ΅„©σ΅„©σ΅„© ] σ΅„©σ΅„© σ΅„© σ΅„© + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 , which hence implies that σ΅„© σ΅„©2 2𝛿𝑛 [πœ‡2 (𝛽̂2 βˆ’ πœ…2 πœ‡2 ) 󡄩󡄩󡄩𝐡2 π‘₯𝑛 βˆ’ 𝐡2 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 + πœ‡1 (𝛽̂1 βˆ’ πœ…2 πœ‡1 ) 󡄩󡄩󡄩𝐡1 𝑒𝑛 βˆ’ 𝐡1 π‘žσ΅„©σ΅„©σ΅„© ] (197)

σ΅„© σ΅„© lim 󡄩󡄩𝐡 𝑒 βˆ’ 𝐡1 π‘žσ΅„©σ΅„©σ΅„© = 0. (198) π‘›β†’βˆž σ΅„© 1 𝑛

Utilizing Proposition 6 and Lemma 9, we have

σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„© σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ≀ 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© βˆ’ 𝑔2 (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 .

(201)

= ⟨π‘₯𝑛 βˆ’ 𝑝, 𝐽 (𝑒𝑛 βˆ’ π‘ž)⟩ + πœ‡2 ⟨𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 , 𝐽 (𝑒𝑛 βˆ’ π‘ž)⟩

(199)

(202)

Substituting (200) for (202), we get σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝑔1 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 .

≀ ⟨π‘₯𝑛 βˆ’ πœ‡2 𝐡2 π‘₯𝑛 βˆ’ (𝑝 βˆ’ πœ‡2 𝐡2 𝑝) , 𝐽 (𝑒𝑛 βˆ’ π‘ž)⟩ 1 σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 [σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© 2 σ΅„© σ΅„© βˆ’π‘”1 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) ] σ΅„© σ΅„©σ΅„© σ΅„© + πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© ,

1 σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 [󡄩𝑒 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 2 σ΅„© 𝑛 σ΅„© σ΅„© βˆ’π‘”2 (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) ] σ΅„© σ΅„©σ΅„© σ΅„© + πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ,

σ΅„© σ΅„© βˆ’ 𝑔2 (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©)

σ΅„©σ΅„© σ΅„©2 󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 = 󡄩󡄩󡄩Π𝐢 (π‘₯𝑛 βˆ’ πœ‡2 𝐡2 π‘₯𝑛 ) βˆ’ Π𝐢 (𝑝 βˆ’ πœ‡2 𝐡2 𝑝)σ΅„©σ΅„©σ΅„©

≀

= βŸ¨π‘’π‘› βˆ’ π‘ž, 𝐽 (V𝑛 βˆ’ 𝑝)⟩ + πœ‡1 ⟨𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 , 𝐽 (V𝑛 βˆ’ 𝑝)⟩

which implies that

Since β€–π‘₯𝑛 βˆ’ π‘₯𝑛+1 β€– β†’ 0, 0 < πœ‡π‘– < 𝛽̂𝑖 /πœ…2 for 𝑖 = 1, 2, and {π‘₯𝑛 } is bounded, we obtain from conditions (i), (ii) that σ΅„© σ΅„© lim 󡄩󡄩𝐡 π‘₯ βˆ’ 𝐡2 𝑝󡄩󡄩󡄩 = 0, π‘›β†’βˆž σ΅„© 2 𝑛

≀ βŸ¨π‘’π‘› βˆ’ πœ‡1 𝐡1 𝑒𝑛 βˆ’ (π‘ž βˆ’ πœ‡1 𝐡1 π‘ž) , 𝐽 (V𝑛 βˆ’ 𝑝)⟩

≀ (196)

σ΅„©σ΅„© σ΅„© σ΅„© + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 .

(200)

In the same way, we derive

σ΅„© σ΅„©2 βˆ’ 2𝛿𝑛 [πœ‡2 (𝛽̂2 βˆ’ πœ…2 πœ‡2 ) 󡄩󡄩󡄩𝐡2 π‘₯𝑛 βˆ’ 𝐡2 𝑝󡄩󡄩󡄩

σ΅„©2 σ΅„© σ΅„©2 σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩

σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝑔1 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„©σ΅„© + 2πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© .

By Lemma 8, we have from (196) and (203) σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 𝜌2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 + 𝛿𝑛 σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 2𝛼𝑛 βŸ¨π‘“ (𝑝) βˆ’ 𝑝, 𝐽 (π‘₯𝑛+1 βˆ’ 𝑝)⟩

(203)

Abstract and Applied Analysis

29

σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 πœŒσ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩

Utilizing the properties of 𝑔1 and 𝑔2 , we deduce that

σ΅„© σ΅„©2 σ΅„© σ΅„© + 𝛿𝑛 [σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝑔1 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©)

+ 2𝛼𝑛 βŸ¨π‘“ (𝑝) βˆ’ 𝑝, 𝐽 (π‘₯𝑛+1 βˆ’ 𝑝)⟩

σ΅„©σ΅„©

σ΅„© βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„© = 0,

σ΅„©σ΅„©

σ΅„© βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„© = 0.

lim σ΅„©π‘₯ π‘›β†’βˆž σ΅„© 𝑛

σ΅„© σ΅„© σ΅„© σ΅„© βˆ’ 𝑔2 (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) + 2πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© Γ— 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ]

lim 󡄩𝑒 π‘›β†’βˆž σ΅„© 𝑛

(207)

From (207), we get

σ΅„© σ΅„©2 ≀ (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩

σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛 βˆ’ V𝑛 σ΅„©σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„© 󳨀→ 0

σ΅„© σ΅„© βˆ’ 𝛿𝑛 [𝑔1 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© + 𝑔2 (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©)] σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© + 2πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝛿𝑛 [𝑔1 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© + 𝑔2 (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©)] σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© + 2πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 ,

as 𝑛 β†’ ∞. (208)

That is, σ΅„©σ΅„©

lim σ΅„©π‘₯ π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(209)

Next, let us show that σ΅„©σ΅„©

lim 󡄩𝑆 𝐺π‘₯𝑛 π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0,

σ΅„©σ΅„©

lim 󡄩𝐡 π‘₯ π‘›β†’βˆž σ΅„© 𝑛 𝑛

σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0. (210)

(204) Indeed, observe that π‘₯𝑛+1 can be rewritten as follows: which hence leads to

π‘₯𝑛+1 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛

σ΅„© σ΅„© σ΅„© σ΅„© 𝛿𝑛 [𝑔1 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) + 𝑔2 (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©)] σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩 βˆ’ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩 σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© σ΅„© σ΅„© + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡2 󡄩󡄩󡄩𝐡2 𝑝 βˆ’ 𝐡2 π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󡄩󡄩󡄩𝑒𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©σ΅„© σ΅„© + 2πœ‡1 󡄩󡄩󡄩𝐡1 π‘ž βˆ’ 𝐡1 𝑒𝑛 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©V𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© σ΅„© σ΅„© + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 . σ΅„©σ΅„©2

= 𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + (𝛾𝑛 + 𝛿𝑛 ) Γ—

σ΅„©σ΅„©2

𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 𝛾𝑛 + 𝛿𝑛

(211)

= 𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + 𝑒𝑛 𝑧̂𝑛 , where 𝑒𝑛 = 𝛾𝑛 + 𝛿𝑛 and 𝑧̂𝑛 = (𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 )/(𝛾𝑛 + 𝛿𝑛 ). Utilizing Lemma 11 and (211), we have σ΅„©2 σ΅„©σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 = 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ 𝑝) + 𝛽𝑛 (π‘₯𝑛 βˆ’ 𝑝) + 𝑒𝑛 (̂𝑧𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 (205)

σ΅„© σ΅„© σ΅„©2 σ΅„© + 𝑒𝑛 󡄩󡄩󡄩𝑧̂𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝛽𝑛 𝑒𝑛 𝑔3 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„©2 σ΅„©2 = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩

From (198), (205), conditions (i), (ii) and the boundedness of {π‘₯𝑛 }, {𝑒𝑛 }, and {V𝑛 }, we deduce that

lim 𝑔 π‘›β†’βˆž 1

σ΅„© σ΅„© (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑒𝑛 βˆ’ (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) = 0,

lim 𝑔 π‘›β†’βˆž 2

σ΅„© σ΅„© (󡄩󡄩󡄩𝑒𝑛 βˆ’ V𝑛 + (𝑝 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„©) = 0.

(206)

σ΅„©σ΅„©2 σ΅„©σ΅„© 𝛾 𝐡 π‘₯ + 𝛿 𝑆 𝐺π‘₯ σ΅„© σ΅„© σ΅„© σ΅„© 𝑛 𝑛 𝑛 βˆ’ 𝛽𝑛 𝑒𝑛 𝑔3 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) + 𝑒𝑛 σ΅„©σ΅„©σ΅„© 𝑛 𝑛 𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛 σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 σ΅„© = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝛽𝑛 𝑒𝑛 𝑔3 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„©σ΅„©2 σ΅„©σ΅„©σ΅„© 𝛾𝑛 𝛿𝑛 σ΅„© + 𝑒𝑛 σ΅„©σ΅„©σ΅„© (𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝) + (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛 σ΅„©σ΅„© σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛

30

Abstract and Applied Analysis σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝛽𝑛 𝑒𝑛 𝑔3 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) + 𝑒𝑛 [

≀

𝛾𝑛 σ΅„©σ΅„© 𝛿𝑛 σ΅„©σ΅„© σ΅„©2 σ΅„©2 󡄩𝐡 π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩𝑆 𝐺π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 ] 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛 𝑛 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛 𝑛

βˆ’

σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝛽𝑛 𝑒𝑛 𝑔3 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) ≀

𝛾𝑛 σ΅„©σ΅„© 𝛿𝑛 σ΅„©σ΅„© σ΅„© σ΅„©2 + 𝑒𝑛 [ σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩𝐺π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 ] 𝛾𝑛 + 𝛿𝑛 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛 σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝛽𝑛 𝑒𝑛 𝑔3 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) + 𝑒𝑛 [

𝛾𝑛 σ΅„©σ΅„© 𝛿𝑛 σ΅„©σ΅„© σ΅„© σ΅„©2 σ΅„©π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 ] 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛

σ΅„© σ΅„© 𝑔4 (󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©)

𝛾𝑛 σ΅„©σ΅„© 𝛿𝑛 σ΅„©σ΅„© σ΅„©2 σ΅„©2 σ΅„©π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛 βˆ’

𝛾𝑛 𝛿𝑛

2

(𝛾𝑛 + 𝛿𝑛 )

σ΅„© σ΅„© 𝑔4 (󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) 𝛾𝑛 𝛿𝑛

2

(𝛾𝑛 + 𝛿𝑛 )

σ΅„© σ΅„© 𝑔4 (󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) , (216)

which hence yields 𝛾𝑛 𝛿𝑛 (𝛾𝑛 + 𝛿𝑛 )

2

σ΅„© σ΅„© 𝑔4 (󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) (217)

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝑧̂𝑛 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑧̂𝑛 σ΅„©σ΅„©σ΅„© .

σ΅„© σ΅„© 𝛽𝑛 𝑒𝑛 𝑔3 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„© σ΅„© Γ— σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© .

(213)

σ΅„© σ΅„© lim 𝑔 (󡄩󡄩𝑧̂ βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) = 0. π‘›β†’βˆž 3 σ΅„© 𝑛

(214)

σ΅„©σ΅„©

σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(215)

Utilizing Lemma 15 and the definition of 𝑧̂𝑛 , we have σ΅„©2 σ΅„©σ΅„©Μ‚ 󡄩󡄩𝑧𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© 𝛾 𝐡 π‘₯ + 𝛿 𝑆 𝐺π‘₯ σ΅„©σ΅„©2 σ΅„© σ΅„© 𝑛 𝑛 𝑛 = σ΅„©σ΅„©σ΅„© 𝑛 𝑛 𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛 σ΅„©σ΅„© 𝛾 σ΅„©σ΅„©2 𝛿𝑛 σ΅„© σ΅„© 𝑛 = σ΅„©σ΅„©σ΅„© (𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝) + (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛 σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛 𝛾𝑛 σ΅„©σ΅„© 𝛿𝑛 σ΅„©σ΅„© σ΅„©2 σ΅„©2 󡄩𝐡 π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩𝑆 𝐺π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛 𝑛 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛 𝑛 2

(𝛾𝑛 + 𝛿𝑛 )

σ΅„© σ΅„© 𝑔4 (󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©)

(β€– 𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 β€–) = 0.

(218)

From the properties of 𝑔4 , we have σ΅„©σ΅„©

lim 󡄩𝑆 𝐺π‘₯𝑛 π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(219)

On the other hand, π‘₯𝑛+1 can also be rewritten as follows:

From the properties of 𝑔3 , we have lim 󡄩𝑧̂ π‘›β†’βˆž σ΅„© 𝑛

Since {π‘₯𝑛 } and {̂𝑧𝑛 } are bounded and ‖̂𝑧𝑛 βˆ’ π‘₯𝑛 β€– β†’ 0 as 𝑛 β†’ ∞, we deduce from condition (ii) that lim 𝑔 π‘›β†’βˆž 4

Utilizing (184), conditions (i), (ii), (iv), and the boundedness of {π‘₯𝑛 } and {𝑓(π‘₯𝑛 )}, we get

𝛾𝑛 𝛿𝑛

2

(𝛾𝑛 + 𝛿𝑛 )

σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 󡄩󡄩󡄩𝑧̂𝑛 βˆ’ 𝑝󡄩󡄩󡄩

which hence implies that

βˆ’

𝛾𝑛 𝛿𝑛

σ΅„© σ΅„©2 = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’

σ΅„© σ΅„©2 σ΅„© σ΅„©2 = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ 𝛼𝑛 ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝑒𝑛 𝑔3 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝛽𝑛 𝑒𝑛 𝑔3 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) , (212)

≀

𝛾𝑛 σ΅„©σ΅„© 𝛿𝑛 σ΅„©σ΅„© σ΅„©2 σ΅„©2 σ΅„©π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩𝐺π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛

π‘₯𝑛+1 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 = 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + (𝛼𝑛 + 𝛿𝑛 )

𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 𝛼𝑛 + 𝛿𝑛

= 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝑑𝑛 𝑧̃𝑛 , (220) where 𝑑𝑛 = 𝛼𝑛 + 𝛿𝑛 and 𝑧̃𝑛 = (𝛼𝑛 𝑓(π‘₯𝑛 ) + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 )/(𝛼𝑛 + 𝛿𝑛 ). Utilizing Lemma 11 and the convexity of β€– β‹… β€–2 , we have σ΅„©2 σ΅„©σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 = 󡄩󡄩󡄩𝛽𝑛 (π‘₯𝑛 βˆ’ 𝑝) + 𝛾𝑛 (𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝) + 𝑑𝑛 (̃𝑧𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„©2 σ΅„© + 𝑑𝑛 󡄩󡄩󡄩𝑧̃𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝛽𝑛 𝛾𝑛 𝑔5 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©)

Abstract and Applied Analysis

31

σ΅„© σ΅„© σ΅„©2 σ΅„©2 = 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩

which, together with (219), implies that

σ΅„©σ΅„©2 σ΅„©σ΅„© 𝛼 𝑓 (π‘₯ ) + 𝛿 𝑆 𝐺π‘₯ σ΅„© σ΅„© 𝑛 𝑛 𝑛 𝑛 + 𝑑𝑛 σ΅„©σ΅„©σ΅„©σ΅„© 𝑛 βˆ’ 𝑝󡄩󡄩󡄩󡄩 𝛼𝑛 + 𝛿𝑛 σ΅„©σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝛾𝑛 𝑔5 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„©2 σ΅„©2 = 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©2 σ΅„©σ΅„© 𝛼 𝛿𝑛 σ΅„© σ΅„© 𝑛 + 𝑑𝑛 σ΅„©σ΅„©σ΅„© (𝑓 (π‘₯𝑛 ) βˆ’ 𝑝) + (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© 𝛼𝑛 + 𝛿𝑛 𝛼𝑛 + 𝛿𝑛 σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝛾𝑛 𝑔5 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 𝛼𝑛 σ΅„©σ΅„© 𝛿𝑛 σ΅„©σ΅„© σ΅„©2 σ΅„©2 󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩𝑆 𝐺π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 ] 𝛼𝑛 + 𝛿𝑛 σ΅„© 𝛼𝑛 + 𝛿𝑛 σ΅„© 𝑛 𝑛 σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝛾𝑛 𝑔5 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝑑𝑛 [

𝛼𝑛 σ΅„©σ΅„© 𝛿𝑛 σ΅„©σ΅„© σ΅„©2 σ΅„©2 󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩𝐺π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 ] 𝛼𝑛 + 𝛿𝑛 𝛼𝑛 + 𝛿𝑛 σ΅„© 𝑛 σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝛾𝑛 𝑔5 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©)

+ 𝑑𝑛 [

σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 𝛼𝑛 σ΅„©σ΅„© 𝛿𝑛 σ΅„©σ΅„© σ΅„©2 σ΅„©2 󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ] 𝛼𝑛 + 𝛿𝑛 𝛼𝑛 + 𝛿𝑛 σ΅„©σ΅„© σ΅„©σ΅„© βˆ’ 𝛽𝑛 𝛾𝑛 𝑔5 (σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©) σ΅„© σ΅„©2 σ΅„© σ΅„©2 = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ 𝛼𝑛 ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝛾𝑛 𝑔5 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝛽𝑛 𝛾𝑛 𝑔5 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) , (221) + 𝑑𝑛 [

as 𝑛 β†’ ∞. (225) That is, σ΅„©σ΅„©

lim 󡄩𝑆 𝐺π‘₯𝑛 π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(226)

We note that σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© . (227) So, in terms of (209), (226), and Lemma 12, we have σ΅„©σ΅„©

lim σ΅„©π‘₯ π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(228)

Suppose that 𝛽𝑛 ≑ 𝛽 for some fixed 𝛽 ∈ (0, 1) such that 𝛼𝑛 + 𝛽 + 𝛾𝑛 + 𝛿𝑛 = 1 for all 𝑛 β‰₯ 0. Define a mapping 𝑉π‘₯ = (1 βˆ’ πœƒ1 βˆ’ πœƒ2 )𝑆π‘₯ + πœƒ1 𝐡π‘₯ + πœƒ2 𝐺π‘₯, where πœƒ1 , πœƒ2 ∈ (0, 1) are two constants with πœƒ1 + πœƒ2 < 1. Then by Lemmas 14 and 17, we have that Fix(𝑉) = Fix(𝑆) ∩ Fix(𝐡) ∩ Fix(𝐺) = 𝐹. For each π‘˜ β‰₯ 1, let {π‘π‘˜ } be a unique element of 𝐢 such that π‘π‘˜ =

1 1 𝑓 (π‘π‘˜ ) + (1 βˆ’ ) π‘‰π‘π‘˜ . π‘˜ π‘˜

(229)

From Lemma 13, we conclude that π‘π‘˜ β†’ π‘ž ∈ Fix(𝑉) = 𝐹 as π‘˜ β†’ ∞. Observe that for every 𝑛, π‘˜

which hence implies that σ΅„© σ΅„© 𝛽𝑛 𝛾𝑛 𝑔5 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„© σ΅„© Γ— σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© .

(222)

σ΅„© σ΅„© (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) = 0.

(223)

Utilizing the properties of 𝑔5 , we have σ΅„©σ΅„©

lim σ΅„©π‘₯ π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0,

σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© = 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ π΅π‘π‘˜ ) + 𝛽 (π‘₯𝑛 βˆ’ π΅π‘π‘˜ )

σ΅„© + 𝛾𝑛 (𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ ) + 𝛿𝑛 (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π΅π‘π‘˜ )σ΅„©σ΅„©σ΅„©

From (184), conditions (i), (ii), (iv), and the boundedness of {π‘₯𝑛 } and {𝑓(π‘₯𝑛 )}, we have lim 𝑔 π‘›β†’βˆž 5

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© ≀ 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󳨀→ 0

(224)

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + 𝛿𝑛 (󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„© σ΅„© = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + (𝛾𝑛 + 𝛿𝑛 ) 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ 𝛽) 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©

32

Abstract and Applied Analysis σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ 𝛽)

σ΅„©2 σ΅„©σ΅„© σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„©2 σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 + 𝑆π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„©2 σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— (2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„©) .

σ΅„© σ΅„© σ΅„© σ΅„© Γ— [󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝐡𝑛 π‘π‘˜ σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©]

σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛼𝑛 βˆ’ 𝛽) [σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + (1 βˆ’ 𝛽) [σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© = πœƒπ‘› + 𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© ,

(233) It is easy to see from (209) and (228) that σ΅„© σ΅„©2 σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„© ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© , σ΅„© σ΅„©2 σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„© ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© . (230)

where πœƒπ‘› = 𝛼𝑛 ‖𝑓(π‘₯𝑛 ) βˆ’ π΅π‘π‘˜ β€– + (1 βˆ’ 𝛽)‖𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ β€– + 𝛿𝑛 ‖𝑆𝑛 𝐺π‘₯𝑛 βˆ’π΅π‘› π‘₯𝑛 β€–. Since lim𝑛 β†’ ∞ 𝛼𝑛 = lim𝑛 β†’ ∞ ‖𝐡𝑛 π‘π‘˜ βˆ’π΅π‘π‘˜ β€– = lim𝑛 β†’ ∞ ‖𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 β€– = 0, we know that πœƒπ‘› β†’ 0 as 𝑛 β†’ ∞. From (230), we obtain σ΅„©2 σ΅„©σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„© ≀ (𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„© σ΅„© + πœƒπ‘› [2 (𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„©) + πœƒπ‘› ] 2σ΅„© σ΅„© σ΅„©2 σ΅„©2 = 𝛽2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© + 2𝛽 (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœπ‘› 2σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛽2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© + 𝛽 (1 βˆ’ 𝛽) (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© ) + πœπ‘› σ΅„©2 σ΅„©2 σ΅„© σ΅„© = 𝛽󡄩󡄩󡄩π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœπ‘› ,

In addition, note that σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„©2 σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 + 𝐺π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„©2 σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— (2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©) ,

Utilizing (232) and (234), we deduce that σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘‰π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© = πœ‡π‘› σ΅„©σ΅„©σ΅„©(1 βˆ’ πœƒ1 βˆ’ πœƒ2 ) (π‘₯𝑛 βˆ’ π‘†π‘π‘˜ ) σ΅„©2 + πœƒ1 (π‘₯𝑛 βˆ’ π΅π‘π‘˜ ) + πœƒ2 (π‘₯𝑛 βˆ’ πΊπ‘π‘˜ )σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 ≀ (1 βˆ’ πœƒ1 βˆ’ πœƒ2 ) πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„©

(235)

σ΅„© σ΅„©2 σ΅„© σ΅„©2 + πœƒ1 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœƒ2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© . Also, observe that 1 1 (π‘₯ βˆ’ 𝑓 (π‘π‘˜ )) + (1 βˆ’ ) (π‘₯𝑛 βˆ’ π‘‰π‘π‘˜ ) ; π‘˜ 𝑛 π‘˜

(236)

1 1 (1 βˆ’ ) (π‘₯𝑛 βˆ’ π‘‰π‘π‘˜ ) = π‘₯𝑛 βˆ’ π‘π‘˜ βˆ’ (π‘₯𝑛 βˆ’ 𝑓 (π‘π‘˜ )) . π‘˜ π‘˜

(237)

π‘₯𝑛 βˆ’ π‘π‘˜ = that is,

It follows from Lemma 8(ii) and (237) that (231)

where πœπ‘› = πœƒπ‘› [2(𝛽‖π‘₯𝑛 βˆ’ π΅π‘π‘˜ β€– + (1 βˆ’ 𝛽)β€–π‘₯𝑛 βˆ’ π‘π‘˜ β€–) + πœƒπ‘› ] β†’ 0 as 𝑛 β†’ ∞. For any Banach limit πœ‡, from (231) we derive σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© = πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© .

(234)

(232)

2

1 σ΅„© σ΅„©2 (1 βˆ’ ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘‰π‘π‘˜ σ΅„©σ΅„©σ΅„© π‘˜ σ΅„©2 2 σ΅„© β‰₯ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© βˆ’ ⟨π‘₯𝑛 βˆ’ π‘π‘˜ + π‘π‘˜ βˆ’ 𝑓 (π‘π‘˜ ) , 𝐽 (π‘₯𝑛 βˆ’ π‘π‘˜ )⟩ π‘˜ 2 σ΅„© σ΅„©2 2 = (1 βˆ’ ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + βŸ¨π‘“ (π‘π‘˜ ) βˆ’ π‘π‘˜ , 𝐽 (π‘₯𝑛 βˆ’ π‘π‘˜ )⟩ . π‘˜ π‘˜ (238) So by (235) and (238), we have 1 2 σ΅„© σ΅„©2 (1 βˆ’ ) πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© π‘˜ 2 σ΅„© σ΅„©2 β‰₯ (1 βˆ’ ) πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© π‘˜ 2 + πœ‡π‘› βŸ¨π‘“ (π‘π‘˜ ) βˆ’ π‘π‘˜ , 𝐽 (π‘₯𝑛 βˆ’ π‘π‘˜ )⟩ , π‘˜

(239)

Abstract and Applied Analysis

33 σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘ž)σ΅„©σ΅„©σ΅„© + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

and hence 1 σ΅„©σ΅„© σ΅„©2 2 πœ‡π‘› σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© β‰₯ πœ‡π‘› βŸ¨π‘“ (π‘π‘˜ ) βˆ’ π‘π‘˜ , 𝐽 (π‘₯𝑛 βˆ’ π‘π‘˜ )⟩ . (240) 2 π‘˜ π‘˜

(241)

Since π‘π‘˜ β†’ π‘ž ∈ Fix(𝑉) = 𝐹 as π‘˜ β†’ ∞, by the uniform Frechet differentiability of the norm of 𝑋, we have πœ‡π‘› βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛 βˆ’ π‘ž)⟩ ≀ 0.

(242)

On the other hand, from (184) and the norm-to-norm uniform continuity of 𝐽 on bounded subsets of 𝑋, it follows that 󡄨 󡄨 lim σ΅„¨σ΅„¨βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ βˆ’ βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛 βˆ’ π‘ž)βŸ©σ΅„¨σ΅„¨σ΅„¨ = 0. (243)

π‘›β†’βˆž 󡄨

So, utilizing Lemma 18 we deduce from (242) and (243) that lim sup βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛 βˆ’ π‘ž)⟩ ≀ 0, π‘›β†’βˆž

(244)

which, together with (184) and the norm-to-norm uniform continuity of 𝐽 on bounded subsets of 𝑋, implies that lim sup βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ ≀ 0. π‘›β†’βˆž

(245)

Finally, let us show that π‘₯𝑛 β†’ π‘ž as 𝑛 β†’ ∞. Utilizing Lemma 8 (i), from (178) and the convexity of β€– β‹… β€–2 , we get σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© = 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘ž)) + 𝛽𝑛 (π‘₯𝑛 βˆ’ π‘ž) + 𝛾𝑛 (𝐡𝑛 π‘₯𝑛 βˆ’ π‘ž) σ΅„©2 + 𝛿𝑛 (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘ž) + 𝛼𝑛 (𝑓 (π‘ž) βˆ’ π‘ž)σ΅„©σ΅„©σ΅„© σ΅„© ≀ 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘ž)) + 𝛽𝑛 (π‘₯𝑛 βˆ’ π‘ž) σ΅„©2 + 𝛾𝑛 (𝐡𝑛 π‘₯𝑛 βˆ’ π‘ž) + 𝛿𝑛 (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„© + 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘ž)σ΅„©σ΅„©σ΅„© + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩

+ 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 πœŒσ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

This implies that 1 σ΅„©σ΅„© σ΅„©2 πœ‡ σ΅„©π‘₯ βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© β‰₯ πœ‡π‘› βŸ¨π‘“ (π‘π‘˜ ) βˆ’ π‘π‘˜ , 𝐽 (π‘₯𝑛 βˆ’ π‘π‘˜ )⟩ . 2π‘˜ 𝑛 σ΅„© 𝑛

σ΅„© σ΅„© σ΅„©2 σ΅„©2 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

σ΅„© σ΅„©2 + 𝛿𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ σ΅„© σ΅„©2 = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ σ΅„© σ΅„©2 = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛼𝑛 (1 βˆ’ 𝜌)

2 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ . 1βˆ’πœŒ (246)

Applying Lemma 7 to (246), we obtain that π‘₯𝑛 β†’ π‘ž as 𝑛 β†’ ∞. This completes the proof. Corollary 32. Let 𝐢 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢. Let {πœŒπ‘› }∞ 𝑛=0 be a sequence of positive numbers in (0, 𝑏] for some 𝑏 ∈ (0, 1) and 𝐴 𝑖 : 𝐢 β†’ 𝑋 an 𝛼̂𝑖 -inverse strongly accretive mapping for each 𝑖 = 0, 1, . . .. Define a mapping 𝐺𝑖 : 𝐢 β†’ 𝐢 by Π𝐢(𝐼 βˆ’ πœ† 𝑖 𝐴 𝑖 )π‘₯ = 𝐺𝑖 π‘₯ for all π‘₯ ∈ 𝐢 and 𝑖 = 0, 1, . . ., where πœ† 𝑖 ∈ (0, 𝛼̂𝑖 /πœ…2 ] and πœ… is the 2-uniformly smooth constant of 𝑋. Let 𝐡𝑛 : 𝐢 β†’ 𝐢 be the π‘Š-mapping generated by 𝐺𝑛 , πΊπ‘›βˆ’1 , . . . , 𝐺0 , and πœŒπ‘› , πœŒπ‘›βˆ’1 , . . . , 𝜌0 . Let 𝑉 : 𝐢 β†’ 𝐢 be an 𝛼-strictly pseudocontractive mapping. Let 𝑓 : 𝐢 β†’ 𝐢 be a contraction with coefficient 𝜌 ∈ (0, 1). Let {𝑆𝑖 }∞ 𝑖=0 be a countable family of nonexpansive mappings of 𝐢 into itself such ∞ that 𝐹 = (β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 )) ∩ Fix(𝑉) ∩ (⋂𝑖=0 VI(𝐢, 𝐴 𝑖 )) =ΜΈ 0. For arbitrarily given π‘₯0 ∈ 𝐢, let {π‘₯𝑛 } be the sequence generated by π‘₯𝑛+1 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 ((1 βˆ’ 𝑙) 𝐼 + 𝑙𝑉) π‘₯𝑛 ,

βˆ€π‘› β‰₯ 0,

(247)

where 0 < 𝑙 < 𝛼/πœ…2 and {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, and {𝛿𝑛 } are the sequences in (0, 1) such that 𝛼𝑛 + 𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 for all 𝑛 β‰₯ 0. Suppose that the following conditions hold: (i) lim𝑛 β†’ ∞ 𝛼𝑛 = 0 and βˆ‘βˆž 𝑛=0 𝛼𝑛 = ∞; (ii) {𝛾𝑛 }, {𝛿𝑛 } βŠ‚ [𝑐, 𝑑] for some 𝑐, 𝑑 ∈ (0, 1); (iii) lim𝑛 β†’ ∞ (|𝛽𝑛 βˆ’ π›½π‘›βˆ’1 | + |𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 | + |𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 |) = 0; (iv) 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1. Assume that βˆ‘βˆž 𝑛=1 supπ‘₯∈𝐷 ‖𝑆𝑛 π‘₯ βˆ’ π‘†π‘›βˆ’1 π‘₯β€– < ∞ for any bounded subset 𝐷 of 𝐢 and let 𝑆 be a mapping of 𝐢 into itself defined

34

Abstract and Applied Analysis

by 𝑆π‘₯ = lim𝑛 β†’ ∞ 𝑆𝑛 π‘₯ for all π‘₯ ∈ 𝐢 and suppose that Fix(𝑆) = β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 ). Then, there hold the following: (I) lim𝑛 β†’ ∞ β€–π‘₯𝑛+1 βˆ’ π‘₯𝑛 β€– = 0;

(II) the sequence {π‘₯𝑛 }∞ 𝑛=0 converges strongly to some π‘ž ∈ 𝐹 which is the unique solution of the variational inequality problem (VIP) ⟨(𝐼 βˆ’ 𝑓) π‘ž, 𝐽 (π‘ž βˆ’ 𝑝)⟩ ≀ 0,

βˆ€π‘ ∈ 𝐹,

(248)

Proof. In Theorem 31, we put 𝐡1 = 𝐼 βˆ’ 𝑉, 𝐡2 = 0 and πœ‡1 = 𝑙 where 0 < 𝑙 < 𝛼/πœ…2 . Then GSVI (13) is equivalent to the VIP of finding π‘₯βˆ— ∈ 𝐢 such that βˆ€π‘₯ ∈ 𝐢.

(249)

In this case, 𝐡1 : 𝐢 β†’ 𝑋 is 𝛼-inverse strongly accretive. Repeating the same arguments as those in the proof of Corollary 25, we can infer that Fix(𝑉) = VI(𝐢, 𝐡1 ). βˆ’1 Accordingly, we know that 𝐹 = β‹‚βˆž 𝑖=0 Fix(𝑇𝑖 ) ∩ Ξ© ∩ 𝐴 0 = ∞ βˆ’1 ⋂𝑖=0 Fix(𝑇𝑖 ) ∩ Fix(𝑉) ∩ 𝐴 0, and Π𝐢 (𝐼 βˆ’ πœ‡1 𝐡1 ) Π𝐢 (𝐼 βˆ’ πœ‡2 𝐡2 ) π‘₯𝑛 = Π𝐢 (𝐼 βˆ’ πœ‡1 𝐡1 ) π‘₯𝑛 = Π𝐢 ((1 βˆ’ 𝑙) π‘₯𝑛 + 𝑙𝑉π‘₯𝑛 )

(i) lim𝑛 β†’ ∞ 𝛼𝑛 = 0 and βˆ‘βˆž 𝑛=0 𝛼𝑛 = ∞; (ii) {𝛾𝑛 }, {𝛿𝑛 } βŠ‚ [𝑐, 𝑑] for some 𝑐, 𝑑 ∈ (0, 1); (iii) βˆ‘βˆž 𝑛=1 (|πœŽπ‘› βˆ’ πœŽπ‘›βˆ’1 | + |𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 | + |𝛽𝑛 βˆ’ π›½π‘›βˆ’1 | + |𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 | + |𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 |) < ∞;

provided 𝛽𝑛 ≑ 𝛽 for some fixed 𝛽 ∈ (0, 1).

⟨𝐡1 π‘₯βˆ— , 𝐽 (π‘₯ βˆ’ π‘₯βˆ— )⟩ β‰₯ 0,

where {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, {𝛿𝑛 }, and {πœŽπ‘› } are the sequences in (0, 1) such that 𝛼𝑛 + 𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 for all 𝑛 β‰₯ 0. Suppose that the following conditions hold:

(250)

(iv) 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1 and 0 < lim inf 𝑛 β†’ ∞ πœŽπ‘› ≀ lim sup𝑛 β†’ ∞ πœŽπ‘› < 1. Assume that βˆ‘βˆž 𝑛=1 supπ‘₯∈𝐷 ‖𝑆𝑛 π‘₯ βˆ’ π‘†π‘›βˆ’1 π‘₯β€– < ∞ for any bounded subset 𝐷 of 𝐢 and let 𝑆 be a mapping of 𝐢 into itself defined by 𝑆π‘₯ = lim𝑛 β†’ ∞ 𝑆𝑛 π‘₯ for all π‘₯ ∈ 𝐢 and suppose that Fix(𝑆) = β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 ). Then, there hold the following: (I) lim𝑛 β†’ ∞ β€–π‘₯𝑛+1 βˆ’ π‘₯𝑛 β€– = 0; (II) the sequence {π‘₯𝑛 }∞ 𝑛=0 converges strongly to some π‘ž ∈ 𝐹 which is the unique solution of the variational inequality problem (VIP) ⟨(𝐼 βˆ’ 𝑓) π‘ž, 𝐽 (π‘ž βˆ’ 𝑝)⟩ ≀ 0,

= ((1 βˆ’ 𝑙) 𝐼 + 𝑙𝑉) π‘₯𝑛 .

βˆ€π‘ ∈ 𝐹,

(252)

So, scheme (178) reduces to (247). Therefore, the desired result follows from Theorem 31. Theorem 33. Let 𝐢 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢. Let {πœŒπ‘› }∞ 𝑛=0 be a sequence of positive numbers in (0, 𝑏] for some 𝑏 ∈ (0, 1) and 𝐴 𝑖 : 𝐢 β†’ π‘‹πœ‰π‘– strictly pseudocontractive and 𝛼̂𝑖 -strongly accretive with πœ‰π‘– + 𝛼̂𝑖 β‰₯ 1 for each 𝑖 = 0, 1, . . .. Define a mapping 𝐺𝑖 : 𝐢 β†’ 𝐢 by Π𝐢(𝐼 βˆ’ πœ† 𝑖 𝐴 𝑖 )π‘₯ = 𝐺𝑖 π‘₯ for all π‘₯ ∈ 𝐢 and 𝑖 = 0, 1, . . ., where 1βˆ’(πœ‰π‘– /(1+πœ‰π‘– ))(1βˆ’βˆš(1 βˆ’ 𝛼̂𝑖 )/πœ‰π‘– ) ≀ πœ† 𝑖 ≀ 1 for all 𝑖 = 0, 1, . . .. Let 𝐡𝑛 : 𝐢 β†’ 𝐢 be the π‘Š-mapping generated by 𝐺𝑛 , πΊπ‘›βˆ’1 , . . . , 𝐺0 and πœŒπ‘› , πœŒπ‘›βˆ’1 , . . . , 𝜌0 . Let the mapping 𝐡𝑖 : 𝐢 β†’ 𝑋 be πœπ‘– -strictly pseudocontractive and 𝛽̂𝑖 -strongly accretive with πœπ‘– + 𝛽̂𝑖 β‰₯ 1 for 𝑖 = 1, 2. Let 𝑓 : 𝐢 β†’ 𝐢 be a contraction with coefficient 𝜌 ∈ (0, 1). Let {𝑆𝑖 }∞ 𝑖=0 be a countable family of nonexpansive mappings of 𝐢 into itself such that 𝐹 = (β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 )) ∩ VI(𝐢, 𝐴 )) = ΜΈ 0, where Ξ© is the fixed point set of the Ξ© ∩ (β‹‚βˆž 𝑖 𝑖=0 mapping 𝐺 = Π𝐢(𝐼 βˆ’ πœ‡1 𝐡1 )Π𝐢(𝐼 βˆ’ πœ‡2 𝐡2 ) with 1 βˆ’ (πœπ‘– /(1 + 𝜁 ))(1 βˆ’ √(1 βˆ’ 𝛽̂ )/𝜁 ) ≀ πœ‡ ≀ 1 for 𝑖 = 1, 2. For arbitrarily 𝑖

𝑖

𝑖

𝑖

given π‘₯0 ∈ 𝐢, let {π‘₯𝑛 } be the sequence generated by

Proof. First of all, it is easy to see that (251) can be rewritten as follows: 𝑦𝑛 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 , π‘₯π‘₯+1 = πœŽπ‘› 𝐺π‘₯𝑛 + (1 βˆ’ πœŽπ‘› ) 𝑦𝑛 ,

βˆ€π‘› β‰₯ 0.

βˆ€π‘› β‰₯ 0, (251)

(253)

Take a fixed 𝑝 ∈ 𝐹 arbitrarily. Then, we obtain 𝑝 = 𝐺𝑝, 𝑝 = 𝐡𝑛 𝑝 and 𝑆𝑛 𝑝 = 𝑝 for all 𝑛 β‰₯ 0. Thus, we get from (253) σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 (𝜌 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩)

π‘₯𝑛+1 = πœŽπ‘› 𝐺π‘₯𝑛 + (1 βˆ’ πœŽπ‘› ) Γ— [𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 ] ,

provided 𝛽𝑛 ≑ 𝛽 for some fixed 𝛽 ∈ (0, 1).

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 ,

(254)

Abstract and Applied Analysis

35

and hence σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„© σ΅„© ≀ πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ πœŽπ‘› ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© ≀ πœŽπ‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ πœŽπ‘› ) σ΅„© σ΅„© σ΅„© σ΅„© Γ— [(1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩] σ΅„© σ΅„© = (1 βˆ’ (1 βˆ’ πœŽπ‘› ) 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© + (1 βˆ’ πœŽπ‘› ) 𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© = (1 βˆ’ (1 βˆ’ πœŽπ‘› ) 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©σ΅„© 󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ πœŽπ‘› ) 𝛼𝑛 (1 βˆ’ 𝜌) σ΅„© 1βˆ’πœŒ σ΅„© σ΅„© σ΅„© 󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© ≀ max {σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 , σ΅„© }. 1βˆ’πœŒ (255)

σ΅„©σ΅„© 𝑦 βˆ’ 𝛽 π‘₯ 𝑦 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©σ΅„© σ΅„© 𝑛 𝑛 βˆ’ π‘›βˆ’1 ≀ σ΅„©σ΅„©σ΅„© 𝑛 σ΅„©σ΅„© σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 σ΅„©σ΅„© 𝑦 βˆ’ 𝛽 π‘₯ 𝑦 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©σ΅„© σ΅„© π‘›βˆ’1 π‘›βˆ’1 + σ΅„©σ΅„©σ΅„© π‘›βˆ’1 βˆ’ π‘›βˆ’1 σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 1 βˆ’ π›½π‘›βˆ’1 =

=

=

1 σ΅„©σ΅„© σ΅„© 󡄩𝑦 βˆ’ 𝛽𝑛 π‘₯𝑛 βˆ’ (π‘¦π‘›βˆ’1 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 σ΅„© 𝑛 󡄨󡄨 1 󡄨󡄨 1 󡄨 󡄨󡄨 σ΅„©σ΅„© σ΅„© + 󡄨󡄨󡄨 βˆ’ 󡄨 󡄩𝑦 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 󡄨󡄨 1 βˆ’ 𝛽𝑛 1 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 σ΅„© π‘›βˆ’1 1 σ΅„©σ΅„© σ΅„© 󡄩𝑦 βˆ’ 𝛽𝑛 π‘₯𝑛 βˆ’ (π‘¦π‘›βˆ’1 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 σ΅„© 𝑛 󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 σ΅„© σ΅„©σ΅„© + 󡄩𝑦 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© (1 βˆ’ π›½π‘›βˆ’1 ) (1 βˆ’ 𝛽𝑛 ) σ΅„© π‘›βˆ’1 1 σ΅„©σ΅„© 󡄩𝛼 𝑓 (π‘₯𝑛 ) + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 1 βˆ’ 𝛽𝑛 σ΅„© 𝑛 βˆ’ π›Όπ‘›βˆ’1 𝑓 (π‘₯π‘›βˆ’1 ) βˆ’ π›Ύπ‘›βˆ’1 π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„© βˆ’π›Ώπ‘›βˆ’1 π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©

By induction, we have σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© 󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© }, σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 ≀ max {σ΅„©σ΅„©σ΅„©π‘₯0 βˆ’ 𝑝󡄩󡄩󡄩 , σ΅„© 1βˆ’πœŒ

+ βˆ€π‘› β‰₯ 0. (256)

which implies that {π‘₯𝑛 } is bounded and so are the sequences {𝑦𝑛 }, {𝐺π‘₯𝑛 } and {𝑓(π‘₯𝑛 )}. Let us show that σ΅„©σ΅„©

lim σ΅„©π‘₯ 𝑛 β†’ ∞ σ΅„© 𝑛+1

σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(257)

As a matter of fact, observe that 𝑦𝑛 can be rewritten as follows: 𝑦𝑛 = 𝛽𝑛 π‘₯𝑛 + (1 βˆ’ 𝛽𝑛 ) 𝑧𝑛 ,

(258)

where 𝑧𝑛 = (𝛼𝑛 𝑓(π‘₯𝑛 ) + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 )/(1 βˆ’ 𝛽𝑛 ). Observe that σ΅„©σ΅„© σ΅„© 󡄩󡄩𝑧𝑛 βˆ’ π‘§π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© 𝛼 𝑓 (π‘₯ ) + 𝛾 𝐡 π‘₯ + 𝛿 𝑆 𝐺π‘₯ σ΅„© 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 = σ΅„©σ΅„©σ΅„©σ΅„© 𝑛 1 βˆ’ 𝛽𝑛 σ΅„©σ΅„© βˆ’

π›Όπ‘›βˆ’1 𝑓 (π‘₯π‘›βˆ’1 ) + π›Ύπ‘›βˆ’1 π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 + π›Ώπ‘›βˆ’1 π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© 1 βˆ’ π›½π‘›βˆ’1 σ΅„©

σ΅„©σ΅„© 𝑦 βˆ’ 𝛽 π‘₯ 𝑦 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©σ΅„© σ΅„© 𝑛 𝑛 = σ΅„©σ΅„©σ΅„© 𝑛 βˆ’ π‘›βˆ’1 σ΅„©σ΅„© σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 σ΅„©σ΅„© 1 βˆ’ π›½π‘›βˆ’1 σ΅„©σ΅„© 𝑦 βˆ’ 𝛽 π‘₯ 𝑦 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„© 𝑛 𝑛 = σ΅„©σ΅„©σ΅„© 𝑛 βˆ’ π‘›βˆ’1 1 βˆ’ 𝛽𝑛 σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 +

π‘¦π‘›βˆ’1 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 π‘¦π‘›βˆ’1 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©σ΅„© βˆ’ σ΅„©σ΅„© σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 1 βˆ’ π›½π‘›βˆ’1

≀

󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 σ΅„© σ΅„©σ΅„© σ΅„©σ΅„©π‘¦π‘›βˆ’1 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© (1 βˆ’ π›½π‘›βˆ’1 ) (1 βˆ’ 𝛽𝑛 )

1 σ΅„© σ΅„© [𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 σ΅„© σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© 󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©] 󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 σ΅„© σ΅„©σ΅„© + σ΅„©σ΅„©π‘¦π‘›βˆ’1 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© . (1 βˆ’ 𝛽 ) (1 βˆ’ 𝛽 ) π‘›βˆ’1

𝑛

(259)

On the other hand, repeating the same arguments as those of (52) and (54) in the proof of Theorem 24, we can deduce that for all 𝑛 β‰₯ 1 σ΅„©σ΅„© σ΅„© 󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© , π‘›βˆ’1

(260)

σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© 󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + π‘€βˆπœ† 𝑖 , 𝑖=0

for some constant 𝑀 > 0. Taking into account 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1, we may assume,

36

Abstract and Applied Analysis

Μ‚ Utilizing (259)without loss of generality, that {𝛽𝑛 } βŠ‚ [̂𝑐, 𝑑]. (260) we have σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑧𝑛 βˆ’ π‘§π‘›βˆ’1 σ΅„©σ΅„©σ΅„© ≀

󡄨󡄩 󡄨 σ΅„© Γ— [󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„©

󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©

󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 (1 βˆ’ π›½π‘›βˆ’1 ) (1 βˆ’ 𝛽𝑛 ) σ΅„© σ΅„©σ΅„© Γ— σ΅„©σ΅„©π›Όπ‘›βˆ’1 𝑓 (π‘₯π‘›βˆ’1 ) + π›Ύπ‘›βˆ’1 π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 + π›Ώπ‘›βˆ’1 π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©] +

1 1 βˆ’ 𝛽𝑛 π‘›βˆ’1

σ΅„© σ΅„© σ΅„© σ΅„© Γ— {𝛼𝑛 𝜌 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 𝛾𝑛 [σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + π‘€βˆπœ† 𝑖 ]

≀ (1 βˆ’

𝑖=0

σ΅„© σ΅„© σ΅„© σ΅„© + 𝛿𝑛 [σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©]

π‘›βˆ’1

󡄨 󡄨 󡄨 󡄨 󡄨 󡄨 + 𝑀1 [βˆπœ† 𝑖 + 󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 + 󡄨󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 𝑖=0

󡄨 󡄨󡄩 󡄨󡄩 σ΅„© σ΅„© 󡄨 + 󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©

󡄨 σ΅„© 󡄨 σ΅„© + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 ] + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© ,

󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© } +

(261)

󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 σ΅„© σ΅„©σ΅„© 󡄩𝑦 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© (1 βˆ’ π›½π‘›βˆ’1 ) (1 βˆ’ 𝛽𝑛 ) σ΅„© π‘›βˆ’1

1 σ΅„© σ΅„© = { (1 βˆ’ 𝛽𝑛 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 π‘›βˆ’1

σ΅„© σ΅„© + 𝛾𝑛 π‘€βˆπœ† 𝑖 + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 𝑖=0

󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© 󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 󡄨 󡄨󡄩 σ΅„© + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© } 󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 σ΅„© σ΅„©σ΅„© + 󡄩𝑦 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© (1 βˆ’ π›½π‘›βˆ’1 ) (1 βˆ’ 𝛽𝑛 ) σ΅„© π‘›βˆ’1 = (1 βˆ’

𝛼𝑛 (1 βˆ’ 𝜌) σ΅„©σ΅„© σ΅„© ) σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛

Μ‚ 2 )(‖𝑓(π‘₯ )β€– + ‖𝐡 π‘₯ β€– + ‖𝑆 𝐺π‘₯ β€– + where sup𝑛β‰₯0 {(1/(1 βˆ’ 𝑑) 𝑛 𝑛 𝑛 𝑛 𝑛 𝑀)} ≀ 𝑀1 for some 𝑀1 > 0. In the meantime, observe that π‘₯𝑛+1 βˆ’ π‘₯𝑛 = πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ 𝐺π‘₯π‘›βˆ’1 ) + (πœŽπ‘› βˆ’ πœŽπ‘›βˆ’1 ) (𝐺π‘₯π‘›βˆ’1 βˆ’ π‘§π‘›βˆ’1 ) + (1 βˆ’ πœŽπ‘› ) (𝑧𝑛 βˆ’ π‘§π‘›βˆ’1 ) . (262) This together with (261), implies that σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© 󡄨 󡄨󡄩 σ΅„© ≀ πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + σ΅„¨σ΅„¨σ΅„¨πœŽπ‘› βˆ’ πœŽπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝐺π‘₯π‘›βˆ’1 βˆ’ π‘§π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + (1 βˆ’ πœŽπ‘› ) 󡄩󡄩󡄩𝑧𝑛 βˆ’ π‘§π‘›βˆ’1 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© 󡄨 󡄨󡄩 σ΅„© ≀ πœŽπ‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + σ΅„¨σ΅„¨σ΅„¨πœŽπ‘› βˆ’ πœŽπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝐺π‘₯π‘›βˆ’1 βˆ’ π‘§π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + (1 βˆ’ πœŽπ‘› ) {(1 βˆ’

π‘›βˆ’1

󡄨 󡄨 󡄨 󡄨 + 𝑀1 [βˆπœ† 𝑖 + 󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 + 󡄨󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨

π‘›βˆ’1

𝛼𝑛 (1 βˆ’ 𝜌) σ΅„©σ΅„© σ΅„© 𝛾𝑀 ) σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 𝑛 βˆπœ† 𝑖 1 βˆ’ 𝛽𝑛 1 βˆ’ 𝛽𝑛 𝑖=0

𝑖=0

󡄨 󡄨 󡄨 󡄨 + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 ]

𝛿𝑛 σ΅„©σ΅„© σ΅„© + 󡄩𝑆 𝐺π‘₯ βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 σ΅„© 𝑛 π‘›βˆ’1 1 󡄨󡄩 󡄨 󡄨 σ΅„© 󡄨 [󡄨󡄨𝛼 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝑓 (π‘₯π‘›βˆ’1 )σ΅„©σ΅„©σ΅„© + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 1 βˆ’ 𝛽𝑛 󡄨 𝑛 σ΅„© σ΅„© 󡄨 󡄨󡄩 σ΅„© Γ— σ΅„©σ΅„©σ΅„©π΅π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 σ΅„©σ΅„©σ΅„©π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„©] 󡄨󡄨 󡄨 󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝑦 βˆ’ π›½π‘›βˆ’1 π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + σ΅„© (1 βˆ’ 𝛽 ) (1 βˆ’ 𝛽 ) σ΅„© π‘›βˆ’1

σ΅„© σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© }

+

π‘›βˆ’1

𝑛

π‘›βˆ’1 𝛼 (1 βˆ’ 𝜌) σ΅„©σ΅„© σ΅„© ≀ (1 βˆ’ 𝑛 ) σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© + π‘€βˆπœ† 𝑖 1 βˆ’ 𝛽𝑛 𝑖=0

σ΅„© σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© +

1 1 βˆ’ 𝛽𝑛

𝛼𝑛 (1 βˆ’ 𝜌) σ΅„©σ΅„© σ΅„© ) σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛

≀ (1 βˆ’

(1 βˆ’ πœŽπ‘› ) 𝛼𝑛 (1 βˆ’ 𝜌) σ΅„©σ΅„© σ΅„© ) σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛

󡄨 󡄨󡄩 σ΅„© + σ΅„¨σ΅„¨σ΅„¨πœŽπ‘› βˆ’ πœŽπ‘›βˆ’1 󡄨󡄨󡄨 󡄩󡄩󡄩𝐺π‘₯π‘›βˆ’1 βˆ’ π‘§π‘›βˆ’1 σ΅„©σ΅„©σ΅„© π‘›βˆ’1

󡄨 󡄨 󡄨 󡄨 + 𝑀1 [βˆπœ† 𝑖 + 󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 + 󡄨󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 𝑖=0

󡄨 󡄨 󡄨 󡄨 + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 ]

Abstract and Applied Analysis

37

σ΅„© σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© ≀ (1 βˆ’

Utilizing Lemma 15 we get from (253) and (265)

(1 βˆ’ πœŽπ‘› ) 𝛼𝑛 (1 βˆ’ 𝜌) σ΅„©σ΅„© σ΅„© ) σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© 1 βˆ’ 𝛽𝑛 π‘›βˆ’1

󡄨 󡄨 󡄨 󡄨 + 𝑀2 [βˆπœ† 𝑖 + σ΅„¨σ΅„¨σ΅„¨πœŽπ‘› βˆ’ πœŽπ‘›βˆ’1 󡄨󡄨󡄨 + 󡄨󡄨󡄨𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 󡄨󡄨󡄨 𝑖=0

󡄨 󡄨 󡄨 󡄨 󡄨 󡄨 + 󡄨󡄨󡄨𝛽𝑛 βˆ’ π›½π‘›βˆ’1 󡄨󡄨󡄨 + 󡄨󡄨󡄨𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 󡄨󡄨󡄨 + 󡄨󡄨󡄨𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 󡄨󡄨󡄨 ] σ΅„© σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯π‘›βˆ’1 βˆ’ π‘†π‘›βˆ’1 𝐺π‘₯π‘›βˆ’1 σ΅„©σ΅„©σ΅„© , (263)

where sup𝑛β‰₯0 {𝑀1 + ‖𝐺π‘₯𝑛 βˆ’ 𝑧𝑛 β€–} ≀ 𝑀2 for some 𝑀2 > 0. Since βˆ‘βˆž 𝑛=0 𝛼𝑛 = ∞ and (1βˆ’πœŽπ‘› )𝛼𝑛 (1βˆ’πœŒ)/(1βˆ’π›½π‘› ) β‰₯ (1βˆ’πœŽπ‘› )𝛼𝑛 (1βˆ’πœŒ), we obtain from conditions (i) and (iv) that βˆ‘βˆž 𝑛=0 ((1βˆ’πœŽπ‘› )𝛼𝑛 (1βˆ’ 𝜌)/(1 βˆ’ 𝛽𝑛 )) = ∞. Thus, applying Lemma 7 to (263), we deduce from condition (iii) and the assumption on {𝑆𝑛 } that (noting that 0 < πœ† 𝑖 ≀ 𝑏 < 1, for all 𝑖 β‰₯ 0) σ΅„©σ΅„©

lim σ΅„©π‘₯ 𝑛 β†’ ∞ σ΅„© 𝑛+1

σ΅„© βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

σ΅„©2 σ΅„©σ΅„© σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 = σ΅„©σ΅„©σ΅„©πœŽπ‘› (𝐺π‘₯𝑛 βˆ’ 𝑝) + (1 βˆ’ πœŽπ‘› ) (𝑦𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ πœŽπ‘› ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© βˆ’ πœŽπ‘› (1 βˆ’ πœŽπ‘› ) 𝑔 (󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©) σ΅„©2 σ΅„© ≀ πœŽπ‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ πœŽπ‘› ) σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„© Γ— [σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩] σ΅„© σ΅„© βˆ’ πœŽπ‘› (1 βˆ’ πœŽπ‘› ) 𝑔 (󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©) σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© βˆ’ πœŽπ‘› (1 βˆ’ πœŽπ‘› ) 𝑔 (󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©) , which hence yields σ΅„© σ΅„© πœŽπ‘› (1 βˆ’ πœŽπ‘› ) 𝑔 (󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩

σ΅„©σ΅„© σ΅„© σ΅„© + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩

(264)

(267)

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„©

Next, we show that β€–π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 β€– β†’ 0 as 𝑛 β†’ ∞. Indeed, according to Lemma 8 we have from (253)

σ΅„©σ΅„© σ΅„© σ΅„© + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 .

σ΅„©2 σ΅„©σ΅„© 󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© = 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (𝑝)) + 𝛽𝑛 (π‘₯𝑛 βˆ’ 𝑝) + 𝛾𝑛 (𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝) σ΅„©2 + 𝛿𝑛 (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝) + 𝛼𝑛 (𝑓 (𝑝) βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„© ≀ 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (𝑝)) + 𝛽𝑛 (π‘₯𝑛 βˆ’ 𝑝) σ΅„©2 + 𝛾𝑛 (𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝) + 𝛿𝑛 (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„©

Since 𝛼𝑛 β†’ 0 and β€–π‘₯𝑛+1 βˆ’ π‘₯𝑛 β€– β†’ 0, from condition (iv) and the boundedness of {π‘₯𝑛 }, it follows that σ΅„© σ΅„© lim 𝑔 (󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©) = 0.

π‘›β†’βˆž

(268)

Utilizing the properties of 𝑔, we have σ΅„© σ΅„© lim 󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© = 0,

π‘›β†’βˆž σ΅„©

(269)

which, together with (253) and (257), implies that σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘₯𝑛+1 σ΅„©σ΅„©σ΅„© + πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© 󳨀→ 0

+ 2𝛼𝑛 βŸ¨π‘“ (𝑝) βˆ’ 𝑝, 𝐽 (π‘₯𝑛+1 βˆ’ 𝑝)⟩ σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (𝑝)σ΅„©σ΅„©σ΅„© + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„©2 σ΅„©2 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩

(270)

as 𝑛 󳨀→ ∞.

+ 2𝛼𝑛 βŸ¨π‘“ (𝑝) βˆ’ 𝑝, 𝐽 (π‘₯𝑛+1 βˆ’ 𝑝)⟩ That is,

σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 πœŒσ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„© + 𝛿𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© σ΅„© σ΅„© + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 2𝛼𝑛 󡄩󡄩󡄩𝑓 (𝑝) βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑝󡄩󡄩󡄩 .

(266)

σ΅„©σ΅„©

lim σ΅„©π‘₯ π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© = 0.

(271)

Since σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© ,

(272)

it immediately follows from (269) and (271) that (265)

σ΅„©σ΅„©

lim σ΅„©π‘₯ π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(273)

38

Abstract and Applied Analysis

On the other hand, observe that 𝑦𝑛 can be rewritten as follows: 𝑦𝑛 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 𝛾 𝐡 π‘₯ + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + (𝛾𝑛 + 𝛿𝑛 ) 𝑛 𝑛 𝑛 (274) 𝛾𝑛 + 𝛿𝑛 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + 𝑒𝑛 𝑧̂𝑛 , where 𝑒𝑛 = 𝛾𝑛 + 𝛿𝑛 and 𝑧̂𝑛 = (𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 )/(𝛾𝑛 + 𝛿𝑛 ). Utilizing Lemma 11, we have σ΅„©2 σ΅„©σ΅„© 󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„©2 = 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ 𝑝) + 𝛽𝑛 (π‘₯𝑛 βˆ’ 𝑝) + 𝑒𝑛 (̂𝑧𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„©2 σ΅„© + 𝑒𝑛 󡄩󡄩󡄩𝑧̂𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝛽𝑛 𝑒𝑛 𝑔1 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„©2 σ΅„©2 = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝑒𝑛 𝑔1 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„©σ΅„©2 σ΅„©σ΅„© 𝛾 𝐡 π‘₯ + 𝛿 𝑆 𝐺π‘₯ σ΅„© σ΅„© 𝑛 𝑛 𝑛 + 𝑒𝑛 σ΅„©σ΅„©σ΅„© 𝑛 𝑛 𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛 σ΅„© σ΅„© σ΅„©2 σ΅„©2 = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝑒𝑛 𝑔1 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„©σ΅„©2 σ΅„©σ΅„© 𝛾 𝛿𝑛 σ΅„© σ΅„© 𝑛 + 𝑒𝑛 σ΅„©σ΅„©σ΅„© (𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝) + (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛 σ΅„©σ΅„© σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛 σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝑒𝑛 𝑔1 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) + 𝑒𝑛 [

𝛾𝑛 σ΅„©σ΅„© 𝛿𝑛 σ΅„©σ΅„© σ΅„©2 σ΅„©2 󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩𝑆 𝐺π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 ] 𝛾𝑛 + 𝛿𝑛 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛 𝑛

σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝑒𝑛 𝑔1 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©)

σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© . (276) Utilizing (271), conditions (i), (ii), (iv), and the boundedness of {π‘₯𝑛 }, {𝑦𝑛 } and {𝑓(π‘₯𝑛 )}, we get σ΅„© σ΅„© lim 𝑔 (󡄩󡄩𝑧̂ βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) = 0. (277) π‘›β†’βˆž 1 σ΅„© 𝑛 From the properties of 𝑔1 , we have σ΅„© σ΅„© lim 󡄩󡄩𝑧̂ βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0. π‘›β†’βˆž σ΅„© 𝑛

(278)

Utilizing Lemma 15 and the definition of 𝑧̂𝑛 , we have σ΅„©σ΅„©Μ‚ σ΅„©2 󡄩󡄩𝑧𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© 𝛾 𝐡 π‘₯ + 𝛿 𝑆 𝐺π‘₯ σ΅„©σ΅„©2 σ΅„© σ΅„© 𝑛 𝑛 𝑛 = σ΅„©σ΅„©σ΅„© 𝑛 𝑛 𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„© σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛 σ΅„©σ΅„© 𝛾 σ΅„©σ΅„©σ΅„©2 𝛿𝑛 σ΅„© 𝑛 = σ΅„©σ΅„©σ΅„© (𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝) + (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛 σ΅„©σ΅„© 𝛾𝑛 + 𝛿𝑛 ≀

𝛾𝑛 σ΅„©σ΅„© 𝛿𝑛 σ΅„©σ΅„© σ΅„©2 σ΅„©2 󡄩𝐡 π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩𝑆 𝐺π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛 𝑛 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛 𝑛 βˆ’

𝛾𝑛 𝛿𝑛

2

(𝛾𝑛 + 𝛿𝑛 )

σ΅„© σ΅„©2 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ which leads to 𝛾𝑛 𝛿𝑛

2

(𝛾𝑛 + 𝛿𝑛 )

σ΅„© σ΅„© 𝑔2 (󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) 𝛾𝑛 𝛿𝑛

2

(𝛾𝑛 + 𝛿𝑛 )

σ΅„© σ΅„© 𝑔2 (󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) , (279)

σ΅„© σ΅„© 𝑔2 (󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©)

σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 󡄩󡄩󡄩𝑧̂𝑛 βˆ’ 𝑝󡄩󡄩󡄩

(280)

σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝑧̂𝑛 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑧̂𝑛 σ΅„©σ΅„©σ΅„© .

Since {π‘₯𝑛 } and {̂𝑧𝑛 } are bounded, we deduce from (278) and condition (ii) that σ΅„© σ΅„© lim 𝑔 (󡄩󡄩𝐡 π‘₯ βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©) = 0. (281) π‘›β†’βˆž 2 σ΅„© 𝑛 𝑛 From the properties of 𝑔2 , we have σ΅„© σ΅„© lim 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

𝛾𝑛 σ΅„©σ΅„© 𝛿𝑛 σ΅„©σ΅„© σ΅„©2 σ΅„©2 + 𝑒𝑛 [ σ΅„©π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©π‘₯ βˆ’ 𝑝󡄩󡄩󡄩 ] 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛 𝛾𝑛 + 𝛿𝑛 σ΅„© 𝑛

π‘›β†’βˆž

(282)

Furthermore, 𝑦𝑛 can also be rewritten as follows:

σ΅„©2 σ΅„© σ΅„©2 σ΅„© = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ 𝛼𝑛 ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩

𝑦𝑛 = 𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛

σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝑒𝑛 𝑔1 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©)

σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝑒𝑛 𝑔1 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) ,

which hence implies that σ΅„© σ΅„© 𝛽𝑛 𝑒𝑛 𝑔1 (󡄩󡄩󡄩𝑧̂𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩

= 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + (𝛼𝑛 + 𝛿𝑛 )

𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 𝛼𝑛 + 𝛿𝑛

= 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝑑𝑛 𝑧̃𝑛 , (275)

(283)

Abstract and Applied Analysis

39

where 𝑑𝑛 = 𝛼𝑛 + 𝛿𝑛 and 𝑧̃𝑛 = (𝛼𝑛 𝑓(π‘₯𝑛 ) + 𝛿𝑛 𝑆𝑛 𝐺π‘₯𝑛 )/(𝛼𝑛 + 𝛿𝑛 ). Utilizing Lemma 11 and the convexity of β€– β‹… β€–2 , we have 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩2 σ΅„© σ΅„© σ΅„© σ΅„©2 = 󡄩󡄩󡄩𝛽𝑛 (π‘₯𝑛 βˆ’ 𝑝) + 𝛾𝑛 (𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝) + 𝑑𝑛 (̃𝑧𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„©2 σ΅„© + 𝑑𝑛 󡄩󡄩󡄩𝑧̃𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝛽𝑛 𝛾𝑛 𝑔3 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©)

Thus, from (282) and (287), we get σ΅„© σ΅„© σ΅„© σ΅„©σ΅„© σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©

σ΅„© σ΅„© + 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󳨀→ 0

That is,

σ΅„© σ΅„© σ΅„©2 σ΅„©2 = 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩

σ΅„©σ΅„©

σ΅„©σ΅„© 𝛼 𝑓 (π‘₯ ) + 𝛿 𝑆 𝐺π‘₯ σ΅„©σ΅„©σ΅„©2 σ΅„©σ΅„© 𝑛 𝑛 𝑛 𝑛 𝑛 σ΅„© + 𝑑𝑛 σ΅„©σ΅„© βˆ’ 𝑝󡄩󡄩󡄩󡄩 𝛼𝑛 + 𝛿𝑛 σ΅„©σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝛾𝑛 𝑔3 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„©2 σ΅„©2 = 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„©σ΅„©2 σ΅„©σ΅„© 𝛼 𝛿𝑛 σ΅„© σ΅„© 𝑛 + 𝑑𝑛 σ΅„©σ΅„©σ΅„© (𝑓 (π‘₯𝑛 ) βˆ’ 𝑝) + (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑝)σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© 𝛼𝑛 + 𝛿𝑛 𝛼𝑛 + 𝛿𝑛 σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝛾𝑛 𝑔3 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 𝛼𝑛 σ΅„©σ΅„© 𝛿𝑛 σ΅„©σ΅„© σ΅„©2 σ΅„©2 󡄩𝑓 (π‘₯𝑛 )βˆ’π‘σ΅„©σ΅„©σ΅„© + 󡄩𝑆 𝐺π‘₯ βˆ’π‘σ΅„©σ΅„© ] 𝛼𝑛 + 𝛿𝑛 σ΅„© 𝛼𝑛 + 𝛿𝑛 σ΅„© 𝑛 𝑛 σ΅„© σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝛾𝑛 𝑔3 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + (𝛽𝑛 + 𝛾𝑛 ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© σ΅„©2 σ΅„© + 𝛿𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 𝛽𝑛 𝛾𝑛 𝑔3 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„©2 σ΅„© σ΅„©2 = 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + (1 βˆ’ 𝛼𝑛 ) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝛾𝑛 𝑔3 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 σ΅„© σ΅„© βˆ’ 𝛽𝑛 𝛾𝑛 𝑔3 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) , (284) + 𝑑𝑛 [

as 𝑛 󳨀→ ∞. (288)

lim σ΅„©π‘₯ π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(289)

Therefore, from Lemma 12, (273), and (289), it follows that σ΅„©σ΅„© σ΅„© σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝑆𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝑆𝑛 π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆𝑛 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© + 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© + 󡄩󡄩󡄩𝑆𝑛 π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© 󳨀→ 0 as 𝑛 󳨀→ ∞. (290) That is, σ΅„©σ΅„©

lim σ΅„©π‘₯ π‘›β†’βˆž σ΅„© 𝑛

σ΅„© βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0.

(291)

Suppose that 𝛽𝑛 ≑ 𝛽 for some fixed 𝛽 ∈ (0, 1) such that 𝛼𝑛 + 𝛽 + 𝛾𝑛 + 𝛿𝑛 = 1 for all 𝑛 β‰₯ 0. Define a mapping 𝑉π‘₯ = (1 βˆ’ πœƒ1 βˆ’ πœƒ2 )𝑆π‘₯ + πœƒ1 𝐡π‘₯ + πœƒ2 𝐺π‘₯, where πœƒ1 , πœƒ2 ∈ (0, 1) are two constants with πœƒ1 + πœƒ2 < 1. Then, by Lemmas 14 and 17, we have that Fix(𝑉) = Fix(𝑆) ∩ Fix(𝐡) ∩ Fix(𝐺) = 𝐹. For each π‘˜ β‰₯ 1, let {π‘π‘˜ } be a unique element of 𝐢 such that π‘π‘˜ =

1 1 𝑓 (π‘π‘˜ ) + (1 βˆ’ ) π‘‰π‘π‘˜ . π‘˜ π‘˜

(292)

which hence implies that σ΅„© σ΅„© 𝛽𝑛 𝛾𝑛 𝑔3 (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 βˆ’ 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩 (285) σ΅„© σ΅„©2 σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑝󡄩󡄩󡄩 + (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑝󡄩󡄩󡄩 + 󡄩󡄩󡄩𝑦𝑛 βˆ’ 𝑝󡄩󡄩󡄩) σ΅„© σ΅„© Γ— σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© . Utilizing (271), conditions (i), (ii), (iv), and the boundedness of {π‘₯𝑛 }, {𝑦𝑛 } and {𝑓(π‘₯𝑛 )}, we get σ΅„© σ΅„© lim 𝑔 (σ΅„©σ΅„©π‘₯ βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„©) = 0. (286) π‘›β†’βˆž 3 σ΅„© 𝑛

From Lemma 13, we conclude that π‘π‘˜ β†’ π‘ž ∈ Fix(𝑉) = 𝐹 as π‘˜ β†’ ∞. Repeating the same arguments as those of (81) in the proof of Theorem 24, we can conclude that for every 𝑛, π‘˜

From the properties of 𝑔3 , we have σ΅„© σ΅„© lim σ΅„©σ΅„©π‘₯ βˆ’ 𝐡𝑛 π‘₯𝑛 σ΅„©σ΅„©σ΅„© = 0. π‘›β†’βˆž σ΅„© 𝑛

where πœƒπ‘› = 𝛼𝑛 ‖𝑓(π‘₯𝑛 ) βˆ’ π΅π‘π‘˜ β€– + (1 βˆ’ 𝛽)‖𝐡𝑛 π‘π‘˜ βˆ’ π΅π‘π‘˜ β€– + 𝛿𝑛 ‖𝑆𝑛 𝐺π‘₯𝑛 βˆ’π΅π‘› π‘₯𝑛 β€–. Since lim𝑛 β†’ ∞ 𝛼𝑛 = lim𝑛 β†’ ∞ ‖𝐡𝑛 π‘π‘˜ βˆ’π΅π‘π‘˜ β€– = lim𝑛 β†’ ∞ ‖𝑆𝑛 𝐺π‘₯𝑛 βˆ’ 𝐡𝑛 π‘₯𝑛 β€– = 0, we know that πœƒπ‘› β†’ 0 as

(287)

σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑦𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© ≀ πœƒπ‘› + 𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© ,

(293)

40

Abstract and Applied Analysis

𝑛 β†’ ∞. So, it immediately follows that

Utilizing (295) and (297), we deduce that

σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„© σ΅„© = 󡄩󡄩󡄩𝑦𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— [2 󡄩󡄩󡄩𝑦𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©] σ΅„© σ΅„© σ΅„© σ΅„©2 ≀ (𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„© σ΅„© σ΅„© σ΅„© + πœƒπ‘› [2 (𝛽 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„©) + πœƒπ‘› ] σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„© [2 󡄩󡄩󡄩𝑦𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ 𝑦𝑛 σ΅„©σ΅„©σ΅„©] 2σ΅„© σ΅„© σ΅„©2 σ΅„©2 = 𝛽2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„©σ΅„© σ΅„© σ΅„© + 2𝛽 (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœπ‘› 2σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛽2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© + 𝛽 (1 βˆ’ 𝛽) (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© ) + πœπ‘› σ΅„©2 σ΅„©2 σ΅„© σ΅„© = 𝛽󡄩󡄩󡄩π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© + (1 βˆ’ 𝛽) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœπ‘› ,

(294)

σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘‰π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© = πœ‡π‘› σ΅„©σ΅„©σ΅„©(1 βˆ’ πœƒ1 βˆ’ πœƒ2 ) (π‘₯𝑛 βˆ’ π‘†π‘π‘˜ ) σ΅„©2 + πœƒ1 (π‘₯𝑛 βˆ’ π΅π‘π‘˜ ) + πœƒ2 (π‘₯𝑛 βˆ’ πΊπ‘π‘˜ )σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ (1 βˆ’ πœƒ1 βˆ’ πœƒ2 ) πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„© + πœƒ1 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 + πœƒ2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© .

Repeating the same arguments as those of (99) in the proof of Theorem 24, we can obtain that 1 σ΅„©σ΅„© σ΅„©2 πœ‡ σ΅„©π‘₯ βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© β‰₯ πœ‡π‘› βŸ¨π‘“ (π‘π‘˜ ) βˆ’ π‘π‘˜ , 𝐽 (π‘₯𝑛 βˆ’ π‘π‘˜ )⟩ . 2π‘˜ 𝑛 σ΅„© 𝑛

(295)

In addition, note that σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„©2 σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 + 𝐺π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„©2 σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— (2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝐺π‘₯𝑛 σ΅„©σ΅„©σ΅„©) , σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„©2 σ΅„© ≀ σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 + 𝑆π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„©2 σ΅„© ≀ (σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„©) σ΅„©2 σ΅„© σ΅„© σ΅„© = σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„© σ΅„© σ΅„© σ΅„© σ΅„© Γ— (2 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© + σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ 𝑆π‘₯𝑛 σ΅„©σ΅„©σ΅„©) .

σ΅„© σ΅„©2 σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘†π‘π‘˜ σ΅„©σ΅„©σ΅„© ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© .

πœ‡π‘› βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛 βˆ’ π‘ž)⟩ ≀ 0.

(300)

On the other hand, from (257) and the norm-to-weakβˆ— uniform continuity of 𝐽 on bounded subsets of 𝑋, it follows that 󡄨 󡄨 lim σ΅„¨σ΅„¨βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛+1 βˆ’ π‘ž)⟩ βˆ’ βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛 βˆ’ π‘ž)βŸ©σ΅„¨σ΅„¨σ΅„¨ = 0. (301)

π‘›β†’βˆž 󡄨

So, utilizing Lemma 18, we deduce from (300) and (301) that lim sup βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (π‘₯𝑛 βˆ’ π‘ž)⟩ ≀ 0, π‘›β†’βˆž

(296)

(302)

which together with (271) and the norm-to-weakβˆ— uniform continuity of 𝐽 on bounded subsets of 𝑋, implies that lim sup βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (𝑦𝑛 βˆ’ π‘ž)⟩ ≀ 0. π‘›β†’βˆž

(303)

Finally, let us show that π‘₯𝑛 β†’ π‘ž as 𝑛 β†’ ∞. Utilizing Lemma 8 (i), from (253) and the convexity of β€– β‹… β€–, we get σ΅„©σ΅„© σ΅„©2 󡄩󡄩𝑦𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© ≀ 󡄩󡄩󡄩𝛼𝑛 (𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘ž)) + 𝛽𝑛 (π‘₯𝑛 βˆ’ π‘ž)

It is easy to see from (273) and (291) that σ΅„© σ΅„©2 σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ πΊπ‘π‘˜ σ΅„©σ΅„©σ΅„© ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© ,

(299)

Since π‘π‘˜ β†’ π‘ž ∈ Fix(𝑉) = 𝐹 as π‘˜ β†’ ∞, by the uniform Gateaux differentiability of the norm of 𝑋 we have

where πœπ‘› = πœƒπ‘› [2(𝛽‖π‘₯𝑛 βˆ’π΅π‘π‘˜ β€–+(1βˆ’π›½)β€–π‘₯𝑛 βˆ’π‘π‘˜ β€–)+πœƒπ‘› ]+β€–π‘₯𝑛+1 βˆ’ 𝑦𝑛 β€–[2‖𝑦𝑛 βˆ’ π΅π‘π‘˜ β€– + β€–π‘₯𝑛+1 βˆ’ 𝑦𝑛 β€–] β†’ 0 as 𝑛 β†’ ∞. For any Banach limit πœ‡, from (294), we derive σ΅„© σ΅„©2 σ΅„© σ΅„©2 σ΅„© σ΅„©2 πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© = πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π΅π‘π‘˜ σ΅„©σ΅„©σ΅„© ≀ πœ‡π‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘π‘˜ σ΅„©σ΅„©σ΅„© .

(298)

(297)

σ΅„©2 + 𝛾𝑛 (𝐡𝑛 π‘₯𝑛 βˆ’ π‘ž) + 𝛿𝑛 (𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘ž)σ΅„©σ΅„©σ΅„© + 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (𝑦𝑛 βˆ’ π‘ž)⟩

Abstract and Applied Analysis

41

σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 󡄩󡄩󡄩𝑓 (π‘₯𝑛 ) βˆ’ 𝑓 (π‘ž)σ΅„©σ΅„©σ΅„© + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

arbitrarily given π‘₯0 ∈ 𝐢, let {π‘₯𝑛 } be the sequence generated by

σ΅„© σ΅„© σ΅„©2 σ΅„©2 + 𝛾𝑛 󡄩󡄩󡄩𝐡𝑛 π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛿𝑛 󡄩󡄩󡄩𝑆𝑛 𝐺π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

π‘₯𝑛+1

+ 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (𝑦𝑛 βˆ’ π‘ž)⟩

= πœŽπ‘› ((1 βˆ’ 𝑙) 𝐼 + 𝑙𝑉) π‘₯𝑛 + (1 βˆ’ πœŽπ‘› )

σ΅„© σ΅„© σ΅„©2 σ΅„©2 ≀ 𝛼𝑛 πœŒσ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛽𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

Γ— [𝛼𝑛 𝑓 (π‘₯𝑛 ) + 𝛽𝑛 π‘₯𝑛 + 𝛾𝑛 𝐡𝑛 π‘₯𝑛 + 𝛿𝑛 𝑆𝑛 ((1βˆ’π‘™) 𝐼+𝑙𝑉) π‘₯𝑛 ] ,

σ΅„© σ΅„© σ΅„©2 σ΅„©2 + 𝛾𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 𝛿𝑛 σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

βˆ€π‘› β‰₯ 0, (306)

+ 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (𝑦𝑛 βˆ’ π‘ž)⟩ σ΅„© σ΅„©2 = (1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (𝑦𝑛 βˆ’ π‘ž)⟩ , (304)

where 1 βˆ’ (𝜁/(1 + 𝜁))(1 βˆ’ √(1 βˆ’ πœƒ)/𝜁) ≀ 𝑙 ≀ 1 and {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, {𝛿𝑛 }, and {πœŽπ‘› } are the sequences in (0, 1) such that 𝛼𝑛 +𝛽𝑛 + 𝛾𝑛 + 𝛿𝑛 = 1 for all 𝑛 β‰₯ 0. Suppose that the following conditions hold: (i) lim𝑛 β†’ ∞ 𝛼𝑛 = 0 and βˆ‘βˆž 𝑛=0 𝛼𝑛 = ∞;

and hence

(ii) {𝛾𝑛 }, {𝛿𝑛 } βŠ‚ [𝑐, 𝑑] for some 𝑐, 𝑑 ∈ (0, 1);

(iii) βˆ‘βˆž 𝑛=1 (|πœŽπ‘› βˆ’ πœŽπ‘›βˆ’1 | + |𝛼𝑛 βˆ’ π›Όπ‘›βˆ’1 | + |𝛽𝑛 βˆ’ π›½π‘›βˆ’1 | + |𝛾𝑛 βˆ’ π›Ύπ‘›βˆ’1 | + |𝛿𝑛 βˆ’ π›Ώπ‘›βˆ’1 |) < ∞;

σ΅„©σ΅„© σ΅„©2 σ΅„©σ΅„©π‘₯𝑛+1 βˆ’ π‘žσ΅„©σ΅„©σ΅„© σ΅„© σ΅„©2 σ΅„© σ΅„©2 ≀ πœŽπ‘› 󡄩󡄩󡄩𝐺π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + (1 βˆ’ πœŽπ‘› ) 󡄩󡄩󡄩𝑦𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„©

(iv) 0 < lim inf 𝑛 β†’ ∞ 𝛽𝑛 ≀ lim sup𝑛 β†’ ∞ 𝛽𝑛 < 1 and 0 < lim inf 𝑛 β†’ ∞ πœŽπ‘› ≀ lim sup𝑛 β†’ ∞ πœŽπ‘› < 1.

σ΅„© σ΅„©2 ≀ πœŽπ‘› σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + (1 βˆ’ πœŽπ‘› ) σ΅„© σ΅„©2 Γ— [(1 βˆ’ 𝛼𝑛 (1 βˆ’ 𝜌)) σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (𝑦𝑛 βˆ’ π‘ž)⟩ ]

(305)

σ΅„© σ΅„©2 = [1 βˆ’ (1 βˆ’ πœŽπ‘› ) 𝛼𝑛 (1 βˆ’ 𝜌)] σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© + 2 (1 βˆ’ πœŽπ‘› ) 𝛼𝑛 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (𝑦𝑛 βˆ’ π‘ž)⟩ σ΅„© σ΅„©2 = [1 βˆ’ (1 βˆ’ πœŽπ‘› ) 𝛼𝑛 (1 βˆ’ 𝜌)] σ΅„©σ΅„©σ΅„©π‘₯𝑛 βˆ’ π‘žσ΅„©σ΅„©σ΅„© 2 βŸ¨π‘“ (π‘ž) βˆ’ π‘ž, 𝐽 (𝑦𝑛 βˆ’ π‘ž)⟩ + (1 βˆ’ πœŽπ‘› ) 𝛼𝑛 (1 βˆ’ 𝜌) . 1βˆ’πœŒ From conditions (i) and (iv), it is easy to see that βˆ‘βˆž 𝑛=0 (1 βˆ’ πœŽπ‘› )𝛼𝑛 (1 βˆ’ 𝜌) = ∞. Applying Lemma 7 to (305), we infer that π‘₯𝑛 β†’ π‘ž as 𝑛 β†’ ∞. This completes the proof. Corollary 34. Let 𝐢 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢. Let {πœŒπ‘› }∞ 𝑛=0 be a sequence of positive numbers in (0, 𝑏] for some 𝑏 ∈ (0, 1) and 𝐴 𝑖 : 𝐢 β†’ π‘‹πœ‰π‘– strictly pseudocontractive and 𝛼̂𝑖 -strongly accretive with πœ‰π‘– + 𝛼̂𝑖 β‰₯ 1 for each 𝑖 = 0, 1, . . .. Define a mapping 𝐺𝑖 : 𝐢 β†’ 𝐢 by Π𝐢(𝐼 βˆ’ πœ† 𝑖 𝐴 𝑖 )π‘₯ = 𝐺𝑖 π‘₯ for all π‘₯ ∈ 𝐢 and 𝑖 = 0, 1, . . ., where 1βˆ’(πœ‰π‘– /(1+πœ‰π‘– ))(1βˆ’βˆš(1 βˆ’ 𝛼̂𝑖 )/πœ‰π‘– ) ≀ πœ† 𝑖 ≀ 1 for all 𝑖 = 0, 1, . . .. Let 𝐡𝑛 : 𝐢 β†’ 𝐢 be the π‘Š-mapping generated by 𝐺𝑛 , πΊπ‘›βˆ’1 , . . . , 𝐺0 and πœŒπ‘› , πœŒπ‘›βˆ’1 , . . . , 𝜌0 . Let 𝑉 : 𝐢 β†’ 𝐢 be a self-mapping such that 𝐼 βˆ’ 𝑉 : 𝐢 β†’ 𝑋 is 𝜁-strictly pseudocontractive and πœƒstrongly accretive with πœƒ + 𝜁 β‰₯ 1. Let 𝑓 : 𝐢 β†’ 𝐢 be a contraction with coefficient 𝜌 ∈ (0, 1). Let {𝑆𝑖 }∞ 𝑖=0 be a countable family of nonexpansive mappings of 𝐢 into itself such that ∞ 𝐹 = (β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 )) ∩ Fix(𝑉) ∩ (⋂𝑖=0 VI(𝐢, 𝐴 𝑖 )) =ΜΈ 0. For

Assume that βˆ‘βˆž 𝑛=1 supπ‘₯∈𝐷 ‖𝑆𝑛 π‘₯ βˆ’ π‘†π‘›βˆ’1 π‘₯β€– < ∞ for any bounded subset 𝐷 of 𝐢 and let 𝑆 be a mapping of 𝐢 into itself defined by 𝑆π‘₯ = lim𝑛 β†’ ∞ 𝑆𝑛 π‘₯ for all π‘₯ ∈ 𝐢 and suppose that Fix(𝑆) = β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 ). Then, there hold the following: (I) lim𝑛 β†’ ∞ β€–π‘₯𝑛+1 βˆ’ π‘₯𝑛 β€– = 0;

(II) the sequence {π‘₯𝑛 }∞ 𝑛=0 converges strongly to some π‘ž ∈ 𝐹 which is the unique solution of the variational inequality problem (VIP) ⟨(𝐼 βˆ’ 𝑓) π‘ž, 𝐽 (π‘ž βˆ’ 𝑝)⟩ ≀ 0,

βˆ€π‘ ∈ 𝐹,

(307)

provided 𝛽𝑛 ≑ 𝛽 for some fixed 𝛽 ∈ (0, 1). Proof. In Theorem 33, we put 𝐡1 = 𝐼 βˆ’ 𝑉, 𝐡2 = 0 and πœ‡1 = 𝑙 where 1 βˆ’ (𝜁/(1 + 𝜁))(1 βˆ’ √(1 βˆ’ πœƒ)/𝜁) ≀ 𝑙 ≀ 1. Then, GSVI (13) is equivalent to the VIP of finding π‘₯βˆ— ∈ 𝐢 such that ⟨𝐡1 π‘₯βˆ— , 𝐽 (π‘₯ βˆ’ π‘₯βˆ— )⟩ β‰₯ 0,

βˆ€π‘₯ ∈ 𝐢.

(308)

In this case, 𝐡1 : 𝐢 β†’ 𝑋 is 𝜁-strictly pseudocontractive and πœƒ-strongly accretive. Repeating the same arguments as those in the proof of Corollary 25, we can infer that Fix(𝑉) = VI(𝐢, 𝐡1 ). Accordingly, 𝐹 = (β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 )) ∩ Ξ© ∩ ∞ VI(𝐢, 𝐴 )) = (β‹‚ Fix(𝑆 )) ∩ Fix(𝑉) ∩ (β‹‚βˆž (β‹‚βˆž 𝑖 𝑖 𝑖=0 𝑖=0 𝑖=0 VI(𝐢, 𝐴 𝑖 )), 𝐺π‘₯𝑛 = ((1 βˆ’ 𝑙) 𝐼 + 𝑙𝑉) π‘₯𝑛 ,

(309)

So, the scheme (251) reduces to (306). Therefore, the desired result follows from Theorem 33. Remark 35. Our Theorems 31 and 33 improve, extend, supplement and develop Ceng and Yao’s [10, Theorem 3.2], Cai and Bu’s [11, Theorem 3.1], Kangtunyakarn’s [38, Theorem 3.1], and Ceng and Yao’s [8, Theorem 3.1], in the following aspects.

42

Abstract and Applied Analysis (i) The problem of finding a point π‘ž ∈ (β‹‚βˆž 𝑖=0 Fix(𝑆𝑖 )) ∩ Ξ© ∩ (β‹‚βˆž 𝑖=0 VI(𝐢, 𝐴 𝑖 )) in our Theorems 31 and 33 is more general and more subtle than every one of the problem of finding a point π‘ž ∈ β‹‚βˆž 𝑖=0 Fix(𝑇𝑖 ) in [10, Theorem 3.2], the problem of finding a point π‘ž ∈ β‹‚βˆž 𝑖=1 Fix(𝑇𝑖 ) ∩ Ξ© in [11, Theorem 3.1], the problem of finding a point π‘ž ∈ Fix(𝑆) ∩ Fix(𝑉) ∩ (⋂𝑁 𝑖=1 VI(𝐢, 𝐴 𝑖 )) in [38, Theorem 3.1], and the problem of finding a point π‘ž ∈ Fix(𝑇) in [8, Theorem 3.1]. (ii) The iterative scheme in [38, Theorem 3.1] is extended to develop the iterative scheme (178) of our Theorem 31, and the iterative scheme in [11, Theorem 3.1] is extended to develop the iterative scheme (251) of our Theorem 33. Iterative schemes (178) and (181) in our Theorems 31 and 33 are more advantageous and more flexible than the iterative scheme of [11, Theorem 3.1] because they both are one-step iteration schemes and involve several parameter sequences {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, {𝛿𝑛 }, (and {πœŽπ‘› }).

argument techniques in [14], the inequality in 2uniformly smooth Banach spaces (see Lemma 4), and the inequality in smooth and uniform convex Banach spaces (see Proposition 6). It depends on only the inequalities in uniform convex Banach spaces (see Lemmas 11 and 15 in Section 2 of this paper) and the properties of the π‘Š-mapping and the Banach limit (see Lemmas 16–18 in Section 2 of this paper). (vi) The assumption of the uniformly convex and 2uniformly smooth Banach space 𝑋 in [11, Theorem 3.1] is weakened to the one of the uniformly convex Banach space 𝑋 having a uniformly Gateaux differentiable norm in our Theorem 33. Moreover, the assumption of the uniformly smooth Banach space 𝑋 in [8, Theorem 3.1] is replaced with the one of the uniformly convex Banach space 𝑋 having a uniformly Gateaux differentiable norm in our Theorem 33.

Conflict of Interests

(iii) Our Theorems 31 and 33 extend and generalize Ceng and Yao’s [8, Theorem 3.1] from a nonexpansive mapping to a countable family of nonexpansive mappings, and Ceng and Yao’s [10, Theorems 3.2] to the setting of the GSVI (13) and infinitely many VIPs, Kangtunyakarn’s [38, Theorem 3.1] from finitely many VIPs to infinitely many VIPs, from a nonexpansive mapping to a countable family of nonexpansive mappings and from a strict pseudocontraction to the GSVI (13). In the meantime, our Theorems 31 and 33 extend and generalize Cai and Bu’s [11, Theorem 3.1] to the setting of infinitely many VIPs.

The authors declare that there is no conflict of interests regarding the publication of this paper.

(iv) The iterative schemes (178) and (251) in our Theorems 31 and 33 are very different from every one in [10, Theorem 3.2], [11, Theorem 3.1], [38, Theorem 3.1], and [8, Theorem 3.1] because the mappings 𝐺 and 𝑇𝑛 in [11, Theorem 3.1] and the mapping 𝑇 in [8, Theorem 3.1] are replaced with the same composite mapping 𝑆𝑛 𝐺 in the iterative schemes (42) and (130) and the mapping π‘Šπ‘› in [10, Theorem 3.2] is replaced by 𝐡𝑛 .

References

(v) Cai and Bu’s proof in [11, Theorem 3.1] depends on the argument techniques in [14], the inequality in 2uniformly smooth Banach spaces (see Lemma 4), and the inequality in smooth and uniform convex Banach spaces (see Proposition 6). Because the composite mapping 𝑆𝑛 𝐺 appears in the iterative scheme (178) of our Theorem 31, the proof of our Theorem 31 depends on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), the inequality in smooth and uniform convex Banach spaces (see Proposition 6), the inequalities in uniform convex Banach spaces (see Lemmas 11 and 15 in Section 2 of this paper), and the properties of the π‘Š-mapping and the Banach limit (see Lemmas 16, 17, and 18 in Section 2 of this paper). However, the proof of our Theorem 33 does not depend on the

Acknowledgments This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under Grant no. HiCi/15130-1433. The authors, therefore, acknowledge technical and financial support of KAU. The authors would like to thank Professor J. C. Yao for motivation and many fruitful discussions regarding this work.

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