Viscosity Projection Algorithms for Pseudocontractive Mappings in ...

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 791209, 8 pages http://dx.doi.org/10.1155/2014/791209

Research Article Viscosity Projection Algorithms for Pseudocontractive Mappings in Hilbert Spaces Xiujuan Pan,1,2 Shin Min Kang,3 and Young Chel Kwun4 1

School of Science, Tianjin Polytechnic University, Tianjin 300387, China College of Management and Economics, Tianjin University, Tianjin 300072, China 3 Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea 4 Department of Mathematics, Dong-A University, Busan 614-714, Republic of Korea 2

Correspondence should be addressed to Young Chel Kwun; [email protected] Received 6 December 2013; Accepted 28 December 2013; Published 9 February 2014 Academic Editor: Chong Li Copyright © 2014 Xiujuan Pan 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. An explicit projection algorithm with viscosity technique is constructed for finding the fixed points of the pseudocontractive mapping in Hilbert spaces. Strong convergence theorem is demonstrated. Consequently, as an application, we can approximate to the minimum-norm fixed point of the pseudocontractive mapping.

1. Introduction Let H be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖, respectively. Let C be a nonempty closed convex subset of H. Recall that a mapping T : C → C is said to be (i) 𝐿-Lipschitz ⇔ there exists a constant 𝐿 > 0 such that ‖T𝑢 − TV‖ ≤ 𝐿‖𝑢 − V‖ for all 𝑢, V ∈ C; if 𝐿 ∈ (0, 1), then T is said to be contractive; if 𝐿 = 1, then T is said to be nonexpansive; (ii) pseudocontractive 2

⇔ ⟨T𝑢 − TV, 𝑢 − V⟩ ≤ ‖𝑢 − V‖ ; ⇔ ‖T𝑢 − TV‖2 ≤ ‖𝑢 − V‖2 +‖(𝐼 − T)𝑢 − (𝐼 − T)V)‖2 ; ⇔ ⟨𝑢 − V, (𝐼 − T)𝑢 − (𝐼 − T)V⟩ ≥ 0, for all 𝑢, V ∈ C. Interest in pseudocontractive mappings stems mainly from their firm connection with the class of nonlinear accretive operators. It is a classical result, see Deimling [1], that if T is an accretive operator, then the solutions of the equations T𝑥 = 0 correspond to the equilibrium points of some evolution systems. This explains the importance, from this point of view, of the improvement brought by the

Ishikawa iteration which was introduced by Ishikawa [2] in 1974. The original result of Ishikawa is stated in the following. Theorem 1. Let C be a convex compact subset of a Hilbert space H and let T : C → C be an 𝐿-Lipschitzian pseudocontractive mapping with Fix(T) ≠ 0. For any 𝑥0 ∈ C, define the sequence {𝑥𝑛 } iteratively by 𝑦𝑛 = (1 − 𝜂𝑛 ) 𝑥𝑛 + 𝜂𝑛 T𝑥𝑛 , 𝑥𝑛+1 = (1 − 𝜉𝑛 ) 𝑥𝑛 + 𝜉𝑛 T𝑦𝑛 ,

(1)

for all 𝑛 ∈ N, where {𝜉𝑛 } ⊂ [0, 1] and {𝜂𝑛 } ⊂ [0, 1] satisfy the conditions: lim𝑛 → ∞ 𝜂𝑛 = 0 and ∑∞ 𝑛=1 𝜉𝑛 𝜂𝑛 = ∞. Then the sequence {𝑥𝑛 } generated by (1) converges strongly to a fixed point of T. The iteration (1) is now referred to as the Ishikawa iterative sequence. However, we note that 𝐶 is compact subset. Now, we know that strong convergence has not been achieved without compactness assumption on the involved operation or the underlying spaces. A counter example can be found in Chidume and Mutangadura [3]. In order to obtain a strong convergence result for pseudocontractive mappings without the compactness assumption, in [4], Zhou coupled the Ishikawa algorithm with the

2

Abstract and Applied Analysis

hybrid technique and presented the following algorithm for Lipschitz pseudocontractive mappings: 𝑦𝑛 = (1 − 𝜉𝑛 ) 𝑥𝑛 + 𝜉𝑛 T𝑥𝑛 , 𝑧𝑛 = (1 − 𝜂𝑛 ) 𝑥𝑛 + 𝜂𝑛 T𝑦𝑛 , 󵄩 󵄩2 󵄩 󵄩2 C𝑛 = {𝑧 ∈ C : 󵄩󵄩󵄩𝑧𝑛 − 𝑧󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑧󵄩󵄩󵄩 󵄩 󵄩2 − 𝜉𝑛 𝜂𝑛 (1 − 2𝜉𝑛 − 𝜉𝑛2 𝐿2 ) 󵄩󵄩󵄩𝑥𝑛 − T𝑥𝑛 󵄩󵄩󵄩 } ,

(2)

2. Preliminaries

Q𝑛 = {𝑧 ∈ C : ⟨𝑥𝑛 − 𝑧, 𝑥0 − 𝑥𝑛 ⟩ ≥ 0} , 𝑥𝑛+1 = projC𝑛 ∩Q𝑛 𝑥0 ,

𝑛 ∈ N.

Zhou proved that the sequence {𝑥𝑛 } generated by (2) converges strongly to the fixed point of T. Further, in [5], Yao et al. introduced the hybrid Mann algorithm as follows. Let C be a nonempty closed convex subset of a real Hilbert space H. Let {𝜉𝑛 } be a sequence in (0, 1). Let 𝑥0 ∈ H. For C1 = C and 𝑥1 = projC1 𝑥0 , define a sequence {𝑥𝑛 } of C as follows: 𝑦𝑛 = (1 − 𝜉𝑛 ) 𝑥𝑛 + 𝜉𝑛 T𝑥𝑛 , 󵄩2 󵄩 C𝑛+1 = {𝑧 ∈ C𝑛 : 󵄩󵄩󵄩𝜉𝑛 (𝐼 − T) 𝑦𝑛 󵄩󵄩󵄩 ≤ 2𝜉𝑛 ⟨𝑥𝑛 − 𝑧, (𝐼 − T) 𝑦𝑛 ⟩} , 𝑥𝑛+1 = projC𝑛+1 𝑥0 ,

(3)

Note that, in iterations (2) and (3), we need to compute the half-spaces C𝑛 (and/or Q𝑛 ). Very recently, Zegeye et al. [6] further studied the convergence analysis of the Ishikawa iteration (1). They proved ingeniously the strong convergence of the Ishikawa iteration (1). However, we have to assume that the interior of Fix(T) is nonempty. This appears very restrictive since even in R with the usual norm, Lipschitz pseudocontractive maps with finite number of fixed points do not enjoy this condition that intFix(T) ≠ 0. For some related works, please refer to [7–19]. On the other hand, we notice that it is quite often to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. For instance, given a closed convex subset C of a Hilbert space H1 and a bounded linear operator B : H1 → H2 , where H2 is another Hilbert space. The C-constrained pseudoinverse of B, B†C , is then defined as the minimum-norm solution of the constrained minimization problem 𝑥∈C

(4)

which is equivalent to the fixed point problem 𝑢 = projC (𝑢 − 𝜇B∗ (B𝑢 − 𝑏)) ,

Recall that the metric projection projC : H → C satisfies ‖𝑢− projC 𝑢‖ = inf{‖𝑢 − V‖ : V ∈ C}. The metric projection projC is a typical firmly nonexpansive mapping. The characteristic inequality of the projection is ⟨𝑢 − projC 𝑢, V − projC 𝑢⟩ ≤ 0 for all 𝑢 ∈ H, V ∈ C. In the sequel we will use the following expressions: (i) Fix(T) denotes the set of fixed points of T; (ii) 𝑥𝑛 ⇀ 𝑥† denotes the weak convergence of 𝑥𝑛 to 𝑥† ; (iii) 𝑥𝑛 → 𝑥† denotes the strong convergence of 𝑥𝑛 to 𝑥† . The following lemmas will be useful for the next section.

𝑛 ∈ N.

B†C (𝑏) := arg min ‖B𝑥 − 𝑏‖

point problem. The purpose of this paper is to solve such a problem for pseudocontractions. More precisely, we will introduce an explicit projection algorithm with viscosity technique for finding the fixed points of a Lipschitzian pseudocontractive mapping. Strong convergence theorem is demonstrated. Consequently, as an application, we can find the minimum-norm fixed point of the pseudocontractive mapping.

(5)

where B∗ is the adjoint of B, 𝜇 > 0 is a constant, and 𝑏 ∈ H2 is such that projB(C) (𝑏) ∈ B(C). It is, therefore, an interesting problem to invent iterative algorithms that can generate sequences which converge strongly to the minimum-norm solution of a given fixed

Lemma 2 (see [20]). Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. Then, C ⊃ 𝑢𝑛 ⇀ 𝑢‡ } 󳨐⇒ (I − T) 𝑢‡ = ]. { (I − T) 𝑢𝑛 󳨀→ ]

(6)

Lemma 3 (see [21]). Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that a mapping A : C → H is monotone and weakly continuous along segments (i.e., A(𝑥 + 𝑡𝑦) → A(𝑥) weakly, as 𝑡 → 0, whenever 𝑥 + 𝑡𝑦 ∈ C for 𝑥, 𝑦 ∈ C). Then the variational inequality 𝑥‡ ∈ 𝐶,

⟨A𝑥‡ , 𝑥 − 𝑥‡ ⟩ ≥ 0,

∀𝑥 ∈ C,

(7)

is equivalent to the dual variational inequality 𝑥‡ ∈ C,

⟨A𝑥, 𝑥 − 𝑥‡ ⟩ ≥ 0,

∀𝑥 ∈ C.

(8)

Lemma 4 (see [22]). Assume that the sequence {𝑎𝑛 } satisfies 𝑎𝑛 ≥ 0 and 𝑎𝑛+1 ≤ (1 − ]𝑛 )𝑎𝑛 + 𝜍𝑛 ]𝑛 where {]𝑛 } is a sequence in (0, 1) and {𝜍𝑛 } is a sequence such that ∑∞ 𝑛=1 ]𝑛 = ∞ and |𝜍 ] | < ∞). Then lim𝑛 → ∞ 𝑎𝑛 = lim sup𝑛 → ∞ 𝜍𝑛 ≤ 0 (or ∑∞ 𝑛=1 𝑛 𝑛 0.

3. Main Results In order to prove our main result, we need the following proposition. Proposition 5. Let C be a nonempty closed convex subset of a real Hilbert space H. Let W : C → C be a nonexpansive mapping with Fix(W) ≠ 0. Let 󰜚 : C → H be a 𝜅-contraction. For each 𝑡 ∈ (0, 1), let the net {𝑢𝑡 } be defined by 𝑢𝑡 = W projC [𝑡󰜚 (𝑢𝑡 ) + (1 − 𝑡) 𝑢𝑡 ] .

(9)


3

Then, as 𝑡 → 0+ , the net {𝑢𝑡 } converges strongly to a point 𝑥‡ ∈ Fix(W) which solves the following variational inequality: 𝑥‡ ∈ Fix (W) ,

⟨(𝐼 − 󰜚) 𝑥‡ , 𝑧 − 𝑥‡ ⟩ ≥ 0,

𝑧 ∈ Fix (W) . (10)

Proof. For 𝑡 ∈ (0, 1), define a mapping W𝑡 : C → C by W𝑡 𝑢 = W projC [𝑡󰜚 (𝑢) + (1 − 𝑡) 𝑢] ,

𝑢 ∈ C.

(11)

For any 𝑢, V ∈ C, we have 󵄩󵄩 󵄩 󵄩 󵄩󵄩W𝑡 𝑢 − W𝑡 V󵄩󵄩󵄩 = 󵄩󵄩󵄩W projC [𝑡󰜚 (𝑢) + (1 − 𝑡) 𝑢]

󵄩 −W projC [𝑡󰜚 (V) + (1 − 𝑡) V]󵄩󵄩󵄩 󵄩 󵄩 ≤ 𝑡 󵄩󵄩󵄩󰜚 (𝑢) − 󰜚 (V)󵄩󵄩󵄩 + (1 − 𝑡) ‖𝑢 − V‖

󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩󰜚 (𝑢) − 𝑢󵄩󵄩󵄩 (14) . 󵄩󵄩𝑢𝑡 − 𝑢󵄩󵄩󵄩 ≤ 󵄩 1−𝜅 Thus, {𝑢𝑡 } and {󰜚(𝑢𝑡 )} are bounded. Again from (9), we get 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑢𝑡 − W𝑢𝑡 󵄩󵄩󵄩 = 󵄩󵄩󵄩W projC [𝑡󰜚 (𝑢𝑡 ) + (1 − 𝑡) 𝑢𝑡 ] − W projC 𝑢𝑡 󵄩󵄩󵄩 󵄩 󵄩 ≤ 𝑡 󵄩󵄩󵄩󰜚 (𝑢𝑡 ) − 𝑢𝑡 󵄩󵄩󵄩 󳨀→ 0 as 𝑡 󳨀→ 0+ . (15) +

Let {𝑡𝑛 } ⊂ (0, 1) be a sequence such that 𝑡𝑛 → 0 as 𝑛 → ∞. Put 𝑢𝑛 := 𝑢𝑡𝑛 . From (15), we have 󵄩󵄩󵄩𝑢𝑛 − W𝑢𝑛 󵄩󵄩󵄩 󳨀→ 0. (16) 󵄩 󵄩 From (9), we obtain 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩𝑢𝑡 − 𝑢󵄩󵄩󵄩 = 󵄩󵄩󵄩W projC [𝑡󰜚(𝑢𝑡 ) + (1 − 𝑡)𝑢𝑡 ] − W projC 𝑢󵄩󵄩󵄩 󵄩 󵄩2 ≤ 󵄩󵄩󵄩𝑢𝑡 − 𝑢 + 𝑡 (󰜚 (𝑢𝑡 ) − 𝑢𝑡 )󵄩󵄩󵄩 󵄩 󵄩2 + 𝑡2 󵄩󵄩󵄩󰜚(𝑢𝑡 ) − 𝑢𝑡 󵄩󵄩󵄩

󵄩 󵄩2 + 2𝑡 ⟨󰜚 (𝑢) − 𝑢, 𝑢𝑡 − 𝑢⟩ + 𝑡2 󵄩󵄩󵄩󰜚(𝑢𝑡 ) − 𝑢𝑡 󵄩󵄩󵄩 .

1 󵄩2 󵄩 sup {󵄩󵄩󵄩󰜚(𝑢𝑡 ) − 𝑢𝑡 󵄩󵄩󵄩 : 𝑡 ∈ (0, 1)} . 2 (1 − 𝜅)

(19)

𝑢 ∈ Fix (W) . (20)

Noting that {𝑢𝑛 } is bounded, without loss of generality, we assume that 𝑢𝑛 ⇀ 𝑥‡ . It is obvious that 𝑥‡ ∈ C. From (16) and Lemma 2, we deduce 𝑥‡ ∈ Fix(W). Substitute 𝑥‡ for 𝑢 in (20) to get 1 󵄩󵄩 󵄩2 󵄩󵄩𝑢𝑛 − 𝑥‡ 󵄩󵄩󵄩 ≤ ⟨󰜚 (𝑥‡ ) − 𝑥‡ , 𝑥𝑛 − 𝑥‡ ⟩ + 𝑡𝑛 𝑀. 󵄩 󵄩 1−𝜅

(21)

Since 𝑢𝑛 ⇀ 𝑥‡ , we deduce from (21) that 𝑢𝑛 → 𝑥‡ . The net {𝑢𝑡 } is, therefore, relatively compact, as 𝑡 → 0+ , in the norm topology. In (20), letting 𝑛 → ∞, we get 1 󵄩󵄩 ‡ 󵄩2 󵄩󵄩𝑥 − 𝑢󵄩󵄩󵄩 ≤ ⟨󰜚 (𝑢) − 𝑢, 𝑥‡ − 𝑢⟩ , 󵄩 󵄩 1−𝜅

𝑢 ∈ Fix (W) . (22)

Therefore, 𝑥‡ solves the variational inequality 𝑥‡ ∈ Fix (W) ,

⟨(𝐼 − 󰜚) 𝑢, 𝑢 − 𝑥‡ ⟩ ≥ 0,

𝑢 ∈ Fix (W) . (23)

By Lemma 3, (23) equals its dual variational inequality 𝑥‡ ∈ Fix (W) ,

⟨(I − 󰜚) 𝑥‡ , 𝑢 − 𝑥‡ ⟩ ≥ 0,

𝑢 ∈ Fix (W) . (24)

This indicates that 𝑥‡ = (projFix(W) 󰜚)𝑥‡ . That is, 𝑥‡ is the unique fixed point in Fix(W) of the contraction projFix(W) 󰜚. So, the entire net {𝑢𝑡 } converges in norm to 𝑥‡ as 𝑡 → 0+ . This completes the proof. Corollary 6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let W : C → C be a nonexpansive mapping with Fix(W) ≠ 0. For each 𝑡 ∈ (0, 1), let the net {𝑢𝑡 } be defined by

󵄩 󵄩2 = 󵄩󵄩󵄩𝑢𝑡 − 𝑢󵄩󵄩󵄩 + 2𝑡 ⟨󰜚 (𝑢𝑡 ) − 𝑢𝑡 , 𝑢𝑡 − 𝑢⟩

󵄩 󵄩2 ≤ [1 − 2 (1 − 𝜅) 𝑡] 󵄩󵄩󵄩𝑢𝑡 − 𝑢󵄩󵄩󵄩

𝑀>

(12)

It follows that

󵄩 󵄩2 + 2𝑡⟨𝑢 − 𝑢𝑡 , 𝑢𝑡 − 𝑢⟩ + 𝑡2 󵄩󵄩󵄩󰜚(𝑢𝑡 ) − 𝑢𝑡 󵄩󵄩󵄩

(18)

where 𝑀 > 0 is a constant such that

1 󵄩2 󵄩󵄩 ⟨󰜚 (𝑢) − 𝑢, 𝑢𝑛 − 𝑢⟩ + 𝑡𝑛 𝑀, 󵄩󵄩𝑢𝑛 − 𝑢󵄩󵄩󵄩 ≤ 1−𝜅

Hence, W𝑡 is a 1 − (1 − 𝜅)𝑡-contraction on C with 𝑢𝑡 ∈ C as its unique fixed point. So, {𝑢𝑡 } is well defined. Let 𝑢 ∈ Fix(W). From (9), we have 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑢𝑡 − 𝑢󵄩󵄩󵄩 = 󵄩󵄩󵄩W projC [𝑡󰜚 (𝑢𝑡 ) + (1 − 𝑡) 𝑢𝑡 ] − W projC 𝑢󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 𝑡 󵄩󵄩󵄩󰜚 (𝑢𝑡 ) − 󰜚 (𝑢)󵄩󵄩󵄩 + 𝑡 󵄩󵄩󵄩󰜚 (𝑢) − 𝑢󵄩󵄩󵄩 + (1 − 𝑡) 󵄩󵄩󵄩𝑢𝑡 − 𝑢󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ [1 − (1 − 𝜅) 𝑡] 󵄩󵄩󵄩𝑢𝑡 − 𝑢󵄩󵄩󵄩 + 𝑡 󵄩󵄩󵄩󰜚 (𝑢) − 𝑢󵄩󵄩󵄩 . (13)

+ 2𝑡 ⟨󰜚 (𝑢) − 𝑢, 𝑢𝑡 − 𝑢⟩

1 󵄩2 󵄩󵄩 ⟨󰜚 (𝑢) − 𝑢, 𝑢𝑡 − 𝑢⟩ + 𝑡𝑀, 󵄩󵄩𝑢𝑡 − 𝑢󵄩󵄩󵄩 ≤ 1−𝜅

In particular, we have

≤ [1 − (1 − 𝜅) 𝑡] ‖𝑢 − V‖ .

󵄩 󵄩2 = 󵄩󵄩󵄩𝑢𝑡 − 𝑢󵄩󵄩󵄩 + 2𝑡 ⟨󰜚 (𝑢𝑡 ) − 󰜚 (𝑢) , 𝑢𝑡 − 𝑢⟩

It follows that

(17)

𝑢𝑡 = W projC [(1 − 𝑡) 𝑢𝑡 ] .

(25)

Then, as 𝑡 → 0+ , the net {𝑢𝑡 } converges strongly to the minimum-norm fixed point of W. Proof. As a matter of fact, in (9), we choose 󰜚 = 0, and then (9) reduces to (25). From Proposition 5, (24) is reduced to 0 ≤ ⟨𝑥‡ , 𝑢 − 𝑥‡ ⟩ ,

𝑢 ∈ Fix (W) .

(26)

4

Abstract and Applied Analysis Since T is pseudocontractive, I − T is monotone. So, we have

Equivalently, 󵄩󵄩 ‡ 󵄩󵄩2 󵄩󵄩𝑥 󵄩󵄩 ≤ ⟨𝑥‡ , 𝑢⟩ , 󵄩 󵄩

𝑢 ∈ Fix (W) .

(27)

This implies that

⟨(I − T) 𝑥𝑚+1 − (I − T) 𝑝, 𝑥𝑚+1 − 𝑝⟩ ≥ 0. From (29), (33), and (34), we obtain

󵄩󵄩 ‡ 󵄩󵄩 󵄩󵄩𝑥 󵄩󵄩 ≤ ‖𝑢‖ , 󵄩 󵄩

𝑢 ∈ Fix (W) .

(28)

Therefore, 𝑥‡ is the minimum-norm fixed point of W. This completes the proof. Algorithm 7. Let C be a nonempty closed subset of a real Hilbert space H. Let T : C → C be a pseudocontraction and let 󰜚 : C → H be a 𝜅-contraction. Let {𝜉𝑛 } ⊂ [0, 1] and {𝜂𝑛 } ⊂ [0, 1] be two real number sequences. For 𝑥0 ∈ C, we define a sequence {𝑥𝑛 } iteratively by 𝑥𝑛+1 = projC [𝜉𝑛 󰜚 (𝑥𝑛 ) + (1 − 𝜉𝑛 − 𝜂𝑛 ) 𝑥𝑛 + 𝜂𝑛 T𝑥𝑛 ] ,

𝑛 ≥ 0. (29)

󵄩2 󵄩󵄩 󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩 = ⟨𝑥𝑚+1 − 𝑝, 𝑥𝑚+1 − 𝑝⟩ = ⟨𝑥𝑚+1 − 𝑦𝑚 , 𝑥𝑚+1 − 𝑝⟩ + ⟨𝑦𝑚 − 𝑝, 𝑥𝑚+1 − 𝑝⟩ ≤ ⟨𝑦𝑚 − 𝑝, 𝑥𝑚+1 − 𝑝⟩ = ⟨𝑥𝑚 − 𝑝, 𝑥𝑚+1 − 𝑝⟩ − 𝜉𝑚 ⟨𝑥𝑚 − 󰜚 (𝑥𝑚 ) , 𝑥𝑚+1 − 𝑝⟩ + 𝜂𝑚 ⟨T𝑥𝑚 − 𝑥𝑚 , 𝑥𝑚+1 − 𝑝⟩ = ⟨𝑥𝑚 − 𝑝, 𝑥𝑚+1 − 𝑝⟩ + 𝜉𝑚 ⟨𝑥𝑚+1 − 𝑥𝑚 , 𝑥𝑚+1 − 𝑝⟩

Theorem 8. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be an 𝐿-Lipschitzian and pseudocontraction with Fix(T) ≠ 0 and 󰜚 : C → H a 𝜅contraction. Suppose the following conditions are satisfied:

+ 𝜉𝑚 ⟨󰜚 (𝑥𝑚 ) − 󰜚 (𝑝) , 𝑥𝑚+1 − 𝑝⟩ + 𝜉𝑚 ⟨𝑓 (𝑝) − 𝑝, 𝑥𝑚+1 − 𝑝⟩

(C1) lim𝑛 → ∞ 𝜉𝑛 = 0 and ∑∞ 𝑛=0 𝜉𝑛 = ∞; (C2) (C3)

− 𝜉𝑚 ⟨𝑥𝑚+1 − 𝑝, 𝑥𝑚+1 − 𝑝⟩

lim𝑛 → ∞ (𝜉𝑛 /𝜂𝑛 ) = lim𝑛 → ∞ (𝜂𝑛2 /𝜉𝑛 ) = 0; lim𝑛 → ∞ ((𝜉𝑛 𝜂𝑛−1 − 𝜉𝑛−1 𝜂𝑛 )/𝜉𝑛2 𝜂𝑛−1 ) = 0.

+ 𝜂𝑚 ⟨T𝑥𝑚 − T𝑥𝑚+1 , 𝑥𝑚+1 − 𝑝⟩ + 𝜂𝑚 ⟨𝑥𝑚+1 − 𝑥𝑚 , 𝑥𝑚+1 − 𝑝⟩

Then the sequence {𝑥𝑛 } generated by the algorithm (29) converges strongly to 𝑥‡ = (projFix(T) 󰜚)𝑥‡ . Proof. First, we prove that the sequence {𝑥𝑛 } is bounded. We will show this fact by induction. According to conditions (C1) and (C2), there exists a sufficiently large positive integer 𝑚 such that 1−

(34)

𝜂2 2 (𝐿 + 1) (𝐿 + 3) (𝜉𝑛 + 2𝜂𝑛 + 𝑛 ) > 0, 1−𝜅 𝜉𝑛

𝑛 ≥ 𝑚. (30)

Fix a 𝑝 ∈ Fix(T) and take a constant 𝑀1 > 0 such that 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 max {󵄩󵄩󵄩𝑥0 − 𝑝󵄩󵄩󵄩 , 󵄩󵄩󵄩𝑥1 − 𝑝󵄩󵄩󵄩 , . . . , 󵄩󵄩󵄩𝑥𝑚 − 𝑝󵄩󵄩󵄩 , 2 󵄩󵄩󵄩󰜚 (𝑝) − 𝑝󵄩󵄩󵄩} ≤ 𝑀1 . (31) Next, we show that ‖𝑥𝑚+1 − 𝑝‖ ≤ 𝑀1 . Set

󵄩 󵄩 󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑥𝑚 − 𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩 + 𝜉𝑚 󵄩󵄩󵄩𝑥𝑚+1 − 𝑥𝑚 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 × 󵄩󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩 + 𝜉𝑚 𝜅 󵄩󵄩󵄩𝑥𝑚 − 𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩2 + 𝜉𝑚 󵄩󵄩󵄩󰜚 (𝑝) − 𝑝󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩 − 𝜉𝑚 󵄩󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 + 𝜂𝑚 (󵄩󵄩󵄩T𝑥𝑚 − T𝑥𝑚+1 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑚+1 − 𝑥𝑚 󵄩󵄩󵄩) 󵄩 󵄩 × 󵄩󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩 󵄩 󵄩 ≤ (1 + 𝜉𝑚 𝜅) 󵄩󵄩󵄩𝑥𝑚 − 𝑝󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 × 󵄩󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩 + 𝜉𝑚 󵄩󵄩󵄩󰜚 (𝑝) − 𝑝󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩2 × 󵄩󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩 − 𝜉𝑚 󵄩󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩 + (𝐿 + 1) (𝜉𝑚 + 𝜂𝑚 ) 󵄩󵄩󵄩𝑥𝑚+1 − 𝑥𝑚 󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩 . (35) It follows that

𝑦𝑚 = 𝜉𝑚 󰜚 (𝑥𝑚 ) + (1 − 𝜉𝑚 − 𝜂𝑚 ) 𝑥𝑚 + 𝜂𝑚 T𝑥𝑚 ; (32) thus, 𝑥𝑚+1 = projC [𝑦𝑚 ] . Then we have ⟨𝑥𝑚+1 − 𝑦𝑚 , 𝑥𝑚+1 − 𝑝⟩ ≤ 0.

− 𝜂𝑚 ⟨𝑥𝑚+1 − T𝑥𝑚+1 , 𝑥𝑚+1 − 𝑝⟩

(33)

󵄩 󵄩 󵄩 󵄩 (1 + 𝜉𝑚 ) 󵄩󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩 ≤ (1 + 𝜉𝑚 𝜅) 󵄩󵄩󵄩𝑥𝑚 − 𝑝󵄩󵄩󵄩 󵄩 󵄩 +𝜉𝑚 󵄩󵄩󵄩󰜚 (𝑝) − 𝑝󵄩󵄩󵄩 󵄩 󵄩 + (𝐿 + 1) (𝜉𝑚 + 𝜂𝑚 ) 󵄩󵄩󵄩𝑥𝑚+1 − 𝑥𝑚 󵄩󵄩󵄩 . (36)


5

By (29), we have 󵄩 󵄩󵄩 󵄩󵄩𝑥𝑚+1 − 𝑥𝑚 󵄩󵄩󵄩 󵄩 = 󵄩󵄩󵄩projC [𝜉𝑚 󰜚 (𝑥𝑚 ) + (1 − 𝜉𝑚 − 𝜂𝑚 ) 𝑥𝑚 + 𝜂𝑚 T𝑥𝑚 ] 󵄩 −projC [𝑥𝑚 ]󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝜉𝑚 󰜚 (𝑥𝑚 ) + (1 − 𝜉𝑚 − 𝜂𝑚 ) 𝑥𝑚 + 𝜂𝑚 T𝑥𝑚 − 𝑥𝑚 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 𝜉𝑚 (󵄩󵄩󵄩󰜚 (𝑥𝑚 ) − 󰜚 (𝑝)󵄩󵄩󵄩 + 󵄩󵄩󵄩󰜚 (𝑝) − 𝑝󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑚 − 𝑝󵄩󵄩󵄩) (37) 󵄩 󵄩 󵄩 󵄩 + 𝜂𝑚 (󵄩󵄩󵄩T𝑥𝑚 − 𝑝󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑚 − 𝑝󵄩󵄩󵄩) 󵄩 󵄩 󵄩 󵄩 ≤ 𝜉𝑚 (󵄩󵄩󵄩󰜚 (𝑝) − 𝑝󵄩󵄩󵄩 + (1 + 𝜅) 󵄩󵄩󵄩𝑥𝑚 − 𝑝󵄩󵄩󵄩) 󵄩 󵄩 + (𝐿 + 1) 𝜂𝑚 󵄩󵄩󵄩𝑥𝑚 − 𝑝󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ (𝐿 + 1 + 𝜅) (𝜉𝑚 + 𝜂𝑚 ) 󵄩󵄩󵄩𝑥𝑚 − 𝑝󵄩󵄩󵄩 + 𝜉𝑚 󵄩󵄩󵄩󰜚 (𝑝) − 𝑝󵄩󵄩󵄩 ≤ (𝐿 + 3) (𝜉𝑚 + 𝜂𝑚 ) 𝑀1 .

Set S = (2I − T)−1 (i.e., S is a resolvent of the monotone operator I − T). We then have that S is a nonexpansive self mapping of C and Fix(S) = Fix(T). By Proposition 5, we know that, whenever {𝛾𝑛 } ⊂ (0, 1) and 𝛾𝑛 → 0+ , the sequence {𝑧𝑛 } defined by 𝑧𝑛 = S projC [𝛾𝑛 󰜚 (𝑧𝑛 ) + (1 − 𝛾𝑛 ) 𝑧𝑛 ]

(42)

converges strongly to the fixed point 𝑥‡ of S (and of T as Fix(S) = Fix(T)). Without loss of generality, we may assume that ‖𝑧𝑛 ‖ ≤ 𝑀2 for all 𝑛. It suffices to prove that ‖𝑥𝑛+1 − 𝑧𝑛 ‖ → 0 as 𝑛 → ∞ (for some 𝛾𝑛 → 0+ ). To this end, we rewrite (42) as (2I − T) 𝑧𝑛 = projC [𝛾𝑛 󰜚 (𝑧𝑛 ) + (1 − 𝛾𝑛 ) 𝑧𝑛 ] ,

𝑛 ≥ 0. (43)

By using the property of the metric projection, we have

Substitute (37) into (36) to obtain 󵄩 󵄩 (1 + 𝜉𝑚 ) 󵄩󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩

⟨𝛾𝑛 󰜚 (𝑧𝑛 ) + (1 − 𝛾𝑛 ) 𝑧𝑛 − (2𝑧𝑛 − T𝑧𝑛 ) , 𝑥𝑛+1

󵄩 󵄩 󵄩 󵄩 ≤ (1 + 𝜉𝑚 𝜅) 󵄩󵄩󵄩𝑥𝑚 − 𝑝󵄩󵄩󵄩 + 𝜉𝑚 󵄩󵄩󵄩󰜚 (𝑝) − 𝑝󵄩󵄩󵄩

− (2𝑧𝑛 − T𝑧𝑛 )⟩ ≤ 0

2

+ (𝐿 + 1) (𝐿 + 3) (𝜉𝑚 + 𝜂𝑚 ) 𝑀1 1+𝜅 2 ≤ (1 + 𝜉 ) 𝑀1 + (𝐿 + 1) (𝐿 + 3) (𝜉𝑚 + 𝜂𝑚 ) 𝑀1 ; 2 𝑚 (38) that is,

󳨐⇒ ⟨−𝛾𝑛 (𝑧𝑛 − 󰜚 (𝑧𝑛 )) , 𝑥𝑛+1 − 𝑧𝑛 − (𝑧𝑛 − T𝑧𝑛 )⟩ + ⟨T𝑧𝑛 − 𝑧𝑛 , 𝑥𝑛+1 − 𝑧𝑛 − (𝑧𝑛 − T𝑧𝑛 )⟩ ≤ 0 󳨐⇒ ⟨−𝛾𝑛 (𝑧𝑛 − 󰜚 (𝑧𝑛 )) + T𝑧𝑛 − 𝑧𝑛 , 𝑥𝑛+1 − 𝑧𝑛 ⟩ 󵄩2 󵄩 + 󵄩󵄩󵄩𝑧𝑛 − T𝑧𝑛 󵄩󵄩󵄩 ≤ ⟨𝛾𝑛 (𝑧𝑛 − 󰜚 (𝑧𝑛 )) , T𝑧𝑛 − 𝑧𝑛 ⟩

󵄩󵄩 󵄩 󵄩󵄩𝑥𝑚+1 − 𝑝󵄩󵄩󵄩 2

((1 − 𝜅) 𝜉𝑚 /2) − (𝐿 + 1) (𝐿 + 3) (𝜉𝑚 + 𝜂𝑚 ) ≤ [1 − ] 𝑀1 1 + 𝜉𝑚

2 𝜂𝑚 ))] ) 𝜉𝑚

−1

× (1 + 𝜉𝑚 ) ) } 𝑀1 ≤ 𝑀1 . (39) By induction, we get 󵄩󵄩 󵄩 󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 ≤ 𝑀1 ,

∀𝑛 ≥ 0,

(40)

which implies that {𝑥𝑛 } is bounded and so is {T𝑥𝑛 }. Now, we take a constant 𝑀2 > 0 such that 󵄩 󵄩 󵄩 󵄩 𝑀2 = sup {󵄩󵄩󵄩𝑥𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩T𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩} . 𝑛

󳨐⇒ ⟨−𝛾𝑛 (𝑧𝑛 − 󰜚 (𝑧𝑛 )) + T𝑧𝑛 − 𝑧𝑛 , 𝑥𝑛+1 − 𝑧𝑛 ⟩ 󵄩 󵄩 󵄩󵄩 ≤ 𝛾𝑛 󵄩󵄩󵄩𝑧𝑛 − 󰜚 (𝑧𝑛 )󵄩󵄩󵄩 󵄩󵄩󵄩T𝑧𝑛 − 𝑧𝑛 󵄩󵄩󵄩 T𝑧𝑛 − 𝑧𝑛 , 𝑥𝑛+1 − 𝑧𝑛 ⟩ 𝛾𝑛 󵄩 󵄩 󵄩󵄩 ≤ 󵄩󵄩󵄩𝑧𝑛 − 󰜚 (𝑧𝑛 )󵄩󵄩󵄩 󵄩󵄩󵄩T𝑧𝑛 − 𝑧𝑛 󵄩󵄩󵄩 .

󳨐⇒ ⟨− (𝑧𝑛 − 󰜚 (𝑧𝑛 )) +

(1 − 𝜅) 𝜉𝑚 2 = {1 − ((( ) [1 − (𝐿 + 1) (𝐿 + 3) 2 1−𝜅 × (𝜉𝑚 + 2𝜂𝑚 + (

(44)

(41)

Note that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝑧𝑛 − T𝑧𝑛 󵄩󵄩󵄩 = 󵄩󵄩󵄩projC [𝛾𝑛 󰜚 (𝑧𝑛 ) + (1 − 𝛾𝑛 ) 𝑧𝑛 ] − 𝑧𝑛 󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝛾𝑛 󰜚 (𝑧𝑛 ) + (1 − 𝛾𝑛 ) 𝑧𝑛 − 𝑧𝑛 󵄩󵄩󵄩 󵄩 󵄩 = 𝛾𝑛 󵄩󵄩󵄩𝑧𝑛 − 󰜚 (𝑧𝑛 )󵄩󵄩󵄩 .

(45)

Hence, we get

⟨− (𝑧𝑛 − 󰜚 (𝑧𝑛 )) +

T𝑧𝑛 − 𝑧𝑛 󵄩 󵄩2 , 𝑥𝑛+1 − 𝑧𝑛 ⟩ ≤ 𝛾𝑛 󵄩󵄩󵄩𝑧𝑛 − 󰜚(𝑧𝑛 )󵄩󵄩󵄩 . 𝛾𝑛 (46)

6

Abstract and Applied Analysis = (1 − 𝜉𝑛 − 𝜂𝑛 ) ⟨𝑥𝑛 − 𝑧𝑛 , 𝑥𝑛+1 − 𝑧𝑛 ⟩

From (42), we have 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑧𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 = 󵄩󵄩󵄩S projC [𝛾𝑛+1 󰜚 (𝑧𝑛+1 ) + (1 − 𝛾𝑛+1 ) 𝑧𝑛+1 ] 󵄩 −S projC [𝛾𝑛 󰜚 (𝑧𝑛 ) + (1 − 𝛾𝑛 ) 𝑧𝑛 ]󵄩󵄩󵄩 󵄩 ≤ 󵄩󵄩󵄩𝛾𝑛+1 󰜚 (𝑧𝑛+1 ) + (1 − 𝛾𝑛+1 ) 𝑧𝑛+1 󵄩 −𝛾𝑛 󰜚 (𝑧𝑛 ) − (1 − 𝛾𝑛 ) 𝑧𝑛 󵄩󵄩󵄩 󵄩 = 󵄩󵄩󵄩(1 − 𝛾𝑛+1 ) (𝑧𝑛+1 − 𝑧𝑛 ) + (𝛾𝑛 − 𝛾𝑛+1 )

+ 𝜂𝑛 ⟨T𝑥𝑛 − T𝑥𝑛+1 , 𝑥𝑛+1 − 𝑧𝑛 ⟩ + 𝜂𝑛 ⟨T𝑥𝑛+1 − T𝑧𝑛 , 𝑥𝑛+1 − 𝑧𝑛 ⟩ + ⟨𝜉𝑛 (󰜚 (𝑥𝑛 ) − 𝑧𝑛 ) + 𝜂𝑛 (T𝑧𝑛 − 𝑧𝑛 ) , 𝑥𝑛+1 − 𝑧𝑛 ⟩ . (52)

󵄩 × (𝑧𝑛 − 󰜚 (𝑧𝑛 )) + 𝛾𝑛+1 (󰜚 (𝑧𝑛+1 ) − 󰜚 (𝑧𝑛 ))󵄩󵄩󵄩

󵄩 󵄩 ≤ [1 − (1 − 𝜅) 𝛾𝑛+1 ] 󵄩󵄩󵄩𝑧𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 󵄨 󵄨󵄩 󵄩 + 󵄨󵄨󵄨𝛾𝑛+1 − 𝛾𝑛 󵄨󵄨󵄨 󵄩󵄩󵄩𝑧𝑛 − 󰜚 (𝑧𝑛 )󵄩󵄩󵄩 .

It follows that 󵄩2 󵄩󵄩 󵄩󵄩𝑥𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩

󵄩󵄩 󵄩 󵄩 ≤ (1 − 𝜉𝑛 − 𝜂𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩

(47)

󵄩󵄩 󵄩 󵄩 + 𝜂𝑛 𝐿 󵄩󵄩󵄩𝑥𝑛 − 𝑥𝑛+1 󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩

It follows that

󵄨󵄨 󵄨 󵄨𝛾𝑛+1 − 𝛾𝑛 󵄨󵄨󵄨 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝑧𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 ≤ 󵄨 󵄩𝑧 − 󰜚 (𝑧𝑛 )󵄩󵄩󵄩 . (1 − 𝜅) 𝛾𝑛+1 󵄩 𝑛

󵄩 󵄩2 + 𝜂𝑛 󵄩󵄩󵄩𝑥𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩

(48)

+ 𝜉𝑛 ⟨󰜚 (𝑧𝑛 ) − 𝑧𝑛 +

Set 𝛾𝑛 :=

𝜉𝑛 . 𝜂𝑛

(49)

≤

+

By condition (C2), 𝛾𝑛 → 0 and 𝛾𝑛 ∈ (0, 1), for 𝑛 large enough. Hence, by (46) and (48), we have ⟨− (𝑧𝑛 − 󰜚 (𝑧𝑛 )) +

≤

𝜉𝑛 󵄩󵄩 󵄩2 𝜉 󵄩𝑧 − 󰜚(𝑧𝑛 )󵄩󵄩󵄩 ≤ 𝑛 𝑀22 , 𝜂𝑛 󵄩 𝑛 𝜂𝑛 󵄨 󵄨 󵄩󵄩 󵄩 󵄨󵄨𝜉𝑛 𝜂𝑛−1 − 𝜉𝑛−1 𝜂𝑛 󵄨󵄨󵄨 𝑀2 . 󵄩󵄩𝑧𝑛 − 𝑧𝑛−1 󵄩󵄩󵄩 ≤ 󵄨 𝜉𝑛 𝜂𝑛−1

1 − 𝜉𝑛 − 𝜂𝑛 󵄩󵄩 󵄩2 󵄩 󵄩2 (󵄩󵄩𝑥𝑛 − 𝑧𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 ) 2 +

𝜂𝑛 (T𝑧𝑛 − 𝑧𝑛 ) , 𝑥𝑛+1 − 𝑧𝑛 ⟩ 𝜉𝑛

2 𝜂𝑛2 󵄩󵄩 󵄩2 𝐿 󵄩 󵄩2 󵄩󵄩𝑥𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛 − 𝑥𝑛+1 󵄩󵄩󵄩 2 2

2 󵄩 󵄩2 𝜉 󵄩 󵄩2 + 𝜂𝑛 󵄩󵄩󵄩𝑥𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 + 𝑛 󵄩󵄩󵄩𝑧𝑛 − 󰜚(𝑧𝑛 )󵄩󵄩󵄩 . 𝜂𝑛

(50)

By (29), we have 󵄩󵄩 󵄩 󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩 󵄩 = 󵄩󵄩󵄩projC [𝜉𝑛 󰜚 (𝑥𝑛 ) + (1 − 𝜉𝑛 − 𝜂𝑛 ) 𝑥𝑛 + 𝜂𝑛 T𝑥𝑛 ] 󵄩 −projC 𝑥𝑛 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 𝜉𝑛 󵄩󵄩󵄩𝑥𝑛 − 󰜚 (𝑥𝑛 )󵄩󵄩󵄩 + 𝜂𝑛 󵄩󵄩󵄩T𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩

Thus, 󵄩󵄩 󵄩2 1 − 𝜉𝑛 − 𝜂𝑛 󵄩󵄩 󵄩2 󵄩󵄩𝑥𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 ≤ 󵄩𝑥 − 𝑧𝑛 󵄩󵄩󵄩 1 + 𝜉𝑛 − 𝜂𝑛 󵄩 𝑛

(51)

+

𝐿2 󵄩󵄩 󵄩2 󵄩𝑥 − 𝑥𝑛 󵄩󵄩󵄩 1 + 𝜉𝑛 − 𝜂𝑛 󵄩 𝑛+1

+

2𝜉𝑛2 󵄩󵄩 󵄩2 󵄩𝑧 − 󰜚(𝑧𝑛 )󵄩󵄩󵄩 (1 + 𝜉𝑛 − 𝜂𝑛 ) 𝜂𝑛 󵄩 𝑛

+

𝜂𝑛2 󵄩󵄩 󵄩2 󵄩𝑥 − 𝑧𝑛 󵄩󵄩󵄩 1 + 𝜉𝑛 − 𝜂𝑛 󵄩 𝑛+1

≤ (𝜉𝑛 + 𝜂𝑛 ) 𝑀2 . Next, we estimate ‖𝑥𝑛+1 − 𝑧𝑛+1 ‖. Since 𝑥𝑛+1 = projC [𝑦𝑛 ], ⟨𝑥𝑛+1 − 𝑦𝑛 , 𝑥𝑛+1 − 𝑧𝑛 ⟩ ≤ 0. Then, we have 󵄩󵄩 󵄩2 󵄩󵄩𝑥𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 = ⟨𝑥𝑛+1 − 𝑧𝑛 , 𝑥𝑛+1 − 𝑧𝑛 ⟩ = ⟨𝑥𝑛+1 − 𝑦𝑛 , 𝑥𝑛+1 − 𝑧𝑛 ⟩ + ⟨𝑦𝑛 − 𝑧𝑛 , 𝑥𝑛+1 − 𝑧𝑛 ⟩ ≤ ⟨𝑦𝑛 − 𝑧𝑛 , 𝑥𝑛+1 − 𝑧𝑛 ⟩ = ⟨[𝜉𝑛 󰜚 (𝑥𝑛 ) + (1 − 𝜉𝑛 − 𝜂𝑛 ) 𝑥𝑛 + 𝜂𝑛 T𝑥𝑛 ] −𝑧𝑛 , 𝑥𝑛+1 − 𝑧𝑛 ⟩

𝜂𝑛 (T𝑧𝑛 − 𝑧𝑛 ) , 𝑥𝑛+1 − 𝑧𝑛 ⟩ (53) 𝜉𝑛

≤ (1 −

2𝜉𝑛 󵄩2 󵄩 ) 󵄩󵄩𝑥 − 𝑧𝑛 󵄩󵄩󵄩 1 + 𝜉𝑛 − 𝜂𝑛 󵄩 𝑛 2

2𝜉𝑛2 (𝜉 + 𝜂𝑛 ) 2 2 + 𝑛 𝐿 𝑀2 + 𝑀2 1 + 𝜉𝑛 − 𝜂𝑛 (1 + 𝜉𝑛 − 𝜂𝑛 ) 𝜂𝑛 2 +

𝜂𝑛2 4𝑀22 1 + 𝜉𝑛 − 𝜂𝑛

≤ (1 −

2𝜉𝑛 ) 1 + 𝜉𝑛 − 𝜂𝑛

󵄩 󵄩 󵄩 󵄩2 × (󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛−1 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑧𝑛 − 𝑧𝑛−1 󵄩󵄩󵄩)


7 2

From Lemma 4 and (57), we get ‖𝑥𝑛+1 − 𝑧𝑛 ‖2 → 0 as 𝑛 → ∞. This completes the proof.

2𝜉𝑛2 (𝜉 + 𝜂𝑛 ) +{ 𝑛 + 1 + 𝜉𝑛 − 𝜂𝑛 (1 + 𝜉𝑛 − 𝜂𝑛 ) 𝜂𝑛 + ≤ (1 −

Algorithm 9. Let C be a nonempty closed subset of a real Hilbert space H. Let T : C → C be a pseudocontraction. Let {𝜉𝑛 } ⊂ [0, 1] and {𝜂𝑛 } ⊂ [0, 1] be two real number sequences. For 𝑥0 ∈ C, we define a sequence {𝑥𝑛 } iteratively by

𝜂𝑛2 }𝑀 1 + 𝜉𝑛 − 𝜂𝑛

2𝜉𝑛 󵄩2 󵄩 ) 󵄩󵄩𝑥 − 𝑧𝑛−1 󵄩󵄩󵄩 1 + 𝜉𝑛 − 𝜂𝑛 󵄩 𝑛

𝑥𝑛+1 = projC [(1 − 𝜉𝑛 − 𝜂𝑛 ) 𝑥𝑛 + 𝜂𝑛 T𝑥𝑛 ] ,

1 󵄩󵄩 󵄩 󵄩𝑧 − 𝑧𝑛−1 󵄩󵄩󵄩 1 + 𝜉𝑛 − 𝜂𝑛 󵄩 𝑛 󵄩 󵄩 󵄩 󵄩 × (2 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛−1 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑧𝑛 − 𝑧𝑛−1 󵄩󵄩󵄩)

(C1) lim𝑛 → ∞ 𝜉𝑛 = 0 and ∑∞ 𝑛=0 𝜉𝑛 = ∞;

2

2𝜉𝑛2 (𝜉 + 𝜂𝑛 ) +{ 𝑛 + 1 + 𝜉𝑛 − 𝜂𝑛 (1 + 𝜉𝑛 − 𝜂𝑛 ) 𝜂𝑛

(C2) lim𝑛 → ∞ (𝜉𝑛 /𝜂𝑛 ) = lim𝑛 → ∞ (𝜂𝑛2 /𝜉𝑛 ) = 0;

(C3) lim𝑛 → ∞ ((𝜉𝑛 𝜂𝑛−1 − 𝜉𝑛−1 𝜂𝑛 )/𝜉𝑛2 𝜂𝑛−1 ) = 0.

𝜂𝑛2 + }𝑀 1 + 𝜉𝑛 − 𝜂𝑛

Then the sequence {𝑥𝑛 } generated by the algorithm (59) converges strongly to the minimum-norm fixed point of T.

2𝜉𝑛 󵄩2 󵄩 ) 󵄩󵄩𝑥 − 𝑧𝑛−1 󵄩󵄩󵄩 1 + 𝜉𝑛 − 𝜂𝑛 󵄩 𝑛 󵄨󵄨 󵄨 󵄨󵄨𝜉𝑛 𝜂𝑛−1 − 𝜉𝑛−1 𝜂𝑛 󵄨󵄨󵄨 1 + 𝑀 1 + 𝜉𝑛 − 𝜂𝑛 𝜉𝑛 𝜂𝑛−1

≤ (1 −

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

2

2𝜉𝑛2 (𝜉𝑛 + 𝜂𝑛 ) + 1 + 𝜉𝑛 − 𝜂𝑛 (1 + 𝜉𝑛 − 𝜂𝑛 ) 𝜂𝑛

Acknowledgment The first author was supported by the National Natural Science Foundation of China, Grant no. 11226125.

𝜂𝑛2 + } 𝑀, 1 + 𝜉𝑛 − 𝜂𝑛 (54) where the finite constant 𝑀 > 0 is given by 𝑀 := max {𝐿2 𝑀22 , 4𝑀22 , (55) 󵄩 󵄩 󵄩 󵄩 𝑀2 sup (2 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛−1 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑧𝑛 − 𝑧𝑛−1 󵄩󵄩󵄩)} . 𝑛 Set 𝛿𝑛 = 𝜃𝑛 = {

2𝜉𝑛 ≈ 2𝜉𝑛 1 + 𝜉𝑛 − 𝜂𝑛

(as 𝑛 󳨀→ ∞) ,

𝜉𝑛 𝜂𝑛−1 − 𝜉𝑛−1 𝜂𝑛 2𝜉𝑛2 𝜂𝑛−1

(56)

𝜂2 𝜂2 𝜉 1 + (𝜉𝑛 + 2𝜂𝑛 + 𝑛 ) + 𝑛 + 𝑛 } 𝑀. 2 𝜉𝑛 𝜂𝑛 2𝜉𝑛 Then (54) can be rewritten as 󵄩2 󵄩2 󵄩󵄩 󵄩 󵄩󵄩𝑥𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 ≤ (1 − 𝛿𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑧𝑛−1 󵄩󵄩󵄩 + 𝛿𝑛 𝜃𝑛 .

(57)

By the conditions (C1)–(C3), we deuce that lim 𝛿𝑛 = 0,

𝑛→∞

∞

∑ 𝛿𝑛 = ∞,

𝑛=1

lim 𝜃 𝑛→∞ 𝑛

(59)

Corollary 10. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be an 𝐿-Lipschitzian and pseudocontraction with Fix(T) ≠ 0. Suppose the following conditions are satisfied:

+

+{

𝑛 ≥ 0.

= 0.

(58)

References [1] K. Deimling, “Zeros of accretive operators,” Manuscripta Mathematica, vol. 13, no. 4, pp. 365–374, 1974. [2] S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974. [3] C. E. Chidume and S. A. Mutangadura, “An example on the mann iteration method for lipschitz pseudocontractions,” Proceedings of the American Mathematical Society, vol. 129, no. 8, pp. 2359–2363, 2001. [4] H. Zhou, “Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 546–556, 2008. [5] Y. Yao, Y.-C. Liou, and G. Marino, “A hybrid algorithm for pseudo-contractive mappings,” Nonlinear Analysis, Theory, Methods and Applications, vol. 71, no. 10, pp. 4997–5002, 2009. [6] H. Zegeye, N. Shahzad, and M. A. Alghamdi, “Convergence of Ishikawa’s iteration method for pseudocontractive mappings,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 18, pp. 7304–7311, 2011. [7] L.-C. Ceng, A. Petrus¸el, and J.-C. Yao, “Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings,” Applied Mathematics and Computation, vol. 209, no. 2, pp. 162–176, 2009. [8] L.-C. Ceng, A. Petrus¸el, and J.-C. Yao, “Iterative approximation of fixed points for asymptotically strict pseudocontractive type mappings in the intermediate sense,” Taiwanese Journal of Mathematics, vol. 15, no. 2, pp. 587–606, 2011.

8 [9] L.-C. Ceng and J.-C. Yao, “Strong convergence theorems for variational inequalities and fixed point problems of asymptotically strict pseudocontractive mappings in the intermediate sense,” Acta Applicandae Mathematicae, vol. 115, no. 2, pp. 167– 191, 2011. [10] J. C. Yao and L. C. Zeng, “Strong convergence of averaged approximants for asymptotically pseudocontractive mappings in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, pp. 451–462, 2007. [11] L. C. Ceng, Q. H. Ansari, and C. F. Wen, “Implicit relaxed and hybrid methods with regularization for minimization problems and asymptotically strict pseudocontractive mappings in the intermediate sense,” Abstract and Applied Analysis, vol. 2013, Article ID 854297, 14 pages, 2013. [12] C. E. Chidume, M. Abbas, and B. Ali, “Convergence of the Mann iteration algorithm for a class of pseudocontractive mappings,” Applied Mathematics and Computation, vol. 194, no. 1, pp. 1–6, 2007. [13] C. E. Chidume and H. Zegeye, “Iterative solution of nonlinear equations of accretive and pseudocontractive types,” Journal of Mathematical Analysis and Applications, vol. 282, no. 2, pp. 756– 765, 2003. ´ c, A. Rafiq, N. Caki´c, and J. S. Ume, “Implicit Mann [14] L. Ciri´ fixed point iterations for pseudo-contractive mappings,” Applied Mathematics Letters, vol. 22, no. 4, pp. 581–584, 2009. [15] C. Moore and B. V. C. Nnoli, “Strong convergence of averaged approximants for Lipschitz pseudocontractive maps,” Journal of Mathematical Analysis and Applications, vol. 260, no. 1, pp. 269– 278, 2001. [16] A. Udomene, “Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 67, no. 8, pp. 2403–2414, 2007. [17] H. Zegeye, N. Shahzad, and T. Mekonen, “Viscosity approximation methods for pseudocontractive mappings in Banach spaces,” Applied Mathematics and Computation, vol. 185, no. 1, pp. 538–546, 2007. [18] Q.-B. Zhang and C.-Z. Cheng, “Strong convergence theorem for a family of Lipschitz pseudocontractive mappings in a Hilbert space,” Mathematical and Computer Modelling, vol. 48, no. 3-4, pp. 480–485, 2008. [19] H. Zhou, “Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 11, pp. 4039–4046, 2009. [20] K. Geobel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1990. [21] X. Lu, H.-K. Xu, and X. Yin, “Hybrid methods for a class of monotone variational inequalities,” Nonlinear Analysis, Theory, Methods and Applications, vol. 71, no. 3-4, pp. 1032–1041, 2009. [22] H. K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 2, pp. 1–17, 2002.


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