On Faster Implicit Hybrid Kirk-Multistep Schemes for Contractive-Type ...

Hindawi Publishing Corporation International Journal of Analysis Volume 2016, Article ID 3791506, 10 pages http://dx.doi.org/10.1155/2016/3791506

Research Article On Faster Implicit Hybrid Kirk-Multistep Schemes for Contractive-Type Operators O. T. Wahab1 and K. Rauf2 1

Department of Statistics and Mathematical Sciences, Kwara State University, Malete, Nigeria Department of Mathematics, University of Ilorin, Ilorin 240001, Nigeria

2

Correspondence should be addressed to K. Rauf; [email protected] Received 21 March 2016; Revised 9 July 2016; Accepted 26 July 2016 Academic Editor: Lianwen Wang Copyright © 2016 O. T. Wahab and K. Rauf. 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. The purpose of this paper is to prove strong convergence and T-stability results of some modified hybrid Kirk-Multistep iterations for contractive-type operator in normed linear spaces. Our results show through analytical and numerical approach that the modified hybrid schemes are better in terms of convergence rate than other hybrid Kirk-Multistep iterative schemes in the literature.

1. Introduction In the recent years, numerous papers have been published on the strong convergence and T-stability of various iterative approximations of fixed points for contractive-type operators. See [1–9]. The Picard iterative scheme defined for 𝑥0 ∈ 𝑋 𝑥𝑛 = 𝑇𝑥𝑛−1 , 𝑛 ≥ 1,

𝑥𝑛 = (1 − 𝛼𝑛 ) 𝑥𝑛−1 + 𝛼𝑛 𝑇𝑦𝑛−1 𝑦𝑛−1 = (1 − 𝛽𝑛 ) 𝑥𝑛−1 + 𝛽𝑛 𝑇𝑥𝑛−1 ,

(1)

was the first iteration to be proved by Banach [10] for a selfmap 𝑇 in a complete metric space (𝑋, 𝑑) satisfying 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑐𝑑 (𝑥, 𝑦)

(2)

(called strict contraction), for all 𝑥, 𝑦 ∈ 𝑋 and 𝑐 ∈ (0, 1). Picard iteration (1) which obeys (2) is said to have a fixed point in 𝐹𝑇 , where 𝐹𝑇 is the set of all fixed points. The Picard iteration will no longer converge to a fixed point of the operator if contractive condition (2) is weaker. Hence, there is a need to consider other iterative procedures. Mann [11] defined a more general iteration in a Banach space 𝐸 satisfying quasi-nonexpansive operators. For 𝑥0 ∈ 𝐸, the Mann iteration is given by 𝑥𝑛 = (1 − 𝛼𝑛 ) 𝑥𝑛−1 + 𝛼𝑛 𝑇𝑥𝑛−1 ,

A double Mann iteration, called Ishikawa iteration, was introduced by Ishikawa [12]. It is defined for 𝑥0 ∈ 𝐸 as

𝑛 ≥ 1,

(3)

where {𝛼𝑛 } is a sequence of positive numbers in [0, 1]. Putting 𝛼𝑛 = 1 in (3) yields Picard iteration (1).

(4) 𝑛 ≥ 1,

where {𝛼𝑛 } and {𝛽𝑛 } are sequences of positive numbers in [0, 1]. The three-step iteration, which is more general than Mann and Ishikawa iterations, was defined by Noor [13]. For 𝑥0 ∈ 𝐸, the Noor iteration is given as 𝑥𝑛 = (1 − 𝛼𝑛 ) 𝑥𝑛−1 + 𝛼𝑛 𝑇𝑦𝑛−1 𝑦𝑛−1 = (1 − 𝛽𝑛 ) 𝑥𝑛−1 + 𝛽𝑛 𝑇𝑧𝑛−1 𝑧𝑛−1 = (1 − 𝛾𝑛 ) 𝑥𝑛−1 + 𝛾𝑛 𝑇𝑥𝑛−1 ,

(5)

𝑛 ≥ 1, where {𝛼𝑛 }, {𝛽𝑛 }, and {𝛾𝑛 } are sequences in [0, 1] with ∑ 𝛼𝑛 = ∞.

2

International Journal of Analysis

Rhoades and Soltuz [14] defined a multistep iteration in a normed linear space 𝐸 as follows. For 𝑥0 ∈ 𝐸, (1) 𝑥𝑛 = (1 − 𝛼𝑛 ) 𝑥𝑛−1 + 𝛼𝑛 𝑇𝑦𝑛−1 (𝑙) 𝑙+1 𝑦𝑛−1 = (1 − 𝛽𝑛(𝑙) ) 𝑥𝑛−1 + 𝛽𝑛(𝑙) 𝑇𝑦𝑛−1 ,

scheme of fixed points for different types of contractivelike operators in various spaces. See [3, 6, 18, 19]. The stability results of explicit and SP-iterative schemes have been discussed in [2, 4, 9, 20, 21]. Kirk’s iterative procedure was defined by Kirk [22]. For 𝑥0 ∈ 𝐸, where 𝐸 is a Banach space and 𝑇 is a self-map of 𝐸,

𝑙 = 1, 2, . . . , 𝑘 − 2 (6) (𝑘−1) 𝑦𝑛−1

𝑞

𝑥𝑛+1 = ∑𝛼𝑖 𝑇𝑖 𝑥𝑛 , 𝑛 = 0, 1, 2, . . . ,

= (1 − 𝛽𝑛 ) 𝑥𝑛−1 + 𝛽𝑛 𝑇𝑥𝑛−1 ,

(10)

𝑖=0

𝑛 ≥ 1, where {𝛼𝑛 } and {𝛽𝑛(𝑙) }, 𝑙 = 1, 2, . . . , 𝑘 − 1, are sequences of positive numbers in [0, 1] with ∑ 𝛼𝑛 = ∞. Iteration (6) generalized (3), (4), and (5); for example, if 𝑘 = 3 in (6), we recover the form (5); if 𝑘 = 2, we have (4); on putting 𝑘 = 2 and 𝛽𝑛(𝑙) = 0 for each 𝑙, we have (3). Another two-step scheme, which is independent of (4), was introduced by Thianwan [15]. Let 𝑥0 ∈ 𝐸; the sequence {𝑥𝑛 } ⊂ 𝐸 is defined as 𝑥𝑛 = (1 − 𝛼𝑛 ) 𝑦𝑛−1 + 𝛼𝑛 𝑇𝑦𝑛−1

where 𝑞 is a fixed integer with 𝑞 ≥ 1, 𝛼𝑖 ≥ 0 for each 𝑖 and 𝑞 ∑𝑖=0 𝛼𝑖 = 1. In order to reduce the cost of computations, Olatinwo [8] introduced two hybrid schemes, namely, Kirk-Mann and Kirk-Ishikawa iterative schemes in a normed linear space. For 𝑥0 ∈ 𝐸, 𝑞

𝑞

𝑖=0

𝑖=0

𝑥𝑛+1 = ∑𝛼𝑛,𝑖 𝑇𝑖 𝑥𝑛 ; ∑𝛼𝑛,𝑖 = 1, 𝑛 = 0, 1, 2, . . . ,

𝑦𝑛−1 = (1 − 𝛽𝑛 ) 𝑥𝑛−1 + 𝛽𝑛 𝑇𝑥𝑛−1 ,

(7)

𝑞

𝑞

𝑖=1

𝑖=0

𝑥𝑛+1 = 𝛼𝑛,0 𝑥𝑛 + ∑𝛼𝑛,𝑖 𝑇𝑖 𝑦𝑛 , ∑𝛼𝑛,𝑖 = 1,

𝑛 ≥ 1, where {𝛼𝑛 } and {𝛽𝑛 } are sequences of positive numbers in [0, 1] with ∑ 𝛼𝑛 = ∞. The three-step iteration of (7) called SP iteration was introduced by Phuengrattana and Suantai [16] and it was defined as follows. For 𝑥0 ∈ 𝐸,

𝑟

𝑦𝑛 = ∑𝛽𝑛,𝑖 𝑇𝑖 𝑥𝑛 , 𝑖=0

𝑟

∑𝛽𝑛,𝑖 = 1; 𝑖=0

𝑛 = 0, 1, 2 . . . ,

(1) (1) 𝑥𝑛 = (1 − 𝛼𝑛 ) 𝑦𝑛−1 + 𝛼𝑛 𝑇𝑦𝑛−1 (1) (2) (2) 𝑦𝑛−1 = (1 − 𝛽𝑛 ) 𝑦𝑛−1 + 𝛽𝑛 𝑇𝑦𝑛−1

(8)

(2) 𝑦𝑛−1 = (1 − 𝛾𝑛 ) 𝑥𝑛−1 + 𝛾𝑛 𝑇𝑥𝑛−1 ,

𝑛 ≥ 1, where {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 } ⊂ [0, 1] with ∑ 𝛼𝑛 = ∞. The two-step iteration of (8) can be easily obtained when 𝛾𝑛 = 0. In [17], Gürsoy et al. defined a generalized scheme of forms (7) and (8) in a Banach space. For 𝑥0 ∈ 𝐸, 𝑥𝑛 = (1 −

(1) 𝛼𝑛 ) 𝑦𝑛−1

+

(1) 𝛼𝑛 𝑇𝑦𝑛−1

(11)

respectively, where 𝑞 and 𝑟 are fixed integers with 𝑞 ≥ 𝑟 and {𝛼𝑛,𝑖 } and {𝛽𝑛,𝑖 } are sequences in [0, 1] satisfying 𝛼𝑛,𝑖 ≥ 0, 𝛼𝑛,0 ≠ 0, 𝛽𝑛,𝑖 ≥ 0, and 𝛽𝑛,0 ≠ 0. The Kirk-Noor iteration was introduced by Chugh and Kumar [23] as follows: for 𝑥0 ∈ 𝐸, 𝑞

𝑥𝑛+1 = 𝛼𝑛,0 𝑥𝑛 + ∑𝛼𝑛,𝑖 𝑇𝑖 𝑦𝑛 , 𝑖=1

𝑞

∑𝛼𝑛,𝑖 = 1, 𝑖=0

𝑟

𝑟

𝑖=1

𝑖=0

𝑦𝑛 = 𝛽𝑛,0 𝑥𝑛 + ∑𝛽𝑛,𝑖 𝑇𝑖 𝑧𝑛 , ∑𝛽𝑛,𝑖 = 1,

(𝑙) (𝑙+1) (𝑙+1) 𝑦𝑛−1 = (1 − 𝛽𝑛(𝑙) ) 𝑦𝑛−1 + 𝛽𝑛(𝑙) 𝑇𝑦𝑛−1 ,

𝑙 = 1, 2, 3, . . . , 𝑘 − 2 (9) (𝑘−1) = (1 − 𝛽𝑛(𝑘−1) ) 𝑥𝑛−1 + 𝛽𝑛(𝑘−1) 𝑇𝑥𝑛−1 , 𝑦𝑛−1

𝑠

𝑠

𝑖=0

𝑖=0

(12)

𝑧𝑛 = ∑𝛾𝑛,𝑖 𝑇𝑖 𝑥𝑛 , ∑𝛾𝑛,𝑖 = 1; 𝑛 = 0, 1, 2 . . . ,

𝑛 ≥ 1, where {𝛼𝑛 }, {𝛽𝑛(𝑙) }, ⊂ [0, 1], 𝑙 = 1, 2, . . . , 𝑘 − 1, with ∑ 𝛼𝑛 = ∞. Several results have been proved for the strong convergence of the explicit iteration as well as the SP-iterative

where 𝑞, 𝑟, and 𝑠 are fixed integers with 𝑞 ≥ 𝑟 ≥ 𝑠, and {𝛼𝑛,𝑖 }, {𝛽𝑛,𝑖 }, and {𝛾𝑛,𝑖 } are sequences in [0, 1] satisfying 𝛼𝑛,𝑖 ≥ 0, 𝛼𝑛,0 ≠ 0, 𝛽𝑛,𝑖 ≥ 0, 𝛽𝑛,0 ≠ 0, 𝛾𝑛,𝑖 ≥ 0, and 𝛾𝑛,0 ≠ 0.


3

In an attempt to generalize (11) and (12), Gürsoy et al. [17] introduced the Kirk-Multistep iteration in an arbitrary Banach space 𝐸. For 𝑥0 ∈ 𝐸, 𝑞1

𝑞1

𝑖=1

𝑖=0

𝑥𝑛+1 = 𝛼𝑛,0 𝑥𝑛 + ∑𝛼𝑛,𝑖 𝑇𝑖 𝑦𝑛 , ∑𝛼𝑛,𝑖 = 1, 𝑞𝑙+1

(𝑙) (𝑙) 𝑖 (𝑙+1) 𝑥𝑛 + ∑ 𝛽𝑛,𝑖 𝑇 𝑦𝑛 , 𝑦𝑛(𝑙) = 𝛽𝑛,0 𝑖=1

𝑞𝑙+1

(𝑙) = 1, 𝑙 = 1 (1) 𝑘 − 2 ∑ 𝛽𝑛,𝑖

(13)

𝑞𝑘

𝑖=0

𝑖=0

𝑦𝑛−1 = 𝛽𝑛 𝑥𝑛−1 + (1 − 𝛽𝑛 ) 𝑇𝑦𝑛−1 ,

(𝑘−1) 𝑖 (𝑘−1) 𝑇 𝑥𝑛 , ∑𝛽𝑛,𝑖 = 1; 𝑘 ≥ 2, 𝑦𝑛(𝑘−1) = ∑𝛽𝑛,𝑖

where 𝑞1 , 𝑞2 , 𝑞3 , . . . , 𝑞𝑘 are fixed integers with 𝑞1 ≥ 𝑞2 ≥ ⋅ ⋅ ⋅ ≥ (𝑙) ∞ 𝑞𝑘 ; {𝛼𝑛,𝑖 }∞ 𝑛=1 and {𝛽𝑛,𝑖 }𝑛=1 are sequences in [0, 1] satisfying (𝑙) (𝑙) 𝛼𝑛,𝑖 ≥ 0, 𝛼𝑛,0 ≠ 0, 𝛽𝑛,𝑖 ≥ 0, and 𝛽𝑛,0 ≠ 0 for each 𝑙. The Kirk-SP iteration was defined by Hussain et al. [24] as follows: for 𝑥0 ∈ 𝐸, 𝑞

𝑖=1

𝑖=0

𝑥𝑛 = 𝛼𝑛 𝑦𝑛−1 + (1 − 𝛼𝑛 ) 𝑇𝑥𝑛 𝑦𝑛−1 = 𝛽𝑛 𝑧𝑛−1 + (1 − 𝛽𝑛 ) 𝑇𝑦𝑛−1 𝑧𝑛−1 = 𝛾𝑛 𝑥𝑛−1 + (1 − 𝛾𝑛 ) 𝑇𝑧𝑛−1 ,

𝑦𝑛 = 𝛽𝑛,0 𝑧𝑛 + ∑𝛽𝑛,𝑖 𝑇 𝑧𝑛 , ∑𝛽𝑛,𝑖 = 1, 𝑖=1

𝑖=0

𝑠

𝑠

𝑖=0

𝑖=0

(14)

𝑧𝑛 = ∑𝛾𝑛,𝑖 𝑇𝑖 𝑥𝑛 , ∑𝛾𝑛,𝑖 = 1; 𝑛 = 0, 1, 2 . . . , where 𝑞, 𝑟, and 𝑠 are fixed integers with 𝑞 ≥ 𝑟 ≥ 𝑠 and {𝛼𝑛,𝑖 }, {𝛽𝑛,𝑖 }, and {𝛾𝑛,𝑖 } are sequences in [0, 1] satisfying 𝛼𝑛,𝑖 ≥ 0, 𝛼𝑛,0 ≠ 0, 𝛽𝑛,𝑖 ≥ 0, 𝛽𝑛,0 ≠ 0, 𝛾𝑛,𝑖 ≥ 0, and 𝛾𝑛,0 ≠ 0. The KirkThianwan iteration can be obtained if 𝑠 = 0 in (14). The Kirk-Multistep-SP, which generalized both Kirk-SP and Kirk-Thianwan schemes, was introduced by Akewe et al. [1] and it was defined as follows: for 𝑥0 ∈ 𝐸, 𝑞1

𝑥𝑛+1 = 𝛼𝑛,0 𝑦𝑛(1) + ∑𝛼𝑛,𝑖 𝑇𝑖 𝑦𝑛(1) , 𝑖=1

𝑞1

∑𝛼𝑛,𝑖 = 1,

𝑞𝑙+1

𝑞𝑘

𝑞𝑘

𝑖=0

𝑖=0

𝑍1 : 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑎𝑑 (𝑥, 𝑦) 𝑍2 : 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑏 [𝑑 (𝑥, 𝑇𝑥) + 𝑑 (𝑦, 𝑇𝑦)]

= 1, 𝑙 = 1 (1) 𝑘 − 2

(𝑘−1) 𝑖 (𝑘−1) 𝑇 𝑥𝑛 , ∑𝛽𝑛,𝑖 = 1; 𝑘 ≥ 2, 𝑦𝑛(𝑘−1) = ∑𝛽𝑛,𝑖

𝑛 = 0, 1, 2 . . . ,

(19)

𝑍3 : 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑐 [𝑑 (𝑥, 𝑇𝑦) + 𝑑 (𝑦, 𝑇𝑥)] , where 𝑎, 𝑏, and 𝑐 are nonnegative constants satisfying 𝑎 ∈ [0, 1), 𝑏, 𝑐 ≤ 1/2. The equivalence form of (19) is

1 [𝑑 (𝑥, 𝑇𝑥) + 𝑑 (𝑦, 𝑇𝑦)] , 2

𝑖=1

(𝑙) ∑ 𝛽𝑛,𝑖 𝑖=0

where {𝛼𝑛 }, {𝛽𝑛 }, and {𝛾𝑛 } are sequences in [0, 1] with ∑(1 − 𝛼𝑛 ) = ∞. The implicit Noor iteration (18) is more general than (16) and (17). The most generalized Banach operator used by several authors is the one proved by Zamfirescu [25]. Let 𝑋 be a complete metric space and let 𝑇 be a self-map of 𝑋. The operator 𝑇 is Zamfirescu operator if for each pair of points 𝑥, 𝑦 ∈ 𝑋, at least one of the following is true:

𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑎 max {𝑑 (𝑥, 𝑦) ,

𝑖=0

(𝑙) (𝑙+1) (𝑙) 𝑖 (𝑙+1) 𝑦𝑛 + ∑ 𝛽𝑛,𝑖 𝑇 𝑦𝑛 , 𝑦𝑛(𝑙) = 𝛽𝑛,0 𝑞𝑙+1

(18)

𝑛 ≥ 1,

𝑟

𝑖

(17)

where {𝛼𝑛 } and {𝛽𝑛 } are sequences in [0, 1]. The implicit Noor iteration was defined by Chugh et al. [5] as follows: for 𝑥0 ∈ 𝐶

𝑥𝑛+1 = 𝛼𝑛,0 𝑦𝑛 + ∑𝛼𝑛,𝑖 𝑇𝑖 𝑦𝑛 , ∑𝛼𝑛,𝑖 = 1, 𝑟

(16)

𝑛 ≥ 1,

𝑛 = 0, 1, 2, . . . ,

𝑞

𝑥𝑛 = 𝛼𝑛 𝑥𝑛−1 + (1 − 𝛼𝑛 ) 𝑇𝑥𝑛 , 𝑛 ≥ 1, 𝑥𝑛 = 𝛼𝑛 𝑦𝑛−1 + (1 − 𝛼𝑛 ) 𝑇𝑥𝑛

𝑖=0

𝑞𝑘

where 𝑞1 , 𝑞2 , 𝑞3 , . . . , 𝑞𝑘 are fixed integers with 𝑞1 ≥ 𝑞2 ≥ ⋅ ⋅ ⋅ ≥ (𝑙) ∞ 𝑞𝑘 ; {𝛼𝑛,𝑖 }∞ 𝑛=1 and {𝛽𝑛,𝑖 }𝑛=1 are sequences in [0, 1] satisfying (𝑙) (𝑙) ≥ 0, and 𝛽𝑛,0 ≠ 0 for each 𝑙. 𝛼𝑛,𝑖 ≥ 0, 𝛼𝑛,0 ≠ 0, 𝛽𝑛,𝑖 Another modified form of explicit iteration is the implicit iteration. The implicit Mann iteration and implicit Ishikawa ´ c et al. [7] and Xue and Zhang iteration were discussed by Ciri´ [19], respectively. For 𝑥0 ∈ 𝐶, 𝐶 being a closed subset of normed linear space, the implicit Mann and implicit Ishikawa iterations are, respectively,

(15)

(20)

1 [𝑑 (𝑥, 𝑇𝑦) + 𝑑 (𝑦, 𝑇𝑥)]} 2 for 𝑥, 𝑦 ∈ 𝑋 and 𝑎 ∈ [0, 1). Berinde [2] observed that condition (20) implies 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 2ℎ𝑑 (𝑥, 𝑇𝑥) + ℎ𝑑 (𝑥, 𝑦) , where ℎ = max{𝑎, 𝑎/(2 − 𝑎)}.

(21)

4


In [26], Rhoades used a more general contractive condition than (21): for 𝑥, 𝑦 ∈ 𝑋, there exists 𝑎 ∈ [0, 1) such that 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑎 max {𝑑 (𝑥, 𝑦) , 1 [𝑑 (𝑥, 𝑇𝑥) + 𝑑 (𝑦, 𝑇𝑦)] , 𝑑 (𝑥, 𝑇𝑦) , 𝑑 (𝑦, 𝑇𝑥)} . 2

(22) 𝑢𝑛+1 ≤ 𝛿𝑢𝑛 + 𝜖𝑛 , ∀𝑛 ∈ N

Osilike [20] extended and generalized the contractive condition (22): for 𝑥, 𝑦 ∈ 𝑋, there exists 𝑎 ∈ [0, 1) and 𝐿 ≥ 0 such that 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝐿𝑑 (𝑥, 𝑇𝑥) + 𝑎𝑑 (𝑥, 𝑦) .

(23)

Imoru and Olatinwo [21] employed a more general class of operators 𝑇 than (23) satisfying the following contractive conditions: 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑎𝑑 (𝑥, 𝑦) + 𝜑 (𝑑 (𝑥, 𝑇𝑥))

for 𝑥, 𝑦 ∈ 𝑋, (24)

where 𝑎 ∈ [0, 1) and 𝜑 : R+ → R+ is a monotone increasing function with 𝜑(0) = 0. The equivalence form of (24) in a normed linear space is 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ≤ 𝑎 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 + 𝜑 (‖𝑥 − 𝑇𝑥‖)

for 𝑥, 𝑦 ∈ 𝑋 (25)

We will need the following definitions and lemmas to prove our main results. Definition 1 (see [9]). Let (𝑋, 𝑑) be a metric space and 𝑇 : 𝑋 → 𝑋 a self-mapping. Suppose that 𝐹𝑇 = {𝑝 ∈ 𝑋 : 𝑇𝑝 = 𝑝} is the set of fixed points of 𝑇. Let {𝑥𝑛 }∞ 𝑛=0 ⊂ 𝑋 be the sequence generated by an iterative procedure involving 𝑇 which is defined by 𝑥

𝑥𝑛+1 = 𝑓𝑇,𝛼𝑛 𝑛 ,

𝑛 ≥ 0,

Lemma 3 (see [2]). Let 𝛿 be a real number such that 𝛿 ∈ [0, 1) and {𝜖𝑛 }∞ 𝑛=0 is a sequence of nonnegative numbers such that lim𝑛→∞ 𝜖𝑛 = 0; then, for any sequence of positive numbers {𝑢𝑛 }∞ 𝑛=0 satisfying

(26) 𝑥

where 𝑥0 ∈ 𝑋 is the initial approximation and 𝑓𝑇,𝛼𝑛 𝑛 is a function such that 𝛼𝑛 ∈ [0, 1]. Suppose that {𝑥𝑛 } converges to a fixed point 𝑝 of 𝑇. Let {𝑦𝑛 }∞ 𝑛=0 ⊂ 𝑋 and set 𝜖𝑛 = 𝑦 𝑑(𝑦𝑛+1 , 𝑓𝑇,𝛼𝑛 𝑛 ), 𝑛 = 0, 1, 2, . . .. Then, iterative procedure (26) is said to be 𝑇-stable or stable with respect to 𝑇 if and only if lim𝑛→∞ 𝜖𝑛 = 0 implies lim𝑛→∞ 𝑦𝑛 = 𝑝. ∞ Definition 2 (see [2]). Let {𝑎𝑛 }∞ 𝑛=0 and {𝑏𝑛 }𝑛=0 be two nonnegative real sequences which converge to 𝑎 and 𝑏, respectively. Let 󵄨󵄨󵄨𝑎 − 𝑎󵄨󵄨󵄨 󵄨; (27) 𝑙 = lim 󵄨󵄨󵄨 𝑛 𝑛→∞ 󵄨𝑏 − 𝑏󵄨󵄨󵄨 󵄨𝑛 󵄨 ∞ (1) if 𝑙 = 0, then {𝑎𝑛 }∞ 𝑛=0 converges to 𝑎 faster than {𝑏𝑛 }𝑛=0 to 𝑏; ∞ (2) if 0 < 𝑙 < ∞, then both {𝑎𝑛 }∞ 𝑛=0 and {𝑏𝑛 }𝑛=0 have the same convergence rate; ∞ (3) if 𝑙 = ∞, then {𝑏𝑛 }∞ 𝑛=0 converges to 𝑏 faster than {𝑎𝑛 }𝑛=0 to 𝑎.

(28)

we have lim𝑛→∞ 𝑢𝑛 = 0. Lemma 4 (see [8]). Let (𝐸, ‖ ⋅ ‖) be a normed linear space and 𝑇 : 𝐸 → 𝐸 a map satisfying (25). Let 𝜑 : R+ → R+ be a subadditive, monotone increasing function such that 𝜑(0) = 0, 𝜑(𝐿𝑢) = 𝐿𝜑(𝑢), for 𝑢 ∈ R+ , 𝐿 ≥ 0. Then, for all 𝑖 ∈ N, 𝐿 ≥ 0 and for all 𝑥, 𝑦 ∈ 𝐸 󵄩 󵄩󵄩 𝑖 󵄩󵄩𝑇 𝑥 − 𝑇𝑖 𝑦󵄩󵄩󵄩 󵄩 󵄩 𝑖 𝑖 󵄩 󵄩 ≤ ∑ ( ) 𝑎𝑖−𝑗 𝜑𝑗 (‖𝑥 − 𝑇𝑥‖) + 𝑎𝑖 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 . 𝑗 𝑗=1

(29)

Note that 𝑎 ∈ [0, 1) in (29).

2. Main Results We present our main results as follows. Let 𝐸 be an arbitrary Banach space and 𝑇 : 𝐸 → 𝐸 a selfmap. Let 𝑥0 ∈ 𝐸; we define the following iteration, namely, implicit hybrid Kirk-Multistep iterative scheme, as follows: 𝑞1

(1) 𝑥𝑛 = 𝛼𝑛,0 𝑥𝑛−1 + ∑𝛼𝑛,𝑖 𝑇𝑖 𝑥𝑛 , 𝑖=1

𝑞1

∑𝛼𝑛,𝑖 = 1 𝑖=0

𝑞𝑙+1

(𝑙) (𝑙) (𝑙+1) (𝑙) 𝑖 (𝑙) = 𝛽𝑛,0 𝑥𝑛−1 + ∑ 𝛽𝑛,𝑖 𝑇 𝑥𝑛−1 , 𝑥𝑛−1 𝑖=1

𝑞𝑙+1

(𝑙) = 1, 𝑙 = 1 (1) 𝑘 − 2 ∑ 𝛽𝑛,𝑖 𝑖=0

(30)

𝑞𝑘

(𝑘−1) (𝑘−1) (𝑘−1) 𝑖 (𝑘−1) = 𝛽𝑛,0 𝑥𝑛−1 + ∑𝛽𝑛,𝑖 𝑇 𝑥𝑛−1 , 𝑥𝑛−1 𝑖=1

𝑞𝑘

(𝑘−1) = 1, 𝑘 ≥ 2, ∑𝛽𝑛,𝑖 𝑖=0

𝑛 ≥ 1, where 𝑞1 , 𝑞2 , 𝑞3 , . . . , 𝑞𝑘 are fixed integers with 𝑞1 ≥ 𝑞2 ≥ (𝑙) ∞ 𝑞3 ≥ ⋅ ⋅ ⋅ ≥ 𝑞𝑘 ; {𝛼𝑛,𝑖 }∞ 𝑛=1 and {𝛽𝑛,𝑖 }𝑛=1 are sequences in [0, 1] (𝑙) (𝑙) ≠ 0 for each 𝑙 satisfying 𝛼𝑛,𝑖 ≥ 0, 𝛼𝑛,0 ≠ 0, 𝛽𝑛,𝑖 ≥ 0, and 𝛽𝑛,0 ∞ with ∑𝑛=1 (1 − 𝛼𝑛,0 ) = ∞.


5

If we let 𝑘 = 3 in (30), we obtain the implicit Kirk-Noor iteration defined by 𝑞1

𝑞1

𝑖=1

𝑖=0

This implies that 𝛼𝑛,0 󵄩 󵄩󵄩 󵄩󵄩󵄩𝑥(1) − 𝑝󵄩󵄩󵄩 . 󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 ≤ 󵄩 󵄩󵄩 𝑞1 1 − ∑𝑖=1 𝛼𝑛,𝑖 𝑎𝑖 󵄩 𝑛−1

(1) 𝑥𝑛 = 𝛼𝑛,0 𝑥𝑛−1 + ∑𝛼𝑛,𝑖 𝑇𝑖 𝑥𝑛 , ∑𝛼𝑛,𝑖 = 1 𝑞2

𝑞2

𝑖=1

𝑖=0

𝑞3

𝑞3

Also, from (30), we have

(1) (1) (2) (1) 𝑖 (1) (1) = 𝛽𝑛,0 𝑥𝑛−1 + ∑𝛽𝑛,𝑖 𝑇 𝑥𝑛−1 , ∑𝛽𝑛,𝑖 =1 𝑥𝑛−1

(2) 𝑥𝑛−1

=

(2) 𝛽𝑛,0 𝑥𝑛−1

+

(2) 𝑖 (2) 𝑇 𝑥𝑛−1 , ∑𝛽𝑛,𝑖 𝑖=1

(2) ∑𝛽𝑛,𝑖 𝑖=0

(31)

= 1,

(1) 󵄩 󵄩󵄩𝑥(2) − 𝑝󵄩󵄩󵄩 ≤ 𝛽𝑛,0 󵄩󵄩 𝑛−1 󵄩󵄩

Theorem 5. Let (𝐸, ‖ ⋅ ‖) be a normed linear space. Assume 𝑇 is self-map of 𝐸 satisfying the contractive condition (29) with 𝐹𝑇 ≠ 𝜙. Then, for 𝑥0 ∈ 𝐸, the sequence {𝑥𝑛 } defined by (30) with ∑∞ 𝑛=1 (1 − 𝛼𝑛,0 ) = ∞ converges strongly to the fixed point 𝑝 ∈ 𝐹𝑇 . Proof. Let 𝑥0 ∈ 𝑋 and 𝑝 ∈ 𝐹𝑇 ; then using (29) and (30) we have 𝑞

1 󵄩 (1) 󵄩 𝑖 󵄩 󵄩 󵄩󵄩 𝑖 󵄩 󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 ≤ 𝛼𝑛,0 󵄩󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩󵄩 + ∑𝛼𝑛,𝑖 󵄩󵄩󵄩󵄩𝑇 𝑥𝑛 − 𝑇 𝑝󵄩󵄩󵄩󵄩

𝑖=1

󵄩 󵄩 + ∑𝛼𝑛,𝑖 [ ∑ ( ) 𝑎 𝜑 (󵄩󵄩󵄩𝑝 − 𝑇𝑝󵄩󵄩󵄩) 𝑗 𝑖=1 [𝑗=1 𝑖−𝑗 𝑗

𝑞

1 󵄩 (1) 󵄩 󵄩 󵄩 󵄩 + 𝑎𝑖 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩] = 𝛼𝑛,0 󵄩󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩󵄩 + ∑𝛼𝑛,𝑖 𝑎𝑖 󵄩󵄩󵄩𝑥𝑛 𝑖=1 ]

󵄩 − 𝑝󵄩󵄩󵄩 .

(34)

󵄩 (1) 󵄩 (1) 󵄩 󵄩󵄩𝑥(2) − 𝑝󵄩󵄩󵄩 + 𝑎𝑖 󵄩󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩󵄩] = 𝛽𝑛,0 󵄩󵄩 𝑛−1 󵄩󵄩 ] 𝑞

2 󵄩 (1) 𝑖 󵄩 (1) + ∑𝛽𝑛,𝑖 𝑎 󵄩󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩󵄩 .

𝑖=1

This becomes 𝛽𝑛(1) 󵄩󵄩 (1) 󵄩 󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 ≤ 󵄩 󵄩 1 − ∑𝑞2 𝛽(1) 𝑎𝑖 𝑖=1 𝑛,𝑖

󵄩󵄩 (2) 󵄩 󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 . 󵄩 󵄩

(35)

From (30) again, we have 𝑞

3 󵄩󵄩 (2) 󵄩 (2) 󵄩 󵄩󵄩𝑥(3) − 𝑝󵄩󵄩󵄩 + ∑𝛽(2) 󵄩󵄩󵄩𝑇𝑖 𝑥(2) − 𝑇𝑖 𝑝󵄩󵄩󵄩 󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 ≤ 𝛽𝑛,0 󵄩 󵄩 󵄩󵄩 𝑛,𝑖 󵄩 󵄩 󵄩 𝑛−1 󵄩 𝑛−1 󵄩 󵄩

𝑖=1

(2) 󵄩 󵄩󵄩𝑥(3) − 𝑝󵄩󵄩󵄩 ≤ 𝛽𝑛,0 󵄩󵄩 𝑛−1 󵄩󵄩 𝑞3 𝑖 𝑖 󵄩 󵄩 (2) [ + ∑𝛽𝑛,𝑖 ∑ ( ) 𝑎𝑖−𝑗 𝜑𝑗 (󵄩󵄩󵄩𝑝 − 𝑇𝑝󵄩󵄩󵄩) 𝑖=1 [𝑗=1 𝑗

(36)

󵄩 (2) 󵄩 (2) 󵄩 󵄩󵄩𝑥(3) − 𝑝󵄩󵄩󵄩 + 𝑎𝑖 󵄩󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩󵄩] = 𝛽𝑛,0 󵄩󵄩 𝑛−1 󵄩󵄩 ]

󵄩 (1) 󵄩 ≤ 𝛼𝑛,0 󵄩󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩󵄩 𝑖

𝑖 𝑖 󵄩 󵄩 (1) [ + ∑𝛽𝑛,𝑖 ∑ ( ) 𝑎𝑖−𝑗 𝜑𝑗 (󵄩󵄩󵄩𝑝 − 𝑇𝑝󵄩󵄩󵄩) 𝑖=1 [𝑗=1 𝑗 𝑞2

By setting 𝑘 = 2 in (30), we have the implicit Kirk-Ishikawa iteration and we can also obtain the implicit Kirk-Mann iteration when 𝑘 = 2 and 𝑞2 = 0 in (30). If 𝑞1 = 𝑞2 = 𝑞3 = ⋅ ⋅ ⋅ = 𝑞𝑘 = 1 in (30), then we obtain the (𝑙) implicit multistep iteration (19) with 𝛼𝑛,1 = 𝛼𝑛 , 𝛽𝑛,1 = 𝛽𝑛(𝑙) . If 𝑘 = 3, 𝑞1 = 𝑞2 = 𝑞3 = 1, and 𝑞4 = 𝑞5 = ⋅ ⋅ ⋅ = 𝑞𝑘 = 0 in (30), we have the implicit Noor scheme (18) with 𝛼𝑛,1 = 𝛼𝑛 , (1) (2) = 𝛽𝑛(1) , and 𝛽𝑛,1 = 𝛽𝑛(2) . 𝛽𝑛,1 If 𝑘 = 2, 𝑞1 = 𝑞2 = 1, and 𝑞3 = 𝑞4 = ⋅ ⋅ ⋅ = 𝑞𝑘 = 0 in (30), we have the implicit Ishikawa scheme (17) with 𝛼𝑛,1 = 𝛼𝑛 , (1) = 𝛽𝑛(1) . 𝛽𝑛,1 If 𝑘 = 2, 𝑞1 = 1, and 𝑞2 = 𝑞3 = 𝑞4 = ⋅ ⋅ ⋅ = 𝑞𝑘 = 0 in (30), we have the implicit Mann scheme (16) with 𝛼𝑛,1 = 𝛼𝑛 . Throughout, the operator 𝑇 will be assumed as fixed and a fixed point 𝑝 ∈ 𝐹𝑇 with the condition (29) is unique.

𝑖

𝑞

2 󵄩󵄩 (1) 󵄩 (1) 󵄩 󵄩󵄩𝑥(2) − 𝑝󵄩󵄩󵄩 + ∑𝛽(1) 󵄩󵄩󵄩𝑇𝑖 𝑥(1) − 𝑇𝑖 𝑝󵄩󵄩󵄩 󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 ≤ 𝛽𝑛,0 󵄩 󵄩 󵄩󵄩 𝑛−1 𝑛,𝑖 󵄩 󵄩 󵄩 󵄩 𝑛−1 󵄩 󵄩

𝑖=1

𝑛 ≥ 1.

𝑞1

(33)

𝑞

(32)

3 󵄩 (2) 𝑖 󵄩 (2) + ∑𝛽𝑛,𝑖 𝑎 󵄩󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩󵄩

𝑖=1

which implies 𝛽𝑛(2) 󵄩󵄩 (2) 󵄩 󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 ≤ 󵄩 󵄩 1 − ∑𝑞3 𝛽(2) 𝑎𝑖 𝑖=1 𝑛,𝑖

󵄩󵄩 (3) 󵄩 󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 . 󵄩 󵄩

(37)

6


Continuing this way up to 𝑘 − 1 in (30), we have

Similarly, we can easily obtain the following from (40): (1) 𝛽𝑛,0

𝑞𝑘

󵄩󵄩 (𝑘−1) 󵄩 (𝑘−1) 󵄩 󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 + ∑𝛽(𝑘−1) 󵄩󵄩󵄩󵄩𝑇𝑖 𝑥(𝑘−1) 󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 ≤ 𝛽𝑛,0 𝑛,𝑖 󵄩 󵄩 󵄩 󵄩 𝑛−1 󵄩

𝑞

(1) 𝑖 2 1 − ∑𝑖=1 𝛽𝑛,𝑖 𝑎

𝑖=1

(2) 𝛽𝑛,0

󵄩 (𝑘−1) 󵄩 󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 − 𝑇 𝑝󵄩󵄩󵄩󵄩 ≤ 𝛽𝑛,0 󵄩 󵄩 𝑖

1−

𝑞𝑘 𝑖 𝑖 󵄩 󵄩 (𝑘−1) [ + ∑𝛽𝑛,𝑖 ∑ ( ) 𝑎𝑖−𝑗 𝜑𝑗 (󵄩󵄩󵄩𝑝 − 𝑇𝑝󵄩󵄩󵄩) 𝑗 𝑖=1 [𝑗=1

𝑞3 ∑𝑖=1

(2) 𝑖 𝛽𝑛,𝑖 𝑎

(3) 𝛽𝑛,0

(38)

1−

𝑞4 ∑𝑖=1

(3) 𝑖 𝛽𝑛,𝑖 𝑎

𝑞2

𝑞2

𝑖=1

𝑖=1

𝑞3

𝑞3

𝑖=1

𝑖=1

𝑞4

𝑞4

𝑖=1

𝑖=1

(1) 𝑖 (1) ≤ ∑𝛽𝑛,𝑖 𝑎 < ∑𝛽𝑛,𝑖 =1

(2) 𝑖 (2) ≤ ∑𝛽𝑛,𝑖 𝑎 < ∑𝛽𝑛,𝑖 =1

(3) 𝑖 (3) ≤ ∑𝛽𝑛,𝑖 𝑎 < ∑𝛽𝑛,𝑖 =1

(43)

.. .

󵄩 (𝑘−1) 󵄩 (𝑘−1) 󵄩 󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 + 𝑎𝑖 󵄩󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩󵄩] = 𝛽𝑛,0 󵄩 󵄩 ]

(𝑘−1) 𝛽𝑛,0 𝑞

(𝑘−1) 𝑖 𝑘 1 − ∑𝑖=1 𝛽𝑛,𝑖 𝑎

𝑞

𝑘 󵄩 (𝑘−1) 𝑖 󵄩 (𝑘−1) + ∑𝛽𝑛,𝑖 𝑎 󵄩󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩󵄩

𝑞𝑘

𝑞𝑘

𝑖=1

𝑖=1

(𝑘−1) 𝑖 (𝑘−1) ≤ ∑𝛽𝑛,𝑖 𝑎 < ∑𝛽𝑛,𝑖 = 1.

Applying (42) and (43) in (40) and letting 𝑎𝑖 ≤ 𝑎 < 1 for each 𝑖, we have

𝑖=1

𝑞1

implying that

󵄩󵄩 󵄩 󵄩 󵄩 𝑖 󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 ≤ (∑𝛼𝑛,𝑖 𝑎 + 𝛼𝑛,0 ) 󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 𝑖=1

󵄩󵄩 (𝑘−1) 󵄩 󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 ≤ 󵄩 󵄩 1−

𝛽𝑛(𝑘−1) 𝑞𝑘 (𝑘−1) 𝑖 𝛽𝑛,𝑖 𝑎 ∑𝑖=1

󵄩󵄩 󵄩 󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩 .

(39)

󵄩 󵄩 = [1 − (1 − 𝛼𝑛,0 ) (1 − 𝑎)] 󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩

Substituting (35)–(39) into (33) it becomes

≤ [1 − (1 − 𝛼𝑛,0 ) (1 − 𝑎)] [1 − (1 − 𝛼𝑛−1,0 ) (1 − 𝑎)]

𝛽𝑛(2) 𝑞

(2) 𝑖 3 1 − ∑𝑖=1 𝛽𝑛,𝑖 𝑎

× ⋅⋅⋅(

1−

)

.. . (40)

𝛽𝑛(𝑘−1) 󵄩 ) 󵄩󵄩󵄩𝑥𝑛−1 𝑞𝑘 (𝑘−1) 𝑖 ∑𝑖=1 𝛽𝑛,𝑖 𝑎

󵄩 − 𝑝󵄩󵄩󵄩 .

𝑛 󵄩 󵄩 ≤ 𝑒−(1−𝑎) ∑𝑟=1 (1−𝛼𝑟,0 ) 󵄩󵄩󵄩𝑥0 − 𝑝󵄩󵄩󵄩 .

As 𝑛 → ∞, ∑∞ 𝑟=1 (1 − 𝛼𝑟,0 ) = ∞. Hence, lim𝑛→∞ ‖𝑥𝑛 − 𝑝‖ = 0. Therefore, implicit Kirk-Multistep scheme (30) converges strongly to 𝑝 ∈ 𝐹𝑇 .

𝑞

1 − 𝜆𝑛 = 1 −

𝑞

1 1 − ∑𝑖=1 𝛼𝑛,𝑖 𝑎𝑖

𝑞

=

𝑛

󵄩 󵄩 ≤ ∏ [1 − (1 − 𝛼𝑟,0 ) (1 − 𝑎)] 󵄩󵄩󵄩𝑥0 − 𝑝󵄩󵄩󵄩 𝑟=1

1 Let 𝜆 𝑛 = 𝛼𝑛,0 /(1 − ∑𝑖=1 𝛼𝑛,𝑖 𝑎𝑖 ); then

𝛼𝑛,0

(44)

󵄩 󵄩 ⋅ 󵄩󵄩󵄩𝑥𝑛−2 − 𝑝󵄩󵄩󵄩

𝛼𝑛,0 𝛽𝑛(1) 󵄩󵄩 󵄩 ) ( ) 󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 ≤ ( 𝑞1 𝑞2 (1) 𝑖 1 − ∑𝑖=1 𝛼𝑛,𝑖 𝑎𝑖 1 − ∑𝑖=1 𝛽𝑛,𝑖 𝑎 ⋅(

󵄩 󵄩 ≤ [(1 − 𝛼𝑛,0 ) 𝑎 + 𝛼𝑛,0 ] 󵄩󵄩󵄩𝑥𝑛−1 − 𝑝󵄩󵄩󵄩

1 𝛼𝑛,𝑖 𝑎𝑖 − 𝛼𝑛,0 1 − ∑𝑖=1

𝑞

1 1 − ∑𝑖=1 𝛼𝑛,𝑖 𝑎𝑖

(41)

𝑞1

≥ 1 − (∑𝛼𝑛,𝑖 𝑎𝑖 + 𝛼𝑛,0 ) .

Corollary 6. Let (𝐸, ‖⋅‖) be a normed linear space. Assume 𝑇 is self-map of 𝐸 satisfying the contractive condition (29) with 𝐹𝑇 ≠ 𝜙. Then, for 𝑥0 ∈ 𝐸, the implicit Kirk-Noor, the implicit KirkIshikawa, and the implicit Kirk-Mann schemes with ∑∞ 𝑛=1 (1 − 𝛼𝑛,0 ) = ∞ converge strongly to the fixed point 𝑝 ∈ 𝐹𝑇 . Remark 7. The strong convergence results for implicit Noor, implicit Ishikawa, and implicit Mann schemes are obvious from Theorem 5.

𝑖=1

Therefore, 𝑞1

𝑞1

𝑞1

𝑖=1

𝑖=0

𝑖=0

𝜆 𝑛 ≤ ∑𝛼𝑛,𝑖 𝑎𝑖 + 𝛼𝑛,0 = ∑𝛼𝑛,𝑖 𝑎𝑖 < ∑𝛼𝑛,𝑖 = 1.

(42)

Theorem 8. Let (𝐸, ‖ ⋅ ‖) be a normed linear space and 𝑇 is a self-map of 𝐸 satisfying contractive condition (29) with 𝐹𝑇 ≠ 𝜙. Then, for 𝑥0 ∈ 𝐸 and 𝑝 ∈ 𝐹𝑇, the sequence {𝑥𝑛 } defined by (30) is T-stable.


7

Proof. Let {𝑦𝑛 } ∈ 𝐸 be an arbitrary sequence and let 𝜖𝑛 = 𝑞1 (1) ‖𝑦𝑛 − 𝛼𝑛,0 𝑧𝑛−1 − ∑𝑖=1 𝛼𝑛,𝑖 𝑇𝑖 𝑦𝑛 ‖, where 𝑞2

(1) (1) (2) (1) 𝑖 (1) 𝑧𝑛−1 = 𝛽𝑛,0 𝑧𝑛−1 + ∑𝛽𝑛,𝑖 𝑇 𝑧𝑛−1 , 𝑖=1

𝑞2

(1) = 1, ∑𝛽𝑛,𝑖 𝑖=0

𝑞𝑙+1

(𝑙) (𝑙) (𝑙+1) (𝑙) 𝑖 (𝑙) = 𝛽𝑛,0 𝑧𝑛−1 + ∑ 𝛽𝑛,𝑖 𝑇 𝑧𝑛−1 , 𝑧𝑛−1

Conversely, suppose ‖𝑦𝑛 − 𝑝‖ = 0 for 𝑝 ∈ 𝐹𝑇 ; then 󵄩󵄩 󵄩󵄩 𝑞1 󵄩󵄩 󵄩󵄩 (1) − ∑𝛼𝑛,𝑖 𝑇𝑖 𝑦𝑛 󵄩󵄩󵄩 𝜖𝑛 = 󵄩󵄩󵄩𝑦𝑛 − 𝛼𝑛,0 𝑧𝑛−1 󵄩󵄩 󵄩󵄩 𝑖=1 󵄩 󵄩 󵄩󵄩 󵄩󵄩 𝑞1 󵄩󵄩 󵄩 󵄩 󵄩󵄩 (1) ≤ 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑝 − (𝛼𝑛,0 𝑧𝑛−1 + ∑𝛼𝑛,𝑖 𝑇𝑖 𝑦𝑛 )󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 𝑖=1 󵄩 󵄩 𝑞

𝑖=1

1 󵄩 (1) 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩 + 𝛼𝑛,0 󵄩󵄩󵄩󵄩𝑧𝑛−1 − 𝑝󵄩󵄩󵄩󵄩 + ∑𝛼𝑛,𝑖 󵄩󵄩󵄩󵄩𝑇𝑖 𝑦𝑛 − 𝑇𝑖 𝑝󵄩󵄩󵄩󵄩 (51)

(45)

𝑞𝑙+1

𝑖=1

(𝑙) = 1, 𝑙 = 1 (1) 𝑘 − 2 ∑ 𝛽𝑛,𝑖

𝑞1

𝑖=0

𝑞𝑘

(𝑘−1) (𝑘−1) (𝑙) 𝑖 (𝑘−1) = 𝛽𝑛,0 𝑦𝑛−1 + ∑𝛽𝑛,𝑖 𝑇 𝑧𝑛−1 , 𝑧𝑛−1 𝑖=1

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩 + 𝛼𝑛,0 󵄩󵄩󵄩𝑦𝑛−1 − 𝑝󵄩󵄩󵄩 + ∑𝛼𝑛,𝑖 𝑎𝑖 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩

𝑞𝑘

𝑖=1

(𝑘−1) = 1. ∑𝛽𝑛,𝑖

𝑞1

𝑖=0

󵄩 󵄩 󵄩 󵄩 ≤ (1 + ∑𝛼𝑛,𝑖 𝑎𝑖 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩 + 𝛼𝑛,0 󵄩󵄩󵄩𝑦𝑛−1 − 𝑝󵄩󵄩󵄩 .

Suppose lim𝑛→∞ 𝜖𝑛 = 0 and 𝑝 ∈ 𝐹𝑇 ; by (29) we have 󵄩󵄩 󵄩󵄩 𝑞1 󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 (1) 𝑖 󵄩 󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩 ≤ 󵄩󵄩𝑦𝑛 − 𝛼𝑛,0 𝑧𝑛−1 − ∑𝛼𝑛,𝑖 𝑇 𝑦𝑛 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 𝑖=1 󵄩 󵄩 󵄩󵄩 󵄩󵄩 𝑞1 󵄩󵄩 󵄩󵄩 (1) + 󵄩󵄩󵄩𝛼𝑛,0 𝑧𝑛−1 + ∑𝛼𝑛,𝑖 𝑇𝑖 𝑦𝑛 − 𝑝󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 𝑖=1 󵄩 󵄩 𝑞

1 󵄩 (1) 󵄩 󵄩 󵄩 ≤ 𝜖𝑛 + 𝛼𝑛,0 󵄩󵄩󵄩󵄩𝑧𝑛−1 − 𝑝󵄩󵄩󵄩󵄩 + ∑𝛼𝑛,𝑖 󵄩󵄩󵄩󵄩𝑇𝑖 𝑦𝑛 − 𝑝󵄩󵄩󵄩󵄩

𝑖=1

Since ‖𝑦𝑛 − 𝑝‖ → 0, then lim𝑛→∞ 𝜖𝑛 = 0. Therefore, iterative scheme (30) is T-stable.

(46)

𝑖=1

Corollary 9. Let (𝐸, ‖ ⋅ ‖) be a normed linear space and 𝑇 is a self-map of 𝐸 satisfying the contractive condition (29) with 𝐹𝑇 ≠ 𝜙. Then, for 𝑥0 ∈ 𝐸 and 𝑝 ∈ 𝐹𝑇 , the sequence {𝑥𝑛 } defined by implicit Kirk-Mann, implicit Kirk-Ishikawa, and implicit KirkNoor schemes are T-stable. Remark 10. The stability results for implicit Mann, implicit Ishikawa, and implicit Noor schemes with contractive condition (29) are special cases of Corollary 9.

𝑞1

󵄩 (1) 󵄩 󵄩 󵄩 ≤ 𝜖𝑛 + 𝛼𝑛,0 󵄩󵄩󵄩󵄩𝑧𝑛−1 − 𝑝󵄩󵄩󵄩󵄩 + ∑𝛼𝑛,𝑖 𝑎𝑖 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩 . 𝑖=1

2.1. Comparison of Several Iterative Schemes. We compare our iterative schemes with others by using the following example.

This implies that 𝜖𝑛 󵄩󵄩 󵄩 󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩 ≤ 𝑞1 1 − ∑𝑖=1 𝛼𝑛,𝑖 𝑎𝑖 𝛼𝑛,0

󵄩󵄩 (1) 󵄩 󵄩󵄩𝑧𝑛−1 − 𝑝󵄩󵄩󵄩 . + 𝑞1 𝑖 󵄩 󵄩 1 − ∑𝑖=1 𝛼𝑛,𝑖 𝑎

(47)

Example 11. Let 𝑇 : [0, 1] → [0, 1] and 𝑇𝑥 = 𝑥/2 with 𝑥0 ≠ 0 (𝑙) = 1 − 2/√𝑛, 𝛼𝑛,𝑖 = and fixed point 𝑝 = 0 using 𝛼𝑛,0 = 𝛽𝑛,0 (𝑙) 𝛽𝑛,𝑖 = 4/√𝑛, for each 𝑙, 𝑛 ≥ 25, and 𝑞1 = 𝑞2 = 𝑞3 = 𝑞4 = 2. For the implicit Kirk-Mann iteration (IKM), we have

From inequalities (42) and (43), one can easily obtain the following: 󵄩󵄩 (1) 󵄩 󵄩 (2) 󵄩 󵄩 (3) 󵄩 󵄩󵄩𝑧𝑛−1 − 𝑝󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑧𝑛−1 − 𝑝󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩𝑧𝑛−1 − 𝑝󵄩󵄩󵄩󵄩 ≤ ⋅ ⋅ ⋅ 󵄩 󵄩 󵄩 󵄩 (𝑘−1) 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩󵄩𝑧𝑛−1 − 𝑝󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑦𝑛−1 − 𝑝󵄩󵄩󵄩 .

2

𝑥𝑛 = 𝛼𝑛,0 𝑥𝑛−1 + ∑𝛼𝑛,𝑖 𝑇𝑖 𝑥𝑛 𝑖=1

2 2 1 ) 𝑥𝑛−1 + 𝑥𝑛 + 𝑥 . = (1 − √𝑛 √𝑛 √𝑛 𝑛

(48)

This implies that

Then, inequality (47) becomes 𝜖𝑛 󵄩󵄩 󵄩 󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩 ≤ 𝑞1 1 − ∑𝑖=1 𝛼𝑛,𝑖 𝑎𝑖 𝛼𝑛,0

󵄩󵄩 󵄩 + 󵄩𝑦𝑛−1 − 𝑝󵄩󵄩󵄩 . 𝑞1 1 − ∑𝑖=1 𝛼𝑛,𝑖 𝑎𝑖 󵄩

𝑥𝑛 (IKM) = (49)

𝑛 2√𝑟 − 4 2√𝑛 − 4 )𝑥 . 𝑥𝑛−1 = ∏ ( 2√𝑛 − 3 2√𝑟 −3 0 𝑟=25

𝑞

(50)

(53)

Also, for implicit Kirk-Ishikawa iteration (IKI), we have 𝑥𝑛 =

1 Letting 𝛿 = 𝛼𝑛,0 /(1 − ∑𝑖=1 𝛼𝑛,𝑖 𝑎𝑖 ) < 1 and by Lemma 3, we have

󵄩 󵄩󵄩 󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩 = 0.

(52)

2√𝑛 − 4 (1) 𝑥 2√𝑛 − 3 𝑛−1

(54)

2√𝑛 − 4 𝑥 . 2√𝑛 − 3 𝑛−1

(55)

with (1) 𝑥𝑛−1 =

8


Hence,

Similarly, using Definition 2, we have that 2

𝑥𝑛 (IKI) = (

󵄨 󵄨󵄨 󵄨𝑥 (IKI) − 0󵄨󵄨󵄨 =0 lim 󵄨󵄨󵄨 𝑛 𝑛→∞ 󵄨𝑥 (IKM) − 0󵄨󵄨󵄨 󵄨 𝑛 󵄨

2

𝑛 2√𝑟 − 4 2√𝑛 − 4 ) 𝑥𝑛−1 = ∏ ( ) 𝑥0 . (56) 2√𝑛 − 3 2√𝑟 −3 𝑟=25

Similarly, implicit Kirk-Noor iteration (IKN) implies 3

𝑥𝑛 (IKN) = (

3

𝑛 2√𝑛 − 4 2√𝑟 − 4 ) 𝑥𝑛−1 = ∏ ( ) 𝑥0 (57) 2√𝑛 − 3 2√𝑟 −3 𝑟=25

while the implicit multistep Kirk iteration (IMK) gives

which implies that the implicit Kirk-Ishikawa iteration 𝑥𝑛 (IKI) converges faster than the implicit Kirk-Mann iteration 𝑥𝑛 (IKM). For the Kirk-Mann iteration (KM), we have the following estimate: 2

𝑥𝑛 = 𝛼𝑛,0 𝑥𝑛−1 + ∑𝛼𝑛,𝑖 𝑇𝑖 𝑥𝑛−1

𝑘

𝑛

𝑥𝑛 (IMK) = ∏ ( 𝑟=25

2√𝑟 − 4 ) 𝑥0 . 2√𝑟 − 3

(58)

Now, using Definition 2, we compare the implicit Kirk type iterations as follows: for 𝑘 ≥ 4, we have 󵄨󵄨 󵄨 𝑘−3 𝑛 󵄨󵄨𝑥𝑛 (IMK) − 0󵄨󵄨󵄨 2√𝑟 − 4 = ) ( ∏ 󵄨󵄨󵄨𝑥𝑛 (IKN) − 0󵄨󵄨󵄨 󵄨 𝑟=25 2√𝑟 − 3 󵄨 (59) 𝑘−3 𝑛 1 ) = ∏ (1 − 2√𝑟 − 3 𝑟=25

𝑖=1

2 2 1 )𝑥 + 𝑥 + 𝑥 . = (1 − √𝑛 𝑛−1 √𝑛 𝑛−1 √𝑛 𝑛−1

𝑥𝑛 (KM) = (1 +

𝑛 √𝑟 √𝑛 ) 𝑥𝑛−1 = ∏ (1 + ) 𝑥0 . 𝑛 𝑟 𝑟=25

2

𝑛

𝑘−3 1 0 ≤ lim ∏ (1 − ) 𝑛→∞ 2√𝑟 − 3 𝑟=25

𝑥𝑛 (KT) = ∏ (1 + 𝑟=25

𝑥𝑛 (KSP) = ∏ (1 + 𝑟=25

(60)

𝑥𝑛 (KMSP) = ∏ (1 + 𝑟=25

24 𝑘−3 ) = 0, ∀𝑘 ≥ 4. 𝑛→∞ 𝑛

Remark 12. The implicit multistep Kirk iteration (IMK) converges faster than the implicit Kirk-Noor iteration (IKN) for 𝑘 = 4, 5, . . ..

𝑛

1 = ∏ (1 − ) 2√𝑟 −3 𝑟=25

√𝑟 ) 𝑥0 . 𝑟

󵄨 󵄨󵄨 𝑛 󵄨󵄨𝑥𝑛 (IKM) − 0󵄨󵄨󵄨 𝑟 2√𝑟 − 4 = )( ) ( ∏ 󵄨󵄨󵄨𝑥𝑛 (KM) − 0󵄨󵄨󵄨 √𝑟 − 3 +𝑟 2√𝑟 󵄨 𝑟=25 󵄨 𝑛

= ∏( 𝑟=25

(61)

2𝑟3/2 − 4𝑟 ) − 3𝑟1/2 − 𝑟 3 (𝑟 − 𝑟1/2 )

𝑛

𝑟=25

(67)

2𝑟3/2

= ∏ (1 −

with

(66)

We compare Kirk-Mann (KM), Kirk-Thianwan (KT), KirkSP (KSP), and Kirk-Multistep-SP (KMSP) iterations with our iterative schemes as follows. Again, using Definition 2, we have

Also, 󵄨󵄨 󵄨 𝑛 󵄨󵄨𝑥𝑛 (IKN) − 0󵄨󵄨󵄨 2√𝑟 − 4 ) 󵄨󵄨 󵄨󵄨 = ∏ ( 2√𝑟 −3 𝑥 − 0 (IKI) 󵄨󵄨 𝑛 󵄨󵄨 𝑟=25

√𝑟 ) 𝑥0 , 𝑟 𝑘

𝑛

𝑘−3

√𝑟 ) 𝑥0 , 𝑟 3

𝑛

𝑛

= ( lim

(65)

The estimates for Kirk-Thianwan (KT), Kirk-SP (KSP), and Kirk-Multistep-SP (KMSP) iterations are, respectively,

𝑛

24 25 𝑛−2 𝑛−1 = ( lim ⋅ ⋅⋅⋅ ⋅ ) 𝑛→∞ 25 26 𝑛−1 𝑛

(64)

This implies that

with

1 𝑘−3 ≤ lim ∏ (1 − ) 𝑛→∞ 𝑟 𝑟=25

(63)

2𝑟3/2 − 3𝑟1/2 − 𝑟

)

with 𝑛

0 ≤ lim ∏ (1 − 𝑛→∞

= lim

𝑟=25

𝑛

1 1 ) ≤ lim ∏ (1 − ) 𝑛→∞ 𝑟 2√𝑟 − 3 𝑟=25

24 25 24 𝑛−2 𝑛−1 ⋅ ⋅⋅⋅ ⋅ = lim = 0. 𝑛→∞ 26 𝑛−1 𝑛 𝑛

𝑛

(62)

𝑛→∞ 25

Remark 13. The implicit Kirk-Noor iteration 𝑥𝑛 (IKN) converges to 𝑝 = 0 faster than the implicit Kirk-Ishikawa iteration 𝑥𝑛 (IKI) to 𝑝 = 0.

0 ≤ lim ∏ (1 − 𝑛→∞ 𝑟=25

3 (𝑟 − 𝑟1/2 ) 2𝑟3/2 − 3𝑟1/2 − 𝑟

)

𝑛 24 25 1 𝑛 − 2 𝑛 − 1 (68) ≤ lim ∏ (1 − ) = lim ⋅ ⋅⋅⋅ ⋅ 𝑛→∞ 𝑛→∞ 25 26 𝑟 𝑛 −1 𝑛 𝑟=25

= lim

𝑛→∞

24 = 0. 𝑛


9

Remark 14. The implicit Kirk-Mann iteration 𝑥𝑛 (IKM) converges to 𝑝 = 0 faster than the Kirk-Mann iteration 𝑥𝑛 (KM) to 𝑝 = 0. For the comparison of implicit Kirk-Ishikawa iteration 𝑥𝑛 (IKI) and Kirk-Thianwan iteration 𝑥𝑛 (KT), we have 󵄨󵄨 󵄨 2 𝑛 󵄨󵄨𝑥𝑛 (IKI) − 0󵄨󵄨󵄨 2√𝑟 − 4 𝑟 )( )] 󵄨󵄨 󵄨󵄨 = ∏ [( √𝑟 + 𝑟 󵄨󵄨𝑥𝑛 (KT) − 0󵄨󵄨 𝑟=25 2√𝑟 − 3 𝑛

= [ ∏ (1 − 𝑟=25

3 (𝑟 − 𝑟1/2 ) 2𝑟3/2 − 3𝑟1/2 − 𝑟

(69)

2

For the comparison of implicit Kirk-Multistep iteration 𝑥𝑛 (IKM) and Kirk-Multistep-SP iteration 𝑥𝑛 (KMSP), we have 󵄨 󵄨󵄨 𝑘 𝑛 󵄨󵄨𝑥𝑛 (IKM) − 0󵄨󵄨󵄨 𝑟 2√𝑟 − 4 = ) ( )] [( ∏ 󵄨󵄨󵄨𝑥𝑛 (KMSP) − 0󵄨󵄨󵄨 √𝑟 + 𝑟 󵄨 󵄨 𝑟=25 2√𝑟 − 3 (73) 𝑘 𝑛 3 (𝑟 − 𝑟1/2 ) = [ ∏ (1 − 3/2 )] 2𝑟 − 3𝑟1/2 − 𝑟 𝑟=25 with

)]

𝑛

0 ≤ [ lim ∏ (1 − 𝑛→∞ 𝑟=25

with

3 (𝑟 − 𝑟1/2 ) 2𝑟3/2 − 3𝑟1/2 − 𝑟

𝑛

0 ≤ [ lim ∏ (1 − 𝑛→∞ 𝑟=25

2

3 (𝑟 − 𝑟1/2 )

𝑛

2𝑟3/2 − 3𝑟1/2 − 𝑟

𝑛 1 ≤ [ lim ∏ (1 − )] 𝑛→∞ 𝑟 𝑟=25

1 ≤ [ lim ∏ (1 − )] 𝑛→∞ 𝑟 𝑟=25

)]

(74)

= 0. Remark 15. The implicit Kirk-Ishikawa iteration 𝑥𝑛 (IKI) converges faster than the Kirk-Thianwan iteration 𝑥𝑛 (KT). For the comparison of implicit Kirk-Noor iteration 𝑥𝑛 (IKN) and Kirk-SP iteration 𝑥𝑛 (KSP), we have 󵄨󵄨 󵄨 3 𝑛 󵄨󵄨𝑥𝑛 (IKN) − 0󵄨󵄨󵄨 2√𝑟 − 4 𝑟 = [( ) ( )] ∏ 󵄨󵄨󵄨𝑥𝑛 (KSP) − 0󵄨󵄨󵄨 √𝑟 + 𝑟 󵄨 𝑟=25 2√𝑟 − 3 󵄨 3 (𝑟 − 𝑟1/2 ) 2𝑟3/2 − 3𝑟1/2 − 𝑟

3

= 0. Remark 17. The implicit Kirk-Multistep iteration 𝑥𝑛 (IKM) has better convergence rate than the Kirk-Multistep-SP iteration 𝑥𝑛 (KMSP) for 𝑘 ≥ 4.

24 25 24 2 𝑛−2 𝑛−1 2 ⋅ ⋅⋅⋅ ⋅ ) = ( lim ) 𝑛→∞ 25 26 𝑛→∞ 𝑛 𝑛−1 𝑛

= ( lim

𝑟=25

𝑘

24 25 24 𝑘 𝑛−2 𝑛−1 𝑘 ⋅ ⋅⋅⋅ ⋅ ) = ( lim ) 𝑛→∞ 25 26 𝑛→∞ 𝑛 𝑛−1 𝑛

(70)

= [ ∏ (1 −

)]

= ( lim

2

𝑛

𝑘

(71)

)]

with

3. Conclusion We have established and proved strong convergence and Tstability results for implicit Kirk-Multistep, implicit KirkNoor, implicit Kirk-Ishikawa, and implicit Kirk-Mann iterative schemes of fixed points with contractive-type operators in normed linear spaces. These iterative schemes have better convergence rate when compared with other iterative schemes, namely, multistep Kirk-SP iteration, Kirk-Multistep scheme, Kirk-SP iteration, Kirk-Thianwan scheme, KirkNoor scheme, Kirk-Ishikawa scheme, Kirk-Mann scheme, implicit Noor iteration, implicit Ishikawa iteration, implicit Mann iteration, and many more iterative schemes of fixed point in the literature.

Disclosure 𝑛

0 ≤ [ lim ∏ (1 − 𝑛→∞ 𝑟=25

2𝑟3/2 − 3𝑟1/2 − 𝑟

𝑛 1 ≤ [ lim ∏ (1 − )] 𝑛→∞ 𝑟 𝑟=25

= ( lim

3

3 (𝑟 − 𝑟1/2 )

Authors agreed to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

)]

3

(72) 3

24 25 24 𝑛−2 𝑛−1 ⋅ ⋅⋅⋅ ⋅ ) = ( lim ) 𝑛→∞ 26 𝑛−1 𝑛 𝑛

3

𝑛→∞ 25

= 0. Remark 16. The implicit Kirk-Noor iteration 𝑥𝑛 (IKN) has better convergence rate than the Kirk-SP iteration 𝑥𝑛 (KSP).

Competing Interests The authors hereby declare that there are no competing interests.

References [1] H. Akewe, G. A. Okeke, and A. F. Olayiwola, “Strong convergence and stability of Kirk-multistep-type iterative schemes for

10

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10] [11] [12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

International Journal of Analysis contractive-type operators,” Fixed Point Theory and Applications, vol. 2014, no. 1, article 45, 2014. V. Berinde, “On the stability of some fixed point procedures,” Buletinul Stiintific al Universitatii Baia Mare, Seria B, Matematica-Informatica, vol. 18, no. 1, pp. 7–14, 2002. V. Berinde, “On the convergence of the Ishikawa iteration in the class of quasi contractive operators,” Acta Mathematica Universitatis Comenianae, vol. 73, no. 1, pp. 119–126, 2004. V. Berinde, “Stability of Picard iteration for contractive mappings satisfying an implicit relation,” Carpathian Journal of Mathematics, vol. 27, no. 1, pp. 13–23, 2011. R. Chugh, P. Malik, and V. Kumar, “On a new faster implicit fixed point iterative scheme in convex metric spaces,” Journal of Function Spaces, vol. 2015, Article ID 905834, 11 pages, 2015. ´ c, “Convergence theorems for a sequence of Ishikawa L. Ciri´ iterations for nonlinear quasi-contractive mappings,” Indian Journal of Pure and Applied Mathematics, vol. 30, no. 4, pp. 425– 433, 1999. ´ c, A. Rafiq, S. Radenović, M. Rajović, and J. S. Ume, L. Ciri´ “On Mann implicit iterations for strongly accretive and strongly pseudo-contractive mappings,” Applied Mathematics and Computation, vol. 198, no. 1, pp. 128–137, 2008. M. O. Olatinwo, “Some stability results for two hybrid fixed point iterative algorithms in normed linear space,” Matematichki Vesnik, vol. 61, no. 4, pp. 247–256, 2009. M. O. Olatinwo, “Stability results for some fixed point iterative processes in convex metric space,” International Journal of Engineering, vol. 9, pp. 103–106, 2011. S. Banach, Theorie des Operations Lineires, Warsaw Publishing House, Warsaw, Poland, 1932. W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953. S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, no. 1, pp. 147– 150, 1974. M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217–229, 2000. B. E. Rhoades and S. M. Soltuz, “The equivalence between Mann-Ishikawa iterations and multistep iteration,” Nonlinear Analysis: Theory, Methods & Applications, vol. 58, no. 1-2, pp. 219–228, 2004. S. Thianwan, “Common fixed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 688–695, 2009. W. Phuengrattana and S. Suantai, “On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval,” Journal of Computational and Applied Mathematics, vol. 235, no. 9, pp. 3006–3014, 2011. F. Gürsoy, V. Karakaya, and B. E. Rhoades, “Data dependence results of new multi-step and Siterative schemes for contractivelike operators,” Fixed Point Theory and Applications, vol. 2013, article 76, 2013. V. Berinde, “Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators,” Fixed Point Theory and Applications, no. 2, pp. 97–105, 2004. Z. Xue and F. Zhang, “The convergence of implicit Mann and Ishikawa iterations for weak generalized 𝜑-hemicontractive mappings in real Banach spaces,” Journal of Inequalities and Applications, vol. 2013, article 231, 2013.

[20] M. O. Osilike, “Stability results for fixed point iteration procedures,” Journal of the Nigerian Mathematical Society, vol. 14, pp. 17–29, 1995. [21] C. O. Imoru and M. O. Olatinwo, “On the stability of Picard and Mann iteration processes,” Carpathian Journal of Mathematics, vol. 19, no. 2, pp. 155–160, 2003. [22] W. A. Kirk, “A fixed point theorem for mappings which do not increase distances,” The American Mathematical Monthly, vol. 72, pp. 1004–1006, 1965. [23] R. Chugh and V. Kumar, “Data dependence of noor and SP iterative schemes when dealing with quasi-contractive operators,” International Journal of Computer Applications, vol. 40, no. 15, pp. 41–46, 2012. [24] N. Hussain, R. Chugh, V. Kumar, and A. Rafiq, “On the rate of convergence of Kirk-type iterative schemes,” Journal of Applied Mathematics, vol. 2012, Article ID 526503, 22 pages, 2012. [25] T. Zamfirescu, “Fix point theorems in metric spaces,” Archiv der Mathematik, vol. 23, no. 1, pp. 292–298, 1972. [26] B. E. Rhoades, “Fixed point theorems and stability results for fixed point iteration procedures,” Indian Journal of Pure and Applied Mathematics, vol. 24, no. 11, pp. 691–703, 1993.

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