1.IntroductionandPreliminaries H.Zegeye andN.Shahzad

Volume 2013, Article ID 821737,7pages http://dx.doi.org/10.1155/2013/821737

Research Article

Approximation Analysis for a Common Fixed Point of

Finite Family of Mappings Which Are Asymptotically 𝑘 -Strict Pseudocontractive in the Intermediate Sense

H. Zegeye

and N. Shahzad

1Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana

2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to N. Shahzad; [email protected] Received 16 February 2013; Accepted 18 April 2013

Academic Editor: Luigi Muglia

Copyright © 2013 H. Zegeye and N. Shahzad. 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 an iterative process which converges strongly to a common fixed point of a finite family of uniformly continuous asymptotically𝑘_𝑖-strict pseudocontractive mappings in the intermediate sense for𝑖 = 1, 2, . . . , 𝑁. The projection of𝑥₀onto the intersection of closed convex sets𝐶_𝑛and𝑄_𝑛for each𝑛 ≥ 1is not required. Moreover, the restriction that the interior of common fixed points is nonempty is not required. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

1. Introduction and Preliminaries

Let 𝐶be a nonempty subset of a real Hilbert space 𝐻. A mapping𝑇 : 𝐶 → 𝐻is called Lipschitzianif there exists 𝐿 > 0such that‖𝑇𝑥 − 𝑇𝑦‖ ≤ 𝐿‖𝑥 − 𝑦‖, for all𝑥,𝑦 ∈ 𝐶.

If𝐿 = 1, then 𝑇is called nonexpansive,and if𝐿 ∈ [0, 1), 𝑇is called contraction.𝑇is called uniformly𝐿-Lipschitzian if there exists 𝐿 > 0such that ‖𝑇^𝑛𝑥 − 𝑇^𝑛𝑦‖ ≤ 𝐿‖𝑥 − 𝑦‖, for all 𝑥, 𝑦 ∈ 𝐶 and all 𝑛 ≥ 1. Clearly, every con- traction mapping is nonexpansive and every nonexpansive mapping is uniformly𝐿-Lipschitzian with𝐿 = 1and hence Lipschitzian.

A mapping 𝑇 : 𝐶 → 𝐻is said to be asymptotically nonexpansiveif there exists a sequence{𝛾_𝑛} ⊂ [0, ∞)with 𝛾_𝑛 → 0such that‖𝑇^𝑛𝑥 − 𝑇^𝑛𝑦‖ ≤ (1 + 𝛾_𝑛)‖𝑥 − 𝑦‖for all integers𝑛 ≥ 1and all𝑥, 𝑦 ∈ 𝐶.𝑇is said to be asymptotically nonexpansive in the intermediate senseif there exist sequences {𝛾_𝑛}, {𝑐_𝑛} ⊂ [0, ∞) with 𝛾_𝑛 → 0, 𝑐_𝑛 → 0 such that

‖𝑇^𝑛𝑥 − 𝑇^𝑛𝑦‖ ≤ (1 + 𝛾_𝑛)‖𝑥 − 𝑦‖ + 𝑐_𝑛 for all integers𝑛 ≥ 1 and all𝑥, 𝑦 ∈ 𝐶.

A mapping𝑇 : 𝐶 → 𝐻is said to be asymptotically𝑘- strict pseudocontractiveif there exist a constant𝑘 ∈ [0, 1)and a sequence{𝛾_𝑛} ⊂ [0, ∞)with𝛾_𝑛 → 0, as𝑛 → ∞, such that

󵄩󵄩󵄩󵄩𝑇^𝑛𝑥 − 𝑇^𝑛𝑦󵄩󵄩󵄩󵄩²

≤ (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩²

+ 𝑘󵄩󵄩󵄩󵄩(𝐼 − 𝑇^𝑛)𝑥 − (𝐼 − 𝑇^𝑛)𝑦󵄩󵄩󵄩󵄩², ∀𝑥, 𝑦 ∈ 𝐶.

(1)

The class of asymptotically𝑘-strict pseudocontractive map- pings which includes the class of asymptotically nonex- pansive, and hence the class of nonexpansive mappings was introduced by Liu [1] in 1996 (see, also [2]). Kim and Xu [3] proved that the fixed point set of asymptotically𝑘-strict pseudocontractions is closed and convex. Recall that a fixed point of a map𝑇 : 𝐶 → 𝐻is a set {𝑥 ∈ 𝐶 : 𝑇𝑥 = 𝑥}, and it is denoted by 𝐹(𝑇). In addition, it is noted in [3]

that every asymptotically𝑘-strict pseudocontractive mapping with sequence {𝛾_𝑛}is a uniformly 𝐿-Lipschitzian mapping with𝐿 :=sup{(𝑘 + √1 + (1 − 𝑘)𝛾_𝑛)/(1 − 𝑘) : 𝑛 ∈N}.

A mapping 𝑇 is said to be an asymptotically 𝑘-strict pseudocontractive in the intermediate senseif

lim sup

𝑛 → ∞ sup

𝑥,𝑦∈𝐶(󵄩󵄩󵄩󵄩𝑇^𝑛𝑥 − 𝑇^𝑛𝑦󵄩󵄩󵄩󵄩²− (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩²

−𝑘 󵄩󵄩󵄩󵄩(𝐼 − 𝑇^𝑛) 𝑥 − (𝐼 − 𝑇^𝑛) 𝑦󵄩󵄩󵄩󵄩) ≤ 0, (2)

where𝑘 ∈ [0, 1)and{𝛾_𝑛} ⊂ [0, ∞)such that𝛾_𝑛 → 0, as 𝑛 → ∞. If we put

𝑐_𝑛 := max{0, sup

𝑥,𝑦∈𝐶(󵄩󵄩󵄩󵄩𝑇^𝑛𝑥 − 𝑇^𝑛𝑦󵄩󵄩󵄩󵄩²− (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩²

−𝑘󵄩󵄩󵄩󵄩(𝐼 − 𝑇^𝑛) 𝑥 − (𝐼 − 𝑇^𝑛) 𝑦󵄩󵄩󵄩󵄩²) } .

(3)

It follows that𝑐_𝑛 → 0, as𝑛 → ∞, and (2) is reduced to the following:

󵄩󵄩󵄩󵄩𝑇^𝑛𝑥 − 𝑇^𝑛𝑦󵄩󵄩󵄩󵄩²

≤ (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩²

+ 𝑘󵄩󵄩󵄩󵄩(𝐼 − 𝑇^𝑛) 𝑥 − (𝐼 − 𝑇^𝑛) 𝑦󵄩󵄩󵄩󵄩²+ 𝑐_𝑛, ∀𝑥, 𝑦 ∈ 𝐶.

(4) We note that the class of asymptotically 𝑘-strict pseudo- contractive mappings is properly contained in a class of asymptotically 𝑘-strict pseudocontractive mapping in the intermediate sense (see examples in [4]). The class of asymptotically 𝑘-strict pseudocontractive mappings in the intermediate sense was introduced by Sahu et al. [4]. They obtained a weak convergence theorem of modified Mann iterative processes for these class of mappings. In [4], Sahu et al. established the following classical result.

Theorem SXY1 (see [4]). Let 𝐻 be a real Hilbert space, and let 𝐶 ⊂ 𝐻 be a nonempty closed and convex set.

Let 𝑇be a uniformly continuous and asymptotically𝑘-strict pseudocontractive mapping in the intermediate sense with sequences {𝛾_𝑛} and {𝑐_𝑛} such that 𝐹(𝑇) is nonempty and

∑^𝑛_𝑛=1𝛾_𝑛 < ∞. Assume that{𝛼_𝑛}is a sequence in (0, 1)such that0 < 𝛿 ≤ 𝛼_𝑛 ≤ 1 − 𝑘 − 𝛿 < 1and∑^𝑛_𝑛=1𝛼_𝑛𝑐_𝑛 < ∞. Let{𝑥_𝑛} be a sequence defined by𝑥₁= 𝑥 ∈ 𝐶and

𝑥_𝑛+1= (1 − 𝛼_𝑛) 𝑥_𝑛+ 𝛼_𝑛𝑇^𝑛𝑥_𝑛, 𝑛 ≥ 1. (5) Then,{𝑥_𝑛}converges weakly to a fixed point of𝑇.

But it is worth mentioning that the convergence obtained is a weak convergence. Furthermore, we observe from the proof of Theorem SXY1 that if, in addition,𝐶or𝑇has some compactness assumption, we obtain that the sequence{𝑥_𝑛} given by (5) converges strongly to a fixed point of𝑇.

Attempts to modify the Mann iteration method (5) so that strong convergence is guaranteed, without compactness assumption on 𝐶or 𝑇, have recently been made. Sahu et al. [4] established the following hybrid Mann algorithm for approximating fixed points of asymptotically 𝑘-strict pseudocontractive mappings in the intermediate sense.

Theorem SXY2 (see [4]). Let𝐻be a real Hilbert space, and let𝐶 ⊂ 𝐻be a nonempty, closed, and convex set. Let𝑇be a uniformly continuous asymptotically𝑘-strict pseudocontrac- tive mapping in the intermediate sense with sequences{𝛾_𝑛}and {𝑐_𝑛}such that𝐹(𝑇)is nonempty and bounded. Assume that{𝛼_𝑛} is a sequence in[0, 1]such that0 < 𝛿 ≤ 𝛼_𝑛 ≤ 1 − 𝑘, for all 𝑛 ∈ 𝑁. Let{𝑥_𝑛}be a sequence in𝐶defined by𝑢 = 𝑥₁∈ 𝐶and

𝑦_𝑛= (1 − 𝛼_𝑛) 𝑥_𝑛+ 𝛼_𝑛𝑇^𝑛𝑥_𝑛, 𝑛 ≥ 1, 𝐶_𝑛 = {𝑧 ∈ 𝐶 : 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑧󵄩󵄩󵄩󵄩²≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑧󵄩󵄩󵄩󵄩²+ 𝜃_𝑛} ,

𝑄_𝑛= {𝑧 ∈ 𝐶 : ⟨𝑥_𝑛− 𝑧, 𝑢 − 𝑥_𝑛⟩ ≥ 0} , 𝑥_𝑛+1= 𝑃_{𝐶𝑛∩ 𝑄𝑛}(𝑢) , 𝑛 ≥ 1,

(6)

where𝜃_𝑛= 𝑐_𝑛+ 𝛾_𝑛𝛿_𝑛and𝛿_𝑛 =sup{‖𝑥_𝑛− 𝑧‖ : 𝑧 ∈ 𝐹(𝑇)} < ∞.

Then,{𝑥_𝑛}converges strongly to𝑃_𝐹(𝑇)(𝑢).

Recently, Hu and Cai [5] studied the strong convergence of the modified Mann iteration process (5) for a finite family of asymptotically 𝑘-strict pseudocontractive mappings in the intermediate sense. More precisely, they obtained the following theorem.

Theorem HC (see [5]). Let 𝐻 be a real Hilbert space, let 𝐶 ⊂ 𝐻 be a nonempty, closed, and convex set. Let 𝑇_𝑖 : 𝐶 → 𝐶 be uniformly continuous asymptotically 𝑘_𝑖-strict pseudocontractive mappings in the intermediate sense for some 0 ≤ 𝑘_𝑖 < 1 with sequences {𝛾_𝑛,𝑖} and {𝑐_𝑛,𝑖} such that lim_{𝑛 → ∞}𝛾_𝑛,𝑖 = 0andlim_{𝑛 → ∞}𝑐_𝑛,𝑖 = 0for 𝑖 = 1,2,. . .,𝑁.

Let𝑘 = max{𝑘_𝑖 : 1 ≤ 𝑖 ≤ 𝑁},𝛾_𝑛 = max{𝛾_𝑛,𝑖 : 1 ≤ 𝑖 ≤ 𝑁}, and𝑐_𝑛 =max{𝑐_𝑛,𝑖: 1 ≤ 𝑖 ≤ 𝑁}. Assume that𝐹 := ⋂^𝑁_𝑛=1𝐹(𝑇_𝑖) is nonempty and bounded. Let{𝛽_𝑛}be a sequence in[0, 1]such that0 < 𝛿 ≤ 𝛽_𝑛 ≤ 1 − 𝑘for all𝑛 ∈ 𝑁. Let{𝑥_𝑛}be a sequence defined by𝑥₀∈ 𝐶and

𝑦_𝑛= (1 − 𝛽_𝑛) 𝑥_𝑛+ 𝛽_𝑛𝑇_𝑖(𝑛)^𝑘(𝑛)𝑥_𝑛, 𝑛 ≥ 1, 𝐶_𝑛 = {𝑧 ∈ 𝐶 : 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑧󵄩󵄩󵄩󵄩²≤ 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑧󵄩󵄩󵄩󵄩²+ 𝜃_𝑛} ,

𝑄_𝑛= {𝑧 ∈ 𝐶 : ⟨𝑥_𝑛− 𝑧, 𝑥₀− 𝑥_𝑛⟩ ≥ 0} , 𝑥_𝑛+1= 𝑃_{𝐶𝑛∩ 𝑄𝑛}(𝑥₀) , 𝑛 ≥ 1,

(7)

where𝜃_𝑛 = 𝛾_𝑘(𝑛)𝜌_𝑛² + 𝑐_𝑘(𝑛) → 0, as 𝑛 → ∞, for 𝜌_𝑛 = sup{‖𝑥_𝑛 − 𝑧‖ : 𝑧 ∈ 𝐹} < ∞,𝑛 = (ℎ − 1)𝑁 + 𝑖, where 𝑖 = 𝑖(𝑛) ∈ {1, 2, . . . , 𝑁},ℎ = ℎ(𝑛) ≥ 1a positive integer such thatℎ(𝑛) → ∞, as𝑛 → ∞. Then,{𝑥_𝑛}converges strongly to 𝑃_𝐹(𝑥₀).

But we observe that the iterative algorithms (6) and (7) generate a sequence {𝑥_𝑛} by projecting 𝑥₀ onto the intersection of closed convex sets𝐶_𝑛 and𝑄_𝑛for each𝑛 ≥ 1 which is not easy to compute.

Attempts to remove projection mapping onto the inter- section of closed convex sets𝐶_𝑛and𝑄_𝑛for each𝑛 ≥ 1have recently been made. In [6], Zegeye et al. studied the strong convergence of the modified Mann iteration process for the class of asymptotically𝑘-strict pseudocontractive mappings

in the intermediate sense. More precisely, they proved the following theorem.

Theorem ZRS (see [6]). Let 𝐶be a nonempty, closed, and convex subset of a real Hilbert space𝐻, and let𝑇 : 𝐶 → 𝐶 be a uniformly 𝐿-Lipschitzian and asymptotically 𝑘-strict pseudocontractive mapping in the intermediate sense with sequences{𝛾_𝑛} ⊂ [0, ∞)and{𝑐_𝑛} ⊂ [0, ∞). Assume that the interior of𝐹(𝑇)is nonempty. Let{𝑥_𝑛}be a sequence defined by 𝑥₁= 𝑥 ∈ 𝐶and

𝑦_𝑛 = 𝛽_𝑛𝑇^𝑛𝑥_𝑛+ (1 − 𝛽_𝑛) 𝑥_𝑛,

𝑥_𝑛+1= 𝛼_𝑛𝑇^𝑛𝑦_𝑛+ (1 − 𝛼_𝑛) 𝑥_𝑛, 𝑛 ≥ 1, (8) where {𝛼_𝑛} and {𝛽_𝑛} satisfy certain conditions. Then, {𝑥_𝑛} converges strongly to a fixed point of𝑇.

But it is worth to mention that the assumptioninterior of 𝐹(𝑇)is nonemptyis severe restriction.

It is our purpose, in this paper, to construct an iteration scheme which converges strongly to a common fixed point of a finite family of uniformly continuous asymptotically𝑘_𝑖- strict pseudocontractive mappings in the intermediate sense for𝑖 = 1,2,. . .,𝑁. The projection of𝑥₀onto the intersection of closed convex sets 𝐶_𝑛 and 𝑄_𝑛 for each 𝑛 ≥ 1 is not required. Furthermore, the restriction that the interior of 𝐹(𝑇) is nonempty is not required. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

In order to prove our results, we need the following lemmas.

Lemma 1. Let𝐻be a real Hilbert space. Then, for any given𝑥, 𝑦 ∈ 𝐻, the following inequality holds:

󵄩󵄩󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩󵄩²≤ ‖𝑥‖²+ 2 ⟨𝑦, 𝑥 + 𝑦⟩ . (9) Lemma 2 (see [7]). Let H be a real Hilbert space. Then, for all 𝑥,𝑦 ∈ 𝐻and𝛼_𝑖, ∈ [0, 1]for𝑖 = 1,2,. . .,𝑛such that𝛼₀+ 𝛼₁+ ⋅ ⋅ ⋅ + 𝛼_𝑛= 1, the following equality holds:

󵄩󵄩󵄩󵄩𝛼⁰𝑥₀+ 𝛼₁𝑥₁+ ⋅ ⋅ ⋅ + 𝛼_𝑛𝑥_𝑛󵄩󵄩󵄩󵄩²

=∑^𝑛

𝑖=0

𝛼_𝑖󵄩󵄩󵄩󵄩𝑥^𝑖󵄩󵄩󵄩󵄩²− ∑

0≤𝑖,𝑗≤𝑛

𝛼_𝑖𝛼_𝑗󵄩󵄩󵄩󵄩󵄩𝑥^𝑖− 𝑥_𝑗󵄩󵄩󵄩󵄩󵄩². (10) Lemma 3 (see [8]). Let{𝑎_𝑛}be sequences of real numbers such that there exists a subsequence{𝑛_𝑖}of{𝑛}such that𝑎_𝑛𝑖 < 𝑎_𝑛𝑖+1 for all𝑖 ∈ 𝑁. Then, there exists a nondecreasing sequence {𝑚_𝑘} ⊂ 𝑁such that𝑚_𝑘 → ∞, and the following properties are satisfied by all (sufficiently large) numbers𝑘 ∈ 𝑁:

𝑎_𝑚𝑘≤ 𝑎_𝑚𝑘+1, 𝑎_𝑘≤ 𝑎_𝑚𝑘+1. (11) In fact,𝑚_𝑘=max{𝑗 ≤ 𝑘 : 𝑎_𝑗 < 𝑎_𝑗+1}.

Lemma 4 (see [9]). Let{𝑎_𝑛}be a sequence of nonnegative real numbers satisfying the following relation:

𝑎_𝑛+1≤ (1 − 𝛼_𝑛) 𝑎_𝑛+ 𝛼_𝑛𝛿_𝑛, 𝑛 ≥ 𝑛₀, (12)

where{𝛼_𝑛} ⊂ (0, 1)and{𝛿_𝑛} ⊂ 𝑅satisfying the following condi- tions:lim_{𝑛 → ∞}𝛼_𝑛 = 0,∑^∞_𝑛=1𝛼_𝑛 = ∞andlim sup_{𝑛 → ∞}𝛿_𝑛 ≤ 0.

Then,lim_{𝑛 → ∞}𝑎_𝑛= 0.

Lemma 5 (see [4]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, and let 𝑇 : 𝐶 → 𝐶 be a continuous asymptotically𝑘-strict pseudocontractive mapping in the intermediate sense. Then,𝐹(𝑇)is closed and convex.

Lemma 6 (see [4]). Let𝐶be a nonempty closed convex subset of a Hilbert space 𝐻, and let𝑇 : 𝐶 → 𝐶 be a uniformly continuous asymptotically𝑘-strict pseudocontractive mapping in the intermediate sense. Let{𝑥_𝑛}be a sequence in𝐶such that

‖𝑥_𝑛− 𝑇^𝑛𝑥_𝑛‖ → 0and‖𝑥_𝑛− 𝑥_𝑛+1‖ → 0, as𝑛 → ∞. Then,

‖𝑥_𝑛− 𝑇𝑥_𝑛‖ → 0, as𝑛 → ∞.

Lemma 7 (see [4]). Let𝐶be a nonempty closed convex subset of a Hilbert space 𝐻, and 𝑇 : 𝐶 → 𝐶 be a continuous asymptotically𝑘-strict pseudocontractive mapping in the inter- mediate sense. Then if {𝑥_𝑛} is a sequence in𝐶such that 𝑥_𝑛 converges weakly𝑥 ∈ 𝐶andlim sup_{𝑚 → ∞}lim sup_{𝑛 → ∞}‖𝑥_𝑛− 𝑇^𝑚𝑥_𝑛‖ = 0, then(𝐼 − 𝑇)𝑥 = 0.

Lemma 8 (see [10]). Let 𝐶 be a nonempty closed, convex subset of a Hilbert space𝐻and let𝑃_𝐶be the metric projection mapping from𝐻onto𝐶. Given𝑧 = 𝑃_𝐶𝑥if and only if⟨𝑦 − 𝑧, 𝑥 − 𝑧⟩ ≤ 0, for all𝑦 ∈ 𝐶.

2. Main Result

Theorem 9. Let𝐶be a nonempty, closed and convex subset of a real Hilbert space𝐻. Let𝑇_𝑖: 𝐶 → 𝐶be uniformly continuous asymptotically 𝑘_𝑖-strict pseudocontractive mappings in the intermediate sense for some0 ≤ 𝑘_𝑖 < 1with sequences{𝛾_𝑛,𝑖} and{𝑐_𝑛,𝑖}, for𝑖 = 1,2,. . .,𝑁. Assume that𝐹 := ⋂^𝑁_𝑖=1𝐹(𝑇_𝑖)is nonempty. Let{𝑥_𝑛}be a sequence generated by

𝑥₀= 𝑤 ∈ 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,

𝑦_𝑛= 𝛽_𝑛,0𝑥_𝑛+∑^𝑁

𝑖=1𝛽_𝑛,𝑖𝑇_𝑖^𝑛𝑥_𝑛, 𝑥_𝑛+1= 𝛼_𝑛𝑤 + (1 − 𝛼_𝑛) 𝑦_𝑛,

(13)

where0 < 𝛼_𝑛 ≤ 𝑎₀ < 1such thatlim_{𝑛 → ∞}𝛼_𝑛 = 0,∑^∞_𝑛=1𝛼_𝑛 =

∞,lim_{𝑛 → ∞}[𝛾_𝑛,𝑖/𝛼_𝑛] = 0 = lim_{𝑛 → ∞}[𝑐_𝑛,𝑖/𝛼_𝑛], and0 < 𝛿 ≤ 𝛽_𝑛,𝑖≤ 1−𝑘−𝛿 < 1satisfying𝛽_𝑛,0+𝛽_𝑛,1+⋅ ⋅ ⋅+𝛽_𝑛,𝑁= 1for each 𝑛 ≥ 1and𝑘 :=max{𝑘_𝑖 : 𝑖 = 1, 2, . . . , 𝑁}. Then,{𝑥_𝑛}converges strongly to an element of𝐹.

Proof. Let𝑥^∗= 𝑃_𝐹𝑤. Let𝛾_𝑛 :=max{𝛾_𝑛,𝑖: 𝑖 = 1, 2, . . . , 𝑁}and 𝑐_𝑛 := max{𝑐_𝑛,𝑖 : 𝑖 = 1, 2, . . . , 𝑁}. Then, from (13), Lemma2,

and asymptotically𝑘_𝑖-strict pseudocontractiveness of𝑇_𝑖, for each𝑖 ∈ {1, 2, . . . , 𝑁}, we get that

󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²

=󵄩󵄩󵄩󵄩

󵄩󵄩󵄩󵄩󵄩󵄩𝛽_𝑛,0𝑥_𝑛+∑^𝑁

𝑖=1

𝛽_𝑛,𝑖𝑇_𝑖^𝑛𝑥_𝑛− 𝑥^∗󵄩󵄩󵄩󵄩

󵄩󵄩󵄩󵄩󵄩󵄩

2

≤ 𝛽_𝑛,0󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²+∑^𝑁

𝑖=1

𝛽_𝑛,𝑖󵄩󵄩󵄩󵄩𝑇^𝑖𝑛𝑥_𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²

−∑^𝑁

𝑖=1

𝛽_𝑛,0𝛽_𝑛,𝑖󵄩󵄩󵄩󵄩𝑇𝑖^𝑛𝑥_𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩²

≤ 𝛽_𝑛,0󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²+ (1 − 𝛽_𝑛,0) (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩² +∑^𝑁

𝑖=1

𝛽_𝑛,𝑖𝑘󵄩󵄩󵄩󵄩𝑇𝑖^𝑛𝑥_𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩²+ (1 − 𝛽_𝑛,0) 𝑐_𝑛

−∑^𝑁

𝑖=1

𝛽_𝑛,0𝛽_𝑛,𝑖󵄩󵄩󵄩󵄩𝑇^𝑖𝑛𝑥_𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩²

≤ (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²

−∑^𝑁

𝑖=1

𝛽_𝑛,𝑖(1 − 𝑘 − 𝛿) 󵄩󵄩󵄩󵄩𝑇𝑖^𝑛𝑥_𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩²+ (1 − 𝛽_𝑛,0) 𝑐_𝑛

≤(1+𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛−𝑥^∗󵄩󵄩󵄩󵄩²−𝛿²∑^𝑁

𝑖=1󵄩󵄩󵄩󵄩𝑇^𝑖𝑛𝑥_𝑛−𝑥_𝑛󵄩󵄩󵄩󵄩²+(1−𝛽_𝑛,0) 𝑐_𝑛, (14)

󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑥^∗󵄩󵄩󵄩󵄩²

= 󵄩󵄩󵄩󵄩𝛼^𝑛𝑤 + (1 − 𝛼_𝑛) 𝑦_𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²

≤ 𝛼_𝑛󵄩󵄩󵄩󵄩𝑤 − 𝑥^∗󵄩󵄩󵄩󵄩²+ (1 − 𝛼_𝑛) 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²

≤ 𝛼_𝑛󵄩󵄩󵄩󵄩𝑤 − 𝑥^∗󵄩󵄩󵄩󵄩²+ (1 − 𝛼_𝑛)

× [ (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²− 𝛿²

×∑^𝑁

𝑖=1󵄩󵄩󵄩󵄩𝑇^𝑖𝑛𝑥_𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩²+ (1 − 𝛽_𝑛,0) 𝑐_𝑛]

≤ 𝛼_𝑛󵄩󵄩󵄩󵄩𝑤 − 𝑥^∗󵄩󵄩󵄩󵄩²+ (1 − 𝛼_𝑛) (1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²

− (1 − 𝛼_𝑛) 𝛿²∑^𝑁

𝑖=1󵄩󵄩󵄩󵄩𝑇^𝑖𝑛𝑥_𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩²+ (1 − 𝛼_𝑛) (1 − 𝛽_𝑛,0) 𝑐_𝑛

≤ 𝛼_𝑛[󵄩󵄩󵄩󵄩𝑤 − 𝑥^∗󵄩󵄩󵄩󵄩²+ 1] + (1 − 𝛼_𝑛(1 − 𝜖)) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²

− (1 − 𝛼_𝑛) 𝛿²∑^𝑁

𝑖=1󵄩󵄩󵄩󵄩𝑇^𝑖𝑛𝑥_𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩²

≤ 𝛼_𝑛[󵄩󵄩󵄩󵄩𝑤 − 𝑥^∗󵄩󵄩󵄩󵄩²+ 1] + (1 − 𝛼_𝑛(1 − 𝜖)) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩² (15)

since there exists𝑁₀ > 0such that𝛾_𝑛/𝛼_𝑛 ≤ 𝜖and𝑐_𝑛/𝛼_𝑛 ≤ 1 for all𝑛 ≥ 𝑁₀and for some𝜖 > 0satisfying(1 − 𝜖)𝛼_𝑛 ≤ 1.

Thus, by induction,

󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑥^∗󵄩󵄩󵄩󵄩²

≤max{(1 − 𝜖)⁻¹[󵄩󵄩󵄩󵄩𝑤 − 𝑥^∗󵄩󵄩󵄩󵄩²+ 1] , 󵄩󵄩󵄩󵄩𝑥⁰− 𝑥^∗󵄩󵄩󵄩󵄩²} ,

∀𝑛 ≥ 𝑁₀, (16)

which implies that {𝑥_𝑛}, and hence {𝑇_𝑖^𝑛𝑥_𝑛} and {𝑦_𝑛} are bounded. Moreover, from (13), (14) and Lemma1, we obtain that

󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑥^∗󵄩󵄩󵄩󵄩²

= 󵄩󵄩󵄩󵄩𝛼^𝑛(𝑤 − 𝑥^∗) + (1 − 𝛼_𝑛) (𝑦_𝑛− 𝑥^∗)󵄩󵄩󵄩󵄩²

≤ (1 − 𝛼_𝑛) 󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²+ 2𝛼_𝑛⟨𝑤 − 𝑥^∗, 𝑥_𝑛+1− 𝑥^∗⟩

≤ (1 − 𝛼_𝑛) [(1 + 𝛾_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²− 𝛿²∑^𝑁

𝑖=1󵄩󵄩󵄩󵄩𝑇^𝑖𝑛𝑥_𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩² +(1−𝛽_𝑛,0) 𝑐_𝑛]+2𝛼_𝑛⟨𝑤−𝑥^∗, 𝑥_𝑛+1−𝑥^∗⟩

≤ (1 − 𝛼_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²+ (1 − 𝛼_𝑛) 𝛾_𝑛󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²

− 𝛿²(1 − 𝛼_𝑛)∑^𝑁

𝑖=1󵄩󵄩󵄩󵄩𝑇^𝑖𝑛𝑥_𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩²+ (1 − 𝛼_𝑛) (1 − 𝛽_𝑛,0) 𝑐_𝑛 + 2𝛼_𝑛⟨𝑤 − 𝑥^∗, 𝑥_𝑛+1− 𝑥^∗⟩

≤ (1 − 𝛼_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²+ 2𝛼_𝑛⟨𝑤 − 𝑥^∗, 𝑥_𝑛+1− 𝑥^∗⟩ + (𝛾_𝑛+ 𝑐_𝑛) 𝑀 − 𝛿²(1 − 𝑎₀)∑^𝑁

𝑖=1󵄩󵄩󵄩󵄩𝑇𝑖^𝑛𝑥_𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩² (17)

≤ (1 − 𝛼_𝑛) 󵄩󵄩󵄩󵄩𝑥^𝑛− 𝑥^∗󵄩󵄩󵄩󵄩²+ 2𝛼_𝑛⟨𝑤 − 𝑥^∗, 𝑥_𝑛+1− 𝑥^∗⟩

+ (𝛾_𝑛+ 𝑐_𝑛) 𝑀 (18)

for some𝑀 > 0and for all𝑛 ∈ 𝑁.

Now, following the method of proof of Lemma 3.2 of Maing´e [8], we consider two cases.

Case 1. Suppose that there exists𝑛₀∈ 𝑁such that{‖𝑥_𝑛−𝑥^∗‖}

is nonincreasing for all𝑛 ≥ 𝑛₀. In this situation,{‖𝑥_𝑛− 𝑥^∗‖)}

is convergent. Then, from (17), we have that

𝑥_𝑛− 𝑇_𝑖^𝑛𝑥_𝑛 󳨀→ 0, as𝑛 󳨀→ ∞, (19) for𝑖 = 1,2,. . .,𝑁. Moreover, from (13) and (19) and the fact that𝛼_𝑛 → 0, we get that

󵄩󵄩󵄩󵄩𝑦^𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩 ≤∑^𝑁

𝑖=1

𝛽_𝑛,𝑖󵄩󵄩󵄩󵄩𝑇^𝑖𝑛𝑥_𝑛− 𝑥_𝑛󵄩󵄩󵄩󵄩 󳨀→ 0, 𝑥_𝑛+1− 𝑦_𝑛= 𝛼_𝑛(𝑤 − 𝑦_𝑛) 󳨀→ 0,

(20)

as𝑛 → ∞, and hence

󵄩󵄩󵄩󵄩𝑥^𝑛+1− 𝑥_𝑛󵄩󵄩󵄩󵄩 󳨀→ 0, as𝑛 󳨀→ ∞. (21) Furthermore, from (19), (21), and Lemma6, we obtain that

𝑥_𝑛− 𝑇_𝑖𝑥_𝑛 󳨀→ 0, as𝑛 󳨀→ ∞. (22) Let {𝑥_𝑛𝑘+1} be a subsequence of {𝑥_𝑛+1} such that lim sup_{𝑛 → ∞}⟨𝑤−𝑥^∗, 𝑥_𝑛+1−𝑥^∗⟩ = lim_{𝑘 → ∞}⟨𝑤−𝑥^∗, 𝑥_𝑛𝑘+1−𝑥^∗⟩ and{𝑥_𝑛𝑘+1}converges weakly toV. Then, from (21), we also get that {𝑥_𝑛𝑘} converges weakly to V. Moreover, since 𝑇_𝑖 is uniformly continuous and ‖𝑥_𝑛 − 𝑇_𝑖𝑥_𝑛‖ → 0, for all 𝑖 ∈ {1, 2, . . . , 𝑁}, we get that‖𝑥_𝑛− 𝑇_𝑖^𝑚𝑥_𝑛‖ → 0, as𝑛 → ∞, for all 𝑚 ∈ 𝑁. Therefore, by Lemma 7, we obtain that V∈ ⋂^𝑁_𝑖=1𝐹(𝑇_𝑖). Now, from Lemma8, we have that

lim sup

𝑛 → ∞ ⟨𝑤 − 𝑥^∗, 𝑥_𝑛+1− 𝑥^∗⟩ = ⟨𝑤 − 𝑥^∗,V− 𝑥^∗⟩ ≤ 0. (23) Then, from (18), (23), and Lemma4, we obtain that ‖𝑥_𝑛 − 𝑥^∗‖ → 0, as𝑛 → ∞. Consequently,𝑥_𝑛 → 𝑥^∗.

Case 2. Suppose that there exists a subsequence{𝑛_𝑖}of{𝑛}

such that

󵄩󵄩󵄩󵄩󵄩𝑥^𝑛𝑖− 𝑥^∗󵄩󵄩󵄩󵄩󵄩 <󵄩󵄩󵄩󵄩󵄩𝑥^𝑛𝑖+1− 𝑥^∗󵄩󵄩󵄩󵄩󵄩 (24) for all𝑖 ∈ 𝑁. Then, by Lemma3, there exist a nondecreasing sequence{𝑚_𝑘} ⊂ 𝑁such that𝑚_𝑘 → ∞,

󵄩󵄩󵄩󵄩󵄩𝑥^𝑚𝑘− 𝑥^∗󵄩󵄩󵄩󵄩󵄩 ≤󵄩󵄩󵄩󵄩󵄩𝑥^𝑚𝑘+1− 𝑥^∗󵄩󵄩󵄩󵄩󵄩 (25) and‖𝑥_𝑘− 𝑥^∗‖ ≤ ‖𝑥_𝑚𝑘+1− 𝑥^∗‖, for all𝑘 ∈ 𝑁. Then, from (17) and the fact that𝛼_𝑛 → 0, we have

𝛿²(1 − 𝑎₀)∑^𝑁

𝑖=1󵄩󵄩󵄩󵄩󵄩𝑥^𝑚𝑘− 𝑇_𝑖^𝑚𝑘𝑥_𝑚𝑘󵄩󵄩󵄩󵄩󵄩²

≤ 󵄩󵄩󵄩󵄩󵄩𝑥^𝑚𝑘− 𝑥^∗󵄩󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩󵄩𝑥^𝑚𝑘+1− 𝑥^∗󵄩󵄩󵄩󵄩󵄩²

+ 𝛼_𝑚𝑘󵄩󵄩󵄩󵄩󵄩𝑥^𝑚𝑘− 𝑥^∗󵄩󵄩󵄩󵄩󵄩²+ 2𝛼_𝑚𝑘⟨𝑤 − 𝑥^∗, 𝑥_𝑚𝑘+1− 𝑥^∗⟩ + (𝛾_𝑚𝑘+ 𝑐_𝑚𝑘) 𝑀 󳨀→ 0, as𝑘 󳨀→ ∞.

(26) Then, we get that𝑥_𝑚𝑘− 𝑇_𝑖^𝑚𝑘𝑥_𝑚𝑘 → 0, as𝑛 → ∞, for each 𝑖 = 1,2,. . .,𝑁. Thus, as in Case 1, we obtain that𝑥_𝑚𝑘−𝑦_𝑚𝑘 → 0and𝑥_𝑚𝑘+1− 𝑥_𝑚𝑘 → 0, as𝑘 → ∞and

lim sup

𝑘 → ∞ ⟨𝑤 − 𝑥^∗, 𝑥_𝑚𝑘+1− 𝑥^∗⟩ ≤ 0. (27) Now, from (18), we have that

󵄩󵄩󵄩󵄩󵄩𝑥^𝑚𝑘+1− 𝑥^∗󵄩󵄩󵄩󵄩󵄩²

≤ (1 − 𝛼_𝑚𝑘) 󵄩󵄩󵄩󵄩󵄩𝑥^𝑚𝑘− 𝑥^∗󵄩󵄩󵄩󵄩󵄩

+ 2𝛼_𝑚𝑘⟨𝑤 − 𝑥^∗, 𝑥_𝑚𝑘+1− 𝑥^∗⟩ + (𝛾_𝑚𝑘+ 𝑐_𝑚𝑘) 𝑀.

(28)

This implies that 𝛼_𝑚𝑘󵄩󵄩󵄩󵄩󵄩𝑥^𝑚𝑘− 𝑥^∗󵄩󵄩󵄩󵄩󵄩²

≤ 󵄩󵄩󵄩󵄩󵄩𝑥^𝑚𝑘− 𝑥^∗󵄩󵄩󵄩󵄩󵄩²− 󵄩󵄩󵄩󵄩󵄩𝑥^𝑚𝑘+1− 𝑥^∗󵄩󵄩󵄩󵄩󵄩²

+ 2𝛼_𝑚𝑘⟨𝑤 − 𝑥^∗, 𝑥_𝑚𝑘+1− 𝑥^∗⟩ + (𝛾_𝑚𝑘+ 𝑐_𝑚𝑘) 𝑀.

(29) Then, using (25), inequality (29) implies that

𝛼_𝑚𝑘󵄩󵄩󵄩󵄩󵄩𝑥^𝑚𝑘− 𝑥^∗󵄩󵄩󵄩󵄩󵄩²

≤ 2𝛼_𝑚𝑘⟨𝑤 − 𝑥^∗, 𝑥_𝑚𝑘+1− 𝑥^∗⟩ + (𝛾_𝑚𝑘+ 𝑐_𝑚𝑘) 𝑀.

(30)

In particular, since𝛼_𝑚𝑘 > 0, we get

󵄩󵄩󵄩󵄩󵄩𝑥^𝑚𝑘− 𝑥^∗󵄩󵄩󵄩󵄩󵄩²≤ 2 ⟨𝑤 − 𝑥^∗, 𝑥_𝑚𝑘+1− 𝑥^∗⟩ +𝛾_𝑚𝑘+ 𝑐_𝑚𝑘 𝛼_𝑚𝑘 . (31) Furthermore, using (27) and the fact that(𝛾_𝑚𝑘+ 𝑐_𝑚𝑘)/𝛼_𝑚𝑘 → 0, we obtain that‖𝑥_𝑚𝑘− 𝑥^∗‖ → 0, as𝑘 → ∞. This together with (28) give‖𝑥_𝑚𝑘+1−𝑥^∗‖ → 0, as𝑘 → ∞. But‖𝑥_𝑘−𝑥^∗‖ ≤

‖𝑥_𝑚𝑘+1 − 𝑥^∗‖for all𝑘 ∈ 𝑁. Thus we obtain that𝑥_𝑘 → 𝑥^∗. Therefore, from the above two cases, we can conclude that {𝑥_𝑛}converges strongly to an element of𝐹, and the proof is complete.

If in Theorem9, we assume that𝑁 = 1, then we get the following corollary.

Corollary 10. Let𝐶be a nonempty, closed, and convex subset of a real Hilbert space𝐻. Let 𝑇 : 𝐶 → 𝐶be a uniformly continuous asymptotically𝑘-strict pseudocontractive mapping in the intermediate sense for some0 ≤ 𝑘 < 1with sequences {𝛾_𝑛}and{𝑐_𝑛}. Assume that𝐹 := 𝐹(𝑇)is nonempty. Let{𝑥_𝑛}be a sequence generated by

𝑥₀= 𝑤 ∈ 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦, 𝑦_𝑛 = 𝛽_𝑛𝑥_𝑛+ (1 − 𝛽_𝑛) 𝑇^𝑛𝑥_𝑛,

𝑥_𝑛+1= 𝛼_𝑛𝑤 + (1 − 𝛼_𝑛) 𝑦_𝑛,

(32)

where0 < 𝛼_𝑛 ≤ 𝑎₀ < 1such thatlim_{𝑛 → ∞}𝛼_𝑛 = 0,∑^∞_𝑛=1𝛼_𝑛 =

∞,lim_{𝑛 → ∞}[𝛾_𝑛/𝛼_𝑛] = 0 =lim_{𝑛 → ∞}[𝑐_𝑛/𝛼_𝑛], and0 < 𝛿 ≤ 𝛽_𝑛≤ 1 − 𝑘 − 𝛿 < 1for each𝑛 ≥ 1. Then,{𝑥_𝑛}converges strongly to an element of𝐹.

If in Theorem9, we assume that each𝑇_𝑖is asymptotically 𝑘_𝑖-strict pseudocontractive mapping, then we get the follow- ing corollary.

Corollary 11. Let𝐶be a nonempty, closed, and convex subset of a real Hilbert space𝐻. Let𝑇_𝑖 : 𝐶 → 𝐶be asymptotically 𝑘_𝑖-strict pseudocontractive mappings for some0 ≤ 𝑘_𝑖 < 1

with sequences{𝛾_𝑛,𝑖}, for𝑖 = 1,2,. . .,𝑁. Assume that𝐹 :=

⋂^𝑁_𝑖=1𝐹(𝑇_𝑖)is nonempty. Let{𝑥_𝑛}be a sequence generated by 𝑥₀= 𝑤 ∈ 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,

𝑦_𝑛= 𝛽_𝑛,0𝑥_𝑛+∑^𝑁

𝑖=1

𝛽_𝑛,𝑖𝑇_𝑖^𝑛𝑥_𝑛,

𝑥_𝑛+1= 𝛼_𝑛𝑤 + (1 − 𝛼_𝑛) 𝑦_𝑛,

(33)

where0 < 𝛼_𝑛 ≤ 𝑎₀ < 1such thatlim_{𝑛 → ∞}𝛼_𝑛 = 0,∑^∞_𝑛=1𝛼_𝑛 =

∞,lim_{𝑛 → ∞}[𝛾_𝑛,𝑖/𝛼_𝑛] = 0, and0 < 𝛿 ≤ 𝛽_𝑛,𝑖 ≤ 1 − 𝑘 − 𝛿 < 1 satisfying𝛽_𝑛,0+ 𝛽_𝑛,1+ ⋅ ⋅ ⋅ + 𝛽_𝑛,𝑁 = 1for each𝑛 ≥ 1and𝑘 :=

max{𝑘_𝑖 : 𝑖 = 1, 2, . . . , 𝑁}. Then,{𝑥_𝑛}converges strongly to an element of𝐹.

Proof. Since every asymptotically𝑘_𝑖-strict pseudocontractive mapping is uniformly continuous and asymptotically𝑘_𝑖-strict pseudocontractive mapping in the intermediate sense with 𝑐_𝑛,𝑖≡ 0, for all𝑛 ≥ 1and each𝑖 = 1, 2, . . . , 𝑁, the conclusion follows from Theorem9.

If in Theorem9, we assume that each𝑇_𝑖is asymptotically nonexpansive in the intermediate sense we obtain the follow- ing corollary.

Corollary 12. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝑇_𝑖 : 𝐶 → 𝐶 be uniformly continuous asymptotically nonexpansive mappings in the intermediate sense with sequences{𝛾_𝑛,𝑖} and{𝑐_𝑛,𝑖}, for 𝑖 = 1,2,. . .,𝑁. Assume that𝐹 := ⋂^𝑁_𝑖=1𝐹(𝑇_𝑖)is nonempty.

Let{𝑥_𝑛}be a sequence generated by

𝑥₀= 𝑤 ∈ 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,

𝑦_𝑛= 𝛽_𝑛,0𝑥_𝑛+∑^𝑁

𝑖=1

𝛽_𝑛,𝑖𝑇_𝑖^𝑛𝑥_𝑛,

𝑥_𝑛+1= 𝛼_𝑛𝑤 + (1 − 𝛼_𝑛) 𝑦_𝑛,

(34)

where0 < 𝛼_𝑛 ≤ 𝑎₀ < 1such thatlim_{𝑛 → ∞}𝛼_𝑛 = 0,∑^∞_𝑛=1𝛼_𝑛 =

∞,lim_{𝑛 → ∞}[𝛾_𝑛,𝑖/𝛼_𝑛] = 0 = lim_{𝑛 → ∞}[𝑐_𝑛,𝑖/𝛼_𝑛], and0 < 𝛿 ≤ 𝛽_𝑛,𝑖≤ 1 − 𝛿 < 1satisfying𝛽_𝑛,0+ 𝛽_𝑛,1+ ⋅ ⋅ ⋅ + 𝛽_𝑛,𝑁= 1for each 𝑛 ≥ 1. Then,{𝑥_𝑛}converges strongly to an element of𝐹.

Proof. Since every asymptotically nonexpansive mapping in the intermediate sense is asymptotically𝑘_𝑖-strict pseudocon- tractive mapping in the intermediate sense with 𝑘_𝑖 = 0, for each 𝑖 = 1, 2, . . ., 𝑁, the conclusion follows from Theorem9.

If in Theorem9, we assume that each𝑇_𝑖is asymptotically nonexpansive, we obtain the following corollary.

Corollary 13. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝑇_𝑖 : 𝐶 → 𝐶 be asymptotically nonexpansive mappings with sequences{𝛾_𝑛,𝑖},

for𝑖 = 1,2,. . .,𝑁. Assume that𝐹 := ⋂^𝑁_𝑖=1𝐹(𝑇_𝑖)is nonempty.

Let{𝑥_𝑛}be a sequence generated by

𝑥₀= 𝑤 ∈ 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,

𝑦_𝑛= 𝛽_𝑛,0𝑥_𝑛+∑^𝑁

𝑖=1

𝛽_𝑛,𝑖𝑇_𝑖^𝑛𝑥_𝑛,

𝑥_𝑛+1= 𝛼_𝑛𝑤 + (1 − 𝛼_𝑛) 𝑦_𝑛,

(35)

where0 < 𝛼_𝑛 ≤ 𝑎₀ < 1such thatlim_{𝑛 → ∞}𝛼_𝑛 = 0,∑^∞_𝑛=1𝛼_𝑛 =

∞,lim_{𝑛 → ∞}[𝛾_𝑛,𝑖/𝛼_𝑛] = 0, and0 < 𝛿 ≤ 𝛽_𝑛,𝑖 ≤ 1 − 𝛿 < 1 satisfying𝛽_𝑛,0+ 𝛽_𝑛,1+ ⋅ ⋅ ⋅ + 𝛽_𝑛,𝑁= 1for each𝑛 ≥ 1. Then,{𝑥_𝑛} converges strongly to an element of𝐹.

Proof. Since every asymptotically nonexpansive mapping is uniformly continuous and asymptotically 𝑘_𝑖-strict pseudo- contractive mapping with𝑘_𝑖= 0and𝑐_𝑛,𝑖= 0, for all𝑛 ≥ 1and 𝑖 = 1,2,. . .,𝑁, the conclusion follows from Theorem9.

Remark 14. Our results extend and unify most of the results that have been proved for this important class of nonlinear mappings. In particular, Theorem9extends Theorem SXY1, SXY2, HC, and Theorem ZRS in the sense that our conver- gence is either strong, does not require computation of closed convex sets𝐶_𝑛and𝑄_𝑛for each𝑛 ≥ 1, or does not require the assumption that interior of set of fixed points is nonempty.

Remark 15. We also remark that Corollary11is more general than Theorem 3.1 of Kim and Xu [3] and Corollary 13 is more general than Theorem 2.2 of Kim and Xu [11] in the sense that our convergence is either strong, does not require computation of closed convex sets𝐶_𝑛and𝑄_𝑛for each𝑛 ≥ 1, or does not require the assumption that interior of set of fixed points is nonempty.

Acknowledgment

The research of N. Shahzad was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

References

[1] Q. H. Liu, “Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemi-contractive map- pings,”Nonlinear Analysis, vol. 26, pp. 1838–1842, 1996.

[2] Y. X. Tian, S.-S. Chang, J. Huang, X. Wang, and J. K. Kim,

“Implicit iteration process for common fixed points of strictly asymptotically pseudocontractive mappings in Banach spaces,”

Fixed Point Theory and Applications, vol. 2008, Article ID 324575, 12 pages, 2008.

[3] T.-H. Kim and H.-K. Xu, “Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions,”

Nonlinear Analysis, vol. 68, no. 9, pp. 2828–2836, 2008.

[4] D. R. Sahu, H.-K. Xu, and J.-C. Yao, “Asymptotically strict pseu- docontractive mappings in the intermediate sense,”Nonlinear Analysis, vol. 70, no. 10, pp. 3502–3511, 2009.

[5] C. S. Hu and G. Cai, “Convergence theorems for equilibrium problems and fixed point problems of a finite family of asymp- totically𝑘-strictly pseudocontractive mappings in the interme- diate sense,”Computers & Mathematics with Applications, vol.

61, no. 1, pp. 79–93, 2011.

[6] H. Zegeye, M. Robdera, and B. Choudhary, “Convergence theorems for asymptotically pseudocontractive mappings in the intermediate sense,”Computers & Mathematics with Applica- tions, vol. 62, no. 1, pp. 326–332, 2011.

[7] H. Zegeye and N. Shahzad, “Convergence of Manns type iteration method for generalized asymptotically nonexpansive mappings,”Computers and Mathematics With Applications, vol.

62, no. 11, pp. 4007–4014, 2011.

[8] P.-E. Maing´e, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,”

Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008.

[9] J. G. O’Hara, P. Pillay, and H.-K. Xu, “Iterative approaches to convex feasibility problems in Banach spaces,”Nonlinear Analysis, vol. 64, no. 9, pp. 2022–2042, 2006.

[10] W. Takahashi,Nonlinear Functional Analysis-Fixed Point The- ory and Applications, Yokohama Publishers, Yokohama, Japan, 2000.

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