Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities

Abstract

Let {𝑡_𝑛}⊂(0,1) be such that 𝑡_𝑛→1 as 𝑛→∞, let 𝛼 and 𝛽 be two positive numbers such that 𝛼+𝛽=1, and let 𝑓 be a contraction. If 𝑇 be a continuous asymptotically pseudocontractive self-mapping of a nonempty bounded closed convex subset 𝐾 of a real reflexive Banach space with a uniformly Gateaux differentiable norm, under suitable conditions on the sequence {𝑡_𝑛}, we show the existence of a sequence {𝑥_𝑛}_𝑛 satisfying the relation 𝑥_𝑛=(1−𝑡_𝑛/𝑘_𝑛)𝑓(𝑥_𝑛)+(𝑡_𝑛/𝑘_𝑛)𝑇^𝑛𝑥_𝑛 and prove that {𝑥_𝑛} converges strongly to the fixed point of 𝑇, which solves some variational inequality provided 𝑇 is uniformly asymptotically regular. As an application, if 𝑇 be an asymptotically nonexpansive self-mapping of a nonempty bounded closed convex subset 𝐾 of a real Banach space with a uniformly Gateaux differentiable norm and which possesses uniform normal structure, we prove that the iterative process defined by 𝑧₀∈𝐾,𝑧_𝑛+1=(1−𝑡_𝑛/𝑘_𝑛)𝑓(𝑧_𝑛)+(𝛼𝑡_𝑛/𝑘_𝑛)𝑇^𝑛𝑧_𝑛+(𝛽𝑡_𝑛/𝑘_𝑛)𝑧_𝑛 converges strongly to the fixed point of 𝑇.

1. Introduction

Let 𝐸 be a real Banach space with dual 𝐸^∗ and 𝐾 a nonempty closed convex subset of 𝐸. Recall that a mapping 𝑇∶𝐾→𝐾 is said to be asymptotically pseudocontractive if, for each 𝑛∈𝑁 and 𝑥,𝑦∈𝐾, there exist 𝑗∈𝐽(𝑥−𝑦) and a constant 𝑘_𝑛≥1 with lim_𝑛→∞𝑘_𝑛=1 such that ⟨𝑇^𝑛𝑥−𝑇^𝑛𝑦⟩≤𝑘_𝑛‖𝑥−𝑦‖²,(1.1) where 𝐽∶𝐸→2^𝐸∗ denote the normalized duality mapping defined by 𝐽(𝑥)=𝑥^∗∈𝐸^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖²,‖𝑥^∗‖=‖𝑥‖,𝑥∈𝐸.(1.2) The class of asymptotically pseudocontractive mappings is essentially wider than the class of asymptotically nonexpansive mappings. A mapping 𝑇 is called asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}⊂[1,∞) with lim_𝑛→∞𝑘_𝑛=1 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖(1.3) for all integers 𝑛≥0 and all 𝑥,𝑦∈𝐾. A mapping 𝑓∶𝐾→𝐾 is called a contraction if there exists a constant 𝛾∈[0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛾‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐾.(1.4) It is clear that every contraction is nonexpansive, every nonexpansive mapping is asymptotically nonexpansive, and every asymptotically nonexpansive mapping is asymptotically pseudocontractive. The converses do not hold. The asymptotically nonexpansive mappings are important generalizations of nonexpansive mappings. For details, you may see [1].

The mapping 𝑇 is called uniformly asymptotically regular (in short u.a.r.) if for each 𝜖>0 there exists 𝑛₀∈𝑁 such that ‖‖𝑇^𝑛+1𝑥−𝑇^𝑛𝑥‖‖≤𝜖,(1.5) for all 𝑛≥𝑛₀ and 𝑥∈𝐾 and it is called uniformly asymptotically regular with sequence {𝜖_𝑛} (in short u.a.r.s.) if ‖‖𝑇^𝑛+1𝑥−𝑇^𝑛𝑥‖‖≤𝜖_𝑛,(1.6) for all integers 𝑛≥1 and all 𝑥∈𝐾, where 𝜖_𝑛→0 as 𝑛→∞.

The viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping was proposed by Moudafi [2] who proved the strong convergence of both the implicit and explicit methods in Hilbert spaces, see [2, Theorems 2.1 and 2.2]. Xu [3] studied the viscosity approximation methods proposed by Moudafi [2] for a nonexpansive mapping in a uniformly smooth Banach space.

Very recently, Shahzad and Udomene [4] obtained fixed point solutions of variational inequalities for an asymptotically nonexpansive mapping defined on a real Banach space with uniformly Gateaux differentiable norm possessing uniform normal structure. They proved the following theorem.

Theorem 1.1. Let 𝐸 be a real Banach space with a uniformly Gateaux differentiable norm possessing uniform normal structure, let 𝐾 be a nonempty closed convex and bounded subset of 𝐸, let 𝑇∶𝐾→𝐾 be an asymptotically nonexpansive mapping with sequence {𝑘_𝑛}_𝑛⊂[1,∞), and let 𝑓∶𝐾→𝐾 be a contraction with constant 𝛼∈[0,1). Let {𝑡_𝑛}_𝑛⊂(0,𝜉_𝑛) be such that lim_𝑛→∞𝑡_𝑛=1,  ∑^∞𝑛=1𝑡_𝑛(1−𝑡_𝑛)=∞, and lim_𝑛→∞((𝑘_𝑛−1)/(𝑘_𝑛−𝑡_𝑛))=0, where 𝜉_𝑛=min{(1−𝛼)𝑘_𝑛/(𝑘_𝑛−𝛼),1/𝑘_𝑛}. For an arbitrary 𝑧₀∈𝐾 let the sequence {𝑧_𝑛} be iteratively defined by 𝑧_𝑛+1=1−𝑡^𝑛𝑘_𝑛𝑓𝑧^𝑛+𝑘𝑡^𝑛_𝑛𝑇^𝑛𝑧_𝑛,𝑛∈𝑁.(1.7) Then(i) for each integer 𝑛≥0, there is a unique 𝑥_𝑛∈𝐾 such that 𝑥_𝑛=1−𝑡^𝑛𝑘_𝑛𝑓𝑥^𝑛+𝑘𝑡^𝑛_𝑛𝑇^𝑛𝑥_𝑛;(1.8) if in addition lim_𝑛→∞‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖=0,lim_𝑛→∞‖‖𝑧_𝑛−𝑇𝑧_𝑛‖‖=0,(1.9) then(ii) the sequence {𝑧_𝑛}_𝑛 converges strongly to the unique solution of the variational inequality: 𝑝∈𝐹(𝑇)suchthat(𝐼−𝑓)𝑝,𝑗𝑝−𝑥^∗≤0,∀𝑥^∗∈𝐹(𝑇).(1.10)

Remark 1.2. We note that ‖𝑇^𝑛+1𝑥−𝑇^𝑛𝑥‖≤𝑘_𝑛‖𝑇𝑥−𝑥‖, then the condition (1.9) lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑧_𝑛−𝑇𝑧_𝑛‖=0 imply that lliimm_{𝑛→∞𝑛→∞}‖‖𝑇^𝑛+1𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖=0,‖‖𝑇^𝑛+1𝑧_𝑛−𝑇^𝑛𝑧_𝑛‖‖=0,(1.11) respectively. In other words, if an asymptotically nonexpansive mapping 𝑇 satisfies the condition (1.9) then 𝑇 must be u.a.r.s.

Inspired by the works in [4–8], in this paper, we suggest and analyze a modification of the iterative algorithm.

Let {𝑡_𝑛}⊂(0,1), let 𝛼 and 𝛽 be two positive numbers such that 𝛼+𝛽=1, and let 𝑓 be a contraction on 𝐾, a sequence {𝑧_𝑛} iteratively defined by: 𝑧₀∈𝐾, 𝑧_𝑛+1=1−𝑡^𝑛𝑘_𝑛𝑓𝑧^𝑛+𝛼𝑘𝑡^𝑛_𝑛𝑇^𝑛𝑧_𝑛+𝛽𝑡^𝑛𝑘_𝑛𝑧_𝑛.(1.12)

Remark 1.3. The algorithm (1.12) includes the algorithm (1.7) of Chidume et al. [5] and Shahzad and Udomene [4] as a special case.

We show the convergence of the proposed algorithm (1.12) to the unique solution of some variational inequality (some related works on VI, please see [9–12]). In this respect, our results can be considered as a refinement and improvement of the known results of Chidume et al. [5], Shahzad and Udomene [4], and Lim and Xu [13].

2. Preliminaries

Let 𝑆={𝑥∈𝐸∶‖𝑥‖=1} denote the unit sphere of the Banach space 𝐸. The space 𝐸 is said to have a Gateaux differentiable norm if the limit lim_𝑛→∞‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.1) exists for each 𝑥,𝑦∈𝑆, and we call 𝐸 smooth; 𝐸 is said to have a uniformly Gateaux differentiable norm if for each 𝑦∈𝑆 the limit (2.1) is attained uniformly for 𝑥∈𝑆. Further, 𝐸 is said to be uniformly smooth if the limit (2.1) exists uniformly for (𝑥,𝑦)∈𝑆×𝑆. It is well known [14] that if 𝐸 is smooth then any duality mapping on 𝐸 is single-valued, and if 𝐸 has a uniformly Gateaux differentiable norm then the duality mapping is norm-to-weak* uniformly continuous on bounded sets.

Let 𝐾 be a nonempty closed convex and bounded subset of the Banach space 𝐸 and let the diameter of 𝐾 be defined by 𝑑(𝐾)=sup{‖𝑥−𝑦‖∶𝑥,𝑦∈𝐾}. For each 𝑥∈𝐾, let 𝑟(𝑥,𝐾)=sup{‖𝑥−𝑦‖∶𝑦∈𝐾} and let 𝑟(𝐾)=inf{𝑟(𝑥,𝐾)∶𝑥∈𝐾} denote the Chebyshev radius of 𝐾 relative to itself. The normal structure coefficient 𝑁(𝐸) of 𝐸 is defined by 𝑁(𝐸)=inf𝑑(𝐾)𝑟(𝐾)∶𝐾isaclosedconvexandboundedsubsetof𝐸with𝑑(𝐾)>0.(2.2) A space 𝐸 such that 𝑁(𝐸)>1 is said to have uniform normal structure. It is known that every space with a uniform normal structure is reflexive, and that all uniformly convex and uniformly smooth Banach spaces have uniform normal structure (see [13]).

We will let LIM be a Banach limit. Recall that LIM∈(𝑙^∞)^∗ such that ‖LIM‖=1, liminf_𝑛→∞𝑎_𝑛≤LIM_𝑛𝑎_𝑛≤limsup_𝑛→∞𝑎_𝑛, and LIM_𝑛𝑎_𝑛=LIM_𝑛𝑎_𝑛+1 for all {𝑎_𝑛}_𝑛∈𝑙^∞. Let {𝑥_𝑛} be a bounded sequence of 𝐸. Then we can define the real-valued continuous convex function 𝑔 on 𝐸 by 𝑔(𝑧)=LIM_𝑛‖𝑥_𝑛−𝑧‖² for all 𝑧∈𝐾.

Let 𝑇∶𝐾→𝐾 be a nonlinear mapping and 𝑀={𝑥∈𝐾∶𝑔(𝑥)=min_𝑧∈𝐾𝑔(𝑧)}. 𝑇 is said to satisfy the property (S) if for any bounded sequence {𝑥_𝑛} in 𝐾, lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0 implies 𝑀∩𝐹(𝑇)≠∅.

Lemma 2.1 (see [15]). Let 𝐸 be a Banach space with the uniformly Gateaux differentiable norm and 𝑢∈𝐸. Then 𝑔(𝑢)=inf_𝑧∈𝐸𝑔(𝑧)(2.3) if and only if LIM𝑧,𝐽𝑥_𝑛−𝑢≤0(2.4) for all 𝑧∈𝐸.

Lemma 2.2 (see [16]). Assume {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛𝛾_𝑛,(2.5) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that (1)∑^∞𝑛=1𝛾_𝑛=∞; (2)limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞𝑛=1|𝛿_𝑛𝛾_𝑛|<∞. Then lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.3 (see [17]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose that 𝑥_𝑛+1=1−𝛽_𝑛𝑦_𝑛+𝛽_𝑛𝑥_𝑛(2.6) for all 𝑛≥0 and limsup_𝑛→∞‖‖𝑦_𝑛+1−𝑦_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(2.7) Then lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.4. Let 𝐸 be an arbitrary real Banach space. Then ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑗(𝑥+𝑦)⟩,(2.8) for all 𝑥,𝑦∈𝐸 and for all 𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦).

Lemma 2.5 (see [5]). Let 𝐸 be a Banach space with uniform normal structure, 𝐾 a nonempty closed convex and bounded subset of 𝐸, and 𝑇∶𝐾→𝐾 an asymptotically nonexpansive mapping. Then 𝑇 has a fixed point.

3. Main Results

Theorem 3.1. Let 𝐸 be a real reflexive Banach space with a uniformly Gateaux differentiable norm, 𝐾 a nonempty closed convex and bounded subset of 𝐸, 𝑇∶𝐾→𝐾 a continuous asymptotically pseudocontractive mapping with sequence {𝑘_𝑛}_𝑛⊂[1,∞), and 𝑓∶𝐾→𝐾 a contraction with constant 𝛾∈[0,1). Let {𝑡_𝑛}⊂(0,(1−𝛾)𝑘_𝑛/(𝑘_𝑛−𝛾)) be such that lim_𝑛→∞𝑡_𝑛=1 and lim_𝑛→∞((𝑘_𝑛−1)/(𝑘_𝑛−𝑡_𝑛))=0. Suppose 𝑇 satisfies the property (S). Then (i) for each integer 𝑛≥0, there is a unique 𝑥_𝑛∈𝐾 such that 𝑥_𝑛=1−𝑡^𝑛𝑘_𝑛𝑓𝑥^𝑛+𝑘𝑡^𝑛_𝑛𝑇^𝑛𝑥_𝑛;(3.1) if 𝑇 is u.a.r.s., then (ii) the sequence {𝑥_𝑛}_𝑛 converges strongly to the unique solution of the variational inequality: 𝑝∈𝐹(𝑇)suchthat(𝐼−𝑓)𝑝,𝑗𝑝−𝑥^∗≤0,∀𝑥^∗∈𝐹(𝑇).(3.2)

Proof. By the conditions on {𝑡_𝑛}, 𝑡_𝑛<(1−𝛾)𝑘_𝑛/(𝑘_𝑛−𝛾) implies (1−𝑡_𝑛/𝑘_𝑛)𝛾+𝑡_𝑛<1 for each integer 𝑛≥0, then the mapping 𝑆_𝑛∶𝐾→𝐾 defined for each 𝑥∈𝐾 by 𝑆_𝑛𝑥=(1−𝑡_𝑛/𝑘_𝑛)𝑓(𝑥)+(𝑡_𝑛/𝑘_𝑛)𝑇^𝑛𝑥 is a strictly pseudocontractive mapping.
Indeed, for all 𝑥,𝑦∈𝐾, we have ⟨𝑆_𝑛𝑥−𝑆_𝑛𝑦,𝑗(𝑥−𝑦)⟩=1−𝑡^𝑛+𝑡𝑘_𝑛⟨𝑓(𝑥)−𝑓(𝑦),𝑗(𝑥−𝑦)⟩_𝑛𝑘_𝑛≤1−⟨𝑡𝑇^𝑛_𝑛𝑥−𝑇^𝑛𝑦,𝑗(𝑥−𝑦)⟩𝑘_𝑛‖𝑓(𝑥)−𝑓(𝑦)‖‖𝑥−𝑦‖+𝑡^𝑛‖𝑥−𝑦‖²≤1−𝑡^𝑛𝑘_𝑛𝛾+𝑡^𝑛‖𝑥−𝑦‖².(3.3) It follows [18, Corollary 1] that 𝑆_𝑛 possesses exactly one fixed point 𝑥_𝑛 in 𝐾 such that 𝑆_𝑛𝑥_𝑛=𝑥_𝑛.
From (3.1), we have ‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖=‖‖‖1−𝑡^𝑛𝑘_𝑛𝑓𝑥^𝑛+𝑘𝑡^𝑛_𝑛−1𝑇^𝑛𝑥_𝑛=1−𝑡_𝑛‖‖‖𝑘_𝑛‖‖𝑓𝑥^𝑛−𝑇^𝑛𝑥_𝑛‖‖→0as𝑛→∞.(3.4) Notice that ‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖=‖‖‖1−𝑡^𝑛𝑘_𝑛𝑓𝑥^𝑛−𝑇𝑥_𝑛+𝑘𝑡^𝑛_𝑛𝑇^𝑛≤1−𝑡_𝑛𝑥_𝑛−𝑇𝑥_𝑛‖‖‖𝑘_𝑛‖‖𝑓𝑥^𝑛−𝑇𝑥_𝑛‖‖+𝑡^𝑛𝑘_𝑛+𝑡_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑇^𝑛+1𝑥_𝑛‖‖𝑘_𝑛≤1−‖‖𝑡𝑇^𝑛+1_𝑛𝑥_𝑛−𝑇𝑥_𝑛‖‖𝑘_𝑛‖‖𝑓𝑥^𝑛−𝑇𝑥_𝑛‖‖+𝑡^𝑛𝑘_𝑛+𝑡_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑇^𝑛+1𝑥_𝑛‖‖𝑘_𝑛𝑘₁‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖.(3.5) Therefore, from (3.4), (3.5), and 𝑇 which is u.a.r.s., we obtain ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0 as 𝑛→∞.
Define a function 𝑔∶𝐾→𝑅⁺ by 𝑔(𝑧)=LIM_𝑛‖‖𝑥_𝑛−𝑧‖‖²(3.6) for all 𝑧∈𝐾. Since 𝑔 is continuous and convex, 𝑔(𝑧)→∞ as ‖𝑧‖→∞, and 𝐸 is reflexive, 𝑔 attains it infimum over 𝐾. Let 𝑧₀∈𝐾 such that 𝑔(𝑧₀)=min_𝑧∈𝐾𝑔(𝑧) and let 𝑀={𝑥∈𝐾∶𝑔(𝑥)=min_𝑧∈𝐾𝑔(𝑧)}. Then 𝑀 is nonempty because 𝑧₀∈𝑀. Since 𝑇 satisfies the property (S), it follows that 𝑀∩𝐹(𝑇)≠∅. Suppose that 𝑝∈𝑀∩𝐹(𝑇). Then, by Lemma 2.1, we have LIM_𝑛𝑥−𝑝,𝑗𝑥_𝑛−𝑝≤0(3.7) for all 𝑥∈𝐾. In particular, we have LIM_𝑛𝑓(𝑝)−𝑝,𝑗𝑥_𝑛−𝑝≤0.(3.8) On the other hand, from (3.1), we have 𝑥_𝑛−𝑇^𝑛𝑥_𝑛=1−𝑡^𝑛𝑘_𝑛𝑓𝑥^𝑛−𝑇^𝑛𝑥_𝑛=1−𝑡^𝑛/𝑘_𝑛𝑡_𝑛/𝑘_𝑛𝑓𝑥_𝑛−𝑥_𝑛.(3.9) Now, for any 𝑝∈𝐹(𝑇), we have 𝑥_𝑛−𝑇^𝑛𝑥_𝑛,𝐽𝑥_𝑛−𝑝≥=−𝑥_𝑛𝑘−_𝑛𝑝+𝑇^𝑛𝑝−𝑇^𝑛𝑥_𝑛,𝐽𝑥_𝑛−𝑝−1‖‖𝑥_𝑛−𝑝‖‖²≥−𝑘_𝑛−1𝐵²(3.10) for some 𝐵>0 and it follows from (3.9) that 𝑥_𝑛−𝑓𝑥_𝑛,𝑗𝑥_𝑛−𝑝≤𝑡^𝑛𝑘_𝑛−1𝑘_𝑛−𝑡_𝑛𝐵²,(3.11) which implies that limsup_𝑛→∞𝑥_𝑛−𝑓𝑥_𝑛,𝑗𝑥_𝑛−𝑝≤0.(3.12) Consequently, similar to the lines of the proof of [4, Theorem 3.1], Theorem 3.1 is easily proved. This completes the proof.

Corollary 3.2. Let 𝐸 be a real Banach space with a uniformly Gateaux differentiable norm possessing uniform normal structure, 𝐾 a nonempty closed convex and bounded subset of 𝐸, 𝑇∶𝐾→𝐾 be an asymptotically nonexpansive mapping with sequence {𝑘_𝑛}_𝑛⊂[1,∞), and 𝑓∶𝐾→𝐾 a contraction with constant 𝛾∈[0,1). Let {𝑡_𝑛}⊂(0,(1−𝛾)𝑘_𝑛/(𝑘_𝑛−𝛾)) be such that lim_𝑛→∞𝑡_𝑛=1 and lim_𝑛→∞((𝑘_𝑛−1)/(𝑘_𝑛−𝑡_𝑛))=0. Then (i) for each integer 𝑛≥0, there is a unique 𝑥_𝑛∈𝐾 such that 𝑥_𝑛=1−𝑡^𝑛𝑘_𝑛𝑓𝑥^𝑛+𝑘𝑡^𝑛_𝑛𝑇^𝑛𝑥_𝑛;(3.13) if 𝑇 is u.a.r.s., then (ii) the sequence {𝑥_𝑛}_𝑛 converges strongly to the unique solution of the variational inequality: 𝑝∈𝐹(𝑇)suchthat(𝐼−𝑓)𝑝,𝑗𝑝−𝑥^∗≤0,∀𝑥^∗∈𝐹(𝑇).(3.14)

Theorem 3.3. Let 𝐸 be a real Banach space with a uniformly Gateaux differentiable norm possessing uniform normal structure, 𝐾 a nonempty closed convex and bounded subset of 𝐸, 𝑇∶𝐾→𝐾 an asymptotically nonexpansive mapping with sequence {𝑘_𝑛}_𝑛⊂[1,∞), and 𝑓∶𝐾→𝐾 a contraction with constant 𝛾∈[0,1). Let {𝑡_𝑛}⊂(0,𝜉_𝑛) be such that lim_𝑛→∞𝑡_𝑛=1,∑^∞𝑛=1𝑡_𝑛(1−𝑡_𝑛)=∞, and lim_𝑛→∞((𝑘_𝑛−1)/(𝑘_𝑛−𝑡_𝑛))=0, where 𝜉_𝑛=min{(1−𝛾)𝑘_𝑛/(𝑘_𝑛−𝛾),1/𝑘_𝑛}. For an arbitrary 𝑧₀∈𝐾, let the sequence {𝑧_𝑛}_𝑛 be iteratively defined by (1.12). Then (i) for each integer 𝑛≥0, there is a unique 𝑥_𝑛∈𝐾 such that 𝑥_𝑛=1−𝑡^𝑛𝑘_𝑛𝑓𝑥^𝑛+𝑘𝑡^𝑛_𝑛𝑇^𝑛𝑥_𝑛;(3.15) if 𝑇 is u.a.r.s., then (ii) the sequence {𝑧_𝑛}_𝑛 converges strongly to the unique solution of the variational inequality: 𝑝∈𝐹(𝑇)suchthat(𝐼−𝑓)𝑝,𝑗𝑝−𝑥^∗≤0,∀𝑥^∗∈𝐹(𝑇).(3.16)

Proof. Set 𝛼_𝑛=𝑡_𝑛/𝑘_𝑛, then 𝛼_𝑛→1 as 𝑛→∞. Define 𝑧_𝑛+1=𝛽𝛼_𝑛𝑧_𝑛+1−𝛽𝛼_𝑛𝑦_𝑛.(3.17) Observe that 𝑦_𝑛+1−𝑦_𝑛=𝑧^𝑛+2−𝛽𝛼_𝑛+1𝑧_𝑛+11−𝛽𝛼_𝑛+1−𝑧^𝑛+1−𝛽𝛼_𝑛𝑧_𝑛1−𝛽𝛼_𝑛=1−𝛼^𝑛+1𝑓𝑧_𝑛+1+𝛼𝛼_𝑛+1𝑇^𝑛+1𝑧_𝑛+11−𝛽𝛼_𝑛+1−1−𝛼^𝑛𝑓𝑧_𝑛+𝛼𝛼_𝑛𝑇^𝑛𝑧_𝑛1−𝛽𝛼_𝑛=1−𝛼^𝑛+11−𝛽𝛼_𝑛+1𝑓𝑧_𝑛+1−𝑓𝑧_𝑛+11−−𝛽𝛼𝛼^𝑛+1_𝑛+1−1−𝛼^𝑛+𝛼𝛼_𝑛+11−𝛽𝛼_𝑛𝑓𝑧^𝑛1−𝛽𝛼_𝑛+1𝑇^𝑛+1+𝛼𝛼_𝑛+1𝑧_𝑛+1−𝑇^𝑛+1𝑧_𝑛1−𝛽𝛼_𝑛+1𝑇^𝑛+1+𝛼𝛼_𝑛+1𝑧_𝑛−𝑇^𝑛𝑧_𝑛1−𝛽𝛼_𝑛+1−𝛼𝛼^𝑛1−𝛽𝛼_𝑛𝑇^𝑛𝑧_𝑛.(3.18) It follows that ‖‖𝑦≤_𝑛+1−1𝑦−_𝑛‖‖𝛼−_𝑛+1‖‖𝑧_𝑛+1−𝑧_𝑛‖‖1−𝛽𝛼_𝑛+1𝛾‖‖𝑧_𝑛+1−𝑧_𝑛‖‖+||||1−𝛼^𝑛+11−𝛽𝛼_𝑛+1−1−𝛼^𝑛1−𝛽𝛼_𝑛||||‖‖𝑓𝑧_𝑛+𝛼𝛼_𝑛+1‖‖1−𝛽𝛼_𝑛+1+𝛼𝛼_𝑛+1‖‖𝑇^𝑛+1𝑧_𝑛+1−𝑇^𝑛+1𝑧_𝑛‖‖1−𝛽𝛼_𝑛+1+||||𝛼𝛼_𝑛+1‖‖𝑇^𝑛+1𝑧_𝑛−𝑇^𝑛𝑧_𝑛‖‖1−𝛽𝛼_𝑛+1−𝛼𝛼^𝑛1−𝛽𝛼_𝑛||||‖‖𝑇_𝑛≤||||1−𝛼_𝑛+1𝑧_𝑛‖‖−‖‖𝑧_𝑛+1−𝑧_𝑛‖‖1−𝛽𝛼_𝑛+1−1−𝛼^𝑛1−𝛽𝛼_𝑛||||‖‖𝑓𝑧_𝑛+||||𝛼𝛼_𝑛+1‖‖+1−𝛼𝛼𝛽𝛼^𝑛+1_𝑛+1‖‖𝑇^𝑛+1𝑧_𝑛−𝑇^𝑛𝑧_𝑛‖‖1−𝛽𝛼_𝑛+1−𝛼𝛼^𝑛1−𝛽𝛼_𝑛||||‖‖𝑇_𝑛+1−𝛼_𝑛+1𝑧_𝑛‖‖1−𝛽𝛼_𝑛+1𝛾+𝛼𝛼^𝑛+11−𝛽𝛼_𝑛+1𝑘_𝑛+1−1‖‖𝑧_𝑛+1−𝑧_𝑛‖‖.(3.19) We note that 𝑘_𝑛+1−𝛾−𝛼𝑘_𝑛+1+𝛽−𝛾≥=1(1−−𝛼𝛼−)𝑘𝛽=_𝑛+10−.𝛽(3.20) It follows that 𝑡_𝑛+1≤(1−𝛾)𝑘^𝑛+1𝑘_𝑛+1−𝛾≤𝛼(𝑘1−_𝑛+1𝛾+)𝑘𝛽−^𝑛+1𝛾,(3.21) which implies that 𝑘_𝑛+1𝑡_𝑛+1𝛼+𝑡_𝑛+1𝛽−𝑡_𝑛+1𝛾≤(1−𝛾)𝑘_𝑛+1⇒⇒𝛼𝛼𝑘𝑘_𝑛+1𝑛+1𝛼_𝑛+1+𝛼_𝑛+1𝛽−𝛼_𝑛+1𝛾≤1−𝛾⇒𝛼𝑘_𝑛+1𝛼_𝑛+1+1−𝛼_𝑛+1𝛾≤1−𝛼_𝑛+1𝛽𝛼_𝑛+1+1−𝛼_𝑛+1𝛾1−𝛼_𝑛+1𝛽≤1.(3.22) From (3.19) and (3.22), we obtain limsup_𝑛→∞‖‖𝑦_𝑛+1−𝑦_𝑛‖‖−‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤0.(3.23) Hence, by Lemma 2.3 we know lim_𝑛→∞‖‖𝑦_𝑛−𝑧_𝑛‖‖=0,(3.24) consequently lim_𝑛→∞‖‖𝑧_𝑛+1−𝑧_𝑛‖‖=0.(3.25)
On the other hand, ‖‖𝑧_𝑛−𝑇^𝑛𝑧_𝑛‖‖≤≤‖‖‖‖𝑧𝑧_𝑛+1𝑛+1−𝑧_𝑛‖‖+‖‖𝑧_𝑛+1−𝑇^𝑛𝑧_𝑛‖‖+𝛽𝛼_𝑛−𝑧_𝑛‖‖+1−𝛼_𝑛‖‖𝑓𝑧_𝑛−𝑇^𝑛𝑧_𝑛‖‖‖‖𝑧_𝑛−𝑇^𝑛𝑧_𝑛‖‖,(3.26) which implies that lim_𝑛→∞‖‖𝑧_𝑛−𝑇^𝑛𝑧_𝑛‖‖=0.(3.27) Hence, we have ‖‖𝑧_𝑛−𝑇𝑧_𝑛‖‖≤≤‖‖‖‖𝑧𝑧_𝑛𝑛−𝑇^𝑛𝑧_𝑛‖‖+‖‖𝑇^𝑛𝑧_𝑛−𝑇^𝑛+1𝑧_𝑛‖‖+‖‖𝑇^𝑛+1𝑧_𝑛−𝑇𝑧_𝑛‖‖=1+−𝑘𝑇^𝑛₁𝑧_𝑛‖‖+‖‖𝑇^𝑛𝑧_𝑛−𝑇^𝑛+1𝑧_𝑛‖‖+𝑘₁‖‖𝑧_𝑛−𝑇^𝑛𝑧_𝑛‖‖‖‖𝑧_𝑛−𝑇^𝑛𝑧_𝑛‖‖+‖‖𝑇^𝑛𝑧_𝑛−𝑇^𝑛+1𝑧_𝑛‖‖→0(𝑛→∞).(3.28) From (3.15), 𝑥_𝑚−𝑧_𝑛=(1−𝛼_𝑚)(𝑓(𝑥_𝑚)−𝑧_𝑛)+𝛼_𝑚(𝑇^𝑚𝑥_𝑚−𝑧_𝑛). Applying Lemma 2.4, we estimate as follows: ‖‖𝑥_𝑚−𝑧_𝑛‖‖²≤≤𝛼𝛼^2𝑚_2𝑚‖‖𝑇^𝑚𝑥_𝑚−𝑧_𝑛‖‖²+21−𝛼_𝑚𝑓𝑥_𝑚−𝑧_𝑛,𝑗𝑥_𝑚−𝑧_𝑛‖‖𝑇^𝑚𝑥_𝑚−𝑇^𝑚𝑧_𝑛‖‖+‖‖𝑇^𝑚𝑧_𝑛−𝑧_𝑛‖‖²≤𝛼+_2𝑚21−𝛼_𝑚𝑓𝑥_𝑚−𝑥_𝑚,𝑗𝑥_𝑚−𝑧_𝑛+‖‖𝑥_𝑚−𝑧_𝑛‖‖²𝑘_𝑚‖‖𝑥_𝑚−𝑧_𝑛‖‖+‖‖𝑇^𝑚𝑧_𝑛−𝑧_𝑛‖‖²=𝛼+_2𝑚21−𝛼_𝑚𝑓𝑥_𝑚−𝑥_𝑚,𝑗𝑥_𝑚−𝑧_𝑛+𝑘^2𝑚‖‖𝑥_𝑚−𝑧_𝑛‖‖²+𝑘^2𝑚‖‖‖‖𝑇𝑥_𝑚𝑚−𝑧_𝑛‖‖²+2𝑘_𝑚‖‖𝑥_𝑚−𝑧_𝑛‖‖‖‖𝑇^𝑚𝑧_𝑛−𝑧_𝑛‖‖+21−𝛼_𝑚𝑧_𝑛−𝑧_𝑛‖‖²=1−(1−𝛼_𝑚𝑓𝑥_𝑚−𝑥_𝑚,𝑗𝑥_𝑚−𝑧_𝑛+𝑘^2𝑚‖‖𝑥_𝑚−𝑧_𝑛‖‖²)²𝑘^2𝑚‖‖𝑥_𝑚−𝑧_𝑛‖‖²++‖‖2𝑇1^𝑚𝑧−_𝑛𝛼−_𝑚𝑧_𝑛‖‖2𝑘_𝑚‖‖𝑥_𝑚−𝑧_𝑛‖‖+‖‖𝑇^𝑚𝑧_𝑛−𝑧_𝑛‖‖≤1+1−𝛼_𝑚𝑓𝑥_𝑚−𝑥_𝑚,𝑗𝑥_𝑚−𝑧_𝑛+𝑘^2𝑚‖‖𝑥_𝑚−𝑧_𝑛‖‖²²𝑘^2𝑚‖‖𝑥_𝑚−𝑧_𝑛‖‖²++‖‖2𝑇1^𝑚𝑧−_𝑛𝛼−_𝑚𝑧_𝑛‖‖2𝑘_𝑚‖‖𝑥_𝑚−𝑧_𝑛‖‖+‖‖𝑇^𝑚𝑧_𝑛−𝑧_𝑛‖‖𝑓𝑥_𝑚−𝑥_𝑚,𝑗𝑥_𝑚−𝑧_𝑛.(3.29) Since 𝐾 is bounded, for some constant 𝑀>0, it follows that limsup_𝑛→∞𝑓𝑥_𝑚−𝑥_𝑚,𝑗𝑧_𝑛−𝑥_𝑚≤𝑘^2𝑚−1+𝑘^2𝑚1−𝛼_𝑚²1−𝛼_𝑚+limsup_𝑛→∞𝑀𝑀‖‖𝑧_𝑛−𝑇^𝑚z_𝑛‖‖1−𝛼_𝑚,(3.30) so that limsup_𝑛→∞𝑓𝑥_𝑚−𝑥_𝑚,𝑗𝑧_𝑛−𝑥_𝑚≤𝑘^2𝑚−1+𝑘^2𝑚1−𝛼_𝑚²1−𝛼_𝑚𝑀.(3.31) By Corollary 3.2, 𝑥_𝑚→𝑝∈𝐹(𝑇), which solve the variational inequality (3.16). Since 𝑗 is norm to weak* continuous on bounded sets, in the limit as 𝑚→∞, we obtain that limsup_𝑛→∞𝑓(𝑝)−𝑝,𝑗𝑧_𝑛−𝑝≤0.(3.32) From Lemma 2.4, we estimate as follows: ‖‖𝑧_𝑛+1−𝑝‖‖²=‖‖(1−𝛼_𝑛)(𝑓(𝑧_𝑛)−𝑝)+𝛼𝛼_𝑛(𝑇^𝑛𝑧_𝑛−𝑝)+𝛽𝛼_𝑛(𝑧_𝑛−𝑝)‖‖²≤‖‖𝛼𝛼_𝑛(𝑇^𝑛𝑧_𝑛−𝑝)+𝛽𝛼_𝑛(𝑧_𝑛−𝑝)‖‖²≤𝛼+₂21−𝛼_𝑛𝑓𝑧_𝑛−𝑝,𝑗𝑧_𝑛+1−𝑝+𝛼𝛽^2𝑛₂‖‖𝑇^𝑛𝑧_𝑛−𝑝‖‖²+2𝛼𝛽𝛼^2𝑛‖‖𝑇^𝑛𝑧_𝑛−𝑝‖‖‖‖𝑧_𝑛−𝑝‖‖+2𝛼1^2𝑛−‖‖𝑧𝛼_𝑛𝑛−𝑝‖‖²+21−𝛼_𝑛⟨𝑓𝑧_𝑛−𝑓(𝑝),𝑗𝑧_𝑛+1−𝑝⟩≤𝛼𝑓(𝑝)−𝑝,𝑗𝑧_𝑛+1−𝑝₂𝑘^2𝑛+2𝛼𝛽𝑘_𝑛+𝛽²𝛼^2𝑛‖‖𝑧_𝑛−𝑝‖‖²++2211−−𝛼𝛼_𝑛𝑛𝛾‖‖𝑧_𝑛−𝑝‖‖‖‖𝑧_𝑛+1−𝑝‖‖≤𝛼_2𝑛𝑓(𝑝)−𝑝,𝑗𝑧_𝑛+1−𝑝+𝑘2^2𝑛‖‖1𝑧_𝑛−−𝛼_𝑛𝑝‖‖²+𝛾1−𝛼_𝑛‖‖𝑧_𝑛−𝑝‖‖²+‖‖𝑧_𝑛+1−𝑝‖‖²⟨𝑓(𝑝)−𝑝,𝑗𝑧_𝑛+1−𝑝⟩,(3.33) so that ‖‖𝑧_𝑛+1−𝑝‖‖²≤𝑡^2𝑛+1−𝛼_𝑛𝛾1−1−𝛼_𝑛𝛾‖‖𝑧^𝑛−𝑝‖‖²+21−𝛼^𝑛1−1−𝛼_𝑛𝛾𝑓(𝑝)−𝑝,𝑗𝑧^𝑛+1=1−1−21−𝛼_𝑛−𝑝𝛾−𝑡^2𝑛1−1−𝛼_𝑛𝛾‖‖𝑧^𝑛−𝑝‖‖²+21−𝛼^𝑛1−1−𝛼_𝑛𝛾𝑓(𝑝)−𝑝,𝑗𝑧^𝑛+1−𝑝.(3.34) Let 𝜆_𝑛=1−21−𝛼^𝑛𝛾−𝑡^2𝑛1−1−𝛼_𝑛𝛾.(3.35) Consequently, following the lines of the proof of [4, Theorem 3.3], Theorem 3.3 is easily proved.

From the lines of the proof of Theorem 3.3, we can obtain the following corollary.

Corollary 3.4. Let 𝐸 be a real Banach space with a uniformly Gateaux differentiable norm possessing uniform normal structure, 𝐾 a nonempty closed convex and bounded subset of 𝐸, 𝑇∶𝐾→𝐾 an asymptotically nonexpansive mapping with sequence {𝑘_𝑛}_𝑛⊂[1,∞), and 𝑓∶𝐾→𝐾 a contraction with constant 𝛾∈[0,1). Let {𝑡_𝑛}⊂(0,𝜉_𝑛) be such that lim_𝑛→∞𝑡_𝑛=1,∑^∞𝑛=1𝑡_𝑛(1−𝑡_𝑛)=∞, and lim_𝑛→∞((𝑘_𝑛−1)/(𝑘_𝑛−𝑡_𝑛))=0, where 𝜉_𝑛=min{(1−𝛾)𝑘_𝑛/(𝑘_𝑛−𝛾),1/𝑘_𝑛}. For an arbitrary 𝑧₀∈𝐾, let the sequence {𝑧_𝑛}_𝑛 be iteratively defined by (1.12). Then(i) for each integer 𝑛≥0, there is a unique 𝑥_𝑛∈𝐾 such that 𝑥_𝑛=1−𝑡^𝑛𝑘_𝑛𝑓𝑥^𝑛+𝑘𝑡^𝑛_𝑛𝑇^𝑛x_𝑛;(3.36) if 𝑇 satisfies lim_𝑛→∞‖𝑇^𝑛+1𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑇^𝑛+1𝑧_𝑛−𝑇^𝑛𝑧_𝑛‖=0 then(ii) the sequence {𝑧_𝑛}_𝑛 converges strongly to the unique solution of the variational inequality: 𝑝∈𝐹(𝑇)suchthat(𝐼−𝑓)𝑝,𝑗𝑝−𝑥^∗≤0,∀𝑥^∗∈𝐹(𝑇).(3.37)

Remark 3.5. Since every nonexpansive mapping is asymptotically nonexpansive, our theorems hold for the case when 𝑇 is simply nonexpansive. In this case, the boundedness requirement on 𝐾 can be removed from the above results.

Remark 3.6. Our results can be viewed as a refinement and improvement of the corresponding results by Shahzad and Udomene [4], Chidume et al. [5], and Lim and Xu [13].

Example 3.7. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping. Let the iterative sequence {𝑥_𝑛} be defined by 𝑥_𝑛+1=1𝑛𝑢+1−1𝑛𝑇𝑥^𝑛.(3.38) It is easy to see that {𝑥_𝑛} converges strongly to some fixed point of 𝑇.
In particular, let 𝐻=𝑅² and define 𝑇∶𝑅²→𝑅² by 𝑇𝑟𝑒^𝑖𝜃=𝑟𝑒^{𝑖(𝜃+𝜋/2)},(3.39) and take that 𝑢=𝑒^𝑖𝜋 is a fix element in 𝐶. It is obvious that 𝑇 is a nonexpansive mapping with a unique fixed point 𝑥^∗=0. In this case, (3.38) becomes 𝑥_𝑛+1=1𝑛𝑒^𝑖𝜋+1−1𝑛𝑟^𝑛𝑒^{𝑖(𝜃𝑛+𝜋/2)}.(3.40) It is clear that the complex number sequence {𝑥_𝑛} converges strongly to a fixed point 𝑥^∗=0.

Acknowledgment

Y.-C. Liou was partially supported by NSC 101-2628-E-230-001-MY3.