Approximation of Common Fixed Points of a Sequence of Nearly Nonexpansive Mappings and Solutions of Variational Inequality Problems

Abstract

We introduce an explicit iterative scheme for computing a common fixed point of a sequence of nearly nonexpansive mappings defined on a closed convex subset of a real Hilbert space which is also a solution of a variational inequality problem. We prove a strong convergence theorem for a sequence generated by the considered iterative scheme under suitable conditions. Our strong convergence theorem extends and improves several corresponding results in the context of nearly nonexpansive mappings.

1. Introduction

Let 𝐶 be a nonempty subset of a real Hilbert space 𝐻 with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. A mapping 𝑇∶𝐶→𝐻 is called the following:(1)monotone if ⟨𝑇𝑥−𝑇𝑦,𝑥−𝑦⟩≥0∀𝑥,𝑦∈𝐶,(1.1)(2)𝜂-strongly monotone if there exists a positive real number 𝜂 such that ⟨𝑇𝑥−𝑇𝑦,𝑥−𝑦⟩≥𝜂‖𝑥−𝑦‖²∀𝑥,𝑦∈𝐶,(1.2)(3)𝑘-Lipschitzian if there exists a constant 𝑘>0 such that ‖𝑇𝑥−𝑇𝑦‖≤𝑘‖𝑥−𝑦‖∀𝑥,𝑦∈𝐶,(1.3)(4)nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖∀𝑥,𝑦∈𝐶,(1.4)(5)nearly nonexpansive [1, 2] with respect to a fixed sequence {𝑎_n} in [0,∞) with 𝑎_𝑛→0 if ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤‖𝑥−𝑦‖+𝑎_𝑛∀𝑥,𝑦∈𝐶and𝑛∈ℕ.(1.5)

In [3], Moudafi proposed viscosity approximation methods of selecting a particular fixed point of a given nonexpansive mapping in Hilbert spaces (see [4] for further developments in both Hilbert and Banach spaces). Let 𝑓 be a contraction on 𝐻. Starting with an arbitrary initial 𝑥₁∈𝐻, define a sequence {𝑥_𝑛} recursively by 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛∀𝑛∈ℕ,(1.6) where {𝛼_𝑛} is a sequence in (0,1). It is proved in [4] that under appropriate conditions imposed on {𝛼_𝑛}, the sequence {𝑥_𝑛} generated by (1.6) strongly converges to the unique solution 𝑥^∗∈𝐶 of the variational inequality ⟨(𝐼−𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0∀𝑥∈𝐶,(1.7) where 𝐶=𝐹(𝑇), the set of fixed points of 𝑇.

In 2006, Marino and Xu [5] introduced the viscosity iterative method for nonexpansive mappings. Starting with an arbitrary initial 𝑥₁∈𝐻, define a sequence {𝑥_𝑛} recursively by 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛∀𝑛∈ℕ.(1.8) They proved that the sequence {𝑥_𝑛} generated by (1.8) converges strongly to the unique solution of the variational inequality ⟨(𝐴−𝛾𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0∀𝑥∈𝐶,(1.9) which is the optimality condition for the minimization problem min_𝑥∈𝐶12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥),(1.10) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ′(𝑥)=𝛾𝑓(𝑥) for all 𝑥∈𝐻), and 𝐴 is a strongly positive bounded linear operator on 𝐻; that is, there is a constant 𝛾>0 such that ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²∀𝑥∈𝐻.(1.11) The applications of the iterative method (1.8) have been studied by some researchers (see [6, 7]).

Also, Wang [8, 9] and Wang and Hu [10] introduced the iterative method for nonexpansive mappings.

Recently, Tian [11] proposed an implicit and an explicit schemes on combining the iterative methods of Marino and Xu [5] and Yamada [12]. He also proved the strong convergence of these two schemes to a fixed point of a nonexpansive mapping 𝑇 defined on a real Hilbert space under suitable conditions.

More recently, Ceng et al. [13] introduced an implicit and an explicit schemes using the properties of projection for finding the fixed points of a nonexpansive mapping defined on the closed convex subset of a real Hilbert space. They also proved the strong convergence of the sequences generated by the proposed schemes to a fixed point of a nonexpansive mapping which is also a solution of a variational inequality defined on the set of fixed points.

Aoyama et al. [14] proved strong convergence of an iterative scheme for a sequence of nonexpansive mappings as follows.

Theorem 1.1. Let 𝑋 be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and C be a nonempty closed convex subset of 𝑋. Let {𝑇_𝑛} be a sequence of nonexpansive mappings from 𝐶 into itself such that ⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅. Let 𝑇 be a mapping from 𝐶 into itself defined by 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥 for all 𝑥∈𝐶 and 𝐹(𝑇)=⋂^∞𝑛=1𝐹(𝑇_𝑛). Let {𝑥_𝑛} be a sequence in 𝐶 generated by the following iterative process: 𝑥𝑥₁=𝑥∈𝐶,_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛∀𝑛∈ℕ,(1.12) where {𝛼_𝑛} is a sequence in [0,1] satisfying the following conditions:(a)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞;(b) either ∑^∞𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞ or 𝛼_𝑛∈(0,1] and lim_𝑛→∞𝛼_𝑛+1/𝛼_𝑛=1;(c)∑^∞𝑛=1sup{‖𝑇_𝑛𝑧−𝑇_𝑛+1𝑧‖∶𝑧∈𝐵}<∞ for any bounded subset 𝐵 of 𝐶.
Then, the sequence {𝑥_𝑛} converges strongly to 𝑄𝑥, where 𝑄 is the sunny nonexpansive retraction from 𝑋 onto 𝐹(𝑇).

Let 𝐶 be a nonempty subset of a real Hilbert space 𝐻. Let 𝒯∶={𝑇_𝑛} be a sequence of mappings from 𝐶 into itself. We denote by 𝐹(𝒯) the set of common fixed points of the sequence 𝒯, that is, 𝐹(𝒯)=⋂^∞𝑛=1𝐹(𝑇_𝑛). Fix a sequence {𝑎_𝑛} in [0,∞) with 𝑎_𝑛→0, and let {𝑇_𝑛} be a sequence of mappings from 𝐶 into 𝐻. Then, the sequence {𝑇_𝑛} is called a sequence of nearly nonexpansive mappings [15] with respect to a sequence {𝑎_𝑛} if ‖‖𝑇_𝑛𝑥−𝑇_𝑛𝑦‖‖≤‖𝑥−𝑦‖+𝑎_𝑛∀𝑥,𝑦∈𝐶,𝑛∈ℕ.(1.13)

It is obvious that the sequence of nearly nonexpansive mappings is a wider class of sequence of nonexpansive mappings.

In this paper, inspired by Aoyama et al. [14], Ceng et al. [13], and Sahu et al. [15], we introduce an explicit iterative scheme and prove a strong convergence theorem for computing an element of 𝐹(𝒯), the set of common fixed points of a sequence 𝒯={𝑇_𝑛} of nearly nonexpansive mappings which is also a solution of a variational inequality over 𝐹(𝒯). Our result generalizes and improves the results of Ceng et al. [13], Tian [11], and many other related works.

2. Preliminaries

Throughout this paper, we denote by 𝐼 the identity operator of 𝐻. Also, we denote by → and ⇀ the strong convergence and weak convergence, respectively. The symbol ℕ stands for the set of all natural numbers.

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Then, for any 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶(𝑥), such that ‖‖𝑥−𝑃_𝐶(𝑥)‖‖≤‖𝑥−𝑦‖∀𝑦∈𝐶.(2.1) The mapping 𝑃_𝐶 is called the metric projection from 𝐻 onto 𝐶 (see [1]).

Let 𝐶 be a nonempty subset of a real Hilbert space 𝐻 and 𝑇₁,𝑇₂∶𝐶→𝐻 be two mappings. We denote ℬ(𝐶), the collection of all bounded subsets of 𝐶. The deviation between 𝑇₁ and 𝑇₂ on 𝐵∈ℬ(𝐶), denoted by 𝒟_𝐵(𝑇₁,𝑇₂), is defined by 𝒟_𝐵𝑇₁,𝑇₂=sup‖‖𝑇₁𝑥−𝑇₂𝑥‖‖∶𝑥∈𝐵.(2.2)

The following lemmas will be needed to prove our main result.

Lemma 2.1 (see [16]). The metric projection mapping 𝑃_𝐶 is characterized by the following properties:(a)𝑃_𝐶(𝑥)∈𝐶 for all 𝑥∈𝐻;(b)⟨𝑥−𝑃_𝐶(𝑥),𝑃_𝐶(𝑥)−𝑦⟩≥0 for all 𝑥∈𝐻 and 𝑦∈𝐶;(c)‖𝑥−𝑦‖²≥‖𝑥−𝑃_𝐶(𝑥)‖²+‖𝑦−𝑃_𝐶(𝑥)‖² for all 𝑥∈𝐻 and 𝑦∈𝐶;(d)⟨𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦),𝑥−𝑦⟩≥‖𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦)‖² for all 𝑥,𝑦∈𝐻.

Lemma 2.2 (see [13]). Let 𝑉∶𝐶→𝐻 be an 𝐿-Lipschitzian mapping and 𝐹∶𝐶→𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator. Then, for 0≤𝛾𝐿<𝜇𝜂, ⟨𝑥−𝑦,(𝜇𝐹−𝛾𝑉)𝑥−(𝜇𝐹−𝛾𝑉)𝑦⟩≥(𝜇𝜂−𝛾𝐿)‖𝑥−𝑦‖²∀𝑥,𝑦∈𝐶.(2.3) That is, 𝜇𝐹−𝛾𝑉 is strongly monotone with coefficient 𝜇𝜂−𝛾𝐿.

Lemma 2.3 (see [12]). Let 𝐶 be a nonempty subset of a real Hilbert space 𝐻. Suppose that 𝜆∈(0,1) and 𝜇>0. Let 𝐹∶𝐶→𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator on 𝐶. Define the mapping 𝐺∶𝐶→𝐻 by 𝐺𝑥=𝑥−𝜆𝜇𝐹𝑥∀𝑥∈𝐶.(2.4) Then 𝐺 is a contraction that provided 𝜇<2𝜂/𝑘². More precisely, for 𝜇∈(0,2𝜂/𝑘²), ‖𝐺𝑥−𝐺𝑦‖≤(1−𝜆𝜏)‖𝑥−𝑦‖∀𝑥,𝑦∈𝐶,(2.5) where 𝜏=1−√1−𝜇(2𝜂−𝜇𝑘²)∈(0,1].

Lemma 2.4 (see [1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping. Then 𝐼−𝑇 is demiclosed at zero; that is, if {𝑥_𝑛} is a sequence in 𝐶 weakly converging to some 𝑥∈𝐶 and the sequence {(𝐼−𝑇)𝑥_𝑛} converges strongly to 0, then 𝑥∈𝐹(𝑇).

Lemma 2.5 (see [17]). Assume that {𝑡_𝑛} is a sequence of nonnegative real numbers such that 𝑡_𝑛+1≤1−𝛼_𝑛𝑡_𝑛+𝛼_𝑛𝛽_𝑛∀𝑛∈ℕ,(2.6) where {𝛼_𝑛} and {𝛽_𝑛} are sequences of nonnegative real numbers which satisfy the following conditions:(a){𝛼_𝑛}^∞𝑛=1⊂(0,1) and ∑^∞𝑛=1𝛼_𝑛=∞;(b)limsup_𝑛→∞𝛽_𝑛≤0, or(b’)∑^∞𝑛=1𝛼_𝑛𝛽_𝑛 is convergent. Then lim_𝑛→∞𝑡_𝑛=0.

Lemma 2.6 (see [18]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝜆_𝑖>0.(𝑖=1,2,3,…,𝑁) such that ∑^𝑁𝑖=1𝜆_𝑖=1. Let 𝑇₁,𝑇₂,𝑇₃,…,𝑇_𝑁∶𝐶→𝐶 be nonexpansive mappings such that ⋂^𝑁𝑖=1𝐹(𝑇_𝑖)≠∅, and let 𝑇=∑^𝑁𝑖=1𝜆_𝑖𝑇_𝑖. Then 𝑇 is nonexpansive from 𝐶 into itself and 𝐹(𝑇)=⋂^𝑁𝑖=1𝐹(𝑇_𝑖).

3. Main Result

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹∶𝐶→𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator and 𝑉∶𝐶→𝐻 be an 𝐿-Lipschitzian mapping. Let 𝒯={𝑇_𝑛} be a sequence of nearly nonexpansive mappings from 𝐶 into itself with respect to a sequence {𝑎_𝑛} such that 𝐹(𝒯)≠∅ and 𝑇 be a mapping from 𝐶 into itself defined by 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥 for all 𝑥∈𝐶. Suppose that 𝐹(𝑇)=𝐹(𝒯),0<𝜇<2𝜂/𝑘² and 0≤𝛾𝐿<𝜏, where 𝜏=1−√1−𝜇(2𝜂−𝜇𝑘²). For an arbitrary 𝑥₁∈𝐶, consider the sequence {𝑥_𝑛} in 𝐶 generated by the following iterative process: 𝑥𝑥₁∈𝐶,_𝑛+1=𝑃_𝐶𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛∀𝑛∈ℕ,(3.1) where {𝛼_𝑛} is a sequence in (0,1) satisfying the conditions:(a)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞;(b)either ∑^∞𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞ or lim_𝑛→∞𝛼_𝑛+1/𝛼_𝑛=1;(c)either ∑^∞𝑛=1𝒟_𝐵(𝑇_𝑛,𝑇_𝑛+1)<∞ or lim_𝑛→∞𝒟_𝐵(𝑇_𝑛,𝑇_𝑛+1)/𝛼_𝑛+1=0 for each 𝐵∈ℬ(𝐶);(d)lim_𝑛→∞𝑎_n/𝛼_𝑛=0. Then, the sequence {𝑥_𝑛} converges strongly to 𝑥̃∈𝐹(𝒯), where 𝑥̃ is the unique solution of variational inequality ⟨(𝜇𝐹−𝛾𝑉)𝑥̃,𝑥̃−𝑦⟩≤0∀𝑦∈𝐹(𝒯).(3.2)

Proof. Let 𝑇 be a mapping from 𝐶 into itself defined by 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥 for all 𝑥∈𝐶. It is clear that 𝑇 is a nonexpansive mapping. So, we have 𝐹(𝑇)≠∅. Now, we proceed with the following steps.
Step 1. ({𝑥_𝑛} is bounded). Let 𝑧∈𝐹(𝒯). From (3.1), we have ‖‖𝑥_𝑛+1−𝑧‖‖≤=‖‖‖‖𝑃𝛼_𝐶𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝑃_𝐶(𝑧)_𝑛‖‖≤‖‖𝛼𝛾𝑉𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝑧_𝑛‖‖≤𝛼_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑧+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑧‖‖≤1𝛾−𝐿𝛼‖‖_𝑛𝑥_𝑛−𝑧‖‖+𝛼_𝑛‖(𝛾𝑉−𝜇𝐹)𝑧‖+1−𝛼_𝑛𝜏‖‖𝑥_𝑛−𝑧‖‖+𝑎_𝑛(𝜏−𝛾𝐿)‖‖𝑥_𝑛−𝑧‖‖+𝛼_𝑛‖(𝛾𝑉−𝜇𝐹)𝑧‖+1−𝛼_𝑛𝜏𝑎_𝑛≤1−𝛼_𝑛(𝜏−𝛾𝐿)‖‖𝑥_𝑛−𝑧‖‖+𝛼_𝑛‖(𝛾𝑉−𝜇𝐹)𝑧‖+𝑎_𝑛.(3.3) Note that lim_𝑛→∞𝑎_𝑛/𝛼_𝑛=0, so there exists a constant 𝐾>0 such that 𝛼_𝑛‖(𝛾𝑉−𝜇𝐹)𝑧‖+𝑎_𝑛𝛼_𝑛≤𝐾∀𝑛∈ℕ.(3.4) Thus, we have ‖‖𝑥_𝑛+1−𝑧‖‖≤≤m1a−x𝛼_𝑛‖‖(𝑥𝜏_𝑛−𝛾𝐿)‖‖𝑥_𝑛−𝑧‖‖+𝛼_𝑛𝐾−𝑧‖‖,𝐾𝜏−𝛾𝐿∀𝑛∈ℕ.(3.5) Hence, {𝑥_𝑛} is bounded. So {𝑇_𝑛𝑥_𝑛} and {𝑉𝑥_𝑛} are bounded.
Step 2. (‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞). From (3.1), we have ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤=‖‖‖‖𝑃𝛼_𝐶𝑛𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝑃_𝐶𝛼_𝑛−1𝛾𝑉𝑥_𝑛−1+𝐼−𝛼_𝑛−1𝜇𝐹𝑇_𝑛−1𝑥_𝑛−1‖‖≤‖‖𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝛼_𝑛−1𝛾𝑉𝑥_𝑛−1+𝐼−𝛼_𝑛−1𝜇𝐹𝑇_𝑛−1𝑥_𝑛−1‖‖𝛾𝑉𝑥_𝑛−𝑉𝑥_𝑛−1+𝛾𝛼_𝑛−𝛼_𝑛−1𝑉𝑥_𝑛−1+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−1≤𝛼+_𝑛𝑇_𝑛𝑥_𝑛−1−𝑇_𝑛−1𝑥_𝑛−1+𝛼_𝑛−1𝜇𝐹𝑇_𝑛−1𝑥_𝑛−1−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−1‖‖+𝛾𝐿1‖‖−𝑥𝛼_𝑛𝑛−𝑥_𝑛−1‖‖+‖‖𝛾𝛼_𝑛−𝛼_𝑛−1𝑉𝑥_𝑛−1‖‖+𝜇‖‖𝛼_𝑛−1𝜏‖‖𝑇_𝑛𝑥_𝑛−𝑇_𝑛𝑥_𝑛−1‖‖+‖‖𝑇_𝑛𝑥_𝑛−1−𝑇_𝑛−1𝑥_𝑛−1‖‖≤1−𝛼_𝑛𝐹𝑇_𝑛−1𝑥_𝑛−1−𝛼_𝑛𝐹𝑇_𝑛𝑥_𝑛−1‖‖(𝜏−𝛾𝐿)‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝒟_𝐵𝑇_𝑛,𝑇_𝑛−1+1−𝛼_𝑛𝜏𝑎_𝑛++‖‖𝜇𝛾‖‖𝛼𝛼_𝑛𝑛−1−𝛼_𝑛−1𝑉𝑥_𝑛−1‖‖≤1−𝛼_𝑛𝐹𝑇_𝑛−1𝑥_𝑛−1−𝐹𝑇_𝑛𝑥_𝑛−1−𝛼_𝑛−𝛼_𝑛−1𝐹𝑇_𝑛𝑥_𝑛−1‖‖+𝑀||𝛼_𝑛(𝜏−𝛾𝐿)‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝒟_𝐵𝑇_𝑛,𝑇_𝑛−11+𝜇𝛼_𝑛−1𝑘−𝛼_𝑛−1||+𝑎_𝑛,(3.6) for some constant 𝑀>0. Thus, using Lemma 2.5, we get ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞.
Step 3. We have (‖𝑥_𝑛−𝑇𝑥_𝑛‖→0 as 𝑛→∞). Note that ‖‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖‖=≤‖‖‖‖𝑥𝑥_𝑛𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑇_𝑛𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑃_𝐶𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝑃_𝐶𝑇_𝑛𝑥_𝑛‖‖=‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖‖−𝑥_𝑛+1‖‖+𝛼_𝑛‖‖𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑥_𝑛‖‖⟶0as𝑛⟶∞.(3.7) Since ‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖≤≤‖‖‖‖𝑥𝑥_𝑛𝑛−𝑇_𝑛𝑥_𝑛‖‖+‖‖𝑇_𝑛𝑥_𝑛−𝑇𝑥_𝑛‖‖−𝑇_𝑛𝑥_𝑛‖‖+𝒟_𝐵𝑇_𝑛,𝑇,(3.8) it follows that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0.
Step 4. We have (limsup_𝑛→∞⟨𝑥_𝑛−𝑥̃,(𝛾𝑉−𝜇𝐹)𝑥̃⟩≤0). Let us choose a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that limsup_𝑛→∞⟨𝑥_𝑛−𝑥̃,(𝛾𝑉−𝜇𝐹)𝑥̃⟩=lim_𝑘→∞𝑥_𝑛𝑘−𝑥̃,(𝛾𝑉−𝜇𝐹)𝑥̃.(3.9) Without loss of generality, we may assume that 𝑥_𝑛𝑘⇀𝑧∈𝐶. By using Lemma 2.4, we get that 𝑧∈𝐹(𝑇). Note that 𝐹(𝑇)=𝐹(𝒯), it follows that 𝑧∈𝐹(𝒯). Hence from (3.2), we get the following: limsup_𝑛→∞⟨𝑥_𝑛−𝑥̃,(𝛾𝑉−𝜇𝐹)𝑥̃⟩=⟨𝑧−𝑥̃,(𝛾𝑉−𝜇𝐹)𝑥̃⟩≤0.(3.10)
Step 5. We have (𝑥_𝑛→𝑥̃ as 𝑛→∞). Set 𝑦_𝑛=𝛼_𝑛𝛾𝑉𝑥_𝑛+(𝐼−𝛼_𝑛𝜇𝐹)𝑇_𝑛𝑥_𝑛 and 𝛾_𝑛=𝛼_𝑛(𝜏−𝛾𝐿). Noticing that 𝑥_𝑛+1=𝑃_𝐶(𝑦_𝑛). From (3.1), we have ‖‖𝑥_𝑛+1−𝑥̃‖‖²≤=𝑦𝑦_𝑛−𝑥̃,𝑥_𝑛+1−𝑥̃+𝑃_𝐶𝑦_𝑛−𝑦_𝑛,𝑃_𝐶𝑦_𝑛−𝑥_𝑛̃=𝛼_𝑛−𝑥̃,𝑥_𝑛+1−𝑥̃+𝛾𝑉𝐼𝑥−_𝑛−𝛼_𝑛𝜇𝐹𝑥̃,𝑥_𝑛+1−𝑥̃=𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥̃,𝑥_𝑛+1−𝑥̃+𝛾𝑉𝐼𝑥−_𝑛−𝛼_𝑛𝑉𝑥̃,𝑥_𝑛+1−𝑥̃+𝛼_𝑛𝛾𝑉𝑥̃−𝜇𝐹𝑥̃,𝑥_𝑛+1−𝑥̃≤𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥̃,𝑥_𝑛+1−𝑥̃+𝛾𝐿1‖‖−𝑥𝛼_𝑛𝑛−𝑥̃‖‖‖‖𝑥_𝑛+1−𝑥̃‖‖+𝛼_𝑛(𝛾𝑉−𝜇𝐹)𝑥̃,𝑥_𝑛+1−𝑥̃=1−𝛼_𝑛𝜏‖‖𝑥_𝑛−𝑥̃‖‖+𝑎_𝑛‖‖𝑥_𝑛+1−𝑥̃‖‖+𝛼_𝑛(𝜏−𝛾𝐿)‖‖𝑥_𝑛−𝑥̃‖‖‖‖𝑥_𝑛+1−𝑥̃‖‖≤1−𝛼_𝑛(𝛾𝑉−𝜇𝐹)𝑥̃,𝑥_𝑛+1−𝑥̃+1−𝛼_𝑛𝜏𝑎_𝑛‖‖𝑥_𝑛+1−𝑥̃‖‖(𝜏−𝛾𝐿)12‖‖𝑥^𝑛+𝛼_𝑛−𝑥̃‖‖²+‖‖𝑥_𝑛+1−𝑥̃‖‖²(𝛾𝑉−𝜇𝐹)𝑥̃,𝑥_𝑛+1−𝑥̃+𝑎_𝑛‖‖𝑥_𝑛+1−𝑥̃‖‖.(3.11) Hence, we have ‖‖𝑥_𝑛+1−𝑥̃‖‖²≤1−𝛼^𝑛(𝜏−𝛾𝐿)1+𝛼_𝑛(𝜏−𝛾𝐿)‖‖𝑥^𝑛−𝑥̃‖‖²+2𝛼^𝑛1+𝛾_𝑛(𝛾𝑉−𝜇𝐹)𝑥̃,𝑥_𝑛+1+2𝑎_𝑛−𝑥̃1+𝛾_𝑛≤1−𝛼_𝑛‖‖𝑥_𝑛+1−𝑥̃‖‖(𝜏−𝛾𝐿)‖‖𝑥_𝑛−𝑥̃‖‖²+2𝛼^𝑛1+𝛾_𝑛(𝛾𝑉−𝜇𝐹)𝑥̃,𝑥_𝑛+1+2𝑎_𝑛−𝑥̃1+𝛾_𝑛=1−𝛾_𝑛‖‖𝑥_𝑛+1−𝑥̃‖‖‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛𝛿_𝑛+12+𝑎^𝑛𝛾_𝑛‖‖𝑥_𝑛+1−𝑥̃‖‖,(3.12) where 𝛿_𝑛=21+𝛾_𝑛(𝜏−𝛾𝐿)(𝛾𝑉−𝜇𝐹)𝑥̃,𝑥^𝑛+1−𝑥̃.(3.13) Noticing that lim_𝑛→∞𝑎_𝑛/𝛼_𝑛=0, it follows from Lemma 2.5 that lim_𝑛→∞𝑥_𝑛=𝑥̃. This completes the proof.
Now, we derive the main result of Ceng et al. ([13], Theorem  3.2) as the following corollary.

Corollary 3.2. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹∶𝐶→𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator and 𝑉∶𝐶→𝐻 be an 𝐿-Lipschitzian mapping. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping such that 𝐹(𝑇)≠∅. Suppose that 0<𝜇<2𝜂/𝑘² and 0≤𝛾𝐿<𝜏, where 𝜏=1−√1−𝜇(2𝜂−𝜇𝑘²). For an arbitrary 𝑥₁∈𝐶, consider the sequence {𝑥_𝑛} generated by the following iterative process: 𝑥𝑥₁∈𝐶,_𝑛+1=𝑃_𝐶𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇𝑥_𝑛∀𝑛∈ℕ,(3.14) where {𝛼_𝑛} is a sequence in (0,1) satisfying the conditions (a) and (b) of Theorem 3.1.
Then, the sequence {𝑥_𝑛} converges strongly to 𝑥̃∈𝐹(𝑇), where 𝑥̃ is the unique solution of the following variational inequality: ⟨(𝜇𝐹−𝛾𝑉)𝑥̃,𝑥̃−𝑦⟩≤0∀𝑦∈𝐹(𝑇).(3.15)

Again, we derive the result of Tian ([11], Theorem  3.2) as the following corollary.

Corollary 3.3. Let 𝐻 be a real Hilbert space. Let 𝑓 be an 𝛼-contraction on 𝐻 and 𝐹∶𝐻→𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator. Let 𝑇∶𝐻→𝐻 be a nonexpansive mapping such that 𝐹(𝑇)≠∅. Suppose that 0<𝜇<2𝜂/𝑘² and 0<𝛾𝛼<𝜏, where 𝜏=𝜇(𝜂−𝜇𝑘²/2). For an arbitrary 𝑥₁∈𝐻, consider the sequence {𝑥_𝑛} generated by the following iterative process: 𝑥𝑥₁∈𝐻,_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇𝑥_𝑛∀𝑛∈ℕ,(3.16) where {𝛼_𝑛} is a sequence in (0,1) satisfying the conditions (𝑎) and (𝑏) of Theorem 3.1.
Then, the sequence {𝑥_𝑛} converges strongly to 𝑥̃∈𝐹(𝑇), where 𝑥̃ is the unique solution of the following variational inequality: ⟨(𝜇𝐹−𝛾𝑓)𝑥̃,𝑥̃−𝑦⟩≤0∀𝑦∈𝐹(𝑇).(3.17)

The following result obtains immediately from Theorem 3.1.

Corollary 3.4. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹∶𝐶→𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator and 𝑉∶𝐶→𝐻 be an 𝐿-Lipschitzian mapping. Let {𝑇_𝑛} be a sequence of nonexpansive mappings from 𝐶 into itself such that ⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅ and 𝑇 be a mapping from 𝐶 into itself defined by 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥 for all 𝑥∈𝐶. Suppose that 0<𝜇<2𝜂/𝑘² and 0≤𝛾𝐿<𝜏, where 𝜏=1−√1−𝜇(2𝜂−𝜇𝑘²). For an arbitrary 𝑥₁∈𝐶, consider the sequence {𝑥_𝑛} in 𝐶 generated by the following iterative process: 𝑥𝑥₁∈𝐶,_𝑛+1=𝑃_𝐶𝛼_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛∀𝑛∈ℕ,(3.18) where {𝛼_𝑛} is a sequence in (0,1) satisfying the conditions (a)–(c) of Theorem 3.1.
Then, the sequence {𝑥_𝑛} converges strongly to 𝑥̃∈⋂^∞𝑛=1𝐹(𝑇_𝑛), where 𝑥̃ is the unique solution of the following variational inequality: ⟨(𝜇𝐹−𝛾𝑉)𝑥̃,𝑥̃−𝑦⟩≤0∀𝑦∈^∞𝑛=1𝐹𝑇_𝑛.(3.19)

4. Application

Recall that the so-called problem of image recovery is essentially to find a common element of finitely many nonexpansive retracts 𝐶₁,𝐶₂,…,𝐶_𝑟 of 𝐶 with ⋂^𝑟𝑖=1𝐶_𝑖≠∅. It is easy to see that every nonexpansive retraction 𝑃_𝑖 of 𝐶 onto 𝐶_𝑖 is a nonexpansive mapping of C into itself. There is no doubt that the problem of image recovery is equivalent to finding a common fixed point of finitely many nonexpansive mappings 𝑃₁,𝑃₂,…,𝑃_𝑟 of 𝐶 into itself. Applying our main result, we obtain the following result which improves a number of results connected to the problem of image recovery.

Theorem 4.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹∶𝐶→𝐻 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator and 𝑉∶𝐶→𝐻 be an 𝐿-Lipschitzian mapping. Let 𝜆_𝑖>0(𝑖=1,2,3,…,𝑁) such that ∑^𝑁𝑖=1𝜆_𝑖=1 and 𝑇₁,𝑇₂,𝑇₃,…,𝑇_𝑁∶𝐶→𝐶 be nonexpansive mappings such that ⋂^𝑁𝑖=1𝐹(𝑇_𝑖)≠∅. Suppose that 0<𝜇<2𝜂/𝑘² and 0≤𝛾𝐿<𝜏, where 𝜏=1−√1−𝜇(2𝜂−𝜇𝑘²). For an arbitrary 𝑥₁∈𝐶, consider the sequence {𝑥_𝑛} in 𝐶 generated by the following iterative process:𝑥𝑥₁∈𝐶,_𝑛+1=𝑃_𝐶𝛼^𝑛𝛾𝑉𝑥_𝑛+𝐼−𝛼_𝑛𝜇𝐹^𝑁_𝑖=1𝜆_𝑖𝑇_𝑖𝑥_𝑛∀𝑛∈ℕ,(4.1) where {𝛼_𝑛} is a sequence in (0,1) satisfying the conditions (𝑎) and (𝑏) of Theorem 3.1.
Then, the sequence {𝑥_𝑛} converges strongly to 𝑥̃∈⋂^𝑁𝑖=1𝐹(𝑇_𝑖), where 𝑥̃ is the unique solution of the following variational inequality: ⟨(𝜇𝐹−𝛾𝑉)𝑥̃,𝑥̃−𝑦⟩≤0∀𝑦∈^𝑁𝑖=1𝐹𝑇_𝑖.(4.2)

Proof. Define 𝑇=∑^𝑁𝑖=1𝜆_𝑖𝑇_𝑖. Then 𝑇 is nonexpansive mapping from 𝐶 into itself. Thus, using Lemma 2.6, we get 𝐹(𝑇)=⋂^𝑁𝑖=1𝐹(𝑇_𝑖). Therefore, the proof follows from Corollary 3.2.

5. Numerical Example

For showing the effectiveness and convergence of the sequence generated by the considered iterative scheme, we discuss the following example.

Example 5.1. Let 𝐻=ℝ and 𝐶=[0,1]. Let 𝑇 be a self-mapping defined by 𝑇𝑥=1−𝑥 for all 𝑥∈𝐶. Let 𝐹,𝑉∶𝐶→𝐻 be two mappings defined by 𝐹𝑥=4𝑥 and 𝑉𝑥=2𝑥 for all 𝑥∈𝐶, where 𝐹 is a 𝑘-Lipschitzian and 𝜂-strongly monotone, and 𝑉 is an 𝐿-Lipschitzian mapping. We take 0<𝜇<2𝜂/𝑘² and 0≤𝛾𝐿<𝜏, and we have 𝜇=1/4,𝜏=1 and 𝛾=1/4. Define {𝛼_𝑛} in (0,1) by 𝛼_𝑛=1/𝑛+1. Without loss of generality, we may assume that 𝑎_𝑛=1/𝑛^3/2 for all 𝑛∈ℕ. For each 𝑛∈ℕ, define 𝑇_𝑛∶𝐶→𝐶 by 𝑇_𝑛𝑥=𝑎1−𝑥,if𝑥∈[0,1),_𝑛,if𝑥=1.(5.1)
In [15], it is proved that 𝒯={𝑇_𝑛} is a sequence of nearly nonexpansive mappings from 𝐶 into itself such that 𝐹(𝒯)={1/2} and 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥 for all 𝑥∈𝐶, where 𝑇 is nonexpansive mapping.

It can be observed that all the assumptions of Theorem 3.1 are satisfied and the sequence {𝑥_𝑛} generated by (3.1) converges to a unique solution 1/2 of variational inequality (3.2) over 𝐹(𝒯). The graphical presentation of the convergence of the sequence {𝑥_𝑛} generated by the iterative scheme (3.1) is given in Figure 1.

Figure 1 

Acknowledgments

The authors would like to thank the referees for useful comments and suggestions.