Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems

Abstract

We consider a general variational inequality and fixed point problem, which is to find a point 𝑥^∗ with the property that (GVF): 𝑥^∗∈GVI(𝐶,𝐴) and 𝑔(𝑥^∗)∈Fix(𝑆) where GVI(𝐶,𝐴) is the solution set of some variational inequality Fix(𝑆) is the fixed points set of nonexpansive mapping 𝑆, and 𝑔 is a nonlinear operator. Assume the solution set Ω of (GVF) is nonempty. For solving (GVF), we suggest the following method 𝑔(𝑥_𝑛+1)=𝛽𝑔(𝑥_𝑛)+(1−𝛽)𝑆𝑃_𝐶[𝛼_𝑛𝐹(𝑥_𝑛)+(1−𝛼_𝑛)(𝑔(𝑥_𝑛)−𝜆𝐴𝑥_𝑛)], 𝑛≥0. It is shown that the sequence {𝑥_𝑛} converges strongly to 𝑥^∗∈Ω which is the unique solution of the variational inequality ⟨𝐹(𝑥^∗)−𝑔(𝑥^∗),𝑔(𝑥)−𝑔(𝑥^∗)⟩≤0, for all 𝑥∈Ω.

1. Introduction

Let 𝐴∶𝐶→𝐻 and 𝑔∶𝐶→𝐶 be two nonlinear mappings. We concern the following generalized variational inequality of finding 𝑢∈𝐶,𝑔(𝑢)∈𝐶 such that𝑔(𝑣)−𝑔(𝑢),𝐴𝑢≥0,∀𝑔(𝑣)∈𝐶. (1.1) The solution set of (1.1) is denoted by GVI(𝐶,𝐴,𝑔). It has been shown that a large class of unrelated odd-order and nonsymmetric obstacle, unilateral, contact, free, moving, and equilibrium problems arising in regional, physical, mathematical, engineering, and applied sciences can be studied in the unified and general framework of the general variational inequalities (1.1), see [1–16] and the references therein. Noor [17] has introduced a new type of variational inequality involving two nonlinear operators, which is called the general variational inequality. It is worth mentioning that this general variational inequality is remarkably different from the so-called general variational inequality which was introduced by Noor [18] in 1988. Noor [17] proved that the general variational inequalities are equivalent to nonlinear projection equations and the Wiener-Hopf equations by using the projection technique. Using this equivalent formulation, Noor [17] suggested and analyzed some iterative algorithms for solving the special general variational inequalities and further proved that these algorithms have strong convergence.

For 𝑔=𝐼, where 𝐼 is the identity operator, problem (1.1) is equivalent to finding 𝑢∈𝐶 such that⟨𝑣−𝑢,𝐴𝑢⟩≥0,∀𝑣∈𝐶, (1.2) which is known as the classical variational inequality introduced and studied by Stampacchia [19] in 1964. This field has been extensively studied due to a wide range of applications in industry, finance, economics, social, pure and applied sciences. For related works, please see [20–35]. Our main purposes in the present paper is devoted to study this topic.

Motivated and inspired by the works in this field, in this paper, we consider a general variational inequality and fixed point problem, which is to find a point 𝑥^∗ with the property that𝑥^∗∈GVI(𝐶,𝐴),𝑔𝑥^∗∈Fix(𝑆), (GVF) where Fix(𝑆) is the fixed points set of nonexpansive mapping 𝑆. Assume the solution set Ω of (GVF) is nonempty. For solving (GVF), we suggest the following method𝑔𝑥_𝑛+1=𝛽𝑔𝑥_𝑛+1−𝛽𝑆𝑃_𝐶𝛼_𝑛𝐹𝑥_𝑛+1−𝛼_𝑛𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛,𝑛≥0. (1.3) It is shown that the sequence {𝑥_𝑛} converges strongly to 𝑥^∗∈Ω which is the unique solution of the following variational inequality𝐹𝑥^∗−𝑔𝑥^∗,𝑔(𝑥)−𝑔𝑥^∗≤0,∀𝑥∈Ω. (1.4) Our results contain some interesting results as special cases.

2. Preliminaries

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 𝑆∶𝐶→𝐶 is said to be nonexpansive if𝑆𝑥−𝑆𝑦≤𝑥−𝑦, (2.1) for all 𝑥,𝑦∈𝐶. We denote by Fix(𝑆) the set of fixed points of 𝑆. A mapping 𝐹∶𝐶→𝐻 is said to be 𝐿-Lipschitz continuous, if there exists a constant 𝐿>0 such that ‖𝐹(𝑥)−𝐹(𝑦)‖≤𝐿‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. A mapping 𝐴∶𝐶→𝐻 is said to be 𝛼-inverse strongly 𝑔-monotone if and only if𝐴𝑥−𝐴𝑦,𝑔(𝑥)−𝑔𝑦≥𝛼𝐴𝑥−𝐴𝑦², (2.2) for some 𝛼>0 and for all 𝑥,𝑦∈𝐶. A mapping 𝑔∶𝐶→𝐶 is said to be strongly monotone if there exists a constant 𝛾>0 such that𝑔(𝑥)−𝑔𝑦,𝑥−𝑦≥𝛾𝑥−𝑦², (2.3) for all 𝑥,𝑦∈𝐶.

Let 𝐵 be a mapping of 𝐻 into 2^𝐻. The effective domain of 𝐵 is denoted by dom(𝐵), that is, dom(𝐵)={𝑥∈𝐻∶𝐵𝑥≠∅}. A multivalued mapping 𝐵 is said to be a monotone operator on 𝐻 if and only if𝑥−𝑦,𝑢−𝑣≥0, (2.4) for all 𝑥,𝑦∈dom(𝐵),𝑢∈𝐵𝑥, and 𝑣∈𝐵𝑦. A monotone operator 𝐵 on 𝐻 is said to be maximal if and only if its graph is not strictly contained in the graph of any other monotone operator on 𝐻. Let 𝐵 be a maximal monotone operator on 𝐻 and let 𝐵⁻¹0={𝑥∈𝐻∶0∈𝐵𝑥}.

It is well known that, for any 𝑢∈𝐻, there exists a unique 𝑢₀∈𝐶 such that𝑢−𝑢₀=inf{‖𝑢−𝑥‖∶𝑥∈𝐶}. (2.5) We denote 𝑢₀ by 𝑃_𝐶𝑢, where 𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. The metric projection 𝑃_𝐶 of 𝐻 onto 𝐶 has the following basic properties:(i)‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻;(ii)⟨𝑥−𝑦,𝑃_𝐶𝑥−𝑃_𝐶𝑦⟩≥‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖² for every 𝑥,𝑦∈𝐻;(iii)⟨𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥⟩≤0 for all 𝑥∈𝐻, 𝑦∈𝐶.

It is easy to see that the following is true:𝑢∈GVI𝐶,𝐴,𝑔⟺𝑔(𝑢)=𝑃_𝐶𝑔(𝑢)−𝜆𝐴(𝑢),∀𝜆>0. (2.6)

We use the following notation:(i)𝑥_𝑛⇀𝑥 stands for the weak convergence of (𝑥_𝑛) to 𝑥;(ii)𝑥_𝑛→𝑥 stands for the strong convergence of (𝑥_𝑛) to 𝑥.

We need the following lemmas for the next section.

Lemma 2.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐺∶𝐶→𝐶 be a nonlinear mapping and let the mapping 𝐴∶𝐶→𝐻 be 𝛼-inverse strongly 𝑔-monotone. Then, for any 𝜆>0, one has 𝑃_𝐶𝑔(𝑥)−𝜆𝐴𝑥−𝑃_𝐶𝑔𝑦−𝜆𝐴𝑦²≤𝑔(𝑥)−𝑔𝑦²+𝜆(𝜆−2𝛼)𝐴𝑥−𝐴𝑦²,𝑥,𝑦∈𝐶. (2.7)

Proof. Consider the following:𝑃_𝐶𝑔(𝑥)−𝜆𝐴𝑥−𝑃_𝐶𝑔𝑦−𝜆𝐴𝑦²≤𝑔(𝑥)−𝑔𝑦−𝜆𝐴𝑥−𝐴𝑦²=𝑔(𝑥)−𝑔𝑦²−2𝜆𝐴𝑥−𝐴𝑦,𝑔(𝑥)−𝑔𝑦+𝜆²𝐴𝑥−𝐴𝑦²≤𝑔(𝑥)−𝑔𝑦²−2𝜆𝛼𝐴𝑥−𝐴𝑦²+𝜆²𝐴𝑥−𝐴𝑦²≤𝑔(𝑥)−𝑔𝑦²+𝜆(𝜆−2𝛼)𝐴𝑥−𝐴𝑦². (2.8) If 𝜆∈[0,2𝛼], we have 𝑃_𝐶𝑔(𝑥)−𝜆𝐴𝑥−𝑃_𝐶𝑔𝑦−𝜆𝐴𝑦≤𝑔(𝑥)−𝑔𝑦−𝜆𝐴𝑥−𝐴𝑦≤𝑔(𝑥)−𝑔𝑦. (2.9)

Lemma 2.2 (see [36]). Let 𝐶 be a closed convex subset of a Hilbert space 𝐻. Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping. Then Fix(𝑆) is a closed convex subset of 𝐶 and the mapping 𝐼−𝑆 is demiclosed at 0, that is, whenever {𝑥_𝑛}⊂𝐶 is such that 𝑥_𝑛⇀𝑥 and (𝐼−𝑆)𝑥_𝑛→0, then (𝐼−𝑆)𝑥=0.

Lemma 2.3 (see [37]). 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 𝑥_𝑛+1=(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑥_𝑛 for all 𝑛≥0 and limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.4 (see [38]). Assume {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛𝛾_𝑛, (2.10) 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.

3. Main Results

In this section, we will prove our main results.

Theorem 3.1. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐹∶𝐶→𝐻 be an 𝐿-Lipschitz continuous mapping, 𝑔∶𝐶→𝐶 be a weakly continuous and 𝛾-strongly monotone mapping such that 𝑅(𝑔)=𝐶. Let 𝐴∶𝐶→𝐻 be an 𝛼-inverse strongly 𝑔-monotone mapping and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping. Suppose that Ω≠∅. Let 𝛽∈(0,1) and 𝛾∈(𝐿,2𝛼). For given 𝑥₀∈𝐶, let {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑔𝑥_𝑛+1=𝛽𝑔𝑥_𝑛+1−𝛽𝑆𝑃_𝐶𝛼_𝑛𝐹𝑥_𝑛+1−𝛼_𝑛𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛,𝑛≥0, (3.1) where {𝛼_𝑛}⊂(0,1) satisfies (𝐶1):lim_𝑛→∞𝛼_𝑛=0 and (𝐶2):∑_𝑛𝛼_𝑛=∞. Then the sequence {𝑥_𝑛} generated by (3.1) converges strongly to 𝑥^∗∈Ω which is the unique solution of the following variational inequality: 𝐹𝑥^∗−𝑔𝑥^∗,𝑔(𝑥)−𝑔𝑥^∗≤0,∀𝑥∈Ω. (3.2)

Proof. First, we show the solution set of variational inequality (3.2) is singleton. Assume 𝑥∈Ω also solves (3.2). Then, we have 𝐹𝑥^∗−𝑔𝑥^∗,𝑔(𝑥)−𝑔𝑥^∗≤0,𝐹(𝑥)−𝑔(𝑥),𝑔𝑥^∗−𝑔(𝑥)≤0. (3.3) It follows that 𝐹(𝑥)−𝑔(𝑥)−𝐹𝑥^∗+𝑔𝑥^∗,𝑔𝑥^∗(⟹𝑔𝑥−𝑔𝑥)≤0_∗−𝑔(𝑥)²≤𝐹𝑥^∗−𝐹(𝑥),𝑔𝑥^∗(⟹𝑔𝑥−𝑔𝑥)_∗−𝑔(𝑥)²≤𝐹𝑥^∗−𝐹(𝑥),𝑔𝑥^∗−𝑔(𝑥)≤𝐹𝑥^∗−𝐹(𝑥)𝑔𝑥^∗(⟹𝑔𝑥−𝑔𝑥)_∗−𝑔(𝑥)≤𝐹𝑥^∗−𝐹(𝑥). (3.4) Since 𝑔 is 𝛾-strongly monotone, we have 𝛾𝑥−𝑦²≤𝑔(𝑥)−𝑔𝑦,𝑥−𝑦≤𝑔(𝑥)−𝑔𝑦𝑥−𝑦,∀𝑥,𝑦∈𝐶. (3.5) Hence, 𝛾𝑥−𝑦≤𝑔(𝑥)−𝑔𝑦,∀𝑥,𝑦∈𝐶. (3.6) In particular, 𝛾‖𝑥^∗−𝑥‖≤‖𝑔(𝑥^∗)−𝑔(𝑥)‖. By (3.4), we deduce 𝛾𝑥^∗−𝑥≤𝑔𝑥^∗−𝑔(𝑥)≤𝐹𝑥^∗−𝐹(𝑥)≤𝐿𝑥^∗−𝑥, (3.7) which implies that 𝑥=𝑥^∗ because of 𝐿<𝛾 by the assumption. Therefore, the solution of variational inequality (3.2) is unique.
Pick up any 𝑢∈Ω. It is obvious that 𝑢∈GVI(𝐶,𝐴,𝑔) and 𝑔(𝑢)∈Fix(𝑆). Set 𝑢_𝑛=𝑃_𝐶[𝛼_𝑛𝐹(𝑥_𝑛)+(1−𝛼_𝑛)(𝑔(𝑥_𝑛)−𝜆𝐴𝑥_𝑛)],𝑛≥0. From (2.6), we know 𝑔(𝑢)=𝑃_𝐶[𝑔(𝑢)−𝜇𝐴𝑢] for any 𝜇>0. Hence, we have 𝑔(𝑢)=𝑃_𝐶𝑔(𝑢)−1−𝛼_𝑛𝜆𝐴𝑢=𝑃_𝐶𝛼_𝑛𝑔(𝑢)+1−𝛼_𝑛𝑔(𝑢)−𝜆𝐴𝑢,∀𝑛≥0. (3.8) From (3.6), (3.8), and Lemma 2.1, we get 𝑢_𝑛−𝑔(𝑢)=𝑃_𝐶𝛼_𝑛𝐹𝑥_𝑛+1−𝛼_𝑛𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛−𝑃_𝐶𝛼_𝑛𝑔(𝑢)+1−𝛼_𝑛𝑔(𝑢)−𝜆𝐴𝑢≤𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)+1−𝛼_𝑛𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛−𝑔(𝑢)−𝜆𝐴𝑢≤𝛼_𝑛𝐹𝑥_𝑛−𝐹(𝑢)+𝛼_𝑛𝐹(𝑢)−𝑔(𝑢)+1−𝛼_𝑛𝑔𝑥_𝑛−𝑔(𝑢)≤𝛼_𝑛𝐿𝑥_𝑛−𝑢+𝛼_𝑛𝐹(𝑢)−𝑔(𝑢)+1−𝛼_𝑛𝑔𝑥_𝑛(𝑢≤𝛼−𝑔)_𝑛𝐿𝛾𝑔𝑥_𝑛−𝑔(𝑢)+𝛼_𝑛𝐹(𝑢)−𝑔(𝑢)+1−𝛼_𝑛𝑔𝑥_𝑛(𝑢=𝐿−𝑔)1−1−𝛾𝛼_𝑛𝑔𝑥_𝑛−𝑔(𝑢)+𝛼_𝑛𝐹(𝑢)−𝑔(𝑢). (3.9) It follows from (3.1) that 𝑔𝑥_𝑛+1−𝑔(𝑢)≤𝛽𝑔𝑥_𝑛−𝑔(𝑢)+1−𝛽𝑆𝑢_𝑛(𝑢𝑔𝑥−𝑆𝑔)≤𝛽_𝑛−𝑔(𝑢)+1−𝛽𝑢_𝑛(𝑢𝑔𝑥−𝑔)≤𝛽_𝑛(𝑢+𝐿−𝑔)1−𝛽1−1−𝛾𝛼_𝑛𝑔𝑥_𝑛(𝑢+𝛼−𝑔)1−𝛽_𝑛𝐹(𝑢)(𝑢=𝐿−𝑔)1−1−𝛾1−𝛽𝛼_𝑛𝑔𝑥_𝑛(𝑢+𝐿−𝑔)1−𝛾1−𝛽𝛼_𝑛𝐹(𝑢)−𝑔(𝑢).1−𝐿/𝛾 (3.10) This indicates by induction that 𝑔𝑥_𝑛+1−𝑔(𝑢)≤max𝑔𝑥_𝑛(𝑢,𝐹(𝑢)(𝑢−𝑔)−𝑔).1−𝐿/𝛾 (3.11) Hence, {𝑔(𝑥_𝑛)} is bounded. By (3.6), we have ‖𝑥_𝑛−𝑢‖≤(1/𝛾)‖𝑔(𝑥_𝑛)−𝑔(𝑢)‖. This implies that {𝑥_𝑛} is bounded. Consequently, {𝐹(𝑥_𝑛)},{𝐴𝑥_𝑛},{𝑢_𝑛}, and {𝑆𝑢_𝑛} are all bounded.
Note that we can rewrite (3.1) as 𝑔(𝑥_𝑛+1)=𝛽𝑔(𝑥_𝑛)+(1−𝛽)𝑆𝑢_𝑛 for all 𝑛. Next, we will use Lemma 2.3 to prove that ‖𝑥_𝑛+1−𝑥_𝑛‖→0. In fact, we firstly have 𝑆𝑢_𝑛−𝑆𝑢_𝑛−1=𝑆𝑃_𝐶𝛼_𝑛𝐹𝑥_𝑛+1−𝛼_𝑛𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛−𝑆𝑃_𝐶𝛼_𝑛−1𝐹𝑥_𝑛−1+1−𝛼_𝑛−1𝑔𝑥_𝑛−1−𝜆𝐴𝑥_𝑛−1≤𝛼_𝑛𝐹𝑥_𝑛+1−𝛼_𝑛𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛−𝛼_𝑛−1𝐹𝑥_𝑛−1+1−𝛼_𝑛−1𝑔𝑥_𝑛−1−𝜆𝐴𝑥_𝑛−1≤𝛼_𝑛𝐹𝑥_𝑛−𝐹𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐹𝑥_𝑛−1+1−𝛼_𝑛𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛−𝑔𝑥_𝑛−1−𝜆𝐴𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑔𝑥_𝑛−1−𝜆𝐴𝑥_𝑛−1≤𝛼_𝑛𝐿𝑥_𝑛−𝑥_𝑛−1+1−𝛼_𝑛𝑔𝑥_𝑛−𝑔𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐹𝑥_𝑛−1+𝑔𝑥_𝑛−1−𝜆𝐴𝑥_𝑛−1≤𝛼_𝑛𝐿𝛾𝑔𝑥_𝑛−𝑔𝑥_𝑛−1+1−𝛼_𝑛𝑔𝑥_𝑛−𝑔𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐹𝑥_𝑛−1+𝑔𝑥_𝑛−1−𝜆𝐴𝑥_𝑛−1=𝐿1−1−𝛾𝛼_𝑛𝑔𝑥_𝑛−𝑔𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐹𝑥_𝑛−1+𝑔𝑥_𝑛−1−𝜆𝐴𝑥_𝑛−1. (3.12) It follows that 𝑆𝑢_𝑛−𝑆𝑢_𝑛−1−𝑔𝑥_𝑛−𝑔𝑥_𝑛−1≤𝛼_𝑛−𝛼_𝑛−1𝐹𝑥_𝑛−1+𝑔𝑥_𝑛−1−𝜆𝐴𝑥_𝑛−1. (3.13) Since 𝛼_𝑛→0 and the sequences {𝐹(𝑥_𝑛)},{𝑔(𝑥_𝑛)}, and {𝐴𝑥_𝑛} are bounded, we have limsup_𝑛→∞𝑆𝑢_𝑛−𝑆𝑢_𝑛−1−𝑔𝑥_𝑛−𝑔𝑥_𝑛−1≤0. (3.14) By Lemma 2.3, we obtain lim_𝑛→∞𝑆𝑢_𝑛−𝑔𝑥_𝑛=0. (3.15) Hence, lim_𝑛→∞𝑔𝑥_𝑛+1−𝑔𝑥_𝑛=lim_𝑛→∞1−𝛽𝑆𝑢_𝑛−𝑔𝑥_𝑛=0. (3.16) This together with (3.6) imply that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.17) By the convexity of the norm and (3.9), we have 𝑔𝑥_𝑛+1−𝑔(𝑢)²=𝛽𝑔𝑥_𝑛−𝑔(𝑢)+1−𝛽𝑆𝑢_𝑛−𝑆𝑔(𝑢)²≤𝛽𝑔𝑥_𝑛−𝑔(𝑢)²+1−𝛽𝑆𝑢_𝑛−𝑆𝑔(𝑢)²≤𝛽𝑔𝑥_𝑛−𝑔(𝑢)²+1−𝛽𝑢_𝑛−𝑔(𝑢)²≤𝛽𝑔𝑥_𝑛−𝑔(𝑢)²+1−𝛽𝛼_𝑛𝐹𝑥_𝑛(𝑢+−𝑔)1−𝛼_𝑛𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛−𝑔(𝑢)−𝜆𝐴𝑢²≤𝛽𝑔𝑥_𝑛−𝑔(𝑢)²+1−𝛽𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)²+1−𝛼_𝑛𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛−𝑔(𝑢)−𝜆𝐴𝑢². (3.18) From Lemma 2.1, we derive 𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛−𝑔(𝑢)−𝜆𝐴𝑢²≤𝑔𝑥_𝑛−𝑔(𝑢)²+𝜆(𝜆−2𝛼)𝐴𝑥_𝑛−𝐴𝑢². (3.19) Thus, 𝑔𝑥_𝑛+1−𝑔(𝑢)²≤𝛽𝑔𝑥_𝑛−𝑔(𝑢)²+1−𝛽𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)²+1−𝛼_𝑛𝑔𝑥_𝑛−𝑔(𝑢)²+𝜆(𝜆−2𝛼)𝐴𝑥_𝑛−𝐴𝑢²=1−𝛽𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)²+1−1−𝛽𝛼_𝑛𝑔𝑥_𝑛−𝑔(𝑢)²+1−𝛽1−𝛼_𝑛𝜆(𝜆−2𝛼)𝐴𝑥_𝑛−𝐴𝑢². (3.20) So, 1−𝛽1−𝛼_𝑛𝜆(2𝛼−𝜆)𝐴𝑥_𝑛−𝐴𝑢²≤1−𝛽𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)²+𝑔𝑥_𝑛−𝑔(𝑢)²−𝑔𝑥_𝑛+1−𝑔(𝑢)²≤1−𝛽𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)²+𝑔𝑥_𝑛−𝑔(𝑢)+𝑔𝑥_𝑛+1−𝑔(𝑢)𝑔𝑥_𝑛+1−𝑔𝑥_𝑛. (3.21) Since 𝛼_𝑛→0,‖𝑔(𝑥_𝑛+1)−𝑔(𝑥_𝑛)‖→0 and liminf_𝑛→∞(1−𝛽)(1−𝛼_𝑛)𝜆(2𝛼−𝜆)>0, we obtain lim_𝑛→∞𝐴𝑥_𝑛−𝐴𝑢=0. (3.22) Set 𝑦_𝑛=𝑔(𝑥_𝑛)−𝜆𝐴𝑥_𝑛−(𝑔(𝑢)−𝜆𝐴𝑢) for all 𝑛. By using the property of projection, we get 𝑢_𝑛−𝑔(𝑢)²=𝑃_𝐶𝛼_𝑛𝐹𝑥_𝑛+1−𝛼_𝑛𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛−𝑃_𝐶𝛼_𝑛𝑔(𝑢)+1−𝛼_𝑛𝑔(𝑢)−𝜆𝐴𝑢²≤𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)+1−𝛼_𝑛𝑦_𝑛,𝑢_𝑛(𝑢=1−𝑔)2𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)+1−𝛼_𝑛𝑦_𝑛²+𝑢_𝑛−𝑔(𝑢)²−𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)+1−𝛼_𝑛𝑦_𝑛−𝑢_𝑛+𝑔(𝑢)²≤12𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)²+1−𝛼_𝑛𝑔𝑥_𝑛−𝑔(𝑢)²+𝑢_𝑛−𝑔(𝑢)²−𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)−𝑦_𝑛+𝑔𝑥_𝑛−𝑢_𝑛−𝜆𝐴𝑥_𝑛−𝐴𝑢²=12𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)²+1−𝛼_𝑛𝑔𝑥_𝑛−𝑔(𝑢)²+𝑢_𝑛−𝑔(𝑢)²−𝑔𝑥_𝑛−𝑢_𝑛²−𝜆²𝐴𝑥_𝑛−𝐴𝑢−𝛼²_𝑛𝐹𝑥_𝑛−𝑔(𝑢)−𝑦_𝑛²+2𝜆𝛼_𝑛𝐴𝑥_𝑛−𝐴𝑢,𝐹𝑥_𝑛−𝑔(𝑢)−𝑦_𝑛+2𝜆𝑔𝑥_𝑛−𝑢_𝑛,𝐴𝑥_𝑛−𝐴𝑢−2𝛼_𝑛𝑔𝑥_𝑛−𝑢_𝑛,𝐹𝑥_𝑛−𝑔(𝑢)−𝑦_𝑛. (3.23) It follows that 𝑢_𝑛−𝑔(𝑢)²≤𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)²+1−𝛼_𝑛𝑔𝑥_𝑛−𝑔(𝑢)²−𝑔𝑥_𝑛−𝑢_𝑛²+2𝜆𝛼_𝑛𝐴𝑥_𝑛−𝐴𝑢𝐹𝑥_𝑛−𝑔(𝑢)−𝑦_𝑛+2𝜆𝑔𝑥_𝑛−𝑢_𝑛𝐴𝑥_𝑛−𝐴𝑢+2𝛼_𝑛𝑔𝑥_𝑛−𝑢_𝑛𝐹𝑥_𝑛−𝑔(𝑢)−𝑦_𝑛. (3.24) From (3.18) and (3.24), we have 𝑔𝑥_𝑛+1−𝑔(𝑢)²≤𝛽𝑔𝑥_𝑛−𝑔(𝑢)²+1−𝛽𝑢_𝑛−𝑔(𝑢)²≤𝛽𝑔𝑥_𝑛−𝑔(𝑢)²+1−𝛽𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)²+1−𝛼_𝑛1−𝛽𝑔𝑥_𝑛−𝑔(𝑢)²−1−𝛽𝑔𝑥_𝑛−𝑢_𝑛²+2𝜆1−𝛽𝛼_𝑛𝐴𝑥_𝑛−𝐴𝑢𝐹𝑥_𝑛−𝑔(𝑢)−𝑦_𝑛+2𝜆1−𝛽𝑔𝑥_𝑛−𝑢_𝑛𝐴𝑥_𝑛𝛼−𝐴𝑢+21−𝛽_𝑛𝑔𝑥_𝑛−𝑢_𝑛𝐹𝑥_𝑛−𝑔(𝑢)−𝑦_𝑛≤𝑔𝑥_𝑛−𝑔(𝑢)²+𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)²−1−𝛽𝑔𝑥_𝑛−𝑢_𝑛²+2𝜆𝛼_𝑛𝐴𝑥_𝑛−𝐴𝑢𝐹𝑥_𝑛−𝑔(𝑢)−𝑦_𝑛+2𝜆𝑔𝑥_𝑛−𝑢_𝑛𝐴𝑥_𝑛−𝐴𝑢+2𝛼_𝑛𝑔𝑥_𝑛−𝑢_𝑛𝐹𝑥_𝑛−𝑔(𝑢)−𝑦_𝑛. (3.25) Then, we obtain 1−𝛽𝑔𝑥_𝑛−𝑢_𝑛²≤𝑔𝑥_𝑛−𝑔(𝑢)+𝑔𝑥_𝑛+1−𝑔(𝑢)𝑔𝑥_𝑛+1−𝑔𝑥_𝑛+𝛼_𝑛𝐹𝑥_𝑛−𝑔(𝑢)²+2𝜆𝛼_𝑛𝐴𝑥_𝑛−𝐴𝑢𝐹𝑥_𝑛−𝑔(𝑢)−𝑦_𝑛+2𝜆𝑔𝑥_𝑛−𝑢_𝑛𝐴𝑥_𝑛−𝐴𝑢+2𝛼_𝑛𝑔𝑥_𝑛−𝑢_𝑛𝐹𝑥_𝑛−𝑔(𝑢)−𝑦_𝑛. (3.26) Since lim_𝑛→∞𝛼_𝑛=0, lim_𝑛→∞‖𝑔(𝑥_𝑛+1)−𝑔(𝑥_𝑛)‖=0 and lim_𝑛→∞‖𝐴𝑥_𝑛−𝐴𝑢‖=0, we have lim_𝑛→∞𝑔𝑥_𝑛−𝑢_𝑛=0. (3.27) Next, we prove limsup_𝑛→∞⟨𝐹(𝑥^∗)−𝑔(𝑥^∗),𝑢_𝑛−𝑔(𝑥^∗)⟩≤0 where 𝑥^∗ is the unique solution of (3.2). We take a subsequence {𝑢_𝑛𝑖} of {𝑢_𝑛} such that limsup_𝑛→∞𝐹𝑥^∗−𝑔𝑥^∗,𝑢_𝑛−𝑔𝑥^∗=lim_𝑖→∞𝐹𝑥^∗−𝑔𝑥^∗,𝑢_𝑛𝑖−𝑔𝑥^∗=lim_𝑖→∞𝐹𝑥^∗−𝑔𝑥^∗,𝑔𝑥_𝑛𝑖−𝑔𝑥^∗. (3.28) Since {𝑥_𝑛𝑖} is bounded, there exists a subsequence {𝑥_𝑛𝑖𝑗} of {𝑥_𝑛𝑖} which converges weakly to some point 𝑧∈𝐶. Without loss of generality, we may assume that 𝑥_𝑛𝑖⇀𝑧. This implies that 𝑔(𝑥_𝑛𝑖)⇀𝑔(𝑧) due to the weak continuity of 𝑔. Now, we show 𝑧∈Ω. First, we note that from (3.15) and (3.27) that ‖𝑔(𝑥_𝑛)−𝑆𝑔(𝑥_𝑛)‖→0. Hence, lim_𝑖→∞‖𝑔(𝑥_𝑛𝑖)−𝑆𝑔(𝑥_𝑛𝑖)‖=0. By the demiclosedness principle of the nonexpansive mapping (see Lemma 2.2), we deduce 𝑔(𝑧)∈Fix(𝑆). Next, we only need to prove 𝑧∈GVI(𝐶,𝐴,𝑔). Set 𝑇𝑣=𝐴𝑣+𝑁_𝐶(𝑣),𝑣∈𝐶,∅,𝑣∉𝐶. (3.29) By [39], we know that 𝑇 is maximal 𝑔-monotone. Let (𝑣,𝑤)∈𝐺(𝑇). Since 𝑤−𝐴𝑣∈𝑁_𝐶(𝑣) and 𝑥_𝑛∈𝐶, we have 𝑔(𝑣)−𝑔𝑥_𝑛,𝑤−𝐴𝑣≥0. (3.30) From 𝑢_𝑛=𝑃_𝐶[𝛼_𝑛𝐹(𝑥_𝑛)+(1−𝛼_𝑛)(𝑔(𝑥_𝑛)−𝜆𝐴𝑥_𝑛)], we get 𝑔(𝑣)−𝑢_𝑛,𝑢_𝑛−𝛼_𝑛𝐹𝑥_𝑛+1−𝛼_𝑛𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛≥0. (3.31) It follows that 𝑔(𝑣)−𝑢_𝑛,𝑢_𝑛−𝑔𝑥_𝑛𝜆+𝐴𝑥_𝑛−𝛼_𝑛𝜆𝐹𝑥_𝑛−𝑔𝑥_𝑛+𝜆𝐴𝑥_𝑛≥0. (3.32) Then, 𝑔(𝑣)−𝑔𝑥_𝑛𝑖,𝑤≥𝑔(𝑣)−𝑔𝑥_𝑛𝑖≥𝑔(𝑣)𝑥,𝐴𝑣−𝑔_𝑛𝑖,𝐴𝑣−𝑔(𝑣)−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑔𝑥_𝑛𝑖𝜆−𝑔(𝑣)−𝑢_𝑛𝑖,𝐴𝑥_𝑛𝑖+𝛼_𝑛𝑖𝜆𝑔(𝑣)−𝑢_𝑛𝑖,𝐹𝑥_𝑛𝑖−𝑔𝑥_𝑛𝑖+𝜆𝐴𝑥_𝑛𝑖=𝑔(𝑣)−𝑔𝑥_𝑛𝑖,𝐴𝑣−𝐴𝑥_𝑛𝑖+𝑔(𝑣)−𝑔𝑥_𝑛𝑖,−𝐴𝑥_𝑛𝑖−𝑔(𝑣)−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑔𝑥_𝑛𝑖𝜆−𝑔(𝑣)−𝑢_𝑛𝑖,𝐴𝑥_𝑛𝑖+𝛼_𝑛𝑖𝜆𝑔(𝑣)−𝑢_𝑛𝑖,𝐹𝑥_𝑛𝑖−𝑔𝑥_𝑛𝑖+𝜆𝐴𝑥_𝑛𝑖≥−𝑔(𝑣)−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑔𝑥_𝑛𝑖𝜆−𝑔𝑥_𝑛𝑖−𝑢_𝑛𝑖,𝐴𝑥_𝑛𝑖+𝛼_𝑛𝑖𝜆𝑔(𝑣)−𝑢_𝑛𝑖,𝐹𝑥_𝑛𝑖−𝑔𝑥_𝑛𝑖+𝜆𝐴𝑥_𝑛𝑖. (3.33) Since ‖𝑔(𝑥_𝑛𝑖)−𝑢_𝑛𝑖‖→0 and 𝑔(𝑥_𝑛𝑖)⇀𝑔(𝑧), we deduce that ⟨𝑔(𝑣)−𝑔(𝑧),𝑤⟩≥0 by taking 𝑖→∞ in (3.33). Thus, 𝑧∈𝑇⁻¹0 by the maximal 𝑔-monotonicity of 𝑇. Hence, 𝑧∈GVI(𝐶,𝐴,𝑔). Therefore, 𝑧∈Ω. From (3.28), we obtain limsup_𝑛→∞𝐹𝑥^∗−𝑔𝑥^∗,𝑢_𝑛−𝑔𝑥^∗=lim_𝑖→∞𝐹𝑥^∗−𝑔𝑥^∗,𝑔𝑥_𝑛𝑖−𝑔𝑥^∗=𝐹𝑥^∗−𝑔𝑥^∗,𝑔(𝑧)−𝑔𝑥^∗≤0. (3.34) We take 𝑢=𝑥^∗ in (3.23) to get 𝑢_𝑛−𝑔𝑥^∗2≤𝛼_𝑛𝐹𝑥_𝑛−𝑔𝑥^∗,𝑢_𝑛−𝑔𝑥^∗+1−𝛼_𝑛𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛−𝑔𝑥^∗−𝜆𝐴𝑥^∗,𝑢_𝑛−𝑔𝑥^∗≤𝛼_𝑛𝐹𝑥_𝑛−𝐹𝑥^∗,𝑢_𝑛−𝑔𝑥^∗+𝛼_𝑛𝐹𝑥^∗−𝑔𝑥^∗,𝑢_𝑛−𝑔𝑥^∗+1−𝛼_𝑛𝑔𝑥_𝑛−𝜆𝐴𝑥_𝑛−𝑔𝑥^∗−𝜆𝐴𝑥^∗𝑢_𝑛−𝑔𝑥^∗≤𝛼_𝑛𝐿𝑥_𝑛−𝑥^∗𝑢_𝑛−𝑔𝑥^∗+𝛼_𝑛𝐹𝑥^∗−𝑔𝑥^∗,𝑢_𝑛−𝑔𝑥^∗+1−𝛼_𝑛𝑔𝑥_𝑛−𝑔𝑥^∗𝑢_𝑛−𝑔𝑥^∗≤𝛼_𝑛𝐿𝛾𝑔𝑥_𝑛−𝑔𝑥^∗𝑢_𝑛−𝑔𝑥^∗+𝛼_𝑛𝐹𝑥^∗−𝑔𝑥^∗,𝑢_𝑛−𝑔𝑥^∗+1−𝛼_𝑛𝑔𝑥_𝑛−𝑔𝑥^∗𝑢_𝑛−𝑔𝑥^∗=𝐿1−1−𝛾𝛼_𝑛𝑔𝑥_𝑛−𝑔𝑥^∗𝑢_𝑛−𝑔𝑥^∗+𝛼_𝑛𝐹𝑥^∗−𝑔𝑥^∗,𝑢_𝑛−𝑔𝑥^∗=1−1−𝐿/𝛾𝛼_𝑛2𝑔𝑥_𝑛−𝑔𝑥^∗2+12𝑢_𝑛−𝑔𝑥^∗2+𝛼_𝑛𝐹𝑥^∗−𝑔𝑥^∗,𝑢_𝑛−𝑔(𝑢). (3.35) It follows that 𝑢_𝑛−𝑔𝑥^∗2≤𝐿1−1−𝛾𝛼_𝑛𝑔𝑥_𝑛−𝑔𝑥^∗2+2𝛼_𝑛𝐹𝑥^∗−𝑔𝑥^∗,𝑢_𝑛−𝑔𝑥^∗. (3.36) Therefore, 𝑔𝑥_𝑛+1−𝑔𝑥^∗2≤𝛽𝑔𝑥_𝑛−𝑔𝑥^∗2+1−𝛽𝑢_𝑛−𝑔𝑥^∗2≤𝛽𝑔𝑥_𝑛−𝑔𝑥^∗2+𝐿1−𝛽1−1−𝛾𝛼_𝑛𝑔𝑥_𝑛−𝑔𝑥^∗2+21−𝛽𝛼_𝑛𝐹𝑥^∗−𝑔𝑥^∗,𝑢_𝑛−𝑔𝑥^∗=𝐿1−1−𝛾1−𝛽𝛼_𝑛𝑔𝑥_𝑛−𝑔𝑥^∗2+21−𝛽𝛼_𝑛𝐹𝑥^∗−𝑔𝑥^∗,𝑢_𝑛−𝑔𝑥^∗=𝐿1−1−𝛾1−𝛽𝛼_𝑛𝑔𝑥_𝑛−𝑔𝑥^∗2+𝐿1−𝛾1−𝛽𝛼_𝑛2𝐹𝑥1−𝐿/𝛾^∗−𝑔𝑥^∗,𝑢_𝑛−𝑔𝑥^∗=1−𝛾_𝑛𝑔𝑥_𝑛−𝑔𝑥^∗2+𝛿_𝑛𝛾_𝑛, (3.37) where 𝛾_𝑛=(1−𝐿/𝛾)(1−𝛽)𝛼_𝑛 and 𝛿_𝑛=(2/(1−𝐿/𝛾))⟨𝐹(𝑥^∗)−𝑔(𝑥^∗),𝑢_𝑛−𝑔(𝑥^∗)⟩. From condition (𝐶2), we have ∑_𝑛𝛾_𝑛=∞. By (3.34), we have limsup_𝑛→∞𝛿_𝑛≤0. We can therefore apply Lemma 2.4 to conclude that 𝑔(𝑥_𝑛)→𝑔(𝑥^∗) and 𝑥_𝑛→𝑥^∗. This completes the proof.

Corollary 3.2. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐹∶𝐶→𝐻 be an 𝐿-contraction. Let 𝐴∶𝐶→𝐻 be an 𝛼-inverse strongly monotone mapping and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping. Suppose that Ω≠∅. Let 𝛽∈(0,1) and 𝛾∈(𝐿,2𝛼). For given 𝑥₀∈𝐶, let {𝑥_𝑛}⊂𝐶 be a sequence generated by 𝑥_𝑛+1=𝛽𝑥_𝑛+1−𝛽𝑆𝑃_𝐶𝛼_𝑛𝐹𝑥_𝑛+1−𝛼_𝑛𝑥_𝑛−𝜆𝐴𝑥_𝑛,𝑛≥0, (3.38) where {𝛼_𝑛}⊂(0,1) satisfies (𝐶1):lim_𝑛→∞𝛼_𝑛=0 and (𝐶2)∶∑_𝑛𝛼_𝑛=∞. Then the sequence {𝑥_𝑛} generated by (3.38) converges strongly to 𝑥^∗∈Ω which is the unique solution of the following variational inequality: 𝐹𝑥^∗−𝑥^∗,𝑥−𝑥^∗≤0,∀𝑥∈Ω. (3.39)

Acknowledgments

The authors are very grateful to the referees for their comments and suggestions which improved the presentation of this paper. The first author was partially supported by the Program TH-1-3, Optimization Lean Cycle, of Subprojects TH-1 of Spindle Plan Four in Excellence Teaching and Learning Plan of Cheng Shiu University and was supported in part by NSC 100-2221-E-230-012. The second author was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105.