Iterative Methods for Equilibrium Problems and Monotone Inclusion Problems in Hilbert Spaces

Abstract

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of zeros of the sum of maximal monotone operators, and we obtain strong convergence theorems in Hilbert spaces. We also apply our results to the variational inequality and convex minimization problems. Our results extend and improve the recent result of Takahashi et al. (2012).

1. Introduction

Let 𝐻 be a real Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ, where ℝ is the set of real numbers. The equilibrium problem for 𝑓 is to find 𝑥∈𝐶 such that 𝑓𝑥,𝑦≥0∀𝑦∈𝐶. (1) The set of solutions of (1) is denoted by 𝐸𝑃(𝑓). Numerous problems in physics, optimization, and economics are reduced to find the solution of an equilibrium problem (e.g., see [1]). For solving an equilibrium problem, we assume that the bifunction 𝑓 satisfies the following conditions:(A1)𝑓(𝑥,𝑥)=0 for all 𝑥∈𝐶; (A2)𝑓 is monotone; that is, 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶; (A3) for every 𝑥,𝑦,𝑧∈𝐶, limsup_𝑡↓0𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦); (A4)𝑓(𝑥,⋅) is convex and lower semicontinuous for each 𝑥∈𝐶. Equilibrium problems have been studied extensively (see [1–6]).

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 monotone if 𝑥−𝑦,𝑢−𝑣≥0∀𝑥,𝑦∈dom(𝐵),𝑢∈𝐵𝑥,𝑣∈𝐵𝑦. (2) A monotone operator 𝐵 is said to be maximal if its graph is not properly contained in the graph of any other monotone operator. For a maximal monotone operator 𝐵 on 𝐻 and 𝑟>0, we may define a single-valued operator 𝐽_𝑟=(𝐼+𝑟𝐵)⁻¹∶𝐻→dom(𝐵), which is called the resolvent of 𝐵 for 𝑟. It is known that 𝐽_𝑟 is firmly nonexpansive. A basic problem for maximal monotone operator 𝐵 is to find𝑥∈𝐻suchthat0∈𝐵𝑥. (3) The classical method for solving problem (3) is the proximal point algorithm. Rockafellar [7] established the weak convergence of the proximal point algorithm for maximal monotone operators. The question whether the strong convergence of the proximal point algorithm holds was answered in the negative by Güler [8]. In order to obtain the strong convergence theorem, some researchers have been devoted to modifying the proximal point algorithm (see [9, 10]). Given a positive constant 𝛼, a mapping 𝐴∶𝐶→𝐻 is said to be 𝛼-inverse strongly monotone if 𝑥−𝑦,𝐴𝑥−𝐴𝑦≥𝛼𝐴𝑥−𝐴𝑦²∀𝑥,𝑦∈𝐶. (4) Generally, monotone inclusion problem is to find𝑥∈𝐶suchthat0∈(𝐴+𝐵)𝑥, (5) where the mapping 𝐴∶𝐶→𝐻 is inverse strongly monotone and 𝐵 is maximal monotone. We write (𝐴+𝐵)⁻¹0 for the set of solutions of problem (5); that is, (𝐴+𝐵)⁻¹0={𝑥∈𝐶∶0∈(𝐴+𝐵)𝑥}. It is well known that (𝐴+𝐵)⁻¹0=Fix(𝐽_𝜆(𝐼−𝜆𝐴)) (see [11]). Takahashi et al. [12] constructed the following iterative sequence. Let 𝑢∈𝐶, 𝑥₁=𝑥∈𝐶 and let {𝑥_𝑛} be a sequence generated by 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝛼_𝑛𝑢+1−𝛼_𝑛𝐽_𝜆𝑛𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛. (6) Under appropriate conditions, they proved that the sequence {𝑥_𝑛} converges strongly to a point 𝑧₀∈(𝐴+𝐵)⁻¹0. Lin and Takahashi [13] introduced an iterative sequence that converges strongly to an element of (𝐴+𝐵)⁻¹0∩𝐹⁻¹0, where 𝐹 is another maximal monotone operator.

Motivated by the above results, especially by Chuang et al. [2] and Takahashi et al. [12], we introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of zeros of the sum of maximal monotone operators, and we obtain strong convergence theorems in Hilbert spaces. We also apply our results to the variational inequality and convex minimization problems. Our results extend and improve the recent result of [12].

2. Preliminaries

Throughout this paper, let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖ and let 𝐶 be a nonempty closed convex subset of 𝐻. A mapping 𝑆∶𝐶→𝐻 is nonexpansive if ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. The set of fixed points of 𝑆 is denoted by Fix(𝑆); that is, Fix(𝑆)={𝑥∈𝐶∶𝑆𝑥=𝑥}. It is well known that Fix(𝑆) is closed and convex. We denote the strong convergence and the weak convergence of {𝑥_𝑛} to 𝑥 by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥, respectively. For any 𝑥∈𝐻, there exists a unique point 𝑃_𝐶𝑥∈𝐶 such that 𝑥−𝑃_𝐶𝑥≤𝑥−𝑦,∀𝑦∈𝐶. (7)𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. Note that 𝑃_𝐶 is a nonexpansive mapping. For 𝑥∈𝐻 and 𝑧∈𝐶, we have 𝑧=𝑃_𝐶𝑥⟺𝑥−𝑧,𝑦−𝑧≤0forevery𝑦∈𝐶. (8)

Now consider inverse strongly monotone operators. For 𝛼>0, let 𝐴∶𝐶→𝐻 be an 𝛼-inverse strongly monotone operator. If 0<𝜆≤2𝛼, then 𝐼−𝜆𝐴 is a nonexpansive mapping. Indeed, for 𝑥,𝑦∈𝐶  and 0<𝜆≤2𝛼, we get (𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦²=𝑥−𝑦−𝜆𝐴𝑥−𝐴𝑦²=𝑥−𝑦²−2𝜆𝑥−𝑦,𝐴𝑥−𝐴𝑦+𝜆²𝐴𝑥−𝐴𝑦²≤𝑥−𝑦²−2𝜆𝛼𝐴𝑥−𝐴𝑦²+𝜆²𝐴𝑥−𝐴𝑦²≤𝑥−𝑦²+𝜆(𝜆−2𝛼)𝐴𝑥−𝐴𝑦²≤𝑥−𝑦². (9) Therefore, the operator 𝐼−𝜆𝐴 is a nonexpansive mapping of 𝐶 into 𝐻.

We collect some important facts and tools.

Lemma 1 (see [1]). Let 𝐶 be a nonempty closed convex subset of 𝐻 and let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). If 𝑟>0 and 𝑥∈𝐻, then there exists 𝑧∈𝐶 such that 𝑓+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0∀𝑦∈𝐶. (10)

Lemma 2 (see [4]). Let 𝐶 be a nonempty closed convex subset of 𝐻, and let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). For 𝑟>0, define a mapping 𝑇_𝑟∶𝐻→2^𝐶 as follows: 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶𝑓𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0∀𝑦∈𝐶. (11) Then, the following hold: (i)𝑇_𝑟 is single valued;(ii)𝑇_𝑟 is firmly nonexpansive; that is, for any 𝑥,𝑦∈𝐻, ⟨𝑥−𝑦,𝑇_𝑟𝑥−𝑇_𝑟𝑦⟩≥‖𝑇_𝑟𝑥−𝑇_𝑟𝑦‖²;(iii)Fix(𝑇_𝑟)=𝐸𝑃(𝑓);(iv)𝐸𝑃(𝑓) is closed and convex.

Lemma 3 (see [6]). Suppose that (A1)–(A4) hold. If 𝑥,𝑦∈𝐻 and 𝑟₁,𝑟₂>0, then 𝑇_𝑟2𝑦−𝑇_𝑟1𝑥≤+𝑟𝑦−𝑥²−𝑟₁𝑟₂𝑇_𝑟2𝑦−𝑦. (12)

Lemma 4 (see [14]). Let 𝐻 be a real Hilbert space and let 𝐵 be a maximal monotone operator on 𝐻. Then, the following holds: 𝑠−𝑡𝑠𝐽_𝑠𝑥−𝐽_𝑡𝑥,𝐽_𝑠𝑥−𝑥≥𝐽_𝑠𝑥−𝐽_𝑡𝑥² (13) for all 𝑠,𝑡>0 and 𝑥∈𝐻.

Lemma 5 (see [15]). 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)𝑥_𝑛+𝛽_𝑛𝑦_𝑛for all integers 𝑛≥1 and limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

The following lemma is an immediate consequence of the inner product on 𝐻.

Lemma 6. For all 𝑥,𝑦∈𝐻, the inequality ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩ holds.

Lemma 7 (demiclosedness principle [16]). Let 𝐶 be a nonempty closed convex subset of 𝐻 and 𝑆∶𝐶→𝐻 a nonexpansive mapping. Let 𝑥 be a point in 𝐻 and let {𝑥_𝑛} be a sequence in 𝐶. Suppose that 𝑥_𝑛⇀𝑥 and that 𝑥_𝑛−𝑆𝑥_𝑛→0. Then, 𝑥∈Fix(𝑆).

Lemma 8 (see [17]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying 𝑎_𝑛+1≤(1−𝛼_𝑛)𝑎_𝑛+𝛼_𝑛𝛽_𝑛, where(i){𝛼_𝑛}⊂(0,1), ∑^∞_𝑛=1𝛼_𝑛=∞; (ii)limsup_𝑛→∞𝛽_𝑛≤0. Then, lim_𝑛→∞𝑎_𝑛=0.

3. Strong Convergence Theorems

In this section, we introduce a new iterative method for finding a common element of the set of zeros of the sum of maximal monotone operators and the set of solutions of an equilibrium problem, and we prove the strong convergence theorems in Hilbert spaces.

Theorem 9. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝐴 an 𝛼-inverse strongly monotone operator of 𝐶 into 𝐻 with 𝛼>0. Suppose that 𝐵 is a maximal monotone operator on 𝐻 and 𝑓 is a bifunction from 𝐶×𝐶 to ℝ with (A1)–(A4). Assume that Ω∶=𝐸𝑃(𝑓)∩(𝐴+𝐵)⁻¹0≠∅. Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in (0,1) and let {𝑢_𝑛} be a sequence in 𝐻. Let {𝑥_𝑛} be a sequence generated by 𝑥₁𝑦∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛,𝑧_𝑛∈𝐶,𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑓𝑧_𝑛+1,𝑦𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑦_𝑛𝑥≥0∀𝑦∈𝐶,_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛. (14) Assume the following conditions are satisfied:(c1)lim_𝑛→∞𝑢_𝑛=𝑢 for some 𝑢∈𝐻;(c2)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞; (c3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1; (c4)0<𝑑≤𝑟_𝑛<∞ and lim_𝑛→∞(𝑟_𝑛+1−𝑟_𝑛)=0; (c5)0<𝑎≤𝜆_𝑛≤2𝛼 and lim_𝑛→∞(𝜆_𝑛+1−𝜆_𝑛)=0. Then, the sequence {𝑥_𝑛} converges strongly to 𝑃_Ω𝑢.

Proof. Note that the set Ω is closed and convex since (𝐴+𝐵)⁻¹0 and 𝐸𝑃(𝑓) are closed and convex.
With the help of Lemmas 1 and 2, we have 𝑧_𝑛=𝑇_𝑟𝑛𝑦_𝑛, and, for any 𝜔∈Ω, 𝑧_𝑛−𝜔=𝑇_𝑟𝑛𝑦_𝑛−𝑇_𝑟𝑛𝜔≤𝑦_𝑛−𝜔. (15) Notice that 𝑦_𝑛−𝜔≤𝛼_𝑛𝑢_𝑛−𝜔+1−𝛼_𝑛𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛−𝜔≤𝛼_𝑛𝑢_𝑛−𝜔+1−𝛼_𝑛𝑥_𝑛−𝜔. (16) It follows that 𝑥_𝑛+1−𝜔=𝛽_𝑛𝑥_𝑛−𝜔+1−𝛽_𝑛𝑧_𝑛−𝜔≤𝛽_𝑛𝑥_𝑛−𝜔+1−𝛽_𝑛×𝛼_𝑛𝑢_𝑛−𝜔+1−𝛼_𝑛𝑥_𝑛≤−𝜔1−𝛼_𝑛1−𝛽_𝑛𝑥_𝑛−𝜔+𝛼_𝑛1−𝛽_𝑛𝑢_𝑛𝑥−𝜔≤max_𝑛−𝜔,𝑢_𝑛−𝜔. (17) By condition (c1), the sequence {𝑢_𝑛} is bounded. Hence, there exists a positive number 𝑀 such that sup_𝑛{‖𝑢_𝑛−𝜔‖}≤𝑀. From a simple inductive process, we have 𝑥_𝑛+1−𝜔≤max𝑥₁−𝜔,𝑀, (18) which yields that {𝑥_𝑛} is bounded. So are {𝑦_𝑛} and {𝑧_𝑛}.
We have from Lemmas 3 and 4 that 𝑧_𝑛+1−𝑧_𝑛=𝑇_𝑟𝑛+1𝑦_𝑛+1−𝑇_𝑟𝑛𝑦_𝑛≤𝑦_𝑛+1−𝑦_𝑛+𝑟_𝑛+1−𝑟_𝑛𝑟_𝑛+1𝑇_𝑟𝑛+1𝑦_𝑛+1−𝑦_𝑛+1≤𝛼_𝑛+1𝑢_𝑛+1+1−𝛼_𝑛+1𝐽_𝜆𝑛+1𝐼−𝜆_𝑛+1𝐴𝑥_𝑛+1−𝛼_𝑛𝑢_𝑛−1−𝛼_𝑛𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝑟_𝑛+1𝑇_𝑟𝑛+1𝑦_𝑛+1−𝑦_𝑛+1≤𝐽_𝜆𝑛+1𝐼−𝜆_𝑛+1𝐴𝑥_𝑛+1−𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛+𝛼_𝑛+1𝑢_𝑛+1+𝐽_𝜆𝑛+1𝐼−𝜆_𝑛+1𝐴𝑥_𝑛+1+𝛼_𝑛𝑢_𝑛+𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝑟_𝑛+1𝑇_𝑟𝑛+1𝑦_𝑛+1−𝑦_𝑛+1≤𝐽_𝜆𝑛+1𝐼−𝜆_𝑛+1𝐴𝑥_𝑛+1−𝐽_𝜆𝑛+1𝐼−𝜆_𝑛+1𝐴𝑥_𝑛+𝐽_𝜆𝑛+1𝐼−𝜆_𝑛+1𝐴𝑥_𝑛−𝐽_𝜆𝑛+1𝐼−𝜆_𝑛𝐴𝑥_𝑛+𝐽_𝜆𝑛+1𝐼−𝜆_𝑛𝐴𝑥_𝑛−𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛+𝛼_𝑛+1𝑢_𝑛+1+𝐽_𝜆𝑛+1𝐼−𝜆_𝑛+1𝐴𝑥_𝑛+1+𝛼_𝑛𝑢_𝑛+𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝑟_𝑛+1𝑇_𝑟𝑛+1𝑦_𝑛+1−𝑦_𝑛+1≤𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛+1−𝜆_𝑛𝐴𝑥_𝑛+𝜆_𝑛+1−𝜆_𝑛𝑎×𝐽_𝜆𝑛+1𝐼−𝜆_𝑛𝐴𝑥_𝑛−𝐼−𝜆_𝑛𝐴𝑥_𝑛+𝛼_𝑛+1𝑢_𝑛+1+𝐽_𝜆𝑛+1𝐼−𝜆_𝑛+1𝐴𝑥_𝑛+1+𝛼_𝑛𝑢_𝑛+𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝑑𝑇_𝑟𝑛+1𝑦_𝑛+1−𝑦_𝑛+1. (19) Thus, conditions (c2), (c4), and (c5) yield that limsup_𝑛→∞𝑧_𝑛+1−𝑧_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (20) Recall that 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑧_𝑛. It follows from Lemma 5 that lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0. (21) Consequently, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽_𝑛𝑥_𝑛−𝑧_𝑛=0. (22)
For any 𝜔∈Ω, according to Lemma 2, we have 𝑧_𝑛−𝜔²=𝑇_𝑟𝑛𝑦_𝑛−𝑇_𝑟𝑛𝜔²≤𝑦_𝑛−𝜔,𝑧_𝑛=1−𝜔2𝑦_𝑛−𝜔²+𝑧_𝑛−𝜔²−𝑦_𝑛−𝑧_𝑛², (23) which yields that 𝑧_𝑛−𝜔²≤𝑦_𝑛−𝜔²−𝑧_𝑛−𝑦_𝑛². (24) By (14), one has 𝑥_𝑛+1−𝜔²=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛−𝜔²≤𝛽_𝑛𝑥_𝑛−𝜔²+1−𝛽_𝑛𝑧_𝑛−𝜔²≤𝛽_𝑛𝑥_𝑛−𝜔²+1−𝛽_𝑛×𝑦_𝑛−𝜔²−𝑧_𝑛−𝑦_𝑛²≤𝛽_𝑛𝑥_𝑛−𝜔²+1−𝛽_𝑛×𝛼_𝑛𝑢_𝑛−𝜔²+1−𝛼_𝑛𝑥_𝑛−𝜔²−𝑧_𝑛−𝑦_𝑛²≤1−𝛼_𝑛1−𝛽_𝑛𝑥_𝑛−𝜔²+𝛼_𝑛1−𝛽_𝑛𝑢_𝑛−𝜔²−1−𝛽_𝑛𝑧_𝑛−𝑦_𝑛², (25) which implies that 1−𝛽_𝑛𝑧_𝑛−𝑦_𝑛²≤𝑥_𝑛−𝜔²−𝑥_𝑛+1−𝜔²+𝛼_𝑛1−𝛽_𝑛𝑢_𝑛−𝜔²≤𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝜔+𝑥_𝑛+1−𝜔+𝛼_𝑛1−𝛽_𝑛𝑢_𝑛−𝜔². (26) This together with (22) deduces that lim_𝑛→∞𝑧_𝑛−𝑦_𝑛=0. (27)
Next, we prove that limsup_𝑛→∞𝑢−𝑥,𝑥_𝑛−𝑥≤0, (28) where 𝑥=𝑃_Ω𝑢. In order to show this inequality, we can choose a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that limsup_𝑛→∞𝑢−𝑥,𝑥_𝑛−𝑥=lim_𝑖→∞𝑢−𝑥,𝑥_𝑛𝑖−𝑥. (29) Owing to the boundedness of {𝑥_𝑛𝑖}, without loss of generality, we assume that 𝑥_𝑛𝑖⇀𝑝. Now we show that 𝑝∈Ω. As {𝑥_𝑛} is contained in 𝐶 and 𝐶 is a closed convex set, one has 𝑝∈𝐶.
We firstly prove that 𝑝∈𝐸𝑃(𝑓). By (14), we have 𝑓𝑧_𝑛+1,𝑦𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑦_𝑛≥0,∀𝑦∈𝐶. (30) The monotonicity of 𝑓 yields that 1𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑦_𝑛≥𝑓𝑦,𝑧_𝑛,∀𝑦∈𝐶. (31) Replacing 𝑛 by 𝑛_𝑖, we obtain 𝑦−𝑧_𝑛𝑖,𝑧_𝑛𝑖−𝑦_𝑛𝑖𝑟_𝑛𝑖≥𝑓𝑦,𝑧_𝑛𝑖,∀𝑦∈𝐶. (32) It follows from (21), (27), and (A4) that 𝑓𝑦,𝑝≤0,∀𝑦∈𝐶. (33) For 0<𝑡≤1, 𝑦∈𝐶, set 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑝. Then, 𝑦_𝑡∈𝐶 and 𝑓(𝑦_𝑡,𝑝)≤0. Thus, 0=𝑓𝑦_𝑡,𝑦_𝑡≤𝑡𝑓𝑦_𝑡,𝑦+(1−𝑡)𝑓𝑦_𝑡,𝑝≤𝑡𝑓𝑦_𝑡,𝑦. (34) Dividing by 𝑡, we see that 𝑓𝑦_𝑡,𝑦≥0. (35) Letting 𝑡↓0, we get 𝑓𝑝,𝑦≥0,∀𝑦∈𝐶; (36) namely, 𝑝∈𝐸𝑃(𝑓).
Secondly, we prove that 𝑝∈(𝐴+𝐵)⁻¹0. Thanks to (c5), there is a subsequence {𝜆_𝑛𝑖𝑗} of {𝜆_𝑛𝑖} such that 𝜆_𝑛𝑖𝑗→𝜆₀∈[𝑎,2𝛼]. Without loss of generality, we assume that 𝜆_𝑛𝑖→𝜆₀. Observe that 𝑥_𝑛𝑖−𝐽_𝜆0𝐼−𝜆₀𝐴𝑥_𝑛𝑖≤𝑥_𝑛𝑖−𝑦_𝑛𝑖+𝑦_𝑛𝑖−𝛼_𝑛𝑖𝑢_𝑛𝑖+1−𝛼_𝑛𝑖𝐽_𝜆0𝐼−𝜆₀𝐴𝑥_𝑛𝑖+𝛼_𝑛𝑖𝑢_𝑛𝑖+1−𝛼_𝑛𝑖𝐽_𝜆0𝐼−𝜆₀𝐴𝑥_𝑛𝑖−𝐽_𝜆0𝐼−𝜆₀𝐴𝑥_𝑛𝑖≤𝑥_𝑛𝑖−𝑦_𝑛𝑖+1−𝛼_𝑛𝑖×𝐽_𝜆𝑛𝑖𝐼−𝜆_𝑛𝑖𝐴𝑥_𝑛𝑖−𝐽_𝜆0𝐼−𝜆₀𝐴𝑥_𝑛𝑖+𝛼_𝑛𝑖𝑢_𝑛𝑖−𝐽_𝜆0𝐼−𝜆₀𝐴𝑥_𝑛𝑖≤𝑥_𝑛𝑖−𝑦_𝑛𝑖+1−𝛼_𝑛𝑖×𝐽_𝜆𝑛𝑖𝐼−𝜆_𝑛𝑖𝐴𝑥_𝑛𝑖−𝐽_𝜆𝑛0𝐼−𝜆_𝑛𝑖𝐴𝑥_𝑛𝑖+𝐽_𝜆𝑛0𝐼−𝜆_𝑛𝑖𝐴𝑥_𝑛𝑖−𝐽_𝜆𝑛0𝐼−𝜆_𝑛0𝐴𝑥_𝑛𝑖+𝛼_𝑛𝑖𝑢_𝑛𝑖−𝐽_𝜆0𝐼−𝜆₀𝐴𝑥_𝑛𝑖≤𝑥_𝑛𝑖−𝑦_𝑛𝑖+1−𝛼_𝑛𝑖×𝜆₀−𝜆_𝑛𝑖𝜆₀𝐽_𝜆𝑛0𝐼−𝜆_𝑛𝑖𝐴𝑥_𝑛𝑖−𝐼−𝜆_𝑛𝑖𝐴𝑥_𝑛𝑖+𝜆_𝑛𝑖−𝜆_𝑛0𝐴𝑥_𝑛𝑖+𝛼_𝑛𝑖𝑢_𝑛𝑖−𝐽_𝜆0𝐼−𝜆₀𝐴𝑥_𝑛𝑖. (37) Note that expressions (21) and (27) imply that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (38) Therefore, lim_𝑖→∞𝑥_𝑛𝑖−𝐽_𝜆0𝐼−𝜆₀𝐴𝑥_𝑛𝑖=0. (39) Since 𝐽_𝜆0(𝐼−𝜆₀𝐴) is nonexpansive and 𝑥_𝑛𝑖⇀𝑝, Lemma 7 gives 𝑝∈Fix(𝐽_𝜆0(𝐼−𝜆₀𝐴)); that is, 𝑝∈(𝐴+𝐵)⁻¹0. Thus, we have 𝑝∈Ω.
By (29) and the property of metric projection, we have limsup_𝑛→∞𝑢−𝑥,𝑥_𝑛−𝑥=lim_𝑖→∞𝑢−𝑥,𝑥_𝑛𝑖−𝑥=𝑢−𝑥,𝑝−𝑥≤0. (40) Inequality (28) arrives.
Finally, we prove that 𝑥_𝑛→𝑥. Using (14) again, we obtain 𝑥_𝑛+1−𝑥²≤𝛽_𝑛𝑥_𝑛−𝑥²+1−𝛽_𝑛𝑦_𝑛−𝑥²≤𝛽_𝑛𝑥_𝑛−𝑥²+1−𝛽_𝑛×1−𝛼_𝑛²𝑥_𝑛−𝑥²+2𝛼_𝑛𝑢_𝑛−𝑥,𝑦_𝑛−𝑥≤1−𝛼_𝑛1−𝛽_𝑛𝑥_𝑛−𝑥²+2𝛼_𝑛1−𝛽_𝑛×𝑢_𝑛−𝑢,𝑦_𝑛−𝑥+𝑢−𝑥,𝑦_𝑛−𝑥_𝑛+𝑢−𝑥,𝑥_𝑛−𝑥. (41) Note that limsup_𝑛→∞𝑢−𝑥,𝑥_𝑛−𝑥≤0. (42) It follows from Lemma 8 that {𝑥_𝑛} converges strongly to 𝑥.

The following results are direct consequences of Theorem 9.

Corollary 10. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝐴 an 𝛼-inverse strongly monotone operator of 𝐶 into 𝐻 with 𝛼>0. Suppose that 𝐵 is a maximal monotone operator on 𝐻 with (𝐴+𝐵)⁻¹0≠∅. Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in (0,1) and let {𝑢_𝑛} be a sequence in 𝐻. Let {𝑥_𝑛} be a sequence generated by 𝑥₁𝑦∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑃_𝐶𝑦_𝑛. (43) If conditions (c1)–(c3) and (c5) are satisfied, then the sequence {𝑥_𝑛} converges strongly to 𝑃_(𝐴+𝐵)⁻¹₀𝑢.

Proof. Letting 𝑓(𝑥,𝑦)≡0 for all 𝑥,𝑦∈𝐶 and 𝑟_𝑛=1 in Theorem 9, the desired result follows.

Remark 11. When 𝑢_𝑛≡𝑢∈𝐶, the algorithm in Corollary 10 corresponds to the one in Theorem 10 of Takahashi et al. [12]. Thus, Corollary 10 includes, as a special case, Theorem 10 of [12].

Corollary 12. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ with (A1)–(A4). Assume that the set 𝐸𝑃(𝑓) is nonempty. Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in (0,1) and let {𝑢_𝑛} be a sequence in 𝐻. Let {𝑥_𝑛} be a sequence generated by 𝑥₁𝑦∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑥_𝑛,𝑧_𝑛∈𝐶,𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑓𝑧_𝑛+1,𝑦𝑟_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑦_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛. (44) If conditions (c1)–(c4) are satisfied, then the sequence {𝑥_𝑛} converges strongly to 𝑃_{𝐸𝑃(𝑓)}𝑢.

Proof. Putting 𝐴=0, 𝐵=𝜕𝛿_𝐶 (i.e., the subdifferential of the indicator function of  𝐶) in Theorem 9, we get the result.

4. Applications

In this section, we apply the results in the preceding section to variational inequality and optimization problems.

Now we consider the variational inequality problem. Let 𝐻 be a real Hilbert space and let 𝑓 be a proper lower semicontinuous convex function of 𝐻 into (−∞,+∞]. Then, the subdifferential 𝜕𝑓 of 𝑓 is defined as follows: 𝜕𝑓(𝑥)=𝑧∈𝐻∶𝑓𝑦−𝑓(𝑥)≥𝑧,𝑦−𝑥,∀𝑦∈𝐻, (45) for all 𝑥∈𝐻. Rockafellar [18] claimed that 𝜕𝑓 is a maximal monotone operator. Let 𝛿_𝐶 be the indicator function of 𝐶; that is, 𝛿_𝐶(𝑥)=0,𝑥∈𝐶,+∞,𝑥∉𝐶. (46) Since 𝛿_𝐶 is a proper lower semicontinuous convex function on 𝐻, the subdifferential 𝜕𝛿_𝐶 of 𝛿_𝐶 is a maximal monotone operator. The resolvent 𝐽_𝜆 of 𝜕𝛿_𝐶 for 𝜆>0 is defined by 𝐽_𝜆𝑥=𝐼+𝜆𝜕𝛿_𝐶⁻¹𝑥,∀𝑥∈𝐻. (47) We have 𝑢=𝐽_𝜆𝑥=𝐼+𝜆𝜕𝛿_𝐶⁻¹𝑥⟺𝑥∈𝑢+𝜆𝜕𝛿_𝐶𝑢⟺𝑥∈𝑢+𝜆𝑁_𝐶𝑢⟺𝑥−𝑢∈𝜆𝑁_𝐶𝑢⟺1𝜆⟺𝑥−𝑢,𝑦−𝑢≤0∀𝑦∈𝐶𝑥−𝑢,𝑦−𝑢≤0∀𝑦∈𝐶⟺𝑢=𝑃_𝐶𝑥, (48) where 𝑁_𝐶𝑢={𝑧∈𝐻∶⟨𝑧,𝑦−𝑢⟩≤0, forall𝑦∈𝐶}.

Let 𝐴∶𝐶→𝐻 be a nonlinear mapping. The variational inequality problem for 𝐴 is to find 𝑧∈𝐶 such that 𝐴𝑧,𝑦−𝑧≥0,∀𝑦∈𝐶. (49) The set of its solutions is denoted by 𝑉𝐼(𝐴,𝐶). It is easy to see that 𝑉𝐼(𝐴,𝐶)=(𝐴+𝜕𝛿_𝐶)⁻¹0 (for more details, see [13]). Using Corollary 10, we obtain the strong convergence theorem for the variational inequality problem.

Theorem 13. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝐴 an 𝛼-inverse strongly monotone operator of 𝐶 into 𝐻 with 𝛼>0. Assume that the set 𝑉𝐼(𝐴,𝐶) is nonempty. Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in (0,1) and let {𝑢_𝑛} be a sequence in 𝐻. Let {𝑥_𝑛} be a sequence generated by 𝑥₁𝑦∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑃_𝐶𝐼−𝜆_𝑛𝐴𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑃_𝐶𝑦_𝑛. (50) If conditions (c1)–(c3) and (c5) are satisfied, then the sequence {𝑥_𝑛} converges strongly to 𝑃_{𝑉𝐼(𝐴,𝐶)}𝑢.

Proof. Notice that 𝑉𝐼(𝐴,𝐶)=(𝐴+𝜕𝛿_𝐶)⁻¹0. Corollary 10 yields the conclusion.

Next we study the optimization problem min𝑔(𝑥),𝑥∈𝐶, (51) where 𝑔(𝑥) is a proper lower semicontinuous convex function of 𝐻 into (−∞,+∞] such that 𝐶 is contained in dom𝑔={𝑥∈𝐻∶𝑔(𝑥)<+∞}. We denote by Sol(𝑔,𝐶) the set of solutions of problem (51). Let 𝑓∶𝐶×𝐶→ℝ be a bifunction defined by 𝑓𝑥,𝑦=𝑔𝑦−𝑔(𝑥). (52) It is clear that 𝑓(𝑥,𝑦) satisfies (A1)–(A4) and 𝐸𝑃(𝑓)=Sol(𝑔,𝐶). Therefore, by Corollary 12, the following result is obtained.

Theorem 14. Let 𝑔(𝑥) be a proper lower semicontinuous convex function of 𝐻 into (−∞,+∞] and let 𝐶 be a nonempty closed convex subset of 𝐻 such that 𝐶 is contained in dom𝑔. Suppose that the set Sol(𝑔,𝐶) is nonempty. Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in (0,1) and let {𝑢_𝑛} be a sequence in 𝐻. Let {𝑥_𝑛} be a sequence generated by 𝑥₁𝑦∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑥_𝑛,𝑧_𝑛=argmin_𝑦∈𝐶𝑔𝑦+12𝑟_𝑛𝑦−𝑦_𝑛²,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛. (53) If conditions (c1)–(c4) are satisfied, then the sequence {𝑥_𝑛} converges strongly to 𝑃_{Sol(𝑔,𝐶)}𝑢.

Proof. Letting 𝑓(𝑥,𝑦)=𝑔(𝑦)−𝑔(𝑥) in Corollary 12, we get the required result.

Remark 15. Letting 0=𝑢=lim_𝑛→∞𝑢_𝑛 in our results, we obtain the algorithms for minimal-norm solutions of corresponding problems.

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgments

The authors would like to thank the referees and editors for their valuable comments and suggestions. The work was supported by graduate funds of Beijing University of Technology (no. ykj-2013-9422).