An Iterative Method with Norm Convergence for a Class of Generalized Equilibrium Problems

Abstract

Recently, Takahashi and Takahashi proposed an iterative algorithm for solving a problem for finding common solutions of generalized equilibrium problems governed by inverse strongly monotone mappings and of fixed point problems for nonexpansive mappings. In this paper, we provide a result that allows for the removal of one condition ensuring the strong convergence of the algorithm.

1. Introduction

Let ℋ be a real Hilbert space and 𝐶 a nonempty closed convex subset. A generalized equilibrium problem is formulated as a problem of finding a point 𝑥^∗∈𝐶 with the property 𝐹𝑥^∗,𝑦+⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩≥0,∀𝑦∈𝐶, (1) where 𝐹∶𝐶×𝐶→ℝ is a bifunction and 𝐴∶𝐶→ℋ is a nonlinear mapping. In particular, if 𝐴 is the zero mapping, then problem (1) is reduced to an equilibrium problem; find a point 𝑥^∗∈𝐶 with the property 𝐹𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (2) We will denote by EP(𝐹;𝐴) and EP(𝐹) the solution set of problem (1) and problem (2), respectively. A fixed point problem (FPP) is to find a point 𝑥^∗ with the property 𝑥^∗∈𝐶,𝑆𝑥^∗=𝑥^∗, (3) where 𝑆∶𝐶→𝐶 is a nonlinear mapping. The set of fixed points of 𝑆 is denoted as Fix(𝑆).

The problem under consideration in this paper is to find a common solution of problem (1) and of FPP (3). Namely, we seek a point 𝑥^∗ such that 𝑥^∗∈Fix(𝑆)∩EP(𝐹;𝐴). (4) We consider problem (4) in the case whenever 𝐴 is a 𝜈-inverse strongly monotone mapping and 𝑆 is a nonexpansive mapping. To solve problem (4), Takahashi and Takahashi [1] introduced an algorithm which generates a sequence (𝑥_𝑛) by the iterative procedure 𝐹𝑧_𝑛,𝑦+𝐴𝑥_𝑛,𝑦−𝑧_𝑛+1𝜆_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝛼_𝑛𝑢+1−𝛼_𝑛𝑧_𝑛, (5) where (𝛼_𝑛)⊆[0,1],(𝛽_𝑛)⊆[0,1], and (𝜆_𝑛)⊆[0,2𝜈] are chosen so that 0<𝑎≤𝜆_𝑛≤𝑏<2𝜈,0<𝑐≤𝛽_𝑛≤𝑑<1,lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=0𝛼_𝑛𝜆=∞,_𝑛−𝜆_𝑛+1⟶0. (6) Under these conditions, they proved that the sequence (𝑥_𝑛) generated by (5) can be strongly convergent to a solution of problem (4).

It is the aim of this paper to continue the study of algorithm (5). We will show that problem (4) is in fact a special fixed point problem for a nonexpansive mapping (a composition of a nonexpansive mapping and an averaged mapping). Our approach mainly uses the properties of averaged mappings, which is different from the existing methods invented by Takahashi and Takahashi. Moreover, we shall prove that condition |𝜆_𝑛−𝜆_𝑛+1|→0 sufficient to guarantee the convergence of algorithm (5) is superfluous.

2. Preliminaries and Notations

Notation 1. → strong convergence, ⇀ weak convergence and 𝜔_𝑤(𝑥_𝑛) the set of the weak cluster points of (𝑥_𝑛).
Denote by 𝑃_𝐶 the projection from ℋ onto 𝐶; namely, for 𝑥∈ℋ, 𝑃_𝐶𝑥 is the unique point in 𝐶 with the property 𝑥−𝑃_𝐶𝑥=min_𝑦∈𝐶𝑥−𝑦. (7) It is well known that 𝑃_𝐶𝑥 is characterized by the inequality 𝑃_𝐶𝑥∈𝐶,𝑥−𝑃_𝐶𝑥,𝑧−𝑃_𝐶𝑥≤0,∀𝑧∈𝐶. (8)
We will use the following notions on nonlinear mappings 𝑇∶𝐶→ℋ. (i)𝑇is nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (9)(ii)𝑇 is firmly nonexpansive if 𝑇𝑥−𝑇𝑦,𝑥−𝑦≥𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐶. (10)(iii)𝑇 is 𝛼-averaged if there exist a constant 𝛼∈(0,1) and a nonexpansive mapping 𝑆 such that 𝑇=(1−𝛼)𝐼+𝛼𝑆, where 𝐼 is the identity mapping on ℋ. (iv)𝑇 is 𝜈-inverse strongly monotone if there is a constant 𝜈>0 such that 𝑇𝑥−𝑇𝑦,𝑥−𝑦≥𝜈𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐶. (11)
The next lemma is referred to as the demiclosedness principle for nonexpansive mappings (see [2]).

Lemma 1. Let 𝐶 be a nonempty closed convex subset of ℋ and 𝑇∶𝐶→ℋ a nonexpansive mapping with Fix(𝑇)≠∅. If (𝑥_𝑛) is a sequence in 𝐶 such that 𝑥_𝑛⇀𝑥 and (𝐼−𝑇)𝑥_𝑛→0, then (𝐼−𝑇)𝑥=0; that is, 𝑥∈Fix(𝑇).

Averaged mappings will play important role in our convergence analysis. We therefore collect some useful properties of averaged mappings (see, e.g., [3–5]).

Lemma 2. The following assertions hold. (i)𝑇 is firmly nonexpansive if and only if 𝑇 is 1/2-averaged.(ii)If 𝑇_𝑖 is 𝜈_𝑖-averaged, 𝑖=1,2, then 𝑇₁𝑇₂ is (𝜈₁+𝜈₂−𝜈₁𝜈₂)-averaged. (iii)If 𝑇∶𝐶→ℋ is 𝜈-averaged, then for any 𝑧∈Fix(𝑇) and for all 𝑥∈𝐶, ‖𝑇𝑥−𝑧‖²≤‖𝑥−𝑧‖²−1−𝜈𝜈‖𝑇𝑥−𝑥‖². (12)

From now on, we assume that 𝐹∶𝐶×𝐶→ℝ is a bifunction so that (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)lim_𝑡↓0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦), for all 𝑥,𝑦∈𝐶; (A4) for each 𝑥∈𝐶,𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous. Under these assumptions, the following results hold (see [6, 7]).

Lemma 3. Let 𝐹∶𝐶×𝐶→ℝ satisfy (A1)–(A4). Then for any 𝜆>0 and 𝑥∈ℋ, there exists 𝑧∈𝐶 so that 𝐹+1𝑧,𝑦𝜆⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶. (13) Moreover if 𝑆_𝜆𝑥={𝑧∈𝐶∶𝐹(𝑧,𝑦)+1/𝜆⟨𝑦−𝑧,𝑧−𝑥⟩≥0,𝑓𝑜𝑟𝑎𝑙𝑙𝑦∈𝐶}, then (i)𝑆_𝜆 is single valued and Fix(𝑆_𝜆)=EP(𝐹);(ii)𝑆_𝜆 is firmly nonexpansive; (iii)EP(𝐹) is closed and convex.

We end this section by a useful lemma (see Xu [8]).

Lemma 4. Let (𝑎_𝑛) be a nonnegative real sequence satisfying 𝑎_𝑛+1≤1−𝛼_𝑛𝑎_𝑛+𝛼_𝑛𝑏_𝑛, (14) where (𝛼_𝑛)⊂(0,1) and (𝑏_𝑛) are real sequences. Then 𝑎_𝑛→0 provided that (i)∑_𝑛𝛼_𝑛=∞, lim_𝑛𝛼_𝑛=0;(ii)limsup_𝑛𝑏_𝑛≤0 or ∑𝛼_𝑛|𝑏_𝑛|<∞.

3. Algorithm and Its Convergence

We begin with the following lemma.

Lemma 5. Assume that 𝐴∶𝐶→ℋ is 𝜈-inverse strongly monotone mapping for some 𝜈>0. Given a real number 𝜆 such that 0<𝜆<2𝜈, set 𝑇_𝜆=𝑆_𝜆(𝐼−𝜆𝐴) with 𝑆_𝜆 defined as in Lemma 3. Then the following assertions hold: (a)𝑇_𝜆 is single valued and Fix(𝑇_𝜆)=EP(𝐹;𝐴); (b)𝑇_𝜆 is (2𝜈+𝜆)/4𝜈-averaged; (c)given 𝑧∈EP(𝐹;𝐴), it follows that 𝑇_𝜆𝑥−𝑧²≤‖𝑥−𝑧‖²−2𝜈−𝜆𝑇2𝜈+𝜆^𝜆𝑥−𝑥²; (15)(d)if 0<𝜆≤𝜆^′<2𝜈, then for all 𝑥∈𝐶𝑇_𝜆𝑥−𝑥≤2𝑇_𝜆′𝑥−𝑥. (16)

Proof. (a) It is readily seen that 𝑇_𝜆 is single valued because 𝑆_𝜆 is single valued. The equality follows from the definition of 𝑆_𝜆.
(b) It follows that (𝐼−2𝜈𝐴)𝑥−(𝐼−2𝜈𝐴)𝑦²=(𝑥−𝑦)−2𝜈(𝐴𝑥−𝐴𝑦)²=𝑥−𝑦²+4𝜈²𝐴𝑥−𝐴𝑦²−4𝜈𝑥−𝑦,𝐴𝑥−𝐴𝑦. (17) Since 𝐴 is 𝜈-inverse strongly monotone, 𝐼−2𝜈𝐴 is nonexpansive. Observe that 𝜆𝐼−𝜆𝐴=1−𝜆2𝜈𝐼+()2𝜈𝐼−2𝜈𝐴, (18) which implies that 𝐼−𝜆𝐴 is 𝜆/2𝜈-averaged. Consequently (b) follows from part (ii) of Lemma 2 and (c) follows from part (iii) of Lemma 2.
(d) Let 𝑧₁=𝑇_𝜆𝑥 and 𝑧₂=𝑇_𝜆′𝑥. By definition of 𝑆_𝜆, 𝐹𝑧₁,𝑦+𝐴𝑥,𝑦−𝑧₁+1𝜆𝑦−𝑧₁,𝑧₁−𝑥≥0,∀𝑦∈𝐶. (19) Letting 𝑦=𝑧₂ in (19) yields 𝐹𝑧₁,𝑧₂+𝐴𝑥,𝑧₂−𝑧₁+1𝜆𝑧₂−𝑧₁,𝑧₁−𝑥≥0. (20) Similarly, 𝐹𝑧₂,𝑧₁+𝐴𝑥,𝑧₁−𝑧₂+1𝜆^′𝑧₁−𝑧₂,𝑧₂−𝑥≥0. (21) Adding up these inequalities and using the monotonicity of 𝐹, 1𝜆𝑧₂−𝑧₁,𝑧₁+1−𝑥𝜆^′𝑧₁−𝑧₂,𝑧₂−𝑥≥0, (22) or equivalently, 𝑧₂−𝑧₁²≤𝜆1−𝜆^′𝑧₂−𝑧₁,𝑧₂−𝑥. (23) Hence, ‖𝑧₂−𝑧₁‖≤‖𝑧₂−𝑥‖. By the triangle inequality, 𝑧₁−𝑥≤𝑧₁−𝑧₂+𝑧₂−𝑥≤2𝑧₂−𝑥, (24) which is the result as desired.

For every 𝑛≥0, if we define 𝑇_𝑛=𝑆_𝜆𝑛(𝐼−𝜆_𝑛𝐴), where 𝑆_𝜆𝑛 is defined as in Lemma 3, then we can rewrite algorithm (5) as 𝑦_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑦_𝑛. (25)

Theorem 6. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)–(A4), 𝐴∶𝐶→ℋ a 𝜈-inverse strongly monotone mapping for some 𝜈>0, and 𝑆∶𝐶→𝐶 a nonexpansive mapping so that the solution set Ω∶=Fix(𝑆)∩EP(𝐹;𝐴) is nonempty. If the following conditions hold: 0<𝑎≤𝜆_𝑛≤𝑏<2𝜈,0<𝑐≤𝛽_𝑛≤𝑑<1,lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=0𝛼_𝑛=∞, (26) then the sequence (𝑥_𝑛) generated by (25) converges strongly to 𝑥^∗=𝑃_Ω𝑢.

Before proving the theorem, we need some lemmas.

Lemma 7. Let the conditions in Theorem 6 be satisfied. If (𝑥_𝑛) and (𝑦_𝑛) are the sequences generated by (25), then both (𝑥_𝑛) and (𝑦_𝑛) are bounded.

Proof. Let 𝑧∈Ω be fixed. We have 𝑥_𝑛+1−𝑧≤1−𝛽_𝑛𝑦_𝑛−𝑧+𝛽_𝑛𝑥_𝑛≤−𝑧1−𝛽_𝑛𝑦_𝑛−𝑧+𝛽_𝑛𝑥_𝑛−𝑧; (27) on the other hand, 𝑦_𝑛−𝑧=𝛼_𝑛(𝑢−𝑧)+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛≤−𝑧1−𝛼_𝑛𝑥_𝑛−𝑧+𝛼_𝑛‖𝑢−𝑧‖. (28) Altogether 𝑥_𝑛+1−𝑧≤1−𝛼_𝑛1−𝛽_𝑛𝑥_𝑛−𝑧+𝛼_𝑛1−𝛽_𝑛‖𝑢−𝑧‖. (29) By induction, (𝑥_𝑛) is bounded and so is (𝑦_𝑛).

Lemma 8. Let the conditions in Theorem 6 be satisfied. If ‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖→0 and ‖𝑥_𝑛−𝑆𝑦_𝑛‖→0, then ‖𝑥_𝑛−𝑦_𝑛‖→0 and 𝜔_𝑤(𝑥_𝑛)⊆Ω.

Proof. Let 𝑇_𝑎=𝑆_𝑎(𝐼−𝑎𝐴). By part (d) of Lemma 5, 𝑥_𝑛−𝑇_𝑎𝑥_𝑛≤2𝑥_𝑛−𝑇_𝑛𝑥_𝑛⟶0. (30) Since 𝑇_𝑎 is nonexpansive, applying the demiclosedness principle yields 𝜔_𝑤𝑥_𝑛⊆Fix𝑇_𝑎=EP(𝐹;𝐴). (31) On the other hand, we see that 𝑥_𝑛−𝑦_𝑛=𝛼_𝑛𝑢−𝑥_𝑛+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛−𝑥_𝑛≤𝛼_𝑛𝑢−𝑥_𝑛+𝑇_𝑛𝑥_𝑛−𝑥_𝑛⟶0, (32) which implies that 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆𝑦_𝑛+𝑆𝑦_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆𝑦_𝑛+𝑦_𝑛−𝑥_𝑛⟶0. (33) Using again the demiclosedness principle gets the desired result.

Proof of Theorem 6. Let 𝑥^∗=𝑃_Ω𝑢. Using Lemma 5(c), we have 𝑇_𝑛𝑥_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−2𝜈−𝜆_𝑛2𝜈+𝜆_𝑛𝑇_𝑛𝑥_𝑛−𝑥_𝑛². (34) By the subdifferential inequality, 𝑦_𝑛−𝑥^∗2=𝛼_𝑛𝑢−𝑥^∗+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛−𝑥^∗2≤1−𝛼_𝑛𝑇_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑢−𝑥^∗,𝑦_𝑛−𝑥^∗≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑢−𝑥^∗,𝑦_𝑛−𝑥^∗−1−𝛼_𝑛2𝜈−𝜆_𝑛2𝜈+𝜆_𝑛𝑇_𝑛𝑥_𝑛−𝑥_𝑛², (35) which implies that 𝑥_𝑛+1−𝑥^∗2=𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝑆𝑦_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑆𝑦_𝑛−𝑥_𝑛²≤𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝑦_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑆𝑦_𝑛−𝑥_𝑛²≤𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛×1−𝛼_𝑛𝑥_𝑛−𝑥^∗2−1−𝛼_𝑛1−𝛽_𝑛2𝜈−𝜆_𝑛2𝜈+𝜆_𝑛×𝑇_𝑛𝑥_𝑛−𝑥_𝑛²+2𝛼_𝑛1−𝛽_𝑛×⟨𝑢−𝑥^∗,𝑦_𝑛−𝑥^∗⟩−𝛽_𝑛1−𝛽_𝑛×𝑆𝑦_𝑛−𝑥_𝑛². (36) By our assumption, there exists 𝜀>0 so that for all 𝑛≥0, 1−𝛼_𝑛1−𝛽_𝑛2𝜈−𝜆_𝑛2𝜈+𝜆_𝑛≥𝜀, (37) and 1−𝛽_𝑛≥𝛽_𝑛(1−𝛽_𝑛)≥𝜀. Consequently, 𝑥_𝑛+1−𝑥^∗2≤1−𝜀𝛼_𝑛𝑥_𝑛−𝑥^∗2−𝜀𝑇_𝑛𝑥_𝑛−𝑥_𝑛²+𝑆𝑦_𝑛−𝑥_𝑛²+2𝛼_𝑛1−𝛽_𝑛𝑢−𝑥^∗,𝑦_𝑛−𝑥^∗. (38) Set 𝑠_𝑛=‖𝑥_𝑛+1−𝑥^∗‖², and let (𝑠_𝑛𝑘) be a subsequence so that it includes all elements in {𝑠_𝑛} with the property; each of them is less than or equal to the term after it. Following an idea developed by Maingé [9], we next consider two possible cases on (𝑠_𝑛𝑘).
Case 1. Assume that {𝑠_𝑛𝑘} is finite. Then there exists 𝑁∈ℕ so that 𝑠_𝑛>𝑠_𝑛+1 for all 𝑛≥𝑁, and therefore {𝑠_𝑛} must be convergent. It follows from (38) that 𝜀𝑇_𝑛𝑥_𝑛−𝑥_𝑛²+𝑆𝑦_𝑛−𝑥_𝑛²≤𝑀𝛼_𝑛+𝑠_𝑛−𝑠_𝑛+1, (39) where 𝑀>0 is a sufficiently large real number. Consequently, both ‖𝑇_𝑛𝑥_𝑛−𝑥_𝑛‖ and ‖𝑆𝑦_𝑛−𝑥_𝑛‖ converge to zero, and by Lemma 8 we conclude that ‖𝑦_𝑛−𝑥_𝑛‖→0 and 𝜔_𝑤(𝑥_𝑛)⊆Ω. Hence, limsup_𝑛→∞𝑢−𝑥^∗,𝑦_𝑛−𝑥^∗=limsup_𝑛→∞𝑢−𝑥^∗,𝑥_𝑛−𝑥^∗=max_{𝑤∈𝜔𝑤𝑥𝑛}𝑢−𝑥^∗,𝑤−𝑥^∗≤0, (40) where the inequality uses (8). It then follows from (38) that 𝑠_𝑛+1≤1−𝜀𝛼_𝑛𝑠_𝑛+2𝛼_𝑛1−𝛽_𝑛𝑢−𝑥^∗,𝑦_𝑛−𝑥^∗. (41) We therefore apply Lemma 4 to conclude that 𝑠_𝑛→0.
Case 2. Assume now that {𝑠_𝑛𝑘} is infinite. Let 𝑛∈ℕ be fixed. Then there exists 𝑘∈ℕ so that 𝑛_𝑘≤𝑛≤𝑛_𝑘+1. By the choice of {𝑠_𝑛𝑘}, we see that 𝑠_𝑛𝑘+1 is the largest one among {𝑠_𝑛𝑘,𝑠_𝑛𝑘+1,…,𝑠_𝑛𝑘+1}; in particular 𝑠_𝑛𝑘≤𝑠_𝑛𝑘+1,𝑠_𝑛≤𝑠_𝑛𝑘+1. (42) Then we deduce from (38) that 𝜀𝑇_𝑛𝑘𝑥_𝑛𝑘−𝑥_𝑛𝑘²+𝑆𝑦_𝑛𝑘−𝑥_𝑛𝑘²≤𝑀𝛼_𝑛𝑘⟶0. (43) Applying Lemma 8 yields ‖𝑦_𝑛𝑘−𝑥_𝑛𝑘‖→0 and 𝜔_𝑤(𝑥_𝑛𝑘)⊆Ω. Similarly limsup_𝑛→∞⟨𝑢−𝑥^∗,𝑦_𝑛𝑘−𝑥^∗⟩≤0. (44) It follows again from (38) and inequality (42) that 𝑠_𝑛𝑘≤21−𝛽_𝑛𝑘𝑢−𝑥^∗,𝑦_𝑛𝑘−𝑥^∗. (45) Taking limsup in (44) yields limsup_𝑘→∞𝑠_𝑛𝑘≤0⟹𝑠_𝑛𝑘⟶0. (46) Moreover, we deduce from algorithm (25) that 𝑠_𝑛𝑘+1=𝑥_𝑛𝑘−𝑥^∗−𝑥_𝑛𝑘−𝑥_𝑛𝑘+1≤𝑠_𝑛𝑘+𝑥_𝑛𝑘−𝑥_𝑛𝑘+1≤𝑠_𝑛𝑘+𝑥_𝑛𝑘−𝑆𝑦_𝑛𝑘, (47) which together with (43) implies that 𝑠_𝑛𝑘+1→0. Consequently 𝑠_𝑛→0 immediately follows from (42).

4. Applications

In this section we present several applications. First we consider a problem for finding a common solution of equilibrium problem (2) and fixed point problem (3); namely, find 𝑥^∗∈𝐶 so that 𝑥^∗∈EP(𝐹)∩Fix(𝑆). (48) Taking 𝐴=0 in Theorem 6 and noting that zero mapping is 𝜈-inverse strongly monotone for any positive number 𝜈, one can easily get the following.

Corollary 9. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)–(A4) and 𝑆∶𝐶→𝐶 a nonexpansive mapping so that the solution set of problem (48) is nonempty. Given 𝑢∈𝐶, let (𝑥_𝑛) generated by the iterative algorithm: 𝐹𝑧_𝑛+1,𝑦𝜆_𝑛𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆1−𝛼_𝑛𝑢+𝛼_𝑛𝑧_𝑛. (49) If the following conditions hold: 0<𝑎≤𝜆_𝑛≤𝑏<∞,0<𝑐≤𝛽_𝑛≤𝑑<1,lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=0𝛼_𝑛=∞, (50) then the sequence (𝑥_𝑛) converges strongly to a solution of problem (48).

A variational inequality problem (VIP) is formulated as a problem of finding a point 𝑥^∗ with the property 𝑥^∗∈𝐶,𝐴𝑥^∗,𝑧−𝑥^∗≥0,∀𝑧∈𝐶. (51) We will denote the solution set of VIP (51) by VI(𝐴;𝐶). Next we consider a problem for finding a common solution of variational inequality problem (51) and of fixed point problem (3), namely; find 𝑥^∗∈𝐶 so that 𝑥^∗∈VI(𝐴;𝐶)∩Fix(𝑆). (52) Taking 𝐹=0 in (1), we note that the generalized equilibrium problem is reduced to the variational problem (51). Thus applying Theorem 6 gets the following.

Corollary 10. Let 𝐴∶𝐶→ℋ be 𝜈-inverse strongly monotone mapping and 𝑆∶𝐶→𝐶 a nonexpansive mapping so that the solution set of problem (52) is nonempty. Given 𝑢∈𝐶, let (𝑥_𝑛) generated by the iterative algorithm: 𝑧_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆1−𝛼_𝑛𝑢+𝛼_𝑛𝑧_𝑛. (53) If the following conditions hold: 0<𝑎≤𝜆_𝑛≤𝑏<2𝜈,0<𝑐≤𝛽_𝑛≤𝑑<1,lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=0𝛼_𝑛=∞, (54) then the sequence (𝑥_𝑛) converges strongly to a solution of problem (52).

Consider the optimization problem of finding a point 𝑥^∗∈𝐶 with the property 𝑓𝑥^∗=min_𝑥∈𝐶𝑓(𝑥), (55) where 𝑓∶ℋ→ℝ is a convex and differentiable function. We say that the differential ∇𝑓 is 1/𝜈-Lipschitz continuous, if (𝑥)𝑦≤1∇𝑓−∇𝑓𝜈𝑥−𝑦,∀𝑥,𝑦∈ℋ. (56) Denote by Argmin(𝐶;𝑓) the solution set of problem (55). Finally we consider a problem for finding a common solution of optimization problem (55) and of fixed point problem (3), namely; find 𝑥^∗∈𝐶 so that 𝑥^∗∈Argmin𝐶;𝑓∩Fix(𝑆). (57) By [10, Lemma 5.13], problem (55) is equivalent to the variational inequality problem ∇𝑓𝑥^∗,𝑥^∗−𝑧≥0,∀𝑧∈𝐶. (58) Taking 𝐴=∇𝑓 in Corollary 10, we have the following result.

Corollary 11. Let 𝑓∶ℋ→ℝ be a convex and differentiable function so that ∇𝑓 is 1/𝜈-Lipschitz continuous. Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping so that the solution set of problem (57) is nonempty. Given 𝑢∈𝐶, let (𝑥_𝑛) generated by 𝑧_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆1−𝛼_𝑛𝑢+𝛼_𝑛𝑧_𝑛. (59) If the following conditions hold: 0<𝑎≤𝜆_𝑛≤𝑏<2𝜈,0<𝑐≤𝛽_𝑛≤𝑑<1,lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=0𝛼_𝑛=∞, (60) then the sequence (𝑥_𝑛) converges strongly to a solution of problem (57).

Proof. It suffices to note that if ∇𝑓 is 1/𝜈-Lipschitz continuous, then it is 𝜈-inverse strongly monotone mapping (see [11, Corollary 10]). Consequently Corollary 10 applies and the result immediately follows.

Remark 12. We can further apply the previous method to find a common solution for fixed point and split feasibility problems, as well as for fixed point and convex constrained linear inverse problems (see [12]).

Acknowledgment

This work is supported by the National Natural Science Foundation of China, Tianyuan Foundation (11226227).