Some Results on Strictly Pseudocontractive Nonself-Mappings and Equilibrium Problems in Hilbert Spaces

Abstract

An equilibrium problem and a strictly pseudocontractive nonself-mapping are investigated. Strong convergence theorems of common elements are established based on hybrid projection algorithms in the framework of real Hilbert spaces.

1. Introduction

Bifunction equilibrium problems which were considered by Blum and Oettli [1] have intensively been studied. It has been shown that the bifunction equilibrium problem covers fixed point problems, variational inequalities, inclusion problems, saddle problems, complementarity problem, minimization problem, and the Nash equilibrium problem; see [1–4] and the references therein. Iterative methods have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization; see [5–16] and the references therein. In this paper, we investigate an equilibrium problem and a strictly pseudocontractive nonself-mapping based on hybrid projection algorithms. Strong convergence theorems of common elements lie in the solution set of the equilibrium problem and the fixed point set of the strictly pseudocontractive nonself-mapping.

Throughout this paper, we always assume that 𝐻 is a real Hilbert space with the inner product ⟨⋅,⋅⟩ and the norm ‖⋅‖. Let 𝐶 be a nonempty closed convex subset of 𝐻 and 𝐹 a bifunction of 𝐶×𝐶 into ℝ, where ℝ denotes the set of real numbers. In this paper, we consider the following equilibrium problem: Find𝑥∈𝐶suchthat𝐹𝑥,𝑦≥0,∀𝑦∈𝐶. (1.1) The set of such an 𝑥∈𝐶 is denoted by 𝐸𝑃(𝐹), that is, 𝐸𝑃(𝐹)=𝑥∈𝐶∶𝐹𝑥,𝑦≥0,∀𝑦∈𝐶. (1.2)

Given a mapping 𝑇∶𝐶→𝐻, let 𝐹(𝑥,𝑦)=⟨𝑇𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶. Then 𝑧∈𝐸𝑃(𝐹) if and only if ⟨𝑇𝑧,𝑦−𝑧⟩≥0 for all 𝑦∈𝐶, that is, 𝑧 is a solution of the variational inequality.

To study the equilibrium problem (1.1), we may assume that 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 each 𝑥,𝑦,𝑧∈𝐶, limsup_𝑡↓0𝐹𝑡𝑧+(1−𝑡)𝑥,𝑦≤𝐹𝑥,𝑦; (1.3)(A4) for each 𝑥∈𝐶, 𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous.

If 𝐻 is an Euclidean space, then we see that 𝐹(𝑥,𝑦)=𝑥−𝑦 is a simple example satisfying the above assumptions. See [2] for more details. Let 𝑆∶𝐷(𝑆)→𝑅(𝑆), where 𝐷(𝑆) and 𝑅(𝑆) denote the domain and the range of the mapping 𝑆. If 𝐷(𝑆)=𝑅(𝑆), then the mapping 𝑆 is said to be a self-mapping. If 𝐷(𝑆)≠𝑅(𝑆), then the mapping 𝑆 is said to be a nonself-mapping. Let 𝑆∶𝐶→𝐻 be a nonself-mapping. In this paper, we use 𝐹(𝑆) to denote the fixed point set of 𝑆. Recall the following definitions. 𝑆 is said to be nonexpansive if 𝑆𝑥−𝑆𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.4)

𝑆 is said to be strictly pseudocontractive if there exists a constant 𝜅∈[0,1) such that 𝑆𝑥−𝑆𝑦²≤𝑥−𝑦²+𝜅(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶. (1.5)

For such a case, 𝑆 is also said to be 𝜅-strict pseudocontraction. It is clear that (1.5) is equivalent to 𝑆𝑥−𝑆𝑦,𝑥−𝑦≤𝑥−𝑦²−1−𝜅2(𝑥−𝑆𝑥)−𝑦−𝑆𝑦²,∀𝑥,𝑦∈𝐶. (1.6)

𝑆 is said to be pseudocontractive if 𝑆𝑥−𝑆𝑦²≤𝑥−𝑦²+(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶. (1.7)

It is clear that (1.7) is equivalent to 𝑆𝑥−𝑆𝑦,𝑥−𝑦≤𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (1.8)

The class of 𝜅-strict pseudocontractions which was introduced by Browder and Petryshyn [17] in 1967 has been considered by many authors. It is easy to see that the class of strict pseudocontractions falls into the one between the class of nonexpansive mappings and the class of pseudocontractions. For studying the class of strict pseudocontractions, Zhou [18] proposed the following convex combination method: define a mapping 𝑆_𝑡∶𝐶→𝐻 by 𝑆_𝑡𝑥=𝑡𝑥+(1−𝑡)𝑆𝑥,∀𝑥∈𝐶. (1.9)

He showed that 𝑆_𝑡 is nonexpansive if 𝑡∈[𝜅,1); see [18] for more details.

Recently, many authors considered the problem of finding a common element in the fixed point set of a nonexpansive mapping and in the solution set of the equilibrium problem (1.1) based on iterative methods; see, for instance, [19–27].

In 2007, Tada and Takahashi [23] considered an iterative method for the equilibrium problem (1.1) and a nonexpansive nonself-mapping. To be more precise, they obtained the following results.

Theorem TT. Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻, let 𝑓∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)–(A4), and let 𝑆 be a nonexpansive mapping of 𝐶 into 𝐻 such that 𝐹(𝑆)∩𝐸𝑃(𝑓)≠∅. Let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated by 𝑥₁=𝑥∈𝐻, and let 𝑢_𝑛∈𝐶suchthat𝑓𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑤≥0,∀𝑦∈𝐶,_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑆𝑢_𝑛,𝐶_𝑛=𝑧∈𝐻∶𝑤_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐻∶𝑥_𝑛−𝑧,𝑥−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥, (1.10)
for every 𝑛∈ℕ, where {𝛼_𝑛}⊂[𝑎,1], for some 𝑎∈(0,1) and {𝑟_𝑛}⊂(0,∞) satisfies liminf_𝑛→∞𝑟_𝑛>0. Then the sequence {𝑥_𝑛} converges strongly to 𝑃_{𝐹(𝑆)∩𝐸𝑃(𝑓)}(𝑥).

We remark that the iterative process (1.10) is called the hybrid projection iterative process. Recently, the hybrid projection iterative process which was first considered by Haugazeau [28] in 1968 has been studied for fixed point problem of nonlinear mappings and equilibrium problems by many authors. Since the sequence generated in the hybrid projection iterative process depends on the sets 𝐶_𝑛 and 𝑄_𝑛, the hybrid projection iterative process is also known as “CQ” iterative process; see [29] and the reference therein.

Recently, Takahashi et al. [30] considered the shrinking projection process for the fixed point problem of nonexpansive self-mapping. More precisely, they obtain the iterative sequence monontonely without the help of the set 𝑄_𝑛; see [30] for more details.

In this paper, we reconsider the same shrinking projection process for the equilibrium problem (1.1) and a strictly pseudocontractive nonself-mapping. We show that the sequence generated in the proposed iterative process converges strongly to some common element in the fixed point set of a strictly pseudocontractive nonself-mapping and in the solution set of the equilibrium problem (1.1). The main results presented in this paper mainly improved the corresponding results in Tada and Takahashi [23].

2. Preliminaries

Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝑃_𝐶 be the metric projection from 𝐻 onto 𝐶. That is, for 𝑥∈𝐻,𝑃_𝐶𝑥 is the only point in 𝐶 such that ‖𝑥−𝑃_𝐶𝑥‖=inf{‖𝑥−𝑧‖∶𝑧∈𝐶}. We know that the mapping 𝑃_𝐶 is firmly nonexpansive, that is, 𝑃_𝐶𝑥−𝑃_𝐶𝑦²≤𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦,∀𝑥,𝑦∈𝐻. (2.1)

The following lemma can be found in [1, 2].

Lemma 2.1. Let 𝐶 be a nonempty closed convex subset of 𝐻, and let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)–(A4). Then, for any 𝑟>0 and 𝑥∈𝐻, there exists 𝑧∈𝐶 such that 𝐹+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (2.2)
Further, define 𝑇_𝑟+1𝑥=𝑧∈𝐶∶𝐹𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶 (2.3)
for all 𝑟>0 and 𝑥∈𝐻. Then, the following hold: (a)𝑇_𝑟 is single valued; (b)𝑇_𝑟 is firmly nonexpansive, that is, 𝑇_𝑟𝑥−𝑇_𝑟𝑦²≤𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦,∀𝑥,𝑦∈𝐻; (2.4)(c)𝐹(𝑇_𝑟)=𝐸𝑃(𝐹); (d)𝐸𝑃(𝐹) is closed and convex.

Lemma 2.2 (see [18]). Let 𝐻 be a real Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻, and 𝑆∶𝐶→𝐻 a strict pseudocontraction. Then the mapping 𝐼−𝑆 is demiclosed at zero, that is, if {𝑥_𝑛} is a sequence in 𝐶 such that 𝑥_𝑛⇀𝑥 and 𝑥_𝑛−𝑆𝑥_𝑛→0, then 𝑥∈𝐹(𝑆).

Lemma 2.3 (see [18]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝑆∶𝐶→𝐻 a 𝜅-strict pseudocontraction. Define a mapping 𝑆_𝛼𝑥=𝛼𝑥+(1−𝛼)𝑆𝑥 for all 𝑥∈𝐶. If 𝛼∈[𝜅,1), then the mapping 𝑆_𝛼 is a nonexpansive mapping such that 𝐹(𝑆_𝛼)=𝐹(𝑆).

3. Main Results

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹₁ and 𝐹₂ be two bifunctions from 𝐶×𝐶 to ℝ which satisfy (A1)–(A4), respectively. Let 𝑆∶𝐶→𝐻 be a 𝜅-strict pseudocontraction. Assume that ℱ∶=𝐸𝑃(𝐹₁)∩𝐸𝑃(𝐹₂)∩𝐹(𝑆)≠∅. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be sequences in [0,1]. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₁𝐶∈𝐻,₁𝑧=𝐻,_𝑛=𝛾_𝑛𝑢_𝑛+1−𝛾_𝑛𝑣_𝑛,𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝛽_𝑛𝑧_𝑛+1−𝛽_𝑛𝑆𝑧_𝑛,𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝑦_𝑛−𝑤≤𝑥_𝑛,𝑥−𝑤_𝑛+1=𝑃_𝐶𝑛+1𝑥₁,𝑛≥0, (Δ)
where 𝑢_𝑛 is chosen such that 𝐹₁𝑢_𝑛+1,𝑢𝑟_𝑛𝑢−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0,∀𝑢∈𝐶, (3.1)
and 𝑣_𝑛 is chosen such that 𝐹₂𝑣_𝑛+1,𝑣𝑠_𝑛𝑣−𝑣_𝑛,𝑣_𝑛−𝑥_𝑛≥0,∀𝑣∈𝐶. (3.2)
Assume that the control sequences {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝑟_𝑛}, and {𝑠_𝑛} satisfy the following restrictions:(a)0≤𝛼_𝑛≤𝑎<1,𝜅≤𝛽_𝑛≤𝑏<1,0<𝑐≤𝛾_𝑛≤𝑑<1; (b)0<𝑒≤𝑟_𝑛,0<𝑓≤𝑠_𝑛
for some 𝑎,𝑏,𝑐,𝑑,𝑒,𝑓∈ℝ. Then the sequence {𝑥_𝑛} generated in the (Δ) converges strongly to some point 𝑥, where 𝑥=𝑃_ℱ𝑥₁.

Proof. First, we show that 𝐶_𝑛 is closed and convex for each 𝑛≥1. It is easy to see that 𝐶_𝑛 is closed for each 𝑛≥1. We only show that 𝐶_𝑛 is convex for each 𝑛≥1. Note that 𝐶₁=𝐻 is convex. Suppose that 𝐶_𝑚 is convex for some positive integer 𝑚. Next, we show that 𝐶_𝑚+1 is convex for the same 𝑚. Note that ‖𝑦_𝑛−𝑤‖≤‖𝑥_𝑛−𝑤‖ is equivalent to 2𝑥_𝑛−𝑦_𝑛,𝑤≤𝑥_𝑛²−𝑦_𝑛². (3.3)
Take 𝑤₁ and 𝑤₂ in 𝐶_𝑚+1, and put 𝑤=𝑡𝑤₁+(1−𝑡)𝑤₂. It follows that 𝑤₁∈𝐶_𝑚,𝑤₂∈𝐶_𝑚, 2𝑥_𝑛−𝑦_𝑛,𝑤₁≤𝑥_𝑛²−𝑦_𝑛²,2𝑥_𝑛−𝑦_𝑛,𝑤₂≤𝑥_𝑛²−𝑦_𝑛². (3.4) Combining (3.4), we can obtain that 2⟨𝑥_𝑛−𝑦_𝑛,𝑤⟩≤‖𝑥_𝑛‖²−‖𝑦_𝑛‖², that is, ‖𝑦_𝑛−𝑤‖≤‖𝑥_𝑛−𝑤‖. In view of the convexity of 𝐶_𝑚, we see that 𝑤∈𝐶_𝑚. This shows that 𝑤∈𝐶_𝑚+1. This concludes that 𝐶_𝑛 is closed and convex for each 𝑛≥1.
Define a mapping 𝑆_𝑛∶𝐶→𝐻 by 𝑆_𝑛𝑥=𝛽_𝑛𝑥+(1−𝛽_𝑛)𝑆𝑥 for all 𝑥∈𝐶. It follows from Lemma 2.3 that 𝑆_𝑛 is nonexpansive and 𝐹(𝑆_𝑛)=𝐹(𝑆) for all 𝑛≥1. Next, we show that ℱ⊂𝐶_𝑛 for all 𝑛≥1. It is easy to see that ℱ⊂𝐶₁=𝐻. Suppose that ℱ⊂𝐶_𝑘 for some integer 𝑘≥1. We intend to claim that ℱ⊂𝐶_𝑘+1 for the same 𝑘. For any 𝑝∈ℱ⊂𝐶_𝑘, we have 𝑦_𝑘−𝑝≤𝛼_𝑘𝑥_𝑘−𝑝+1−𝛼_𝑘𝑆_𝑘𝑧_𝑘−𝑝≤𝛼_𝑘𝑥_𝑘−𝑝+1−𝛼_𝑘𝑧_𝑘−𝑝≤𝛼_𝑘𝑥_𝑘−𝑝+1−𝛼_𝑘𝛾_𝑘𝑢_𝑘−𝑝+1−𝛾_𝑘𝑣_𝑘−𝑝=𝛼_𝑘𝑥_𝑘−𝑝+1−𝛼_𝑘𝛾_𝑘𝑇_𝑟𝑘𝑥_𝑘−𝑝+1−𝛾_𝑘𝑇_𝑠𝑘𝑥_𝑘−𝑝≤𝛼_𝑘𝑥_𝑘−𝑝+1−𝛼_𝑘𝛾_𝑘𝑥_𝑘−𝑝+1−𝛾_𝑘𝑥_𝑘=𝑥−𝑝_𝑘−𝑝. (3.5)
This shows that 𝑝∈𝐶_𝑘+1. This proves that ℱ⊂𝐶_𝑛 for all 𝑛≥1.
Since 𝑥_𝑛=𝑃_𝐶𝑛𝑥₁ and 𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₁∈𝐶_𝑛+1⊂𝐶_𝑛, we have that 0≤𝑥₁−𝑥_𝑛,𝑥_𝑛−𝑥_𝑛+1=𝑥₁−𝑥_𝑛,𝑥_𝑛−𝑥₁+𝑥₁−𝑥_𝑛+1≤−𝑥₁−𝑥_𝑛²+𝑥₁−𝑥_𝑛𝑥₁−𝑥_𝑛+1. (3.6)
It follows that 𝑥₁−𝑥_𝑛≤𝑥₁−𝑥_𝑛+1. (3.7)
On the other hand, for any 𝑝∈ℱ⊂𝐶_𝑛, we see that ‖𝑥₁−𝑥_𝑛‖≤‖𝑥₁−𝑝‖. In particular, we have 𝑥₁−𝑥_𝑛≤𝑥₁−𝑃_ℱ𝑥₁. (3.8)
This shows that the sequence {𝑥_𝑛} is bounded. In view of (3.7), we see that lim_𝑛→∞‖𝑥_𝑛−𝑥₁‖ exists. It follows from (3.6) that 𝑥_𝑛−𝑥_𝑛+1²=𝑥_𝑛−𝑥₁+𝑥₁−𝑥_𝑛+1²=𝑥_𝑛−𝑥₁²+2𝑥_𝑛−𝑥₁,𝑥₁−𝑥_𝑛+1+𝑥₁−𝑥_𝑛+1²=𝑥_𝑛−𝑥₁²+2𝑥_𝑛−𝑥₁,𝑥₁−𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+1+𝑥₁−𝑥_𝑛+1²=𝑥_𝑛−𝑥₁²−2𝑥_𝑛−𝑥₁²+2𝑥_𝑛−𝑥₁,𝑥_𝑛−𝑥_𝑛+1+𝑥₁−𝑥_𝑛+1²≤𝑥₁−𝑥_𝑛+1²−𝑥_𝑛−𝑥₁². (3.9)
This yields that lim_𝑛→∞𝑥_𝑛−𝑥_𝑛+1=0. (3.10)
Since 𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₁∈𝐶_𝑛+1, we see that 𝑦_𝑛−𝑥_𝑛+1≤𝑥_𝑛−𝑥_𝑛+1. (3.11)
It follows that 𝑦_𝑛−𝑥_𝑛≤𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛≤2𝑥_𝑛−𝑥_𝑛+1. (3.12)
From (3.10), we obtain that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (3.13)
On the other hand, we have 𝑥_𝑛−𝑦_𝑛=𝑥_𝑛−𝛼_𝑛𝑥_𝑛−1−𝛼_𝑛𝑆_𝑛𝑧_𝑛=1−𝛼_𝑛𝑥_𝑛−𝑆_𝑛𝑧_𝑛. (3.14)
From the restriction (a), we obtain from (3.13) that lim_𝑛→∞𝑥_𝑛−𝑆_𝑛𝑧_𝑛=0. (3.15)
For any 𝑝∈ℱ, we have that 𝑢_𝑛−𝑝²=𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑝²≤𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑝,𝑥_𝑛=𝑢−𝑝_𝑛−𝑝,𝑥_𝑛=1−𝑝2𝑢_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑢_𝑛−𝑥_𝑛². (3.16)
This implies that 𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑢_𝑛−𝑥_𝑛². (3.17)
In a similar way, we get that 𝑣_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑣_𝑛−𝑥_𝑛². (3.18)
It follows from (3.17) and (3.18) that 𝑦_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑆_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝛾_𝑛𝑢_𝑛−𝑝²+1−𝛾_𝑛𝑣_𝑛−𝑝²≤𝑥_𝑛−𝑝²−1−𝛼_𝑛𝛾_𝑛𝑢_𝑛−𝑥_𝑛²−1−𝛼_𝑛1−𝛾_𝑛𝑣_𝑛−𝑥_𝑛². (3.19)
This implies that 1−𝛼_𝑛𝛾_𝑛𝑢_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛. (3.20)
In view of the restriction (a), we obtain from (3.13) that lim_𝑛→∞𝑢_𝑛−𝑥_𝑛=0. (3.21)
It also follows from (3.19) that 1−𝛼_𝑛1−𝛾_𝑛𝑣_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛. (3.22)
In view of the restriction (a), we obtain from (3.13) that lim_𝑛→∞𝑣_𝑛−𝑥_𝑛=0. (3.23)
Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑞. From (3.21) and (3.23), we see that 𝑢_𝑛𝑖⇀𝑞 and 𝑣_𝑛𝑖⇀𝑞, respectively. From (3.21) and the restriction (b), we see that lim_𝑛→∞𝑢_𝑛−𝑥_𝑛𝑟_𝑛=0. (3.24)
Now, we are in a position to show that 𝑞∈𝐸𝑃(𝐹₁). Note that 𝐹₁𝑢_𝑛+1,𝑢𝑟_𝑛𝑢−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0,∀𝑢∈𝐶. (3.25)
From (A2), we see that 1𝑟_𝑛𝑢−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥𝐹₁𝑢,𝑢_𝑛. (3.26)
Replacing 𝑛 by 𝑛_𝑖, we arrive at 𝑢−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖≥𝐹₁𝑢,𝑢_𝑛𝑖. (3.27)
In view of (3.24) and (A4), we get that 𝐹₁𝑢,𝑞≤0,∀𝑢∈𝐶. (3.28) For any 𝑡 with 0<𝑡≤1 and 𝑢∈𝐶, let 𝑢_𝑡=𝑡𝑢+(1−𝑡)𝑞. Since 𝑢∈𝐶 and 𝑞∈𝐶, we have 𝑢_𝑡∈𝐶 and hence 𝐹₁(𝑢_𝑡,𝑞)≤0. It follows that 0=𝐹₁𝑢_𝑡,𝑢_𝑡≤𝑡𝐹₁𝑢_𝑡,𝑢+(1−𝑡)𝐹₁𝑢_𝑡,𝑞≤𝑡𝐹₁𝑢_𝑡,𝑢, (3.29)
which yields that 𝐹₁𝑢_𝑡,𝑢≥0,∀𝑢∈𝐶. (3.30)
Letting 𝑡↓0, we obtain from (A3) that 𝐹₁𝑞,𝑢≥0,∀𝑢∈𝐶. (3.31) This means that 𝑞∈𝐸𝑃(𝐹₁). In the same way, we can obtain that 𝑞∈𝐸𝑃(𝐹₂). Next, we show that 𝑞∈𝐹(𝑆). Note that 𝑧_𝑛−𝑥_𝑛≤𝛾_𝑛𝑢_𝑛−𝑥_𝑛+1−𝛾_𝑛𝑣_𝑛−𝑥_𝑛. (3.32)
It follows from (3.21) and (3.23) that lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0. (3.33)
On the other hand, we have 𝑥_𝑛−𝑆_𝑛𝑥_𝑛≤𝑆_𝑛𝑥_𝑛−𝑆_𝑛𝑧_𝑛+𝑆_𝑛𝑧_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑧_𝑛+𝑆_𝑛𝑧_𝑛−𝑥_𝑛. (3.34)
It follows from (3.15) and (3.33) that lim_𝑛→∞𝑥_𝑛−𝑆_𝑛𝑥_𝑛=0. (3.35)
Note that 𝑆𝑥_𝑛−𝑥_𝑛≤𝑆𝑥_𝑛−𝑆_𝑛𝑥_𝑛+𝑆_𝑛𝑥_𝑛−𝑥_𝑛≤𝛽_𝑛𝑆𝑥_𝑛−𝑥_𝑛+𝑆_𝑛𝑥_𝑛−𝑥_𝑛, (3.36)
which yields that 1−𝛽_𝑛𝑆𝑥_𝑛−𝑥_𝑛≤𝑆_𝑛𝑥_𝑛−𝑥_𝑛. (3.37)
This implies from the restriction (a) and (3.35) that lim_𝑛→∞𝑆𝑥_𝑛−𝑥_𝑛=0. (3.38)
It follows from Lemma 2.2 that 𝑞∈𝐹(𝑆). This shows that 𝑞∈ℱ. Since 𝑥=𝑃_ℱ𝑥₁, we obtain that 𝑥₁−𝑥≤𝑥₁−𝑞≤liminf_𝑖→∞𝑥₁−𝑥_𝑛𝑖≤limsup_𝑖→∞𝑥₁−𝑥_𝑛𝑖≤𝑥₁−𝑥, (3.39)
which yields that lim_𝑖→∞𝑥₁−𝑥_𝑛𝑖=𝑥₁−𝑞=𝑥₁−𝑥. (3.40) It follows that {𝑥_𝑛𝑖} converges strongly to 𝑥. Therefore, we can conclude that the sequence {𝑥_𝑛} converges strongly to 𝑥=𝑃_ℱ𝑥₁. This completes the proof.

From Theorem 3.1, we have the following results.

Corollary 3.2. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹 be a bifunction from 𝐶×𝐶 to ℝ which satisfies (A1)–(A4). Let 𝑆∶𝐶→𝐻 be a 𝜅-strict pseudocontraction. Assume that ℱ∶=𝐸𝑃(𝐹)∩𝐹(𝑆)≠∅. Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in [0,1]. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₁𝐶∈𝐻,₁𝑦=𝐻,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝛽_𝑛𝑢_𝑛+1−𝛽_𝑛𝑆𝑢_𝑛,𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝑦_𝑛−𝑤≤𝑥_𝑛,𝑥−𝑤_𝑛+1=𝑃_𝐶𝑛+1𝑥₁,𝑛≥0, (3.41)
where 𝑢_𝑛 is chosen such that 𝐹𝑢_𝑛+1,𝑢𝑟_𝑛𝑢−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0,∀𝑢∈𝐶. (3.42)
Assume that the control sequences {𝛼_𝑛}, {𝛽_𝑛}, and {𝑟_𝑛} satisfy the following restrictions: (a)0≤𝛼_𝑛≤𝑎<1,𝜅≤𝛽_𝑛≤𝑏<1; (b)0<𝑒≤𝑟_𝑛
for some 𝑎,𝑏,𝑒∈ℝ. Then the sequence {𝑥_𝑛} converges strongly to some point 𝑥, where 𝑥=𝑃_ℱ𝑥₁.

Proof. Putting 𝐹₁≡𝐹₂≡𝐹 and 𝑟_𝑛≡𝑠_𝑛 in Theorem 3.1, we see that 𝑧_𝑛≡𝑢_𝑛. From the proof of Theorem 3.1, we can conclude the desired conclusion immediately.

Remark 3.3. Corollary 3.2 improves Theorem TT in the following aspects.(1)From the viewpoint of mappings, the class of nonexpansive mappings is extended to the class of strict pseudocontractions. (2)From the viewpoint of computation, the set 𝑄_𝑛 is removed.

Corollary 3.4. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑆∶𝐶→𝐻 be a 𝜅-strict pseudocontraction with fixed points. Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in [0,1]. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₁𝐶∈𝐻,₁𝑦=𝐻,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝛽_𝑛𝑃_𝐶𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝑥_𝑛,𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝑦_𝑛−𝑤≤𝑥_𝑛,𝑥−𝑤_𝑛+1=𝑃_𝐶𝑛+1𝑥₁,𝑛≥0. (3.43)
Assume that the control sequences {𝛼_𝑛} and {𝛽_𝑛} satisfy the restrictions 0≤𝛼_𝑛≤𝑎<1 and 𝜅≤𝛽_𝑛≤𝑏<1 for some 𝑎,𝑏∈ℝ. Then the sequence {𝑥_𝑛} converges strongly to some point 𝑥, where 𝑥=𝑃_𝐹(𝑆)𝑥₁.

Proof. Putting 𝐹₁≡𝐹₂≡0 and 𝑟_𝑛≡𝑠_𝑛≡1, we can obtain from Theorem 3.1 the desired conclusion easily.
If 𝑆 is nonexpansive and 𝛽_𝑛≡0, then Corollary 3.4 is reduced to the following.

Corollary 3.5. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑆∶𝐶→𝐻 be a nonexpansive mapping with fixed points. Let {𝛼_𝑛} be a sequence in [0,1]. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₁𝐶∈𝐻,₁𝑦=𝐻,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆𝑃_𝐶𝑥_𝑛,𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝑦_𝑛−𝑤≤𝑥_𝑛,𝑥−𝑤_𝑛+1=𝑃_𝐶𝑛+1𝑥₁,𝑛≥0. (3.44)
Assume that the control sequence {𝛼_𝑛} satisfies the restriction 0≤𝛼_𝑛≤𝑎<1 for some 𝑎∈ℝ. Then the sequence {𝑥_𝑛} converges strongly to some point 𝑥, where 𝑥=𝑃_𝐹(𝑆)𝑥₁.

Recently, many authors studied the following convex feasibility problem (CFP): findingan𝑥∈^𝑁_𝑚=1𝐶_𝑚, (3.45) where 𝑁≥1 is an integer and each 𝐶_𝑚 is a nonempty closed and convex subset of a real Hilbert space 𝐻. There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomography, and radiation therapy treatment planning.

Next, we consider the case that each 𝐶_𝑚 is the solution set of an equilibrium problem.

Theorem 3.6. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹_𝑚 be a bifunction from 𝐶×𝐶 to ℝ which satisfies (A1)–(A4) for each 1≤𝑚≤𝑁. Assume that ℱ∶=∩^𝑁_𝑚=1𝐸𝑃(𝐹_𝑚)≠∅. Let {𝛼_𝑛}, {𝛾_(𝑛,1)},…, and {𝛾_(𝑛,𝑁)} be sequences in [0,1]. Let 𝑟_(𝑛,1),…, and 𝑟_(𝑛,𝑁) be sequences in (0,∞). Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑥₁𝐶∈𝐻,₁𝐹=𝐻,_𝑚𝑢_(𝑛,𝑚)+1,𝑢𝑟_(𝑛,𝑚)𝑢−𝑢_(𝑛,𝑚),𝑢_(𝑛,𝑚)−𝑥_𝑛𝑦≥0,∀𝑢∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛^𝑁_𝑚=1𝛾_(𝑛,𝑚)𝑢_(𝑛,𝑚),𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝑦_𝑛−𝑤≤𝑥_𝑛,𝑥−𝑤_𝑛+1=𝑃_𝐶𝑛+1𝑥₁,𝑛≥0. (3.46)
Assume that the control sequences satisfy the following restrictions:(a)0≤𝛼_𝑛≤𝑎<1,∑^𝑁_𝑚=1𝛾_(𝑛,𝑚)=1, 0<𝑐_𝑚≤𝛾_(𝑛,𝑚)≤𝑑_𝑚<1;(b)0<𝑒_𝑚≤𝑟_(𝑛,𝑚), where 𝑎,𝑐₁,…,𝑐_𝑚,𝑑₁,…,𝑑_𝑚,𝑒₁,…,𝑒_𝑚∈ℝ. Then the sequence {𝑥_𝑛} generated in the above iterative process converges strongly to some point 𝑥, where 𝑥=𝑃_ℱ𝑥₁.

Proof. Let 𝑆 be the identity mapping and 𝛽_𝑛≡0, then we can obtain from Theorem 3.1 the desired conclusion easily.

Acknowledgment

This work was supported by Natural Science Foundation of Zhejiang Province (Y6110270).