An Alternative Regularization Method for Equilibrium Problems and Fixed Point of Nonexpansive Mappings

Abstract

We introduce a new regularization iterative algorithm for equilibrium and fixed point problems of nonexpansive mapping. Then, we prove a strong convergence theorem for nonexpansive mappings to solve a unique solution of the variational inequality and the unique sunny nonexpansive retraction. Our results extend beyond the results of S. Takahashi and W. Takahashi (2007), and many others.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝜙 be a bifunction of 𝐶×𝐶 into ℝ, where ℝ is the set of real numbers. The equilibrium problem for 𝜙∶𝐶×𝐶→ℝ is to find 𝑥∈𝐶 such that 𝜙𝑥,𝑦≥0,∀𝑦∈𝐶. (1.1) The set of solutions of (1.1) is denoted by EP(𝜙). Given a mapping 𝑇∶𝐶→𝐻, let 𝜙(𝑥,𝑦)=⟨𝑇𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶. Then, 𝑧∈EP(𝜙) if and only if ⟨𝑇𝑧,𝑦−𝑧⟩≥0 for all 𝑦∈𝐶, that is, 𝑧 is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, [1–6].

A mapping 𝑆 of 𝐶 into 𝐻 is said to be nonexpansive if 𝑆𝑥−𝑆𝑦≤𝑥−𝑦∀𝑥,𝑦∈𝐶. (1.2)

We denote by 𝐹(𝑆) the set of fixed points of 𝑆. The fixed point equation 𝑇𝑥=𝑥 is ill-posed (it may fail to have a solution, nor uniqueness of solution) in general. Regularization therefore is needed. Contractions can be used to regularize nonexpansive mappings. In fact, the following regularization has widely been implemented ([7–9]). Fixing a point 𝑢∈𝐶 and for each 𝑡∈(0,1), one defines a contraction 𝑇_𝑡∶𝐶→𝐶 by 𝑇_𝑡𝑥=𝑡𝑢+(1−𝑡)𝑇𝑥,𝑥∈𝐶. (1.3) In this paper we provide an alternative regularization method. Our idea is to shrink 𝑥 first and then apply 𝑇 to the convex combination of the shrunk 𝑥 and the anchor 𝑢 (this idea appeared implicitly in [10] where iterative methods for finding zeros of maximal monotone operators were investigated). In other words, we fix an anchor 𝑢∈𝐶 and 𝑡∈(0,1) and define a contraction 𝑇_𝑡∶𝐶→𝐶 by 𝑇_𝑡𝑥=𝑇(𝑡𝑢+(1−𝑡)𝑥),𝑥∈𝐶. (1.4) Compared with (1.1), (1.4) looks slightly more compact in the sense that the mapping 𝑇 is more directly involved in the regularization and thus may be more convenient in manipulations since the nonexpansivity of 𝑇 is utilized first.

In 2000, Moudafi [11] proved the following strong convergence theorem.

Theorem 1.1 (Moudafi [11]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that 𝐹(𝑆) is nonempty. Let 𝑓 be a contraction of 𝐶 into itself and let {𝑥_𝑛} be a sequence defined as follows: 𝑥₁=𝑥∈𝐶 and 𝑥_𝑛+1=11+𝜀_𝑛𝑆𝑥_𝑛+𝜀_𝑛1+𝜀_𝑛𝑓𝑥_𝑛 (1.5) for all 𝑛∈𝑁, where {𝜀_𝑛}⊂(0,1) satisfies lim_𝑛→∞𝜀_𝑛=0,^∞_𝑛=1𝜀_𝑛=∞,lim_𝑛→∞1𝜀_𝑛+1−1𝜀_𝑛=0. (1.6) Then, {𝑥_𝑛} converges strongly to 𝑧∈𝐹(𝑆), where 𝑧=𝑃_𝐹(𝑆)𝑓(𝑧) and 𝑃_𝐹(𝑆) is the metric projection of 𝐻 onto 𝐹(𝑆).

Such a method for approximation of fixed points is called the viscosity approximation method.

In 2007, S. Takahashi and W. Takahashi [5] introduced and considered the following iterative algorithm by the viscosity approximation method in the Hilbert space:𝑥₁𝜙𝑢∈𝐻,_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈C,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆𝑢_𝑛, (1.7) for all 𝑛∈𝑁, where {𝛼_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞) satisfy some appropriate conditions. Furthermore, they proved that {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑧∈𝐹(𝑆)∩EP(𝜙), where 𝑧=𝑃_{𝐹(𝑆)∩EP(𝜙)}𝑓(𝑧).

In this paper, motivated and inspired by the above results, we introduce an iterative scheme by the general iterative method for finding a common element of the set of solutions of (1.1) and the set of fixed points of a nonexpansive mapping in Hilbert space.

Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping. Starting with an arbitrary 𝑥₁,𝑢∈𝐻, define sequences {𝑥_𝑛} and {𝑢_𝑛} by 𝜙𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑇𝛼_𝑛𝑢+1−𝛼_𝑛𝑢_𝑛,𝑛≥1. (1.8) We will prove in Section 3 that if the sequences {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} of parameters satisfy appropriate conditions, then the sequence {𝑥_𝑛} and {𝑢_𝑛} generated by (1.8) converges strongly to the unique solution of the variational inequality 𝐼−𝑓𝑧,𝑥−𝑧≥0,∀𝑥∈𝐹(𝑆)∩EP𝜙, (1.9) which is the optimality condition for the minimization problem min_{(𝑆)𝜙𝑥∈𝐹∩EP}12⟨𝑥,𝑥⟩−ℎ(𝑥), (1.10) where ℎ is a potential function for 𝑓 and at the same time, the sequence {𝑥_𝑛} and {𝑢_𝑛} generated by (1.8) converges in norm to 𝑄(𝑢), where 𝑄∶𝐶→Fix(𝑇) is the sunny nonexpansive retraction.

2. Preliminaries

Throughout this paper, we consider 𝐻 as the Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively, 𝐶 is a nonempty closed convex subset of 𝐻. Consider a subset 𝐷 of 𝐶 and a mapping 𝑄∶𝐶→𝐷. Then we say that(i)𝑄 is a retraction provided 𝑄𝑥=𝑥𝑓𝑜𝑟𝑥∈𝐷;(ii)𝑄 is a nonexpansive retraction provided 𝑄 is a retraction that is also nonexpansive;(iii)𝑄 is a sunny nonexpansive retraction provided 𝑄 is a nonexpansive retraction with the additional property: 𝑄(𝑥+𝑡(𝑥−𝑄𝑥))=𝑄𝑥 whenever 𝑥+𝑡(𝑥−𝑄𝑥)∈𝐶, where 𝑥∈𝐶 and 𝑡≥0.

Let now 𝑇∶𝐶→𝐶 be a nonexpansive mapping. For a fixed anchor 𝑢∈𝐶 and each 𝑡∈(0,1) recall that 𝑧_𝑡∈𝐶 is the unique fixed point of the contraction 𝐶∋𝑥↦𝑇(𝑡𝑢+(1−𝑡)𝑥). Namely, 𝑧_𝑡∈𝐶 is the unique solution in 𝐶 to the fixed point equation 𝑧_𝑡=𝑇𝑡𝑢+(1−𝑡)𝑧_𝑡. (2.1) In the Hilbert space (either uniformly smooth or reflexive with a weakly continuous duality map), then 𝑧_𝑡 is strongly convergent should it is bounded as 𝑡→0⁺.

We also know that for any sequence {𝑥_𝑛}⊂𝐻 with 𝑥_𝑛⇀𝑥, the inequality liminf_𝑛→∞𝑥_𝑛−𝑥<liminf_𝑛→∞𝑥_𝑛−𝑦 (2.2) holds for every 𝑦∈𝐻 with 𝑥≠𝑦, (we usually call it Opial’s condition); see [12, 13] for more details.

For solving the equilibrium problem for a bifunction 𝜙∶𝐶×𝐶→ℝ, let us assume that 𝜙 satisfies the following conditions:(𝐴₁)𝜙(𝑥,𝑥)=0,forall𝑥∈𝐶; (𝐴₂)𝜙 is monotone, that is, 𝜙(𝑥,𝑦)+𝜙(𝑦,𝑥)≤0,forall𝑥,𝑦∈𝐶;(𝐴₃) For each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡↓0𝜙(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝜙(𝑥,𝑦);(𝐴₄) For each 𝑥∈𝐶, the function 𝑦↦𝜙(𝑥,𝑦) is convex and lower semicontinuous.

The following lemma appeared implicitly in [14].

Lemma 2.1 (see [14]). Let 𝐶 be a nonempty closed convex subset of 𝐻 and let 𝜙∶𝐶×𝐶→ℝ be a bifunction satisfying (A₁)–(A₄). Let 𝑟>0 and 𝑥∈𝐻, then, there exists 𝑧∈𝐶 such that 𝜙+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (2.3)

Lemma 2.2 (see [6]). Assume that 𝜙∶𝐶×𝐶→ℝ satisfies (A₁)–(A₄). For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶𝜙𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶, (2.4) for all 𝑧∈𝐻. Then, the following hold:(1)𝑇_𝑟 is single-valued;(2)𝑇_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻,𝑇_𝑟𝑥−𝑇_𝑟𝑦²≤𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦; (2.5)(3)𝐹(𝑇_𝑟)=EP(𝜙);(4)EP(𝜙) is closed and convex.

Lemma 2.3 (see [15]). Let {𝑎_𝑛}⊂[0,∞), {𝑏_𝑛}⊂[0,∞) and {𝑐_𝑛}⊂[0,1) be sequences of real numbers such that 𝑎_𝑛+1≤1−𝑐_𝑛𝑎_𝑛+𝑏_𝑛,∀𝑛∈𝑁,_∞𝑛=1𝑐_𝑛=∞,^∞_𝑛=1𝑏_𝑛<∞𝑜𝑟limsup_𝑛→∞𝑏_𝑛𝑐_𝑛≤0, (2.6) then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.4 (see [9]). Suppose that 𝑋 is a smooth Banach space. Then a retraction 𝑄∶𝐶→𝐷 is sunny nonexpansive if and only if 𝑥−𝑄𝑥,𝐽𝑦−𝑄𝑥≤0,𝑥∈𝐶,𝑦∈𝐷. (2.7)

Lemma 2.5. Let 𝑋 be a uniformly smooth Banach space, 𝐶 a nonempty closed convex subset of 𝑋, and 𝑇∶𝐶→𝐶 a nonexpansive mapping. Let 𝑧_𝑡 be defined by (2.1). Then (𝑧_𝑡) remains bounded as 𝑡→0 if and only if Fix(𝑇)≠. Moreover, if Fix(𝑇)≠, then (𝑧_𝑡) converges in norm, as 𝑡→0⁺, to a fixed point of T; and if one sets 𝑄(𝑢)∶=lim_𝑡→0𝑧_𝑡, (2.8) then 𝑄 defines the unique sunny nonexpansive retraction from 𝐶 onto Fix(𝑇).

Lemma 2.6. In the Hilbert space, the following inequalities always hold (i)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩; (ii)‖𝑡𝑥+(1−𝑡)𝑦‖²≤𝑡‖𝑥‖²+(1−𝑡)‖𝑦‖².

3. Main Results

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of 𝐻, 𝜙∶𝐶×𝐶→∞ be a bifunction satisfying (A₁)–(A₄) and 𝑇∶𝐶→𝐶 be a nonexpansive mapping of 𝐶 into 𝐻 such that 𝐹(𝑇)∩EP(𝜙)≠∅. Let 𝑓 be a contraction of 𝐻 into itself with 𝛼∈(0,1), initially give an arbitrary element 𝑥₁∈𝐻 and let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated by (1.8), where {𝛼_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞) satisfy the following conditions: (I)lim_𝑛→∞𝛼_𝑛=0,∑^∞_𝑛=1𝛼_𝑛=∞𝑎𝑛𝑑∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(II)liminf_𝑛𝑟_𝑛>0𝑎𝑛𝑑∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞;(III)lim_𝑛→∞𝛽_𝑛=0,∑^∞_𝑛=1𝛽_𝑛=∞𝑎𝑛𝑑∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞;(IV)lim_𝑛→∞(𝛼_𝑛/𝛽_𝑛)=0. Then, the sequences {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑧∈𝐹(𝑇)∩EP(𝜙), where 𝑧=𝑃_{𝐹(𝑇)∩EP(𝜙)}𝑓(𝑧) and converge in norm to 𝑄(𝑢), where 𝑄∶𝐶→Fix(𝑇) is the sunny nonexpansive retraction.

Proof. Let 𝑄=𝑃_{𝐹(𝑆)∩EP(𝜙)}. Then 𝑄𝑓 is a contraction of 𝐻 into itself. In fact, there exists 𝑎∈[0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝑎‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻. So, we have that 𝑄𝑓(𝑥)−𝑄𝑓𝑦≤𝑓(𝑥)−𝑓𝑦≤𝑎𝑥−𝑦 (3.1) for all 𝑥,𝑦∈𝐻. So, 𝑄𝑓 is a contraction of 𝐻 into itself. Since 𝐻 is complete, there exists a unique element 𝑧∈𝐻 such that 𝑧=𝑄𝑓(𝑧), such a 𝑧∈𝐻 is an element of 𝐶. We divide the proof into serval steps.
Step 1. {𝑥_𝑛} and {𝑢_𝑛} are all bounded. Let 𝑝∈𝐹(𝑇)∩EP(𝜙), Then from 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛, we have 𝑢_𝑛−𝑝=𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑝≤𝑥_𝑛−𝑝 (3.2) for all 𝑛∈𝑁. Put 𝑦_𝑛=𝛼_𝑛𝑢+(1−𝛼_𝑛)𝑢_𝑛, so {𝑥_𝑛+1} can be rewritten as 𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑇𝑦_𝑛,𝑦_𝑛−𝑝=𝛼_𝑛𝑢+1−𝛼_𝑛𝑢_𝑛=𝛼−𝑝_𝑛𝑢−𝑝+1−𝛼_𝑛𝑢_𝑛−𝑝≤𝛼_𝑛𝑢−𝑝+1−𝛼_𝑛𝑢_𝑛−𝑝. (3.3) Therefore, from (3.2) we get 𝑦_𝑛−𝑝≤𝛼_𝑛𝑢−𝑝+1−𝛼_𝑛𝑥_𝑛−𝑝≤max𝑢−𝑝,𝑥_𝑛−𝑝. (3.4) If ‖𝑥_𝑛−𝑝‖≤‖𝑢−𝑝‖, then {𝑥_𝑛} is bounded. So, we assume that ‖𝑥_𝑛−𝑝‖≥‖𝑢−𝑝‖.
Therefore ‖𝑦_𝑛−𝑝‖≤‖𝑥_𝑛−𝑝‖, 𝑥_𝑛+1−𝑝=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑇𝑦_𝑛=𝛽−𝑝_𝑛𝑓𝑥_𝑛−𝑝+1−𝛽_𝑛𝑇𝑦_𝑛−𝑝≤𝛽_𝑛𝑓𝑥_𝑛−𝑝+1−𝛽_𝑛𝑦_𝑛−𝑝≤𝛽_𝑛𝛼𝑥_𝑛−𝑝+1−𝛽_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑓𝑝=()𝛽−𝑝1−1−𝛼_𝑛𝑥_𝑛−𝑝+𝛽_𝑛()𝑓𝑝1−𝛼−𝑝𝑥1−𝛼≤max_𝑛,𝑓𝑝−𝑝−𝑝.1−𝛼 (3.5) So, by induction, we have 𝑥_𝑛−𝑝≤max𝑥₁,𝑓𝑝−𝑝−𝑝1−𝛼, (3.6) hence {𝑥_𝑛} is bounded. we also obtain that {𝑢_𝑛}, {𝑇𝑢_𝑛}, {𝑇𝑥_𝑛}, {𝑓(𝑥_𝑛)} and {𝑦_𝑛} are bounded.Step 2. ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞, 𝑥_𝑛+1−𝑥_𝑛=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽𝑛𝑇𝑦_𝑛−𝛽_𝑛−1𝑓𝑥_𝑛−1−1−𝛽_𝑛−1𝑇𝑦_𝑛−1=𝛽_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑓𝑥_𝑛−1+𝑇𝑦_𝑛−𝑇𝑦_𝑛−1−𝛽_𝑛𝑇𝑦_𝑛−𝑇𝑦_𝑛−1−𝛽_𝑛−𝛽_𝑛−1𝑇𝑦_𝑛−1≤𝛽_𝑛𝛼𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑓𝑥_𝑛−1−𝑇𝑦_𝑛−1+1−𝛽_𝑛𝑦_𝑛−𝑦_𝑛−1,𝑦_𝑛−𝑦_𝑛−1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑢_𝑛−𝛼_𝑛−1𝑢−1−𝛼_𝑛−1𝑢_𝑛−1=𝛼_𝑛−𝛼_𝑛−1𝑢+𝑢_𝑛−𝑢_𝑛−1−𝛼_𝑛𝑢_𝑛−𝑢_𝑛−1−𝛼_𝑛−𝛼_𝑛−1𝑢_𝑛−1≤𝛼_𝑛−𝛼_𝑛−1𝑢−𝑢_𝑛−1+1−𝛼_𝑛𝑢_𝑛−𝑢_𝑛−1. (3.7)(3.8)
On the other hand, from 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛 and 𝑢_𝑛+1=𝑇_𝑟𝑛+1𝑥_𝑛+1, we have 𝜙𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝜙𝑢≥0,∀𝑦∈𝐶,_𝑛+1+1,𝑦𝑟_𝑛+1𝑦−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1≥0,∀𝑦∈𝐶. (3.9)(3.10) Putting 𝑦=𝑢_𝑛+1 in (3.9) and 𝑦=𝑢_𝑛 in (3.10), we have 𝜙𝑢_𝑛,𝑢_𝑛+1+1𝑟_𝑛𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝜙𝑢≥0,_𝑛+1,𝑢_𝑛+1𝑟_𝑛+1𝑢_𝑛−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1≥0. (3.11) So, from (𝐴₂) we have 𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑟_𝑛−𝑢_𝑛+1−𝑥_𝑛+1𝑟_𝑛+1≥0, (3.12) and hence 𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛+1+𝑢_𝑛+1−𝑥_𝑛−𝑟_𝑛𝑟_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1≥0. (3.13) Without loss of generality, let us assume that there exists a real number 𝑏 such that 𝑟_𝑛>𝑏>0 for all 𝑛∈𝑁. Then, we have 𝑢_𝑛+1−𝑢_𝑛²≤𝑢_𝑛+1−𝑢_𝑛,𝑥_𝑛+1−𝑥_𝑛+𝑟1−^𝑛𝑟_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1≤𝑢_𝑛+1+𝑢_𝑛𝑥_𝑛+1−𝑥_𝑛+𝑟1−^𝑛𝑟_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1, (3.14) and hence 𝑢_𝑛+1−𝑢_𝑛≤𝑥_𝑛+1−𝑥_𝑛+1𝑟_𝑛+1𝑟_𝑛+1−𝑟_𝑛𝑢_𝑛+1−𝑥_𝑛+1≤𝑥_𝑛+1−𝑥_𝑛+1𝑏𝑟_𝑛+1−𝑟_𝑛𝐿, (3.15) where 𝐿=sup{‖𝑢_𝑛−𝑥_𝑛‖∶𝑛∈𝑁}. Then we obtain 𝑢_𝑛−𝑢_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+1𝑏𝑟_𝑛−𝑟_𝑛−1𝐿. (3.16)
So, put (3.8) and (3.16) into (3.7) we have 𝑥_𝑛+1−𝑥_𝑛≤𝛽_𝑛𝛼𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑓𝑥_𝑛−1−𝑇𝑦_𝑛−1+1−𝛽_𝑛𝛼_𝑛−𝛼_𝑛−1𝑢−𝑢_𝑛−1+1−𝛼_𝑛𝑥_𝑛+1−𝑥_𝑛+1𝑏𝑟_𝑛+1−𝑟_𝑛𝐿≤1−(1−𝛼)𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+1𝑏𝑟_𝑛+1−𝑟_𝑛𝐿+𝛼_𝑛−𝛼_𝑛−1𝐾₁+𝛽_𝑛−𝛽_𝑛−1𝐾₂, (3.17) where 𝐾₁∶=sup{‖𝑢−𝑢_𝑛‖,∀𝑛≥1} is a constant; 𝐾₂∶=sup{‖𝑓(𝑥_𝑛−1)‖+‖𝑇𝑦_𝑛−1‖,∀𝑛≥1} is a constant.
Using Lemma 2.3 and conditions (I), (II), (III) we have lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.18) From (3.15) and |𝑟_𝑛+1−𝑟_𝑛|→0, we have lim_𝑛→∞𝑢_𝑛+1−𝑢_𝑛=0. (3.19) Since 𝑦_𝑛=𝛼_𝑛𝑢+(1−𝛼_𝑛)𝑢_𝑛,𝑥_𝑛=𝛽_𝑛𝑓(𝑥_𝑛−1)+(1−𝛽_𝑛)𝑇𝑦_𝑛−1, we have 𝑦_𝑛−𝑢_𝑛=𝛼_𝑛𝑢−𝑢_𝑛𝑥⟶0,as𝑛⟶∞,_𝑛−𝑇𝑢_𝑛≤𝑥_𝑛−𝑇𝑦_𝑛−1+𝑇𝑦_𝑛−1−𝑇𝑢_𝑛−1+𝑇𝑢_𝑛−1−𝑇𝑢_𝑛≤𝛽_𝑛−1𝑓𝑥_𝑛−1+𝑇𝑦_𝑛−1+𝑦_𝑛−1−𝑢_𝑛−1+𝑢_𝑛−𝑢_𝑛−1⟶0,as𝑛⟶∞. (3.20)(3.21) For 𝑝∈𝐹(𝑇)∩EP(𝜙), we have 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,𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑢_𝑛−𝑥_𝑛². (3.22) Therefore, we have 𝑥_𝑛+1−𝑝²=1−𝛽_𝑛𝑇𝑦_𝑛−𝑝+𝛽_𝑛𝑓𝑥_𝑛−𝑝²≤1−𝛽_𝑛²𝑇𝑦_𝑛−𝑝²+2𝛽_𝑛𝑓𝑥_𝑛−𝑝,𝑥_𝑛+1≤−𝑝1−𝛽_𝑛²𝑦_𝑛−𝑢_𝑛+𝑢_𝑛−𝑝²+2𝛽_𝑛𝛼𝑥_𝑛−𝑝𝑥_𝑛+1−𝑝+2𝛽_𝑛𝑓𝑝−𝑝𝑥_𝑛+1=−𝑝1−𝛽_𝑛²𝑢_𝑛−𝑝²+𝑦_𝑛−𝑢_𝑛1−𝛽_𝑛²𝑦_𝑛−𝑢_𝑛+2𝑢_𝑛−𝑝+2𝛽_𝑛𝛼𝑥_𝑛−𝑝𝑥_𝑛+1−𝑝+2𝛽_𝑛𝑓𝑝−𝑝𝑥_𝑛+1=−𝑝1−𝛽_𝑛²𝑥_𝑛−𝑝²−𝑢_𝑛−𝑥_𝑛²+𝑦_𝑛−𝑢_𝑛1−𝛽_𝑛²𝑦_𝑛−𝑢_𝑛+2𝑢_𝑛−𝑝+2𝛽_𝑛𝛼𝑥_𝑛−𝑝𝑥_𝑛+1−𝑝+2𝛽_𝑛𝑓𝑝−𝑝𝑥_𝑛+1=−𝑝1−2𝛽_𝑛+𝛽²_𝑛𝑥_𝑛−𝑝²−1−𝛽_𝑛²𝑢_𝑛−𝑥_𝑛²+𝑦_𝑛−𝑢_𝑛1−𝛽_𝑛²𝑦_𝑛−𝑢_𝑛+2𝑢_𝑛−𝑝+2𝛽_𝑛𝛼𝑥_𝑛−𝑝𝑥_𝑛+1−𝑝+2𝛽_𝑛𝑓𝑝−𝑝𝑥_𝑛+1−𝑝,1−𝛽_𝑛²𝑢_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝛽_𝑛𝛽_𝑛𝑥_𝑛−𝑝²−2𝑥_𝑛−𝑝²−1−𝛽_𝑛²𝑢_𝑛−𝑥_𝑛²+𝑦_𝑛−𝑢_𝑛1−𝛽_𝑛²𝑦_𝑛−𝑢_𝑛+2𝑢_𝑛−𝑝+2𝛽_𝑛𝛼𝑥_𝑛−𝑝𝑥_𝑛+1−𝑝+2𝛽_𝑛𝑓𝑝−𝑝𝑥_𝑛+1≤𝑥−𝑝_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑝+𝑥_𝑛+1−𝑝+𝛽_𝑛𝛽_𝑛𝑥_𝑛−𝑝²−2𝑥_𝑛−𝑝²−1−𝛽_𝑛²𝑢_𝑛−𝑥_𝑛²+𝑦_𝑛−𝑢_𝑛1−𝛽_𝑛²𝑦_𝑛−𝑢_𝑛+2𝑢_𝑛−𝑝+2𝛽_𝑛𝛼𝑥_𝑛−𝑝𝑥_𝑛+1−𝑝+2𝛽_𝑛𝑓𝑝−𝑝𝑥_𝑛+1−𝑝. (3.23) By the above of what we have and the condition of lim_𝑛→∞𝛽_𝑛=0, we get lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0. Since ‖𝑇𝑢_𝑛−𝑢_𝑛‖≤‖𝑇𝑢_𝑛−𝑥_𝑛‖+‖𝑥_𝑛−𝑢_𝑛‖, it follows that ‖𝑇𝑢_𝑛−𝑢_𝑛‖→0.Step 3. we show that limsup_𝑛→∞𝑓(𝑧)−𝑧,𝑥_𝑛−𝑧≤0, (3.24) where 𝑧=𝑃_{𝐹(𝑆)∩EP(𝜙)}𝑓(𝑧). To show this inequality, we choose a subsequence {𝑢_𝑛𝑖} of {𝑢_𝑛} such that limsup_𝑖→∞𝑓(𝑧)−𝑧,𝑥_𝑛𝑖−𝑧=limsup_𝑛→∞𝑓(𝑧)−𝑧,𝑥_𝑛−𝑧. (3.25) Since {𝑢_𝑛𝑖} is bounded, there exists a subsequence {𝑢_𝑛𝑖𝑗} of {𝑢_𝑛𝑖} which converges weakly to 𝑤. Without loss of generality, we can assume that 𝑢_𝑛𝑖⇀𝑤. From ‖𝑇𝑢_𝑛−𝑢_𝑛‖→0, we obtain 𝑇𝑢_𝑛𝑖⇀𝑤. Let us show 𝑤∈EP(𝜙). By 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛, we have 𝜙𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶. (3.26) From (𝐴₂), we also have 1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥𝜙𝑦,𝑢_𝑛, (3.27) and hence 𝑦−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖≥𝜙𝑦,𝑢_𝑛𝑖. (3.28) Since 𝑢_𝑛𝑖−𝑥_𝑛𝑖/𝑟_𝑛𝑖→0 and 𝑢_𝑛𝑖⇀𝑤, from (𝐴₄) we have 0≥𝜙(𝑦,𝑤) for all 𝑦∈𝐶. For 𝑡 with 0<𝑡<1 and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑤. Since 𝑦∈𝐶 and 𝑤∈𝐶, we have 𝑦_𝑡∈𝐶 and hence 𝜙(𝑦_𝑡,𝑤)≤0. So, from (𝐴₁) and 𝐴₄ we have 0=𝜙𝑦_𝑡,𝑦_𝑡≤𝑡𝜙𝑦_𝑡,𝑦+(1−𝑡)𝜙𝑦_𝑡𝑦,𝑤≤𝑡𝜙_𝑡,𝑦, (3.29) and hence 0≤𝜙(𝑦_𝑡,𝑦). From (𝐴₃), we have 0≤𝜙(𝑤,𝑦) for all 𝑦∈𝐶, and hence 𝑤∈EP(𝜙). We will show that 𝑤∈𝐹(𝑇). Assume that 𝑤∉𝐹(𝑇). Since 𝑢_𝑛𝑖⇀𝑤 and 𝑤≠𝑇𝑤, from Opial’s theorem we have liminf_𝑖→∞𝑢_𝑛𝑖−𝑤<liminf_𝑖→∞𝑢_𝑛𝑖−𝑇𝑤≤liminf_𝑖→∞𝑢_𝑛𝑖−𝑇𝑢_𝑛𝑖+𝑇𝑢_𝑛𝑖−𝑇𝑤≤liminf_𝑖→∞𝑢_𝑛𝑖−𝑤. (3.30) This is a contradiction. So, we get 𝑤∈𝐹(𝑇). Therefore, 𝑤∈𝐹(𝑇)∩EP(𝜙). Since 𝑧=𝑃_{𝐹(𝑇)∩EP(𝜙)}𝑓(𝑧), we have limsup_𝑛→∞𝑓(𝑧)−𝑧,𝑥_𝑛−𝑧=lim_𝑖→∞𝑓(𝑧)−𝑧,𝑥_𝑛𝑖=𝑓(𝑧)−𝑧−𝑧,𝑤−𝑧≤0. (3.31)
From 𝑥_𝑛+1−𝑧=(1−𝛽_𝑛)(𝑇𝑦_𝑛−𝑧)+𝛽_𝑛(𝑓(𝑥_n)−𝑧), we have 𝑥_𝑛+1−𝑧²≤1−𝛽_𝑛²𝑦_𝑛−𝑧²+2𝛽_𝑛𝑓𝑥_𝑛−𝑧,𝑥_𝑛+1≤−𝑧1−𝛽_𝑛²𝛼_𝑛(𝑢−𝑧)+1−𝛼_𝑛𝑥_𝑛−𝑧²+2𝛽_𝑛𝛼𝑥_𝑛−𝑧𝑥_𝑛+1−𝑧+2𝛽_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1≤−𝑧1−𝛽_𝑛²𝛼_𝑛‖𝑢−𝑧‖²+1−𝛼_𝑛𝑥_𝑛−𝑧²+𝛽_𝑛𝛼𝑥_𝑛−𝑧²+𝑥_𝑛+1−𝑧²+2𝛽_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1≤−𝑧1−𝛽_𝑛²1−𝛼_𝑛+𝛽_𝑛𝛼𝑥_𝑛−𝑧²+𝛽_𝑛𝛼𝑥_𝑛+1−𝑧²+1−𝛽_𝑛²𝛼_𝑛‖𝑢−𝑧‖²+2𝛽_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1,𝑥−𝑧_𝑛+1−𝑧²≤1−𝛽_𝑛²+𝛽_𝑛𝛼1−𝛼𝛽_𝑛𝑥_𝑛−𝑧²+1−𝛽_𝑛²𝛼_𝑛1−𝛼𝛽_𝑛‖𝑢−𝑧‖²+2𝛽_𝑛1−𝛼𝛽_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1≤2()𝛽−𝑧1−1−𝛼_𝑛1−𝛼𝛽_𝑛𝑥_𝑛−𝑧²+𝛽²_𝑛1−𝛼𝛽_𝑛𝑥_𝑛−𝑧²+1−𝛽_𝑛²𝛼_𝑛1−𝛼𝛽_𝑛‖𝑢−𝑧‖²+2𝛽_𝑛1−𝛼𝛽_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1≤2()𝛽−𝑧1−1−𝛼_𝑛1−𝛼𝛽_𝑛𝑥_𝑛−𝑧²+2(1−𝛼)𝛽_𝑛1−𝛼𝛽_𝑛×𝛽_𝑛2(1−𝛼)𝑀+1−𝛽_𝑛²𝛼_𝑛2(1−𝛼)𝛽_𝑛‖𝑢−𝑧‖²+1𝑓(𝑧)1−𝛼−𝑧,𝑥_𝑛+1=−𝑧1−𝛿_𝑛𝑥_𝑛−𝑧²+𝛿_𝑛𝜁_𝑛, (3.32) where 𝑀=sup{‖𝑥_𝑛−𝑧‖²∶𝑛∈ℕ},   𝛿_𝑛=2(1−𝛼)𝛽_𝑛/1−𝛼𝛽_𝑛 and 𝜁_𝑛∶=𝛽_𝑛/2(1−𝛼)𝑀+(1−𝛽_𝑛)²𝛼_𝑛/2(1−𝛼)𝛽_𝑛‖𝑢−𝑧‖²+1/1−𝛼⟨𝑓(𝑧)−𝑧,𝑥_𝑛+1−𝑧⟩. It is easy to see that 𝛿_𝑛→0,Σ^∞_𝑛=1𝛿_𝑛=∞ and limsup_𝑛→∞𝜁_𝑛/𝛿_𝑛≤0 by (3.31) and the conditions. Hence, by Lemma 2.3, the sequence {𝑥_𝑛} converges strongly to 𝑧.
If 𝑧_𝑡 is definition as (2.1), then, from Lemma 2.5, we have ‖𝑧_𝑡−𝑞‖→0 as 𝑡→0, and if we set 𝑄(𝑢)∶=lim_𝑡→0𝑧_𝑡, then 𝑄 defines the unique sunny nonexpansive retraction from 𝐶 onto Fix(𝑇). So, if we replace 𝑡 with 𝛼_𝑛, the corollary still holds. and it is that 𝑧_𝑛=𝑇(𝛼_𝑛𝑢+(1−𝛼_𝑛)𝑧_𝑛) is a fixed point sequence and ‖𝑧_𝑛−𝑞‖→0 as 𝑛→∞, and if we set 𝑄(𝑢)∶=lim_𝑛→∞𝑧_𝑛, then 𝑄 defines the unique sunny nonexpansive retraction from 𝐶 onto Fix(𝑇). In the iterative algorithm of Theorem 3.1, we can take 𝑧_𝑛 to replace 𝑇𝑦_𝑛 in particular. Then, we have 𝑥_𝑛+1=𝛽_𝑛𝑓(𝑥_𝑛)+(1−𝛽_𝑛)𝑧_𝑛, so ‖𝑥_𝑛+1−𝑧_𝑛‖=𝛽_𝑛‖𝑓(𝑥_𝑛)−𝑧_𝑛‖→0 as 𝑛→∞. By the uniqueness of limit, we have 𝑧=𝑞, that is, 𝑧=𝑄(𝑢), where 𝑄 defines the unique sunny nonexpansive retraction from 𝐶 onto Fix(𝑇).

Remark 3. We notice that 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛 has not influence on 𝑥_𝑛,𝑢_𝑛→𝑧=𝑃_{𝐹(𝑇)∩EP(𝜙)}𝑓(𝑧).

As direct consequences of Theorem 3.1, we obtain corollary.

Corollary 3.2. Let 𝐶 be a nonempty closed convex subset of 𝐻, 𝑆∶𝐶→𝐶 be a nonexpansive mapping of 𝐶 into 𝐻 such that 𝐹(𝑆)≠∅. Let 𝑓 be a contraction of 𝐻 into itself and let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated initially by an arbitrary elements 𝑥₁∈𝐻 and then by 𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑆𝛼_𝑛𝑢+1−𝛼_𝑛𝑃_𝐶𝑥_𝑛 (3.33) for all 𝑛∈𝑁, where {𝛼_𝑛}⊂(0,∞) satisfies the following conditions:(I)lim_𝑛→∞𝛼_𝑛=0,∑^∞_𝑛=1𝛼_𝑛=∞𝑎𝑛𝑑∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(II)lim_𝑛→∞𝛽_𝑛=0,∑^∞_𝑛=1𝛽_𝑛=∞𝑎𝑛𝑑∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞;(III)lim_𝑛→∞𝛼_𝑛/𝛽_𝑛=0. Then, the sequences {𝑥_𝑛} converge strongly to 𝑧∈𝐹(𝑆), where 𝑧=𝑃_𝐹(𝑆)𝑓(𝑧).

Proof. Put 𝜙(𝑥,𝑦)=0, for all 𝑥,𝑦∈𝐶 and 𝑟_𝑛=1, for all 𝑛∈ℕ in Theorem 3.1.
Then we have 𝑢_𝑛=𝑃_𝐶𝑥_𝑛. So, from Theorem 3.1, the sequence 𝑥_𝑛 generated by 𝑥₁∈𝐻 and 𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑆𝛼_𝑛𝑢+1−𝛼_𝑛𝑃_𝐶𝑥_𝑛 (3.34) for all 𝑛∈ℕ converges strongly to 𝑧∈𝐹(𝑆), where 𝑧=𝑃_𝐹(𝑆)𝑓(𝑧).

4. Application for Zeros of Maximal Monotone Operators

We adapt in this section the iterative algorithm (3.1) to find zeros of maximal monotone operators and find EP(𝜙). Let us recall that an operator 𝐴 with domain 𝐷(𝐴) and range 𝑅(𝐴) in a real Hilbert space 𝐻 with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖ is said to be monotone if the graph of 𝐴, 𝐺(𝑇)∶=𝑥,𝑦∈𝐻×𝐻∶𝑥∈𝐷(𝑇),𝑦∈𝑇𝑥 (4.1) is a monotone set. Namely, 𝑥−𝑥^′,𝑦−𝑦^′≥0,𝑥,𝑦,𝑥^′,𝑦^′∈𝐺(𝐴). (4.2)

A monotone operator 𝐴 is said to be maximal monotone of the graph 𝐺(𝑇) is not properly contained in the graph of any other monotone defined in 𝐻. See Brezis [16] for more details on maximal monotone operators.

In this section we always assume that 𝐴 is maximal monotone and the set of zeros of 𝐴, 𝑁(𝐴)={𝑥∈𝐷(𝐴)∶0∈𝐴𝑥}, is nonempty so that the metric projection 𝑃_𝑁(𝐴) from 𝐻 onto 𝑁(𝐴) is well-defined.

One of the major problems in the theory of maximal operators is to find a point in the zero set 𝑁(𝐴) because various problems arising from economics, convex programming, and other applied areas can be formulated as finding a zero of maximal monotone operators. The proximal point algorithm (PPA) of Rockafellar [17] is commonly recognized as the most powerful algorithm in finding a zero of maximal monotone operators. This (PPA) generates, starting with any initial guess 𝑥₀∈𝐻, a sequence {𝑥_𝑛} according to the inclusion: 𝑥_𝑛+𝑒_𝑛∈𝑥_𝑛+1+𝑐_𝑛𝐴𝑥_𝑛+1, (4.3) where {𝑒_𝑛} is a sequence of errors and {𝑐_𝑛} is a sequence of positive regularization parameters. Equivalently, we can write 𝑥_𝑛+1=𝐽^𝐴_𝑐𝑛𝑥_𝑛+𝑒_𝑛, (4.4) where for 𝑐>0,𝐽^𝐴_𝑐 denotes the resolvent of 𝐴, 𝐽^𝐴_𝑐=(𝐼+𝑐𝐴)⁻¹, (4.5) with 𝐼 being the identity operator on the space 𝐻.

Rockafellar [17] proved the weak convergence of his algorithm (4.4) provided the regularization sequence {𝑐_𝑛} remains bounded away from zero and the error sequence {𝑒_𝑛} satisfies the condition ^∞_𝑛=0𝑒_𝑛<∞. (4.6)

The aim of this section is to combine algorithm (3.1) with algorithm (4.4). Our algorithm generates a sequence {𝑥_𝑛} and {𝑢_𝑛} be sequences generated initially by an arbitrary elements 𝑥₁∈𝐻 and then by 𝜙𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝐽^𝐴_𝑐𝑛𝛼_𝑛𝑢+1−𝛼_𝑛𝑢_𝑛+𝑒_𝑛, (4.7) where 𝛼_𝑛 and 𝑐_𝑛 are sequences of positive real numbers. Furthermore, we prove that {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑧∈𝑁(𝐴)∩EP(𝜙), where 𝑧=𝑃_{𝑁(𝐴)∩EP(𝜙)}𝑓(𝑧).

Before stating the convergence theorem of the algorithm (4.7), we list some properties of maximal monotone operators.

Proposition 4.1. Let 𝐴 be a maximal monotone operator in 𝐻 and let 𝐽^𝐴_𝑐=(𝐼+𝑐𝐴)⁻¹ denote the resolvent, where 𝑐>0, (a)𝐽^𝐴_𝑐 is nonexpansive for all 𝑐>0;(b)𝑁(𝐴)=𝐹(𝐽_𝑐) for all 𝑐>0;(c)For 𝑐>𝑐^′>0,‖𝐽^𝐴_𝑐′−𝑥‖≤2‖𝐽^𝐴_𝑐−𝑥‖for𝑥∈𝐻;(d)(The Resolvent Identity) For 𝜆,𝜇>0, there holds the identity:𝐽^𝑇_𝜆𝑥=𝐽^𝑇_𝜇𝜇𝜆𝜇𝑥+1−𝜆𝐽^𝑇_𝜆𝑥,𝑥∈𝐻. (4.8)

Theorem 4.2. Let 𝐶 be a nonempty closed convex subset of 𝐻, 𝜙∶𝐶×𝐶→∞ be a bifunction satisfying (A₁)–(A₄) and 𝐴 be a maximal monotone operator such that 𝑁(𝐴)∩EP(𝜙)≠∅. Let 𝑓 be a contraction of 𝐻 into itself and let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated initially by an arbitrary elements 𝑥₀∈𝐻 and then by 𝜙𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝐽^𝐴_𝑐𝑛𝛼_𝑛𝑢+1−𝛼_𝑛𝑢_𝑛+𝑒_𝑛, (4.9) for all 𝑛∈𝑁, where {𝛼_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞) satisfy the following conditions:(I)lim_𝑛→∞𝛼_𝑛=0, Σ^∞_𝑛=1𝛼_𝑛=∞ and Σ^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(II)liminf_𝑛𝑟_𝑛>0 and Σ^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞;(III)lim_𝑛→∞(𝑐_𝑛+1/𝑐_𝑛)=1;(IV)lim_𝑛→∞𝛽_𝑛=0, Σ^∞_𝑛=1𝛽_𝑛=∞, Σ^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞, and lim_𝑛→∞(𝛼_𝑛/𝛽_𝑛)=0. Then, the sequences {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑧∈𝑁(𝐴)∩EP(𝜙), where 𝑧=𝑃_{𝑁(𝐴)∩EP(𝜙)}𝑓(𝑧).

Proof. Below we write 𝐽_𝑐=𝐽^𝐴_𝑐 for simplicity. Setting 𝑤_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑢_𝑛+𝑒_𝑛,𝑦_𝑛=𝐽_𝑐𝑛𝑤_𝑛, (4.10) we rewrite 𝑥_𝑛+1 of (4.7) as 𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝐽_𝑐𝑛𝑤_𝑛=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛. (4.11) Because the proof is similar to Theorem 3.1, here we just give the main steps as follows:(1){𝑥_𝑛} is bounded;(2)‖𝑥_𝑛+1−𝑥_𝑛‖→0, as 𝑛→0;(3)‖𝑢_𝑛−𝐽_𝑐𝑛𝑢_𝑛‖→0, as 𝑛→0;(4)‖𝑥_𝑛−𝑢_𝑛‖→0, as 𝑛→0;(5)limsup_𝑛→∞⟨𝑓(𝑧)−𝑧,𝑥_𝑛−𝑧⟩≤0;(6)𝑥_𝑛,𝑢_𝑛→𝑧, as 𝑛→𝑧.

5. Application for Optimization Problem

In this section, we study a kind of optimization problem by using the result of this paper. That is, we will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutionsminℎ(𝑥),𝑥∈𝐶, (5.1) where ℎ(𝑥) is a convex and lower semicontinuous functional defined on a closed convex subset 𝐶 of a Hilbert space 𝐻. We denoted by 𝐵 the set of solutions of (5.1). Let 𝜙 be a bifunction from 𝐶×𝐶 to 𝑅 defined by 𝜙(𝑥,𝑦)=ℎ(𝑦)−ℎ(𝑥). We consider the following equilibrium problem, that is, to find 𝑥∈𝐶 such that 𝜙𝑥,𝑦≥0,∀𝑦∈𝐶. (5.2)

It is obvious that EP(𝜙)=𝐵, where EP(𝜙) denotes the set of solutions of equilibrium problem (5.2). In addition, it is easy to see that 𝜙(𝑥,𝑦) satisfies the conditions (A₁)–(A₄) in the Section 2. Therefore, from Theorem 3.1, we know that the following iterative algorithm: ℎ𝑦−ℎ𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑇𝛼_𝑛𝑢+1−𝛼_𝑛𝑢_𝑛, (5.3) for any initial guess 𝑥₁, converges strongly to a solution 𝑧=𝑃_𝐵𝑓(𝑧) of optimization problem (5.1), where {𝛼_𝑛}⊂[0,1], {𝛽_𝑛}⊂[0,1], and {𝑟_𝑛}⊂[0,∞) satisfylim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=1𝛼_𝑛=∞,^∞_𝑛=1𝛼_𝑛+1−𝛼_𝑛<∞,lim_𝑛→∞𝛽_𝑛=0,^∞_𝑛=1𝛽_𝑛=∞,^∞_𝑛=1𝛽_𝑛+1−𝛽_𝑛<∞,lim_𝑛→∞𝛼_𝑛𝛽_𝑛=0liminf𝑟_{𝑛𝑛→∞}>0,^∞_𝑛=1𝑟_𝑛+1−𝑟_𝑛<∞., (5.4)

For a special case, we pick 𝑓(𝑥)=𝜂, for all 𝜂∈𝐻, and 𝑟_𝑛=1,  𝛽_𝑛=1/2 and 𝛼_𝑛=0 for all 𝑛≥1, then 𝑥_𝑛+1=(1/2)𝑇𝑢_𝑛+(1/2)𝜂, from (5.3), we get the special iterative algorithm as follows: ℎ𝑦−ℎ𝑢_𝑛+𝑦−𝑢_𝑛,𝑢_𝑛−121𝜂+2𝑇𝑢_𝑛ℎ𝑦𝑢≥0,∀𝑦∈𝐶,𝑛≥2,−ℎ₁+𝑦−𝑢₁,𝑢₁−121𝜂+2𝑇𝑢₁≥0,∀𝑦∈𝐶. (5.5) Then {𝑢_𝑛} converges strongly to a solution 𝑧=𝑃_𝐵𝜂 of optimization problem (5.1).

In fact, 𝑧 is the minimum norm point from 𝜂 onto the 𝐵, furthermore, if 𝜂=0, then 𝑧 is the minimum norm point on the 𝐵.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grants (11071270), and (10771050).