Composite Algorithms for Minimization over the Solutions of Equilibrium Problems and Fixed Point Problems

Abstract

The purpose of this paper is to solve the minimization problem of finding 𝑥^∗ such that 𝑥^∗=argmin_𝑥∈Γ‖𝑥‖², where Γ stands for the intersection set of the solution set of the equilibrium problem and the fixed points set of a nonexpansive mapping. We first present two new composite algorithms (one implicit and one explicit). Further, we prove that the proposed composite algorithms converge strongly to 𝑥^∗.

1. Introduction

In the present paper, our main purpose is to solve the minimization problem of finding 𝑥^∗ such that𝑥^∗=argmin_𝑥∈Γ‖𝑥‖²,(1.1) where Γ stands for the intersection set of the solution set of the equilibrium problem and the fixed points set of a nonexpansive mapping. This problem is motivated by the following least-squares solution to the constrained linear inverse problem:𝐵𝑥=𝑏,𝑥∈𝐶,(1.2) where 𝐶 is a nonempty closed convex subset of a real Hilbert space 𝐻, 𝐵 is a bounded linear operator from 𝐻 to another real Hilbert space 𝐻₁ and 𝑏 is a given point in 𝐻₁. The least-squares solution to (1.2) is the least-norm minimizer of the minimization problemmin_𝑥∈𝐶‖𝐵𝑥−𝑏‖².(1.3) Let 𝑆_𝑏 denote the solution set of (1.2) (or equivalently (1.3)). It is known that 𝑆_𝑏 is nonempty if and only if 𝑃_𝐵(𝐶)(𝑏)∈𝐵(𝐶). In this case, 𝑆_𝑏 has a unique element with minimum norm (equivalently, (1.2) has a unique least-squares solution); that is, there exists a unique point 𝑥^†∈𝑆_𝑏 satisfying‖𝑥^†‖=min{‖𝑥‖∶𝑥∈𝑆_𝑏}.(1.4) The so-called 𝐶-constrained pseudoinverse of 𝐵 is then defined as the operator 𝐵^†_𝐶 with domain and values given by 𝐷(𝐵^†_𝐶)={𝑏∈𝐻∶𝑃_𝐵(𝐶)(𝑏)∈𝐵(𝐶)},𝐵^†_𝐶(𝑏)=𝑥^†,𝑏∈𝐷(𝐵^†_𝐶),(1.5) where 𝑥^†∈𝑆_𝑏 is the unique solution to (1.4).

Note that the optimality condition for the minimization (1.3) is the variational inequality (VI) 𝑥∈𝐶,⟨𝐵^∗(𝐵𝑥−𝑏),𝑥−𝑥⟩≥0,𝑥∈𝐶,(1.6) where 𝐵^∗ is the adjoint of 𝐵.

If 𝑏∈𝐷(𝐵^†_𝐶), then (1.3) is consistent and its solution set 𝑆_𝑏 coincides with the solution set of VI (1.6). On the other hand, VI (1.6) can be rewritten as 𝑥∈𝐶,⟨(𝑥−𝜆𝐵^∗(𝐵𝑥−𝑏))−𝑥,𝑥−𝑥⟩≤0,𝑥∈𝐶,(1.7) where 𝜆>0 is any positive scalar. In the terminology of projections, (1.7) is equivalent to the fixed point equation 𝑥=𝑃_𝐶(𝑥−𝜆𝐵^∗(𝐵𝑥−𝑏)).(1.8) It is not hard to find that for 0<𝜆<2/‖𝐵‖², the mapping 𝑥↦𝑃_𝐶(𝑥−𝜆𝐵^∗(𝐵𝑥−𝑏)) is nonexpansive. Therefore, finding the least-squares solution of the constrained linear inverse problem is equivalent to finding the minimum-norm fixed point of the nonexpansive mapping 𝑥↦𝑃_𝐶(𝑥−𝜆𝐵^∗(𝐵𝑥−𝑏)).

Based on the above facts, it is an interesting topic of finding the minimum norm fixed point of the nonexpansive mappings. In this paper, we will consider a general problem. We will focus on to solve the minimization problem (1.1). At this point, we first recall some definitions on the fixed point problem and the equilibrium problem as follows.

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Recall that a mapping 𝐴∶𝐶→𝐻 is called 𝛼-inverse strongly monotone if there exists a positive real number 𝛼 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝛼‖𝐴𝑥−𝐴𝑦‖², for all 𝑥,𝑦∈𝐶. It is clear that any 𝛼-inverse strongly monotone mapping is monotone and 1/𝛼-Lipschitz continuous. Let 𝑓∶𝐶→𝐻 be a 𝜌-contraction; that is, there exists a constant 𝜌∈[0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝜌‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. A mapping 𝑆∶𝐶→𝐶 is said to be nonexpansive if ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐶. Denote the set of fixed points of 𝑆 by Fix(𝑆).

Let 𝐴∶𝐶→𝐻 be a nonlinear mapping and 𝐹∶𝐶×𝐶→𝑅 be a bifunction. The equilibrium problem is to find 𝑧∈𝐶 such that 𝐹(𝑧,𝑦)+⟨𝐴𝑧,𝑦−𝑧⟩≥0,∀𝑦∈𝐶.(1.9) The solution set of (1.9) is denoted by EP. If 𝐴=0, then (1.9) reduces to the following equilibrium problem of finding 𝑧∈𝐶 such that 𝐹(𝑧,𝑦)≥0,∀𝑦∈𝐶.(1.10) If 𝐹=0, then (1.9) reduces to the variational inequality problem of finding 𝑧∈𝐶 such that⟨𝐴𝑧,𝑦−𝑧⟩≥0,∀𝑦∈𝐶.(1.11) We note that the problem (1.9) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others, see, for example, [1–4].

We next briefly review some historic approaches which relate to the fixed point problems and the equilibrium problems.

In 2005, Combettes and Hirstoaga [5] introduced an iterative algorithm of finding the best approximation to the initial data and proved a strong convergence theorem. In 2007, by using the viscosity approximation method, S. Takahashi and W. Takahashi [6] introduced another iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of fixed point points of a nonexpansive mapping. Subsequently, algorithms constructed for solving the equilibrium problems and fixed point problems have further developed by some authors. In particular, Ceng and Yao [7] introduced an iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problem (1.9) and the set of common fixed points of finitely many nonexpansive mappings. Maingé and Moudafi [8] introduced an iterative algorithm for equilibrium problems and fixed point problems. Yao et al. [9] considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of common fixed points of an infinite nonexpansive mappings. Noor et al. [10] introduced an iterative method for solving fixed point problems and variational inequality problems. Their results extend and improve many results in the literature. Some works related to the equilibrium problem, fixed point problems, and the variational inequality problem in[1–45] and the references therein.

However, we note that all constructed algorithms in [2, 4, 6–10, 14, 15, 21, 23–40] do not work to find the minimum-norm solution of the corresponding fixed point problems and the equilibrium problems. It is our main purpose in this paper that we devote to construct some algorithms for finding the minimum-norm solution of the fixed point problems and the equilibrium problems. We first suggest two new composite algorithms (one implicit and one explicit) for solving the above minimization problem. Further, we prove that the proposed composite algorithms converge strongly to the minimum norm element 𝑥^∗.

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Throughout this paper, we assume that a bifunction 𝐹∶𝐶×𝐶→𝑅 satisfies the following conditions: (H1)𝐹(𝑥,𝑥)=0, for all 𝑥∈𝐶; (H2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶; (H3)for each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡↓0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦); (H4)for each 𝑥∈𝐶, 𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous.

The metric (or nearest point) projection from 𝐻 onto 𝐶 is the mapping 𝑃_𝐶∶𝐻→𝐶 which assigns to each point 𝑥∈𝐶 the unique point 𝑃_𝐶𝑥∈𝐶 satisfying the property ‖𝑥−𝑃_𝐶𝑥‖=inf_𝑦∈𝐶‖𝑥−𝑦‖=∶𝑑(𝑥,𝐶).(2.1) It is well known that 𝑃_𝐶 is a nonexpansive mapping and satisfies ⟨𝑥−𝑦,𝑃_𝐶𝑥−𝑃_𝐶𝑦⟩≥‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖²,∀𝑥,𝑦∈𝐻.(2.2) We need the following lemmas for proving our main results.

Lemma 2.1 (see [5]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹∶𝐶×𝐶→𝑅 be a bifunction which satisfies conditions (H1)–(H4). Let 𝑟>0 and 𝑥∈𝐶. Then, there exists 𝑧∈𝐶 such that 1𝐹(𝑧,𝑦)+𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶.(2.3) Further, if 𝑇_𝑟(𝑥)={𝑧∈𝐶∶𝐹(𝑧,𝑦)+(1/𝑟)⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶}, then the following hold: (i)𝑇_𝑟 is single-valued and 𝑇_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻, ‖𝑇_𝑟𝑥−𝑇_𝑟𝑦‖²≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦⟩; (ii)EP is closed and convex and 𝐸𝑃=Fix(𝑇_𝑟).

Lemma 2.2 (see [17]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let the mapping 𝐴∶𝐶→𝐻 be 𝛼-inverse strongly monotone and 𝑟>0 be a constant. Then, one has ‖(𝐼−𝑟𝐴)𝑥−(𝐼−𝑟𝐴)𝑦‖²≤‖𝑥−𝑦‖²+𝑟(𝑟−2𝛼)‖𝐴𝑥−𝐴𝑦‖²,∀𝑥,𝑦∈𝐶.(2.4) In particular, if 0≤𝑟≤2𝛼, then 𝐼−𝑟𝐴 is nonexpansive.

Lemma 2.3 (see [28]). Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻 and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping. Then, the mapping 𝐼−𝑆 is demiclosed, that is, if {𝑥_𝑛} is a sequence in 𝐶 such that 𝑥_𝑛→𝑥^∗ weakly and (𝐼−𝑆)𝑥_𝑛→𝑦 strongly, then (𝐼−𝑆)𝑥^∗=𝑦.

Lemma 2.4 (see [22]). Assume {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤(1−𝛾_𝑛)𝑎_𝑛+𝛿_𝑛𝛾_𝑛,(2.5) 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 introduce two algorithms for finding the minimum norm element 𝑥^∗ of Γ∶=𝐸𝑃∩Fix(𝑆). Namely, we want to find the unique point 𝑥^∗ which solves the following minimization problem:𝑥^∗=argmin_𝑥∈Γ‖𝑥‖².(3.1)

Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping and 𝐴∶𝐶→𝐻 be an 𝛼-inverse strongly monotone mapping. Let 𝐹∶𝐶×𝐶→𝑅 be a bifunction which satisfies conditions (H1)–(H4). Let 𝑟 and 𝜇 be two constants such that 𝑟∈(0,2𝛼) and 𝜇∈(0,1). In order to find a solution of the minimization problem (3.1), we construct the following implicit algorithm 𝑥_𝑡=𝜇𝑃_𝐶[(1−𝑡)𝑆𝑥_𝑡]+(1−𝜇)𝑇_𝑟(𝑥_𝑡−𝑟𝐴𝑥_𝑡),∀𝑡∈(0,1),(3.2) where 𝑇_𝑟 is defined as Lemma 2.1. We will show that the net {𝑥_𝑡} defined by (3.2) converges to a solution of the minimization problem (3.1). As matter of fact, in this paper, we will study the following general algorithm.

Let 𝑓∶𝐶→𝐻 be a 𝜌-contraction. For each 𝑡∈(0,1), we consider the following mapping 𝑊_𝑡 given by 𝑊_𝑡𝑥=𝜇𝑃_𝐶[𝑡𝑓(𝑥)+(1−𝑡)𝑆𝑥_𝑡]+(1−𝜇)𝑇_𝑟(𝐼−𝑟𝐴)𝑥_𝑡,∀𝑥∈𝐶.(3.3) Since the mappings 𝑆, 𝑃_𝐶, 𝑇_𝑟 and 𝐼−𝑟𝐴 are nonexpansive, then we can check easily that ‖𝑊_𝑡𝑥−𝑊_𝑡𝑦‖≤[1−(1−𝜌)𝜇𝑡]‖𝑥−𝑦‖ which implies that 𝑊_𝑡 is a contraction. Using the Banach contraction principle, there exists a unique fixed point 𝑥_𝑡 of 𝑊_𝑡 in 𝐶, that is, 𝑥_𝑡=𝜇𝑃_𝐶[𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑆𝑥_𝑡]+(1−𝜇)𝑇_𝑟(𝐼−𝑟𝐴)𝑥_𝑡,𝑡∈(0,1).(3.4)

In this point, we would like to point out that algorithm (3.4) includes algorithm (3.2) as a special case due to the contraction 𝑓 is a possible nonself-mapping.

In the sequel, we assume (1)𝐶 is a nonempty closed convex subset of a real Hilbert space 𝐻; (2)𝑆∶𝐶→𝐶 is a nonexpansive mapping, 𝐴∶𝐶→𝐻 is an 𝛼-inverse strongly monotone mapping and 𝑓∶𝐶→𝐻 is a 𝜌-contraction; (3)𝐹∶𝐶×𝐶→𝑅 is a bifunction which satisfies conditions (H1)–(H4); (4)Γ≠∅.

In order to prove our first main result, we need the following lemmas.

Lemma 3.1. The net {𝑥_𝑡} generated by the implicit method (3.4) is bounded.

Proof. Set 𝑢_𝑡=𝑇_𝑟(𝑥_𝑡−𝑟𝐴𝑥_𝑡) and 𝑦_𝑡=𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑆𝑥_𝑡 for all 𝑡∈(0,1). Take 𝑧∈Γ. It is clear that 𝑆𝑧=𝑧=𝑇_𝑟(𝑧−𝑟𝐴𝑧). Since 𝑇_𝑟 is nonexpansive and 𝐴 is 𝛼-inverse strongly monotone, we have from Lemma 2.2 that ‖𝑢_𝑡−𝑧‖²≤‖𝑥_𝑡−𝑟𝐴𝑥_𝑡−(𝑧−𝑟𝐴𝑧)‖²≤‖𝑥_𝑡−𝑧‖²+𝑟(𝑟−2𝛼)‖𝐴𝑥_𝑡−𝐴𝑧‖²≤‖𝑥_𝑡−𝑧‖².(3.5) So, we have that ‖𝑢_𝑡−𝑧‖≤‖𝑥_𝑡−𝑧‖.(3.6) It follows from (3.4) that ‖𝑥_𝑡−𝑧‖=‖𝜇𝑃_𝐶[𝑦_𝑡]+(1−𝜇)𝑢_𝑡−𝑧‖≤𝜇‖𝑃_𝐶[𝑦_𝑡]−𝑧‖+(1−𝜇)‖𝑢_𝑡−𝑧‖≤𝜇(𝑡‖𝑓(𝑥_𝑡)−𝑧‖+(1−𝑡)‖𝑆𝑥_𝑡−𝑧‖)+(1−𝜇)‖𝑥_𝑡−𝑧‖≤𝜇(𝑡𝜌+1−𝑡)‖𝑥_𝑡−𝑧‖+𝜇𝑡‖𝑓(𝑧)−𝑧‖+(1−𝜇)‖𝑥_𝑡−𝑧‖=[1−(1−𝜌)𝜇𝑡]‖𝑥_𝑡−𝑧‖+𝑡𝜇‖𝑓(𝑧)−𝑧‖,(3.7) that is, ‖𝑥_𝑡−𝑧‖≤‖𝑓(𝑧)−𝑧‖.1−𝜌(3.8) So, {𝑥_𝑡} is bounded. Hence {𝑢_𝑡} and {𝑓(𝑥_𝑡)} are also bounded. This completes the proof.

According to Lemma 3.1, we can choose some appropriate constant 𝑀>0 such that 𝑀 satisfies the following request.

Lemma 3.2. The net {𝑥_𝑡} generated by the implicit method (3.4) is relatively norm compact as 𝑡→0.

Proof. From (3.4) and (3.5), we have ‖𝑥_𝑡−𝑧‖²=‖𝜇(𝑃_𝐶[𝑦_𝑡]−𝑧)+(1−𝜇)(𝑢_𝑡−𝑧)‖²≤𝜇‖𝑃_𝐶[𝑦_𝑡]−𝑧‖²+(1−𝜇)‖𝑢_𝑡−𝑧‖²≤𝜇‖𝑡(𝑓(𝑥_𝑡)−𝑧)+(1−𝑡)(𝑆𝑥_𝑡−𝑧)‖²+(1−𝜇)‖𝑢_𝑡−𝑧‖²≤𝜇[𝑡‖𝑓(𝑥_𝑡)−𝑧‖²+(1−𝑡)‖𝑆𝑥_𝑡−𝑧‖²]+(1−𝜇)‖𝑢_𝑡−𝑧‖²≤𝜇[𝑡𝑀+(1−𝑡)‖𝑥_𝑡−𝑧‖²]+(1−𝜇)‖𝑢_𝑡−𝑧‖².(3.9) It follows that ‖𝑥_𝑡−𝑧‖²≤1−𝜇1−𝜇+𝜇𝑡‖𝑢_𝑡−𝑧‖²+𝜇𝑡𝑀1−𝜇+𝜇𝑡≤‖𝑢_𝑡−𝑧‖²+𝑡𝑀≤‖𝑥_𝑡−𝑧‖²+𝑟(𝑟−2𝛼)‖𝐴𝑥_𝑡−𝐴𝑧‖²+𝑡𝑀,(3.10) that is, 𝑟(2𝛼−𝑟)‖𝐴𝑥_𝑡−𝐴𝑧‖²≤𝑡𝑀→0.(3.11) Since 𝑟(2𝛼−𝑟)>0, we derive lim_𝑡→0‖𝐴𝑥_𝑡−𝐴𝑧‖=0.(3.12)
From Lemmas 2.1 and 2.2, we obtain ‖𝑢_𝑡−𝑧‖²=‖𝑇_𝑟(𝑥_𝑡−𝑟𝐴𝑥_𝑡)−𝑇_𝑟(𝑧−𝑟𝐴𝑧)‖²≤⟨(𝑥_𝑡−𝑟𝐴𝑥_𝑡)−(𝑧−𝑟𝐴𝑧),𝑢_𝑡=1−𝑧⟩2(‖(𝑥_𝑡−𝑟𝐴𝑥_𝑡)−(𝑧−𝑟𝐴𝑧)‖²+‖𝑢_𝑡−𝑧‖²−‖(𝑥_𝑡−𝑧)−𝑟(𝐴𝑥_𝑡−𝐴𝑧)−(𝑢_𝑡−𝑧)‖²)≤12(‖𝑥_𝑡−𝑧‖²+‖𝑢_𝑡−𝑧‖²−‖(𝑥_𝑡−𝑢_𝑡)−𝑟(𝐴𝑥_𝑡−𝐴𝑧)‖²)=12(‖𝑥_𝑡−𝑧‖²+‖𝑢_𝑡−𝑧‖²−‖𝑥_𝑡−𝑢_𝑡‖²+2𝑟⟨𝑥_𝑡−𝑢_𝑡,𝐴𝑥_𝑡−𝐴𝑧⟩−𝑟²‖𝐴𝑥_𝑡−𝐴𝑧‖²),(3.13) which implies that ‖𝑢_𝑡−𝑧‖²≤‖𝑥_𝑡−𝑧‖²−‖𝑥_𝑡−𝑢_𝑡‖²+2𝑟⟨𝑥_𝑡−𝑢_𝑡,𝐴𝑥_𝑡−𝐴𝑧⟩−𝑟²‖𝐴𝑥_𝑡−𝐴𝑧‖²≤‖𝑥_𝑡−𝑧‖²−‖𝑥_𝑡−𝑢_𝑡‖²+2𝑟‖𝑥_𝑡−𝑢_𝑡‖‖𝐴𝑥_𝑡−𝐴𝑧‖≤‖𝑥_𝑡−𝑧‖²−‖𝑥_𝑡−𝑢_𝑡‖²+𝑀‖𝐴𝑥_𝑡−𝐴𝑧‖.(3.14) By (3.10), and (3.14), we have ‖𝑥_𝑡−𝑧‖²≤‖𝑢_𝑡−𝑧‖²+𝑡𝑀≤‖𝑥_𝑡−𝑧‖²−‖𝑥_𝑡−𝑢_𝑡‖²+(‖𝐴𝑥_𝑡−𝐴𝑧‖+𝑡)𝑀.(3.15) It follows that ‖𝑥_𝑡−𝑢_𝑡‖²≤(‖𝐴𝑥_𝑡−𝐴𝑧‖+𝑡)𝑀.(3.16) This together with (3.12) imply that lim_𝑡→0‖𝑥_𝑡−𝑢_𝑡‖=0.(3.17) It follows that lim_𝑡→0‖𝑥_𝑡−𝑃_𝐶[𝑦_𝑡]‖=lim_𝑡→01−𝜇𝜇‖𝑥_𝑡−𝑢_𝑡‖=0.(3.18) Hence, ‖𝑥_𝑡−𝑆𝑥_𝑡‖≤‖𝑥_𝑡−𝑃_𝐶[𝑦_𝑡]‖+‖𝑃_𝐶[𝑦_𝑡]−𝑆𝑥_𝑡‖≤‖𝑥_𝑡−𝑃_𝐶[𝑦_𝑡]‖+‖𝑦_𝑡−𝑆𝑥_𝑡‖≤‖𝑥_𝑡−𝑃_𝐶[𝑦_𝑡]‖+𝑡‖𝑓(𝑥_𝑡)‖→0.(3.19) Next, we show that {𝑥_𝑡} is relatively norm compact as 𝑡→0. Let {𝑡_𝑛}⊂(0,1) be a sequence such that 𝑡_𝑛→0 as 𝑛→∞. Put 𝑥_𝑛∶=𝑥_𝑡𝑛 and 𝑢_𝑛∶=𝑢_𝑡𝑛. From (3.19), we get ‖𝑥_𝑛−𝑆𝑥_𝑛‖→0.(3.20) By (3.4), we deduce ‖𝑥_𝑡−𝑧‖²=‖𝜇(𝑃_𝐶[𝑦_𝑡]−𝑧)+(1−𝜇)(𝑢_𝑡−𝑧)‖²≤𝜇‖𝑃_𝐶[𝑦_𝑡]−𝑧‖²+(1−𝜇)‖𝑢_𝑡−𝑧‖²≤𝜇‖𝑦_𝑡−𝑧‖²+(1−𝜇)‖𝑥_𝑡−𝑧‖²,(3.21) that is, ‖𝑥_𝑡−𝑧‖²≤‖𝑡(𝑓(𝑥_𝑡)−𝑧)+(1−𝑡)(𝑆𝑥_𝑡−𝑧)‖²=(1−𝑡)²‖𝑆𝑥_𝑡−𝑧‖²+2𝑡(1−𝑡)⟨𝑓(𝑥_𝑡)−𝑓(𝑧),𝑆𝑥_𝑡−𝑧⟩+2𝑡(1−𝑡)⟨𝑓(𝑧)−𝑧,𝑆𝑥_𝑡−𝑧⟩+𝑡²‖𝑓(𝑥_𝑡)−𝑧‖²≤(1−𝑡)²‖𝑥_𝑡−𝑧‖²+2𝑡(1−𝑡)𝜌‖𝑥_𝑡−𝑧‖²+2𝑡(1−𝑡)⟨𝑓(𝑧)−𝑧,𝑆𝑥_𝑡−𝑧⟩+𝑡²‖𝑓(𝑥_𝑡)−𝑧‖²≤[1−2(1−𝜌)𝑡]‖𝑥_𝑡−𝑧‖²+2𝑡⟨𝑓(𝑧)−𝑧,𝑆𝑥_𝑡−𝑧⟩+𝑡²𝑀.(3.22) It follows that ‖𝑥_𝑡−𝑧‖²≤11−𝜌⟨𝑧−𝑓(𝑧),𝑧−𝑆𝑥_𝑡𝑡⟩+2(1−𝜌)𝑀.(3.23) In particular, ‖𝑥_𝑛−𝑧‖²≤11−𝜌⟨𝑧−𝑓(𝑧),𝑧−𝑆𝑥_𝑛𝑡⟩+^𝑛2(1−𝜌)𝑀,𝑧∈Γ.(3.24)
Since {𝑥_𝑛} is bounded, without loss of generality, we may assume that {𝑥_𝑛} converges weakly to a point 𝑥^∗∈𝐶. Also 𝑆𝑥_𝑛→𝑥^∗ weakly. Noticing (3.20) we can use Lemma 2.3 to get 𝑥^∗∈Fix(𝑆).
Now, we show 𝑥^∗∈𝐸𝑃. Since 𝑢_𝑛=𝑇_𝑟(𝑥_𝑛−𝑟𝐴𝑥_𝑛), for any 𝑦∈𝐶, we have 𝐹(𝑢_𝑛1,𝑦)+𝑟⟨𝑦−𝑢_𝑛,𝑢_𝑛−(𝑥_𝑛−𝑟𝐴𝑥_𝑛)⟩≥0.(3.25) From the monotonicity of 𝐹, we have 1𝑟⟨𝑦−𝑢_𝑛,𝑢_𝑛−(𝑥_𝑛−𝑟𝐴𝑥_𝑛)⟩≥𝐹(𝑦,𝑢_𝑛),∀𝑦∈𝐶.(3.26) Hence, ⟨𝑦−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝑟+𝐴𝑥_𝑛𝑖⟩≥𝐹(𝑦,𝑢_𝑛𝑖),∀𝑦∈𝐶.(3.27) Put 𝑧_𝑡=𝑡𝑦+(1−𝑡)𝑥^∗ for all 𝑡∈(0,1] and 𝑦∈𝐶. Then, we have 𝑧_𝑡∈𝐶. So, from (3.27), we have ⟨𝑧_𝑡−𝑢_𝑛𝑖,𝐴𝑧_𝑡⟩≥⟨𝑧_𝑡−𝑢_𝑛𝑖,𝐴𝑧_𝑡⟩−⟨𝑧_𝑡−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝑟+𝐴𝑥_𝑛𝑖⟩+𝐹(𝑧_𝑡,𝑢_𝑛𝑖)=⟨𝑧_𝑡−𝑢_𝑛𝑖,𝐴𝑧_𝑡−𝐴𝑢_𝑛𝑖⟩+⟨𝑧_𝑡−𝑢_𝑛𝑖,𝐴𝑢_𝑛𝑖−𝐴𝑥_𝑛𝑖⟩−⟨𝑧_𝑡−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝑟⟩+𝐹(𝑧_𝑡,𝑢_𝑛𝑖).(3.28) Note that ‖𝐴𝑢_𝑛𝑖−𝐴𝑥_𝑛𝑖‖≤(1/𝛼)‖𝑢_𝑛𝑖−𝑥_𝑛𝑖‖→0. Further, from monotonicity of 𝐴, we have ⟨𝑧_𝑡−𝑢_𝑛𝑖,𝐴𝑧_𝑡−𝐴𝑢_𝑛𝑖⟩≥0. Letting 𝑖→∞ in (3.28), we have ⟨𝑧_𝑡−𝑥^∗,𝐴𝑧_𝑡⟩≥𝐹(𝑧_𝑡,𝑥^∗).(3.29) From (H1), (H4), and (3.29), we also have 0=𝐹(𝑧_𝑡,𝑧_𝑡)≤𝑡𝐹(𝑧_𝑡,𝑦)+(1−𝑡)𝐹(𝑧_𝑡,𝑥^∗)≤𝑡𝐹(𝑧_𝑡,𝑦)+(1−𝑡)⟨𝑧_𝑡−𝑥^∗,𝐴𝑧_𝑡⟩=𝑡𝐹(𝑧_𝑡,𝑦)+(1−𝑡)𝑡⟨𝑦−𝑥^∗,𝐴𝑧_𝑡⟩(3.30) and hence 0≤𝐹(𝑧_𝑡,𝑦)+(1−𝑡)⟨𝐴𝑧_𝑡,𝑦−𝑥^∗⟩.(3.31) Letting 𝑡→0 in (3.31), we have, for each 𝑦∈𝐶, 0≤𝐹(𝑥^∗,𝑦)+⟨𝑦−𝑥^∗,𝐴𝑥^∗⟩.(3.32) This implies that 𝑥^∗∈𝐸𝑃. Therefore, 𝑥^∗∈Γ.
We substitute 𝑥^∗ for 𝑧 in (3.24) to get ‖𝑥_𝑛−𝑥^∗‖²≤11−𝜌⟨𝑥^∗−𝑓(𝑥^∗),𝑥^∗−𝑆𝑥_𝑛𝑡⟩+^𝑛2(1−𝜌)𝑀.(3.33) Hence, the weak convergence of {𝑆𝑥_𝑛} to 𝑥^∗ implies that 𝑥_𝑛→𝑥^∗ strongly. This has proved the relative norm compactness of the net {𝑥_𝑡} as 𝑡→0. This completes the proof.

Now, we show our first main result.

Theorem 3.3. The net {𝑥_𝑡} generated by the implicit method (3.4) converges in norm, as 𝑡→0, to the unique solution 𝑥^∗ of the following variational inequality: 𝑥^∗∈Γ,⟨(𝐼−𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0,𝑥∈Γ.(3.34) In particular, if we take 𝑓=0, then the net {𝑥_𝑡} defined by (3.2) converges in norm, as 𝑡→0, to a solution of the minimization problem (3.1).

Proof. Now we return to (3.24) in Lemma 3.2 and take the limit as 𝑛→∞ to get ‖𝑥^∗−𝑧‖²≤11−𝜌⟨𝑧−𝑓(𝑧),𝑧−𝑥^∗⟩,𝑧∈Γ.(3.35) In particular, 𝑥^∗ solves the following variational inequality 𝑥^∗∈Γ,⟨(𝐼−𝑓)𝑧,𝑧−𝑥^∗⟩≥0,𝑧∈Γ(3.36) or the equivalent dual variational inequality: 𝑥^∗∈Γ,⟨(𝐼−𝑓)𝑥^∗,𝑧−𝑥^∗⟩≥0,𝑧∈Γ.(3.37) Therefore, 𝑥^∗=(𝑃_Γ𝑓)𝑥^∗. That is, 𝑥^∗ is the unique fixed point in Γ of the contraction 𝑃_Γ𝑓. Clearly this is sufficient to conclude that the entire net {𝑥_𝑡} converges in norm to 𝑥^∗ as 𝑡→0.
Finally, if we take 𝑓=0, then (3.35) is reduced to‖𝑥^∗−𝑧‖²≤⟨𝑧,𝑧−𝑥^∗⟩,𝑧∈Γ.(3.38) Equivalently, ‖𝑥^∗‖²≤⟨𝑥^∗,𝑧⟩,𝑧∈Γ.(3.39) This clearly implies that ‖𝑥^∗‖≤‖𝑧‖,𝑧∈Γ.(3.40) Therefore, 𝑥^∗ is a solution of minimization problem (3.1). This completes the proof.

Next we introduce an explicit algorithm for finding a solution of minimization problem (3.1). This scheme is obtained by discretizing the implicit scheme (3.4).

Algorithm 3.4. Given that 𝑥₀∈𝐶 arbitrarily, let the sequence {𝑥_𝑛} be generated iteratively by 𝑥_𝑛+1=𝜇_𝑛𝑃_𝐶[𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑆𝑥_𝑛]+(1−𝜇_𝑛)𝑇_𝑟(𝑥_𝑛−𝑟𝐴𝑥_𝑛),𝑛≥0,(3.41) where {𝛼_𝑛} and {𝜇_𝑛} are two sequences in [0,1].

Next, we give several lemmas in order to prove our second main result.

Lemma 3.5. The sequence {𝑥_𝑛} generated by (3.41) is bounded.

Proof. Pick 𝑧∈Γ. Let 𝑢_𝑛=𝑇_𝑟(𝑥_𝑛−𝑟𝐴𝑥_𝑛) and 𝑦_𝑛=𝑡𝑓(𝑥_𝑛)+(1−𝑡)𝑆𝑥_𝑛 for all 𝑛≥0. From (3.41), we get ‖𝑢_𝑛−𝑧‖=‖𝑇_𝑟(𝑥_𝑛−𝑟𝐴𝑥_𝑛)−𝑇_𝑟(𝑧−𝑟𝐴𝑧)‖≤‖𝑥_𝑛−𝑧‖,‖𝑥_𝑛+1−𝑧‖=‖𝜇_𝑛(𝑃_𝐶[𝑦_𝑛]−𝑧)+(1−𝜇_𝑛)(𝑢_𝑛−𝑧)‖≤𝜇_𝑛‖𝑃_𝐶[𝑦_𝑛]−𝑧‖+(1−𝜇_𝑛)‖𝑢_𝑛−𝑧‖≤𝜇_𝑛‖𝑦_𝑛−𝑧‖+(1−𝜇_𝑛)‖𝑥_𝑛−𝑧‖≤𝜇_𝑛𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑧‖+𝜇_𝑛(1−𝛼_𝑛)‖𝑆𝑥_𝑛−𝑧‖+(1−𝜇_𝑛)‖𝑥_𝑛−𝑧‖≤[1−(1−𝜌)𝜇_𝑛𝛼_𝑛]‖𝑥_𝑛−𝑧‖+𝜇_𝑛𝛼_𝑛‖𝑓(𝑧)−𝑧‖.(3.42) By induction, we obtain, for all 𝑛≥0, ‖𝑥_𝑛−𝑧‖≤max{‖𝑥₀−𝑧‖,‖𝑓(𝑧)−𝑧‖1−𝜌}.(3.43) Hence, {𝑥_𝑛} is bounded. Consequently, we deduce that {𝑢_𝑛}, {𝑓(𝑥_𝑛)} and {𝐴𝑥_𝑛} are all bounded. This completes the proof.

Lemma 3.6. Assume the sequences {𝛼_𝑛} and {𝜇_𝑛} satisfy the following conditions: (i)lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞(𝛼_𝑛+1/𝛼_𝑛)=1; (ii)0<liminf_𝑛→∞𝜇_𝑛≤limsup_𝑛→∞𝜇_𝑛<1 and lim_𝑛→∞((𝜇_𝑛+1−𝜇_𝑛)/𝛼_𝑛+1)=1. Then lim_𝑛→∞‖𝑥_𝑛−𝑆𝑥_𝑛‖=0.

Proof. From (3.41), we have ‖𝑥_𝑛+2−𝑥_𝑛+1‖=‖𝜇_𝑛+1𝑃_𝐶[𝑦_𝑛+1]+(1−𝜇_𝑛+1)𝑢_𝑛+1−𝜇_𝑛𝑃_𝐶[𝑦_𝑛]−(1−𝜇_𝑛)𝑢_𝑛‖=‖𝜇_𝑛+1(𝑃_𝐶[𝑦_𝑛+1]−𝑃_𝐶[𝑦_𝑛])+(𝜇_𝑛+1−𝜇_𝑛)𝑃_𝐶[𝑦_𝑛]+(1−𝜇_𝑛+1)(𝑢_𝑛+1−𝑢_𝑛)+(𝜇_𝑛−𝜇_𝑛+1)𝑢_𝑛‖≤𝜇_𝑛+1‖𝑦_𝑛+1−𝑦_𝑛‖+(1−𝜇_𝑛+1)‖𝑢_𝑛+1−𝑢_𝑛‖+|𝜇_𝑛+1−𝜇_𝑛|(‖𝑃_𝐶[𝑦_𝑛]‖+‖𝑢_𝑛‖).(3.44) Next we estimate ‖𝑦_𝑛+1−𝑦_𝑛‖ and ‖𝑢_𝑛+1−𝑢_𝑛‖. We have ‖𝑦_𝑛+1−𝑦_𝑛‖=‖𝛼_𝑛+1𝑓(𝑥_𝑛+1)+(1−𝛼_𝑛+1)𝑆𝑥_𝑛+1−𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑆𝑥_𝑛‖=‖𝛼_𝑛+1(𝑓(𝑥_𝑛+1)−𝑓(𝑥_𝑛))+(𝛼_𝑛+1−𝛼_𝑛)𝑓(𝑥_𝑛)+(1−𝛼_𝑛+1)(𝑆𝑥_𝑛+1−𝑆𝑥_𝑛)+(𝛼_𝑛−𝛼_𝑛+1)𝑆𝑥_𝑛‖≤[1−(1−𝜌)𝛼_𝑛+1]‖𝑥_𝑛+1−𝑥_𝑛‖+|𝛼_𝑛+1−𝛼_𝑛|(‖𝑓(𝑥_𝑛)‖+‖𝑆𝑥_𝑛‖),‖𝑢_𝑛+1−𝑢_𝑛‖=‖𝑇_𝑟(𝑥_𝑛+1−𝑟𝐴𝑥_𝑛+1)−𝑇_𝑟(𝑥_𝑛−𝑟𝐴𝑥_𝑛)‖≤‖(𝑥_𝑛+1−𝑟𝐴𝑥_𝑛+1)−(𝑥_𝑛−𝑟𝐴𝑥_𝑛)‖≤‖𝑥_𝑛+1−𝑥_𝑛‖.(3.45) Then, we obtain ‖𝑥_𝑛+2−𝑥_𝑛+1‖≤[1−(1−𝜌)𝛼_𝑛+1𝜇_𝑛+1]‖𝑥_𝑛+1−𝑥_𝑛‖+(|𝛼_𝑛+1−𝛼_𝑛|+|𝜇_𝑛+1−𝜇_𝑛|)𝑀,(3.46) where 𝑀>0 is a constant satisfying sup_𝑛{‖𝑓(𝑥_𝑛)‖+‖𝑆𝑥_𝑛‖,‖𝑃_𝐶[𝑦_𝑛]‖+‖𝑢_𝑛‖,‖𝑓(𝑥_𝑛)−𝑧‖²,2‖𝑥_𝑛−𝑧‖,2𝑟‖𝑥_𝑛−𝑢_𝑛‖}≤𝑀.(3.47) This together with (i), (ii) and Lemma 2.4 imply that lim_𝑛→∞‖𝑥_𝑛+2−𝑥_𝑛+1‖=0.(3.48) By the convexity of the norm ‖⋅‖, we have ‖𝑥_𝑛+1−𝑧‖²=‖𝜇_𝑛(𝑃_𝐶[𝑦_𝑛]−𝑧)+(1−𝜇_𝑛)(𝑢_𝑛−𝑧)‖²≤𝜇_𝑛‖𝑃_𝐶[𝑦_𝑛]−𝑧‖²+(1−𝜇_𝑛)‖𝑢_𝑛−𝑧‖²≤𝜇_𝑛‖𝛼_𝑛(𝑓(𝑥_𝑛)−𝑧)+(1−𝛼_𝑛)(𝑆𝑥_𝑛−𝑧)‖²+(1−𝜇_𝑛)‖𝑢_𝑛−𝑧‖²≤𝜇_𝑛[𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑧‖²+(1−𝛼_𝑛)‖𝑆𝑥_𝑛−𝑧‖²]+(1−𝜇_𝑛)‖𝑢_𝑛−𝑧‖²≤𝜇_𝑛[𝛼_𝑛𝑀+(1−𝛼_𝑛)‖𝑥_𝑛−𝑧‖²]+(1−𝜇_𝑛)‖𝑢_𝑛−𝑧‖².(3.49) From Lemma 2.2, we get ‖𝑢_𝑛−𝑧‖²=‖𝑇_𝑟(𝑥_𝑛−𝑟𝐴𝑥_𝑛)−𝑇_𝑟(𝑧−𝑟𝐴𝑧)‖²≤‖(𝑥_𝑛−𝑟𝐴𝑥_𝑛)−(𝑧−𝑟𝐴𝑧)‖²≤‖𝑥_𝑛−𝑧‖²+𝑟(𝑟−2𝛼)‖𝐴𝑥_𝑛−𝐴𝑧‖².(3.50) Substituting (3.50) into (3.49), we have ‖𝑥_𝑛+1−𝑧‖²≤(1−𝜇_𝑛𝛼_𝑛)‖𝑥_𝑛−𝑧‖²+𝑟(𝑟−2𝛼)(1−𝜇_𝑛)‖𝐴𝑥_𝑛−𝐴𝑧‖²+𝜇_𝑛𝛼_𝑛𝑀.(3.51) Therefore, 𝑟(2𝛼−𝑟)(1−𝜇_𝑛)‖𝐴𝑥_𝑛−𝐴𝑧‖²≤(1−𝜇_𝑛𝛼_𝑛)‖𝑥_𝑛−𝑧‖²−‖𝑥_𝑛+1−𝑧‖²+𝜇_𝑛𝛼_𝑛𝑀≤(‖𝑥_𝑛−𝑧‖+‖𝑥_𝑛+1−𝑧‖)‖𝑥_𝑛−𝑥_𝑛+1‖+𝛼_𝑛𝑀≤(‖𝑥_𝑛−𝑥_𝑛+1‖+𝛼_𝑛)𝑀.(3.52) Since liminf_𝑛→∞(1−𝜇_𝑛)𝑟(2𝛼−𝑟)>0, ‖𝑥_𝑛−𝑥_𝑛+1‖→0 and 𝛼_𝑛→0, we derive lim_𝑛→∞‖𝐴𝑥_𝑛−𝐴𝑧‖=0.(3.53)
From Lemma 2.1 and (3.41), we obtain‖𝑢_𝑛−𝑧‖²=‖𝑇_𝑟(𝑥_𝑛−𝑟𝐴𝑥_𝑛)−𝑇_𝑟(𝑧−𝑟𝐴𝑧)‖²≤⟨(𝑥_𝑛−𝑟𝐴𝑥_𝑛)−(𝑧−𝑟𝐴𝑧),𝑢_𝑛=1−𝑧⟩2(‖(𝑥_𝑛−𝑟𝐴𝑥_𝑛)−(𝑧−𝑟𝐴𝑧)‖²+‖𝑢_𝑛−𝑧‖²−‖(𝑥_𝑛−𝑧)−𝑟(𝐴𝑥_𝑛−𝐴𝑧)−(𝑢_𝑛−𝑧)‖²)≤12(‖𝑥_𝑛−𝑧‖²+‖𝑢_𝑛−𝑧‖²−‖(𝑥_𝑛−𝑢_𝑛)−𝑟(𝐴𝑥_𝑛−𝐴𝑧)‖²)=12(‖𝑥_𝑛−𝑧‖²+‖𝑢_𝑛−𝑧‖²−‖𝑥_𝑛−𝑢_𝑛‖²+2𝑟⟨𝑥_𝑛−𝑢_𝑛,𝐴𝑥_𝑛−𝐴𝑧⟩−𝑟²‖𝐴𝑥_𝑛−𝐴𝑧‖²).(3.54) Thus, we deduce ‖𝑢_𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²−‖𝑥_𝑛−𝑢_𝑛‖²+2𝑟‖𝑥_𝑛−𝑢_𝑛‖‖𝐴𝑥_𝑛−𝐴𝑧‖≤‖𝑥_𝑛−𝑧‖²−‖𝑥_𝑛−𝑢_𝑛‖²+𝑀‖𝐴𝑥_𝑛−𝐴𝑧‖.(3.55) By (3.49) and (3.55), we have ‖𝑥_𝑛+1−𝑧‖²≤𝜇_𝑛[𝛼_𝑛𝑀+(1−𝛼_𝑛)‖𝑥_𝑛−𝑧‖²]+(1−𝜇_𝑛)‖𝑢_𝑛−𝑧‖²≤𝜇_𝑛[𝛼_𝑛𝑀+(1−𝛼_𝑛)‖𝑥_𝑛−𝑧‖²]+(1−𝜇_𝑛)[‖𝑥_𝑛−𝑧‖²−‖𝑥_𝑛−𝑢_𝑛‖²+𝑀‖𝐴𝑥_𝑛−𝐴𝑧‖]≤‖𝑥_𝑛−𝑧‖²−(1−𝜇_𝑛)‖𝑥_𝑛−𝑢_𝑛‖²+(‖𝐴𝑥_𝑛−𝐴𝑧‖+𝛼_𝑛)𝑀.(3.56) It follows that (1−𝜇_𝑛)‖𝑥_𝑛−𝑢_𝑛‖²≤(‖𝑥_𝑛+1−𝑥_𝑛‖+‖𝐴𝑥_𝑛−𝐴𝑧‖+𝛼_𝑛)𝑀.(3.57) Since liminf_𝑛→∞(1−𝜇_𝑛)>0, 𝛼_𝑛→0, ‖𝑥_𝑛+1−𝑥_𝑛‖→0 and ‖𝐴𝑥_𝑛−𝐴𝑧‖→0, we derive that lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0.(3.58) Note that 𝑥_𝑛+1−𝑥_𝑛=𝜇_𝑛(𝑃_𝐶[𝑦_𝑛]−𝑥_𝑛)+(1−𝜇_𝑛)(𝑢_𝑛−𝑥_𝑛). Hence, ‖𝑃_𝐶[𝑦_𝑛]−𝑥_𝑛‖→0.(3.59) Therefore, ‖𝑆𝑥_𝑛−𝑥_𝑛‖≤‖𝑆𝑥_𝑛−𝑃_𝐶[𝑦_𝑛]‖+‖𝑃_𝐶[𝑦_𝑛]−𝑥_𝑛‖≤‖𝑆𝑥_𝑛−𝑦_𝑛‖+‖𝑃_𝐶[𝑦_𝑛]−𝑥_𝑛‖≤𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑆𝑥_𝑛‖+‖𝑃_𝐶[𝑦_𝑛]−𝑥_𝑛‖→0.(3.60) This completes the proof.

Now, we show the strong convergence of the sequence {𝑥_𝑛} generated by (3.41).

Theorem 3.7. Assume the sequences {𝛼_𝑛} and {𝜇_𝑛} satisfy the following conditions: (i)lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=0𝛼_𝑛=∞ and lim_𝑛→∞(𝛼_𝑛+1/𝛼_𝑛)=1; (ii)0<liminf_𝑛→∞𝜇_𝑛≤limsup_𝑛→∞𝜇_𝑛<1 and lim_𝑛→∞((𝜇_𝑛+1−𝜇_𝑛)/𝛼_𝑛+1)=1. Then the sequence {𝑥_𝑛} generated by (3.41) converges strongly to 𝑥^∗ which is the unique solution of variational inequality (3.34). In particular, if 𝑓=0, then the sequence {𝑥_𝑛} generated by 𝑥_𝑛+1=𝜇_𝑛𝑃_𝐶[(1−𝛼_𝑛)𝑆𝑥_𝑛]+(1−𝜇_𝑛)𝑇_𝑟(𝑥_𝑛−𝑟𝐴𝑥_𝑛),𝑛≥0,(3.61) converges strongly to a solution of the minimization problem (3.1).

Proof. We first prove limsup_𝑛→∞⟨𝑥^∗−𝑓(𝑥^∗),𝑥^∗−𝑆𝑥_𝑛⟩≤0(3.62) where 𝑥^∗=𝑃_Γ𝑓(𝑥^∗).
Indeed, we can choose a subsequence {𝑆𝑥_𝑛𝑖} of {𝑆𝑥_𝑛} such that limsup_𝑛→∞⟨𝑥^∗−𝑓(𝑥^∗),𝑥^∗−𝑆𝑥_𝑛⟩=lim_𝑖→∞⟨𝑥^∗−𝑓(𝑥^∗),𝑥^∗−𝑆𝑥_𝑛𝑖⟩.(3.63) Without loss of generality, we may further assume that 𝑆𝑥_𝑛𝑖→𝑥 weakly. By the same argument as that of Theorem 3.3, we can deduce that 𝑥∈Γ. Therefore, limsup_𝑛→∞⟨𝑥^∗−𝑓(𝑥^∗),𝑥^∗−𝑆𝑥_𝑛⟩=⟨𝑥^∗−𝑓(𝑥^∗),𝑥^∗−𝑥⟩≤0.(3.64) From (3.41), we have ‖𝑥_𝑛+1−𝑥^∗‖²=‖𝜇_𝑛(𝑃_𝐶[𝑦_𝑛]−𝑧)+(1−𝜇_𝑛)(𝑢_𝑛−𝑧)‖²≤𝜇_𝑛‖𝑃_𝐶[𝑦_𝑛]−𝑧‖²+(1−𝜇_𝑛)‖𝑢_𝑛−𝑧‖²≤𝜇_𝑛‖𝑦_𝑛−𝑥^∗‖²+(1−𝜇_𝑛)‖𝑥_𝑛−𝑥^∗‖²=𝜇_𝑛[(1−𝛼_𝑛)²‖𝑆𝑥_𝑛−𝑥^∗‖²+2𝛼_𝑛(1−𝛼_𝑛)⟨𝑓(𝑥_𝑛)−𝑓(𝑥^∗),𝑆𝑥_𝑛−𝑥^∗⟩+2𝛼_𝑛(1−𝛼_𝑛)⟨𝑓(𝑥^∗)−𝑥^∗,𝑆𝑥_𝑛−𝑥^∗⟩+𝛼²_𝑛‖𝑓(𝑥_𝑛)−𝑥^∗‖²]+(1−𝜇_𝑛)‖𝑥_𝑛−𝑥^∗‖²≤𝜇_𝑛[(1−𝛼_𝑛)²‖𝑥_𝑛−𝑥^∗‖²+2𝛼_𝑛(1−𝛼_𝑛)𝜌‖𝑥_𝑛−𝑥^∗‖²+2𝛼_𝑛(1−𝛼_𝑛)⟨𝑓(𝑥^∗)−𝑥^∗,𝑆𝑥_𝑛−𝑥^∗⟩+𝛼²_𝑛𝑀]+(1−𝜇_𝑛)‖𝑥_𝑛−𝑥^∗‖²≤[1−2(1−𝜌)𝜇_𝑛𝛼_𝑛]‖𝑥_𝑛−𝑥^∗‖²+2𝛼_𝑛𝜇_𝑛(1−𝛼_𝑛)⟨𝑓(𝑥^∗)−𝑥^∗,𝑆𝑥_𝑛−𝑥^∗⟩+2𝛼²_𝑛𝜇_𝑛𝑀=(1−𝛾_𝑛)‖𝑥_𝑛−𝑥^∗‖²+𝛿_𝑛𝛾_𝑛,(3.65) where 𝛾_𝑛=2(1−𝜌)𝜇_𝑛𝛼_𝑛 and 𝛿_𝑛=((1−𝛼_𝑛)/(1−𝜌))⟨𝑓(𝑥)^∗−𝑥^∗,𝑆𝑥_𝑛−𝑥^∗⟩+𝛼_𝑛𝑀/(1−𝜌). It is clear that ∑^∞_𝑛=0𝛾_𝑛=∞ and limsup_𝑛→∞𝛿≤0. Hence, all conditions of Lemma 2.4 are satisfied. Therefore, we immediately deduce that 𝑥_𝑛→𝑥^∗.
Finally, if we take 𝑓=0, by the similar argument as that Theorem 3.3, we deduce immediately that 𝑥^∗ is a minimum norm element in Γ. This completes the proof.

4. Conclusions

Iterative methods for finding the common element of the equilibrium problem and the fixed point problem have been extensively studied, see, for example, [2, 4, 6, 7, 9, 14, 15, 21, 23–28]. However, iterative methods for finding the minimum norm solution of the equilibrium problem and the fixed point problem are far less developed than those for only finding the common element of the equilibrium problem and the fixed point problem. In the present paper, we suggest two algorithm, one implicit algorithm (3.4) and one explicit algorithm (3.41). We prove the strong convergence of the algorithms (3.4) and (3.41) to the common element of the equilibrium problem and the fixed points set of a nonexpansive mapping. As special cases, we prove that algorithms (3.2) and (3.61) converges to 𝑥^∗ which solves the minimization problem (3.1). It should be pointed out that our algorithms and our main results are new even if we assume 𝑓 is a self-mapping on 𝐶.

Since in many problems, it is needed to find a solution with minimum norm. Hence, it is a very interesting problem to construct some algorithms for finding the minimum norm solution of some practical problem. The reader can develop iterative algorithms for solving some minimization problems by using our methods and technique contained in the present paper.

Acknowledgments

The authors thank three anonymous referees for their comments which improved the presentation of this paper. Y. Yao was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin and NSFC 11071279. The second author was supported in part by NSC 99-2221-E-230-006.