Algorithmic Approach to a Minimization Problem

Abstract

We first construct an implicit algorithm for solving the minimization problem min_𝑥∈Ω‖𝑥‖ , where Ω is the intersection set of the solution set of some equilibrium problem, the fixed points set of a nonexpansive mapping, and the solution set of some variational inequality. Further, we suggest an explicit algorithm by discretizing this implicit algorithm. We prove that the proposed implicit and explicit algorithms converge strongly to a solution of the above minimization problem.

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. Recall that a mapping 𝐴∶𝐶→𝐻 is called 𝛼-inverse-strongly monotone if there exists a constant 𝛼>0 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶. (1.1) 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 𝐹(𝑆).

Let 𝐵∶𝐶→𝐻 be a nonlinear mapping and 𝐹∶𝐶×𝐶→𝑅 be a bifunction. Now we concern the following equilibrium problem is to find 𝑧∈𝐶 such that 𝐹𝑧,𝑦+𝐵𝑧,𝑦−𝑧≥0,∀𝑦∈𝐶. (1.2) The solution set of (1.2) is denoted by EP(𝐹,𝐵). If 𝐵=0, then (1.2) reduces to the following equilibrium problem of finding 𝑧∈𝐶 such that 𝐹𝑧,𝑦≥0,∀𝑦∈𝐶. (1.3) The solution set of (1.3) is denoted by EP(𝐹). If 𝐹=0, then (1.2) reduces to the variational inequality problem of finding 𝑧∈𝐶 such that 𝐵𝑧,𝑦−𝑧≥0,∀𝑦∈𝐶. (1.4) The solution set of variational inequality (1.4) is denoted by VI(𝐶,𝐵).

Equilibrium problems which were introduced by Blum and Oettli [1] in 1994 have had a great impact and influence in pure and applied sciences. It has been shown that the equilibrium problems theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. Equilibrium problems include variational inequalities, fixed point, Nash equilibrium, and game theory as special cases. The equilibrium problems and the variational inequality problems have been investigated by many authors. Please see [2–35] and the references therein. The problem (1.2) 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.

On the other hand, we also notice that it is quite often to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. For instance, given a closed convex subset 𝐶 of a Hilbert space 𝐻₁ and a bounded linear operator 𝑅∶𝐻₁→𝐻₂, where 𝐻₂ is another Hilbert space. The 𝐶-constrained pseudoinverse of 𝑅, 𝑅^†_𝐶, is then defined as the minimum-norm solution of the constrained minimization problem: 𝑅^†_𝐶(𝑏)∶=argmin_𝑥∈𝐶‖𝑅𝑥−𝑏‖, (1.5) which is equivalent to the fixed point problem: 𝑥=𝑃_𝐶𝑥−𝜆𝑅^∗(𝑅𝑥−𝑏), (1.6) where 𝑃_𝐶 is the metric projection from 𝐻₁ onto 𝐶, 𝑅^∗ is the adjoint of 𝑅, 𝜆>0 is a constant, and 𝑏∈𝐻₂ is such that 𝑃_𝑅(𝐶)(𝑏)∈𝑅(𝐶).

It is therefore an interesting problem to invent some algorithms that can generate schemes which converge strongly to the minimum-norm solution of a given problem.

In this paper, we focus on the following minimization problem: find 𝑥^∗∈Ω such that 𝑥^∗=argmin_𝑥∈Ω‖𝑥‖, (1.7) where Ω is the intersection set of the solution set of some equilibrium problem, the fixed points set of a nonexpansive mapping, and the solution set of some variational inequality. We will suggest and analyze two very simple algorithms for solving the above minimization problem.

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∈𝐶𝑥−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_𝐶𝑦²,∀𝑥,𝑦∈𝐻. (2.2) We need the following well-known lemmas for proving our main results.

Lemma 2.1 (see [13]). 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 𝜇>0 and 𝑥∈𝐶. Then there exists 𝑧∈𝐶 such that 𝐹+1𝑧,𝑦𝜇𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (2.3) Further, if 𝑇_𝜇(𝑥)={𝑧∈𝐶∶𝐹(𝑧,𝑦)+(1/𝜇)⟨𝑦−𝑧,𝑧−𝑥⟩≥0,forall𝑦∈𝐶}, then the following hold:(a)𝑇_𝜇 is single-valued and 𝑇_𝜇 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐶, ‖𝑇_𝜇𝑥−𝑇_𝜇𝑦‖²≤⟨𝑇_𝜇𝑥−𝑇_𝜇𝑦,𝑥−𝑦⟩;(b)EP(𝐹) is closed and convex and EP(𝐹)=𝐹(𝑇_𝜇).

Lemma 2.2 2.2 (see [27]). Let {𝑥_𝑛} and {𝑣_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 that 𝑥_𝑛+1=(1−𝛽_𝑛)𝑣_𝑛+𝛽_𝑛𝑥_𝑛 for all 𝑛≥0 and limsup_𝑛→∞(‖𝑣_𝑛+1−𝑣_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then lim_𝑛→∞‖𝑣_𝑛−𝑥_𝑛‖=0.

Lemma 2.3 (see [29]). 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 [29]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛𝛾_𝑛, (2.4) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that(a)∑^∞_𝑛=1𝛾_𝑛=∞; (b)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 (one implicit and one explicit) for finding the minimum norm element 𝑥^∗ of Ω∶=EP(𝐹,𝐵)∩VI(𝐶,𝐴)∩𝐹(𝑆). Namely, we want to find a point 𝑥^∗ which solves the following minimization problem: 𝑥^∗=argmin_𝑥∈Ω‖𝑥‖. (3.1) Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping and 𝐴,𝐵∶𝐶→𝐻 be 𝛼-inverse-strongly monotone and 𝛽-inverse-strongly monotone mappings, respectively. Let 𝐹∶𝐶×𝐶→𝑅 be a bifunction which satisfies conditions (H1)–(H4). In order to solve the minimization problem (3.1), we first construct the following implicit algorithm by using the projection method: 𝑥_𝑡=𝑆𝑃_𝐶(1−𝑡)(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑥_𝑡,∀𝑡∈(0,1), (3.2) where 𝑇_𝜇 is defined as Lemma 2.1 and 𝜆,𝜇 are two constants such that 𝜆∈(0,2𝛼) and 𝜇∈(0,2𝛽). We will show that the net {𝑥_𝑡} defined by (3.2) converges to a solution of the minimization problem (3.1). First, we show that the net {𝑥_𝑡} is well defined. As matter of fact, for each 𝑡∈(0,1), we consider the following mapping 𝑊_𝑡 given by 𝑊_𝑡𝑥=𝑆𝑃_𝐶(1−𝑡)(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑥,∀𝑥∈𝐶. (3.3) Since the mappings 𝑆, 𝑃_𝐶, 𝐼−𝜆𝐴, 𝑇_𝜇 and 𝐼−𝜇𝐵 are nonexpansive, then we can check easily that ‖𝑊_𝑡𝑥−𝑊_𝑡𝑦‖≤(1−𝑡)‖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=𝑊_𝑡𝑥_𝑡 which is exactly (3.2).

Next we show the first main result of the present paper.

Theorem 3.1. Suppose that Ω≠∅. Then the net {𝑥_𝑡} generated by the implicit method (3.2) converges in norm, as 𝑡→0, to a solution of the minimization problem (3.1).

Proof. Take 𝑧∈Ω. First we need use the following facts:(1)𝑧=𝑆𝑧=𝑇_𝜇(𝑧−𝜇𝐵𝑧)=𝑃_𝐶(𝐼−𝜆𝐴)𝑧=𝑆𝑃_𝐶[(𝐼−𝜆𝐴)𝑇_𝜇(𝐼−𝜇𝐵)𝑧] for all 𝜆>0, 𝜇>0. In particular, for all 𝑡∈(0,1), 𝑧=𝑃_𝐶[𝑧−(1−𝑡)𝜆𝐴𝑧]=𝑃_𝐶[𝑡𝑧+(1−𝑡)(𝐼−𝜆𝐴)𝑧]=𝑆𝑃_𝐶𝑡𝑧+(1−𝑡)(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑧; (3.4)(2)𝐼−𝜆𝐴 and 𝐼−𝜇𝐵 are nonexpansive and for all 𝑥,𝑦∈𝐶(𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦²≤𝑥−𝑦²+𝜆(𝜆−2𝛼)𝐴𝑥−𝐴𝑦²;𝐼−𝜇𝐵𝑥−𝐼−𝜇𝐵𝑦²≤𝑥−𝑦²+𝜇𝜇−2𝛽𝐵𝑥−𝐵𝑦². (3.5)
Set 𝑢_𝑡=𝑇_𝜇(𝐼−𝜇𝐵)𝑥_𝑡 and 𝑦_𝑡=(𝐼−𝜆𝐴)𝑢_𝑡 for all 𝑡∈(0,1). It follows that 𝑢_𝑡−𝑧²=𝑇_𝜇𝑥_𝑡−𝜇𝐵𝑥_𝑡−𝑇_𝜇𝑧−𝜇𝐵𝑧²≤𝐼−𝜇𝐵𝑥_𝑡−𝐼−𝜇𝐵𝑧²≤𝑥_𝑡−𝑧²+𝜇𝜇−2𝛽𝐵𝑥_𝑡−𝐵𝑧². (3.6) From (3.2), we have 𝑥_𝑡−𝑧=𝑆𝑃_𝐶(1−𝑡)(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑥_𝑡−𝑆𝑃_𝐶𝑡𝑧+(1−𝑡)(𝐼−𝜆𝐴)𝑇_𝜇𝑧≤)𝑇𝐼−𝜇𝐵(1−𝑡)(𝐼−𝜆𝐴_𝜇𝐼−𝜇𝐵𝑥_𝑡−𝑡𝑧+(1−𝑡)(𝐼−𝜆𝐴)𝑇_𝜇𝑧≤()𝑇𝐼−𝜇𝐵1−𝑡)(𝐼−𝜆𝐴_𝜇𝐼−𝜇𝐵𝑥_𝑡−(𝐼−𝜆𝐴)𝑇_𝜇𝑧‖𝑧‖≤(𝑥𝐼−𝜇𝐵+𝑡1−𝑡)_𝑡−𝑧+𝑡‖𝑧‖, (3.7) that is, 𝑥_𝑡−𝑧≤‖𝑧‖. (3.8) So, {𝑥_𝑡} is bounded. Hence {𝑢_𝑡}, {𝐴𝑥_𝑡}, and {𝐵𝑥_𝑡} are also bounded. Next we will use 𝑀>0 to denote some possible constant appearing in the following.
From (3.7), we have 𝑥_𝑡−𝑧²≤(1−𝑡)(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑥_𝑡−𝑡𝑧+(1−𝑡)(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑧²=(1−𝑡)(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑥_𝑡−(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑧−𝑡𝑧²=(1−𝑡)²(𝐼−𝜆𝐴)𝑢_𝑡−(𝐼−𝜆𝐴)𝑧²−2𝑡(1−𝑡)𝑧,(𝐼−𝜆𝐴)𝑢_𝑡−(𝐼−𝜆𝐴)𝑧+𝑡²‖𝑧‖²≤𝑢_𝑡−𝑧²+𝜆(𝜆−2𝛼)𝐴𝑢_𝑡−𝐴𝑧²≤𝑥+𝑡𝑀_𝑡−𝑧²+𝜇𝜇−2𝛽𝐵𝑥_𝑡−𝐵𝑧²+𝜆(𝜆−2𝛼)𝐴𝑢_𝑡−𝐴𝑧²+𝑡𝑀, (3.9) that is, 𝜆(2𝛼−𝜆)𝐴𝑢_𝑡−𝐴𝑧²+𝜇2𝛽−𝜇𝐵𝑥_𝑡−𝐵𝑧²≤𝑡𝑀⟶0. (3.10) Since 𝜆(2𝛼−𝜆)>0 and 𝜇(2𝛽−𝜇)>0, we derive lim_𝑡→0𝐴𝑢_𝑡−𝐴𝑧=lim_𝑡→0𝐵𝑥_𝑡−𝐵𝑧=0. (3.11) From Lemma 2.1 and (2.2), we obtain 𝑢_𝑡−𝑧²=𝑇_𝜇𝑥_𝑡−𝜇𝐵𝑥_𝑡−𝑇_𝜇𝑧−𝜇𝐵𝑧²≤𝑥_𝑡−𝜇𝐵𝑥_𝑡−𝑧−𝜇𝐵𝑧,𝑢_𝑡=1−𝑧2𝑥_𝑡−𝜇𝐵𝑥_𝑡−𝑧−𝜇𝐵𝑧²+𝑢_𝑡−𝑧²−𝑥_𝑡−𝑧−𝜇𝐵𝑥_𝑡−𝐵𝑧−𝑢_𝑡−𝑧²≤12𝑥_𝑡−𝑧²+𝑢_𝑡−𝑧²−𝑥_𝑡−𝑢_𝑡−𝜇𝐵𝑥_𝑡−𝐵𝑧²=12𝑥_𝑡−𝑧²+𝑢_𝑡−𝑧²−𝑥_𝑡−𝑢_𝑡²+2𝜇𝑥_𝑡−𝑢_𝑡,𝐵𝑥_𝑡−𝐵𝑧−𝜇²𝐵𝑥_𝑡−𝐵𝑧². (3.12) It follows that 𝑢_𝑡−𝑧²≤𝑥_𝑡−𝑧²−𝑥_𝑡−𝑢_𝑡²+2𝜇𝑥_𝑡−𝑢_𝑡,𝐵𝑥_𝑡−𝐵𝑧−𝜇²𝐵𝑥_𝑡−𝐵𝑧²≤𝑥_𝑡−𝑧²−𝑥_𝑡−𝑢_𝑡²+2𝜇𝑥_𝑡−𝑢_𝑡𝐵𝑥_𝑡≤𝑥−𝐵𝑧_𝑡−𝑧²−𝑥_𝑡−𝑢_𝑡²+𝑀𝐵𝑥_𝑡−𝐵𝑧. (3.13) Set 𝑧_𝑡=𝑃_𝐶[(1−𝑡)𝑦_𝑡] for all 𝑡∈(0,1). By Lemma 2.1 and (2.2), we have 𝑧_𝑡−𝑧²=𝑃_𝐶(1−𝑡)(𝐼−𝜆𝐴)𝑢_𝑡−𝑃_𝐶[(𝐼−𝜆𝐴)𝑧]²≤(1−𝑡)(𝐼−𝜆𝐴)𝑢_𝑡−(𝐼−𝜆𝐴)𝑧,𝑧_𝑡=1−𝑧2(𝐼−𝜆𝐴)𝑢_𝑡−(𝐼−𝜆𝐴)𝑧−𝑡𝑦_𝑡²+𝑧_𝑡−𝑧²−(𝐼−𝜆𝐴)𝑢_𝑡−(𝐼−𝜆𝐴)𝑧−𝑧_𝑡−𝑧−𝑡𝑦_𝑡²≤12(𝐼−𝜆𝐴)𝑢_𝑡−(𝐼−𝜆𝐴)𝑧²+𝑡𝑀+𝑧_𝑡−𝑧²−𝑢_𝑡−𝑧_𝑡−𝜆(𝐴𝑢_𝑡−𝐴𝑧)−𝑡𝑦_𝑡²≤12𝑢_𝑡−𝑧²+𝑡𝑀+𝑧_𝑡−𝑧²−𝑢_𝑡−𝑧_𝑡²+2𝜆𝑢_𝑡−𝑧_𝑡,𝐴𝑢_𝑡𝑦−𝐴𝑧+2𝑡_𝑡,𝑢_𝑡−𝑧_𝑡−𝜆𝐴(𝑢_𝑡−𝐴𝑧)+𝑡𝑦_𝑡²≤12𝑢_𝑡−𝑧²+𝑡𝑀+𝑧_𝑡−𝑧²−𝑢_𝑡−𝑧_𝑡²+2𝜆𝑢_𝑡−𝑧_𝑡𝐴𝑢_𝑡𝑦−𝐴𝑧+2𝑡_𝑡𝑢_𝑡−𝑧_𝑡, (3.14) that is, 𝑧_𝑡−𝑧²≤𝑢_𝑡−𝑧²+𝑡𝑀−𝑢_𝑡−𝑧_𝑡²+2𝜆𝑢_𝑡−𝑧_𝑡𝐴𝑢_𝑡−𝐴𝑧+2𝑡𝑦_𝑡𝑢_𝑡−𝑧_𝑡≤𝑥_𝑡−𝑧²−𝑥_𝑡−𝑢_𝑡²+𝑀𝐵𝑥_𝑡−𝐵𝑧+𝑡𝑀−𝑢_𝑡−𝑧_𝑡²+2𝜆𝑢_𝑡−𝑧_𝑡𝐴𝑢_𝑡−𝐴𝑧+2𝑡𝑦_𝑡𝑢_𝑡−𝑧_𝑡. (3.15) Therefore, we have 𝑥_𝑡−𝑧²=𝑆𝑧_𝑡−𝑧²≤𝑧_𝑡−𝑧²≤𝑥_𝑡−𝑧²−𝑥_𝑡−𝑢_𝑡²+𝑀𝐵𝑥_𝑡−𝐵𝑧+𝑡𝑀−𝑢_𝑡−𝑧_𝑡²+2𝜆𝑢_𝑡−𝑧_𝑡𝐴𝑢_𝑡−𝐴𝑧+2𝑡𝑦_𝑡𝑢_𝑡−𝑧_𝑡. (3.16) Hence, we deduce 𝑥_𝑡−𝑢_𝑡²+𝑢_𝑡−𝑧_𝑡²≤𝑀𝐵𝑥_𝑡−𝐵𝑧+𝑡+2𝜆𝑢_𝑡−𝑧_𝑡𝐴𝑢_𝑡−𝐴𝑧+2𝑡𝑦_𝑡𝑢_𝑡−𝑧_𝑡⟶0. (3.17) This denotes that lim_𝑡→0𝑥_𝑡−𝑢_𝑡=lim_𝑡→0𝑢_𝑡−𝑧_𝑡=0. (3.18) Note that 𝑧_𝑡−𝑦_𝑡=𝑃_𝐶(1−𝑡)𝑦_𝑡−𝑃_𝐶𝑦_𝑡≤𝑡𝑦_𝑡⟶0, (3.19) thus, 𝑦_𝑡−𝑆𝑦_𝑡≤𝑦_𝑡−𝑧_𝑡+𝑧_𝑡−𝑢_𝑡+𝑢_𝑡−𝑥_𝑡+𝑥_𝑡−𝑆𝑦_𝑡≤2𝑦_𝑡−𝑧_𝑡+𝑧_𝑡−𝑢_𝑡+𝑢_𝑡−𝑥_𝑡⟶0. (3.20) 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, 𝑢_𝑛∶=𝑢_𝑡𝑛 and 𝑦_𝑛∶=𝑦_𝑡𝑛. From (3.20), we get 𝑦_𝑛−𝑆𝑦_𝑛⟶0. (3.21) By (3.2), we have 𝑥_𝑡−𝑧²=𝑆𝑃_𝐶(1−𝑡)𝑦_𝑡−𝑆𝑃_𝐶𝑧²≤𝑦_𝑡−𝑧−𝑡𝑦_𝑡²=𝑦_𝑡−𝑧²−2𝑡𝑦_𝑡,𝑦_𝑡−𝑧+𝑡²𝑦_𝑡²=𝑦_𝑡−𝑧²−2𝑡𝑦_𝑡−𝑧,𝑦_𝑡−𝑧−2𝑡𝑧,𝑦_𝑡−𝑧+𝑡²𝑦_𝑡²≤(1−2𝑡)𝑦_𝑡−𝑧²+2𝑡𝑧,𝑧−𝑦_𝑡+𝑡²𝑦_𝑡²≤(1−2𝑡)𝑥_𝑡−𝑧²+2𝑡𝑧,𝑧−𝑦_𝑡+𝑡²𝑦_𝑡². (3.22) It follows that 𝑥_𝑡−𝑧²≤𝑧,𝑧−𝑦_𝑡+𝑡𝑀2. (3.23) In particular, 𝑥_𝑛−𝑧²≤𝑧,𝑧−𝑦_𝑛+𝑡_𝑛𝑀2,𝑧∈Ω. (3.24)
Since {𝑦_𝑛} is bounded, without loss of generality, we may assume that {𝑦_𝑛} converges weakly to a point 𝑥^∗∈𝐶. Hence, 𝑢_𝑛 and 𝑥_𝑛 also converge weakly to 𝑥^∗. Noticing (3.21) we can use Lemma 2.3 to get 𝑥^∗∈𝐹(𝑆).
Now we show 𝑥^∗∈EP(𝐹,𝐵). Since 𝑢_𝑛=𝑇_𝜇(𝑥_𝑛−𝜇𝐵𝑥_𝑛), for any 𝑦∈𝐶 we have 𝐹𝑢_𝑛+1,𝑦𝜇𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝜇𝐵𝑥_𝑛≥0. (3.25) From the monotonicity of 𝐹, we have 1𝜇𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝜇𝐵𝑥_𝑛≥𝐹𝑦,𝑢_𝑛,∀𝑦∈𝐶. (3.26) Hence, 𝑦−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝜇+𝐵𝑥_𝑛𝑖≥𝐹𝑦,𝑢_𝑛𝑖,∀𝑦∈𝐶. (3.27) Put 𝑣_𝑡=𝑡𝑦+(1−𝑡)𝑥^∗ for all 𝑡∈(0,1] and 𝑦∈𝐶. Then, we have 𝑣_𝑡∈𝐶. So, from (3.27) we have 𝑣_𝑡−𝑢_𝑛𝑖,𝐵𝑣_𝑡≥𝑣_𝑡−𝑢_𝑛𝑖,𝐵𝑣_𝑡−𝑣_𝑡−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝜇+𝐵𝑥_𝑛𝑖+𝐹𝑣_𝑡,𝑢_𝑛𝑖=𝑣_𝑡−𝑢_𝑛𝑖,𝐵𝑣_𝑡−𝐵𝑢_𝑛𝑖+𝑣_𝑡−𝑢_𝑛𝑖,𝐵𝑢_𝑛𝑖−𝐵𝑥_𝑛𝑖−𝑣_𝑡−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝜇+𝐹𝑣_𝑡,𝑢_𝑛𝑖. (3.28) Note that ‖𝐵𝑢_𝑛𝑖−𝐵𝑥_𝑛𝑖‖≤(1/𝛽)‖𝑢_𝑛𝑖−𝑥_𝑛𝑖‖→0. Further, from monotonicity of 𝐵, we have ⟨𝑣_𝑡−𝑢_𝑛𝑖,𝐵𝑣_𝑡−𝐵𝑢_𝑛𝑖⟩≥0. Letting 𝑖→∞ in (3.28), we have 𝑣_𝑡−𝑥^∗,𝐵𝑣_𝑡≥𝐹𝑣_𝑡,𝑥^∗. (3.29) From (H1), (H4), and (3.29), we also have 0=𝐹𝑣_𝑡,𝑣_𝑡≤𝑡𝐹𝑣_𝑡,𝑦+(1−𝑡)𝐹𝑣_𝑡,𝑥^∗≤𝑡𝐹𝑣_𝑡,𝑦+(1−𝑡)𝑣_𝑡−𝑥^∗,𝐵𝑣_𝑡=𝑡𝐹𝑣_𝑡,𝑦+(1−𝑡)𝑡𝑦−𝑥^∗,𝐵𝑣_𝑡, (3.30) and hence 0≤𝐹𝑣_𝑡,𝑦+(1−𝑡)𝐵𝑣_𝑡,𝑦−𝑥^∗. (3.31) Letting 𝑡→0 in (3.31), we have, for each 𝑦∈𝐶, 0≤𝐹𝑥^∗,𝑦+𝑦−𝑥^∗,𝐵𝑥^∗. (3.32) This implies that 𝑥^∗∈EP(𝐹,𝐵). By the same argument as that of [13], we have 𝑥^∗∈VI(𝐶,𝐴). Therefore, x^∗∈Ω.
We substitute 𝑥^∗ for 𝑧 in (3.24) to get 𝑥_𝑛−𝑥^∗2≤𝑥^∗,𝑥^∗−𝑦_𝑛+𝑡_𝑛2𝑀. (3.33) Hence, the weak convergence of {𝑦_𝑛} to 𝑥^∗ implies that 𝑥_𝑛→𝑥^∗ strongly. This has proved the relative norm compactness of the net {𝑥_𝑡} as 𝑡→0.
Now we return to (3.24) and take the limit as 𝑛→∞ to get 𝑥^∗−𝑧²≤𝑧,𝑧−𝑥^∗,𝑧∈Ω. (3.34) To show that the entire net {𝑥_𝑡} converges to 𝑥^∗, assume 𝑥_𝑠𝑛→𝑥∈Ω, where 𝑠_𝑛→0. In (3.34), we take 𝑧=𝑥 to get 𝑥^∗−𝑥²≤𝑥,𝑥−𝑥^∗. (3.35) Interchange 𝑥^∗ and 𝑥 to obtain 𝑥−𝑥^∗2≤𝑥^∗,𝑥^∗−𝑥. (3.36) Adding up (3.35) and (3.36) yields 2𝑥^∗−𝑥²≤𝑥^∗−𝑥², (3.37) which implies that 𝑥=𝑥^∗.
We note that (3.34) is equivalent to 𝑥^∗2≤𝑥^∗,𝑧,𝑧∈Ω. (3.38) This clearly implies that 𝑥^∗≤‖𝑧‖,𝑧∈Ω. (3.39) Therefore, 𝑥^∗ solves the minimization problem (3.1). This completes the proof.

Next we introduce an explicit algorithm for finding a solution of the minimization problem (3.1). This scheme is obtained by discretizing the implicit scheme (3.2). We will show the strong convergence of this algorithm.

Theorem 3.2. Suppose that Ω≠∅. For given 𝑥₀∈𝐶 arbitrarily, let the sequence {𝑥_𝑛} be generated iteratively by 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶1−𝛼_𝑛(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑥_𝑛,𝑛≥0, (3.40) where {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) satisfying the following conditions:(a)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(b)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Then the sequence {𝑥_𝑛} converges strongly to a solution of the minimization problem (3.1).

Proof. Take 𝑧∈Ω. First we need use the following fact:
𝑧=𝑆𝑧=𝑇_𝜇(𝐼−𝜇𝐵)𝑧=𝑃_𝐶(𝐼−𝜆𝐴)𝑧=𝑆𝑃_𝐶[(𝐼−𝜆𝐴)𝑇_𝜇(𝐼−𝜇𝐵)𝑧] for all 𝜆>0,𝜇>0. In particular, 𝑧=𝑃_𝐶𝑧−1−𝛼_𝑛𝜆𝐴𝑧=𝑃_𝐶𝛼_𝑛𝑧+1−𝛼_𝑛(𝐼−𝜆𝐴)𝑧=𝑆𝑃_𝐶𝛼_𝑛𝑧+1−𝛼_𝑛(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑧,∀𝑛≥0. (3.41)
Set 𝑢_𝑛=𝑇_𝜇(𝑥_𝑛−𝜇𝐵𝑥_𝑛), 𝑦_𝑛=(𝐼−𝜆𝐴)𝑢_𝑛 and 𝑧_𝑛=𝑃_𝐶[(1−𝛼_𝑛)𝑦_𝑛] for all 𝑛≥0. From (3.40), we get 𝑥_𝑛+1−𝑧=𝛽_𝑛𝑥_𝑛−𝑧+1−𝛽_𝑛𝑆𝑃_𝐶1−𝛼_𝑛(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑥_𝑛−𝑆𝑃_𝐶𝛼_𝑛𝑧+1−𝛼_𝑛(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑧≤𝛽_𝑛𝑥_𝑛−𝑧+1−𝛽_𝑛1−𝛼_𝑛(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑥_𝑛−(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑧−𝛼_𝑛𝑧≤𝛽_𝑛𝑥_𝑛−𝑧+1−𝛽_𝑛1−𝛼_𝑛𝑥_𝑛−𝑧+𝛼_𝑛‖𝑧=‖1−1−𝛽_𝑛𝛼_𝑛𝑥_𝑛−𝑧+𝛼_𝑛1−𝛽_𝑛‖𝑧‖≤max𝑥_𝑛−𝑧,‖𝑧‖. (3.42) By induction, we obtain, for all 𝑛≥0, 𝑥_𝑛−𝑧≤max𝑥₀−𝑧,‖𝑧‖. (3.43) Hence, {𝑥_𝑛} is bounded. Consequently, we deduce that {𝑢_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} are all bounded. We will use 𝑀>0 to denote some possible constant appearing in the following.
Define 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑣_𝑛 for all 𝑛≥0. It follows that 𝑣_𝑛+1−𝑣_𝑛=𝑆𝑃_𝐶1−𝛼_𝑛+1𝑦_𝑛+1−𝑆𝑃_𝐶1−𝛼_𝑛𝑦_𝑛≤1−𝛼_𝑛+1𝑦_𝑛+1−1−𝛼_𝑛𝑦_𝑛≤𝑦_𝑛+1−𝑦_𝑛+𝛼_𝑛+1𝑦_𝑛+1+𝛼_𝑛𝑦_𝑛≤(𝐼−𝜆𝐴)𝑢_𝑛+1−(𝐼−𝜆𝐴)𝑢_𝑛+𝑀𝛼_𝑛+1+𝛼_𝑛≤𝑢_𝑛+1−𝑢_𝑛+𝑀𝛼_𝑛+1+𝛼_𝑛=𝑇_𝜇𝐼−𝜇𝐵𝑥_𝑛+1−𝑇_𝜇𝐼−𝜇𝐵𝑥_𝑛+𝑀𝛼_𝑛+1+𝛼_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑀𝛼_𝑛+1+𝛼_𝑛. (3.44) This together with (a) implies that limsup_𝑛→∞𝑣_𝑛+1−𝑣_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (3.45) Hence by Lemma 2.2, we get lim_𝑛→∞𝑣_𝑛−𝑥_𝑛=0. (3.46) Therefore, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽_𝑛𝑣_𝑛−𝑥_𝑛=0. (3.47) By the convexity of the norm ‖⋅‖, we have 𝑥_𝑛+1−𝑧²=𝛽_𝑛𝑥_𝑛−𝑧+1−𝛽_𝑛𝑣_𝑛−𝑧²≤𝛽_𝑛𝑥_𝑛−𝑧²+1−𝛽_𝑛𝑃_𝐶1−𝛼_𝑛(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑥_𝑛−𝑃_𝐶𝛼_𝑛𝑧+1−𝛼_𝑛(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑧²≤𝛽_𝑛𝑥_𝑛−𝑧²+1−𝛽_𝑛1−𝛼_𝑛(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑥_𝑛−(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑧−𝛼_𝑛𝑧²=𝛽_𝑛𝑥_𝑛−𝑧²+1−𝛽_𝑛1−𝛼_𝑛²(𝐼−𝜆𝐴)𝑢_𝑛−(𝐼−𝜆𝐴)𝑧²−2𝛼_𝑛1−𝛼_𝑛(𝐼−𝜆𝐴)𝑢_𝑛−(𝐼−𝜆𝐴)𝑧,𝑧+𝛼²_𝑛‖𝑧‖²≤𝛽_𝑛𝑥_𝑛−𝑧²+1−𝛽_𝑛𝑢_𝑛−𝑧²+𝜆(𝜆−2𝛼)𝐴𝑢_𝑛−𝐴𝑧²+𝛼_𝑛𝑀=𝛽_𝑛𝑥_𝑛−𝑧²+1−𝛽_𝑛𝑇_𝜇𝐼−𝜇𝐵𝑥_𝑛−𝑇_𝜇𝐼−𝜇𝐵𝑧²+𝜆(𝜆−2𝛼)𝐴𝑢_𝑛−𝐴𝑧²+𝛼_𝑛𝑀≤𝛽_𝑛𝑥_𝑛−𝑧²+1−𝛽_𝑛𝑥_𝑛−𝑧²+𝜇𝜇−2𝛽𝐵𝑥_𝑛−𝐵𝑧²+𝜆(𝜆−2𝛼)𝐴𝑢_𝑛−𝐴𝑧²+𝛼_𝑛𝑀. (3.48) It follows that 1−𝛽_𝑛𝜆(2𝛼−𝜆)𝐴𝑢_𝑛−𝐴𝑧²+1−𝛽_𝑛𝜇2𝛽−𝜇𝐵𝑥_𝑛−𝐵𝑧²≤𝑥_𝑛−𝑧²−𝑥_𝑛+1−𝑧²+𝛼_𝑛𝑀≤𝑥_𝑛−𝑧+𝑥_𝑛+1−𝑧𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑀. (3.49) Since liminf_𝑛→∞(1−𝛽_𝑛)𝜆(2𝛼−𝜆)>0, liminf_𝑛→∞(1−𝛽_𝑛)𝜇(2𝛽−𝜇)>0, ‖𝑥_𝑛−𝑥_𝑛+1‖→0 and 𝛼_𝑛→0, we derive lim_𝑛→∞𝐴𝑢_𝑛−𝐴𝑧=lim_𝑛→∞𝐵𝑥_𝑛−𝐵𝑧=0. (3.50) From Lemma 2.1 and (2.2), we obtain 𝑢_𝑛−𝑧²=𝑇_𝜇𝑥_𝑛−𝜇𝐵𝑥_𝑛−𝑇_𝜇𝑧−𝜇𝐵𝑧²≤𝑥_𝑛−𝜇𝐵𝑥_𝑛−𝑧−𝜇𝐵𝑧,𝑢_𝑛=1−𝑧2𝑥_𝑛−𝜇𝐵𝑥_𝑛−𝑧−𝜇𝐵𝑧²+𝑢_𝑛−𝑧²−𝑥_𝑛−𝑢_𝑛−𝜇𝐵𝑥_𝑛−𝐵𝑧²≤12𝑥_𝑛−𝑧²+𝑢_𝑛−𝑧²−𝑥_𝑛−𝑢_𝑛−𝜇𝐵𝑥_𝑛−𝐵𝑧²=12𝑥_𝑛−𝑧²+𝑢_𝑛−𝑧²−𝑥_𝑛−𝑢_𝑛²+2𝜇𝑥_𝑛−𝑢_𝑛,𝐵𝑥_𝑛−𝐵𝑧−𝜇²𝐵𝑥_𝑛−𝐵𝑧². (3.51) It follows that 𝑢_𝑛−𝑧²≤𝑥_𝑛−𝑧²−𝑥_𝑛−𝑢_𝑛²+2𝜇𝑥_𝑛−𝑢_𝑛,𝐵𝑥_𝑛−𝐵𝑧−𝜇²𝐵𝑥_𝑛−𝐵𝑧²≤𝑥_𝑛−𝑧²−𝑥_𝑛−𝑢_𝑛²+2𝜇𝑥_𝑛−𝑢_𝑛𝐵𝑥_𝑛≤𝑥−𝐵𝑧_𝑛−𝑧²−𝑥_𝑛−𝑢_𝑛²+𝑀𝐵𝑥_𝑛−𝐵𝑧. (3.52) Again, by Lemma 2.1 and (2.2), we have 𝑧_𝑛−𝑧²=𝑃_𝐶1−𝛼_𝑛(𝐼−𝜆𝐴)𝑢_𝑛−𝑃_𝐶[(𝐼−𝜆𝐴)𝑧]²≤1−𝛼_𝑛(𝐼−𝜆𝐴)𝑢_𝑛−(𝐼−𝜆𝐴)𝑧,𝑧_𝑛=1−𝑧2(𝐼−𝜆𝐴)𝑢_𝑛−(𝐼−𝜆𝐴)𝑧−𝛼_𝑛𝑦_𝑛²+𝑧_𝑛−𝑧²−(𝐼−𝜆𝐴)𝑢_𝑛−(𝐼−𝜆𝐴)𝑧−𝑧_𝑛−𝑧−𝛼_𝑛𝑦_𝑛²≤12(𝐼−𝜆𝐴)𝑢_𝑛−(𝐼−𝜆𝐴)𝑧²+𝛼_𝑛𝑀+𝑧_𝑛−𝑧²−𝑢_𝑛−𝑧_𝑛−𝜆𝐴𝑢_𝑛−𝐴𝑧−𝛼_𝑛𝑦_𝑛²≤12𝑢_𝑛−𝑧²+𝛼_𝑛𝑀+𝑧_𝑛−𝑧²−𝑢_𝑛−𝑧_𝑛²+2𝜆𝑢_𝑛−𝑧_𝑛,𝐴𝑢_𝑛−𝐴𝑧+2𝛼_𝑛𝑦_𝑛,𝑢_𝑛−𝑧_𝑛−𝜆𝐴𝑢_𝑛−𝐴𝑧+𝛼_𝑛𝑦_𝑛²≤12𝑢_𝑛−𝑧²+𝛼_𝑛𝑀+𝑧_𝑛−𝑧²−𝑢_𝑛−𝑧_𝑛²+2𝜆𝑢_𝑛−𝑧_𝑛𝐴𝑢_𝑛−𝐴𝑧+2𝛼_𝑛𝑦_𝑛𝑢_𝑛−𝑧_𝑛, (3.53) that is, 𝑧_𝑛−𝑧²≤𝑢_𝑛−𝑧²+𝛼_𝑛𝑀−𝑢_𝑛−𝑧_𝑛²+2𝜆𝑢_𝑛−𝑧_𝑛𝐴𝑢_𝑛−𝐴𝑧+2𝛼_𝑛𝑦_𝑛𝑢_𝑛−𝑧_𝑛≤𝑥_𝑛−𝑧²−𝑥_𝑛−𝑢_𝑛²+𝑀𝐵𝑥_𝑛−𝐵𝑧+𝛼_𝑛𝑀−𝑢_𝑛−𝑧_𝑛²+2𝜆𝑢_𝑛−𝑧_𝑛𝐴𝑢_𝑛−𝐴𝑧+2𝛼_𝑛𝑦_𝑛𝑢_𝑛−𝑧_𝑛. (3.54) Hence, 𝑥_𝑛+1−𝑧²≤𝛽_𝑛𝑥_𝑛−𝑧²+1−𝛽_𝑛𝑧_𝑛−𝑧²≤𝛽_𝑛𝑥_𝑛−𝑧²+1−𝛽_𝑛𝑥_𝑛−𝑧²−𝑥_𝑛−𝑢_𝑛²+𝑀𝐵𝑥_𝑛−𝐵𝑧+𝛼_𝑛𝑀−𝑢_𝑛−𝑧_𝑛²+2𝜆𝑢_𝑛−𝑧_𝑛𝐴𝑢_𝑛−𝐴𝑧+2𝛼_𝑛𝑦_𝑛𝑢_𝑛−𝑧_𝑛≤𝑥_𝑛−𝑧²−1−𝛽_𝑛𝑥_𝑛−𝑢_𝑛²+𝐵𝑥_𝑛−𝐵𝑧+𝛼_𝑛𝑀−1−𝛽_𝑛𝑢_𝑛−𝑧_𝑛²+2𝜆𝑢_𝑛−𝑧_𝑛𝐴𝑢_𝑛−𝐴𝑧+2𝛼_𝑛𝑦_𝑛𝑢_𝑛−𝑧_𝑛. (3.55) It follows that 1−𝛽_𝑛𝑥_𝑛−𝑢_𝑛²+𝑢_𝑛−𝑧_𝑛²≤𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑧+𝑥_𝑛+1−𝑧+𝐵𝑥_𝑛−𝐵𝑧+𝛼_𝑛𝑀+2𝜆𝑢_𝑛−𝑧_𝑛𝐴𝑢_𝑛−𝐴𝑧+2𝛼_𝑛𝑦_𝑛𝑢_𝑛−𝑧_𝑛. (3.56) Since liminf_𝑛→∞(1−𝛽_𝑛)>0, 𝛼_𝑛→0, ‖𝑥_𝑛+1−𝑥_𝑛‖→0, ‖𝐴𝑢_𝑛−𝐴𝑧‖→0 and ‖𝐵𝑥_𝑛−𝐵𝑧‖→0, we derive that lim_𝑛→∞𝑥_𝑛−𝑢_𝑛=lim_𝑛→∞𝑢_𝑛−𝑧_𝑛=0. (3.57) Note that 𝑧_𝑛−𝑦_𝑛=𝑃_𝐶1−𝛼_𝑛𝑦_𝑛−𝑃_𝐶𝑦_𝑛≤𝛼_𝑛𝑦_𝑛⟶0, (3.58) therefore, 𝑆𝑦_𝑛−𝑦_𝑛≤𝑆𝑦_𝑛−𝑆𝑧_𝑛+𝑆𝑧_𝑛−𝑥_𝑛+𝑥_𝑛−𝑢_𝑛+𝑢_𝑛−𝑧_𝑛+𝑧_𝑛−𝑦_𝑛≤2𝑦_𝑛−𝑧_𝑛+𝑆𝑧_𝑛−𝑥_𝑛+𝑥_𝑛−𝑢_𝑛+𝑢_𝑛−𝑧_𝑛⟶0. (3.59) Next we prove limsup_𝑛→∞𝑥^∗,𝑥^∗−𝑦_𝑛≤0, (3.60) where 𝑥^∗=𝑃_Ω(0) is a solution of the minimization problem (3.1).
Indeed, we can choose a subsequence {𝑦_𝑛𝑖} of {𝑦_𝑛} such that limsup_𝑛→∞𝑥^∗,𝑥^∗−𝑦_𝑛=lim_𝑖→∞𝑥^∗,𝑥^∗−𝑦_𝑛𝑖. (3.61) Without loss of generality, we may further assume that 𝑦_𝑛𝑖→𝑥 weakly. By the same argument as that of Theorem 3.1, we can deduce that 𝑥∈Ω. Therefore, limsup_𝑛→∞𝑥^∗,𝑥^∗−𝑦_𝑛=𝑥^∗,𝑥^∗−𝑥≤0. (3.62) Finally, we prove 𝑥_𝑛→𝑥^∗. As a matter of fact, we have 𝑥_𝑛+1−𝑥^∗2=𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝑆𝑃_𝐶1−𝛼_𝑛𝑦_𝑛−𝑆𝑃_𝐶𝑥^∗2≤𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛1−𝛼_𝑛𝑦_𝑛−𝑥^∗−𝛼_𝑛𝑥^∗2=𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛1−𝛼_𝑛²𝑦_𝑛−𝑥^∗2−2𝛼_𝑛1−𝛼_𝑛𝑥^∗,𝑦_𝑛−𝑥^∗+𝛼²_𝑛𝑥^∗2≤𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛1−𝛼_𝑛𝑥_𝑛−𝑥^∗2−2𝛼_𝑛1−𝛼_𝑛𝑥^∗,𝑦_𝑛−𝑥^∗+𝛼²_𝑛𝑥^∗2≤1−21−𝛽_𝑛𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛1−𝛼_𝑛1−𝛽_𝑛𝑥^∗,𝑥^∗−𝑦_𝑛+1−𝛽_𝑛𝛼²_𝑛𝑀=1−𝛾_𝑛𝑥_𝑛−𝑥^∗2+𝛿_𝑛𝛾_𝑛, (3.63) where 𝛾_𝑛=2(1−𝛽_𝑛)𝛼_𝑛 and 𝛿_𝑛=(1−𝛼_𝑛)⟨𝑥^∗,𝑥^∗−𝑦_𝑛⟩+(𝛼_𝑛𝑀/2). 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^∗ strongly. This completes the proof.

Acknowledgments

Y. Yao was supported in part by NSFC 11071279 and NSFC 71161001-G0105. Y. Liou was supported in part by NSC 100-2221-E-230-012.