Hybrid Iterative Scheme by a Relaxed Extragradient Method for Equilibrium Problems, a General System of Variational Inequalities and Fixed-Point Problems of a Countable Family of Nonexpansive Mappings

Abstract

Based on the relaxed extragradient method and viscosity method, we introduce a new iterative method for finding a common element of solution of equilibrium problems, the solution set of a general system of variational inequalities, and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Furthermore, we prove the strong convergence theorem of the studied iterative method. The results of this paper extend and improve the results of Ceng et al., (2008), W. Kumam and P. Kumam, (2009), Yao et al., (2010) and many others.

1. Introduction

Let 𝐻 be a real Hilbert space with the inner product ⟨⋅,⋅⟩ and the norm ‖⋅‖. Let 𝐶 be a closed convex subset of 𝐻. Let 𝐹 be a bifunction of 𝐶×𝐶 into ℝ, where ℝ is the set of real numbers. The equilibrium problem for 𝐹∶𝐶×𝐶→ℝ is to find 𝑥∈𝐶 such that 𝐹(𝑥,𝑦)≥0,∀𝑦∈𝐶.(1.1) The set of solutions of (1.1) is denoted by EP(𝐹). The equilibrium problems covers, as special cases, monotone inclusion problems, saddle point problems, minimization problems, optimization problems, variational inequality problems, Nash equilibria in noncooperative games, and various forms of feasibility problems (see [1–4] and the references therein).

A mapping 𝐴∶𝐶→𝐻 is called 𝛼-inverse-strongly monotone if there exists a positive real number 𝛼 such that ⟨𝐴𝑢−𝐴𝑣,𝑢−𝑣⟩≥𝛼‖𝐴𝑢−𝐴𝑣‖²,∀𝑢,𝑣∈𝐶.(1.2) It is obvious that any 𝛼-inverse-strongly monotone mapping 𝐴 is monotone and Lipschitz continuous. A mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.3) We denote by 𝐹(𝑇) the set of fixed points of 𝑇. Recently, Wang and Guo [5] introduced an iterative scheme for a countable family of nonexpansive mappings.

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. For a given nonlinear operator 𝐴∶𝐶→𝐻, consider the following variational inequality problem of finding 𝑥^∗∈𝐶 such that ⟨𝐴𝑥^∗,𝑥−𝑥^∗⟩≥0,𝑥∈𝐶.(1.4) The set of solutions of the variational inequality (1.4) is denoted by VI(𝐶,𝐴) (see [6–9] and the references therein).

Let 𝐴,𝐵∶𝐶→𝐻 be two mappings. Consider the following problem of finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that ⟨⟨𝜇𝜆𝐴𝐵𝑦𝑥^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶,+𝑦^∗−𝑥^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐶,(1.5) which is called a general system of variational inequalities, where 𝜆>0 and 𝜇>0 are two constants. The set of solutions of (1.5) is denoted by GSVI(𝐴,𝐵,and𝐶). In particular, if 𝐴=𝐵, then problem (1.5) reduces to finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that ⟨⟨𝜇𝜆𝐴𝐴𝑦𝑥^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶,+𝑦^∗−𝑥^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐶,(1.6) which is defined by Verma [7] (see also [10]) and is called the new system of variational inequalities. Further, if we add up the requirement that 𝑥^∗=𝑦^∗, then problem (1.6) reduces to the classical variational inequality problem (1.4). Recently, Yao et al. [11] presented system of variational inequalities in Banach space. For solving problem (1.5), recently, Ceng et al. [12] introduced and studied a relaxed extragradient method. Based on the relaxed extragradient method and the viscosity approximation method, W. Kumam and P. Kumam [13] constructed a new viscosity-relaxed extragradient approximation method. Very recently, based on the extragradient method, Yao et al. [14] proposed an iterative method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real Hilbert space.

Motivated and inspired by the above works, in this paper, we introduce an iterative method based on the extragradient method and viscosity method for finding a common element of solution of equilibrium problems, the solution set of a general system of variational inequalities, and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Furthermore, we prove the strong convergence theorem of the proposed iterative method.

2. Preliminaries

Let 𝐶 be a closed convex subset of 𝐻, and let 𝑇∶𝐶→𝐶 be nonexpansive such that 𝐹(𝑇)≠∅. For all 𝑥̂∈𝐹(𝑇) and all 𝑥∈𝐶, we have ‖𝑥−𝑥̂‖²≥‖𝑇𝑥−𝑇𝑥̂‖²=‖𝑇𝑥−𝑥̂‖²=‖(𝑇𝑥−𝑥)+(𝑥−𝑥̂)‖²=‖𝑇𝑥−𝑥‖²+‖𝑥−𝑥̂‖²+2⟨𝑇𝑥−𝑥,𝑥−𝑥̂⟩,(2.1) and hence ‖𝑇𝑥−𝑥‖²≤2⟨𝑥−𝑇𝑥,𝑥−𝑥̂⟩,∀𝑥̂∈𝐹(𝑇),∀𝑥∈𝐶.(2.2)

Remark 2.1. Let A be 𝛼-inverse-strongly monotone. For all 𝑥,𝑦∈𝐶, 𝜆>0, we have ‖(𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦‖²=‖𝑥−𝑦‖²−2𝜆⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩+𝜆²‖𝐴𝑥−𝐴𝑦‖²≤‖𝑥−𝑦‖²+𝜆(𝜆−2𝛼)‖𝐴𝑥−𝐴𝑦‖².(2.3)So, if 𝜆≤2𝛼, then 𝐼−𝜆𝐴 is a nonexpansive mapping from 𝐶 to 𝐻.

Recall that the (nearest point) projection 𝑃_𝐶 from 𝐻 onto 𝐶 assigns to each 𝑥∈𝐻 the unique point 𝑃_𝐶𝑥∈𝐶 satisfying the property ‖‖𝑥−𝑃_𝐶𝑥‖‖=min_𝑦∈𝐶‖𝑥−𝑦‖.(2.4) The following characterizes the projection 𝑃_𝐶.

Lemma 2.2. Given that 𝑥∈𝐻 and 𝑦∈𝐶, then 𝑃_𝐶𝑥=𝑦 if and only if there holds the inequality ⟨𝑥−𝑦,𝑦−𝑧⟩≥0∀𝑧∈𝐶.(2.5)

Lemma 2.3 (see [15]). Let 𝐻 be a Hilbert space, 𝐶 a closed convex subset of 𝐻, and 𝑇∶𝐶→𝐶 a nonexpansive mapping with 𝐹(𝑇)≠∅. If {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥∈𝐶 and if {(1−𝑇)𝑥_𝑛} converges strongly to 𝑦, then(1−𝑇)𝑥=𝑦.

Lemma 2.4 (see [16]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝛼_𝑛+1≤1−𝛾_𝑛𝛼_𝑛+𝛿_𝑛,∀𝑛≥0,(2.6) where two sequences {𝛾_𝑛}⊂(0,1) and {𝛿_𝑛} satisfy(1)∑^∞𝑛=1𝛾_𝑛=∞; (2)limsup_𝑛→∞𝛿_𝑛/𝛾_𝑛≤0𝑜𝑟∑^∞𝑛=1|𝛿_𝑛|<∞. Then lim_𝑛→∞𝛼_𝑛=0.

Lemma 2.5 (see [12]). For given 𝑥^∗,𝑦^∗∈𝐶, (𝑥^∗,𝑦^∗) is a solution of problem (1.5) if and only if 𝑥^∗ is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺(𝑥)=𝑃_𝐶𝑃_𝐶(𝑥−𝜇𝐵𝑥)−𝜆𝐴𝑃_𝐶(𝑥−𝜇𝐵𝑥),∀𝑥∈𝐶,(2.7) where 𝑦^∗=𝑃_𝐶(𝑥^∗−𝜇𝐵𝑥^∗).

Note that the mapping 𝐺 is nonexpansive provided that 𝜆∈(0,2𝛼) and 𝜇∈(0,2𝛽).

Throughout this paper, the set of fixed points of the mapping 𝐺 is denoted by Γ.

Lemma 2.6 (see [1]). Let 𝐶 be a nonempty closed convex subset of 𝐻 and 𝐹∶𝐶×𝐶→ℝ satisfy following conditions: (A1)𝐹(𝑥,𝑥)=0, ∀𝑥∈𝐶; (A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0, ∀𝑥,𝑦∈𝐶;(A3)limsup_𝑡→0⁺𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦), ∀𝑥,𝑦,𝑧∈𝐶;(A4)for each 𝑥∈𝐶, 𝐹(𝑥,⋅) is convex and lower semicontinuous.For 𝑥∈𝐶 and 𝑟>0, set 𝑇^𝐹𝑟∶𝐻→𝐶 to be 𝑇^𝐹𝑟(𝑥)=𝑧∈𝐶∶𝐹(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶.(2.8) Then 𝑇^𝐹𝑟 is well defined and the following holds: (1)𝑇^𝐹𝑟 is single valued;(2)𝑇^𝐹𝑟 is firmly nonexpansive [17], that is, for any 𝑥,𝑦∈𝐸, ‖‖𝑇^𝐹𝑟𝑥−𝑇^𝐹𝑟𝑦‖‖²≤𝑇^𝐹𝑟𝑥−𝑇^𝐹𝑟𝑦,𝑥−𝑦;(2.9)(3)𝐹(𝑇^𝐹𝑟)=𝐸𝑃(𝐹); (4)𝐸𝑃(𝐹) is closed and convex.

By the proof of Lemma  5 in [2] (also see [3]), we have following lemma.

Lemma 2.7. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and 𝐹∶𝐶×𝐶→ℝ be a bifunction. Let 𝑥∈𝐶 and 𝑟₁,𝑟₂∈(0,∞). Then ‖‖𝑇_𝑟1𝑥−𝑇_𝑟2𝑥‖‖≤||||1−𝑟²𝑟₁||||‖‖𝑇_𝑟1𝑥‖‖+‖𝑥‖.(2.10)

3. Main Results

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let the mappings 𝐴,𝐵∶𝐶→𝐻 be 𝛼-inverse strongly monotone and 𝛽-inverse strongly monotone, respectively. Let 𝐹 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) and {𝑇_𝑛}^∞𝑛=1∶𝐶→𝐶 be a countable family of nonexpansive mappings such that Ω∶=⋂^∞𝑛=1𝐹(𝑇_𝑛)∩𝐸𝑃(𝐹)∩Γ≠∅. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1/2). Set 𝛽₀=1. For given 𝑥₁∈𝐶 arbitrarily, let the sequences {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, and {𝑢_𝑛} be generated by 𝐹𝑢_𝑛,𝑦+1𝑟_𝑛𝑧⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝑦_𝑛=𝑃_𝐶𝑢_𝑛−𝜇𝐵𝑢_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑧_𝑛−𝜆𝐴𝑧_𝑛,=𝛽_𝑛𝑥_𝑛+𝜎_𝑛𝑛_𝑖=1𝛽_𝑖−1−𝛽_𝑖𝑇_𝑖𝑦_𝑛+1−𝛽_𝑛1−𝜎_𝑛𝑃_𝐶𝑧_𝑛−𝜆𝐴𝑧_𝑛,(3.1) where 𝜆∈(0,2𝛼), 𝜇∈(0,2𝛽), and sequences {𝛼_𝑛}⊂[0,1], {𝛽_𝑛}⊂[0,1], {𝜎_𝑛}⊂[0,1], and {𝑟_𝑛}⊂(𝑟,∞), 𝑟>0, are such that
(i){𝛽_𝑛} is strictly decreasing, (ii)0<liminf_𝑛→∞𝛽_n≤limsup_𝑛→∞𝛽_𝑛<1, (iii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞,(iv)𝜎_𝑛>1/2(1−𝜌),∑^∞𝑛=1|𝜎_𝑛−𝜎_𝑛−1|<∞, (v)∑^∞𝑛=1|𝑟_𝑛−𝑟_𝑛−1|<∞.Then the sequence {𝑥_𝑛} generated by (3.1) converges strongly to 𝑥^∗=𝑃_Ω⋅𝑓(𝑥^∗), and (𝑥^∗,𝑦^∗) is a solution of the general system of variational inequalities (1.5), where 𝑦^∗=𝑃_𝐶(𝑥^∗−𝜇𝐵𝑥^∗).

Proof. The proof is divided into several steps.
Step 1. The sequence {𝑥_𝑛} defined by (3.1) is bounded.
For each 𝑥^∗∈Ω, from Lemma 2.6, we have 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛 and hence ‖‖𝑢_𝑛−𝑥^∗‖‖=‖‖𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑥^∗‖‖≤‖‖𝑥_𝑛−𝑥^∗‖‖.(3.2) Since 𝑓 is a 𝜌-contraction mapping, using (2.3) and (3.2), we have ‖‖𝑧_𝑛−𝑦^∗‖‖²=‖‖𝑃_𝐶𝑢_𝑛−𝜇𝐵𝑢_𝑛−𝑃_𝐶𝑥^∗−𝜇𝐵𝑥^∗‖‖²≤‖‖𝑢_𝑛−𝜇𝐵𝑢_𝑛−𝑥^∗−𝜇𝐵𝑥^∗‖‖²≤‖‖𝑢_𝑛−𝑥^∗‖‖²+𝜇(𝜇−2𝛽)‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜇(𝜇−2𝛽)‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖².(3.3) Set 𝑣_𝑛=𝑃_𝐶(𝑧_𝑛−𝜆𝐴𝑧_𝑛). Since 𝑃_𝐶 is nonexpansive, from (2.3), we have ‖‖𝑣_𝑛−𝑥^∗‖‖²=‖‖𝑃_𝐶𝑧_𝑛−𝜆𝐴𝑧_𝑛−𝑃_𝐶𝑦^∗−𝜆𝐴𝑦^∗‖‖²≤‖‖𝑧_𝑛−𝑦^∗‖‖²+𝜆(𝜆−2𝛼)‖‖𝐴𝑧_𝑛−𝐴𝑦^∗‖‖²≤‖‖𝑧_𝑛−𝑦^∗‖‖².(3.4) Hence we get ‖‖𝑦_𝑛−𝑥^∗‖‖≤=𝛼‖‖𝛼_𝑛𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑣_𝑛−𝑥^∗‖‖≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖+1−𝛼_𝑛‖‖𝑧_𝑛−𝑦^∗‖‖≤1−‖‖𝑓𝛼_𝑛𝑥_𝑛−𝑓𝑥^∗‖‖+‖‖𝑓𝑥^∗−𝑥^∗‖‖+1−𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖(1−𝜌)‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛‖‖𝑓𝑥^∗−𝑥^∗‖‖.(3.5) From (3.1) and (3.3)–(3.5), we get ‖‖𝑥_𝑛+1−𝑥^∗‖‖=‖‖‖‖𝛽_𝑛𝑥_𝑛+𝜎_𝑛𝑛_𝑖=1𝛽_𝑖−1−𝛽_𝑖𝑇_𝑖𝑦_𝑛+1−𝛽_𝑛1−𝜎_𝑛𝑣_𝑛−𝑥^∗≤𝛽_𝑛‖‖‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+𝜎_𝑛𝑛_𝑖=1𝛽_𝑖−1≤𝛽_𝑛−𝛽_𝑖‖‖𝑇_𝑖𝑦_𝑛−𝑥^∗‖‖+1−𝛽_𝑛1−𝜎_𝑛‖‖𝑣_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+𝜎_𝑛𝑛_𝑖=1𝛽_𝑖−1≤1−𝜎_𝑛−𝛽_𝑖‖‖𝑦_𝑛−𝑥^∗‖‖+1−𝛽_𝑛1−𝜎_𝑛‖‖𝑧_𝑛−𝑦^∗‖‖≤1−𝜎_𝑛1−𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+𝜎_𝑛1−𝛽_𝑛‖‖𝑦_𝑛−𝑦^∗‖‖+𝜎_𝑛1−𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖≤1−𝜎1_𝑛−𝛽_𝑛1−𝛼_𝑛(1−𝜌)‖‖𝑥_𝑛−𝑥^∗‖‖+𝛼_𝑛‖‖𝑓𝑥^∗−𝑥^∗‖‖+𝜎_𝑛1−𝛽_𝑛𝛼_𝑛(1−𝜌)‖‖𝑥_𝑛−𝑥^∗‖‖≤𝑀,1−𝛽_𝑛𝛼_𝑛(1−𝜌)1−1𝜌‖‖𝑓𝑥^∗−𝑥^∗‖‖(3.6) where 𝑀=max{‖𝑥₁−𝑥^∗‖,1/(1−𝜌)‖𝑓(𝑥^∗)−𝑥^∗‖}. Hence, {𝑥_𝑛} is bounded and therefore {𝑢_𝑛}, {𝑧_𝑛}, {𝑣_𝑛}, and {𝑦_𝑛} are also bounded.Step 2. lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. Since 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛 and 𝑢_𝑛−1=𝑇_𝑟𝑛−1𝑥_𝑛−1, using Lemma 2.7, we have ‖‖𝑢_𝑛−𝑢_𝑛−1‖‖=≤‖‖‖‖𝑇𝑇_{𝑟𝑛𝑟𝑛}𝑥_𝑛−𝑇_𝑟𝑛−1𝑥_𝑛−1‖‖≤‖‖𝑥_𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑥_𝑛−1‖‖+‖‖𝑇_𝑟𝑛𝑥_𝑛−1−𝑇_𝑟𝑛−1𝑥_𝑛−1‖‖−𝑥_𝑛−1‖‖+||||1−𝑟^𝑛𝑟_𝑛−1||||‖‖𝑢_𝑛−1≤‖‖𝑥_𝑛‖‖+‖‖𝑥_𝑛−1‖‖−𝑥_𝑛−1‖‖+1𝑟||𝑟^𝑛−1−𝑟_𝑛||𝐿,(3.7) where 𝐿=sup{‖𝑢_𝑛‖+‖𝑥_𝑛‖∶𝑛=1,2,…}. From (3.1) and (3.7), it follows that ‖‖𝑧_𝑛−𝑧_𝑛−1‖‖≤=‖‖‖‖𝑃𝑢_𝐶𝑛(𝐼−𝜇𝐵)𝑢_𝑛−𝑃_𝐶(𝐼−𝜇𝐵)𝑢_𝑛−1‖‖≤‖‖𝑥_𝑛−𝑢_𝑛−1‖‖−𝑥_𝑛−1‖‖+1𝑟||𝑟^𝑛−1−𝑟_𝑛||𝐿.(3.8) From (3.1) and (3.8), we have ‖‖𝑦_𝑛−𝑦_𝑛−1‖‖≤≤𝛼𝛼_𝑛𝑛‖‖𝑓𝑥_𝑛−𝑓𝑥_𝑛−1‖‖+1−𝛼_𝑛‖‖𝑃_𝐶(𝐼−𝜆𝐴)𝑧_𝑛−𝑃_𝐶(𝐼−𝜆𝐴)𝑧_𝑛−1‖‖≤𝛼_𝑛𝜌‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+1−𝛼_𝑛‖‖𝑧_𝑛−𝑧_𝑛−1‖‖𝜌‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+1−𝛼_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+1𝑟||𝑟^𝑛−1≤1−𝛼_𝑛−𝑟_𝑛||𝐿(1−𝜌)‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+1−𝛼_𝑛1𝑟||𝑟^𝑛−1−𝑟_𝑛||𝐿.(3.9) By definition of scheme (3.1), we have 𝑥_𝑛+1−𝑥_𝑛=𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1+𝜎_𝑛𝑛_𝑖=1𝛽_𝑖−1+𝜎_𝑛−𝛽_𝑖𝑇_𝑖𝑦_𝑛−𝑇_𝑖𝑦_𝑛−1−𝜎_𝑛−1^𝑛−1_𝑖=1𝛽_𝑖−1−𝛽_𝑖𝑇_𝑖𝑦_𝑛−1+𝜎_𝑛𝛽_𝑛−1−𝛽_𝑛𝑇_𝑛𝑦_𝑛−1−+11−−𝛽𝛽_𝑛_𝑛−11−𝜎_𝑛𝑃_𝐶𝑧_𝑛−𝜆𝐴𝑧_𝑛−𝑃_𝐶𝑧_𝑛−1−𝜆𝐴𝑧_𝑛−11−𝜎_𝑛−1𝑃_𝐶𝑧_𝑛−1−𝜆𝐴𝑧_𝑛−1+1−𝛽_𝑛1−𝜎_𝑛𝑃_𝐶𝑧_𝑛−1−𝜆𝐴𝑧_𝑛−1.(3.10) Thus, from (3.8)–(3.10), ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤𝛽_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝛽_𝑛−1−𝛽_𝑛‖‖𝑥_𝑛−1‖‖+𝜎_𝑛𝑛_𝑖=1𝛽_𝑖−1+||𝜎_𝑛−𝛽_𝑖‖‖𝑇_𝑖𝑦_𝑛−𝑇_𝑖𝑦_𝑛−1‖‖−𝜎_𝑛−1||^𝑛−1_𝑖=1𝛽_𝑖−1+1−𝛽_𝑛−𝛽_𝑖‖‖𝑇_𝑖𝑦_𝑛−1‖‖+𝜎_𝑛𝛽_𝑛−1−𝛽_𝑛‖‖𝑇_𝑛𝑦_𝑛−1‖‖+||1−𝛽_𝑛1−𝜎_𝑛‖‖𝑃_𝐶𝑧_𝑛−𝜆𝐴𝑧_𝑛−𝑃_𝐶𝑧_𝑛−1−𝜆𝐴𝑧_𝑛−1‖‖≤𝛽_𝑛1−𝜎_𝑛−1−𝛽_𝑛−11−𝜎_𝑛−1||‖‖𝑃_𝐶𝑧_𝑛−1−𝜆𝐴𝑧_𝑛−1‖‖‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝛽_𝑛−1−𝛽_𝑛‖‖𝑥_𝑛−1‖‖+𝜎_𝑛𝑛_𝑖=1𝛽_𝑖−1+||𝜎_𝑛−𝛽_𝑖‖‖𝑦_𝑛−𝑦_𝑛−1‖‖−𝜎_𝑛−1||^𝑛−1_𝑖=1𝛽_𝑖−1+1−𝛽_𝑛−𝛽_𝑖‖‖𝑇_𝑖𝑦_𝑛−1‖‖+𝛽_𝑛−1−𝛽_𝑛‖‖𝑇_𝑛𝑦_𝑛−1‖‖+1−𝛽_𝑛−11−𝜎_𝑛‖‖𝑧_𝑛−𝑧_𝑛−1‖‖+𝛽_𝑛−1−𝛽_𝑛1−𝜎_𝑛‖‖𝑃_𝐶(𝐼−𝜆𝐴)𝑧_𝑛−1‖‖≤𝛽_𝑛||𝜎_𝑛−𝜎_𝑛−1||‖‖𝑃_𝐶(𝐼−𝜆𝐴)𝑧_𝑛−1‖‖×‖‖𝑥1_𝑛−−𝛼𝑥_𝑛−1𝑛‖‖+𝛽_𝑛−1−𝛽_𝑛‖‖𝑥_𝑛−1‖‖+𝜎_𝑛1−𝛽_𝑛(1−𝜌)‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+1−𝛼_𝑛1𝑟||𝑟^𝑛−1+||𝜎_𝑛−𝑟_𝑛||𝐿−𝜎_𝑛−1||^𝑛−1_𝑖=1𝛽_𝑖−1+𝛽_𝑛−1−𝛽_𝑖‖‖𝑇_𝑖𝑦_𝑛−1‖‖−𝛽_𝑛‖‖𝑇_𝑛𝑦_𝑛−1‖‖+1−𝛽_𝑛1−𝜎_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+1𝑟||𝑟^𝑛−1+𝛽_𝑛−1−𝑟_𝑛||𝐿≤1−𝜎_𝑛−𝛽_𝑛‖‖𝑃_𝐶(𝐼−𝜆𝐴)𝑧_𝑛−1‖‖+||𝜎_𝑛−𝜎_𝑛−1||‖‖𝑃_𝐶(𝐼−𝜆𝐴)𝑧_𝑛−1‖‖+11−𝛽_𝑛𝛼_𝑛(1−𝜌)‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝛽_𝑛−1−𝛽_𝑛𝑀𝑟||𝑟^𝑛−1−𝑟_𝑛||𝐿+||𝜎_𝑛−𝜎_𝑛−1||^𝑛−1_𝑖=1𝛽_𝑖−1+𝛽_𝑛−1−𝛽_𝑖𝑀+𝛽_𝑛−1−𝛽_𝑛𝑀≤1−𝜎_𝑛−𝛽_𝑛+||𝜎_𝑛−𝜎_𝑛−1||𝑀+3𝛽1_𝑛−1−𝛽_𝑛𝛼_𝑛(1−𝜌)‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+1𝑟||𝑟^𝑛−1−𝑟_𝑛||𝐿−𝛽_𝑛𝑀+2||𝜎_𝑛−𝜎_𝑛−1||𝑀,(3.11) where 𝑀=max{sup_𝑛≥1‖𝑥_𝑛‖,sup_{𝑖≥1,𝑛≥1}‖𝑇_𝑖𝑦_𝑛‖,sup_𝑛≥1‖𝑃_𝐶(𝐼−𝜆𝐴)𝑧_𝑛‖}. Since {𝛽_𝑛} is strictly decreasing, we have ∑^∞𝑛=2(𝛽_𝑛−1−𝛽_𝑛)=𝛽₁<∞. Further, by assumption conditions (iv)-(v), we have ^∞_𝑛=21𝑟||𝑟^𝑛+1−𝑟_𝑛||𝐿+3𝛽_𝑛−1−𝛽_𝑛𝑀+2||𝜎_𝑛−𝜎_𝑛−1||𝑀<∞.(3.12) Thus, using Lemma 2.4, we have lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.Step 3. lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0.
For any 𝑥^∗∈Ω, it follows from Lemma 2.6 that ‖‖𝑢_𝑛−𝑥^∗‖‖²==‖‖1𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑥^∗‖‖²≤𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑥^∗,𝑥_𝑛−𝑥^∗=⟨𝑢_𝑛−𝑥^∗,𝑥_𝑛−𝑥^∗⟩2‖‖𝑢^𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑢_𝑛−𝑥_𝑛‖‖².(3.13) From (2.3), (3.1)–(3.4), we can get ‖‖𝑦_𝑛−𝑥^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛‖‖𝑣_𝑛−𝑥^∗‖‖²≤≤𝛼𝛼_𝑛𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛‖‖𝑧_𝑛−𝑦^∗‖‖²+𝜆(𝜆−2𝛼)‖‖𝐴𝑧_𝑛−𝐴𝑦^∗‖‖²‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛‖‖𝑧_𝑛−𝑦^∗‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛‖‖𝑢_𝑛−𝑥^∗‖‖².(3.14) From (3.1), (3.3)-(3.4), and (3.13)-(3.14), it follows that ‖‖𝑥_𝑛+1−𝑥^∗‖‖²=‖‖‖‖𝛽_𝑛𝑥_𝑛+𝜎_𝑛𝑛_𝑖=1𝛽_𝑖−1−𝛽_𝑖𝑇_𝑖𝑦_𝑛+1−𝛽_𝑛1−𝜎_𝑛𝑣_𝑛−𝑥^∗‖‖‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛‖‖‖‖𝜎_𝑛11−𝛽_{𝑛𝑛𝑖=1}𝛽_𝑖−1−𝛽_𝑖𝑇_𝑖𝑦_𝑛+1−𝜎_𝑛𝑣_𝑛−𝑥^∗‖‖‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛𝜎_𝑛‖‖‖‖11−𝛽_{𝑛𝑛𝑖=1}𝛽_𝑖−1−𝛽_𝑖𝑇_𝑖𝑦_𝑛−𝑥^∗‖‖‖‖²+1−𝛽_𝑛1−𝜎_𝑛‖‖𝑣_𝑛−𝑥^∗‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛𝜎_𝑛11−𝛽_{𝑛𝑛𝑖=1}𝛽_𝑖−1−𝛽_𝑖‖‖𝑇_𝑖𝑦_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛1−𝜎_𝑛‖‖𝑣_𝑛−𝑥^∗‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛𝜎_𝑛‖‖𝑦_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛1−𝜎_𝑛‖‖𝑧_𝑛−𝑦^∗‖‖²≤+𝛽_𝑛‖‖1𝑥−^𝑛−𝛽_𝑛𝑥^∗‖‖²+1−𝛽_𝑛𝜎_𝑛1−𝛼^𝑛‖‖𝑢_𝑛−𝑥^∗‖‖²+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²1−𝜎_𝑛‖‖𝑢_𝑛−𝑥^∗‖‖²≤−‖‖𝑥^𝑛1−−𝑥𝛽^∗_𝑛‖‖²+1−𝛽_𝑛𝜎_𝑛𝛼_𝑛‖‖𝑓𝑥^𝑛−𝑥^∗‖‖²−‖‖𝑥_𝑛−𝑥^∗‖‖²1−𝜎_𝑛𝛼_𝑛‖‖𝑢_𝑛−𝑥_𝑛‖‖²(3.15) which implies that 1−𝛽_𝑛1−𝜎_𝑛𝛼_𝑛‖‖𝑢_𝑛−𝑥_𝑛‖‖²≤+‖‖𝛼𝑥_𝑛𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑓𝑥_𝑛−𝑥^∗‖‖².(3.16) Since {𝑥_𝑛} is both bounded, using Step 2 and conditions (ii)–(iv), we conclude the result.Step 4. lim_𝑛→∞‖𝐴𝑧_𝑛−𝐴𝑦^∗‖=0 and lim_𝑛→∞‖𝐵𝑥_𝑛−𝐵𝑥^∗‖=0.Using (3.3), (3.14), and (3.15), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛𝜎_𝑛‖‖𝑦_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛1−𝜎_𝑛‖‖𝑧_𝑛−𝑦^∗‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛𝜎_𝑛×+1𝛼_𝑛−‖‖𝑓𝛽_𝑛𝑥_𝑛−𝑥^∗‖‖²+1−𝛼_𝑛‖‖𝑧_𝑛−𝑦^∗‖‖²+𝜆(𝜆−2𝛼)‖‖𝐴𝑧_𝑛−𝐴𝑦^∗‖‖²1−𝜎_𝑛‖‖𝑧_𝑛−𝑦^∗‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛𝜎_𝑛1−𝛼_𝑛𝜆(𝜆−2𝛼)‖‖𝐴𝑧_𝑛−𝐴𝑦^∗‖‖²+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛‖‖𝑧_𝑛−𝑦^∗‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜆(𝜆−2𝛼)‖‖𝐴𝑧_𝑛−𝐴𝑦^∗‖‖²≤+‖‖𝑥𝛼_𝑛𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜇(𝜇−2𝛽)‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖²−𝑥^∗‖‖²+𝜆(𝜆−2𝛼)‖‖𝐴𝑧_𝑛−𝐴𝑦^∗‖‖²+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+𝜇(𝜇−2𝛽)‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖².(3.17) Therefore, 𝜆(2𝛼−𝜆)‖‖𝐴𝑧_𝑛−𝐴𝑦^∗‖‖²+𝜇(2𝛽−𝜇)‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑦^∗‖‖².(3.18) From Step 2, using condition (iii), we get lim_𝑛→∞‖𝐴𝑧_𝑛−𝐴𝑦^∗‖=0 and lim_𝑛→∞‖𝐵𝑢_𝑛−𝐵𝑥^∗‖=0. From the fact that the 𝐵 is 𝛽-inverse strongly monotone operator, it follows that ‖‖𝐵𝑥_𝑛−𝐵𝑥^∗‖‖≤‖‖𝐵𝑥_𝑛−𝐵𝑢_𝑛‖‖+‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖≤1𝛽‖‖𝑥^𝑛−𝑢_𝑛‖‖+‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖.(3.19) Applying Step 3, we have lim_𝑛→∞‖𝐵𝑥_𝑛−𝐵𝑥^∗‖=0.Step 5. lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0,𝑓𝑜𝑟𝑎𝑙𝑙𝑖=1,2,….
Noting that 𝑃_𝐶 is firmly nonexpansive, we have‖‖𝑧_𝑛−𝑦^∗‖‖²=‖‖𝑃_𝐶𝑢_𝑛−𝜇𝐵𝑢_𝑛−𝑃_𝐶𝑥^∗−𝜇𝐵𝑥^∗‖‖²=≤1𝑢_𝑛−𝜇𝐵𝑢_𝑛−𝑥^∗−𝜇𝐵𝑥^∗,𝑧_𝑛−𝑦^∗2‖‖𝑢^𝑛−𝑥^∗−𝜇𝐵𝑢_𝑛−𝐵𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑦^∗‖‖²≤1−‖‖𝑢_𝑛−𝑧_𝑛−𝜇𝐵𝑢_𝑛−𝐵𝑥^∗−𝑥^∗−𝑦^∗‖‖²2‖‖𝑢^𝑛−𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑦^∗‖‖²−‖‖𝑢_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖²≤1+2𝜇𝑢_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗,𝐵𝑢_𝑛−𝐵𝑥^∗−𝜇²‖‖𝐵𝑢_𝑛−𝐵𝑥^∗‖‖²2‖‖𝑥^𝑛−𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑦^∗‖‖²−‖‖𝑢_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖²‖‖𝑣_𝑛+2𝜇𝑢_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗,𝐵𝑢_𝑛−𝐵𝑥^∗,−𝑥^∗‖‖²=‖‖𝑃_𝐶𝑧_𝑛−𝜆𝐴𝑧_𝑛−𝑃_𝐶𝑦^∗−𝜆𝐴𝑦^∗‖‖²≤=1𝑧_𝑛−𝜆𝐴𝑧_𝑛−𝑦^∗−𝜆𝐴𝑦^∗,𝑣_𝑛−𝑥^∗2‖‖𝑧^𝑛−𝑦^∗−𝜆𝐴𝑧_𝑛−𝐴𝑦^∗‖‖²+‖‖𝑣_𝑛−𝑥^∗‖‖²≤1−‖‖𝑧_𝑛−𝑣_𝑛−𝜆𝐴𝑧_𝑛−𝐴𝑦^∗−𝑦^∗−𝑥^∗‖‖²2‖‖𝑧^𝑛−𝑦^∗‖‖²+‖‖𝑣_𝑛−𝑥^∗‖‖²−‖‖𝑧_𝑛−𝑣_𝑛−𝑦^∗−𝑥^∗‖‖²≤1+2𝜆𝑧_𝑛−𝑣_𝑛−𝑦^∗−𝑥^∗,𝐴𝑧_𝑛−𝐴𝑦^∗−𝜆²‖‖𝐴𝑧_𝑛−𝐴𝑦^∗‖‖²2‖‖𝑥^𝑛−𝑥^∗‖‖²+‖‖𝑣_𝑛−𝑥^∗‖‖²−‖‖𝑧_𝑛−𝑣_𝑛−𝑦^∗−𝑥^∗‖‖²+2𝜆𝑧_𝑛−𝑣_𝑛−𝑦^∗−𝑥^∗𝐴𝑧_𝑛−𝐴𝑦^∗.(3.20) It follows that ‖‖𝑧_𝑛−𝑦^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑢_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖²‖‖𝑣+2𝜇𝑢_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗,𝐵𝑢_𝑛−𝐵𝑥^∗,(3.21)_𝑛−𝑥^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑧_𝑛−𝑣_𝑛−𝑦^∗−𝑥^∗‖‖²+2𝜆𝑧_𝑛−𝑣_𝑛−𝑦^∗−𝑥^∗,𝐴𝑧_𝑛−𝐴𝑦^∗.(3.22) By (3.3), (3.14)-(3.15), and (3.22), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1−1−𝛽_𝑛𝜎_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛𝜎_𝑛‖‖𝑦_𝑛−𝑥^∗‖‖²≤≤11−−11−−𝛽𝛽^𝑛_𝑛𝜎_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛𝜎_𝑛𝛼^𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑣_𝑛−𝑥^∗‖‖²𝜎_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛𝜎_𝑛𝛼^𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑧_𝑛−𝑣_𝑛−𝑦^∗−𝑥^∗‖‖²=+‖‖2𝑥𝜆_𝑛𝑧_𝑛−𝑣_𝑛−𝑦^∗−𝑥^∗𝐴𝑧_𝑛−𝐴𝑦^∗×−−‖‖𝑥𝑧^∗_𝑛‖‖²+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛1−𝜎_𝑛−𝑣_𝑛−𝑦^∗−𝑥^∗‖‖²+2𝜆𝑧_𝑛−𝑣_𝑛−𝑦^∗−𝑥^∗,𝐴𝑧_𝑛−𝐴𝑦^∗.(3.23) It follows that 1−𝛽_𝑛𝜎_𝑛‖‖𝑣_𝑛−𝑧−𝑥^∗−𝑦^∗‖‖²≤‖‖𝑥_𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+2𝜆1−𝛽_𝑛𝜎_𝑛𝑧_𝑛−𝑣_𝑛−𝑦^∗−𝑥^∗,𝐴𝑧_𝑛−𝐴𝑦^∗.(3.24) From conditions (ii)–(iv), Steps 2 and 4, we get the following: lim_𝑛⟶∞‖‖𝑣_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖=0.(3.25) On the other hand, from (3.14)–(3.21), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤+𝛽_𝑛‖‖1𝑥−^𝑛−𝛽_𝑛𝑥^∗‖‖²+1−𝛽_𝑛𝜎_𝑛𝛼^𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑧_𝑛−𝑦^∗‖‖²1−𝜎_𝑛‖‖𝑧_𝑛−𝑦^∗‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛‖‖𝑧_𝑛−𝑦^∗‖‖²≤×𝛽_𝑛‖‖‖‖𝑥𝑥_𝑛𝑛−𝑥^∗‖‖²+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛=‖‖𝑥_𝑛−𝑥^∗‖‖²−‖‖𝑢_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖²+2𝜇𝑢_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗,𝐵𝑢_𝑛−𝐵𝑥^∗−𝑥^∗‖‖²+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖²+1−𝛽_𝑛−‖‖𝑢_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖²+2𝜇𝑢_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗,𝐵𝑢_𝑛−𝐵𝑥^∗,(3.26) which implies that ‖‖1−𝛽_𝑛𝑢_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖²≤+‖‖𝛼𝑥_𝑛𝑛−𝑥^∗‖‖+‖‖𝑥_𝑛+1−𝑥^∗‖‖‖‖𝑥_𝑛−𝑥_𝑛+1‖‖×𝑢‖‖_𝑛𝑓𝑥_𝑛−𝑥^∗‖‖²+2𝜇1−𝛽_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗,𝐵𝑢_𝑛−𝐵𝑥^∗.(3.27) From (ii)-(iii), Steps 2 and 4, we get the following: lim_𝑛⟶∞‖‖𝑢_𝑛−𝑧_𝑛−𝑥^∗−𝑦^∗‖‖=0.(3.28) Combining (3.25) and (3.28), we get the following: lim_𝑛⟶∞‖‖𝑢_𝑛−𝑣_𝑛‖‖=0.(3.29) Using Step 3, we obtain the following: lim_𝑛⟶∞‖‖𝑥_𝑛−𝑣_𝑛‖‖=0.(3.30) This together with ‖𝑦_𝑛−𝑣_𝑛‖=𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑣_𝑛‖→0 implies that lim_𝑛⟶∞‖‖𝑥_𝑛−𝑦_𝑛‖‖=0.(3.31) For any 𝑥^∗∈Ω, we have from (3.1) that 𝜎_𝑛𝑛_𝑖=1𝛽_𝑖−1=𝜎_𝑛𝑛−𝛽_𝑖⟨𝑥_𝑛−𝑇_𝑖𝑥_𝑛,𝑥_𝑛−𝑥^∗⟩+⟨𝑇_𝑖𝑥_𝑛−𝑇_𝑖𝑦_𝑛,𝑥_𝑛−𝑥^∗⟩_𝑖=1𝛽_𝑖−1=𝑥_𝑛−𝛽_𝑖⟨𝑥_𝑛−𝑇_𝑖𝑦_𝑛,𝑥_𝑛−𝑥^∗⟩−𝑥_𝑛+1,𝑥_𝑛−𝑥^∗+1−𝛽_𝑛1−𝜎_𝑛⟨𝑣_𝑛−𝑥_𝑛,𝑥_𝑛−𝑥^∗⟩.(3.32) Since each 𝑇_𝑖 is nonexpansive, from (2.2), we have ‖‖𝑇_𝑖𝑥_𝑛−𝑥_𝑛‖‖²≤2⟨𝑥_𝑛−𝑇_𝑖𝑥_𝑛,𝑥_𝑛−𝑥^∗⟩.(3.33) Hence, combining this inequality with (3.32), we get the following: 12𝜎^𝑛𝑛_𝑖=1𝛽_𝑖−1−𝛽_𝑖‖‖𝑇_𝑖𝑥_𝑛−𝑥_𝑛‖‖²≤−𝜎_𝑛𝑛_𝑖=1𝛽_𝑖−1+𝑥_𝑛−𝛽_𝑖⟨𝑇_𝑖𝑥_𝑛−𝑇_𝑖𝑦_𝑛,𝑥_𝑛−𝑥^∗⟩×⟨𝑣_𝑛−𝑥_𝑛+1,𝑥_𝑛−𝑥^∗+1−𝛽_𝑛1−𝜎_𝑛−𝑥_𝑛,𝑥_𝑛−𝑥^∗⟩,(3.34) that is (noting that 𝛽_𝑛 is strictly decreasing), ‖‖𝑇_𝑖𝑥_𝑛−𝑥_𝑛‖‖²≤21−𝛽^𝑛𝛽_𝑖−1−𝛽_𝑖‖‖𝑥_𝑛−𝑦_𝑛‖‖‖‖𝑥_𝑛−𝑥^∗‖‖+2𝜎_𝑛𝛽_𝑖−1−𝛽_𝑖‖‖𝑥^𝑛+21−𝛽_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑥^∗‖‖1−𝜎_𝑛𝜎_𝑛𝛽_𝑖−1−𝛽_𝑖‖‖𝑣^𝑛−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑥^∗‖‖.(3.35) Now from (3.30)-(3.31) and Step 2, we conclude that lim_𝑛⟶∞‖‖𝑇_𝑖𝑥_𝑛−𝑥_𝑛‖‖=0,∀𝑖=1,2,…,(3.36) which completes the proof.Step 6. limsup_𝑛→∞⟨𝑓(𝑥^∗)−𝑥^∗,𝑥_𝑛−𝑥^∗⟩≤0, where 𝑥^∗=𝑃_Ω⋅𝑓(𝑥^∗).
As {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that {𝑥_𝑛𝑖}→𝑥̃ weakly. First, it is clear from Step 5 and Lemma 2.3 that 𝑥̃∈⋂^∞𝑛=1𝐹(𝑇_𝑛). Next, we prove that 𝑥̃∈Γ. From (iii), Step 3 and (3.31), we note that ‖‖𝑦_𝑛−𝐺𝑦_𝑛‖‖≤+𝛼_𝑛‖‖1𝑓−𝛼𝑥_𝑛𝑛−𝐺𝑦_𝑛‖‖=𝛼_𝑛‖‖𝑃_𝐶𝑃_C𝑢_𝑛−𝜇𝐵𝑢_𝑛−𝜆𝐴𝑃_𝐶𝑢_𝑛−𝜇𝐵𝑢_𝑛−𝐺𝑦_𝑛‖‖≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝐺𝑦_𝑛‖‖+1−𝛼_𝑛‖‖𝐺𝑢_𝑛−𝐺𝑦_𝑛‖‖⟶0‖‖.𝑓𝑥_𝑛−𝐺𝑦_𝑛‖‖+1−𝛼_𝑛‖‖𝑢_𝑛−𝑦_𝑛‖‖(3.37) According to (3.31) and Lemma 2.3, we obtain that 𝑥̃∈Γ. Next, we show that 𝑥̃∈EP(𝐹). Indeed, by 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛, we have 𝐹𝑢_𝑛,𝑦+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶.(3.38) From (A2), we have also 1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥−𝐹𝑢_𝑛,𝑦≥𝐹𝑦,𝑢_𝑛,(3.39) and hence, 𝑦−𝑢^𝑛𝑘,𝑢^𝑛𝑘−𝑥_𝑛𝑘𝑟_𝑛𝑘≥𝐹𝑦,𝑢^𝑛𝑘.(3.40) According to 𝑟_𝑛>𝑟>0 and 𝑢_𝑛−𝑥_𝑛→0, we conclude that (𝑢_𝑛𝑖−𝑥_𝑛𝑖)/𝑟_𝑛𝑖→0 and 𝑢_𝑛𝑘⇀𝑥̃. From (A4), we obtain the following: 𝐹(𝑦,𝑥̃)≤0,∀𝑦∈𝐶.(3.41) For 𝑡 with 0<𝑡≤1 and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑥̃. Since 𝑦∈𝐶 and 𝑥̃∈𝐶 (due to 𝑢_𝑛𝑖→𝑥̃ as 𝑖→∞), we have 𝑦_𝑡∈𝐶 and hence 𝐹(𝑦_𝑡,𝑥̃)≤0. So, from (A1) and (A4), we have 𝑡𝐹𝑦_𝑡,𝑦≥𝑡𝐹𝑦_𝑡,𝑦+(1−𝑡)𝐹𝑦_𝑡,𝑥̃≥𝑡𝐹𝑦_𝑡,𝑦_𝑡=0(3.42) and 𝐹(𝑦_𝑡,𝑦)≥0, for all 𝑡∈(0,1) and 𝑦∈𝐶. From (A3), we obtain the following: 𝐹(𝑥̃,𝑦)≥0∀𝑦∈𝐶,(3.43) and hence 𝑥̃∈EP(𝐹). Therefore, we obtain that 𝑥̃∈Ω. Hence, it follows from Lemma 2.2 that limsup_𝑛→∞𝑓𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗==li𝑓m_𝑖→∞𝑥_∗𝑓𝑥^∗−𝑥^∗,𝑥_𝑛𝑖−𝑥^∗≤0.−𝑥^∗,𝑥̃−𝑥^∗(3.44)Step 7. lim_𝑛→∞𝑥_𝑛=𝑥^∗.
From (3.1), (3.3)-(3.4), and the convexity of ‖⋅‖, we have ‖‖𝑦_𝑛−𝑥^∗‖‖²=‖‖𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑣_𝑛−𝑥^∗‖‖²≤≤11−−𝛼𝛼_𝑛𝑛‖‖𝑣_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗,𝑦_𝑛−𝑥^∗‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗,𝑦_𝑛−𝑥^∗.(3.45) We can also get ‖‖𝑥_𝑛+1−𝑥^∗‖‖²=‖‖‖‖𝛽_𝑛𝑥_𝑛−𝑥^∗+𝜎_𝑛𝑛_𝑖=1𝛽_𝑖−1+1−𝛽_𝑛−𝛽_𝑖𝑇_𝑖𝑦_𝑛−𝑥^∗+1−𝛽_𝑛1−𝜎_𝑛𝑦_𝑛−𝑥^∗1−𝜎_𝑛𝛼_𝑛𝑣_𝑛−𝑓𝑥_𝑛‖‖‖²≤‖‖‖‖𝛽_𝑛𝑥_𝑛−𝑥^∗+𝜎_𝑛𝑛_𝑖=1𝛽_𝑖−1−𝛽_𝑖𝑇_𝑖𝑦_𝑛−𝑥^∗+1−𝛽_𝑛1−𝜎_𝑛𝑦_𝑛−𝑥^∗‖‖‖‖²≤+𝛽_𝑛21−𝛽_𝑛1−𝜎_𝑛𝛼_𝑛𝑣_𝑛−𝑓𝑥_𝑛,𝑥_𝑛+1−𝑥^∗+2‖‖𝑥1_𝑛−−𝛽𝑥^∗_𝑛‖‖²+1−𝛽_𝑛‖‖𝑦_𝑛−𝑥^∗‖‖²+21−𝛽_𝑛1−𝜎_𝑛𝛼_𝑛𝑣_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗1−𝜎_𝑛𝛼_𝑛𝑥^∗−𝑓𝑥_𝑛,𝑥_𝑛+1−𝑥^∗.(3.46) By (3.45), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤+𝛽_𝑛2‖‖𝑥1_𝑛−−𝛽𝑥^∗_𝑛‖‖²+1−𝛽_𝑛1−𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗,𝑦_𝑛−𝑥^∗+21−𝛽_𝑛1−𝜎_𝑛𝛼_𝑛‖‖𝑣_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥^∗‖‖≤1−1−𝛽_𝑛𝛼_𝑛1−𝜎_𝑛𝑥^∗−𝑓𝑥_𝑛,𝑥_𝑛+1−𝑥^∗+21−𝛽_𝑛𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+21−𝛽_𝑛1−𝜎_𝑛𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗,𝑦_𝑛−𝑥_𝑛+1+21−𝛽_𝑛1−𝜎_𝑛𝛼_𝑛‖‖𝑧_𝑛−𝑦^∗‖‖‖‖𝑥_𝑛+1−𝑥^∗‖‖≤1−1−𝛽_𝑛𝛼_𝑛𝜎_𝑛𝑓𝑥_𝑛−𝑥^∗,𝑦_𝑛−𝑥^∗+21−𝛽_𝑛𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+21−𝛽_𝑛1−𝜎_𝑛𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑦_𝑛‖‖+21−𝛽_𝑛1−𝜎_𝑛𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥^∗‖‖+21−𝛽_𝑛𝛼_𝑛𝜎_𝑛𝑓𝑥_𝑛−𝑥^∗,𝑦_𝑛−𝑥_𝑛+21−𝛽_𝑛𝛼_𝑛𝜎_𝑛𝑓𝑥_𝑛−𝑓𝑥^∗,𝑥_𝑛−𝑥^∗≤1−1−𝛽_𝑛𝛼_𝑛𝜎_𝑛𝑓𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+21−𝛽_𝑛1−𝜎_𝑛𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑦_𝑛‖‖+21−𝛽_𝑛1−𝜎_𝑛𝛼_𝑛‖‖𝑥^𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛+1−𝑥^∗‖‖²𝛼_𝑛𝜎_𝑛‖‖𝑓𝑥_𝑛−𝑥^∗‖‖‖‖𝑦_𝑛−𝑥_𝑛‖‖+21−𝛽_𝑛𝛼_𝑛𝜎_𝑛𝜌‖‖𝑥_𝑛−𝑥^∗‖‖²+21−𝛽_𝑛𝛼_𝑛𝜎_𝑛𝑓𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗,(3.47) which implies that ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1−2𝜎^𝑛(1−𝜌)−11−𝛽_𝑛1−1−𝛽_𝑛1−𝜎_𝑛𝛼_𝑛𝛼_𝑛‖‖𝑥^𝑛−𝑥^∗‖‖²+2𝜎^𝑛(1−𝜌)−11−𝛽_𝑛𝛼_𝑛1−1−𝛽_𝑛1−𝜎_𝑛𝛼_𝑛×21−𝜎^𝑛2𝜎_𝑛(1−𝜌)−1‖‖𝑓𝑥^𝑛−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑦_𝑛‖‖+2𝜎^𝑛2𝜎_𝑛(1−𝜌)−1‖‖𝑓𝑥^𝑛−𝑥^∗×‖‖𝑦_𝑛‖‖−𝑥_𝑛‖‖+2𝜎^𝑛2𝜎_𝑛(1−𝜌)−1𝑓𝑥^𝑛−𝑥^∗,𝑥_𝑛−𝑥^∗.(3.48) From liminf_𝑛→∞(2𝜎_𝑛(1−𝜌)−1)(1−𝛽_𝑛)/(1−(1−𝛽_𝑛)(1−𝜎_𝑛))>0, it follows that ∑^∞𝑛=0((2𝜎_𝑛(1−𝜌)−1)(1−𝛽_𝑛)/(1−(1−𝛽_𝑛)(1−𝜎_𝑛)𝛼_𝑛))𝛼_𝑛=∞. It is clear that limsup_𝑛→∞21−𝜎^𝑛2𝜎_𝑛(1−𝜌)−1‖‖𝑓𝑥^𝑛−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑦_𝑛‖‖+2𝜎^𝑛2𝜎_𝑛(1−𝜌)−1‖‖𝑓𝑥^𝑛−𝑥^∗×‖‖𝑦_𝑛‖‖−𝑥_𝑛‖‖+2𝜎^𝑛2𝜎_𝑛(1−𝜌)−1𝑓𝑥^𝑛−𝑥^∗,𝑥_𝑛−𝑥^∗≤0.(3.49) Therefore, all conditions of Lemma 2.4 are satisfied. Therefore, we immediately deduce that 𝑥_𝑛→𝑥^∗. This completes the proof.

Corollary 3.2. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let the mappings 𝐴,𝐵∶𝐶→𝐻 be 𝛼-inverse strongly monotone and 𝛽-inverse strongly monotone, respectively. Let 𝐹 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) and 𝑇∶𝐶→𝐶 be a nonexpansive mapping such that Ω∶=𝐹(𝑇)∩𝐸𝑃(𝐹)∩Γ≠∅.Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1/2). For given 𝑥₀∈𝐶 arbitrarily, let the sequences {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛} and, {𝑢_𝑛} be generated By 𝐹𝑢_𝑛,𝑦+1𝑟_𝑛𝑧⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝑦_𝑛=𝑃_𝐶𝑢_𝑛−𝜇𝐵𝑢_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑧_𝑛−𝜆𝐴𝑧_𝑛,=𝛽_𝑛𝑥_𝑛+𝜎_𝑛1−𝛽_𝑛𝑇𝑦_𝑛+1−𝛽_𝑛1−𝜎_𝑛𝑃_𝐶𝑧_𝑛−𝜆𝐴𝑧_𝑛,(3.50) where 𝜆∈(0,2𝛼), 𝜇∈(0,2𝛽), and sequences {𝛼_𝑛}⊂[0,1], {𝛽_𝑛}⊂[0,1], {𝜎_𝑛}⊂[0,1], and {𝑟_𝑛}⊂(𝑟,∞), 𝑟>0, are such that (i)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (ii)lim_𝑛→∞𝛼_𝑛=0and ∑^∞𝑛=1𝛼_𝑛=∞,(iii)𝜎_𝑛>1/2(1−𝜌),∑^∞𝑛=1|𝜎_𝑛−𝜎_𝑛−1|<∞, (iv)∑^∞𝑛=1|𝑟_𝑛−𝑟_𝑛−1|<∞. Then the sequence {𝑥_𝑛} generated by (3.50) converges strongly to 𝑥^∗=𝑃_Ω⋅𝑓(𝑥^∗), and (𝑥^∗,𝑦^∗) is a solution of the general system of variational inequalities (1.5), where 𝑦^∗=𝑃_𝐶(𝑥^∗−𝜇𝐵𝑥^∗).

Acknowledgments

The authors would like to express their thanks to the referee for the valuable comments and suggestions for improving this article. This paper was supported by the NSFC Tianyuan Youth Foundation of Mathematics of China (no. 11126136) and Fundamental Research Funds for the Central Universities (no. ZXH2011C002).