Viscosity Approximation Method for System of Variational Inclusions Problems and Fixed-Point Problems of a Countable Family of Nonexpansive Mappings

Abstract

We propose new iterative schemes for finding the common element of the set of common fixed points of countable family of nonexpansive mappings, the set of solutions of the variational inequality problem for relaxed cocoercive and Lipschitz continuous, the set of solutions of system of variational inclusions problem, and the set of solutions of equilibrium problems in a real Hilbert space by using the viscosity approximation method. We prove strong convergence theorem under some parameters. The results in this paper unify and generalize some well-known results in the literature.

1. Introduction

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. A mapping 𝑆 of 𝐶 into itself is called nonexpansive if ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. We denote by 𝐹(𝑆) the set of fixed points of 𝑆; that is, 𝐹(𝑆)={𝑥∈𝐶∶𝑆𝑥=𝑥}. If 𝐶⊂𝐻 is nonempty, closed and convex and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping, then 𝐹(𝑆) is closed and convex and 𝐹(𝑆)≠∅, when 𝐶 is bounded; see, for example, [1, 2]. The metric projection, 𝑃_𝐶, onto a given nonempty, closed and convex subset 𝐶, satisfies the nonexpansive with 𝐹(𝑃_𝐶)=𝐶. A mapping 𝐵∶𝐶→𝐶 is called monotone if ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥0 for all 𝑥,𝑦∈𝐶. A mapping 𝐵∶𝐶→𝐶 is called 𝛽-inverse-strongly monotone if there exists a constant 𝛽>0 such that ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥𝛽‖𝑥−𝑦‖², for all 𝑥,𝑦∈𝐶. A mapping 𝐵∶𝐶→𝐶 is called relaxed (𝜙,𝜔)-cocoercive if there exists 𝜙,𝜔>0 such that⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥(−𝜙)‖𝐵𝑥−𝐵𝑦‖²+𝜔‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐶.(1.1)

A mapping 𝐵∶𝐶→𝐶 is said to be 𝜉-Lipschitz continuous if there exists 𝜉≥0 such that‖𝐵𝑥−𝐵𝑦‖≤𝜉‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.2)

Let 𝐵∶𝐻→𝐻 be a single-valued nonlinear mapping and 𝑀∶𝐻→2^𝐻 a multivalued mapping. The variational inclusion problem is to find 𝑥̃∈𝐻 such that𝜃∈𝐵(𝑥̃)+𝑀(𝑥̃),(1.3) where 𝜃 is the zero vector in 𝐻. The set of solutions of problem (1.3) is denoted by 𝐼(𝐵,𝑀). If 𝑀=𝜕𝜓_𝐶, where 𝐶 is a nonempty closed convex subset of 𝐻 and 𝜕𝜓_𝐶∶𝐻→[0,+∞] is the indicator function of 𝐶; that is,𝜓_𝐶(𝑥)=0+,∞,𝑥𝑥∈∉𝐶𝐶,,(1.4)

then, the variational inclusion problem (1.3) is equivalent to the variational inequality problems denoted by VI(𝐶,𝐵) which is to find 𝑥̃∈𝐶 such that⟨𝐵𝑥̃,𝑦−𝑥̃⟩≥0,∀𝑦∈𝐶.(1.5)

In 2003, Takahashi and Toyoda [3] to find 𝑥^∗∈𝐹(𝑆)∩VI(𝐶,𝐵) introduced the following iterative scheme:𝑥𝑥₀∈Cchosenarbitrary,_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜉_𝑛𝐵𝑥_𝑛,∀𝑛≥0,(1.6) where 𝐵 is a 𝛽-inverse-strongly monotone mapping, {𝛼_𝑛} is a sequence in (0, 1), and {𝜉_𝑛} is a sequence in (0,2𝛽). They showed that if 𝐹(𝑆)∩VI(𝐶,𝐵) is nonempty, then the sequence {𝑥_𝑛} generated by (1.6) converges weakly to some 𝑥^∗∈𝐹(𝑆)∩VI(𝐶,𝐵).

In 2008, Zhang et al. [4] to find 𝑥^∗∈𝐹(𝑆)∩𝐼(𝑀,𝐵). They introduced the following new iterative scheme:𝑥𝑦₀∈Cchosenarbitrary,_𝑛𝑥_𝑛+1=𝐽_𝑀,𝜆𝑥_𝑛−𝜆𝐵𝑥_𝑛,=𝛼_𝑛𝑥+1−𝛼_𝑛𝑆𝑦_𝑛,∀𝑛≥0,(1.7) where 𝐽_𝑀,𝜆=(𝐼+𝜆𝑀)⁻¹ is the resolvent operator associated with 𝑀 and a positive number 𝜆,{𝛼_𝑛} is a sequence in the interval [0,1].

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 𝑥̃∈𝐶 such that𝐹𝑥,𝑦≥0,∀𝑦∈𝐶.(1.8) The set of solutions of (1.8) is denoted by EP(𝐹). Many problems in applied sciences, such as monotone inclusion problems, variational inequality problems, saddle point problems, Nash equilibria in noncooperative games, as well as certain fixed-point problems reduce to finding some element to EP(𝐹) in Hilbert and Banach spaces (see [5–14]).

Given any 𝑟>0. The operator 𝑇_𝑟∶𝐻→𝐶 defined by𝑇_𝑟(𝑥)=𝑧∈𝐶∶𝐹(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶,(1.9) is called the resolvent of 𝐹 (see [5, 6]).

It is shown in [6] that, under suitable hypotheses on 𝐹 (to be stated precisely in Section 2), 𝑇_𝑟∶𝐻→𝐶 is single valued and firmly nonexpansive and satisfies𝐹𝑇_𝑟=EP(𝐹),∀𝑟>0.(1.10)

Using this result, for finding an element of 𝐹(𝑆)∩VI(𝐶,𝐵)∩EP(𝐹), Su et al. [15] introduced the following iterative scheme by the viscosity approximation method in Hilbert spaces:𝑥𝑥₀∈𝐶chosenarbitrary,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆𝑃_𝐶𝐼−𝜉_𝑛𝐵𝑇_𝑟𝑛𝑥_𝑛,∀𝑛≥0,(1.11) where 𝑓∶𝐶→𝐶 is a contraction (i.e., ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝜓‖𝑥−𝑦‖,forall𝑥,𝑦∈𝐶and0≤𝜓<1) and {𝛼_𝑛}⊂(0,1), 𝜉_𝑛⊂(0,2𝛽), and 𝑟_𝑛⊂(0,∞) satisfy some appropriate conditions. Furthermore, they prove {𝑥_𝑛} converges strongly to the same point 𝑥^∗∈𝐹(𝑆)∩VI(𝐶,𝐵)∩EP(𝐹), where 𝑥^∗=𝑃_{𝐹(𝑆)∩VI(𝐶,𝐵)∩EP(𝐹)}𝑓(𝑥^∗).

In this paper, motivated and inspired by the above facts, we introduce a new iterative scheme for finding a common element of the set of solutions of the variational inequalities for 𝜇-Lipschitz continuous and relaxed (𝜙,𝜔)-cocoercive mapping, the set of solutions to the variational inclusion for family of 𝛼-inverse strongly monotone mappings, the set of fixed points of a countable family of nonexpansive mappings, and the set of solutions of an equilibrium problem in a real Hilbert space by using the viscosity approximation method. Strong convergence results are derived under suitable conditions in a real Hilbert space.

2. Preliminaries

In this section, we will recall some basic notations and collect some conclusions that will be used in the next section.

Let 𝐻 be a real Hilbert space whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. We denote strong convergence of {𝑥_𝑛} to 𝑥∈𝐻 by 𝑥_𝑛→𝑥 and weak convergence by 𝑥_𝑛⇀𝑥. Let 𝐶 be nonempty closed convex subset of 𝐻. Recall that for all 𝑥∈𝐻 there exists a unique nearest point in 𝐶 to 𝑥 denoted 𝑃_𝐶𝑥; that is, ‖𝑥−𝑃_𝐶𝑥‖≤‖𝑥−𝑦‖,forall𝑦∈𝐶. The mapping 𝑃_𝐶 is nonexpansive; that is, ‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖≤‖𝑥−𝑦‖,forall𝑥,𝑦∈𝐻. The mapping 𝑃_𝐶 is firmly nonexpansive; that is, ‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖²≤⟨𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦⟩,forall𝑥,𝑦∈𝐻. It is well known that𝑥̃∈VI(𝐶,𝐵)⇔𝑥̃=𝑃_𝐶(𝑥̃−𝜆𝐵𝑥̃),∀𝜆>0.(2.1) A set-valued mapping 𝑀∶𝐻→2^𝐻 is called monotone if, for all 𝑥,𝑦∈𝐻, 𝑓∈𝑀𝑥 and 𝑔∈𝑀𝑦 imply ⟨𝑥−𝑦,𝑓−𝑔⟩≥0. A monotone mapping 𝑀∶𝐻→2^𝐻 is called maximal, if its graph of any Graph (𝑀)∶={(𝑥,𝑓)∈𝐻×𝐻|𝑓∈𝑀(𝑥)} of 𝑀 is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping 𝑀 is maximal if and only if for all (𝑥,𝑓)∈𝐻×𝐻,⟨𝑥−𝑦,𝑓−𝑔⟩≥0,forall(𝑦,𝑔)∈ Graph (𝑀) (the graph of mapping 𝑀) implies that 𝑓∈𝑀𝑥.

Definition 2.1. Let 𝑀∶𝐻→2^𝐻 be a multivalued maximal monotone mapping; then the set-valued mapping 𝐽_𝑀,𝜆∶𝐻→𝐻 defined by 𝐽_𝑀,𝜆(𝑥̃)=(𝐼+𝜆𝑀)⁻¹(𝑥̃),∀𝑥̃∈𝐻,(2.2) is called the resolvent operator associated with 𝑀, where 𝜆 is any positive number and 𝐼 is the identity mapping.

Lemma 2.2 (see [16]). Let 𝑀∶𝐻→2^𝐻 be a maximal monotone mapping and let 𝐵∶𝐻→𝐻 be a Lipschitz continuous mapping. Then the mapping 𝑀+𝐵∶𝐻→2^𝐻 is a maximal monotone mapping.

Lemma 2.3 (see [16, 17]). (1)The resolvent operator J_𝑀,𝜆 is single valued and nonexpansive for all 𝜆>0; that is,‖‖𝐽_𝑀,𝜆(𝑥)−𝐽_𝑀,𝜆(𝑦)‖‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐻,∀𝜆>0.(2.3)(2)The resolvent operator 𝐽_𝑀,𝜆 is 1-inverse-strongly monotone; that is,‖‖𝐽_𝑀,𝜆(𝑥)−𝐽_𝑀,𝜆(𝑦)‖‖²≤𝑥−𝑦,𝐽_𝑀,𝜆(𝑥)−𝐽_𝑀,𝜆(𝑦),∀𝑥,𝑦∈𝐻.(2.4)

Lemma 2.4 (see [17]). (1)Let 𝑥̃∈𝐻 is a solution of problem (1.3) if and only if 𝑥̃=𝐽_𝑀,𝜆(𝐼−𝜆𝐵) for all 𝜆>0; that is,𝐼(𝐵,𝑀)=𝐹𝐽_𝑀,𝜆(𝐼−𝜆𝐵),∀𝜆>0.(2.5)(2)If 𝜆∈[0,2𝛽], then 𝐼(𝐵,𝑀) is a closed convex subset in 𝐻.

Lemma 2.5 (see [18]). Each Hilbert space 𝐻 satisfies Opial’s condition; that is, for any sequence {𝑥_𝑛}⊂𝐻 with 𝑥_𝑛⇀𝑥, the inequality liminf_𝑛→∞‖‖𝑥_𝑛−𝑥‖‖<liminf_𝑛→∞‖‖𝑥_𝑛−𝑦‖‖(2.6) holds for each 𝑦∈𝐻 with 𝑦≠𝑥.

Lemma 2.6 (see [19]). Let {𝑥_𝑛} and {𝑧_𝑛} be bounded sequences in a Banach space 𝐸, and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose 𝑥_𝑛+1=(1−𝛽_𝑛)𝑧_𝑛+𝛽_𝑛𝑥_𝑛 for all integers 𝑛≥1 and limsup_𝑛→∞(‖𝑧_𝑛+1−𝑧_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0.

Lemma 2.7 (see [20]). Assume {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝑏_𝑛𝑎_𝑛+𝛿_𝑛,𝑛≥0,(2.7) where {𝑏_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in ℝ such that(1)∑^∞𝑛=1𝑏_𝑛=∞,(2)limsup_𝑛→∞𝛿_𝑛/𝑏_𝑛≤0 or ∑^∞𝑛=1|𝛿_𝑛|<∞. Then lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.8. Let 𝐻 be a real Hilbert space. Then hold the following identities:(i)‖𝑡𝑥+(1−𝑡)𝑦‖²=𝑡‖𝑥‖²+(1−𝑡)‖𝑦‖²−𝑡(1−𝑡)‖𝑥−𝑦‖²,∀𝑡∈[0,1],∀𝑥,𝑦∈𝐻, (ii)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩,∀𝑥,𝑦∈𝐻.

Lemma 2.9 (see [21]). Let 𝐶 be a nonempty closed subset of a Banach space, and let {𝑆_𝑛} be a sequence of mappings of 𝐶 into itself. Suppose that ∑^∞𝑛=1sup{‖𝑆_𝑛+1𝑧−𝑆_𝑛𝑧‖∶𝑧∈𝐶}<∞. Then, for each 𝑦∈𝐶, {𝑆_𝑛𝑦} converges strongly to some point of 𝐶. Moreover, let 𝑆 be a mapping of 𝐶 into itself defined by 𝑆𝑦=lim_𝑛→∞𝑆_𝑛𝑦,∀𝑦∈𝐶.(2.8) Then lim_𝑛→∞sup{‖𝑆𝑧−𝑆_𝑛𝑧‖∶𝑧∈𝐶}=0.

For solving the equilibrium problem for a bifunction 𝐹∶𝐶×𝐶→ℝ, let us 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,∀𝑥,𝑦∈𝐶;(A3) for each 𝑥,𝑦,𝑧∈𝐶,lim_𝑡↓0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦);(A4) for each 𝑥∈𝐶,𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous.

Lemma 2.10 (see [5]). Let 𝐶 be a nonempty closed convex subset of 𝐻, and let 𝐹 be a bifunction of 𝐶×𝐶 into ℝ satisfying (A1)–(A4). Let 𝑟>0 and 𝑥∈𝐻. Then, there exists 𝑧∈𝐶 such that 𝐹(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶.(2.9)

Lemma 2.11 (see [6]). Assume that 𝐹∶𝐶×𝐶→ℝ satisfies (A1)–(A4). For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟(𝑥)=𝑧∈𝐶∶𝐹(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶,(2.10) for all 𝑥∈𝐻. Then, the following hold:(i)𝑇_𝑟 is single valued;(ii)T_𝑟 is firmly nonexpansive; that is, for any 𝑥,𝑦∈𝐻‖𝑇_𝑟𝑥−𝑇_𝑟𝑦‖²≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦⟩;(iii)𝐹(𝑇_𝑟)=EP(𝐹);(iv)EP(𝐹) is closed and convex.

Lemma 2.12 (see [22]). Let 𝐻 be a Hilbert space and 𝑀 a maximal monotone on 𝐻. Then, the following holds: ‖‖𝐽_𝑀,𝑟𝑥−𝐽_𝑀,𝑠𝑥‖‖²≤𝑟−𝑠𝑟𝐽^𝑀,𝑟𝑥−𝐽_𝑀,𝑠𝑥,𝐽_𝑀,𝑟𝑥−𝑥,∀𝑠,𝑟>0,𝑥∈𝐻,(2.11) where 𝐽_𝑀,𝑟=(𝐼+𝑟𝑀)⁻¹ and 𝐽_𝑀,𝑠=(𝐼+𝑠𝑀)⁻¹.

3. Main Results

In this section, we will use the viscosity approximation method to prove a strong convergence theorem for finding a common element of the set of fixed points of a countable family of nonexpansive mappings, the set of solutions of the variational inequality problem for relaxed cocoercive and Lipschitz continuous mappings, the set of solutions of system of variational inclusions, and the set of solutions of equilibrium problem in a real Hilbert space.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝐵∶𝐶→𝐻 be relaxed (𝜙,𝜔)-cocoercive and 𝜇-Lipschitz continuous with 𝜔>𝜙𝜇², for some 𝜙,𝜔,𝜇>0. Let 𝒢={𝐺_𝑘∶𝑘=1,2,3,…,𝑁} be a finite family of 𝛽-inverse strongly monotone mappings from 𝐶 into 𝐻, and let 𝐹 be a bifunction from 𝐶×𝐶→ℝ satisfying (A1)–(A4). Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜓(0≤𝜓<1), and let {𝑆_𝑛} be a sequence of nonexpansive mappings of 𝐶 into itself such that Ω∶^∞_𝑛=1𝐹𝑆_𝑛∩^𝑁_𝑘=1𝐼𝐺_𝑘,𝑀_𝑘∩VI(𝐶,𝐵)∩EP(𝐹)≠∅.(3.1) Let the sequences {𝑥_𝑛} and {𝑦_𝑛} be generated by 𝑥𝑦₁=𝑥∈𝐶chosenarbitrarily,_𝑛𝑥_𝑛+1=𝐽_{𝑀𝑁,𝜆𝑁,𝑛}𝐼−𝜆_𝑁,𝑛𝐺_𝑛…𝐽_{𝑀2,𝜆2,𝑛}𝐼−𝜆_2,𝑛𝐺₂𝐽_{𝑀1,𝜆1,𝑛}𝐼−𝜆_1,𝑛𝐺₁𝑇_𝑟𝑛𝑥_𝑛,=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑃_𝐶𝑦_𝑛−𝜉_𝑛𝐵𝑦_𝑛,∀𝑛≥1,(3.2) where {𝛼_𝑛},{𝛽_𝑛}, {𝛾_𝑛}⊂(0,1) and {𝜉_𝑛},{𝑟_𝑛}⊂(0,∞) satisfy the following conditions:(C1)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,(C2)lim_𝑛→∞𝛼_𝑛=0,∑^∞𝑛=1𝛼_𝑛=∞, (C3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C4){𝜉_𝑛}⊂[𝑎,𝑏] for some 𝑎, 𝑏 with 0≤𝑎≤𝑏≤2(𝜔−𝜙𝜇²)/𝜇² and lim_𝑛→∞|𝜉_𝑛+1−𝜉_𝑛|=0,(C5){𝜆_𝑘,𝑛}^𝑁𝑘=1⊂[𝑐,𝑑]⊂(0,2𝛽) and lim_𝑛→∞|𝜆_𝑘,𝑛+1−𝜆_𝑘,𝑛|=0, for each 𝑘∈{1,2,…,𝑁},(C6)liminf_𝑛→∞𝑟_𝑛>0 and lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0.Suppose that ∑^∞𝑛=1sup{‖𝑆_𝑛+1𝑧−𝑆_𝑛𝑧‖∶𝑧∈𝐾}<∞ for any bounded subset 𝐾 of 𝐶. Let 𝑆 be a mapping of 𝐶 into itself defined by 𝑆𝑦=lim_𝑛→∞𝑆_𝑛𝑦forall𝑦∈𝐶 and suppose that 𝐹(𝑆)=⋂^∞𝑛=1𝐹(𝑆_𝑛). Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to the same point 𝑥^∗∈Ω, where 𝑥^∗=𝑃_Ω𝑓(𝑥^∗).

Proof. First, we prove that the mapping 𝑃_Ω𝑓∶𝐻→𝐶 has a unique fixed point.
In fact, since 𝑓∶𝐶→𝐶 is a contraction with 𝜓∈[0,1) and 𝑃_Ω𝑓∶𝐻→Ω is also a contraction, we obtain ‖‖𝑃_Ω𝑓(𝑥)−𝑃_Ω𝑓(𝑦)‖‖≤‖𝑓(𝑥)−𝑓(𝑦)‖≤𝜓‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(3.3) Therefore, there exists a unique element 𝑥^∗∈𝐶 such that 𝑥^∗=𝑃_Ω𝑓(𝑥^∗), where Ω∶^∞_𝑛=1𝐹𝑆_𝑛∩^𝑁_𝑘=1𝐼𝐺_𝑘,𝑀_𝑘∩VI(𝐶,𝐵)∩EP(𝐹).(3.4) Now, we prove that (𝐼−𝜉_𝑛𝐵) is nonexpansive.
Indeed, for any 𝑥,𝑦∈𝐶, since 𝐵∶𝐶→𝐻 is a 𝜇-Lipschitz continuous and relaxed (𝜙,𝜔)-cocoercive mappings with 𝜔>𝜙𝜇² and 𝜉_𝑛≤2(𝜔−𝜙𝜇²)/𝜇², we obtain ‖‖𝐼−𝜉_𝑛𝐵𝑥−𝐼−𝜉_𝑛𝐵𝑦‖‖²=‖‖(𝑥−𝑦)−𝜉_𝑛(𝐵𝑥−𝐵𝑦)‖‖²=‖𝑥−𝑦‖²−2𝜉_𝑛⟨𝑥−𝑦,𝐵𝑥−𝐵𝑦⟩+𝜉^2𝑛‖𝐵𝑥−𝐵𝑦‖²≤‖𝑥−𝑦‖²−2𝜉_𝑛−𝜙‖𝐵𝑥−𝐵𝑦‖²+𝜔‖𝑥−𝑦‖²+𝜉^2𝑛‖𝐵𝑥−𝐵𝑦‖²≤‖𝑥−𝑦‖²+2𝜉_𝑛𝜙𝜇²‖𝑥−𝑦‖²−2𝜉_𝑛𝜔‖𝑥−𝑦‖²+𝜉^2𝑛𝜇²‖𝑥−𝑦‖²=1+2𝜉_𝑛𝜙𝜇²−2𝜉_𝑛𝜔+𝜉^2𝑛𝜇²‖𝑥−𝑦‖²=1−𝜉_𝑛𝜇²2𝜔−𝜙𝜇²𝜇²−𝜉_𝑛‖𝑥−𝑦‖²≤1−𝜉_𝑛𝜇²2𝜔−𝜙𝜇²𝜇²−𝑏‖𝑥−𝑦‖².(3.5) Setting 𝜁=𝜇²22𝜔−𝜙𝜇²𝜇²−𝑏>0,(3.6) thus, ‖‖𝐼−𝜉_𝑛𝐵𝑥−𝐼−𝜉_𝑛𝐵𝑦‖‖²≤1−2𝜉_𝑛𝜁‖𝑥−𝑦‖²≤1−𝜉_𝑛𝜁²‖𝑥−𝑦‖²,(3.7) which implies that ‖‖𝐼−𝜉_𝑛𝐵𝑥−𝐼−𝜉_𝑛𝐵𝑦‖‖≤1−𝜉_𝑛𝜁‖𝑥−𝑦‖≤‖𝑥−𝑦‖.(3.8) Hence (𝐼−𝜉_𝑛𝐵) is nonexpansive.
We divide the proof of Theorem 3.1 into five steps.
Step 1. We show that the sequence {𝑥_𝑛} is bounded.
Now, let 𝑥̃∈Ω and if {𝑇_𝑟𝑛} is a sequence of mappings defined as in Lemma 2.11, then 𝑥̃=𝑃_𝐶(𝑥̃−𝜆_𝑛𝐵𝑥̃)=𝑇_𝑟𝑛𝑥̃, and let 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛. So, we have ‖‖𝑢_𝑛−𝑥̃‖‖=‖‖𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑥̃‖‖≤‖‖𝑥_𝑛−𝑥̃‖‖.(3.9) For 𝑘∈{1,2,…,𝑁} and for any positive integer number 𝑛, we define the operator Υ^𝑘𝑛∶𝐶→𝐻 as follows: ΥΥ^0𝑛𝑥=𝑥,_𝑘𝑛𝑥=𝐽_{𝑀𝑘,𝜆𝑘,𝑛}𝐼−𝜆_𝑘,𝑛𝐺_𝑘…𝐽_{𝑀2,𝜆2,𝑛}𝐼−𝜆_2,𝑛𝐺₂𝐽_{𝑀1,𝜆1,𝑛}𝐼−𝜆_1,𝑛𝐺₁𝑥,(3.10) for all 𝑛, we get 𝑦_𝑛=Υ^𝑁𝑛𝑢_𝑛. On the other hand, since 𝐺_𝑘∶𝐶→𝐻 is 𝛽-inverse strongly monotone and 𝜆_𝑘,𝑛⊂[𝑐,𝑑]⊂(0,2𝛽), then 𝐽_{𝑀𝑘,𝜆𝑘,𝑛}(𝐼−𝜆_𝑘,𝑛𝐺_𝑘) is nonexpansive. Thus Υ^𝑘𝑛 is nonexpansive. From Lemma 2.4(1), we have 𝑥̃=Υ^𝑁𝑛𝑥̃. It follows that ‖‖𝑦_𝑛−𝑥̃‖‖=‖‖Υ^𝑁𝑛𝑢_𝑛−Υ^𝑁𝑛𝑥̃‖‖≤‖‖𝑢_𝑛−𝑥̃‖‖≤‖‖𝑥_𝑛−𝑥̃‖‖.(3.11) Setting 𝑣_𝑛=𝑃_𝐶(𝑦_𝑛−𝜉_𝑛𝐵𝑦_𝑛) and 𝐼−𝜉_𝑛𝐵 is a nonexpansive mapping, we obtain ‖‖𝑣_𝑛−𝑥̃‖‖=≤‖‖‖‖𝑃_𝐶𝑦𝑦_𝑛−𝜉_𝑛𝐵𝑦_n−𝑃_𝐶𝑥̃−𝜉_𝑛𝐵𝑥_𝑛̃‖‖=‖‖𝐼−−𝜉_𝑛𝜉_𝑛𝐵𝑦_𝑛−𝑥̃−𝜉_𝑛𝐵𝑥̃‖‖≤‖‖𝑦𝐵𝑦_𝑛−𝐼−𝜉_𝑛𝐵𝑥_𝑛̃‖‖−𝑥̃‖‖≤‖‖𝑥_𝑛−𝑥̃‖‖.(3.12) From (3.2) and (3.12), we deduce that ‖‖𝑥_𝑛+1−𝑥̃‖‖=≤𝛼‖‖𝛼_𝑛𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑣_𝑛−𝑥̃‖‖≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖+𝛾_𝑛‖‖𝑣_𝑛−𝑥̃‖‖≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑓(𝑥̃)‖‖+𝛼_𝑛‖𝑓(𝑥̃)−𝑥̃‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖+𝛾_𝑛‖‖𝑥_𝑛−𝑥̃‖‖≤1𝜓−‖‖𝛼𝑥_𝑛𝑛−𝑥̃‖‖+𝛼_𝑛‖𝑓(𝑥̃)−𝑥̃‖+1−𝛼_𝑛‖‖𝑥_𝑛−𝑥̃‖‖=1−𝛼_𝑛(1−𝜓)‖‖𝑥_𝑛−𝑥̃‖‖+𝛼_𝑛‖𝑓(𝑥̃)−𝑥̃‖(1−𝜓)‖‖𝑥_𝑛−𝑥̃‖‖+𝛼_𝑛(1−𝜓)‖𝑓(𝑥̃)−𝑥̃‖≤max‖‖𝑥_𝑛(1−𝜓)−𝑥̃‖‖,‖𝑓(𝑥̃)−𝑥̃‖1−𝜓.(3.13) It follows from induction that ‖‖𝑥_𝑛−𝑥̃‖‖≤max‖‖𝑥₁−𝑥̃‖‖,‖𝑓(𝑥̃)−𝑥̃‖1−𝜓,∀𝑛≥1.(3.14) Therefore, {𝑥_𝑛} is bounded and hence so are {𝑣_𝑛}, {𝑦_𝑛}, {𝑢_𝑛}, {𝐵𝑦_𝑛}, and {𝑆_𝑛𝑣_𝑛}.Step 2. We claim that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
By the definition of 𝑇_𝑟, 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛 and 𝑢_𝑛+1=𝑇_𝑟𝑛+1𝑥_𝑛+1, we get 𝐹𝑢_𝑛,𝑦+1𝑟_𝑛𝐹𝑢⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐻,(3.15)_𝑛+1,𝑦+1𝑟_𝑛+1𝑦−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1≥0,∀𝑦∈𝐻.(3.16) Taking 𝑦=𝑢_𝑛+1 in (3.15) and 𝑦=𝑢_𝑛 in (3.16), we have 𝐹𝑢_𝑛,𝑢_𝑛+1+1𝑟_𝑛𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0,(3.17) and hence 𝐹𝑢_𝑛+1,𝑢_𝑛+1𝑟_𝑛+1𝑢_𝑛−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1≥0.(3.18) So, from (A2) we have 𝑢^𝑛+1−𝑢_𝑛,𝑢^𝑛−𝑥_𝑛𝑟_𝑛−𝑢^𝑛+1−𝑥_𝑛+1𝑟_𝑛+1≥0,(3.19) and hence 𝑢^𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛+1+𝑢_𝑛+1−𝑥_𝑛−𝑟^𝑛𝑟_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1≥0.(3.20) Without loss of generality, let us assume that there exists a real number 𝑐 such that 𝑟_𝑛>𝑐>0 for all 𝑛∈ℕ. Then, we have ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖²≤𝑢_𝑛+1−𝑢_𝑛,𝑥_𝑛+1−𝑥_𝑛+1−𝑟^𝑛𝑟_𝑛+1𝑢^𝑛+1≤‖‖𝑢_𝑛+1−𝑥_𝑛+1−𝑢_𝑛‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||||1−𝑟^𝑛𝑟_𝑛+1||||‖‖𝑢_𝑛+1−𝑥_𝑛+1‖‖,(3.21) and hence ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+1𝑟_𝑛+1≤‖‖𝑥_𝑛+1||𝑟_𝑛+1−𝑟_𝑛||‖‖𝑢_𝑛+1−𝑥_𝑛+1‖‖−𝑥_𝑛‖‖+𝑀¹𝑐||𝑟^𝑛+1−𝑟_𝑛||,(3.22) where 𝑀₁=sup{‖𝑢_𝑛−𝑥_𝑛‖∶𝑛∈ℕ}.
Notice from Lemma 2.12 that ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖=≤‖‖‖‖Υ𝑢^𝑁𝑛+1𝑢_𝑛+1𝑛+1−Υ^𝑁𝑛𝑢_𝑛‖‖+‖‖𝐽_𝑀𝑘−𝜆_𝑘,𝑛+1𝐺_𝑘Υ^𝑘𝑛+1𝑢_𝑛+1−𝑢_𝑛−𝜆_𝑘,𝑛𝐺_𝑘Υ^𝑘𝑛𝑢_𝑛‖‖≤‖‖𝑢_𝑛+1,𝜆_𝑘,𝑛+1𝑢_𝑛−𝜆_𝑘,𝑛𝐺_𝑘Υ^𝑘𝑛𝑢_𝑛−𝐽_𝑀𝑘,𝜆_𝑘,𝑛𝑢_𝑛−𝜆_𝑘,𝑛𝐺_𝑘Υ^𝑘𝑛𝑢_𝑛‖‖+||𝜆−_𝑘,𝑛+1𝑢_𝑛‖‖+||𝜆_𝑘,𝑛+1−𝜆_𝑘,𝑛||‖‖𝐺_𝑘Υ^𝑘𝑛𝑢_𝑛‖‖−𝜆_𝑘,𝑛||𝜆_𝑘,𝑛+1≤‖‖𝑢_𝑛+1‖‖𝐽_𝑀𝑘,𝜆_𝑘,𝑛+1𝑢_𝑛−𝜆_𝑘,𝑛𝐺_𝑘Υ^𝑘𝑛𝑢_𝑛−𝑢_𝑛−𝜆_𝑘,𝑛𝐺_𝑘Υ^𝑘𝑛𝑢_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑢_𝑛‖‖+2𝑀₂||𝜆_𝑘,𝑛+1−𝜆_𝑘,𝑛||−𝑥_𝑛‖‖+𝑀¹𝑐||𝑟^𝑛+1−𝑟_𝑛||+2𝑀₂||𝜆_𝑘,𝑛+1−𝜆_𝑘,𝑛||,(3.23) where 𝑀₂ is an appropriate constant such that 𝑀₂=maxsup_𝑛≥1‖‖𝐺_𝑘Υ^𝑘𝑛𝑢_𝑛‖‖,sup_𝑛≥1‖‖𝐽^𝑀𝑘,𝜆_𝑘,𝑛+1𝑢_𝑛−𝜆_𝑘,𝑛𝐺_𝑘Υ^𝑘𝑛𝑢_𝑛−𝑢_𝑛−𝜆_𝑘,𝑛𝐺_𝑘Υ^𝑘𝑛𝑢_𝑛‖‖𝐽_𝑀𝑘,𝜆_𝑘,𝑛+1(3..24) Since 𝐼−𝜉_𝑛𝐵 is nonexpansive mappings, we have the following estimates: ‖‖𝑣_𝑛+1−𝑣_𝑛‖‖≤≤‖‖‖‖𝑃_𝐶𝑦𝑦_𝑛+1𝑛+1−𝜉_𝑛+1𝐵𝑦_𝑛+1−𝑃_𝐶𝑦_𝑛−𝜉_𝑛𝐵𝑦_𝑛‖‖=‖‖𝑦_𝑛+1−𝜉_𝑛+1𝐵𝑦_𝑛+1−𝑦_𝑛−𝜉_𝑛𝐵𝑦_𝑛‖‖≤‖‖𝑦_𝑛+1−𝜉_𝑛+1𝐵𝑦_𝑛+1−𝑦_𝑛−𝜉_𝑛+1𝐵𝑦_𝑛+𝜉_𝑛−𝜉_𝑛+1𝐵𝑦_𝑛‖‖=‖‖𝐼−𝜉−_𝑛+1𝜉_𝑛+1𝐵𝑦_𝑛+1−𝑦_𝑛−𝜉_𝑛+1𝐵𝑦_𝑛‖‖+||𝜉_𝑛−𝜉_𝑛+1||‖‖𝐵𝑦_𝑛‖‖≤‖‖𝑦_𝑛+1𝐵𝑦_𝑛+1−𝐼−𝜉_𝑛+1𝐵𝑦_𝑛‖‖+||𝜉_𝑛−𝜉_𝑛+1||‖‖𝐵𝑦_𝑛‖‖−𝑦_𝑛‖‖+||𝜉_𝑛−𝜉_𝑛+1||‖‖𝐵𝑦_𝑛‖‖.(3.25) Substituting (3.23) into (3.25), we obtain ‖‖𝑣_𝑛+1−𝑣_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+𝑀¹𝑐||𝑟^𝑛+1+||𝜉_𝑛−𝑟_𝑛||+2𝑀₂||𝜆_𝑘,𝑛+1−𝜆_𝑘,𝑛||−𝜉_𝑛+1||‖‖𝐵𝑦_𝑛‖‖.(3.26) Indeed, define 𝑥_𝑛+1=(1−𝛽_𝑛)𝑧_𝑛+𝛽_𝑛𝑥_𝑛 for all 𝑛∈ℕ. It follows that 𝑧_𝑛=𝑥^𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛=𝛼^𝑛𝑓𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑣_𝑛1−𝛽_𝑛.(3.27) Thus, we have ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖=‖‖‖‖𝛼^𝑛+1𝑓𝑥_𝑛+1+𝛾_𝑛+1𝑆_𝑛+1𝑣_𝑛+11−𝛽_𝑛+1−𝛼^𝑛𝑓𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑣_𝑛1−𝛽_𝑛=‖‖‖𝛼_𝑛+1‖‖‖‖1−𝛽_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+1−𝛾𝛽^𝑛+1_𝑛+1𝑆_𝑛+1+𝛼_𝑛+1𝑣_𝑛+1−𝑆_𝑛𝑣_𝑛1−𝛽_𝑛+1−𝛼^𝑛1−𝛽_𝑛𝑓𝑥^𝑛+1−𝛾𝛽^𝑛+1_𝑛+1−𝛾^𝑛1−𝛽_𝑛𝑆^𝑛𝑣_𝑛≤𝛼_𝑛+1‖‖‖1−𝛽_𝑛+1‖‖𝑓𝑥_𝑛+1−𝑓𝑥_𝑛‖‖+𝛾^𝑛+11−𝛽_𝑛+1+||||𝛼_𝑛+1‖‖𝑆_𝑛+1𝑣_𝑛+1−𝑆_𝑛𝑣_𝑛‖‖1−𝛽_𝑛+1−𝛼^𝑛1−𝛽_𝑛||||‖‖𝑓𝑥_𝑛≤𝜓𝛼_𝑛+1−𝑆_𝑛𝑣_𝑛‖‖1−𝛽_𝑛+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+𝛾^𝑛+11−𝛽_𝑛+1+||||𝛼_𝑛+1‖‖𝑆_𝑛+1𝑣_𝑛+1−𝑆_𝑛𝑣_𝑛‖‖1−𝛽_𝑛+1−𝛼^𝑛1−𝛽_𝑛||||‖‖𝑓𝑥_𝑛−𝑆_𝑛𝑣_𝑛‖‖.(3.28) Now, compute ‖‖𝑆_𝑛+1𝑣_𝑛+1−𝑆_𝑛𝑣_𝑛‖‖≤≤‖‖‖‖𝑣𝑆_𝑛+1𝑛+1𝑣_𝑛+1−𝑆_𝑛+1𝑣_𝑛‖‖+‖‖𝑆_𝑛+1𝑣_𝑛−𝑆_𝑛𝑣_𝑛‖‖≤‖‖x_𝑛+1−𝑣_𝑛‖‖+‖‖𝑆_𝑛+1𝑣_𝑛−𝑆_𝑛𝑣_𝑛‖‖−𝑥_𝑛‖‖+𝑀¹𝑐||𝑟^𝑛+1+2𝑀₂−𝑟_𝑛||+||𝜉_𝑛−𝜉_𝑛+1||‖‖𝐵𝑦_𝑛‖‖||𝜆_𝑘,𝑛+1−𝜆_𝑘,𝑛||+‖‖𝑆_𝑛+1𝑣_𝑛−𝑆_𝑛𝑣_𝑛‖‖.(3.29) Combining (3.28) and (3.29), we have ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤𝜓𝛼^𝑛+11−𝛽_𝑛+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+𝛾^𝑛+11−𝛽_𝑛+1‖‖𝑥^𝑛+1−𝑥_𝑛‖‖+𝑀¹𝑐||𝑟^𝑛+1+2𝑀₂−𝑟_𝑛||+||𝜉_𝑛−𝜉_𝑛+1||‖‖𝐵𝑦_𝑛‖‖+||||𝛼_𝑛+1||𝜆_𝑘,𝑛+1−𝜆_𝑘,𝑛||+‖‖𝑆_𝑛+1𝑣_𝑛−𝑆_𝑛𝑣_𝑛‖‖1−𝛽_𝑛+1−𝛼^𝑛1−𝛽_𝑛||||‖‖𝑓𝑥_𝑛≤‖‖𝑥_𝑛+1−𝑆_𝑛𝑣_𝑛‖‖−𝑥_𝑛‖‖+𝛾^𝑛+11−𝛽_𝑛+1𝑀𝑐¹||𝑟^𝑛+1+2𝑀₂−𝑟_𝑛||+||𝜉_𝑛−𝜉_𝑛+1||‖‖𝐵𝑦_𝑛‖‖||𝜆_𝑘,𝑛+1−𝜆_𝑘,𝑛||+𝛾^𝑛+11−𝛽_𝑛+1+||||𝛼_𝑛+1‖‖𝑆_𝑛+1𝑣_𝑛−𝑆_𝑛𝑣_𝑛‖‖1−𝛽_𝑛+1−𝛼^𝑛1−𝛽_𝑛||||‖‖𝑓𝑥_𝑛−𝑆_𝑛𝑣_𝑛‖‖.(3.30) It follows that ‖‖𝑧≤_𝑛+1−𝑧𝛾_𝑛‖‖_𝑛+1−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖1−𝛽_𝑛+1𝑀𝑐¹||𝑟^𝑛+1+𝛾_𝑛+1−𝑟_𝑛||+||𝜉_𝑛−𝜉_𝑛+1||‖‖𝐵𝑦_𝑛‖‖+2𝑀₂||𝜆_𝑘,𝑛+1−𝜆_𝑘,𝑛||1−𝛽_𝑛+1‖‖𝑆_𝑛+1𝑣_𝑛−𝑆_𝑛𝑣_𝑛‖‖+||||𝛼^𝑛+11−𝛽_𝑛+1−𝛼^𝑛1−𝛽_𝑛||||‖‖𝑓𝑥_𝑛≤𝛾_𝑛+1−𝑆_𝑛𝑣_𝑛‖‖1−𝛽_𝑛+1𝑀𝑐¹||𝑟^𝑛+1+𝛾_𝑛+1−𝑟_𝑛||+||𝜉_𝑛−𝜉_𝑛+1||‖‖𝐵𝑦_𝑛‖‖+2𝑀₂||𝜆_𝑘,𝑛+1−𝜆_𝑘,𝑛||1−𝛽_𝑛+1sup‖‖𝑆_𝑛+1𝑧−𝑆_𝑛𝑧‖‖∶𝑧∈𝑣_𝑛+||||𝛼^𝑛+11−𝛽_𝑛+1−𝛼^𝑛1−𝛽_𝑛||||‖‖𝑓𝑥_𝑛−𝑆_𝑛𝑣_𝑛‖‖.(3.31) This together with conditions (C1)–(C6) and lim_𝑛→∞sup{‖𝑆_𝑛+1𝑧−𝑆_𝑛𝑧‖∶𝑧∈{𝑣_𝑛}}=0 implies that limsup_𝑛→∞‖‖𝑧_𝑛+1−𝑧_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(3.32) Hence, by Lemma 2.6, we obtain ‖𝑧_𝑛−𝑥_𝑛‖→0 as 𝑛→∞. It then follows that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞1−𝛽_𝑛‖‖𝑧_𝑛−𝑥_𝑛‖‖=0.(3.33) By (3.26), we also have lim_𝑛→∞‖‖𝑣_𝑛+1−𝑣_𝑛‖‖=0.(3.34)Step 3. We claim that lim_𝑛→∞‖𝑆𝑣_𝑛−𝑣_𝑛‖=0.
Since {𝐺_𝑘∶𝑘=1,2,3,…,𝑁} is 𝛽-inverse strongly monotone mappings, by the choice of {𝜆_𝑘,𝑛} for given 𝑥̃∈Ω and 𝑘∈{0,1,2,…,𝑁−1}, we also have ‖‖Υ^𝑛𝑘+1𝑢_𝑛−𝑥̃‖‖²=‖‖𝐽_{𝑀𝑘+1,𝜆𝑘+1,𝑛}𝐼−𝜆_𝑘+1,𝑛𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐽_{𝑀𝑘+1,𝜆𝑘+1,𝑛}𝐼−𝜆_𝑘+1,𝑛𝐺_𝑘+1𝑥̃‖‖²≤‖‖𝐼−𝜆_𝑘+1,𝑛𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐼−𝜆_𝑘+1,𝑛𝐺_𝑘+1𝑥̃‖‖²=‖‖Υ^𝑘𝑛𝑢_𝑛−𝜆_𝑘+1,𝑛𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝑥̃−𝜆_𝑘+1,𝑛𝐺_𝑘+1𝑥̃‖‖²=‖‖Υ^𝑘𝑛𝑢_𝑛−𝑥̃−𝜆_𝑘+1,𝑛𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖²=‖‖Υ^𝑘𝑛𝑢_𝑛−𝑥̃‖‖²−2𝜆_𝑘+1,𝑛Υ^𝑘𝑛𝑢_𝑛−𝑥̃,𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃+𝜆^2𝑘+1,𝑛‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖²≤‖‖Υ^𝑘𝑛𝑢_𝑛−𝑥̃‖‖²−2𝜆_𝑘+1,𝑛𝛽‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖+𝜆^2𝑘+1,𝑛‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖²≤‖‖𝑢_𝑛−𝑥̃‖‖²−2𝜆_𝑘+1,𝑛𝛽‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖+𝜆^2𝑘+1,𝑛‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖²≤‖‖𝑥_𝑛−𝑥̃‖‖²+𝜆_𝑘+1,𝑛𝜆_𝑘+1,𝑛−2𝛽‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖².(3.35) Form (3.13), we have ‖‖𝑥_𝑛+1−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑣_𝑛−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑦_𝑛−𝑥̃‖‖²=𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖Υ^𝑁𝑛𝑢_𝑛−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖Υ^𝑛𝑘+1𝑢_𝑛−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²≤+𝛼_𝑛𝛾_𝑛‖‖𝑥^𝑛−𝑥̃‖‖²+𝜆_𝑘+1,𝑛𝜆_𝑘+1,𝑛−2𝛽‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖²‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛𝜆_𝑘+1,𝑛𝜆_𝑘+1,𝑛−2𝛽‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖².(3.36) It follows that 𝛾_𝑛𝜆_𝑘+1,𝑛2𝛽−𝜆_𝑘+1,𝑛‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖²≤𝛾_𝑛𝑐(2𝛽−𝑑)‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖²≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑥̃‖‖+‖‖𝑥_𝑛+1−𝑥̃‖‖+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖².(3.37) By condition (C2), (3.33), and liminf_𝑛→∞𝛾_𝑛>0, we obtain lim_𝑛→∞‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖=0.(3.38) From Lemma 2.3(2) and as 𝐼−𝜆_𝑘+1,𝑛𝐺_𝑘+1 is nonexpansive, we have ‖‖Υ^𝑛𝑘+1𝑢_𝑛−𝑥̃‖‖²=‖‖𝐽_{𝑀𝑘+1,𝜆𝑘+1,𝑛}𝐼−𝜆_𝑘+1,𝑛𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐽_{𝑀𝑘+1,𝜆𝑘+1,𝑛}𝐼−𝜆_𝑘+1,𝑛𝐺_𝑘+1𝑥̃‖‖²≤=1𝐼−𝜆_𝑘+1,𝑛𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐼−𝜆_𝑘+1,𝑛𝐺_𝑘+1𝑥̃,Υ^𝑛𝑘+1𝑢_𝑛−𝑥̃2‖‖𝐼−𝜆^𝑘+1,𝑛𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐼−𝜆_𝑘+1,𝑛𝐺_𝑘+1𝑥̃‖‖²+‖‖Υ^𝑛𝑘+1𝑢_𝑛−𝑥̃‖‖²≤−1‖‖𝐼−𝜆_𝑘+1,𝑛𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐼−𝜆_𝑘+1,𝑛𝐺_𝑘+1𝑥̃−Υ^𝑛𝑘+1𝑢_𝑛−𝑥̃‖‖²2‖‖Υ^𝑘𝑛≤1𝑢_𝑛−𝑥̃‖‖²+‖‖Υ^𝑛𝑘+1𝑢_𝑛−𝑥̃‖‖²−‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛−𝜆_𝑘+1,𝑛𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖²2‖‖Υ^𝑘𝑛𝑢_𝑛−𝑥̃‖‖²+‖‖Υ^𝑛𝑘+1𝑢_𝑛−𝑥̃‖‖²−‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖²−𝜆^2𝑘+1,𝑛‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖²+2𝜆_𝑘+1,𝑛Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛,𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃,(3.39) which yields that ‖‖Υ^𝑛𝑘+1𝑢_𝑛−𝑥̃‖‖²≤≤‖‖‖‖Υ𝑢^𝑘𝑛𝑢_𝑛−𝑥̃‖‖²−‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖²+2𝜆_𝑘+1,𝑛‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥_𝑛̃‖‖≤‖‖𝑥−𝑥̃‖‖²−‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖²+2𝜆_𝑘+1,𝑛‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥_𝑛̃‖‖−𝑥̃‖‖²−‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖²+2𝜆_𝑘+1,𝑛‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖.(3.40) Substituting (3.40) into (3.36), we obtain ‖‖𝑥_𝑛+1−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖Υ^𝑛𝑘+1𝑢_𝑛−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑥^𝑛−𝑥̃‖‖²−‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖²≤𝛼_𝑛+2𝜆_𝑘+1,𝑛‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+‖‖𝑥_𝑛−𝑥̃‖‖²−𝛾_𝑛‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖²+2𝜆_𝑘+1,𝑛𝛾_𝑛‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖.(3.41) It follows that 𝛾_𝑛‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖²≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑥̃‖‖+‖‖𝑥_𝑛+1−𝑥̃‖‖+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+2𝜆_𝑘+1,𝑛𝛾_𝑛‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖‖‖𝐺_𝑘+1Υ^𝑘𝑛𝑢_𝑛−𝐺_𝑘+1𝑥̃‖‖.(3.42) By condition (C2), (3.33), (3.38), and liminf_𝑛→∞𝛾_𝑛>0, we obtain lim_𝑛→∞‖‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖‖=0.(3.43) For 𝑥̃∈Ω, we obtain ‖‖𝑣_𝑛−𝑥̃‖‖²=‖‖𝑃_𝐶𝑦_𝑛−𝜉_𝑛𝐵𝑦_𝑛−𝑃_𝐶𝑥̃−𝜉_𝑛𝐵𝑥̃‖‖²≤‖‖𝑦_𝑛−𝜉_𝑛𝐵𝑦_𝑛−𝑥̃−𝜉_𝑛𝐵𝑥̃‖‖²=‖‖𝑦_𝑛−𝑥̃−𝜉_𝑛𝐵𝑦_𝑛−𝐵𝑥̃‖‖²≤‖‖𝑦_𝑛−𝑥̃‖‖²−2𝜉_𝑛⟨𝑦_𝑛−𝑥̃,𝐵𝑦_𝑛−𝐵𝑥̃⟩+𝜉^2𝑛‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖²≤‖‖𝑦_𝑛−𝑥̃‖‖²−2𝜉_𝑛−𝜙‖‖𝐵𝑦^𝑛−𝐵𝑥̃‖‖²+𝜔‖‖𝑦_𝑛−𝑥̃‖‖²+𝜉^2𝑛‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖²≤‖‖𝑦_𝑛−𝑥̃‖‖²+2𝜉_𝑛𝜙‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖²−2𝜉_𝑛𝜔‖‖𝑦_𝑛−𝑥̃‖‖²+𝜉^2𝑛‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖²≤‖‖𝑦_𝑛−𝑥̃‖‖²+2𝜉_𝑛𝜙‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖²−2𝜉^𝑛𝜔𝜇²‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖²+𝜉^2𝑛‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖²≤‖‖𝑥_𝑛−𝑥̃‖‖²+2𝜉_𝑛𝜙+𝜉^2𝑛−2𝜉^𝑛𝜔𝜇²‖‖𝐵𝑦^𝑛−𝐵𝑥̃‖‖².(3.44) On the other hand, we have ‖‖𝑥_𝑛+1−𝑥̃‖‖²=‖‖𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑣_𝑛−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑆_𝑛𝑣_𝑛−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑣_𝑛−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑥^𝑛−𝑥̃‖‖²+2𝜉_𝑛𝜙+𝜉^2𝑛−2𝜉^𝑛𝜔𝜇²‖‖𝐵𝑦^𝑛−𝐵𝑥̃‖‖²=𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛2𝜉^𝑛𝜙+𝜉^2𝑛−2𝜉^𝑛𝜔𝜇²‖‖𝐵𝑦^𝑛−𝐵𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛2𝜉^𝑛𝜙+𝜉^2𝑛−2𝜉^𝑛𝜔𝜇²‖‖𝐵𝑦^𝑛−𝐵𝑥̃‖‖².(3.45) It follows that 2𝜇𝑎²𝜔−𝑏²−2𝑏𝜙𝛾_𝑛‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖²≤2𝜉^𝑛𝜔𝜇²−𝜉^2𝑛−2𝜉_𝑛𝜙𝛾_𝑛‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖²≤‖‖𝑥_𝑛−𝑥̃‖‖²−‖‖𝑥_𝑛+1−𝑥̃‖‖²+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑥̃‖‖+‖‖𝑥_𝑛+1−𝑥̃‖‖+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖².(3.46) It now follows from the last inequality, conditions (C2), (3.33), and liminf_𝑛→∞𝛾_𝑛>0 that lim_𝑛→∞‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖=0.(3.47) Since 𝑃_𝐶 is firmly nonexpansive, we have ‖‖𝑣_𝑛−𝑥̃‖‖²=‖‖𝑃_𝐶𝑦_𝑛−𝜉_𝑛𝐵𝑦_𝑛−𝑃_𝐶𝑥̃−𝜉_𝑛𝐵𝑥̃‖‖²=‖‖𝑃_𝐶𝐼−𝜉_𝑛𝐵𝑦_𝑛−𝑃_𝐶𝐼−𝜉_𝑛𝐵𝑥̃‖‖²≤=1𝐼−𝜉_𝑛𝐵𝑦_𝑛−𝐼−𝜉_𝑛𝐵𝑥̃,𝑣_𝑛−𝑥̃2‖‖𝐼−𝛼^𝑛𝐵𝑦_𝑛−𝐼−𝜉_𝑛𝐵𝑥̃‖‖²+‖‖𝑣_𝑛−𝑥̃‖‖²≤1−‖‖𝐼−𝜉_𝑛𝐵𝑦_𝑛−𝐼−𝜉_𝑛𝐵𝑥̃−𝑣_𝑛−𝑥̃‖‖²2‖‖𝑦^𝑛≤1−𝑥̃‖‖²+‖‖𝑣_𝑛−𝑥̃‖‖²−‖‖𝑦_𝑛−𝑣_𝑛−𝜉_𝑛𝐵𝑦_𝑛−𝐵𝑥̃‖‖²2‖‖𝑦^𝑛−𝑥̃‖‖²+‖‖𝑣_𝑛−𝑥̃‖‖²−‖‖𝑦_𝑛−𝑣_𝑛‖‖²−𝜉^2𝑛‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖²+2𝜉_𝑛⟨𝑦_𝑛−𝑣_𝑛,𝐵𝑦_𝑛−𝐵𝑥̃⟩,(3.48) which yields that ‖‖𝑣_𝑛−𝑥̃‖‖²≤≤‖‖‖‖𝑦𝑥_𝑛−𝑥̃‖‖²−‖‖𝑦_𝑛−𝑣_𝑛‖‖²+2𝜉_𝑛‖‖𝑦_𝑛−𝑣_𝑛‖‖‖‖𝐵𝑦_𝑛−𝐵𝑥_𝑛̃‖‖−𝑥̃‖‖²−‖‖𝑦_𝑛−𝑣_𝑛‖‖²+2𝜉_𝑛‖‖𝑦_𝑛−𝑣_𝑛‖‖‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖.(3.49) Substituting (3.49) into (3.45), we obtain ‖‖𝑥_𝑛+1−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑣_𝑛−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²=+𝛼_𝑛𝛾_𝑛‖‖𝑥^𝑛−𝑥̃‖‖²−‖‖𝑦_𝑛−𝑣_𝑛‖‖²+2𝜉_𝑛‖‖𝑦_𝑛−𝑣_𝑛‖‖‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²−𝛾_𝑛‖‖𝑦_𝑛−𝑣_𝑛‖‖²≤+𝛼_𝑛2𝛾_𝑛𝜉_𝑛‖‖𝑦_𝑛−𝑣_𝑛‖‖‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+‖‖𝑥_𝑛−𝑥̃‖‖²−𝛾_𝑛‖‖𝑦_𝑛−𝑣_𝑛‖‖²+2𝛾_𝑛𝜉_𝑛‖‖𝑦_𝑛−𝑣_𝑛‖‖‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖.(3.50) It follows that 𝛾_𝑛‖‖𝑦_𝑛−𝑣_𝑛‖‖²≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑥̃‖‖+‖‖𝑥_𝑛+1−𝑥̃‖‖+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+2𝛾_𝑛𝜉_𝑛‖‖𝑦_𝑛−𝑣_𝑛‖‖‖‖𝐵𝑦_𝑛−𝐵𝑥̃‖‖.(3.51) By condition (C2), (3.33), (3.47), and liminf_𝑛→∞𝛾_𝑛>0, we obtain lim_𝑛→∞‖‖𝑦_𝑛−𝑣_𝑛‖‖=0.(3.52) On the other hand, in the light of Lemma 2.11(ii) 𝑇_𝑟𝑛 is firmly nonexpansive; so we have ‖‖𝑢_𝑛−𝑥̃‖‖²=‖‖𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑥̃‖‖²≤=1𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑥̃,𝑥_𝑛−𝑥̃=⟨𝑢_𝑛−𝑥̃,𝑥_𝑛−𝑥̃⟩2‖‖𝑢^𝑛−𝑥̃‖‖²+‖‖𝑥_𝑛−𝑥̃‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖²,(3.53) which implies that ‖‖𝑢_𝑛−𝑥̃‖‖²≤‖‖𝑥_𝑛−𝑥̃‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖².(3.54) Form (3.45), we have ‖‖𝑥_𝑛+1−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑣_𝑛−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑦_𝑛−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑢_𝑛−𝑥̃‖‖²=≤𝛼𝛼^𝑛_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑥^𝑛−𝑥̃‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖²‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛾_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²−𝛾_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+‖‖𝑥_𝑛−𝑥̃‖‖²−𝛾_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖².(3.55) It follows that 𝛾_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖²≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑥̃‖‖+‖‖𝑥_𝑛+1−𝑥̃‖‖+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖².(3.56) By condition (C2), (3.33), and liminf_𝑛→∞𝛾_𝑛>0, we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑢_𝑛‖‖=0.(3.57) Observe that 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑣_𝑛−𝑥_𝑛.(3.58) By condition (C2) and (3.33), we have lim_𝑛→∞𝛾_𝑛‖‖𝑆_𝑛𝑣_𝑛−𝑥_𝑛‖‖=lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖−𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥_𝑛‖‖=0.(3.59) Since ‖‖𝑆𝑣_𝑛−𝑢_𝑛‖‖≤‖‖𝑆𝑣_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑢_𝑛‖‖,(3.60) from (3.57) and (3.59), we have lim_𝑛→∞‖‖𝑆_𝑛𝑣_𝑛−𝑢_𝑛‖‖=0.(3.61) Form (3.55), we have ‖‖𝑥_𝑛+1−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+‖‖𝑥_𝑛−𝑥̃‖‖²−𝛾_𝑛‖‖𝑥_𝑛−𝑢_𝑛‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+‖‖𝑥_𝑛−𝑥̃‖‖²−𝛾_𝑛‖‖𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑢_𝑛‖‖²=≤𝛼𝛼^𝑛_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+‖‖𝑥_𝑛−𝑥̃‖‖²−𝛾_𝑛‖‖𝑥^𝑛−𝑦_𝑛‖‖²+2‖‖𝑥_𝑛−𝑦_𝑛‖‖‖‖𝑦_𝑛−𝑢_𝑛‖‖+‖‖𝑦_𝑛−𝑢_𝑛‖‖²‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+‖‖𝑥_𝑛−𝑥̃‖‖²−𝛾_𝑛‖‖𝑥_𝑛−𝑦_𝑛‖‖²−2𝛾_𝑛‖‖𝑥_𝑛−𝑦_𝑛‖‖‖‖𝑦_𝑛−𝑢_𝑛‖‖−𝛾_𝑛‖‖𝑦_𝑛−𝑢_𝑛‖‖²≤𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖²+‖‖𝑥_𝑛−𝑥̃‖‖²−𝛾_𝑛‖‖𝑥_𝑛−𝑦_𝑛‖‖².(3.62) It follows that 𝛾_𝑛‖‖𝑥_𝑛−𝑦_𝑛‖‖²≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑥̃‖‖+‖‖x_𝑛+1−𝑥̃‖‖+𝛼_𝑛‖‖𝑓𝑥_𝑛−𝑥̃‖‖².(3.63) By condition (C2), (3.33), and liminf_𝑛→∞𝛾_𝑛>0, we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑦_𝑛‖‖=0.(3.64) Since ‖‖𝑢_𝑛−𝑦_𝑛‖‖≤‖‖𝑢_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑦_𝑛‖‖,(3.65) from (3.57) and (3.64), we have lim_𝑛→∞‖‖𝑢_𝑛−𝑦_𝑛‖‖=0.(3.66) Furthermore, by the triangular inequality we also have ‖‖𝑆_𝑛𝑣_𝑛−𝑣_𝑛‖‖≤‖‖𝑆_𝑛𝑣_𝑛−𝑢_𝑛‖‖+‖‖𝑢_𝑛−𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑣_𝑛‖‖.(3.67) From (3.52), (3.61), and (3.66), we have lim_𝑛→∞‖‖𝑆_𝑛𝑣_𝑛−𝑣_𝑛‖‖=0.(3.68) Applying Lemma 2.9 and (3.68), we have ‖‖𝑆𝑣_𝑛−𝑣_𝑛‖‖≤≤s‖‖u𝑆p𝑣_𝑛−‖‖𝑆𝑆𝑧_𝑛−𝑣_𝑛𝑆‖‖_𝑛+‖‖𝑆_𝑛𝑣_𝑛−𝑣_𝑛‖‖𝑧‖‖∶𝑧∈𝑣_𝑛+‖‖𝑆_𝑛𝑣_𝑛−𝑣_𝑛‖‖⟶0.(3.69)Step 4. We claim that limsup_𝑛→∞⟨𝑓(𝑥^∗)−𝑥^∗,𝑥_𝑛−𝑥^∗⟩≤0.
Indeed, we choose a subsequence {𝑣_𝑛𝑖} of {𝑣_𝑛} such that limsup_𝑛→∞𝑓𝑥^∗−𝑥^∗,𝑆𝑣_𝑛−𝑥^∗=lim_𝑖→∞𝑓𝑥^∗−𝑥^∗,𝑆𝑣_𝑛𝑖−𝑥^∗.(3.70) Without loss of generality, let {𝑣_𝑛𝑖}⇀𝑧∈𝐶. From ‖𝑆𝑣_𝑛−𝑣_𝑛‖→0, we obtain 𝑆𝑣_𝑛𝑖⇀𝑧. Then, (3.70) reduces to limsup_𝑛→∞𝑓𝑥^∗−𝑥^∗,𝑆𝑣_𝑛−𝑥^∗=𝑓𝑥^∗−𝑥^∗,𝑧−𝑥^∗.(3.71) In order to show ⟨f(𝑥^∗)−𝑥^∗,𝑧−𝑥^∗⟩≤0, it suffices to show that 𝑧∈Ω∶^∞_𝑛=1𝐹𝑆_𝑛∩^𝑁_𝑘=1𝐼𝐺_𝑘,𝑀_𝑘∩VI(𝐶,𝐵)∩EP(𝐹).(3.72) Firstly, we will show 𝑧∈𝐹(𝑆)=⋂^∞𝑛=1𝐹(𝑆_𝑛).
Assume 𝑧∉𝐹(𝑆). By Opial’s theorem (Lemma 2.5) and ‖𝑆𝑣_𝑛−𝑣_𝑛‖→0, we have liminf_𝑖→∞‖‖𝑣_𝑛𝑖−𝑧‖‖=<lliimmiinnf_𝑖→∞f_𝑖→∞‖‖𝑣_𝑛𝑖−𝑆𝑧‖‖=liminf_𝑖→∞‖‖𝑣_𝑛𝑖−𝑆𝑣_𝑛𝑖+𝑆𝑣_𝑛𝑖−𝑆𝑧‖‖≤liminf_𝑖→∞‖‖𝑆𝑣_𝑛𝑖−𝑆𝑧‖‖‖‖𝑣_𝑛𝑖−𝑧‖‖.(3.73) This is a contradiction. Thus, we obtain 𝑧∈𝐹(𝑆).
Next, we will show that 𝑧∈VI(𝐶,𝐵).
Let 𝑇𝑤₁=𝐵𝑤¹∅,+𝑁_𝐶𝑤₁,𝑤𝑤₁₁∈𝐶;∉𝐶.(3.74) Since 𝐵 is relaxed (𝜙,𝜔)-cocoercive, 𝜇-Lipschitz continuous with 𝜔>𝜙𝜇², we obtain ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥(−𝜙)‖𝐵𝑥−𝐵𝑦‖²+𝜔‖𝑥−𝑦‖²≥𝜔−𝜙𝜇²‖𝑥−𝑦‖²≥0,(3.75) which yields that 𝐵 is monotone. Then 𝑇 is maximal monotone (see [23]). Let (𝑤₁,𝑤₂)∈𝐺(𝑇). Since 𝑤₂−𝐵𝑤₁∈𝑁_𝐶(𝑤₁) and 𝑣_𝑛∈𝐶, we have ⟨𝑤₁−𝑣_𝑛,𝑤₂−𝐵𝑤₁⟩≥0. On the other hand, from 𝑣_𝑛=𝑃_𝐶(𝑦_𝑛−𝜉_𝑛𝐵𝑦_𝑛), we have 𝑤₁−𝑣_𝑛,𝑣_𝑛−𝑦_𝑛−𝜉_𝑛𝐵𝑦_𝑛≥0;(3.76) that is, 𝑤¹−𝑣_𝑛,𝑣^𝑛−𝑦_𝑛𝜉_𝑛+𝐵𝑦_𝑛≥0.(3.77) Therefore, we obtain 𝑤₁−𝑣_𝑛𝑖,𝑤₂≥≥𝑤𝑤₁₁−𝑣_𝑛𝑖,𝐵𝑤₁−𝑣_𝑛𝑖,𝐵𝑤₁−𝑤₁−𝑣_𝑛𝑖,𝑣^𝑛𝑖−𝑦_𝑛𝑖𝜉_𝑛𝑖+𝐵𝑦_𝑛𝑖=𝑤₁−𝑣_𝑛𝑖,𝐵𝑤₁−𝐵𝑦_𝑛𝑖−𝑣^𝑛𝑖−𝑦_𝑛𝑖=𝑤₁𝜉_𝑛𝑖−𝑤−₁𝑣_𝑛𝑖,𝐵𝑤₁−𝐵𝑣_𝑛𝑖+𝑤₁−𝑣_𝑛𝑖,𝐵𝑣_𝑛𝑖−𝐵𝑦_𝑛𝑖−𝑣_𝑛𝑖,𝑣^𝑛𝑖−𝑦_𝑛𝑖≥𝑤₁𝜉_𝑛𝑖−𝑣_𝑛𝑖,𝐵𝑣_𝑛𝑖−𝑤₁−𝑣_𝑛𝑖,𝑣^𝑛𝑖−𝑦_𝑛𝑖𝜉_𝑛𝑖+𝐵𝑦_𝑛𝑖=𝑤₁−𝑣_𝑛𝑖,𝐵𝑣_𝑛𝑖−𝐵𝑦_𝑛𝑖−𝑤₁−𝑣_𝑛𝑖,𝑣^𝑛𝑖−𝑦_𝑛𝑖𝜉_𝑛𝑖.(3.78) Noting that ‖𝑣_𝑛𝑖−𝑦_𝑛𝑖‖→0 and 𝐵 is relaxed (𝜙,𝜔)-cocoercive and (3.78), we obtain ⟨𝑤₁−𝑧,𝑤₂⟩≥0.(3.79) Since 𝑇 is maximal monotone, we have 𝑧∈𝑇⁻¹0, and hence 𝑧∈VI(𝐶,𝐵).
Now, we will show that 𝑧∈⋂^𝑁𝑘=1𝐼(𝐺_𝑘,𝑀_𝑘).
For this purpose, let 𝑘∈{1,2,3,…,𝑁} and 𝐺_𝑘 is 𝛽-inverse strongly monotone, 𝐺_𝑘 is an 1/𝛽-Lipschitz continuous monotone mapping. From Lemma 2.2, we know that 𝑀_𝑘+𝐺_𝑘 is maximal monotone. Let (𝑣,𝑔)∈𝐺(𝑀_𝑘+𝐺_𝑘); that is, 𝑔−𝐺_𝑘𝑣∈𝑀_𝑘(𝑣). On the other hand, since Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖=𝐽_{𝑀𝑘,𝜆𝑖𝑘,𝑛}(Υ^𝑛𝑘−1_i𝑢_𝑛𝑖−𝜆_{𝑘,𝑛𝑖}𝐺_𝑘Υ^𝑛𝑘−1_𝑖𝑢_𝑛𝑖), we have Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖−𝜆_{𝑘,𝑛𝑖}𝐺_𝑘Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖∈𝐼+𝜆_{𝑘,𝑛𝑖}𝑀_𝑘Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖;(3.80) that is, 1𝜆_{𝑘,𝑛𝑖}Υ^𝑛𝑘−1_𝑖𝑢_𝑛𝑖−Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖−𝜆_{𝑘,𝑛𝑖}𝐺_𝑘Υ^𝑛𝑘−1_𝑖𝑢_𝑛𝑖∈𝑀_𝑘Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖.(3.81) By virtue of the maximal monotonicity of 𝑀_𝑘+𝐺_𝑘, we have 𝑣−Υ^𝑘𝑛𝑖𝑢_𝑛𝑖,𝑔−𝐺_𝑘𝑣−1𝜆_{𝑘,𝑛𝑖}Υ^𝑛𝑘−1_𝑖𝑢_𝑛𝑖−Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖−𝜆_{𝑘,𝑛𝑖}𝐺_𝑘Υ^𝑛𝑘−1_𝑖𝑢_𝑛𝑖≥0,(3.82) and so 𝑣−Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖,𝑔≥=𝑣𝑣−−ΥΥ^𝑘𝑛_{𝑖𝑘𝑛𝑖}𝑢_𝑛𝑖,𝐺_𝑘𝑣+𝜆1_{𝑘,𝑛𝑖}Υ^𝑛𝑘−1_𝑖𝑢_𝑛𝑖−Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖−𝜆_{𝑘,𝑛𝑖}𝐺_𝑘Υ^𝑛𝑘−1_𝑖𝑢_𝑛𝑖𝑢_𝑛𝑖,𝐺_𝑘𝑣−𝐺_𝑘Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖+𝐺_𝑘Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖−𝐺_𝑘Υ^𝑛𝑘−1_𝑖𝑢_𝑛𝑖+1𝜆_{𝑘,𝑛𝑖}Υ^𝑛𝑘−1_𝑖≥0+𝑣−Υ_𝑘𝑛𝑖𝑢_𝑛𝑖−Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖+𝑣−Υ_𝑘𝑛𝑖𝑢_𝑛𝑖,𝐺_𝑘Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖−𝐺_𝑘Υ^𝑛𝑘−1_𝑖𝑢_𝑛𝑖𝑢_𝑛𝑖,1𝜆_{𝑘,𝑛𝑖}Υ^𝑛𝑘−1_𝑖𝑢_𝑛𝑖−Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖.(3.83) From ‖Υ^𝑘𝑛𝑢_𝑛−Υ^𝑛𝑘+1𝑢_𝑛‖→0, we also obtain that Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖⇀𝑧 and {𝐺_𝑘∶𝑘=1,2,3,…,𝑁} are Lipschitz continuous; we have lim_𝑛→∞𝑣−Υ^𝑘𝑛_𝑖𝑢_𝑛𝑖,𝑔=⟨𝑣−𝑧,𝑔⟩≥0.(3.84) Since 𝑀_𝑘+𝐺_𝑘 is maximal monotone, we have 𝜃∈(𝑀_𝑘+𝐺_𝑘)(𝑧); that is, 𝑧∈⋂^𝑁𝑘=1𝐼(𝐺_𝑘,𝑀_𝑘).
Finally, we will show that 𝑧∈EP(𝐹).
Since 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛, we have 𝐹𝑢_𝑛,𝑦+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶.(3.85) If follows from (A2) that 1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥−𝐹𝑢_𝑛,𝑦≥𝐹𝑦,𝑢_𝑛,(3.86) and hence 𝑦−𝑢^𝑛𝑖,𝑢^𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖≥𝐹𝑦,𝑢^𝑛𝑖.(3.87) Since (𝑢_𝑛𝑖−𝑥_𝑛𝑖)/𝑟_𝑛𝑖→0 and 𝑢_𝑛𝑖⇀𝑧, it follows by (A4) that 𝐹(𝑦,𝑧)≤0 for all 𝑦∈𝐻. For 𝑡 with 0<𝑡≤1 and 𝑦∈𝐻, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑧. Since 𝑦∈𝐻 and 𝑧∈𝐻, we have 𝑦_𝑡∈𝐻, and hence 𝐹(𝑦_𝑡,𝑧)≤0. So, from (A1) and (A4) we have 0=𝐹𝑦_𝑡,𝑦_𝑡≤𝑡𝐹𝑦_𝑡,𝑦+(1−𝑡)𝐹𝑦_𝑡,𝑧≤𝑡𝐹𝑦_𝑡,𝑦,(3.88) and hence 𝐹(𝑦_𝑡,𝑦)≥0. From (A3), we have 𝐹(𝑧,𝑦)≥0 for all 𝑦∈𝐻 and hence 𝑧∈EP(𝐹). Therefore, it follows that 𝑧∈Ω.
Since 𝑥^∗=𝑃_Ω𝑓(𝑥^∗), we have limsup_𝑛→∞𝑓𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗==lliimm_𝑖→∞sup_𝑛→∞𝑓𝑥^∗−𝑥^∗,𝑆𝑣_𝑛−𝑥^∗=𝑓𝑥_∗𝑓𝑥^∗−𝑥^∗,𝑆𝑣_𝑛𝑖−𝑥^∗−𝑥^∗,𝑧−𝑥^∗≤0.(3.89) On the other hand, we have limsup_𝑛→∞𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤+limlimsusp_𝑛→∞up_𝑛→∞𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥_𝑛𝑓𝑥^∗−𝑥^∗,𝑥_𝑛−𝑥^∗.(3.90) Since ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞ and (3.91), we have limsup_𝑛→∞𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤0.(3.91)Step 5. We claim that lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖=0.
Indeed, from (3.2) and (3.12), we obtain ‖‖𝑥_𝑛+1−𝑥^∗‖‖²==𝛼_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑣_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1𝑓𝑥_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛾_𝑛𝑆_𝑛𝑣_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗2𝛽^𝑛‖‖𝑥^𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛+1−𝑥^∗‖‖²+12𝛾^𝑛‖‖𝑣^𝑛+𝛼_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1𝑓𝑥_𝑛−𝑓𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗21−𝛼^𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛+1−𝑥^∗‖‖²+12𝛼^𝑛‖‖𝑓𝑥^𝑛−𝑓𝑥^∗‖‖²+𝛼_𝑛+‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗21−𝛼^𝑛1−𝜓²‖‖𝑥_𝑛−𝑥^∗‖‖²+121−𝛼^𝑛‖‖𝑥_𝑛+1−𝑥^∗‖‖²+12𝛼^𝑛‖‖𝑥_𝑛+1−𝑥^∗‖‖²+𝛼_𝑛𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗,(3.92) which implies that ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤=11−−𝑏𝛼_𝑛𝑛1−𝜓²‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛿_𝑛,(3.93) where 𝑏_𝑛=𝛼_𝑛(1−𝜓²) and 𝛿_𝑛=2𝛼_𝑛⟨𝑓(𝑥^∗)−𝑥^∗,𝑥_𝑛+1−𝑥^∗⟩. It is easy to see that 𝑏_𝑛→0, ∑^∞𝑛=1𝑏_𝑛=∞, and limsup_𝑛→∞𝛿_𝑛/𝑏_𝑛≤0. Applying Lemma 2.7 to (3.93), we conclude that 𝑥_𝑛→𝑥^∗=𝑃_Ω𝑓𝑥^∗.(3.94) Consequently, also {𝑦_𝑛} converges strongly to 𝑥^∗. The proof is now complete.

As in [21, Theorem  4.1], we can generate a sequence {𝑆_𝑛} of nonexpansive mappings satisfying condition ∑^∞𝑛=1sup{‖𝑆_𝑛+1𝑧−𝑆_𝑛𝑧‖∶𝑧∈𝐾}<∞ for any bounded subset 𝐾 of 𝐶 by using convex combination of general sequence {𝑇_𝑘} of nonexpansive mappings with a common fixed point.

Corollary 3.2. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝐵∶𝐶→𝐻 be relaxed (𝜙,𝜔)-cocoercive and 𝜇-Lipschitz continuous with 𝜔>𝜙𝜇², for some 𝜙,𝜔,𝜇>0. Let 𝒢={𝐺_𝑘∶𝑘=1,2,3,…,𝑁} be a finite family of 𝛽-inverse strongly monotone mappings from 𝐶 into 𝐻, and let 𝐹 be a bifunction from 𝐶×𝐶→ℝ satisfying (A1)–(A4). Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜓(0≤𝜓<1), and let {𝛿^𝑘𝑛} be a family of nonnegative numbers with indices 𝑛,𝑘∈ℕ with 𝑘≤𝑛 such that Ω∶𝐹^∞_𝑘=1𝐹𝑇_𝑘∩^𝑁_𝑘=1𝐼𝐺_𝑘,𝑀_𝑘∩VI(𝐶,𝐵)∩EP(𝐹)≠∅.(3.95) Let the sequences {𝑥_𝑛} and {𝑦_𝑛} be generated by 𝑥𝑦₁=𝑥∈Cchosenarbitrarily,_𝑛𝑥_𝑛+1=𝐽_{𝑀𝑁,𝜆𝑁,𝑛}𝐼−𝜆_𝑁,𝑛𝐺_𝑛⋅⋅⋅𝐽_{𝑀2,𝜆2,𝑛}𝐼−𝜆_2,𝑛𝐺₂𝐽_{𝑀1,𝜆1,𝑛}𝐼−𝜆_1,𝑛𝐺₁𝑇_𝑟𝑛𝑥_𝑛,=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑛_𝑘=1𝛿^𝑘𝑛𝑇_𝑘𝑃_𝐶𝑦_𝑛−𝜉_𝑛𝐵𝑦_𝑛,∀𝑛≥1,(3.96) where {𝛼_𝑛},{𝛽_𝑛}, {𝛾_𝑛}⊂(0,1) and {𝜉_𝑛},{𝑟_𝑛}⊂(0,∞) satisfy the following conditions:(C1)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,(C2)lim_𝑛→∞𝛼_𝑛=0,∑^∞𝑛=1𝛼_𝑛=∞, (C3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C4){𝜉_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0≤𝑎≤𝑏≤2(𝜔−𝜙𝜇²)/𝜇² and lim_𝑛→∞|𝜉_𝑛+1−𝜉_𝑛|=0,(C5){𝜆_𝑘,𝑛}^𝑁𝑘=1⊂[𝑐,𝑑]⊂(0,2𝛽) and lim_𝑛→∞|𝜆_𝑘,𝑛+1−𝜆_𝑘,𝑛|=0, for each 𝑘∈{1,2,…,𝑁},(C6)liminf_𝑛→∞𝑟_𝑛>0 and lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0,(C7)∑^𝑛𝑘=1𝛿^𝑘𝑛,forall𝑛∈ℕ, lim_𝑛→∞𝛿^𝑘𝑛>0,forall𝑘∈ℕ and ∑^𝑘𝑛=1∑^𝑛𝑘=1|𝛿^𝑘𝑛+1−𝛿^𝑘𝑛|<∞.Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to the same point 𝑥^∗∈Ω, where 𝑥^∗=𝑃_Ω𝑓(𝑥^∗).

In Theorem 3.1, taking 𝑁=1 and 𝑆_𝑛=𝑆, then we have the following corollary.

Corollary 3.3. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝐵∶𝐶→𝐻 be relaxed (𝜙,𝜔)-cocoercive and 𝜇-Lipschitz continuous with 𝜔>𝜙𝜇², for some 𝜙,𝜔,𝜇>0. Let 𝐺 be an 𝛽-inverse strongly monotone mappings from 𝐶 into 𝐻, and let 𝐹 be a bifunction from 𝐶×𝐶→ℝ satisfying (A1)–(A4). Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜓(0≤𝜓<1), and let 𝑆 be a nonexpansive mappings of 𝐶 into itself such that Ω∶𝐹(𝑆)∩𝐼(𝐺,𝑀)∩VI(𝐶,𝐵)∩EP(𝐹)≠∅.(3.97) Let the sequences {𝑥_𝑛} and {𝑦_𝑛} be generated by 𝐹𝑢𝑥₁=𝑥∈Cchosenarbitrarily,_𝑛,𝑦+1𝑟_𝑛𝑦⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝑥_𝑛+1=𝐽_{𝑀,𝜆𝑛}𝐼−𝜆_𝑛𝐺𝑢_𝑛,=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑃_𝐶𝑦_𝑛−𝜉_𝑛𝐵𝑦_𝑛,∀𝑛≥1,(3.98) where {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}⊂(0,1) and {𝜉_𝑛},{𝑟_𝑛}⊂(0,∞) satisfy the following conditions:(C1)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,(C2)lim_𝑛→∞𝛼_𝑛=0,∑^∞𝑛=1𝛼_𝑛=∞, (C3)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C4){𝜉_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0≤𝑎≤𝑏≤2(𝜔−𝜙𝜇²)/𝜇² and lim_𝑛→∞|𝜉_𝑛+1−𝜉_𝑛|=0,(C5){𝜆_𝑛}⊂[𝑐,𝑑]⊂(0,2𝛽) and lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=0,(C6)liminf_𝑛→∞𝑟_𝑛>0 and lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0. Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to the same point 𝑥^∗∈Ω, where 𝑥^∗=𝑃_Ω𝑓(𝑥^∗).

Acknowledgments

This research was supported by the Rajamangala University of Technology Rattanakosin Research and Development Institute, the Thailand Research Fund, and the Commission on Higher Education under Grant no. MRG5480206. The second author would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office ofthe Higher Education Commission (under the Project NRU-CSEC no. 54000267) for financial support. The authors are very grateful to the referees for their careful reading, comments, and suggestions which improved the presentation of this article.