Composite Iterative Algorithms for Variational Inequality and Fixed Point Problems in Real Smooth and Uniformly Convex Banach Spaces

Abstract

We introduce composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich's extragradient method and viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.

1. Introduction

Let 𝑋 be a real Banach space whose dual space is denoted by 𝑋^∗. The normalized duality mapping 𝐽∶𝑋→2^𝑋∗ is defined by 𝐽(𝑥)=𝑥^∗∈𝑋^∗∶𝑥,𝑥^∗=‖𝑥‖²=𝑥^∗2,∀𝑥∈𝑋, (1) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach theorem that 𝐽(𝑥) is nonempty for each 𝑥∈𝑋. Let 𝐶 be a nonempty, closed, and convex subset of 𝑋. A mapping 𝑇∶𝐶→C is called nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for every 𝑥,𝑦∈𝐶. The set of fixed points of 𝑇 is denoted by Fix(𝑇). We use the notation ⇀ to indicate the weak convergence and the one → to indicate the strong convergence. A mapping 𝐴∶𝐶→𝑋 is said to be accretive if for each 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥0. (2) It is said to be 𝛼-strongly accretive if for each 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥𝛼𝑥−𝑦², (3) for some 𝛼∈(0,1). The mapping is called 𝛽-inverse strongly-accretive if for each 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥𝛽𝐴𝑥−𝐴𝑦², (4) for some 𝛽>0 and is said to be 𝜆-strictly pseudocontractive if for each 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≤𝑥−𝑦²−𝜆𝑥−𝑦−𝐴𝑥−𝐴𝑦² (5) for some 𝜆∈(0,1).

Let 𝑈={𝑥∈𝑋∶‖𝑥‖=1} denote the unite sphere of 𝑋. A Banach space 𝑋 is said to be uniformly convex if for each 𝜖∈(0,2], there exists 𝛿>0 such that for all 𝑥,𝑦∈𝑈, 𝑥−𝑦≥𝜖⟹𝑥+𝑦2≤1−𝛿. (6) It is known that a uniformly convex Banach space is reflexive and strict convex. A Banach space 𝑋 is said to be smooth if the limit lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡 (7) exists for all 𝑥,𝑦∈𝑈; in this case, 𝑋 is also said to have a Gâteaux differentiable norm. 𝑋 is said to have a uniformly Gâteaux differentiable norm if for each 𝑦∈𝑈, the limit is attained uniformly for 𝑥∈𝑈. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for 𝑥,𝑦∈𝑈. The norm of 𝑋 is said to be the Fréchet differential if for each 𝑥∈𝑈, this limit is attained uniformly for 𝑦∈𝑈. In addition, we define a function 𝜌∶[0,∞)→[0,∞) called the modulus of smoothness of 𝑋 as follows: 𝜌(𝜏)1=sup2+‖𝑥‖𝑦.𝑥+𝑦𝑥−𝑦−1∶𝑥,𝑦∈𝑋,=1,=𝜏 (8) It is known that 𝑋 is uniformly smooth if and only if lim_𝜏→0𝜌(𝜏)/𝜏=0. Let 𝑞 be a fixed real number with 1<𝑞≤2. Then a Banach space 𝑋 is said to be 𝑞-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌(𝜏)≤𝑐𝜏^𝑞 for all 𝜏>0. As pointed out in [1], no Banach space is 𝑞-uniformly smooth for 𝑞>2. In addition, it is also known that 𝐽 is single-valued if and only if 𝑋 is smooth, whereas if 𝑋 is uniformly smooth, then the mapping 𝐽 is norm-to-norm uniformly continuous on bounded subsets of 𝑋. If 𝑋 has a uniformly Gâteaux differentiable norm, then the duality mapping 𝐽 is norm-to-weak* uniformly continuous on bounded subsets of 𝑋.

Very recently, Cai and Bu [2] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space 𝑋, which involves finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜇₁𝐵₁𝑦^∗+𝑥^∗−𝑦^∗,𝐽𝑥−𝑥^∗𝜇≥0,∀𝑥∈𝐶,₂𝐵₂𝑥^∗+𝑦^∗−𝑥^∗,𝐽𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (9) where 𝐶 is a nonempty, closed, and convex subset of 𝑋, 𝐵₁, and 𝐵₂∶𝐶→𝑋 are two nonlinear mappings, and 𝜇₁ and 𝜇₂ are two positive constants. Here the set of solutions of GSVI (9) is denoted by GSVI(𝐶,𝐵₁,𝐵₂). In particular, if 𝑋=𝐻, a real Hilbert space, then GSVI (9) reduces to the following GSVI of finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that ⟨𝜇₁𝐵₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶,⟨𝜇₂𝐵₂𝑥^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐶, (10) which 𝜇₁ and 𝜇₂ are two positive constants. The set of solutions of problem (10) is still denoted by GSVI(𝐶,𝐵₁,𝐵₂). It is clear that the problem (10) covers as special case the classical variational inequality problem (VIP) of finding 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (11) The solution set of the VIP (11) is denoted by VI(𝐶,𝐴).

Recently, Ceng et al. [3] transformed problem (10) into a fixed point problem in the following way.

Lemma 1 (see [3]). For given 𝑥,𝑦∈𝐶,(𝑥,𝑦) is a solution of problem (10) if and only if 𝑥 is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺(𝑥)=𝑃_𝐶𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁𝑃_𝐶×𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶, (12) where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥) and 𝑃_𝐶 is the the projection of 𝐻 onto 𝐶.

In particular, if the mappings 𝐵_𝑖∶𝐶→𝐻 is 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2, then the mapping 𝐺 is nonexpansive provided 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2.

Let 𝐶 be a nonempty, closed, and convex subset of a real smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶, and let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). In this paper we introduce composite implicit and explicit iterative algorithms for solving GSVI (9) and the common fixed point problem of an infinite family {𝑆_𝑛} of nonexpansive mappings of 𝐶 into itself. These composite iterative algorithms are based on Korpelevich’s extragradient method [4] and viscosity approximation method [5]. Let the mapping 𝐺 be defined by 𝐺(𝑥)∶=Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶. (13) We first propose a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space 𝑋: 𝑦_𝑛=𝛼_𝑛𝑓𝑦_𝑛+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛,∀𝑛≥0, (14) where 𝐵_𝑖∶𝐶→𝑋 is 𝛼_𝑖-inverse-strongly accretive with 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2 and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are the sequences in (0,1) such that 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 for all 𝑛≥0. It is proven that under appropriate conditions, {𝑥_𝑛} converges strongly to 𝑞∈𝐹=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Ω, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (15) On the other hand, we also propose another composite explicit iterative algorithm in a uniformly convex Banach space 𝑋 with a uniformly Gateaux differentiable norm: 𝑦_𝑛=𝛼_𝑛𝐺𝑥_𝑛+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛,∀𝑛≥0, (16) where 𝐵_𝑖∶𝐶→𝑋 is 𝜆_𝑖-strictly pseudocontractive and 𝛼_𝑖-strongly accretive with 𝛼_𝑖+𝜆_𝑖≥1 for 𝑖=1,2 and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are the sequences in (0,1) such that 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 for all 𝑛≥0. It is proven that under mild conditions, {𝑥_𝑛} also converges strongly to 𝑞∈𝐹=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Ω, which solves the VIP (15). The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.

2. Preliminaries

We list some lemmas that will be used in the sequel. Lemma 2 can be found in [6]. Lemma 3 is an immediate consequence of the subdifferential inequality of the function (1/2)‖⋅‖².

Lemma 2. Let {𝑠_𝑛} be a sequence of nonnegative real numbers satisfying 𝑠_𝑛+1≤1−𝛼_𝑛𝑠_𝑛+𝛼_𝑛𝛽_𝑛+𝛾_𝑛,∀𝑛≥0, (17) where {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} satisfy the conditions:(i){𝛼_𝑛}⊂[0,1] and ∑^∞_𝑛=0𝛼_𝑛=∞,(ii)limsup_𝑛→∞𝛽_𝑛≤0,(iii)𝛾_𝑛≥0, ∀𝑛≥0, and ∑^∞_𝑛=0𝛾_𝑛<∞.Then limsup_𝑛→∞𝑠_𝑛=0.

Lemma 3. In a smooth Banach space 𝑋, there holds the inequality 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽𝑥+𝑦,∀𝑥,𝑦∈𝑋. (18)

Lemma 4 (see [7]). Let {𝑥_𝑛} and {𝑧_𝑛} be bounded sequences in a Banach space 𝑋, and let {𝛼_𝑛} be a sequence in [0,1] which satisfies the following condition: 0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1. (19) Suppose that 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑧_𝑛, ∀𝑛≥0, and limsup_𝑛→∞(‖𝑧_𝑛+1−𝑧_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0.

Let 𝐷 be a subset of 𝐶, and let Π be a mapping of 𝐶 into 𝐷. Then Π is said to be sunny if Π[Π(𝑥)+𝑡(𝑥−Π(𝑥))]=Π(𝑥), (20) whenever Π(𝑥)+𝑡(𝑥−Π(𝑥))∈𝐶 for 𝑥∈𝐶 and 𝑡≥0. A mapping Π of 𝐶 into itself is called a retraction if Π²=Π. If a mapping Π of 𝐶 into itself is a retraction, then Π(𝑧)=𝑧 for every 𝑧∈𝑅(Π) where 𝑅(Π) is the range of Π. A subset 𝐷 of 𝐶 is called a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction from 𝐶 onto 𝐷. The following lemma concerns the sunny nonexpansive retraction.

Lemma 5 (see [8]). Let 𝐶 be a nonempty, closed, and convex subset of a real smooth Banach space 𝑋. Let 𝐷 be a nonempty subset of 𝐶. Let Π be a retraction of 𝐶 onto 𝐷. Then the following are equivalent:(i)Π is sunny and nonexpansive;(ii)‖Π(𝑥)−Π(𝑦)‖²≤⟨𝑥−𝑦,𝐽(Π(𝑥)−Π(𝑦))⟩, ∀𝑥,𝑦∈𝐶;(iii)⟨𝑥−Π(𝑥),𝐽(𝑦−Π(𝑥))⟩≤0, ∀𝑥∈𝐶, 𝑦∈𝐷.

It is well known that if 𝑋=𝐻 a Hilbert space, then a sunny nonexpansive retraction Π_𝐶 is coincident with the metric projection from 𝑋 onto 𝐶; that is, Π_𝐶=𝑃_𝐶. If 𝐶 is a nonempty, closed, and convex subset of a strictly convex and uniformly smooth Banach space 𝑋 and if 𝑇∶𝐶→𝐶 is a nonexpansive mapping with the fixed point set Fix(𝑇)≠∅, then the set Fix(𝑇) is a sunny nonexpansive retract of 𝐶.

Lemma 6 (see [9]). Given a number 𝑟>0. A real Banach space 𝑋 is uniformly convex if and only if there exists a continuous strictly increasing function 𝑔∶[0,∞)→[0,∞), 𝑔(0)=0, such that 𝜆𝑥+(1−𝜆)𝑦²≤𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝜆(1−𝜆)𝑔𝑥−𝑦 (21) for all 𝜆∈[0,1] and 𝑥,𝑦∈𝑋 such that ‖𝑥‖≤𝑟 and ‖𝑦‖≤𝑟.

Lemma 7 (see [10]). Let 𝐶 be a nonempty, closed, and convex subset of a Banach space 𝑋. Let 𝑆₀,𝑆₁,…, be a sequence of mappings of 𝐶 into itself. Suppose that ∑^∞_𝑛=1sup{‖𝑆_𝑛𝑥−𝑆_𝑛−1𝑥‖∶𝑥∈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 for all 𝑦∈𝐶. Then lim_𝑛→∞sup{‖𝑆𝑥−𝑆_𝑛𝑥‖∶𝑥∈𝐶}=0.

Let 𝐶 be a nonempty, closed, and convex subset of a Banach space 𝑋, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix(𝑇)≠∅. As previous, let Ξ_𝐶 be the set of all contractions on 𝐶. For 𝑡∈(0,1) and 𝑓∈Ξ_𝐶, let 𝑥_𝑡∈𝐶 be the unique fixed point of the contraction 𝑥↦𝑡𝑓(𝑥)+(1−𝑡)𝑇𝑥 on 𝐶; that is, 𝑥_𝑡=𝑡𝑓𝑥_𝑡+(1−𝑡)𝑇𝑥_𝑡. (22)

Lemma 8 (see [11, 12]). Let 𝑋 be a uniformly smooth Banach space or a reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm. Let 𝐶 be a nonempty, closed, and convex subset of 𝑋, let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix(𝑇)≠∅, and 𝑓∈Ξ_𝐶. Then the net {𝑥_𝑡} defined by 𝑥_𝑡=𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑇𝑥_𝑡 converges strongly to a point in Fix(𝑇). If we define a mapping 𝑄∶Ξ_𝐶→Fix(𝑇) by 𝑄(𝑓)∶=𝑠−lim_𝑡→0𝑥_𝑡, ∀𝑓∈Ξ_𝐶, then 𝑄(𝑓) solves the VIP: ⟨𝐼−𝑓𝑄𝑓,𝐽𝑄𝑓−𝑝⟩≤0,∀𝑓∈Ξ_𝐶,𝑝∈Fix(𝑇). (23)

Lemma 9 (see [13]). Let 𝐶 be a nonempty, closed, and convex subset of a strictly convex Banach space 𝑋. Let {𝑇_𝑛}^∞_𝑛=0 be a sequence of nonexpansive mappings on 𝐶. Suppose that ⋂^∞_𝑛=0Fix(𝑇_𝑛) is nonempty. Let {𝜆_𝑛} be a sequence of positive numbers with ∑^∞_𝑛=0𝜆_𝑛=1. Then a mapping 𝑆 on 𝐶 defined by 𝑆𝑥=∑^∞_𝑛=0𝜆_𝑛𝑇_𝑛𝑥 for 𝑥∈𝐶 is defined well, nonexpansive, and Fix(𝑆)=⋂^∞_𝑛=0Fix(𝑇_𝑛) holds.

3. Implicit Iterative Schemes

In this section, we introduce our implicit iterative schemes and show the strong convergence theorems. We will use the following useful lemmas in the sequel.

Lemma 10 (see [2, Lemma 2.8]). Let 𝐶 be a nonempty, closed, and convex subset of a real 2-uniformly smooth Banach space 𝑋. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛼_𝑖-inverse-strongly accretive. Then, one has 𝐼−𝜇_𝑖𝐵_𝑖𝑥−𝐼−𝜇_𝑖𝐵_𝑖𝑦²≤𝑥−𝑦²+2𝜇_𝑖𝜇_𝑖𝜅²−𝛼_𝑖×𝐵_𝑖𝑥−𝐵_𝑖𝑦²,∀𝑥,𝑦∈𝐶, (24) for 𝑖=1,2, where 𝜇_𝑖>0. In particular, if 0<𝜇_𝑖≤𝛼_𝑖/𝜅², then 𝐼−𝜇_𝑖𝐵_𝑖 is nonexpansive for 𝑖=1,2.

Lemma 11 (see [2, Lemma 2.9]). Let 𝐶 be a nonempty, closed, and convex subset of a real 2-uniformly smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛼_𝑖-inverse-strongly accretive for 𝑖=1,2. Let 𝐺∶𝐶→𝐶 be the mapping defined by 𝐺(𝑥)=Π_𝐶Π_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁Π_𝐶𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶. (25) If  0<𝜇_𝑖≤𝛼_𝑖/𝜅² for 𝑖=1,2, then 𝐺∶𝐶→𝐶 is nonexpensive.

Lemma 12 (see [2, Lemma 2.10]). Let 𝐶 be a nonempty, closed, and convex subset of a real 2-uniformly smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝐵₁,𝐵₂∶𝐶→𝑋 be two nonlinear mappings. For given 𝑥^∗,𝑦^∗∈𝐶,(𝑥^∗,𝑦^∗) is a solution of GSVI (9) if and only if 𝑥^∗=Π_𝐶(𝑦^∗−𝜇₁𝐵₁𝑦^∗) where 𝑦^∗=Π_𝐶(𝑥^∗−𝜇₂𝐵₂𝑥^∗).

Remark 13. By Lemma 12, we observe that 𝑥^∗=Π_𝐶Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗−𝜇₁𝐵₁Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗, (26) which implies that 𝑥^∗ is a fixed point of the mapping 𝐺.

We now state and prove our first result on the implicit iterative scheme.

Theorem 14. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛼_𝑖-inverse-strongly accretive for 𝑖=1,2. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let {𝑆_𝑛}^∞_𝑛=0 be an infinite family of nonexpansive mappings of 𝐶 into itself such that 𝐹=⋂^∞_𝑛=0Fix(𝑆_𝑛)∩Ω≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂). For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛼_𝑛𝑓𝑦_𝑛+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛,∀𝑛≥0, (27) where 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2 and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are the sequences in (0,1) such that 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1, ∀𝑛≥0. Suppose that the following conditions hold:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞,(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞(𝛽_𝑛+𝛾_𝑛)<1 and liminf_𝑛→∞𝛾_𝑛>0,(iii)lim_𝑛→∞|𝛾_𝑛/(1−𝛽_𝑛)−𝛾_𝑛−1/(1−𝛽_𝑛−1)|=0.Assume that ∑^∞_𝑛=1sup_𝑥∈𝐷‖𝑆_𝑛𝑥−𝑆_𝑛−1𝑥‖<∞ for any bounded subset 𝐷 of 𝐶, and let 𝑆 be a mapping of 𝐶 into itself defined by 𝑆𝑥=lim_𝑛→∞𝑆_𝑛𝑥 for all 𝑥∈𝐶. Suppose that Fix(𝑆)=⋂^∞_𝑛=0Fix(𝑆_𝑛). Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (28)

Proof. Take a fixed 𝑝∈𝐹 arbitrarily. Then by Lemma 12, we know that 𝑝=𝐺(𝑝) and 𝑝=𝑆_𝑛𝑝 for all 𝑛≥0. Moreover, by Lemma 11, we have 𝑦_𝑛=𝛼−𝑝_𝑛𝑓𝑦_𝑛−𝑝+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝≤𝛼_𝑛𝑓𝑦_𝑛−𝑓𝑝+𝛼_𝑛𝑓𝑝+−𝑝1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝≤𝛼_𝑛𝜌𝑦_𝑛−𝑝+𝛼_𝑛𝑓𝑝+−𝑝1−𝛼_𝑛𝐺𝑥_𝑛−𝑝≤𝛼_𝑛𝜌𝑦_𝑛−𝑝+𝛼_𝑛𝑓𝑝+−𝑝1−𝛼_𝑛𝑥_𝑛−𝑝, (29) which hence implies that 𝑦_𝑛≤−𝑝1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛𝑥_𝑛+1−𝑝1−𝛼_𝑛𝜌𝛼_𝑛𝑓𝑝≤𝑥−𝑝_𝑛+𝛼−𝑝^𝑛1−𝛼_𝑛𝜌𝑓𝑝−𝑝. (30) Thus, from (27), we have 𝑥_𝑛+1=𝛽−𝑝_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝐺𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑦_𝑛−𝑝=𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛×1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛𝑥_𝑛+1−𝑝1−𝛼_𝑛𝜌𝛼_𝑛𝑓𝑝=−𝑝1−1−𝛽_𝑛1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛𝑥_𝑛+−𝑝1−𝛽_𝑛1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛𝑓𝑝−𝑝𝑥1−𝜌≤max₀,𝑓𝑝−𝑝−𝑝.1−𝜌 (31) It immediately follows that {𝑥_𝑛} is bounded, and so are the sequences {𝑦_𝑛}, {𝐺(𝑥_𝑛)}, and {𝐺(𝑦_𝑛)} due to (30) and the nonexpansivity of 𝐺.
Let us show that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞. As a matter of fact, from (27), we have 𝑦_𝑛=𝛼_𝑛𝑓𝑦_𝑛+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛,𝑦_𝑛−1=𝛼_𝑛−1𝑓𝑦_𝑛−1+1−𝛼_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1,∀𝑛≥1. (32) Simple calculations show that 𝑦_𝑛−𝑦_𝑛−1=𝛼_𝑛𝑓𝑦_𝑛−𝑓𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛−1𝐺𝑥_𝑛−1. (33) It follows that 𝑦_𝑛−𝑦_𝑛−1≤𝛼_𝑛𝑓𝑦_𝑛−𝑓𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1×𝑓𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛−1𝐺𝑥_𝑛−1≤𝛼_𝑛𝜌𝑦_𝑛−𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1×𝑓𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+1−𝛼_𝑛×𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1≤𝛼_𝑛𝜌𝑦_𝑛−𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1×𝑓𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+1−𝛼_𝑛×‖𝑥_𝑛−𝑥_𝑛−1‖+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1, (34) which hence yields 𝑦_𝑛−𝑦_𝑛−1≤1−𝛼_𝑛1−𝛼_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−11−𝛼_𝑛𝜌𝑓𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−11−𝛼_𝑛𝜌𝑓𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1. (35) Now, we write 𝑥_𝑛=𝛽_𝑛−1𝑥_𝑛−1+(1−𝛽_𝑛−1)𝑧_𝑛−1, ∀𝑛≥1, where 𝑧_𝑛−1=(𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−1)/(1−𝛽_𝑛−1). It follows that for all 𝑛≥1, 𝑧_𝑛−𝑧_𝑛−1=𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛−𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1=𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛1−𝛽_𝑛−𝛾_𝑛−1𝑦_𝑛−1+𝛿_𝑛−1𝑆_𝑛−1𝐺𝑦_𝑛−11−𝛽_𝑛−1=𝛾_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑆_𝑛−1𝐺𝑦_𝑛−11−𝛽_𝑛+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1+𝛿_𝑛1−𝛽_𝑛−𝛿_𝑛−11−𝛽_𝑛−1𝑆_𝑛−1𝐺𝑦_𝑛−1. (36) This together with (35) implies that 𝑧_𝑛−𝑧_𝑛−1≤𝛾_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑆_𝑛−1𝐺𝑦_𝑛−11−𝛽_𝑛+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1+𝛿_𝑛1−𝛽_𝑛−𝛿_𝑛−11−𝛽_𝑛−1𝑆_𝑛−1𝐺𝑦_𝑛−1≤𝛾_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛−1+𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑦_𝑛−1×1−𝛽_𝑛⁻¹+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑆_𝑛−1𝐺𝑦_𝑛−1≤𝛾_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛿_𝑛𝑦_𝑛−𝑦_𝑛−1𝛾_𝑛+𝛿_𝑛+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑦_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1+𝑆_𝑛−1𝐺𝑦_𝑛−1≤𝑦_𝑛−𝑦_𝑛−1+𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑦_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1+𝑆_𝑛−1𝐺𝑦_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−11−𝛼_𝑛𝜌𝑓𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑦_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1+𝑆_𝑛−1𝐺𝑦_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−11−𝛼_𝑛𝜌𝑀+𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑦_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_n−1𝑀=𝑥_𝑛−𝑥_𝑛−11+𝑀1−𝛼_𝑛𝜌𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑦_𝑛−1, (37) where sup_𝑛≥0{‖𝑓(𝑦_𝑛)‖+‖𝑆_𝑛𝐺(𝑥_𝑛)‖+‖𝑦_𝑛‖+‖𝑆_𝑛𝐺(𝑦_𝑛)‖}≤𝑀 for some 𝑀>0. So, from 𝛼_𝑛→0, condition (iii), and the assumption on {𝑆_𝑛}, it immediately follows that limsup_𝑛→∞𝑧_𝑛−𝑧_𝑛−1−𝑥_𝑛−𝑥_𝑛−1≤0. (38) In terms of condition (ii) and Lemma 4, we get lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0. (39) Hence we obtain lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛=0. (40)
Next we show that ‖𝑥_𝑛−𝐺(𝑥_𝑛)‖→0 as 𝑛→∞.
For simplicity, put 𝑞=Π_𝐶(𝑝−𝜇₂𝐵₂𝑝), 𝑢_𝑛=Π_𝐶(𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛), and 𝑣_𝑛=Π_𝐶(𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛). Then 𝑣_𝑛=𝐺(𝑥_𝑛). From Lemma 10, we have 𝑢_𝑛−𝑞²=Π_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−Π_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²≤𝑥_𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²,𝑣_𝑛−𝑝²=Π_𝐶𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛−Π_𝐶𝑞−𝜇₁𝐵₁𝑞²≤𝑢_𝑛−𝑞−𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²≤𝑢_𝑛−𝑞²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞². (41)(42) Substituting (41) into (42), we obtain 𝑣_𝑛−𝑝²≤𝑥_𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞². (43) According to Lemma 3, we have from (27) 𝑦_𝑛−𝑝²=𝛼_𝑛𝑓𝑦_𝑛−𝑓𝑝+1−𝛼_𝑛𝑆_𝑛𝑣_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝²≤𝛼_𝑛𝑓𝑦_𝑛−𝑓𝑝+1−𝛼_𝑛𝑆_𝑛𝑣_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛−𝑝≤𝛼_𝑛𝑓𝑦_𝑛−𝑓𝑝²+1−𝛼_𝑛𝑆_𝑛𝑣_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛−𝑝≤𝛼_𝑛𝜌𝑦_𝑛−𝑝²+1−𝛼_𝑛𝑣_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝𝑦_𝑛−𝑝, (44) which hence yields 𝑦_𝑛−𝑝²≤1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛𝑣_𝑛−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝜌𝑓𝑝−𝑝𝑦_𝑛−𝑝. (45) This together with (43) and the convexity of ‖⋅‖², we have 𝑥_𝑛+1−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑦_𝑛−𝑝²+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑦_𝑛−𝑝²+𝛿_𝑛𝑦_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛×1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛𝑣_𝑛−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝜌𝑓𝑝−𝑝𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛×1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛𝑣_𝑛−𝑝²+𝛼_𝑛𝑀₁≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛×𝑥_𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²+𝛼_𝑛𝑀₁=1−1−𝛽_𝑛1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛𝑥_𝑛−𝑝²−21−𝛽_𝑛×1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛×𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²+𝛼_𝑛𝑀₁≤𝑥_𝑛−𝑝²−21−𝛽_𝑛1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛×𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²+𝛼_𝑛𝑀₁, (46) where sup_𝑛≥0{2(1−𝛽_𝑛)/(1−𝛼_𝑛𝜌)‖𝑓(𝑝)−𝑝‖‖𝑦_𝑛−𝑝‖}≤𝑀₁ for some 𝑀₁>0. So, it follows that 21−𝛽_𝑛1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛×𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁u_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝛼_𝑛𝑀₁≤𝑥_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑀₁. (47) Since 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2, from conditions (i), (ii), and (40), we obtain lim_𝑛→∞𝐵₂𝑥_𝑛−𝐵₂𝑝=0,lim_𝑛→∞𝐵₁𝑢_𝑛−𝐵₁𝑞=0. (48) Utilizing [14, Proposition 1] and Lemma 5, we have 𝑢_𝑛−𝑞²=Π_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−Π_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑝,𝐽𝑢_𝑛=𝑥−𝑞_𝑛−𝑝,𝐽𝑢_𝑛−𝑞+𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛,𝐽𝑢_𝑛≤1−𝑞2𝑥_𝑛−𝑝²+𝑢_𝑛−𝑞²−𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞, (49) which implies that 𝑢_𝑛−𝑞²≤𝑥_𝑛−𝑝²−𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞. (50) In the same way, we derive 𝑣_𝑛−𝑝²=Π_𝐶𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛−Π_𝐶𝑞−𝜇₁𝐵₁𝑞²≤𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛−𝑞−𝜇₁𝐵₁𝑞,𝐽𝑣_𝑛=𝑢−𝑝_𝑛−𝑞,𝐽𝑣_𝑛−𝑝+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛,𝐽𝑣_𝑛≤1−𝑝2𝑢_𝑛−𝑞²+𝑣_𝑛−𝑝²−𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝, (51) which implies that 𝑣_𝑛−𝑝²≤𝑢_𝑛−𝑞²−𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝. (52) Substituting (50) into (52), we get 𝑣_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞−𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝. (53) From (46) and (53), we have 𝑥_𝑛+1−𝑝²≤𝛼_𝑛𝑀₁+𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛×𝑥_𝑛−𝑝²−𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞−𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝≤𝛼_𝑛𝑀₁+1−1−𝛽_𝑛1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛𝑥_𝑛−𝑝²−1−𝛽_𝑛1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛×𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝≤𝛼_𝑛𝑀₁+𝑥_𝑛−𝑝²−1−𝛽_𝑛×1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛×𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝, (54) which implies that 1−𝛽_𝑛1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛×𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞≤𝛼_𝑛𝑀₁+𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝≤𝛼_𝑛𝑀₁+𝑥_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝. (55) Utilizing conditions (i), (ii), from (40) and (48), we have lim_𝑛→∞𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞=0,lim_𝑛→∞𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞=0. (56) Utilizing the properties of 𝑔₁ and 𝑔₂, we deduce that lim_𝑛→∞𝑥_𝑛−𝑢_𝑛−𝑝−𝑞=0,lim_𝑛→∞𝑢_𝑛−𝑣_𝑛+𝑝−𝑞=0. (57) From (57), we obtain 𝑥_𝑛−𝑣_𝑛≤𝑥_𝑛−𝑢_𝑛−+𝑢𝑝−𝑞_𝑛−𝑣_𝑛+𝑝−𝑞⟶0as𝑛→∞. (58) That is, lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0. (59) On the other hand, since {𝑦_𝑛} and {𝑆_𝑛𝐺(𝑦_𝑛)} are bounded, by Lemma 6, there exists a continuous strictly increasing function 𝑔₃∶[0,∞)→[0,∞), 𝑔₃(0)=0 such that for 𝑝∈𝐹𝑥_𝑛+1−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑝²=𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑦_𝑛+𝛿−𝑝_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝²≤𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑦_𝑛+𝛿−𝑝^𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²≤𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺(𝑦_𝑛)−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₃𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛+𝛽_𝑛𝑥_𝑛−𝑝²≤𝛾_𝑛𝑦_𝑛−𝑝²+𝛿_𝑛𝑦_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑔₃𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛+𝛽_𝑛𝑥_𝑛−𝑝²=1−𝛽_𝑛𝑦_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛×𝑔₃𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛+𝛽_𝑛𝑥_𝑛−𝑝², (60) which together with (30) implies that 𝑥_𝑛+1−𝑝²≤1−𝛽_𝑛𝑥_𝑛+𝛼−𝑝^𝑛1−𝛼_𝑛𝜌𝑓𝑝−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑔₃𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛+𝛽_𝑛𝑥_𝑛−𝑝²≤𝑥_𝑛+𝛼−𝑝^𝑛1−𝛼_𝑛𝜌𝑓𝑝−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑔₃𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛. (61) It immediately follows that 𝛾_𝑛𝛿_𝑛𝑔₃𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛≤𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑔₃𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛≤𝑥_𝑛+𝛼−𝑝^𝑛1−𝛼_𝑛𝜌𝑓𝑝−𝑝²−𝑥_𝑛+1−𝑝²≤𝑥_𝑛−𝑝+𝑥_𝑛+1+𝛼−𝑝_𝑛1−𝛼_𝑛𝜌𝑓𝑝×𝑥−𝑝_𝑛−𝑥_𝑛+1+𝛼_𝑛1−𝛼_𝑛𝜌𝑓𝑝−𝑝. (62) According to condition (ii), we get liminf_𝑛→∞𝛿_𝑛=liminf_𝑛→∞1−𝛽_𝑛−𝛾_𝑛=1−limsup_𝑛→∞𝛽_𝑛+𝛾_𝑛>0. (63) Since 𝛼_𝑛→0, ‖𝑥_𝑛+1−𝑥_𝑛‖→0, and liminf_𝑛→∞𝛾_𝑛>0, we conclude that lim_𝑛→∞𝑔₃𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛=0. (64) Utilizing the property of 𝑔₃, we have lim_𝑛→∞𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛=0. (65) We note that 𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛=𝑥_𝑛+1−𝑦_𝑛=𝛽_𝑛𝑥_𝑛−𝑦_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑦_𝑛. (66) So, 1−𝛽_𝑛𝑥_𝑛−𝑦_𝑛=𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑦_𝑛+𝑥_𝑛+1−𝑥_𝑛≤𝑆_𝑛𝐺𝑦_𝑛−𝑦_𝑛+𝑥_𝑛+1−𝑥_𝑛⟶0as𝑛⟶∞. (67) That is, lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (68) We observe that 𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛≤𝐺𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛+𝑆_𝑛𝐺𝑦_𝑛−𝑆_𝑛𝐺𝑥_𝑛≤𝐺𝑥_𝑛−𝑥_𝑛+2𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛. (69) Thus, from (59)–(68), we obtain that lim_𝑛→∞𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛=0. (70) By (70) and Lemma 7, we have 𝑆𝐺𝑥_𝑛−𝐺𝑥_𝑛≤𝑆𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝑆_𝑛G𝑥_𝑛−𝐺𝑥_𝑛⟶0as𝑛⟶∞. (71) In terms of (59) and (71), we have 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝐺𝑥_𝑛+𝐺𝑥_𝑛−𝑆𝐺𝑥_𝑛+𝑆𝐺𝑥_𝑛−𝑆𝑥_𝑛≤2𝑥_𝑛−𝐺𝑥_𝑛+𝐺𝑥_𝑛−𝑆𝐺𝑥_𝑛⟶0as𝑛⟶∞. (72) Define a mapping 𝑊𝑥=(1−𝜃)𝑆𝑥+𝜃𝐺(𝑥), where 𝜃∈(0,1) is a constant. Then by Lemma 9, we have that Fix(𝑊)=Fix(𝑆)∩Fix(𝐺)=𝐹. We observe that 𝑥_𝑛−𝑊𝑥_𝑛=(1−𝜃)𝑥_𝑛−𝑆𝑥_𝑛+𝜃𝑥_𝑛−𝐺𝑥_𝑛≤(1−𝜃)𝑥_𝑛−𝑆𝑥_𝑛+𝜃𝑥_𝑛−𝐺𝑥_𝑛. (73) From (59) and (72), we obtain lim_𝑛→∞𝑥_𝑛−𝑊𝑥_𝑛=0. (74)
Now, we claim that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0, (75) where 𝑞=𝑠−lim_𝑡→0𝑥_𝑡 with 𝑥_𝑡 being the fixed point of the contraction 𝑥⟼𝑡𝑓(𝑥)+(1−𝑡)𝑊𝑥. (76) Then 𝑥_𝑡 solves the fixed point equation 𝑥_𝑡=𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑊𝑥_𝑡. Thus we have 𝑥_𝑡−𝑥_𝑛=(1−𝑡)𝑊𝑥_𝑡−𝑥_𝑛+𝑡𝑓𝑥_𝑡−𝑥_𝑛. (77) By Lemma 3, we conclude that 𝑥_𝑡−𝑥_𝑛²=(1−𝑡)(𝑊𝑥_𝑡−𝑥_𝑛)+𝑡(𝑓(𝑥_𝑡)−𝑥_𝑛)²≤(1−𝑡)²𝑊𝑥_𝑡−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤(1−𝑡)²𝑊𝑥_𝑡−𝑊𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤(1−𝑡)²𝑥_𝑡−𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛=(1−𝑡)²𝑥_𝑡−𝑥_𝑛²+2𝑥_𝑡−𝑥_𝑛×𝑊𝑥_𝑛−𝑥_𝑛+‖𝑊𝑥_𝑛−𝑥_𝑛‖²+2𝑡𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛+2𝑡𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛=1−2𝑡+𝑡²𝑥_𝑡−𝑥_𝑛²+𝑓_𝑛(𝑡)+2𝑡𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛+2𝑡𝑥_𝑡−𝑥_𝑛², (78) where 𝑓_𝑛(𝑡)=(1−𝑡)²2𝑥_𝑡−𝑥_𝑛+𝑥_𝑛−𝑊𝑥_𝑛×𝑥_𝑛−𝑊𝑥_𝑛⟶0,as𝑛⟶∞. (79) It follows from (78) that 𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤𝑡2𝑥_𝑡−𝑥_𝑛²+1𝑓2𝑡^𝑛(𝑡). (80) Letting 𝑛→∞ in (80) and noticing (79), we derive limsup_𝑛→∞𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤𝑡2𝑀₂, (81) where 𝑀₂>0 is a constant such that ‖𝑥_𝑡−𝑥_𝑛‖²≤𝑀₂ for all 𝑡∈(0,1) and 𝑛≥0. Taking 𝑡→0 in (81), we have limsup_𝑡→0limsup_𝑛→∞𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤0. (82) On the other hand, we have 𝑓𝑞−𝑞,𝐽𝑥_𝑛=𝑓𝑞𝑥−𝑞−𝑞,𝐽_𝑛−𝑞−𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑥_𝑡−𝑓𝑞−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑞−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡−𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡=𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞−𝐽𝑥_𝑛−𝑥_𝑡+𝑥_𝑡−𝑞,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑞−𝑓𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡. (83) It follows that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞−𝐽𝑥_𝑛−𝑥_𝑡+𝑥_𝑡−𝑞limsup_𝑛→∞𝑥_𝑛−𝑥_𝑡+𝜌𝑞−𝑥_𝑡limsup_𝑛→∞𝑥_𝑛−𝑥_𝑡+limsup_𝑛→∞𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡. (84) Taking into account that 𝑥_𝑡→𝑞 as 𝑡→0, we have from (82) limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞=limsup_𝑡→0limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤limsup_𝑡→0limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞−𝐽𝑥_𝑛−𝑥_𝑡. (85) Since 𝑋 has a uniformly Fréchet differentiable norm, the duality mapping 𝐽 is norm-to-norm uniformly continuous on bounded subsets of 𝑋. Consequently, the two limits are interchangeable, and hence (75) holds. From (68), we get (𝑦_𝑛−𝑞)−(𝑥_𝑛−𝑞)→0. Noticing that 𝐽 is norm-to-norm uniformly continuous on bounded subsets of 𝑋, we deduce from (75) that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞=limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛+𝑓𝑞𝑦−𝑞−𝑞,𝐽_𝑛−𝑞−𝐽𝑥_𝑛−𝑞=limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0. (86)
Finally, let us show that 𝑥_𝑛→𝑞 as 𝑛→∞. We observe that 𝑦_𝑛−𝑞²=𝛼_𝑛𝑓𝑦_𝑛−𝑓𝑞+1−𝛼_𝑛×𝑆_𝑛𝐺𝑥_𝑛−𝑞+𝛼_𝑛𝑓𝑞−𝑞²≤𝛼_𝑛𝑓𝑦_𝑛𝑞+−𝑓1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞≤𝛼_𝑛𝑓𝑦_𝑛−𝑓𝑞²+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞≤𝛼_𝑛𝜌𝑦_𝑛−𝑞²+1−𝛼_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞, (87) which implies that 𝑦_𝑛−𝑞²≤1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛𝑥_𝑛−𝑞²+𝛼_𝑛1−𝜌1−𝛼_𝑛𝜌⋅2𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞.1−𝜌 (88) By (27) and the convexity of ‖⋅‖², we get 𝑥_𝑛+1−𝑞²≤𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝑦_𝑛−𝑞²+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑞²≤𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝑦_𝑛−𝑞²+𝛿_𝑛𝑦_𝑛−𝑞²=𝛽_𝑛𝑥_𝑛−𝑞²+1−𝛽_𝑛𝑦_𝑛−𝑞², (89) which together with (88) leads to 𝑥_𝑛+1−𝑞²≤𝛽_𝑛𝑥_𝑛−𝑞²+1−𝛽_𝑛×1−1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛𝑥_𝑛−𝑞²+𝛼_𝑛1−𝜌1−𝛼_𝑛𝜌⋅2𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞=1−𝜌1−1−𝛽_𝑛1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛𝑥_𝑛−𝑞²+1−𝛽_𝑛1−𝜌1−𝛼_𝑛𝜌𝛼_𝑛⋅2𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞.1−𝜌 (90) Applying Lemma 2 to (88), we obtain that 𝑥_𝑛→𝑞 as 𝑛→∞. This completes the proof.

Corollary 15. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛼_𝑖-inverse-strongly accretive for 𝑖=1,2. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that 𝐹=Fix(𝑆)∩Ω≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂). For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛼_𝑛𝑓𝑦_𝑛+1−𝛼_𝑛𝑆𝐺𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆𝐺𝑦_𝑛,∀𝑛≥0, (91) where 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2 and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are the sequences in (0,1) such that 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1, ∀𝑛≥0. Suppose that the following conditions hold:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞,(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞(𝛽_𝑛+𝛾_𝑛)<1 and liminf_𝑛→∞𝛾_𝑛>0,(iii)lim_𝑛→∞|(𝛾_𝑛/(1−𝛽_𝑛))−(𝛾_𝑛−1/(1−𝛽_𝑛−1))|=0.Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (92)

Further, we illustrate Theorem 14 by virtue of an example, that is, the following corollary.

Corollary 16. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let 𝑇 be an 𝜂-strictly pseudocontractive mapping of 𝐶 into itself, and let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that 𝐹=Fix(𝑆)∩Fix(𝑇)≠∅. For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛼_𝑛𝑓𝑦_𝑛+1−𝛼_𝑛𝑆(𝐼−𝜆(I−𝑇))𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆(𝐼−𝜆(𝐼−𝑇))𝑦_𝑛,∀𝑛≥0, (93) where 0<𝜆<max{1,𝜂/𝜅²} and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are the sequences in (0,1) such that 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1, ∀𝑛≥0. Suppose that the following conditions hold:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞,(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞(𝛽_𝑛+𝛾_𝑛)<1 and liminf_𝑛→∞𝛾_𝑛>0,(iii)lim_𝑛→∞|(𝛾_𝑛/(1−𝛽_𝑛))−(𝛾_𝑛−1/(1−𝛽_𝑛−1))|=0.Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (94)

Proof. In Corollary 15, put 𝐵₁=𝐼−𝑇, 𝐵₂=0, 𝜇₁=𝜆, and 𝛼₁=𝜂. Since 𝑇 is an 𝜂-strictly pseudocontractive mapping, it is clear that 𝐵₁=𝐼−𝑇 is an 𝜂-inverse strongly accretive mapping. Hence, the GSVI (9) is equivalent to the following VIP of finding 𝑥^∗∈𝐶 such that 𝐵₁𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (95) which leads to Ω=VI(𝐶,𝐵₁). In the meantime, we have 𝐺𝑥_𝑛=Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑥_𝑛=Π_𝐶𝐼−𝜇₁𝐵₁𝑥_𝑛=Π_𝐶(1−𝜆)𝑥_𝑛+𝜆𝑇𝑥_𝑛=𝑥_𝑛−𝜆(𝐼−𝑇)𝑥_𝑛. (96) In the same way, we get 𝐺𝑦_𝑛=𝑦_𝑛−𝜆(𝐼−𝑇)𝑦_𝑛. In this case, it is easy to see that (91) reduces to (93). We claim that Fix(𝑇)=VI(𝐶,𝐵₁). As a matter of fact, we have, for 𝜆>0, 𝑢∈VI𝐶,𝐵₁⟺⟨𝐵₁𝑢,𝐽𝑦−𝑢⟩≥0∀𝑦∈𝐶⟺⟨𝑢−𝜆𝐵₁𝑢−𝑢,𝐽𝑢−𝑦⟩≥0∀𝑦∈𝐶⟺𝑢=Π_𝐶𝑢−𝜆𝐵₁𝑢⟺𝑢=Π_𝐶()(𝑇).𝑢−𝜆𝑢+𝜆𝑇𝑢⟺⟨𝑢−𝜆𝑢+𝜆𝑇𝑢−𝑢,𝐽𝑢−𝑦⟩≥0∀𝑦∈𝐶⟺⟨𝑢−𝑇𝑢,𝐽𝑢−𝑦⟩≤0∀𝑦∈𝐶⟺𝑢=𝑇𝑢⟺𝑢∈Fix (97) So, we conclude that 𝐹=Fix(𝑆)∩Ω=Fix(𝑆)∩Fix(𝑇). Therefore, the desired result follows from Corollary 15.

Remark 17. Theorem 14 improves, extends, supplements, and develops Cai and Bu [2, Theorem  3.1 and Corollary  3.2] and Jung [5, Theorem  3.1] in the following aspects.(i)The problem of finding a point 𝑞∈⋂_𝑛Fix(𝑆_𝑛)∩Ω in Theorem 14 is more general and more subtle than the problem of finding a point 𝑞∈Fix(𝑆)∩VI(𝐶,𝐴) in Jung [5, Theorem  3.1].(ii)The iterative scheme in [2, Theorem  3.1] is extended to develop the iterative scheme (27) of Theorem 14 by virtue of the iterative scheme of [5, Theorem  3.1]. The iterative scheme (27) of Theorem 14 is more advantageous and more flexible than the iterative scheme of [2, Theorem  3.1] because it involves several parameter sequences {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛}.(iii)The iterative scheme (27) in Theorem 14 is very different from everyone in both [2, Theorem  3.1] and [5, Theorem  3.1] because the mappings 𝑆_𝑛 and 𝐺 in the iterative scheme of [2, Theorem  3.1] and the mapping 𝑆𝑃_𝐶(𝐼−𝜆_𝑛𝐴) in the iterative scheme of [5, Theorem  3.1] are replaced by the same composite mapping 𝑆_𝑛𝐺 in the iterative scheme (27) of Theorem 14.(iv)The proof in [2, Theorem  3.1] depends on the argument techniques in [3], the inequality in 2-uniformly smooth Banach spaces ([9]), and the inequality in smooth and uniform convex Banach spaces ([14, Proposition 1]). Because the composite mapping 𝑆_𝑛𝐺 appears in the iterative scheme (27) of Theorem 14, the proof of Theorem 14 depends on the argument techniques in [3], the inequality in 2-uniformly smooth Banach spaces, the inequality in smooth and uniform convex Banach spaces, and the inequality in uniform convex Banach spaces (Lemma 6).(v)The iterative scheme in [2, Corollary 3.2] is extended to develop the new iterative scheme in Corollary 15 because the mappings 𝑆 and 𝐺 are replaced by the same composite mapping 𝑆𝐺 in Corollary 15.

4. Explicit Iterative Schemes

In this section, we introduce our explicit iterative schemes and show the strong convergence theorems. First, we give several useful lemmas.

Lemma 18. Let 𝐶 be a nonempty, closed, and convex subset of a smooth Banach space 𝑋, and let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝜆_𝑖-strictly pseudocontractive and 𝛼_𝑖-strongly accretive with 𝛼_𝑖+𝜆_𝑖≥1 for 𝑖=1,2. Then, for 𝜇_𝑖∈(0,1], we have 𝐼−𝜇_𝑖𝐵_𝑖𝑥−𝐼−𝜇_𝑖𝐵_𝑖𝑦≤1−𝛼_𝑖𝜆_𝑖+1−𝜇_𝑖11+𝜆_𝑖𝑥−𝑦,∀𝑥,𝑦∈𝐶, (98) for 𝑖=1,2. In particular, if 1−(𝜆_𝑖/(1+𝜆_𝑖))(1−(1−𝛼_𝑖)/𝜆_𝑖)≤𝜇_𝑖≤1, then 𝐼−𝜇_𝑖𝐵_𝑖 is nonexpansive for 𝑖=1,2.

Proof. Taking into account the 𝜆_𝑖-strict pseudocontractivity of 𝐵_𝑖, we derive for every 𝑥,𝑦∈𝐶𝜆_𝑖𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖𝑦²≤𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖≤𝑦,𝐽𝑥−𝑦𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖𝑦𝑥−𝑦, (99) which implies that 𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖𝑦≤1𝜆_𝑖𝑥−𝑦. (100) Hence, 𝐵_𝑖𝑥−𝐵_𝑖𝑦≤𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖𝑦+≤1𝑥−𝑦1+𝜆_𝑖𝑥−𝑦. (101) Utilizing the 𝛼_𝑖-strong accretivity and 𝜆_𝑖-strict pseudocontractivity of 𝐵_𝑖, we get 𝜆_𝑖𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖𝑦²≤𝑥−𝑦²−𝐵_𝑖𝑥−𝐵_𝑖≤𝑦,𝐽𝑥−𝑦1−𝛼_𝑖𝑥−𝑦². (102) So, we have 𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖𝑦≤1−𝛼_𝑖𝜆_𝑖𝑥−𝑦. (103) Therefore, for 𝜇_𝑖∈(0,1], we have 𝐼−𝜇_𝑖𝐵_𝑖𝑥−𝐼−𝜇_𝑖𝐵_𝑖𝑦≤𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖𝑦+1−𝜇_𝑖𝐵_𝑖𝑥−𝐵_𝑖𝑦≤1−𝛼_𝑖𝜆_𝑖𝑥−𝑦+1−𝜇_𝑖11+𝜆_𝑖=𝑥−𝑦1−𝛼_𝑖𝜆_𝑖+1−𝜇_𝑖11+𝜆_𝑖𝑥−𝑦. (104) Since 1−(𝜆_𝑖/(1+𝜆_𝑖))(1−(1−𝛼_i)/𝜆_𝑖)≤𝜇_𝑖≤1, it follows immediately that 1−𝛼_𝑖𝜆_𝑖+1−𝜇_𝑖11+𝜆_𝑖≤1. (105) This implies that 𝐼−𝜇_𝑖𝐵_𝑖 is nonexpansive for 𝑖=1,2.

Lemma 19. Let 𝐶 be a nonempty, closed, and convex subset of a smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶, and let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝜆_𝑖-strictly pseudocontractive and 𝛼_𝑖-strongly accretive with 𝛼_𝑖+𝜆_𝑖≥1 for 𝑖=1,2. Let 𝐺∶𝐶→𝐶 be the mapping defined by 𝐺(𝑥)=Π_𝐶Π_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁Π_𝐶𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶. (106) If 1−(𝜆_𝑖/(1+𝜆_𝑖))(1−(1−𝛼_𝑖)/𝜆_𝑖)≤𝜇_𝑖≤1, then 𝐺∶𝐶→𝐶 is nonexpansive.

Proof. According to Lemma 10, we know that 𝐼−𝜇_𝑖𝐵_𝑖 is nonexpansive for 𝑖=1,2. Hence, for all 𝑥,𝑦∈𝐶, we have 𝐺(𝑥)𝑦=Π−𝐺_𝐶Π_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁Π_𝐶𝑥−𝜇₂𝐵₂𝑥−Π_𝐶Π_𝐶𝑦−𝜇₂𝐵₂𝑦−𝜇₁𝐵₁Π_𝐶𝑦−𝜇₂𝐵₂𝑦=Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑥−Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑦≤𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑥−𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑦≤Π_𝐶𝐼−𝜇₂𝐵₂𝑥−Π_𝐶𝐼−𝜇₂𝐵₂𝑦≤𝐼−𝜇₂𝐵₂𝑥−𝐼−𝜇₂𝐵₂𝑦≤𝑥−𝑦. (107) This shows that 𝐺∶𝐶→𝐶 is nonexpansive. This completes the proof.

Lemma 20. Let 𝐶 be a nonempty, closed, and convex subset of a smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶, and let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝜆_𝑖-strictly pseudocontractive and 𝛼_𝑖-strongly accretive for 𝑖=1,2. For given 𝑥^∗,𝑦^∗∈𝐶, (𝑥^∗,𝑦^∗) is a solution of GSVI (9) if and only if 𝑥^∗=Π_𝐶(𝑦^∗−𝜇₁𝐵₁𝑦^∗) where 𝑦^∗=Π_𝐶(𝑥^∗−𝜇₂𝐵₂𝑥^∗).

Proof. We can rewrite GSVI (9) as 𝑥^∗−𝑦^∗−𝜇₁𝐵₁𝑦^∗,𝐽𝑥−𝑥^∗𝑦≥0,∀𝑥∈𝐶,_∗−𝑥^∗−𝜇₂𝐵₂𝑥^∗,𝐽𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (108) which is obviously equivalent to 𝑥^∗=Π_𝐶𝑦^∗−𝜇₁𝐵₁𝑦^∗,𝑦^∗=Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗, (109) because of Lemma 5. This completes the proof.

Remark 21. By Lemma 20, we observe that 𝑥^∗=Π_𝐶Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗−𝜇₁𝐵₁Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗, (110) which implies that 𝑥^∗ is a fixed point of the mapping 𝐺. Throughout this paper, the set of fixed points of the mapping 𝐺 is denoted by Ω.

We are now in a position to state and prove our result on the explicit iterative scheme.

Theorem 22. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gâteaux differentiable norm. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝜆_𝑖-strictly pseudocontractive and 𝛼_𝑖-strongly accretive with 𝛼_𝑖+𝜆_𝑖≥1 for 𝑖=1,2. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let {𝑆_𝑛}^∞_𝑛=0 be an infinite family of nonexpansive mappings of 𝐶 into itself such that 𝐹=⋂^∞_𝑛=0Fix(𝑆_𝑛)∩Ω≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂). For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛼_𝑛𝐺𝑥_𝑛+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛,∀𝑛≥0, (111) where 1−(𝜆_𝑖/(1+𝜆_𝑖))(1−(1−𝛼_𝑖)/𝜆_𝑖)≤𝜇_𝑖≤1 for 𝑖=1,2 and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are the sequences in (0,1) such that 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1, ∀𝑛≥0. Suppose that the following conditions hold:(i)0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1,(ii)lim_𝑛→∞𝛽_𝑛=0 and ∑^∞_𝑛=0𝛽_𝑛=∞,(iii)∑^∞_𝑛=1|𝛼_𝑛−𝛼_𝑛−1|<∞ or lim_𝑛→∞|𝛼_𝑛−𝛼_𝑛−1|/𝛽_𝑛=0,(iv)∑^∞_𝑛=1|𝛽_𝑛−𝛽_𝑛−1|<∞ or lim_𝑛→∞𝛽_𝑛−1/𝛽_𝑛=1,(v)∑^∞_𝑛=1|(𝛾_𝑛/(1−𝛽_𝑛))−(𝛾_𝑛−1/(1−𝛽_𝑛−1))|<∞ or lim_𝑛→∞(1/𝛽_𝑛)|(𝛾_𝑛/(1−𝛽_𝑛))−(𝛾_𝑛−1/(1−𝛽_𝑛−1))|=0,(vi)0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1.Assume that ∑^∞_𝑛=1sup_𝑥∈𝐷‖𝑆_𝑛𝑥−𝑆_𝑛−1𝑥‖<∞ for any bounded subset 𝐷 of 𝐶, and let 𝑆 be a mapping of 𝐶 into itself defined by 𝑆𝑥=lim_𝑛→∞𝑆_𝑛𝑥 for all 𝑥∈𝐶. Suppose that Fix(𝑆)=⋂^∞_𝑛=0Fix(𝑆_𝑛). Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (112)

Proof. Take a fixed 𝑝∈𝐹 arbitrarily. Then by Lemma 20, we know that 𝑝=𝐺(𝑝) and 𝑝=𝑆_𝑛𝑝 for all 𝑛≥0. Moreover, by Lemma 19, we have 𝑦_𝑛−𝑝≤𝛼_𝑛𝐺𝑥_𝑛−𝑝+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝≤𝛼_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑥_𝑛=𝑥−𝑝_𝑛−𝑝. (113) From (113) we obtain 𝑥_𝑛+1−𝑝≤𝛽_𝑛𝑓𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑝≤𝛽_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝑓𝑝−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑦_𝑛−𝑝≤𝛽_𝑛𝜌𝑥_𝑛−𝑝+𝛽_𝑛𝑓𝑝+−𝑝1−𝛽_𝑛𝑦_𝑛−𝑝≤𝛽_𝑛𝜌𝑥_𝑛−𝑝+𝛽_𝑛𝑓𝑝+−𝑝1−𝛽_𝑛𝑥_𝑛=−𝑝1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑝+𝛽_𝑛𝑓𝑝=−𝑝1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑝+𝛽_𝑛⋅𝑓𝑝1−𝜌−𝑝𝑥1−𝜌≤max₀,𝑓𝑝−𝑝−𝑝,1−𝜌 (114) which implies that {𝑥_𝑛} is bounded. By Lemma 19 we know from (113) that {𝑦_𝑛}, {𝐺(𝑥_𝑛)}, and {𝐺(𝑦_𝑛)} are bounded.
Let us show that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 and ‖𝑥_𝑛−𝑦_𝑛‖→0 as 𝑛→∞. As a matter of fact, from (113), we have 𝑦_𝑛=𝛼_𝑛𝐺𝑥_𝑛+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛,𝑦_𝑛−1=𝛼_𝑛−1𝐺𝑥_𝑛−1+1−𝛼_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1,∀𝑛≥1. (115) Simple calculations show that 𝑦_𝑛−𝑦_𝑛−1=𝛼_𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_n−𝑆_𝑛−1𝐺𝑥_𝑛−1. (116) It follows that 𝑦_𝑛−𝑦_𝑛−1≤𝛼_𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛−1𝐺𝑥_𝑛−1≤𝛼_𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1≤𝛼_𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+1−𝛼_𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1≤𝐺𝑥_𝑛−𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1×𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1. (117) Now, we write 𝑥_𝑛=𝛽_𝑛−1𝑓(𝑥_𝑛−1)+(1−𝛽_𝑛−1)𝑣_𝑛−1, ∀𝑛≥1, where 𝑣_𝑛−1=(𝑥_𝑛−𝛽_𝑛−1𝑓(𝑥_𝑛−1))/(1−𝛽_𝑛−1). It follows that, for all 𝑛≥1, 𝑣_𝑛−𝑣_𝑛−1=𝑥_𝑛+1−𝛽_𝑛𝑓𝑥_𝑛1−𝛽_𝑛−𝑥_𝑛−𝛽_𝑛−1𝑓𝑥_𝑛−11−𝛽_𝑛−1=𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛1−𝛽_𝑛−𝛾_𝑛−1𝑦_𝑛−1+𝛿_𝑛−1𝑆_𝑛−1𝐺𝑦_𝑛−11−𝛽_𝑛−1=𝛾_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑆_𝑛−1𝐺𝑦_𝑛−11−𝛽_𝑛+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1+𝛿_𝑛1−𝛽_𝑛−𝛿_𝑛−11−𝛽_𝑛−1𝑆_𝑛−1𝐺𝑦_𝑛−1. (118) This together with (117) implies that 𝑣_𝑛−𝑣_𝑛−1≤𝛾_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑆_𝑛−1𝐺𝑦_𝑛−11−𝛽_𝑛+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1+𝛿_𝑛1−𝛽_𝑛−𝛿_𝑛−11−𝛽_𝑛−1𝑆_𝑛−1𝐺𝑦_𝑛−1≤𝛾_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛−1+𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑦_𝑛−1×1−𝛽_𝑛⁻¹+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑆_𝑛−1𝐺𝑦_𝑛−1≤𝛾_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛿_𝑛𝑦_𝑛−𝑦_𝑛−1𝛾_𝑛+𝛿_𝑛+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑦_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1×𝑦_𝑛−1+𝑆_𝑛−1𝐺𝑦_𝑛−1≤𝑦_𝑛−𝑦_𝑛−1+𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑦_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1+𝑆_𝑛−1𝐺𝑦_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑦_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1+𝑆_𝑛−1𝐺𝑦_𝑛−1. (119) Furthermore, we note that 𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑣_𝑛,𝑥_𝑛=𝛽_𝑛−1𝑓𝑥_𝑛−1+1−𝛽_𝑛−1𝑣_𝑛−1,∀𝑛≥1. (120) Also, simple calculations show that 𝑥_𝑛+1−𝑥_𝑛=𝛽_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1×𝑓𝑥_𝑛−1−𝑣_𝑛−1+1−𝛽_𝑛𝑣_𝑛−𝑣_𝑛−1. (121) This together with (119) implies that 𝑥_𝑛+1−𝑥_𝑛≤𝛽_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1×𝑓𝑥_𝑛−1−𝑣_𝑛−1+1−𝛽_𝑛𝑣_𝑛−𝑣_𝑛−1≤𝛽_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑓𝑥_𝑛−1−𝑣_𝑛−1+1−𝛽_𝑛×𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑦_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1×𝑦_𝑛−1+𝑆_𝑛−1𝐺𝑦_𝑛−1≤1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑓𝑥_𝑛−1−𝑣_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑦_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1+𝑆_𝑛−1𝐺𝑦_𝑛−1≤1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑀+𝛼_𝑛−𝛼_𝑛−1𝑀+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺𝑦_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1𝑀=1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝑀𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛−11−𝛽_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑦_𝑛−1−𝑆_𝑛−1𝐺y_𝑛−1, (122) where sup_𝑛≥0{‖𝑓(𝑥_𝑛)‖+‖𝑣_𝑛‖+‖𝐺(𝑥_𝑛)‖+‖𝑆_𝑛𝐺(𝑥_𝑛)‖+‖𝑦_𝑛‖+‖𝑆_𝑛𝐺(𝑦_𝑛)‖}≤𝑀 for some 𝑀>0. Utilizing Lemma 2, from conditions (ii)–(v) and the assumption on {𝑆_𝑛}, we deduce that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (123) Taking into account the boundedness of {𝐺(𝑥_𝑛)} and {𝑆_𝑛𝐺(𝑥_𝑛)}, by Lemma 6, we know that there exists a continuous strictly increasing function 𝑔₁∶[0,∞)→[0,∞), 𝑔₁(0)=0 such that for 𝑝∈𝐹𝑦_𝑛−𝑝²≤𝛼_𝑛𝐺𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²−𝛼_𝑛1−𝛼_𝑛𝑔₁𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²−𝛼_𝑛1−𝛼_𝑛𝑔₁𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛=𝑥_𝑛−𝑝²−𝛼_𝑛1−𝛼_𝑛𝑔₁𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛. (124) Since {𝑦_𝑛} and {𝑆_𝑛𝐺(𝑦_𝑛)} are bounded, by Lemma 6, there exists a continuous strictly increasing function 𝑔₂∶[0,∞)→[0,∞), 𝑔₂(0)=0 such that for 𝑝∈𝐹𝑥_𝑛+1−𝑝²=𝛽_𝑛𝑓𝑥_𝑛−𝑝+𝛾_𝑛𝑦_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑝²=𝛾_𝑛+𝛿_𝑛×𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑦_𝑛+𝛿−𝑝_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑝+𝛽_𝑛𝑓𝑥_𝑛−𝑝²≤𝛾_𝑛+𝛿_𝑛²𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑦_𝑛+𝛿−𝑝_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑝²+2𝛽_𝑛𝑓𝑥_𝑛−𝑝,𝐽𝑥_𝑛+1≤𝛾−𝑝_𝑛+𝛿_𝑛²𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₂𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛+2𝛽_𝑛𝑓𝑥_𝑛−𝑝𝑥_𝑛+1−𝑝≤𝛾_𝑛𝑦_𝑛−𝑝²+𝛿_𝑛𝑦_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝑔₂𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛+2𝛽_𝑛𝑓𝑥_𝑛−𝑝𝑥_𝑛+1≤𝑦−𝑝_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝑔₂𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛+2𝛽_𝑛𝑓𝑥_𝑛−𝑝𝑥_𝑛+1−𝑝, (125) which together with (124) implies that 𝑥_𝑛+1−𝑝²≤𝑥_𝑛−𝑝²−𝛼_𝑛1−𝛼_𝑛𝑔₁𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛−𝛾_𝑛𝛿_𝑛𝑔₂𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛+2𝛽_𝑛𝑓𝑥_𝑛−𝑝𝑥_𝑛+1−𝑝. (126) It immediately follows that 𝛼_𝑛1−𝛼_𝑛𝑔₁𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝛾_𝑛𝛿_𝑛𝑔₂𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+2𝛽_𝑛𝑓𝑥_𝑛−𝑝𝑥_𝑛+1≤𝑥−𝑝_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+2𝛽_𝑛𝑓𝑥_𝑛−𝑝𝑥_𝑛+1−𝑝. (127) According to condition (vi), we get liminf_𝑛→∞𝛿_𝑛=liminf_𝑛→∞1−𝛽_𝑛−𝛾_𝑛=1−limsup_𝑛→∞𝛽_𝑛+𝛾_𝑛>0. (128) Since 𝛽_𝑛→0 and ‖𝑥_𝑛+1−𝑥_𝑛‖→0, we conclude from conditions (i) and (vi) that lim_𝑛→∞𝑔₁𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛=0,lim_𝑛→∞𝑔₂𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛=0. (129) Utilizing the properties of 𝑔₁ and 𝑔₂, we have lim_𝑛→∞𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛=0,lim_𝑛→∞𝑦_𝑛−𝑆_𝑛𝐺𝑦_𝑛=0. (130) Note that 𝑦_𝑛−𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛−𝛽_𝑛𝑓𝑥_𝑛−𝑦_𝑛−𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑦_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛𝑓𝑥_𝑛−𝑦_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑦_𝑛. (131) Thus, from (123), (130), and 𝛽_𝑛→0, it follows that lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (132) On the other hand, from (130), we get lim_𝑛→∞𝑦_𝑛−𝐺𝑥_𝑛=lim_𝑛→∞1−𝛼_𝑛×𝑆_𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛=0. (133) This together with (132) implies that lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0. (134) By (130) and Lemma 7, we have 𝑆𝐺𝑥_𝑛−𝐺𝑥_𝑛≤𝑆𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝑆_𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛⟶0as𝑛⟶∞. (135) In terms of (134) and (135), we have 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝐺𝑥_𝑛+𝐺𝑥_𝑛−𝑆𝐺𝑥_𝑛+𝑆𝐺𝑥_𝑛−𝑆𝑥_𝑛≤2𝑥_𝑛−𝐺𝑥_𝑛+𝐺𝑥_𝑛−𝑆𝐺𝑥_𝑛⟶0as𝑛⟶∞. (136) Define a mapping 𝑊𝑥=(1−𝜃)𝑆𝑥+𝜃𝐺(𝑥), where 𝜃∈(0,1) is a constant. Then by Lemma 9, we have that Fix(𝑊)=Fix(𝑆)∩Fix(𝐺)=𝐹. We observe that 𝑥_𝑛−𝑊𝑥_𝑛=(1−𝜃)𝑥_𝑛−𝑆𝑥_𝑛+𝜃𝑥_𝑛−𝐺𝑥_𝑛≤(1−𝜃)𝑥_𝑛−𝑆𝑥_𝑛+𝜃𝑥_𝑛−𝐺𝑥_𝑛. (137) From (134) and (136), we obtain lim_𝑛→∞‖𝑥_𝑛−𝑊𝑥_𝑛‖=0. (138)
Now, we claim that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0, (139) where 𝑞=𝑠−lim_𝑡→0𝑥_𝑡 with 𝑥_𝑡 being the fixed point of the contraction 𝑥⟼𝑡𝑓(𝑥)+(1−𝑡)𝑊𝑥. (140) Then 𝑥_𝑡 solves the fixed point equation 𝑥_𝑡=𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑊𝑥_𝑡. Thus we have 𝑥_𝑡−𝑥_𝑛=(1−𝑡)𝑊𝑥_𝑡−𝑥_𝑛+𝑡𝑓𝑥_𝑡−𝑥_𝑛. (141) By Lemma 3, we conclude that 𝑥_𝑡−𝑥_𝑛²≤(1−𝑡)²𝑊𝑥_𝑡−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤(1−𝑡)²𝑊𝑥_𝑡−𝑊𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤(1−𝑡)²𝑥_𝑡−𝑥_𝑛²+2𝑥_𝑡−𝑥_𝑛×𝑊𝑥_𝑛−𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛+2𝑡𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛=1−2𝑡+𝑡²𝑥_𝑡−𝑥_𝑛²+𝑓_𝑛(𝑡)+2𝑡𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛+2𝑡𝑥_𝑡−𝑥_𝑛², (142) where 𝑓_𝑛(𝑡)=(1−𝑡)²2𝑥_𝑡−𝑥_𝑛+𝑥_𝑛−𝑊𝑥_𝑛×𝑥_𝑛−𝑊𝑥_𝑛⟶0as𝑛⟶∞. (143) It follows from (142) that 𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤𝑡2𝑥_𝑡−𝑥_𝑛²+1𝑓2𝑡^𝑛(𝑡). (144) Letting 𝑛→∞ in (144) and noticing (143), we derive limsup_𝑛→∞𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤𝑡2𝑀₂, (145) where 𝑀₂>0 is a constant such that ‖𝑥_𝑡−𝑥_𝑛‖²≤𝑀₂ for all 𝑡∈(0,1) and 𝑛≥0. Taking 𝑡→0 in (145), we have limsup_𝑡→0limsup_𝑛→∞𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤0. (146) On the other hand, we have 𝑓𝑞−𝑞,𝐽𝑥_𝑛=𝑓𝑞𝑥−𝑞−𝑞,𝐽_𝑛−𝑞−𝐽𝑥_𝑛−𝑥_𝑡+𝑥_𝑡−𝑞,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑞−𝑓𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑥_𝑡−𝑥_𝑡,𝐽x_𝑛−𝑥_𝑡. (147) Hence it follows that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞−𝐽𝑥_𝑛−𝑥_𝑡+𝑥_𝑡−𝑞limsup_𝑛→∞𝑥_𝑛−𝑥_𝑡+𝜌𝑞−𝑥_𝑡limsup_𝑛→∞𝑥_𝑛−𝑥_𝑡+limsup_𝑛→∞𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡. (148) Taking into account that 𝑥_𝑡→𝑞 as 𝑡→0, we have from (146) limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞=limsup_𝑡→0limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤limsup_𝑡→0limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞−𝐽𝑥_𝑛−𝑥_𝑡. (149) Since 𝑋 has a uniformly Gâteaux differentiable norm, the duality mapping 𝐽 is norm-to-weak* uniformly continuous on bounded subsets of 𝑋. Consequently, the two limits are interchangeable, and hence (139) holds. From (123), we get (𝑥_𝑛+1−𝑞)−(𝑥_𝑛−𝑞)→0. Noticing the norm-to-weak* uniform continuity of 𝐽 on bounded subsets of 𝑋, we deduce from (139) that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞=limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞−𝐽𝑥_𝑛+𝑓𝑞𝑥−𝑞−𝑞,𝐽_𝑛+1−𝑞=limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0. (150)
Finally, let us show that 𝑥_𝑛→𝑞 as 𝑛→∞. We observe that 𝑦_𝑛−𝑞=𝛼_𝑛𝐺𝑥_𝑛+−𝑞1−𝛼_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑞≤𝛼_𝑛𝑥_𝑛−𝑞+1−𝛼_𝑛𝑥_𝑛=𝑥−𝑞_𝑛,𝑥−𝑞_𝑛+1−𝑞²=𝛽_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝑓𝑞−𝑞,𝐽𝑥_𝑛+1+𝛾−𝑞_𝑛𝑦_𝑛−𝑞+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑞,𝐽𝑥_𝑛+1−𝑞≤𝛽_𝑛𝑓𝑥_𝑛−𝑓𝑞𝑥_𝑛+1−𝑞+𝛽_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1+𝛾−𝑞_𝑛𝑦_𝑛−𝑞+𝛿_𝑛𝑆_𝑛𝐺𝑦_𝑛−𝑞𝑥_𝑛+1−𝑞≤𝛽_𝑛𝜌𝑥_𝑛−𝑞𝑥_𝑛+1−𝑞+𝛽_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1+𝛾−𝑞_𝑛𝑦_𝑛−𝑞+𝛿_𝑛𝑦_𝑛−𝑞𝑥_𝑛+1−𝑞=𝛽_𝑛𝜌𝑥_𝑛−𝑞𝑥_𝑛+1−𝑞+𝛽_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1+𝛾−𝑞_𝑛+𝛿_𝑛𝑦_𝑛−𝑞𝑥_𝑛+1−𝑞≤𝛽_𝑛𝜌𝑥_𝑛−𝑞𝑥_𝑛+1−𝑞+𝛽_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1+−𝑞1−𝛽_𝑛𝑥_𝑛−𝑞𝑥_𝑛+1=−𝑞1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑞𝑥_𝑛+1−𝑞+𝛽_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1≤−𝑞1−𝛽_𝑛1−𝜌2𝑥_𝑛−𝑞²+𝑥_𝑛+1−𝑞²+𝛽_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1≤−𝑞1−𝛽_𝑛1−𝜌2𝑥_𝑛−𝑞²+12𝑥_𝑛+1−𝑞²+𝛽_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞. (151)(152) So, we have 𝑥_𝑛+1−𝑞²≤1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑞²+2𝛽_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1=−𝑞1−𝛽_𝑛1−𝜌𝑥_𝑛−𝑞²+𝛽_𝑛2𝑓𝑞𝑥1−𝜌−𝑞,𝐽_𝑛+1−𝑞.1−𝜌 (153) Since ∑^∞_𝑛=0𝛽_𝑛=∞ and limsup_𝑛→∞⟨𝑓(𝑞)−𝑞, 𝐽(𝑥_𝑛+1−𝑞)⟩≤0, by Lemma 2, we conclude from (153) that 𝑥_𝑛→𝑞 as 𝑛→∞. This completes the proof.

Corollary 23. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gâteaux differentiable norm. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝜆_𝑖-strictly pseudocontractive and 𝛼_𝑖-strongly accretive with 𝛼_𝑖+𝜆_𝑖≥1 for 𝑖=1,2. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that 𝐹=Fix(𝑆)∩Ω≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂). For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛼_𝑛𝐺𝑥_𝑛+1−𝛼_𝑛𝑆𝐺𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆𝐺𝑦_𝑛,∀𝑛≥0, (154) where 1−(𝜆_𝑖/(1+𝜆_𝑖))(1−(1−𝛼_𝑖)/𝜆_𝑖)≤𝜇_𝑖≤1 for 𝑖=1,2. Suppose that {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are the sequences in (0,1) satisfying the following conditions:(i)0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1,(ii)lim_𝑛→∞𝛽_𝑛=0 and ∑^∞_𝑛=0𝛽_𝑛=∞,(iii)∑^∞_𝑛=1|𝛼_𝑛−𝛼_𝑛−1|<∞ or lim_𝑛→∞|𝛼_𝑛−𝛼_𝑛−1|/𝛽_𝑛=0,(iv)∑^∞_𝑛=1|𝛽_𝑛−𝛽_𝑛−1|<∞ or lim_𝑛→∞𝛽_𝑛−1/𝛽_𝑛=1,(v)∑^∞_𝑛=1|(𝛾_𝑛/(1−𝛽_𝑛))−(𝛾_𝑛−1/(1−𝛽_𝑛−1))|<∞ or lim_𝑛→∞(1/𝛽_𝑛)|(𝛾_𝑛/(1−𝛽_𝑛))−(𝛾_𝑛−1/(1−𝛽_𝑛−1))|=0,(vi)0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1.Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (155)

Further, we illustrate Theorem 22 by virtue of an example, that is, the following corollary.

Corollary 24. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gâteaux differentiable norm. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝑓∶𝐶→C be a contraction with coefficient 𝜌∈(0,1). Let 𝑇∶𝐶→𝐶 be a self-mapping on 𝐶 such that 𝐼−𝑇 is 𝜁-strictly pseudocontractive and 𝜃-strongly accretive with 𝜁+𝜃≥1, and let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that 𝐹=Fix(𝑆)∩Fix(T)≠∅. For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛼_𝑛𝐺𝑥_𝑛+1−𝛼_𝑛𝑆(𝐼−𝜆(𝐼−𝑇))𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆(𝐼−𝜆(𝐼−𝑇))𝑦_𝑛,∀𝑛≥0, (156) where 1−(𝜁/(1+𝜁))(1−(1−𝜃)/𝜁)≤𝜆≤1. Suppose that {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are the sequences in (0,1) satisfying the following conditions:(i)0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1,(ii)lim_𝑛→∞𝛽_𝑛=0 and ∑^∞_𝑛=0𝛽_𝑛=∞,(iii)∑^∞_𝑛=1|𝛼_𝑛−𝛼_𝑛−1|<∞ or lim_𝑛→∞|𝛼_𝑛−𝛼_𝑛−1|/𝛽_𝑛=0,(iv)∑^∞_𝑛=1|𝛽_𝑛−𝛽_𝑛−1|<∞ or lim_𝑛→∞𝛽_𝑛−1/𝛽_𝑛=1,(v)∑^∞_𝑛=1|(𝛾_𝑛/(1−𝛽_𝑛))−(𝛾_𝑛−1/(1−𝛽_𝑛−1))|<∞ or lim_𝑛→∞(1/𝛽_𝑛)|(𝛾_𝑛/(1−𝛽_𝑛))−(𝛾_𝑛−1/(1−𝛽_𝑛−1))|=0,(vi)0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1.Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (157)

Proof. Utilizing the arguments similar to those in the proof of Corollary 16, we can obtain the desired result.

Remark 25. As previous, we emphasize that our composite iterative algorithms (i.e., the iterative schemes (27) and (111)) are based on Korpelevich’s extragradient method and viscosity approximation method. It is well known that the so-called viscosity approximation method must contain a contraction 𝑓 on 𝐶. In the meantime, it is worth pointing out that our proof of Theorems 14 and 22 must make use of Lemma 8 for implicit viscosity approximation method; that is, Lemma 8 plays a key role in our proof of Theorems 14 and 22. Therefore, there is no doubt that the contraction 𝑓 in Theorems 14 and 22 cannot be replaced by a general 𝑘-Lipschitzian mapping with constant 𝑘≥0.

Remark 26. Theorem 22 improves, extends, supplements, and develops [2, Theorem  3.1 and Corollary  3.2] and [5, Theorems  3.1] in the following aspects.(i)The problem of finding a point 𝑞∈⋂_𝑛Fix(𝑆_𝑛)∩Ω in Theorem 22 is more general and more subtle than the problem of finding a point 𝑞∈Fix(𝑆)∩VI(𝐶,𝐴) in Jung [5, Theorem  3.1].(ii)The iterative scheme in [2, Theorem  3.1] is extended to develop the iterative scheme (111) of Theorem 22 by virtue of the iterative scheme of [5, Theorem  3.1]. The iterative scheme (111) in Theorem 22 is more advantageous and more flexible than the iterative scheme in [2, Theorem  3.1] because it involves several parameter sequences {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛}.(iii)The iterative scheme (111) in Theorem 22 is very different from everyone in both [2, Theorem  3.1] and [5, Theorem  3.1] because the mappings 𝑆_𝑛 and 𝐺 in the iterative scheme of [2, Theorem  3.1] and the mapping 𝑆𝑃_𝐶(𝐼−𝜆_𝑛𝐴) in the iterative scheme of [5, Theorem  3.1] are replaced by the same composite mapping 𝑆_𝑛𝐺 in the iterative scheme (111) of Theorem 22.(iv)The proof in [2, Theorem  3.1] depends on the argument techniques in [3], the inequality in 2-uniformly smooth Banach spaces, and the inequality in smooth and uniform convex Banach spaces. However, the proof of Theorem 22 does not depend on the argument techniques in [3], the inequality in 2-uniformly smooth Banach spaces, and the inequality in smooth and uniform convex Banach spaces. It depends on only the inequality in uniform convex Banach spaces.(v)The assumption of the uniformly convex and 2-uniformly smooth Banach space 𝑋 in [2, Theorem  3.1] is weakened to the one of the uniformly convex Banach space 𝑋 having a uniformly Gateaux differentiable norm in Theorem 22.(vi)The iterative scheme in [2, Corollary  3.2] is extended to develop the new iterative scheme in Corollary 15 because the mappings 𝑆 and 𝐺 are replaced by the same composite mapping 𝑆𝐺 in Corollary 23.
Finally, we observe that related results can be found in recent papers, for example, [15–24] and the references therein.

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). This research was partially supported by a Grant from NSC 101-2115-M-037-001.