Hybrid and Relaxed Mann Iterations for General Systems of Variational Inequalities and Nonexpansive Mappings

Abstract

We introduce hybrid and relaxed Mann iteration methods for a general system of variational inequalities with solutions being also common solutions of a countable family of variational inequalities and common fixed points of a countable family of nonexpansive mappings in real smooth and uniformly convex Banach spaces. Here, the hybrid and relaxed Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method, and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for hybrid and relaxed Mann iteration algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gateaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

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 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(i) accretive if for each 𝑥,𝑦∈𝐶 there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥0; (2)(ii) 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);(iii) 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;(iv) 𝜆-strictly pseudocontractive [1] (see also [2]) 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).

It is worth emphasizing that the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, for example, [3–5]. 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→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (7)

exists for all 𝑥,𝑦∈𝑈; in this case, 𝑋 is also said to have a Gateaux differentiable norm. 𝑋 is said to have a uniformly, Gateaux 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 Frechet differential if for each 𝑥∈𝑈, this limit is attained uniformly for 𝑦∈𝑈. In the meantime, 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 [6], 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 Gateaux differentiable norm, then the duality mapping 𝐽 is norm-to-weak* uniformly continuous on bounded subsets of 𝑋.

Recently, Yao et al. [7] combined the viscosity approximation method and Mann iteration method and gave the following hybrid viscosity approximation method.

Let 𝐶 be a nonempty closed convex subset of a real uniformly smooth Banach space 𝑋, 𝑇∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑇)≠∅, and 𝑓∶𝐶→𝐶 a contraction with coefficient 𝜌∈(0,1). For an arbitrary 𝑥₀∈𝐶, define {𝑥_𝑛} in the following way: 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛,∀𝑛≥0, (YCY)

where {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1).

They proved under certain control conditions on the sequences {𝛼_𝑛} and {𝛽_𝑛} that {𝑥_𝑛} converges strongly to a fixed point of 𝑇. Subsequently, under the following control conditions on {𝛼_𝑛} and {𝛽_𝑛}:(i)1≤𝛽_𝑛≤1−𝜌, for all 𝑛≥𝑛₀ for some integer 𝑛₀≥1,(ii)∑^∞_𝑛=0𝛽_𝑛=∞,(iii)0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1,(iv)lim_𝑛→∞(𝛽_𝑛+1/(1−(1−𝛽_𝑛+1)𝛼_𝑛+1)−𝛽_𝑛/(1−(1−𝛽_𝑛)𝛼_𝑛))=0.

Ceng and Yao [8] proved that 𝑥_𝑛⟶𝑞⟺𝛽_𝑛𝑓𝑥_𝑛−𝑥_𝑛⟶0, (9)

where 𝑞∈Fix(𝑇) solves the variational inequality problem (VIP): 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈Fix(𝑇). (10)

Such a result includes [7, Theorem  1] as a special case.

Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝑋 and 𝑓∈Ξ_𝐶 with a contractive coefficient 𝜌∈(0,1), where Ξ_𝐶 is the set of all contractive self-mappings on 𝐶. Let {𝑇_𝑛}^∞_𝑛=0 be a sequence of nonexpansive self-mappings on 𝐶 and {𝜆_𝑛}^∞_𝑛=0 a sequence of nonnegative numbers in [0,1]. For any 𝑛≥0, define a self-mapping 𝑊_𝑛 on 𝐶 as follows: 𝑈_𝑛,𝑛+1𝑈=𝐼,_𝑛,𝑛=𝜆_𝑛𝑇_𝑛𝑈_𝑛,𝑛+1+1−𝜆_𝑛𝑈𝐼,_{𝑛,𝑛−1}=𝜆_𝑛−1𝑇_𝑛−1𝑈_𝑛,𝑛+1−𝜆_𝑛−1⋮𝑈𝐼,_𝑛,𝑘=𝜆_𝑘𝑇_𝑘𝑈_𝑛,𝑘+1+1−𝜆_𝑘𝑈𝐼,_{𝑛,𝑘−1}=𝜆_𝑘−1𝑇_𝑘−1𝑈_𝑛,𝑘+1−𝜆_𝑘−1⋮𝑈𝐼,_𝑛,1=𝜆₁𝑇₁𝑈_𝑛,2+1−𝜆₁𝑊𝐼,_𝑛=𝑈_𝑛,0=𝜆₀𝑇₀𝑈_𝑛,1+1−𝜆₀𝐼. (CY)

Such a mapping 𝑊_𝑛 is called the 𝑊-mapping generated by 𝑇_𝑛,𝑇_𝑛−1,…,𝑇₀, and 𝜆_𝑛,𝜆_𝑛−1,…,𝜆₀; see [9].

In 2008, Ceng and Yao [10] introduced and analyzed the following relaxed viscosity approximation method for finding a common fixed point of an infinite family of nonexpansive mappings in a strictly convex and reflexive Banach space with a uniformly Gateaux differentiable norm.

Theorem 1 (see [10]). Let 𝑋 be a strictly convex and reflexive Banach space with a uniformly Gateaux differentiable norm, 𝐶 a nonempty closed convex subset of 𝑋, {𝑇_𝑛}^∞_𝑛=0 a sequence of nonexpansive self-mappings on 𝐶 such that the common fixed point set 𝐹∶=⋂^∞_𝑛=0Fix(𝑇_𝑛)≠∅, and 𝑓∈Ξ_𝐶 with a contractive coefficient 𝜌∈(1/2,1). For any given 𝑥₀∈𝐶, let {𝑥_𝑛}^∞_𝑛=0 be the iterative sequence defined by 𝑦_𝑛=1−𝛾_𝑛𝑥_𝑛+𝛾_𝑛𝑊_𝑛𝑥_𝑛,𝑥_𝑛+1=1−𝛼_𝑛−𝛽_𝑛𝑥_𝑛+𝛼_𝑛𝑓𝑦_𝑛+𝛽_𝑛𝑊_𝑛𝑦_𝑛,∀𝑛≥0, (11)
where {𝛼_𝑛}^∞_𝑛=0 and {𝛽_𝑛}^∞_𝑛=0 are two sequences in (0,1) with 𝛼_𝑛+𝛽_𝑛≤1(𝑛≥0), {𝛾_𝑛}^∞_𝑛=0 is a sequence in [0,1], and 𝑊_𝑛 is the 𝑊-mapping generated by (CY). Assume that(i)lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=0𝛼_𝑛=∞ and 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;(ii)lim_𝑛→∞|𝛾_𝑛−𝛾_𝑛−1|=0 and limsup_𝑛→∞𝛾_𝑛<1.
Then, there hold the following:(i)lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0;(ii)the sequence {𝑥_𝑛}^∞_𝑛=0 converges strongly to some 𝑝∈𝐹 which is the unique solution of the variational inequality problem (VIP) 𝐼−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑓∈Ξ_𝐶,𝑝∈𝐹, (12) provided lim_𝑛→∞𝛾_𝑛=0 and 𝛽_𝑛≡𝛽 for some fixed 𝛽∈(0,1).

On the other hand, Cai and Bu [11] 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, (13)

where 𝐶 is a nonempty, closed, and convex subset of 𝑋, 𝐵₁,𝐵₂∶𝐶→𝑋 are two nonlinear mappings, and 𝜇₁ and 𝜇₂ are two positive constants. Here, the set of solutions of GSVI (13) is denoted by GSVI(𝐶,𝐵₁,𝐵₂). In particular, if 𝑋=𝐻, a real Hilbert space, then GSVI (13) reduces to the following GSVI of finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜇₁𝐵₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝜇≥0,∀𝑥∈𝐶,₂𝐵₂𝑥^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (14)

in which 𝜇₁ and 𝜇₂ are two positive constants. The set of solutions of problem (14) is still denoted by GSVI(𝐶,𝐵₁,𝐵₂). In particular, if 𝐵₁=𝐵₂=𝐴, then problem (14) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [12]. Further, if 𝑥^∗=𝑦^∗ additionally, then the NSVI reduces to the classical variational inequality problem (VIP) of finding 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (15)

The solution set of the VIP (15) is denoted by VI(𝐶,𝐴). Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems. It is now well known that the variational inequalities are equivalent to the fixed point problems, the origin of which can be traced back to Lions and Stampacchia [13]. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous.

Recently, Ceng et al. [14] transformed problem (14) into a fixed point problem in the following way.

Lemma 2 (see [14]). For given 𝑥,𝑦∈𝐶, (𝑥,𝑦) is a solution of problem (14) if and only if 𝑥 is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺(𝑥)=𝑃_𝐶𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶, (16)
where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥) and 𝑃_𝐶 is the projection of 𝐻 onto 𝐶.

In particular, if the mapping 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.

In 1976, Korpelevič [15] proposed an iterative algorithm for solving the VIP (15) in Euclidean space 𝐑^𝑛: 𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝜏𝐴𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝜏𝐴𝑦_𝑛,𝑛≥0 (17)

with 𝜏>0 a given number, which is known as the extragradient method (see also [16]). The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [3, 11, 13, 17–33] and references therein, to name but a few.

In particular, whenever 𝑋 is still a real smooth Banach space, 𝐵₁=𝐵₂=𝐴 and 𝑥^∗=𝑦^∗, then GSVI (13) reduces to the variational inequality problem (VIP) of finding 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (18)

which was considered by Aoyama et al. [34]. Note that VIP (18) is connected with the fixed point problem for nonlinear mapping (see, e.g., [35]), the problem of finding a zero point of a nonlinear operator (see, e.g., [36]), and so on. It is clear that VIP (18) extends VIP (15) from Hilbert spaces to Banach spaces.

In order to find a solution of VIP (18), Aoyama et al. [34] introduced the following Mann-type iterative scheme for an accretive operator 𝐴: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛Π_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,∀𝑛≥1, (19)

where Π_𝐶 is a sunny nonexpansive retraction from 𝑋 onto 𝐶. Then, they proved a weak convergence theorem. For the related work, see [37] and the references therein.

Let 𝐶 be a nonempty convex subset of a real Banach space 𝑋. Let {𝑇_𝑖}^𝑁_𝑖=1 be a finite family of nonexpansive mappings of 𝐶 into itself and let 𝜆₁,…,𝜆_𝑁 be real numbers such that 0≤𝜆_𝑖≤1 for every 𝑖=1,…,𝑁. Define a mapping 𝐾∶𝐶→𝐶 as follows: 𝑈₁=𝜆₁𝑇₁+1−𝜆₁𝑈𝐼,₂=𝜆₂𝑇₂𝑈₁+1−𝜆₂𝑈₁,𝑈₃=𝜆₃𝑇₃𝑈₂+1−𝜆₃𝑈₂,⋮𝑈_𝑁−1=𝜆_𝑁−1𝑇_𝑁−1𝑈_𝑁−2+1−𝜆_𝑁−1𝑈_𝑁−2,𝐾=𝑈_𝑁=𝜆_𝑁𝑇_𝑁𝑈_𝑁−1+1−𝜆_𝑁𝑈_𝑁−1. (20)

Such a mapping 𝐾 is called the 𝐾-mapping generated by 𝑇₁,…,𝑇_𝑁 and 𝜆₁,…,𝜆_𝑁.

Very recently, Kangtunyakarn [38] introduced and analyzed an iterative algorithm by the modification of Mann’s iteration process for finding a common element of the set of solutions of a finite family of variational inequalities and the set of fixed points of an 𝜂-strictly pseudocontractive mapping and a nonexpansive mapping in uniformly convex and 2-uniformly smooth Banach spaces.

Theorem 3 (see [38]). Let 𝐶 be a nonempty closed 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 𝐴_𝑖∶𝐶→𝑋 be an 𝛼_𝑖-inverse-strongly accretive mapping for each 𝑖=1,…,𝑁. Define the mapping 𝐺_𝑖∶𝐶→𝐶 by 𝐺_𝑖=Π_𝐶(𝐼−𝜆_𝑖𝐴_𝑖) for 𝑖=1,…,𝑁, where 𝜆_𝑖∈(0,𝛼_𝑖/𝜅²) and 𝜅 is the 2-uniformly smooth constant of 𝑋. Let 𝐵∶𝐶→𝐶 be the 𝐾-mapping generated by 𝐺₁,…,𝐺_𝑁 and 𝜌₁,…,𝜌_𝑁, where 𝜌_𝑖∈(0,1), for all 𝑖=1,…,𝑁−1, and 𝜌_𝑁∈(0,1]. Let 𝑓∶𝐶→𝐶 a contraction with coefficient 𝜌∈(0,1). Let 𝑉∶𝐶→𝐶 be an 𝜂-strictly pseudocontractive mapping and 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that 𝐹=Fix(𝑆)∩Fix(𝑉)∩(⋂^𝑁_𝑖=1VI(𝐶,𝐴_𝑖))≠∅. For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆((1−𝛼)𝐼+𝛼𝑉)𝑥_𝑛,∀𝑛≥0, (21)
where 𝛼∈(0,𝜂/𝜅²). Suppose that {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are the sequences in [0,1], 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and satisfy the following conditions:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(ii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iii)∑^∞_𝑛=1(|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|)<∞;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.
Then, {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (22)

Beyond doubt, it is an interesting and valuable problem of constructing some algorithms with strong convergence for solving GSVI (13) which contains VIP (18) as a special case. Very recently, Cai and Bu [11] constructed an iterative algorithm for solving GSVI (13) and a common fixed point problem of a countable family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a 2-uniformly smooth Banach space 𝑋.

Lemma 4 (see [39]). Let 𝑋 be a 2-uniformly smooth Banach space. Then, 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽(𝑥)+2𝜅𝑦²,∀𝑥,𝑦∈𝑋, (23)
where 𝜅 is the 2-uniformly smooth constant of 𝑋 and 𝐽 is the normalized duality mapping from 𝑋 into 𝑋^∗.

Define the mapping 𝐺∶𝐶→𝐶 as follows: 𝐺(𝑥)∶=Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶. (24)

The fixed point set of 𝐺 is denoted by Ω. Then, their strong convergence theorem on the proposed method is stated as follows.

Theorem 5 (see [11]). Let 𝐶 be a nonempty closed 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 𝛽_𝑖-inverse-strongly accretive with 0<𝜇_𝑖<𝛽_𝑖/𝜅² for 𝑖=1,2. Let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛿∈(0,1). Let {𝑇_𝑛}^∞_𝑛=1 be a countable family of nonexpansive mappings of 𝐶 into itself such that 𝐹=⋂^∞_𝑖=1Fix(𝑇_𝑖)∩Ω≠∅, where Ω is the fixed point set of the mapping 𝐺 defined by (24). For arbitrarily given 𝑥₁∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆_𝑛𝑦_𝑛,𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑧_𝑛,𝑧_𝑛=Π_𝐶𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛,𝑢_𝑛=Π_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛,∀𝑛≥1. (25)

Suppose that {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) satisfying the following conditions:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.

Assume that ∑^∞_𝑛=1sup_𝑥∈𝐷‖𝑇_𝑛+1𝑥−𝑇_𝑛𝑥‖<∞ for any bounded subset 𝐷 of 𝐶 and let 𝑇 be a mapping of 𝐶 into 𝑋 defined by 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥 for all 𝑥∈𝐶 and suppose that Fix(𝑇)=⋂^∞_𝑛=1Fix(𝑇_𝑛). Then, {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (26)

It is easy to see that the iterative scheme in Theorem 5 is essentially equivalent to the following two-step iterative scheme: 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐺𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇_𝑛𝑦_𝑛,∀𝑛≥1. (27)

For the convenience of implementing the argument techniques in [14], the authors of [11] have used the following inequality in a real smooth and uniform convex Banach space 𝑋.

Proposition 6 (see [40]). Let 𝑋 be a real smooth and uniform convex Banach space and let 𝑟>0. Then, there exists a strictly increasing, continuous, and convex function 𝑔∶[0,2𝑟]→𝐑, 𝑔(0)=0 such that 𝑔𝑥−𝑦≤‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝐵_𝑟, (28)
where 𝐵_𝑟={𝑥∈𝑋∶‖𝑥‖≤𝑟}.

Let 𝐶 be a nonempty closed convex subset of a real smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶 and 𝑓∶𝐶→𝐶 a contraction with coefficient 𝜌∈(0,1). Motivated and inspired by the research going on this area, we consider and introduce hybrid and relaxed Mann iteration methods for finding solutions of the GSVI (13) which are also common solutions of a countable family of variational inequalities and common fixed points of a countable family of nonexpansive mappings in 𝑋. Here, the hybrid and relaxed Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method, and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for hybrid and relaxed Mann iteration algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gateaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature; see, for example, [8, 10, 11, 14, 33, 38].

2. Preliminaries

We list some lemmas that will be used in the sequel.

Lemma 7 (see [41]). 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, (29)
where {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} satisfy the following conditions:(i){𝛼_𝑛}⊂[0,1] and ∑^∞_𝑛=0𝛼_𝑛=∞;(ii)limsup_𝑛→∞𝛽_𝑛≤0;(iii)𝛾_𝑛≥0, for all 𝑛≥0, and ∑^∞_𝑛=0𝛾_𝑛<∞.
Then, limsup_𝑛→∞𝑠_𝑛=0.

The following lemma is an immediate consequence of the subdifferential inequality of the function (1/2)‖⋅‖².

Lemma 8 (see [42]). Let 𝑋 be a real Banach space 𝑋. Then, for all 𝑥,𝑦∈𝑋(i)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑗(𝑥+𝑦)⟩ for all 𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦);(ii)‖𝑥+𝑦‖²≥‖𝑥‖²+2⟨𝑦,𝑗(𝑥)⟩ for all 𝑗(𝑥)∈𝐽(𝑥).

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), (30)

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 9 (see [43]). Let 𝐶 be a nonempty closed 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))⟩, for all 𝑥,𝑦∈𝐶;(iii)⟨𝑥−Π(𝑥),𝐽(𝑦−Π(𝑥))⟩≤0, for all 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 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 10. Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶 and let 𝐵₁,𝐵₂∶𝐶→𝑋 be nonlinear mappings. For given 𝑥^∗,𝑦^∗∈𝐶, (𝑥^∗,𝑦^∗) is a solution of GSVI (13) if and only if 𝑥^∗=Π_𝐶(𝑦^∗−𝜇₁𝐵₁𝑦^∗), where 𝑦^∗=Π_𝐶(𝑥^∗−𝜇₂𝐵₂𝑥^∗).

Proof. We can rewrite GSVI (13) as 𝑥^∗−𝑦^∗−𝜇₁𝐵₁𝑦^∗,𝐽𝑥−𝑥^∗𝑦≥0,∀𝑥∈𝐶,_∗−𝑥^∗−𝜇₂𝐵₂𝑥^∗,𝐽𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (31)
which is obviously equivalent to 𝑥^∗=Π_𝐶𝑦^∗−𝜇₁𝐵₁𝑦^∗,𝑦^∗=Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗, (32)
because of Lemma 9. This completes the proof.

In terms of Lemma 10, we observe that 𝑥^∗=Π_𝐶Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗−𝜇₁𝐵₁Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗, (33)

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.

Lemma 11 (see [44]). Let 𝑋 be a uniformly convex Banach space and 𝐵_𝑟={𝑥∈𝑋∶‖𝑥‖≤𝑟}, 𝑟>0. Then, there exists a continuous, strictly increasing, and convex function 𝑔∶[0,∞]→[0,∞], 𝑔(0)=0 such that 𝛼𝑥+𝛽𝑦+𝛾𝑧²≤𝛼‖𝑥‖²+𝛽𝑦²+𝛾‖𝑧‖²−𝛼𝛽𝑔𝑥−𝑦 (34)
for all 𝑥,𝑦,𝑧∈𝐵_𝑟, and all 𝛼,𝛽,𝛾∈[0,1] with 𝛼+𝛽+𝛾=1.

Lemma 12 (see [45]). Let 𝐶 be a nonempty closed 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 convex subset of a Banach space 𝑋 and 𝑇∶𝐶→𝐶 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. (35)

Lemma 13 (see [35, 46]). 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 convex subset of 𝑋, 𝑇∶𝐶→𝐶 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, for all 𝑓∈Ξ_𝐶, then 𝑄(𝑓) solves the VIP: 𝐼−𝑓𝑄𝑓,𝐽𝑄𝑓−𝑝≤0,∀𝑓∈Ξ_𝐶,𝑝∈Fix(𝑇). (36)

Lemma 14 (see [47]). Let 𝐶 be a nonempty closed 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.

Lemma 15 (see [39]). 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 (37)
for all 𝜆∈[0,1] and 𝑥,𝑦∈𝑋 such that ‖𝑥‖≤𝑟 and ‖𝑦‖≤𝑟.

Lemma 16 (see [48, Lemma  3.2]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space 𝑋. Let {𝑇_𝑛}^∞_𝑛=0 be a sequence of nonexpansive self-mappings on 𝐶 such that ⋂^∞_𝑛=0Fix(𝑇_𝑛)≠∅ and let {𝜆_𝑛}^∞_𝑛=0 be a sequence of positive numbers in (0,𝑏] for some 𝑏∈(0,1). Then, for every 𝑥∈𝐶 and 𝑘≥0, the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists.

Using Lemma 16, one can define a mapping 𝑊∶𝐶→𝐶 as follows: 𝑊𝑥=lim_𝑛→∞𝑊_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,0𝑥 (38)

for every 𝑥∈𝐶. Such a 𝑊 is called the 𝑊-mapping generated by the sequences {𝑇_𝑛}^∞_𝑛=0 and {𝜆_𝑛}^∞_𝑛=0. Throughout this paper, we always assume that {𝜆_𝑛}^∞_𝑛=0 is a sequence of positive numbers in (0,𝑏] for some 𝑏∈(0,1).

Lemma 17 (see [48]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space 𝑋. Let {𝑇_𝑛}^∞_𝑛=0 be a sequence of nonexpansive self-mappings on 𝐶 such that ⋂^∞_𝑛=0Fix(𝑇_𝑛)≠∅ and let {𝜆_𝑛}^∞_𝑛=0 be a sequence of positive numbers in (0,𝑏] for some 𝑏∈(0,1). Then, Fix(𝑊)=⋂^∞_𝑛=0Fix(𝑇_𝑛).
Let 𝜇 be a continuous linear functional on 𝑙^∞ and 𝑠=(𝑎₀,𝑎₁,…)∈𝑙^∞. One writes 𝜇_𝑛(𝑎_𝑛) instead of 𝜇(𝑠). 𝜇 is called a Banach limit if 𝜇 satisfies ‖𝜇‖=𝜇_𝑛(1)=1 and 𝜇_𝑛(𝑎_𝑛+1)=𝜇_𝑛(𝑎_𝑛) for all (𝑎₀,𝑎₁,…)∈𝑙^∞. If 𝜇 is a Banach limit, then, there hold the following:(i)for all 𝑛≥0, 𝑎_𝑛≤𝑐_𝑛 implies 𝜇_𝑛(𝑎_𝑛)≤𝜇_𝑛(𝑐_𝑛);(ii)𝜇_𝑛(𝑎_𝑛+𝑟)=𝜇_𝑛(𝑎_𝑛) for any fixed positive integer 𝑟;(iii)liminf_𝑛→∞𝑎_𝑛≤𝜇_𝑛(𝑎_𝑛)≤limsup_𝑛→∞𝑎_𝑛 for all (𝑎₀,𝑎₁,…)∈𝑙^∞.

Lemma 18 (see [49]). Let 𝑎∈𝐑 be a real number and a sequence {𝑎_𝑛}∈𝑙^∞ satisfy the condition 𝜇_𝑛(𝑎_𝑛)≤𝑎 for all Banach limit 𝜇. If limsup_𝑛→∞(𝑎_𝑛+𝑟−𝑎_𝑛)≤0, then limsup_𝑛→∞𝑎_𝑛≤𝑎.

In particular, if 𝑟=1 in Lemma 18, then we immediately obtain the following corollary.

Corollary 19 (see [50]). Let 𝑎∈𝐑 be a real number and a sequence {𝑎_𝑛}∈𝑙^∞ satisfy the condition 𝜇_𝑛(𝑎_𝑛)≤𝑎 for all Banach limit 𝜇. If limsup_𝑛→∞(𝑎_𝑛+1−𝑎_𝑛)≤0, then, limsup_𝑛→∞𝑎_𝑛≤𝑎.

Lemma 20 (see [51]). Let {𝑥_𝑛} and {𝑧_𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽_𝑛} be a sequence of nonnegative numbers in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose that 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑧_𝑛 for all integers 𝑛≥0 and limsup_𝑛→∞(‖𝑧_𝑛+1−𝑧_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑥_𝑛−𝑧_𝑛‖=0.

Lemma 21 (see [34]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶 and 𝐴∶𝐶→𝑋 an accretive mapping. Then for all 𝜆>0, VI(𝐶,𝐴)=FixΠ_𝐶(𝐼−𝜆𝐴). (39)

Lemma 22 (see [11]). Let 𝐶 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝑋. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛽_𝑖-inverse-strongly accretive. Then, one has 𝐼−𝜇_𝑖𝐵_𝑖𝑥−𝐼−𝜇_𝑖𝐵_𝑖𝑦²≤𝑥−𝑦²+2𝜇_𝑖𝜇_𝑖𝜅²−𝛽_𝑖𝐵_𝑖𝑥−𝐵_𝑖𝑦²,∀𝑥,𝑦∈𝐶, (40)
for 𝑖=1,2 where 𝜇_𝑖>0. In particular, if 0<𝜇_𝑖≤𝛽_𝑖/𝜅², then 𝐼−𝜇_𝑖𝐵_𝑖 is nonexpansive for 𝑖=1,2.

Lemma 23 (see [11]). Let 𝐶 be a nonempty closed 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 𝛽_𝑖-inverse-strongly accretive for 𝑖=1,2. Let 𝐺∶𝐶→𝐶 be the mapping defined by 𝐺𝑥=Π_𝐶Π_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁Π_𝐶𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶. (41)
If 0<𝜇_𝑖≤𝛽_𝑖/𝜅² for 𝑖=1,2, then 𝐺∶𝐶→𝐶 is nonexpansive.

3. Hybrid Mann Iterations and Their Convergence Criteria

In this section, we introduce our hybrid Mann iteration algorithms in real smooth and uniformly convex Banach spaces and present their convergence criteria.

Theorem 24. Let 𝐶 be a nonempty closed 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 {𝜌_𝑛}^∞_𝑛=0 be a sequence of positive numbers in (0,𝑏] for some 𝑏∈(0,1) and 𝐴_𝑖∶𝐶→𝐸 an 𝛼_𝑖-inverse strongly accretive mapping for each 𝑖=0,1,…. Define a mapping 𝐺_𝑖∶𝐶→𝐶 by Π_𝐶(𝐼−𝜆_𝑖𝐴_𝑖)𝑥=𝐺_𝑖𝑥 for all 𝑥∈𝐶 and 𝑖=0,1,…, where 𝜆_𝑖∈(0,𝛼_𝑖/𝜅²], 𝜅 is the 2-uniformly smooth constant of 𝑋. Let 𝐵_𝑛∶𝐶→𝐶 be the 𝑊-mapping generated by 𝐺_𝑛,𝐺_𝑛−1,…,𝐺₀ and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₀. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛽_𝑖-inverse strongly accretive for 𝑖=1,2. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let {𝑆_𝑖}^∞_𝑖=0 be a countable family of nonexpansive mappings of 𝐶 into itself such that 𝐹=(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Ω∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖))≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) with 0<𝜇_𝑖<𝛽_𝑖/𝜅² for 𝑖=1,2. For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛,∀𝑛≥0, (42)
where {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. Suppose that the following conditions hold:(i)∑^∞_𝑛=0𝛼_𝑛=∞ and 0≤𝛼_𝑛≤1−𝜌, for all 𝑛≥𝑛₀ for some integer 𝑛₀≥0;(ii)liminf_𝑛→∞𝛾_𝑛>0 and liminf_𝑛→∞𝛿_𝑛>0;(iii)lim_𝑛→∞(|𝛼_𝑛+1/(1−(1−𝛼_𝑛+1)𝛽_𝑛+1)−𝛼_𝑛/(1−(1−𝛼_𝑛)𝛽_𝑛)|+|𝛿_𝑛+1/(1−𝛽_𝑛+1)−𝛿_𝑛/(1−𝛽_𝑛)|)=0;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.
Assume that ∑^∞_𝑛=0sup_𝑥∈𝐷‖𝑆_𝑛+1𝑥−𝑆_𝑛𝑥‖<∞ 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 and suppose that Fix(𝑆)=⋂^∞_𝑖=0Fix(𝑆_𝑖). Then, there hold the following:(I)lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0;(II)𝑥_𝑛→𝑞⇔𝛼_𝑛(𝑓(𝑥_𝑛)−𝑥_𝑛)→0 provided 𝛽_𝑛≡𝛽 for some fixed 𝛽∈(0,1), where 𝑞∈𝐹 solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (43)

Proof. First of all, since 0<𝜆_𝑖<𝛼_𝑖/𝜅² for 𝑖=0,1,…, it is easy to see that 𝐺_𝑖 is a nonexpansive mapping for each 𝑖=0,1,…. Since 𝐵_𝑛∶𝐶→𝐶 is the 𝑊-mapping generated by 𝐺_𝑛,𝐺_𝑛−1,…,𝐺₀ and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₀, by Lemma 16 we know that, for each 𝑥∈𝐶 and 𝑘≥0, the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists. Moreover, one can define a mapping 𝐵∶𝐶→𝐶 as follows: 𝐵𝑥=lim_𝑛→∞𝐵_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,0𝑥 (44)
for every 𝑥∈𝐶. That is, such a 𝐵 is the 𝑊-mapping generated by the sequences {𝐺_𝑛}^∞_𝑛=0 and {𝜌_𝑛}^∞_𝑛=0. According to Lemma 17, we know that Fix(𝐵)=⋂^∞_𝑖=0Fix(𝐺_𝑖). From Lemma 15 and the definition of 𝐺_𝑖, we have Fix(𝐺_𝑖)=VI(𝐶,𝐴_𝑖) for each 𝑖=0,1,…. Hence, we have Fix(𝐵)=^∞_𝑖=0Fix𝐺_𝑖=^∞_𝑖=0VI𝐶,𝐴_𝑖. (45)
Next, let us show that the sequence {𝑥_𝑛} is bounded. Indeed, take a fixed 𝑝∈𝐹 arbitrarily. Then, we get 𝑝=𝐺𝑝, 𝑝=𝐵_𝑛𝑝, and 𝑝=𝑆_𝑛𝑝 for all 𝑛≥0. By Lemma 23 we know that 𝐺 is nonexpansive. Then, from (42), we have 𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝐺𝑥_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑥_𝑛=𝑥−𝑝_𝑛−𝑝, (46)
and hence 𝑥_𝑛+1−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑝+1−𝛼_𝑛𝑦_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛𝑝+𝑓𝑝+−𝑓−𝑝1−𝛼_𝑛𝑦_𝑛−𝑝≤𝛼_𝑛𝜌𝑥_𝑛+𝑓𝑝+−𝑝−𝑝1−𝛼_𝑛𝑥_𝑛=−𝑝1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝1−𝜌−𝑝𝑥1−𝜌≤max_𝑛,𝑓𝑝−𝑝−𝑝.1−𝜌 (47)
By induction, we obtain 𝑥_𝑛−𝑝≤max𝑥₀,𝑓𝑝−𝑝−𝑝1−𝜌,∀𝑛≥0. (48)
Thus, {𝑥_𝑛} is bounded, and so are the sequences {𝑦_𝑛}, {𝐺𝑥_𝑛} and {𝑓(𝑥_𝑛)}.
Let us show that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (49)
As a matter of fact, put 𝜎_𝑛=(1−𝛼_𝑛)𝛽_𝑛, for all 𝑛≥0. Then, it follows from (i) and (iv) that 𝛽_𝑛≥𝜎_𝑛=1−𝛼_𝑛𝛽_𝑛≥1−1−𝜌𝛽_𝑛=𝜌𝛽_𝑛,∀𝑛≥𝑛₀, (50)
and hence 0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1. (51)
Define 𝑥_𝑛+1=𝜎_𝑛𝑥_𝑛+1−𝜎_𝑛𝑧_𝑛. (52)
Observe that 𝑧_𝑛+1−𝑧_𝑛=𝑥_𝑛+2−𝜎_𝑛+1𝑥_𝑛+11−𝜎_𝑛+1−𝑥_𝑛+1−𝜎_𝑛𝑥_𝑛1−𝜎_𝑛=𝛼_𝑛+1𝑓𝑥_𝑛+1+1−𝛼_𝑛+1𝑦_𝑛+1−𝜎_𝑛+1𝑥_𝑛+11−𝜎_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛−𝜎_𝑛𝑥_𝑛1−𝜎_𝑛=𝛼_𝑛+1𝑓𝑥_𝑛+11−𝜎_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛1−𝜎_𝑛−1−𝛼_𝑛𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝜎_𝑛𝑥_𝑛1−𝜎_𝑛+1−𝛼_𝑛+1𝛽_𝑛+1𝑥_𝑛+1+𝛾_𝑛+1𝐵_𝑛+1𝑥_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛+1−𝜎_𝑛+1𝑥_𝑛+1×1−𝜎_𝑛+1⁻¹=𝛼_𝑛+1𝑓𝑥_𝑛+11−𝜎_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛1−𝜎_𝑛+1−𝛼_𝑛+11−𝜎_𝑛+1×𝛾_𝑛+1𝐵_𝑛+1𝑥_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛+1−1−𝛼_𝑛1−𝜎_𝑛𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛=𝛼_𝑛+1𝑓𝑥_𝑛+11−𝜎_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛1−𝜎_𝑛+1−𝛼_𝑛+11−𝛽_𝑛+11−𝜎_𝑛+1×𝛾_𝑛+1𝐵_𝑛+1𝑥_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛+11−𝛽_𝑛+1−𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛1−𝛽_𝑛+1−𝛼_𝑛+11−𝛽_𝑛+11−𝜎_𝑛+1−1−𝛼_𝑛1−𝛽_𝑛1−𝜎_𝑛×𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛1−𝛽_𝑛=𝛼_𝑛+11−𝜎_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝛼_𝑛+11−𝜎_𝑛+1−𝛼_𝑛1−𝜎_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛+11−𝛽_𝑛+11−𝜎_𝑛+1×𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝐵_𝑛+1𝑥_𝑛+1−𝐵_𝑛𝑥_𝑛+𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛+1−𝑆_𝑛𝐺𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝛼_𝑛+11−𝜎_𝑛+1−𝛼_𝑛1−𝜎_𝑛𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛=𝛼_𝑛+11−𝜎_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝛼_𝑛+11−𝜎_𝑛+1−𝛼_𝑛1−𝜎_𝑛×𝑓𝑥_𝑛−𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛+1−𝜎_𝑛+1−𝛼_𝑛+11−𝜎_𝑛+1×𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝐵_𝑛+1𝑥_𝑛+1−𝐵_𝑛𝑥_𝑛+𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛+1−𝑆_𝑛𝐺𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛, (53)
and hence 𝑧_𝑛+1−𝑧_𝑛≤𝛼_𝑛+11−𝜎_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝛼_𝑛+11−𝜎_𝑛+1−𝛼_𝑛1−𝜎_𝑛𝑓𝑥_𝑛−𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛+1−𝜎_𝑛+1−𝛼_𝑛+11−𝜎_𝑛+1×𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝐵_𝑛+1𝑥_𝑛+1−𝐵_𝑛𝑥_𝑛+𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛+1−𝑆_𝑛𝐺𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛≤𝜌𝛼_𝑛+11−𝜎_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+11−𝜎_𝑛+1−𝛼_𝑛1−𝜎_𝑛×𝑓𝑥_𝑛+𝐵_𝑛𝑥_𝑛+𝑆_𝑛𝐺𝑥_𝑛+1−𝜎_𝑛+1−𝛼_𝑛+11−𝜎_𝑛+1×𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝐵_𝑛+1𝑥_𝑛+1−𝐵_𝑛𝑥_𝑛+𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛+1−𝑆_𝑛𝐺𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛. (54)
On the other hand, we note that, for all 𝑛≥0, 𝑆_𝑛+1𝐺𝑥_𝑛+1−𝑆_𝑛𝐺𝑥_𝑛≤𝑆_𝑛+1𝐺𝑥_𝑛+1−𝑆_𝑛+1𝐺𝑥_𝑛+𝑆_𝑛+1𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛≤𝐺𝑥_𝑛+1−𝐺𝑥_𝑛+𝑆_𝑛+1𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑆_𝑛+1𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛. (55)
Furthermore, by (CY), since 𝐺_𝑖 and 𝑈_𝑛,𝑖 are nonexpansive, we deduce that for each 𝑛≥0𝐵_𝑛+1𝑥_𝑛+1−𝐵_𝑛𝑥_𝑛≤𝐵_𝑛+1𝑥_𝑛+1−𝐵_𝑛+1𝑥_𝑛+𝐵_𝑛+1𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝐵_𝑛+1𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛+𝜆₀𝐺₀𝑈_𝑛+1,1𝑥_𝑛−𝜆₀𝐺₀𝑈_𝑛,1𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆₀𝑈_𝑛+1,1𝑥_𝑛−𝑈_𝑛,1𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛+𝜆₀𝜆₁𝐺₁𝑈_𝑛+1,2𝑥_𝑛−𝜆₁𝐺₁𝑈_𝑛,2𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆₀𝜆₁𝑈_𝑛+1,2𝑥_𝑛−𝑈_𝑛,2𝑥_𝑛⋮≤𝑥_𝑛+1−𝑥_𝑛+^𝑛_𝑖=0𝜆_𝑖𝑈_{𝑛+1,𝑛+1}𝑥_𝑛−𝑈_𝑛,𝑛+1𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑀₀^𝑛_𝑖=0𝜆_𝑖, (56)
for some constant 𝑀₀>0. Utilizing (54)–(56), we have 𝑧_𝑛+1−𝑧_𝑛≤𝜌𝛼_𝑛+11−𝜎_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+11−𝜎_𝑛+1−𝛼_𝑛1−𝜎_𝑛×𝑓𝑥_𝑛+𝐵_𝑛𝑥_𝑛+𝑆_𝑛𝐺𝑥_𝑛+1−𝜎_𝑛+1−𝛼_𝑛+11−𝜎_𝑛+1×𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝑀₀^𝑛_𝑖=0𝜆_𝑖+𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝑆_𝑛+1𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛=1−𝜎_𝑛+1−𝛼_𝑛+11−𝜌1−𝜎_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+11−𝜎_𝑛+1−𝛼_𝑛1−𝜎_𝑛𝑓𝑥_𝑛+𝐵_𝑛𝑥_𝑛+𝑆_𝑛𝐺𝑥_𝑛+1−𝜎_𝑛+1−𝛼_𝑛+11−𝜎_𝑛+1×𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑀₀^𝑛_𝑖=0𝜆_𝑖+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝑆_𝑛𝐺𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+11−𝜎_𝑛+1−𝛼_𝑛1−𝜎_𝑛𝑀+𝑀^𝑛_𝑖=0𝜆_𝑖+𝑆_𝑛+1𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑀=𝑥_𝑛+1−𝑥_𝑛𝛼+𝑀^𝑛+11−𝜎_𝑛+1−𝛼_𝑛1−𝜎_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛+^𝑛_𝑖=0𝜆_𝑖+𝑆_𝑛+1𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛, (57)
which hence yields 𝑧_𝑛+1−𝑧_𝑛−𝑥_𝑛+1−𝑥_𝑛𝛼≤𝑀^𝑛+11−𝜎_𝑛+1−𝛼_𝑛1−𝜎_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛+^𝑛_𝑖=0𝜆_𝑖+𝑆_𝑛+1𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛, (58)
where sup_𝑛≥0{‖𝑓(𝑥_𝑛)‖+‖𝐵_𝑛𝑥_𝑛‖+‖𝑆_𝑛𝐺𝑥_𝑛‖+𝑀₀}≤𝑀 for some 𝑀>0. So, from (58), condition (iii), and the assumption on {𝑆_𝑛}, it follows that (noting that 0<𝜆_𝑖≤𝑏<1, for all 𝑖≥0) limsup_𝑛→∞𝑧_𝑛+1−𝑧_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (59)
Consequently, by Lemma 20, we have lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0. (60)
It follows from (51) and (52) that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝜎_𝑛𝑧_𝑛−𝑥_𝑛=0. (61)
From (42), we have 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛−𝑥_𝑛, (62)
which hence implies that 𝜌𝑦_𝑛−𝑥_𝑛=1−1−𝜌𝑦_𝑛−𝑥_𝑛≤1−𝛼_𝑛𝑦_𝑛−𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛−𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛. (63)
Since 𝑥_𝑛+1−𝑥_𝑛→0 and 𝛼_𝑛(𝑓(𝑥_𝑛)−𝑥_𝑛)→0, we get lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (64)
Next, we show that ‖𝑥_𝑛−𝐺𝑥_𝑛‖→0 as 𝑛→∞.
Indeed, 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=𝐺𝑥_𝑛 for all 𝑛≥0. From Lemma 22, we have 𝑢_𝑛−𝑞²=Π_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−Π_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²≤𝑥_𝑛−𝑝²−2𝜇₂𝛽₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²,𝑣_𝑛−𝑝²=Π_𝐶𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛−Π_𝐶𝑞−𝜇₁𝐵₁𝑞²≤𝑢_𝑛−𝑞−𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²≤𝑢_𝑛−𝑞²−2𝜇₁𝛽₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞². (65)(66)
Substituting (65) for (66), we obtain 𝑣_𝑛−𝑝²≤𝑥_𝑛−𝑝²−2𝜇₂𝛽₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−2𝜇₁𝛽₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞². (67)
From (42) and (67), we have 𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑣_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑥_𝑛−𝑝²−2𝜇₂𝛽₂−𝜅²𝜇₂×𝐵₂𝑥_𝑛−𝐵₂𝑝²−2𝜇₁𝛽₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²=𝑥_𝑛−𝑝²−2𝛿_𝑛𝜇₂𝛽₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+2𝜇₁𝛽₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞², (68)
which hence implies that 2𝛿_𝑛𝜇₂𝛽₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝜇₁𝛽₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛. (69)
Since 0<𝜇_𝑖<𝛽_𝑖/𝜅² for 𝑖=1,2, and {𝑥_𝑛}, {𝑦_𝑛} are bounded, we obtain from (64), (69), and condition (ii) that lim_𝑛→∞𝐵₂𝑥_𝑛−𝐵₂𝑝=0,lim_𝑛→∞𝐵₁𝑢_𝑛−𝐵₁𝑞=0. (70)
Utilizing Proposition 6 and Lemma 9, 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, (71)
which implies that 𝑢_𝑛−𝑞²≤𝑥_𝑛−𝑝²−𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞. (72)
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, (73)
which implies that 𝑣_𝑛−𝑝²≤𝑢_𝑛−𝑞²−𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝. (74)
Substituting (72) for (74), 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. (75)
By Lemma 8(i), we have from (68) and (75) 𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑣_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑥_𝑛−𝑝²−𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞−𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛×𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛≤𝑥−𝑝_𝑛−𝑝²−𝛿_𝑛𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝, (76)
which hence leads to 𝛿_𝑛𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝑔₂𝑢_𝑛−𝑣_𝑛+≤𝑥𝑝−𝑞_𝑛−𝑝²−𝑦_𝑛−𝑝²+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛≤𝑥−𝑝_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝. (77)
From (70), (77), condition (ii), and the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, {𝑢_𝑛}, and {𝑣_𝑛}, we deduce that lim_𝑛→∞𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞=0,lim_𝑛→∞𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞=0. (78)
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. (79)
From (79), we get 𝑥_𝑛−𝑣_𝑛≤𝑥_𝑛−𝑢_𝑛−+𝑢𝑝−𝑞_𝑛−𝑣_𝑛+𝑝−𝑞⟶0as𝑛⟶∞. (80)
That is, lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0. (81)
Next, let us show that lim_𝑛→∞𝐵_𝑛𝑥_𝑛−𝑥_𝑛=0,lim_𝑛→∞𝑆𝑥_𝑛−𝑥_𝑛=0. (82)
Indeed, utilizing Lemma 15 and (42), we have 𝑦_𝑛−𝑝²=𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝+𝛽_𝑛+𝛾_𝑛𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛𝛽_𝑛+𝛾_𝑛−𝑝²≤𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²+𝛽_𝑛+𝛾_𝑛×𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛𝛽_𝑛+𝛾_𝑛−𝑝²=𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²+𝛽_𝑛+𝛾_𝑛×𝛽_𝑛𝛽_𝑛+𝛾_𝑛𝑥_𝑛+𝛾−𝑝^𝑛𝛽_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝²≤𝛿_𝑛𝐺𝑥_𝑛−𝑝²+𝛽_𝑛+𝛾_𝑛×𝛽_𝑛𝛽_𝑛+𝛾_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝛽_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝛽_𝑛+𝛾_𝑛²𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛿_𝑛𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝛽_𝑛+𝛾_𝑛𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝛽_𝑛+𝛾_𝑛𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛, (83)
which immediately implies that 𝛽_𝑛𝛾_𝑛𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛽_𝑛𝛾_𝑛𝛽_𝑛+𝛾_𝑛𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛. (84)
So, from (64), the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, and conditions (ii), (iv), it follows that lim_𝑛→∞𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛=0. (85)
From the properties of 𝑔₃, we have lim_𝑛→∞𝑥_𝑛−𝐵_𝑛𝑥_𝑛=0. (86)
Taking into account that 𝑦_𝑛−𝑥_𝑛=𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑥_𝑛, (87)
we have 𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑥_𝑛=𝑦_𝑛−𝑥_𝑛−𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑥_𝑛≤𝑦_𝑛−𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑥_𝑛≤𝑦_𝑛−𝑥_𝑛+𝐵_𝑛𝑥_𝑛−𝑥_𝑛. (88)
From (64), (86), and condition (ii), it follows that lim_𝑛→∞𝑆_𝑛𝐺𝑥_𝑛−𝑥_𝑛=0,lim_𝑛→∞𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛=0. (89)
Note that 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛𝑥_𝑛+𝑆_𝑛𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝐺𝑥_𝑛−𝑥_𝑛+𝑆_𝑛𝑥_𝑛−𝑆𝑥_𝑛. (90)
So, in terms of (81), (89), and Lemma 12, we have lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (91)
Suppose that 𝛽_𝑛≡𝛽 for some fixed 𝛽∈(0,1) such that 𝛽+𝛾_𝑛+𝛿_𝑛=1 for all 𝑛≥0. Define a mapping 𝑉𝑥=(1−𝜃₁−𝜃₂)𝑆𝑥+𝜃₁𝐵𝑥+𝜃₂𝐺𝑥, where 𝜃₁,𝜃₂∈(0,1) are two constants with 𝜃₁+𝜃₂<1. Then, by Lemmas 14 and 17, we have that Fix(𝑉)=Fix(𝑆)∩Fix(𝐵)∩Fix(𝐺)=𝐹. For each 𝑘≥1, let {𝑝_𝑘} be a unique element of 𝐶 such that 𝑝_𝑘=1𝑘𝑓𝑝_𝑘+11−𝑘𝑉𝑝_𝑘. (92)
From Lemma 13, we conclude that 𝑝_𝑘→𝑞∈Fix(𝑉)=𝐹 as 𝑘→∞. Observe that for every 𝑛,𝑘𝑦_𝑛−𝐵𝑝_𝑘=𝛽𝑥_𝑛−𝐵𝑝_𝑘+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵𝑝_𝑘≤𝛽𝑥_𝑛−𝐵𝑝_𝑘+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛+𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘=𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛, (93)
and hence 𝑥_𝑛+1−𝐵𝑝_𝑘≤𝛼_𝑛𝑓𝑥_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛𝑦_𝑛−𝐵𝑝_𝑘≤𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛𝑦_𝑛−𝐵𝑝_𝑘≤𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝛼_𝑛𝑥_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛×𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝛼_𝑛𝑥_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛×𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝐵_𝑛𝑥_𝑛−𝐵_𝑛𝑝_𝑘+𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝛼_𝑛𝑥_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝑥_𝑛−𝑝_𝑘+𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝛽+𝛼_𝑛×𝑥1−𝛽_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛1−𝛽𝑥_𝑛−𝑝_𝑘+1−𝛼_𝑛1−𝛽𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝛽+𝛼_𝑛×𝑥1−𝛽_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝐵𝑝_𝑘+1−𝛼_𝑛1−𝛽𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑝_𝑘+1−𝛼_𝑛1−𝛽𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝛽+𝛼_𝑛×𝑥1−𝛽_𝑛+1−𝐵𝑝_𝑘+1−𝛼_𝑛1−𝛽𝑥_𝑛+1−𝑝_𝑘+1−𝛼_𝑛1−𝛽𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+1. (94)
So, it immediately follows from 0≤𝛼_𝑛≤1−𝜌, for all 𝑛≥𝑛₀, that 𝑥_𝑛+1−𝐵𝑝_𝑘≤𝑥_𝑛+1−𝑝_𝑘+𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+11−𝛼_𝑛×𝛼1−𝛽_𝑛𝑥_𝑛−𝑓𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+1+𝛿_𝑛𝑆1−𝛽^𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑝_𝑘+𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛+1𝜌×𝛼1−𝛽_𝑛𝑥_𝑛−𝑓𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+1=𝑥_𝑛+1−𝑝_𝑘+𝜃_𝑛,∀𝑛≥𝑛₀, (95)
where 𝜃_𝑛=‖𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘‖+‖𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛‖+(1/𝜌(1−𝛽))(‖𝛼_𝑛(𝑥_𝑛−𝑓(𝑥_𝑛))‖+‖𝑥_𝑛−𝑥_𝑛+1‖). Since lim_𝑛→∞‖𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘‖=lim_𝑛→∞‖𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛‖=lim_𝑛→∞‖𝛼_𝑛(𝑥_𝑛−𝑓(𝑥_𝑛))‖=lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0, we know that 𝜃_𝑛→0 as 𝑛→∞.
From (95), we obtain 𝑥_𝑛+1−𝐵𝑝_𝑘²≤𝑥_𝑛+1−𝑝_𝑘²+𝜃_𝑛2𝑥_𝑛+1−𝑝_𝑘+𝜃_𝑛,∀𝑛≥𝑛₀. (96)
For any Banach limit 𝜇, from (96), we derive 𝜇_𝑛𝑥_𝑛−𝐵𝑝_𝑘²=𝜇_𝑛𝑥_𝑛+1−𝐵𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛+1−𝑝_𝑘²=𝜇_𝑛𝑥_𝑛−𝑝_𝑘². (97)
In addition, note that 𝑥_𝑛−𝐺𝑝_𝑘²≤𝑥_𝑛−𝐺𝑥_𝑛+𝐺𝑥_𝑛−𝐺𝑝_𝑘²≤𝑥_𝑛−𝐺𝑥_𝑛+𝑥_𝑛−𝑝_𝑘²=𝑥_𝑛−𝑝_𝑘²+𝑥_𝑛−𝐺𝑥_𝑛×2𝑥_𝑛−𝑝_𝑘+𝑥_𝑛−𝐺𝑥_𝑛,𝑥_𝑛−𝑆𝑝_𝑘²≤𝑥_𝑛−𝑆𝑥_𝑛+𝑆𝑥_𝑛−𝑆𝑝_𝑘²≤𝑥_𝑛−𝑆𝑥_𝑛+𝑥_𝑛−𝑝_𝑘²=𝑥_𝑛−𝑝_𝑘²+𝑥_𝑛−𝑆𝑥_𝑛×2𝑥_𝑛−𝑝_𝑘+𝑥_𝑛−𝑆𝑥_𝑛. (98)
It is easy to see from (81) and (91) that 𝜇_𝑛𝑥_𝑛−𝐺𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘²,𝜇_𝑛𝑥_𝑛−𝑆𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘². (99)
Utilizing (97) and (99), we deduce that 𝜇_𝑛𝑥_𝑛−𝑉𝑝_𝑘²=𝜇_𝑛1−𝜃₁−𝜃₂𝑥_𝑛−𝑆𝑝_𝑘+𝜃₁𝑥_𝑛−𝐵𝑝_𝑘+𝜃₂𝑥_𝑛−𝐺𝑝_𝑘²≤1−𝜃₁−𝜃₂𝜇_𝑛𝑥_𝑛−𝑆𝑝_𝑘²+𝜃₁𝜇_𝑛𝑥_𝑛−𝐵𝑝_𝑘²+𝜃₂𝜇_𝑛𝑥_𝑛−𝐺𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘². (100)
Also, observe that 𝑥_𝑛−𝑝_𝑘=1𝑘𝑥_𝑛−𝑓𝑝_𝑘+11−𝑘𝑥_𝑛−𝑉𝑝_𝑘; (101)
that is, 11−𝑘𝑥_𝑛−𝑉𝑝_𝑘=𝑥_𝑛−𝑝_𝑘−1𝑘𝑥_𝑛−𝑓𝑝_𝑘. (102)
It follows from Lemma 8 (ii) and (102) that 11−𝑘²𝑥_𝑛−𝑉𝑝_𝑘²≥𝑥_𝑛−𝑝_𝑘²−2𝑘𝑥_𝑛−𝑝_𝑘+𝑝_𝑘−𝑓𝑝_𝑘,𝐽𝑥_𝑛−𝑝_𝑘=21−𝑘𝑥_𝑛−𝑝_𝑘²+2𝑘𝑓𝑝_𝑘−𝑝_𝑘,𝐽𝑥_𝑛−𝑝_𝑘. (103)
So by (100) and (103), we have 11−𝑘²𝜇_𝑛𝑥_𝑛−𝑝_𝑘²≥21−𝑘𝜇_𝑛𝑥_𝑛−𝑝_𝑘²+2𝑘𝜇_𝑛𝑓𝑝_𝑘−𝑝_𝑘,𝐽𝑥_𝑛−𝑝_𝑘, (104)
and hence 1𝑘²𝜇_𝑛𝑥_𝑛−𝑝_𝑘²≥2𝑘𝜇_𝑛𝑓𝑝_𝑘−𝑝_𝑘,𝐽𝑥_𝑛−𝑝_𝑘. (105)
This implies that 1𝜇2𝑘^𝑛𝑥_𝑛−𝑝_𝑘²≥𝜇_𝑛𝑓𝑝_𝑘−𝑝_𝑘,𝐽𝑥_𝑛−𝑝_𝑘. (106)
Since 𝑝_𝑘→𝑞∈Fix(𝑉)=𝐹 as 𝑘→∞, by the uniform Frechet differentiability of the norm of 𝑋 we have 𝜇_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0. (107)
On the other hand, from (49) and the norm-to-norm uniform continuity of 𝐽 on bounded subsets of 𝑋, it follows that lim_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞−𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞=0. (108)
So, utilizing Lemma 18 we deduce from (107) and (108) that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0, (109)
which together with (49) and the norm-to-norm uniform continuity of 𝐽 on bounded subsets of 𝑋, implies that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞≤0. (110)
Finally, let us show that 𝑥_𝑛→𝑞 as 𝑛→∞. Utilizing Lemma 8 (i), from (42) and the convexity of ‖⋅‖², we get 𝑦_𝑛−𝑞²≤𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑞²+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑞²≤𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝑥_𝑛−𝑞²+𝛿_𝑛𝑥_𝑛−𝑞²=𝑥_𝑛−𝑞²,𝑥_𝑛+1−𝑞²=𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+1−𝛼_𝑛𝑦_𝑛−𝑞+𝛼_𝑛𝑓𝑞−𝑞²≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+1−𝛼_𝑛𝑦_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞²+1−𝛼_𝑛𝑦_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞≤𝛼_𝑛𝜌𝑥_𝑛−𝑞²+1−𝛼_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1=−𝑞1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1=−𝑞1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+𝛼_𝑛2𝑓𝑞𝑥1−𝜌−𝑞,𝐽_𝑛+1−𝑞.1−𝜌 (111)(112)
Applying Lemma 7 to (112), we obtain that 𝑥_𝑛→𝑞 as 𝑛→∞.
Conversely, if 𝑥_𝑛→𝑞∈𝐹 as 𝑛→∞, then from (42) it follows that 𝑦_𝑛−𝑞≤𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑞+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑞≤𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝑥_𝑛−𝑞+𝛿_𝑛𝑥_𝑛=𝑥−𝑞_𝑛−𝑞⟶0as𝑛⟶∞, (113)
that is, 𝑦_𝑛→𝑞. Again from (42) we obtain that 𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛−1−𝛼_𝑛𝑦_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑞+𝑥_𝑛+−𝑞1−𝛼_𝑛𝑦_𝑛−𝑞+𝑥_𝑛≤𝑥−𝑞_𝑛+1−𝑞+2𝑥_𝑛−𝑞+𝑦_𝑛−𝑞. (114)
Since 𝑥_𝑛→𝑞 and 𝑦_𝑛→𝑞, we get 𝛼_𝑛(𝑓(𝑥_𝑛)−𝑥_𝑛)→0. This completes the proof.

Corollary 25. Let 𝐶 be a nonempty closed 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 {𝜌_𝑛}^∞_𝑛=0 be a sequence of positive numbers in (0,𝑏] for some 𝑏∈(0,1) and 𝐴_𝑖∶𝐶→𝐸 an 𝛼_𝑖-inverse strongly accretive mapping for each 𝑖=0,1,…. Define a mapping 𝐺_𝑖∶𝐶→𝐶 by Π_𝐶(𝐼−𝜆_𝑖𝐴_𝑖)𝑥=𝐺_𝑖𝑥 for all 𝑥∈𝐶 and 𝑖=0,1,…, where 𝜆_𝑖∈(0,𝛼_𝑖/𝜅²] and 𝜅 is the 2-uniformly smooth constant of 𝑋. Let 𝐵_𝑛∶𝐶→𝐶 be the 𝑊-mapping generated by 𝐺_𝑛,𝐺_𝑛−1,…,𝐺₀ and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₀. Let 𝑉∶𝐶→𝐶 be an 𝛼-strictly pseudocontractive mapping. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let {𝑆_𝑖}^∞_𝑖=0 be a countable family of nonexpansive mappings of 𝐶 into itself such that 𝐹=(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Fix(𝑉)∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖))≠∅. For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛((1−𝑙)𝐼+𝑙𝑉)𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛,∀𝑛≥0, (115)
where 0<𝑙<𝛼/𝜅², {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. Suppose that the following conditions hold:(i)∑^∞_𝑛=0𝛼_𝑛=∞ and 0≤𝛼_𝑛≤1−𝜌, for all 𝑛≥𝑛₀ for some integer 𝑛₀≥0;(ii)liminf_𝑛→∞𝛾_𝑛>0 and liminf_𝑛→∞𝛿_𝑛>0;(iii)lim_𝑛→∞(|𝛼_𝑛+1/(1−(1−𝛼_𝑛+1)𝛽_𝑛+1)−𝛼_𝑛/(1−(1−𝛼_𝑛)𝛽_𝑛)|+|𝛿_𝑛+1/(1−𝛽_𝑛+1)−𝛿_𝑛/(1−𝛽_𝑛)|)=0;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.
Assume that ∑^∞_𝑛=0sup_𝑥∈𝐷‖𝑆_𝑛+1𝑥−𝑆_𝑛𝑥‖<∞ 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 and suppose that Fix(𝑆)=⋂^∞_𝑖=0Fix(𝑆_𝑖). Then, there hold the following:(I)lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0;(II)𝑥_𝑛→𝑞⇔𝛼_𝑛(𝑓(𝑥_𝑛)−𝑥_𝑛)→0 provided 𝛽_𝑛≡𝛽 for some fixed 𝛽∈(0,1), where 𝑞∈𝐹 solves the following VIP 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (116)

Proof. In Theorem 24, we put 𝐵₁=𝐼−𝑉, 𝐵₂=0, and 𝜇₁=𝑙, where 0<𝑙<𝛼/𝜅². Then, GSVI (13) is equivalent to the VIP of finding 𝑥^∗∈𝐶 such that 𝐵₁𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (117)
In this case, 𝐵₁∶𝐶→𝑋 is 𝛼-inverse strongly accretive. It is not hard to see that Fix(𝑉)=VI(𝐶,𝐵₁). As a matter of fact, we have, for 𝑙>0, 𝑢∈VI𝐶,𝐵₁⟺𝐵₁⟺𝑢,𝐽𝑦−𝑢≥0∀𝑦∈𝐶𝑢−𝑙𝐵₁𝑢−𝑢,𝐽𝑢−𝑦≥0∀𝑦∈𝐶⟺𝑢=Π_𝐶𝑢−𝑙𝐵₁𝑢⟺𝑢=Π_𝐶()⟺⟺(𝑉).𝑢−𝑙𝑢+𝑙𝑉𝑢𝑢−𝑙𝑢+𝑙𝑉𝑢−𝑢,𝐽𝑢−𝑦≥0∀𝑦∈𝐶𝑢−𝑉𝑢,𝐽𝑢−𝑦≤0∀𝑦∈𝐶⟺𝑢=𝑉𝑢⟺𝑢∈Fix (118)
Accordingly, we know that 𝐹=(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Ω∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖))=(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Fix(𝑉)∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖)), and Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑥_𝑛=Π_𝐶𝐼−𝜇₁𝐵₁𝑥_𝑛=Π_𝐶(1−𝑙)𝑥_𝑛+𝑙𝑉𝑥_𝑛=((1−𝑙)𝐼+𝑙𝑉)𝑥_𝑛. (119)
So, the scheme (42) reduces to (115). Therefore, the desired result follows from Theorem 24.

Here, we prove the following important lemmas which will be used in the sequel.

Lemma 26. Let 𝐶 be a nonempty closed 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] one has 𝐼−𝜇_𝑖𝐵_𝑖𝑥−𝐼−𝜇_𝑖𝐵_𝑖𝑦≤1−𝛼_𝑖𝜆_𝑖+1−𝜇_𝑖11+𝜆_𝑖𝑥−𝑦,∀𝑥,𝑦∈𝐶, (120)
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, (121)
which implies that 𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖𝑦≤1𝜆_𝑖𝑥−𝑦. (122)
Hence, 𝐵_𝑖𝑥−𝐵_𝑖𝑦≤𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖𝑦+≤1𝑥−𝑦1+𝜆_𝑖𝑥−𝑦. (123)
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. (124)
So, we have 𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖𝑦≤1−𝛼_𝑖𝜆_𝑖𝑥−𝑦. (125)
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. (126)
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. (127)
This implies that 𝐼−𝜇_𝑖𝐵_𝑖 is nonexpansive for 𝑖=1,2.

Lemma 27. Let 𝐶 be a nonempty closed 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. (128)
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 26, 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. (129)
This shows that 𝐺∶𝐶→𝐶 is nonexpansive. This completes the proof.

Theorem 28. Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let {𝜌_𝑛}^∞_𝑛=0 be a sequence of positive numbers in (0,𝑏] for some 𝑏∈(0,1) and 𝐴_𝑖∶𝐶→𝑋 be 𝜉_𝑖-strictly pseudocontractive and 𝛼_𝑖-strongly accretive with 𝜉_𝑖+𝛼_𝑖≥1 for each 𝑖=0,1,…. Define a mapping 𝐺_𝑖∶𝐶→𝐶 by Π_𝐶(𝐼−𝜆_𝑖𝐴_𝑖)𝑥=𝐺_𝑖𝑥 for all 𝑥∈𝐶 and 𝑖=0,1,…, where 1−(𝜉_𝑖/(1+𝜉_𝑖))(1−(1−𝛼_𝑖)/𝜉_𝑖)≤𝜆_𝑖≤1 for all 𝑖=0,1,…. Let 𝐵_𝑛∶𝐶→𝐶 be the 𝑊-mapping generated by 𝐺_𝑛,𝐺_𝑛−1,…,𝐺₀ and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₀. Let the mapping 𝐵_𝑖∶𝐶→𝑋  𝜁_𝑖-strictly pseudocontractive and 𝛽_𝑖-strongly accretive with 𝜁_𝑖+𝛽_𝑖≥1 for 𝑖=1,2. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let {𝑆_𝑖}^∞_𝑖=0 be a countable family of nonexpansive mappings of 𝐶 into itself such that 𝐹=(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Ω∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖))≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) with 1−(𝜁_𝑖/(1+𝜁_𝑖))(1−(1−𝛽_𝑖)/𝜁_𝑖)≤𝜇_𝑖≤1 for 𝑖=1,2. For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛+1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛,∀𝑛≥0, (130)
where {𝜎_𝑛}, {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are the sequences in [0,1] such that 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and 𝛼_𝑛+𝜎_𝑛≤1 for all 𝑛≥0. Suppose that the following conditions hold:(i)∑^∞_𝑛=0𝛼_𝑛=∞ and 0≤𝛼_𝑛+𝜎_𝑛≤1−𝜌, for all 𝑛≥𝑛₀ for some integer 𝑛₀≥0;(ii)liminf_𝑛→∞𝜎_𝑛>0, liminf_𝑛→∞𝛾_𝑛>0 and liminf_𝑛→∞𝛿_𝑛>0;(iii)lim_𝑛→∞(|𝛼_𝑛+1/(1−(1−𝛼_𝑛+1−𝜎_𝑛+1)𝛽_𝑛+1)−(𝛼_𝑛/(1−(1−𝛼_𝑛−𝜎_𝑛)𝛽_𝑛))|+  |𝜎_𝑛+1/(1−(1−𝛼_𝑛+1−𝜎_𝑛+1)𝛽_𝑛+1)−(𝜎_𝑛/(1−(1−𝛼_𝑛−𝜎_𝑛)𝛽_𝑛))|+  |𝛿_𝑛+1/(1−𝛽_𝑛+1)−𝛿_𝑛/(1−𝛽_𝑛)|)=0;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.
Assume that ∑^∞_𝑛=0sup_𝑥∈𝐷‖𝑆_𝑛+1𝑥−𝑆_𝑛𝑥‖<∞ 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 and suppose that Fix(𝑆)=⋂^∞_𝑖=0Fix(𝑆_𝑖). Then there hold the following:(I)lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0;(II)𝑥_𝑛→𝑞⇔𝛼_𝑛(𝑓(𝑥_𝑛)−𝑥_𝑛)→0 provided 𝛽_𝑛≡𝛽 for some fixed 𝛽∈(0,1), where 𝑞∈𝐹 solves the following VIP 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (131)

Proof. First of all, take a fixed 𝑝∈𝐹 arbitrarily. Then we obtain 𝑝=𝐺𝑝, 𝑝=𝐵_𝑛𝑝 and 𝑆_𝑛𝑝=𝑝 for all 𝑛≥0. By Lemma 27, we get from (130) 𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑥_𝑛=𝑥−𝑝_𝑛−𝑝, (132)
and hence 𝑥_𝑛+1−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑝+𝜎_𝑛𝐺𝑥_𝑛+−𝑝1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝑓𝑝−𝑝+𝜎_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛−𝜎_𝑛𝑥_𝑛−𝑝≤𝛼_𝑛𝜌𝑥_𝑛−𝑝+𝑓𝑝−𝑝+𝜎_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛−𝜎_𝑛𝑥_𝑛=−𝑝1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝=−𝑝1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝1−𝜌−𝑝𝑥1−𝜌≤max_𝑛,𝑓𝑝−𝑝−𝑝.1−𝜌 (133)
By induction, we have 𝑥_𝑛−𝑝≤max𝑥₀,𝑓𝑝−𝑝−𝑝1−𝜌,∀𝑛≥0, (134)
which implies that {𝑥_𝑛} is bounded and so are the sequences {𝑦_𝑛}, {𝐺𝑥_𝑛}, and {𝑓(𝑥_𝑛)}.
Let us show that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (135)
As a matter of fact, put 𝜃_𝑛=(1−𝛼_𝑛−𝜎_𝑛)𝛽_𝑛, for all 𝑛≥0. Then, it follows from (i) and (iv) that 𝛽_𝑛≥𝜃_𝑛=1−𝛼_𝑛−𝜎_𝑛𝛽_𝑛≥1−1−𝜌𝛽_𝑛=𝜌𝛽_𝑛,∀𝑛≥𝑛₀, (136)
and hence 0<liminf_𝑛→∞𝜃_𝑛≤limsup_𝑛→∞𝜃_𝑛<1. (137)
Define 𝑥_𝑛+1=𝜃_𝑛𝑥_𝑛+1−𝜃_𝑛𝑧_𝑛. (138)
Observe that 𝑧_𝑛+1−𝑧_𝑛=𝑥_𝑛+2−𝜃_𝑛+1𝑥_𝑛+11−𝜃_𝑛+1−𝑥_𝑛+1−𝜃_𝑛𝑥_𝑛1−𝜃_𝑛=𝛼_𝑛+1𝑓𝑥_𝑛+1+𝜎_𝑛+1𝐺𝑥_𝑛+1+1−𝛼_𝑛+1−𝜎_𝑛+1𝑦_𝑛+1−𝜃_𝑛+1𝑥_𝑛+1×1−𝜃_𝑛+1⁻¹−𝛼_𝑛𝑓𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛+1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝜃_𝑛𝑥_𝑛1−𝜃_𝑛=𝛼_𝑛+1𝑓𝑥_𝑛+1+𝜎_𝑛+1𝐺𝑥_𝑛+11−𝜃_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛1−𝜃_𝑛−1−𝛼_𝑛−𝜎_𝑛𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝜃_𝑛𝑥_𝑛1−𝜃_𝑛+1−𝛼_𝑛+1−𝜎_𝑛+1×𝛽_𝑛+1𝑥_𝑛+1+𝛾_𝑛+1𝐵_𝑛+1𝑥_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛+1−𝜃_𝑛+1𝑥_𝑛+1×1−𝜃_𝑛+1⁻¹=𝛼_𝑛+1𝑓𝑥_𝑛+1+𝜎_𝑛+1𝐺𝑥_𝑛+11−𝜃_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛1−𝜃_𝑛+1−𝛼_𝑛+1−𝜎_𝑛+11−𝜃_𝑛+1𝛾_𝑛+1𝐵_𝑛+1𝑥_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛+1−1−𝛼_𝑛−𝜎_𝑛1−𝜃_𝑛𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛=𝛼_𝑛+1𝑓𝑥_𝑛+1+𝜎_𝑛+1𝐺𝑥_𝑛+11−𝜃_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛1−𝜃_𝑛+1−𝛼_𝑛+1−𝜎_𝑛+11−𝛽_𝑛+11−𝜃_𝑛+1×𝛾_𝑛+1𝐵_𝑛+1𝑥_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛+11−𝛽_𝑛+1−𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛1−𝛽_𝑛+1−𝛼_𝑛+1−𝜎_𝑛+11−𝛽_𝑛+11−𝜃_𝑛+1−1−𝛼_𝑛−𝜎_𝑛1−𝛽_𝑛1−𝜃_𝑛×𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛1−𝛽_𝑛=𝛼_𝑛+11−𝜃_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝜎_𝑛+11−𝜃_𝑛+1𝐺𝑥_𝑛+1−𝐺𝑥_𝑛+𝛼_𝑛+11−𝜃_𝑛+1−𝛼_𝑛1−𝜃_𝑛𝑓𝑥_𝑛+𝜎_𝑛+11−𝜃_𝑛+1−𝜎_𝑛1−𝜃_𝑛𝐺𝑥_𝑛+1−𝛼_𝑛+1−𝜎_𝑛+11−𝛽_𝑛+11−𝜃_𝑛+1×𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝐵_𝑛+1𝑥_𝑛+1−𝐵_𝑛𝑥_𝑛+𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛+1−𝑆_𝑛𝐺𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝛼_𝑛+1+𝜎_𝑛+11−𝜃_𝑛+1−𝛼_𝑛+𝜎_𝑛1−𝜃_𝑛𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛=𝛼_𝑛+11−𝜃_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝜎_𝑛+11−𝜃_𝑛+1𝐺𝑥_𝑛+1−𝐺𝑥_𝑛+𝛼_𝑛+11−𝜃_𝑛+1−𝛼_𝑛1−𝜃_𝑛𝑓𝑥_𝑛−𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛+𝜎_𝑛+11−𝜃_𝑛+1−𝜎_𝑛1−𝜃_𝑛𝐺𝑥_𝑛−𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛+1−𝛼_𝑛+1−𝜎_𝑛+1−𝜃_𝑛+11−𝜃_𝑛+1×𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝐵_𝑛+1𝑥_𝑛+1−𝐵_𝑛𝑥_𝑛+𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛+1−𝑆_𝑛𝐺𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛, (139)
and hence 𝑧_𝑛+1−𝑧_𝑛≤𝛼_𝑛+11−𝜃_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝜎_𝑛+11−𝜃_𝑛+1𝐺𝑥_𝑛+1−𝐺𝑥_𝑛+𝛼_𝑛+11−𝜃_𝑛+1−𝛼_𝑛1−𝜃_𝑛𝑓𝑥_𝑛−𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛+𝜎_𝑛+11−𝜃_𝑛+1−𝜎_𝑛1−𝜃_𝑛𝐺𝑥_𝑛−𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛+1−𝛼_𝑛+1−𝜎_𝑛+1−𝜃_𝑛+11−𝜃_𝑛+1×𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝐵_𝑛+1𝑥_𝑛+1−𝐵_𝑛𝑥_𝑛+𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛+1−𝑆_𝑛𝐺𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛. (140)
On the other hand, repeating the same arguments as those of (55) and (56) in the proof of Theorem 24, we can get 𝑆_𝑛+1𝐺𝑥_𝑛+1−𝑆_𝑛𝐺𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑆_𝑛+1𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛,𝐵_𝑛+1𝑥_𝑛+1−𝐵_𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑀₀^𝑛_𝑖=0𝜆_𝑖, (141)
for some constant 𝑀₀>0. Utilizing (140)-(141), we have 𝑧_𝑛+1−𝑧_𝑛≤𝛼_𝑛+11−𝜃_𝑛+1𝜌𝑥_𝑛+1−𝑥_𝑛+𝜎_𝑛+11−𝜃_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+11−𝜃_𝑛+1−𝛼_𝑛1−𝜃_𝑛×𝑓𝑥_𝑛+𝐵_𝑛𝑥_𝑛+𝑆_𝑛𝐺𝑥_𝑛+𝜎_𝑛+11−𝜃_𝑛+1−𝜎_𝑛1−𝜃_𝑛×𝐺𝑥_𝑛+𝐵_𝑛𝑥_𝑛+𝑆_𝑛𝐺𝑥_𝑛+1−𝛼_𝑛+1−𝜎_𝑛+1−𝜃_𝑛+11−𝜃_𝑛+1×𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝑀₀^𝑛_𝑖=0𝜆_𝑖+𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝑆_𝑛+1𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛=1−𝛼_𝑛+11−𝜌−𝜃_𝑛+11−𝜃_𝑛+1𝑥_𝑛+1−𝑥_𝑛+1−𝛼_𝑛+1−𝜎_𝑛+1−𝜃_𝑛+11−𝜃_𝑛+1×𝛾_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑀₀^𝑛_𝑖=0𝜆_𝑖+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1−𝛿_𝑛𝛾_𝑛+𝛿_𝑛×𝐵_𝑛𝑥_𝑛+𝑆_𝑛𝐺𝑥_𝑛+𝛿_𝑛+1𝛾_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝛼_𝑛+11−𝜃_𝑛+1−𝛼_𝑛1−𝜃_𝑛×𝑓𝑥_𝑛+𝐵_𝑛𝑥_𝑛+𝑆_𝑛𝐺𝑥_𝑛+𝜎_𝑛+11−𝜃_𝑛+1−𝜎_𝑛1−𝜃_𝑛×𝐺𝑥_𝑛+𝐵_𝑛𝑥_𝑛+𝑆_𝑛𝐺𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑀^𝑛_𝑖=0𝜆_𝑖+𝛿_𝑛+11−𝛽_𝑛+1−𝛿_𝑛1−𝛽_𝑛+𝛼_𝑛+11−𝜃_𝑛+1−𝛼_𝑛1−𝜃_𝑛+𝜎_𝑛+11−𝜃_𝑛+1−𝜎_𝑛1−𝜃_𝑛+𝑆_𝑛+1𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛, (142)
where sup_𝑛≥0{‖𝑓(𝑥_𝑛)‖+‖𝐺𝑥_𝑛‖+‖𝐵_𝑛𝑥_𝑛‖+‖𝑆_𝑛𝐺𝑥_𝑛‖+𝑀₀}≤𝑀 for some 𝑀>0. So, from (142), condition (iii), and the assumption on {𝑆_𝑛} it follows that (noting that 0<𝜆_𝑖≤𝑏<1, for all 𝑖≥0) limsup_𝑛→∞𝑧_𝑛+1−𝑧_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (143)
Consequently, by Lemma 20, we have lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0. (144)
It follows from (137) and (138) that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝜃_𝑛𝑧_𝑛−𝑥_𝑛=0. (145)
Next, we show that ‖𝑥_𝑛−𝐺𝑥_𝑛‖→0 as 𝑛→∞.
Indeed, in terms of Lemma 11, from (130), we have 𝑦_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑥_𝑛−𝑝²=𝑥_𝑛−𝑝²,𝑥_𝑛+1−𝑝²=𝛼_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝐺𝑥_𝑛+−𝑝1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑝+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛²≤𝛼_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝐺𝑥_𝑛−𝑝+1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛+𝛼−𝑝_𝑛𝑓𝑥_𝑛−𝑥_𝑛²=𝛼_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝐺𝑥_𝑛−𝑝+1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑝²+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛×2𝛼_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝐺𝑥_𝑛+−𝑝1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑝+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛=𝛼_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛×𝜎_𝑛1−𝛼_𝑛𝐺𝑥_𝑛−𝑝+1−𝛼_𝑛−𝜎_𝑛1−𝛼_𝑛𝑦_𝑛−𝑝²+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛×2𝛼_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝐺𝑥_𝑛+−𝑝1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛+𝛼−𝑝_𝑛𝑓𝑥_𝑛−𝑥_𝑛≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝜎_𝑛1−𝛼_𝑛𝐺𝑥_𝑛−𝑝+1−𝛼_𝑛−𝜎_𝑛1−𝛼_𝑛𝑦_𝑛−𝑝²+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛×2𝛼_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝐺𝑥_𝑛+−𝑝1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑝+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛×𝜎_𝑛1−𝛼_𝑛𝐺𝑥_𝑛−𝑝²+1−𝛼_𝑛−𝜎_𝑛1−𝛼_𝑛𝑦_𝑛−𝑝²−𝜎_𝑛1−𝛼_𝑛−𝜎_𝑛1−𝛼_𝑛²𝑔𝐺𝑥_𝑛−𝑦_𝑛+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛×2𝛼_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛−𝜎_𝑛𝑥_𝑛+𝛼−𝑝_𝑛𝑓𝑥_𝑛−𝑥_𝑛≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛×𝜎_𝑛1−𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛−𝜎_𝑛1−𝛼_𝑛𝑥_𝑛−𝑝²−𝜎_𝑛1−𝛼_𝑛−𝜎_𝑛1−𝛼_𝑛²𝑔𝐺𝑥_𝑛−𝑦_𝑛+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛2𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛=𝑥_𝑛−𝑝²−𝜎_𝑛1−𝛼_𝑛−𝜎_𝑛1−𝛼_𝑛𝑔𝐺𝑥_𝑛−𝑦_𝑛+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛2𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛. (146)
Then, it immediately follows from 0≤𝛼_𝑛+𝜎_𝑛≤1−𝜌, for all 𝑛≥𝑛₀ that 𝜌𝜎_𝑛𝑔𝐺𝑥_𝑛−𝑦_𝑛≤𝜎_𝑛1−𝛼_𝑛−𝜎_𝑛1−𝛼_𝑛𝑔𝐺𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛2𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛×2𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛, (147)
for all 𝑛≥𝑛₀. Since ‖𝛼_𝑛(𝑓(𝑥_𝑛)−𝑥_𝑛)‖→0 and {𝑥_𝑛} is bounded, we deduce from (145) and condition (ii) that lim_𝑛→∞𝑔𝐺𝑥_𝑛−𝑦_𝑛=0. (148)
Utilizing the properties of 𝑔, we have lim_𝑛→∞𝐺𝑥_𝑛−𝑦_𝑛=0. (149)
Also, from (130) we have 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑥_𝑛+1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥_𝑛+1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛+1−𝛼_𝑛𝑦_𝑛−𝑥_𝑛, (150)
which hence leads to 𝜌𝑦_𝑛−𝑥_𝑛≤1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑥_𝑛≤1−𝛼_𝑛𝑦_𝑛−𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛−𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛−𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝐺𝑥_𝑛−𝑦_𝑛. (151)
So, it is easy to see from (145), (149), and ‖𝛼_𝑛(𝑓(𝑥_𝑛)−𝑥_𝑛)‖→0 that lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (152)
We note that 𝐺𝑥_𝑛−𝑥_𝑛≤𝐺𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥_𝑛. (153)
Therefore, from (149) and (152) it follows that lim_𝑛→∞𝐺𝑥_𝑛−𝑥_𝑛=0. (154)
Repeating the same arguments as those of (86), (89), and (91) in the proof of Theorem 24, we can obtain lim_𝑛→∞𝑥_𝑛−𝐵_𝑛𝑥_𝑛=lim_𝑛→∞𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛=lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (155)
Suppose that 𝛽_𝑛≡𝛽 for some fixed 𝛽∈(0,1) such that 𝛽+𝛾_𝑛+𝛿_𝑛=1 for all 𝑛≥0. Define a mapping 𝑉𝑥=(1−𝜃₁−𝜃₂)𝑆𝑥+𝜃₁𝐵𝑥+𝜃₂𝐺𝑥, where 𝜃₁,𝜃₂∈(0,1) are two constants with 𝜃₁+𝜃₂<1. Then, by Lemmas 14 and 17, we have that Fix(𝑉)=Fix(𝑆)∩Fix(𝐵)∩Fix(𝐺)=𝐹. For each 𝑘≥1, let {𝑝_𝑘} be a unique element of 𝐶 such that 𝑝_𝑘=1𝑘𝑓𝑝_𝑘+11−𝑘𝑉𝑝_𝑘. (156)
From Lemma 13, we conclude that 𝑝_𝑘→𝑞∈Fix(𝑉)=𝐹 as 𝑘→∞. Observe that for every 𝑛,𝑘𝑦_𝑛−𝐵𝑝_𝑘≤𝛽𝑥_𝑛−𝐵𝑝_𝑘+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛+𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘=𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛, (157)
and hence 𝑥_𝑛+1−𝐵𝑝_𝑘≤𝛼_𝑛𝑓𝑥_𝑛−𝐵𝑝_𝑘+𝜎_𝑛𝐺𝑥_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝐵𝑝_𝑘≤𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝐵𝑝_𝑘+𝜎_𝑛𝐺𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛−𝜎_𝑛×𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝐵𝑝_𝑘+𝜎_𝑛𝐺𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛−𝜎_𝑛×𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝐵_𝑛𝑥_𝑛−𝐵_𝑛𝑝_𝑘+𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝐵𝑝_𝑘+𝜎_𝑛𝐺𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛−𝜎_𝑛×𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝑥_𝑛−𝑝_𝑘+𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑥_𝑛+𝛽+𝛼_𝑛+𝜎_𝑛1−𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛−𝜎_𝑛1−𝛽𝑥_𝑛−𝑝_𝑘+1−𝛼_𝑛−𝜎_𝑛×1−𝛽𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑥_𝑛+𝛽+𝛼_𝑛+𝜎_𝑛×𝑥1−𝛽_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝐵𝑝_𝑘+1−𝛼_𝑛−𝜎_𝑛×𝑥1−𝛽_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑝_𝑘+1−𝛼_𝑛−𝜎_𝑛×1−𝛽𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑥_𝑛+𝛽+𝛼_𝑛+𝜎_𝑛1−𝛽𝑥_𝑛+1−𝐵𝑝_𝑘+1−𝛼_𝑛−𝜎_𝑛1−𝛽𝑥_𝑛+1−𝑝_𝑘+1−𝛼_𝑛−𝜎_𝑛×1−𝛽𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+1. (158)
So, it immediately follows from 0≤𝛼_𝑛≤1−𝜌, for all 𝑛≥𝑛₀ that 𝑥_𝑛+1−𝐵𝑝_𝑘≤𝑥_𝑛+1−𝑝_𝑘+𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+11−𝛼_𝑛−𝜎_𝑛×𝛼1−𝛽_𝑛𝑥_𝑛−𝑓𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+1+𝛿_𝑛𝑆1−𝛽^𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑝_𝑘+𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛+1𝜌×𝛼1−𝛽_𝑛𝑥_𝑛−𝑓𝑥_𝑛+𝐺𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+1=𝑥_𝑛+1−𝑝_𝑘+𝜏_𝑛,∀𝑛≥𝑛₀, (159)
where 𝜃_𝑛=‖𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘‖+  ‖𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛‖+  1/(𝜌(1−𝛽))(‖𝛼_𝑛(𝑥_𝑛−𝑓(𝑥_𝑛))‖+  ‖𝐺𝑥_𝑛−𝑥_𝑛‖+  ‖𝑥_𝑛−𝑥_𝑛+1‖). Since lim_𝑛→∞‖𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘‖=  lim_𝑛→∞‖𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛‖=  lim_𝑛→∞‖𝛼_𝑛(𝑥_𝑛−𝑓(𝑥_𝑛))‖=  lim_𝑛→∞‖𝐺𝑥_𝑛−𝑥_𝑛‖=  lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=  0, we know that 𝜏_𝑛→0 as 𝑛→∞.
From (159), we obtain 𝑥_𝑛+1−𝐵𝑝_𝑘²≤𝑥_𝑛+1−𝑝_𝑘²+𝜏_𝑛2𝑥_𝑛+1−𝑝_𝑘+𝜏_𝑛,∀𝑛≥𝑛₀. (160)
For any Banach limit 𝜇, from (160) we derive 𝜇_𝑛𝑥_𝑛−𝐵𝑝_𝑘²=𝜇_𝑛𝑥_𝑛+1−𝐵𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛+1−𝑝_𝑘²=𝜇_𝑛𝑥_𝑛−𝑝_𝑘². (161)
Repeating the same arguments as those of (99), in the proof of Theorem 24, we can get 𝜇_𝑛𝑥_𝑛−𝐺𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘²,𝜇_𝑛𝑥_𝑛−𝑆𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘². (162)
Utilizing (161) and (162), we deduce that 𝜇_𝑛𝑥_𝑛−𝑉𝑝_𝑘²≤1−𝜃₁−𝜃₂𝜇_𝑛𝑥_𝑛−𝑆𝑝_𝑘²+𝜃₁𝜇_𝑛𝑥_𝑛−𝐵𝑝_𝑘²+𝜃₂𝜇_𝑛𝑥_𝑛−𝐺𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘². (163)
Also, observe that 11−𝑘𝑥_𝑛−𝑉𝑝_𝑘=𝑥_𝑛−𝑝_𝑘−1𝑘𝑥_𝑛−𝑓𝑝_𝑘. (164)
Repeating the same arguments as those of (106) in the proof of Theorem 24, we can get 1𝜇2𝑘^𝑛𝑥_𝑛−𝑝_𝑘²≥𝜇_𝑛𝑓𝑝_𝑘−𝑝_𝑘,𝐽𝑥_𝑛−𝑝_𝑘. (165)
Since 𝑝_𝑘→𝑞∈Fix(𝑉)=𝐹 as 𝑘→∞, by the uniform Gateaux differentiability of the norm of 𝑋, we have 𝜇_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0. (166)
On the other hand, from (135) and the norm-to-weak* uniform continuity of 𝐽 on bounded subsets of 𝑋, it follows that lim_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞−𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞=0. (167)
So, utilizing Lemma 18, we deduce from (166) and (167) that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0, (168)
which, together with (135) and the norm-to-norm uniform continuity of 𝐽 on bounded subsets of 𝑋, implies that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞≤0. (169)
Finally, let us show that 𝑥_𝑛→𝑞 as 𝑛→∞. Utilizing Lemma 8 (i), from (130) and the convexity of ‖⋅‖², we get 𝑦_𝑛−𝑞²≤𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑞²+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑞²≤𝑥_𝑛−𝑞²,𝑥_𝑛+1−𝑞²=𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝜎_𝑛𝐺𝑥_𝑛+−𝑞1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑞+𝛼_𝑛𝑓𝑞−𝑞²≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝜎_𝑛𝐺𝑥_𝑛+−𝑞1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞²+𝜎_𝑛𝐺𝑥_𝑛−𝑞²+1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞≤𝛼_𝑛𝜌𝑥_𝑛−𝑞²+𝜎_𝑛𝑥_𝑛−𝑞²+1−𝛼_𝑛−𝜎_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1=−𝑞1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1=−𝑞1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+𝛼_𝑛2𝑓𝑞𝑥1−𝜌−𝑞,𝐽_𝑛+1−𝑞.1−𝜌 (170)(171)
Applying Lemma 7 to (171), we obtain that 𝑥_𝑛→𝑞 as 𝑛→∞.
Conversely, if 𝑥_𝑛→𝑞∈𝐹 as 𝑛→∞, then from (130) it follows that 𝑦_𝑛−𝑞≤𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑞+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑞≤𝑥_𝑛−𝑞⟶0 (172)
as 𝑛→∞; that is, 𝑦_𝑛→𝑞. Again from (130) we obtain that 𝛼_𝑛𝑓𝑥_𝑛−𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛−𝜎_𝑛𝐺𝑥_𝑛−𝑥_𝑛−1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑥_𝑛+1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑞+𝑥_𝑛−𝑞+𝜎_𝑛𝐺𝑥_𝑛−𝑞+𝑥_𝑛+−𝑞1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑞+𝑥_𝑛≤𝑥−𝑞_𝑛+1−𝑞+𝑥_𝑛−𝑞+𝜎_𝑛𝑥_𝑛−𝑞+𝑥_𝑛+−𝑞1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛−𝑞+𝑥_𝑛≤𝑥−𝑞_𝑛+1−𝑞+3𝑥_𝑛−𝑞+𝑦_𝑛−𝑞. (173)
Since 𝑥_𝑛→𝑞 and 𝑦_𝑛→𝑞, we get 𝛼_𝑛(𝑓(𝑥_𝑛)−𝑥_𝑛)→0. This completes the proof.

Corollary 29. Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let {𝜌_𝑛}^∞_𝑛=0 be a sequence of positive numbers in (0,𝑏] for some 𝑏∈(0,1) and 𝐴_𝑖∶𝐶→𝑋 𝜉_𝑖-strictly pseudocontractive and 𝛼_𝑖-strongly accretive with 𝜉_𝑖+𝛼_𝑖≥1 for each 𝑖=0,1,…. Define a mapping 𝐺_𝑖∶𝐶→𝐶 by Π_𝐶(𝐼−𝜆_𝑖𝐴_𝑖)𝑥=𝐺_𝑖𝑥 for all 𝑥∈𝐶 and 𝑖=0,1,…, where 1−(𝜉_𝑖/(1+𝜉_𝑖))(1−(1−𝛼_𝑖)/𝜉_𝑖)≤𝜆_𝑖≤1 for all 𝑖=0,1,…. Let 𝐵_𝑛∶𝐶→𝐶 be the 𝑊-mapping generated by 𝐺_𝑛,𝐺_𝑛−1,…,𝐺₀ and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₀. Let 𝑉∶𝐶→𝐶 be a self-mapping such that 𝐼−𝑉∶𝐶→𝑋 is 𝜆-strictly pseudocontractive and 𝛼-strongly accretive with 𝛼+𝜆≥1. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let {𝑆_𝑖}^∞_𝑖=0 be a countable family of nonexpansive mappings of 𝐶 into itself such that 𝐹=(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Fix(𝑉)∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖))≠∅. For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛((1−𝑙)𝐼+𝑙𝑉)𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝜎_𝑛((1−𝑙)𝐼+𝑙𝑉)𝑥_𝑛+1−𝛼_𝑛−𝜎_𝑛𝑦_𝑛,∀𝑛≥0, (174)
where 1−(𝜆/(1+𝜆))(1−(1−𝛼)/𝜆)≤𝑙≤1 and {𝜎_𝑛}, {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are the sequences in [0,1] such that 𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and 𝛼_𝑛+𝜎_𝑛≤1 for all 𝑛≥0. Suppose that the following conditions hold:(i)∑^∞_𝑛=0𝛼_𝑛=∞ and 0≤𝛼_𝑛+𝜎_𝑛≤1−𝜌, for all 𝑛≥𝑛₀ for some integer 𝑛₀≥0;(ii)liminf_𝑛→∞𝜎_𝑛>0, liminf_𝑛→∞𝛾_𝑛>0 and liminf_𝑛→∞𝛿_𝑛>0;(iii)lim_𝑛→∞(|𝛼_𝑛+1/(1−(1−𝛼_𝑛+1−𝜎_𝑛+1)𝛽_𝑛+1)−𝛼_𝑛/(1−(1−𝛼_𝑛−𝜎_𝑛)𝛽_𝑛)|+  |𝜎_𝑛+1/(1−(1−𝛼_𝑛+1−𝜎_𝑛+1)𝛽_𝑛+1)−𝜎_𝑛/(1−(1−𝛼_𝑛−𝜎_𝑛)𝛽_𝑛)|+  |𝛿_𝑛+1/(1−𝛽_𝑛+1)−𝛿_𝑛/(1−𝛽_𝑛)|)=  0;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.
Assume that ∑^∞_𝑛=0sup_𝑥∈𝐷‖𝑆_𝑛+1𝑥−𝑆_𝑛𝑥‖<∞ 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 and suppose that Fix(𝑆)=⋂^∞_𝑖=0Fix(𝑆_𝑖). Then there hold the following:(I)lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0;(II)𝑥_𝑛→𝑞⇔𝛼_𝑛(𝑓(𝑥_𝑛)−𝑥_𝑛)→0 provided 𝛽_𝑛≡𝛽 for some fixed 𝛽∈(0,1), where 𝑞∈𝐹 solves the following VIP 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (175)

Proof. In Theorem 28, we put 𝐵₁=𝐼−𝑉, 𝐵₂=0, and 𝜇₁=𝑙, where 1−(𝜆/(1+𝜆))(1−(1−𝛼)/𝜆)≤𝑙≤1. Then, GSVI (13) is equivalent to the VIP of finding 𝑥^∗∈𝐶 such that 𝐵₁𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (176)
In this case, 𝐵₁∶𝐶→𝑋 is 𝜆-strictly pseudocontractive and 𝛼-strongly accretive. Repeating the same arguments as those in the proof of Corollary 25, we can infer that Fix(𝑉)=VI(𝐶,𝐵₁). Accordingly, 𝐹=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Ω∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖))=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Fix(𝑉)∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖)), and 𝐺𝑥_𝑛=((1−𝑙)𝐼+𝑙𝑉)𝑥_𝑛,∀𝑛≥0. (177)
So, scheme (130) reduces to (174). Therefore, the desired result follows from Theorem 31.

Remark 30. Our Theorems 24 and 28 improve, extend, supplement and develop Ceng and Yao’s [10, Theorem  3.2], Cai and Bu’s [11, Theorem  3.1], Kangtunyakarn’s [38, Theorem  3.1], and Ceng and Yao’s [8, Theorem  3.1], in the following aspects.(i)The problem of finding a point 𝑞∈(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Ω∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖)) in our Theorems 24 and 28 is more general and more subtle than every one of the problem of finding a point 𝑞∈⋂^∞_𝑖=0Fix(𝑇_𝑖) in [10, Theorem  3.2], the problem of finding a point 𝑞∈⋂^∞_𝑖=1Fix(𝑇_𝑖)∩Ω in [11, Theorem  3.1], the problem of finding a point 𝑞∈Fix(𝑆)∩Fix(𝑉)∩(⋂^𝑁_𝑖=1VI(𝐶,𝐴_𝑖)) in [38, Theorem  3.1], and the problem of finding a point 𝑞∈Fix(𝑇) in [8, Theorem  3.1].(ii)The iterative scheme in [8, Theorem  3.1] is extended to develop the iterative schemes (42) and (130) in our Theorems 24 and 28 by virtue of the iterative schemes of [11, Theorem  3.1] and [10, Theorems  3.2]. The iterative schemes (42) and (130) in our Theorems 24 and 28 are more advantageous and more flexible than the iterative scheme of [8, Theorem  3.1] because they can be applied to solving three problems (i.e., GSVI (13), fixed point problem and infinitely many VIPs), and involve several parameter sequences {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛿_𝑛}, (and {𝜎_𝑛}).(iii)Our Theorems 24 and 28 extend and generalize Ceng and Yao [8, Theorem  3.1] from a nonexpansive mapping to a countable family of nonexpansive mappings, and Ceng and Yao’s [10, Theorems  3.2], to the setting of the GSVI (13) and infinitely many VIPs, Kangtunyakarn [38, Theorem  3.1], from finitely many VIPs to infinitely many VIPs, from a nonexpansive mapping to a countable family of nonexpansive mappings and from a strict pseudocontraction to the GSVI (13). In the meantime, our Theorems 24 and 28 extend and generalize Cai and Bu’s [11, Theorem  3.1], to the setting of infinitely many VIPs.(iv)The iterative schemes (42) and (130) in our Theorems 24 and 28 are very different from every one in [10, Theorem  3.2], [11, Theorem  3.1], [38, Theorem  3.1], and [8, Theorem  3.1] because the mappings 𝐺 and 𝑇_𝑛 in [11, Theorem  3.1] and the mapping 𝑇 in [8, Theorem  3.1] are replaced with the same composite mapping 𝑆_𝑛𝐺 in the iterative schemes (42) and (130) and the mapping 𝑊_𝑛 in [10, Theorem  3.2] is replaced with 𝐵_𝑛.(v)Cai and Bu’s proof in [11, Theorem  3.1] depends on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), and the inequality in smooth and uniform convex Banach spaces (see Proposition 6). Because the composite mapping 𝑆_𝑛𝐺 appears in the iterative scheme (42) of our Theorem 24, the proof of our Theorem 24 depends on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), the inequality in smooth and uniform convex Banach spaces (see Proposition 6), the inequality in uniform convex Banach spaces (see Lemma 15 in Section 2 of this paper), and the properties of the 𝑊-mapping and the Banach limit (see Lemmas 16–18 in Section 2 of this paper). However, the proof of our Theorem 28 does not depend on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), and the inequality in smooth and uniform convex Banach spaces (see Proposition 6). It depends on only the inequality in uniform convex Banach spaces (see Lemma 15 in Section 2 of this paper) and the properties of the 𝑊-mapping and the Banach limit (see Lemmas 16–18 in Section 2 of this paper).(vi)The assumption of the uniformly convex and 2-uniformly smooth Banach space 𝑋 in [11, Theorem  3.1] is weakened to the one of the uniformly convex Banach space 𝑋 having a uniformly Gateaux differentiable norm in our Theorem 28. Moreover, the assumption of the uniformly smooth Banach space 𝑋 in [8, Theorem  3.1] is replaced with the one of the uniformly convex Banach space 𝑋 having a uniformly Gateaux differentiable norm in our Theorem 28. It is worth emphasizing that there is no assumption on the convergence of parameter sequences {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} (and {𝜎_𝑛}) to zero in our Theorems 24 and 28.

4. Relaxed Mann Iterations and Their Convergence Criteria

In this section, we introduce our relaxed Mann iteration algorithms in real smooth and uniformly convex Banach spaces and present their convergence criteria.

Theorem 31. Let 𝐶 be a nonempty closed 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 {𝜌_𝑛}^∞_𝑛=0 be a sequence of positive numbers in (0,𝑏] for some 𝑏∈(0,1) and 𝐴_𝑖∶𝐶→𝑋 an 𝛼_𝑖-inverse strongly accretive mapping for each 𝑖=0,1,…. Define a mapping 𝐺_𝑖∶𝐶→𝐶 by Π_𝐶(𝐼−𝜆_𝑖𝐴_𝑖)𝑥=𝐺_𝑖𝑥 for all 𝑥∈𝐶 and 𝑖=0,1,…, where 𝜆_𝑖∈(0,𝛼_𝑖/𝜅²] and 𝜅 is the 2-uniformly smooth constant of 𝑋. Let 𝐵_𝑛∶𝐶→𝐶 be the 𝑊-mapping generated by 𝐺_𝑛,𝐺_𝑛−1,…,𝐺₀ and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₀. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛽_𝑖-inverse strongly accretive for 𝑖=1,2. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let {𝑆_𝑖}^∞_𝑖=0 be a countable family of nonexpansive mappings of 𝐶 into itself such that 𝐹=(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Ω∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖))≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) with 0<𝜇_𝑖<𝛽_𝑖/𝜅² for 𝑖=1,2. For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛,∀𝑛≥0, (178)
where {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are the sequences in (0,1) such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 for all 𝑛≥0. Suppose that the following conditions hold:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(ii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iii)lim_𝑛→∞(|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|)=0;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_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 and suppose that Fix(𝑆)=⋂^∞_𝑖=0Fix(𝑆_𝑖). Then, there hold the following:(I)lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0;(II)the sequence {𝑥_𝑛}^∞_𝑛=0 converges strongly to some 𝑞∈𝐹 which is the unique solution of the variational inequality problem (VIP) 𝐼−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹, (179)

Proof. First of all, since 0<𝜆_𝑖<[𝛼_𝑖/𝜅²] for 𝑖=0,1,…, it is easy to see that 𝐺_𝑖 is a nonexpansive mapping for each 𝑖=0,1,…. Since 𝐵_𝑛∶𝐶→𝐶 is the 𝑊-mapping generated by 𝐺_𝑛,𝐺_𝑛−1,…,𝐺₀, and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₀, by Lemma 16 we know that, for each 𝑥∈𝐶 and 𝑘≥0, the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists. Moreover, one can define a mapping 𝐵∶𝐶→𝐶 as follows: 𝐵𝑥=lim_𝑛→∞𝐵_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,0𝑥 (180)
for every 𝑥∈𝐶. That is, such a 𝐵 is the 𝑊-mapping generated by the sequences {𝐺_𝑛}^∞_𝑛=0 and {𝜌_𝑛}^∞_𝑛=0. According to Lemma 17, we know that Fix(𝐵)=⋂^∞_𝑖=0Fix(𝐺_𝑖). From Lemma 21 and the definition of 𝐺_𝑖, we have Fix(𝐺_𝑖)=VI(𝐶,𝐴_𝑖) for each 𝑖=0,1,…. Hence, we have Fix(𝐵)=^∞_𝑖=0Fix𝐺_𝑖=^∞_𝑖=0VI𝐶,𝐴_𝑖. (181)
Next, let us show that the sequence {𝑥_𝑛} is bounded. Indeed, take a fixed 𝑝∈𝐹 arbitrarily. Then, we get 𝑝=𝐺𝑝, 𝑝=𝐵_𝑛𝑝, and 𝑝=𝑆_𝑛𝑝 for all 𝑛≥0. By Lemma 23, we know that 𝐺 is nonexpansive. Then, from (178), we have 𝑥_𝑛+1−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝑓𝑝−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝐺𝑥_𝑛−𝑝≤𝛼_𝑛𝜌𝑥_𝑛−𝑝+𝑓𝑝−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑥_𝑛=−𝑝1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝1−𝜌−𝑝𝑥1−𝜌≤max_𝑛,𝑓𝑝−𝑝−𝑝.1−𝜌 (182)
By induction, we obtain 𝑥_𝑛−𝑝≤max𝑥₀,𝑓𝑝−𝑝−𝑝1−𝜌,∀𝑛≥0. (183)
Hence, {𝑥_𝑛} is bounded, and so are the sequences {𝐺𝑥_𝑛} and {𝑓(𝑥_𝑛)}.
Let us show that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (184)
As a matter of fact, observe that 𝑥_𝑛+1 can be rewritten as follows: 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛, (185)
where 𝑧_𝑛=(𝛼_𝑛𝑓(𝑥_𝑛)+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛)/(1−𝛽_𝑛). Observe that 𝑧_𝑛−𝑧_𝑛−1=𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛1−𝛽_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−11−𝛽_𝑛−1=𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛−𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1=𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛−𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛+𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1≤𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛−𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛+𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1=11−𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−1+11−𝛽_𝑛−11−𝛽_𝑛−1𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−1=11−𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−1=11−𝛽_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1−𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1−𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−1≤11−𝛽_𝑛𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−1. (186)
On the other hand, we note that, for all 𝑛≥1, 𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛−1𝐺𝑥_𝑛−1≤𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1≤𝐺𝑥_𝑛−𝐺𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1. (187)
Furthermore, by (CY), since 𝐺_𝑖 and 𝑈_𝑛,𝑖 are nonexpansive, we deduce that for each 𝑛≥1𝐵_𝑛𝑥_𝑛−𝐵_𝑛−1𝑥_𝑛−1≤𝐵_𝑛𝑥_𝑛−𝐵_𝑛𝑥_𝑛−1+𝐵_𝑛𝑥_𝑛−1−𝐵_𝑛−1𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝐵_𝑛𝑥_𝑛−1−𝐵_𝑛−1𝑥_𝑛−1=𝑥_𝑛−𝑥_𝑛−1+𝜆₀𝐺₀𝑈_𝑛,1𝑥_𝑛−1−𝜆₀𝐺₀𝑈_𝑛−1,1𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝜆₀𝑈_𝑛,1𝑥_𝑛−1−𝑈_𝑛−1,1𝑥_𝑛−1=𝑥_𝑛−𝑥_𝑛−1+𝜆₀𝜆₁𝐺₁𝑈_𝑛,2𝑥_𝑛−1−𝜆₁𝐺₁𝑈_𝑛−1,2𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝜆₀𝜆₁𝑈_𝑛,2𝑥_𝑛−1−𝑈_𝑛−1,2𝑥_𝑛−1⋮≤𝑥_𝑛−𝑥_𝑛−1+^𝑛−1_𝑖=0𝜆_𝑖𝑈_𝑛,𝑛𝑥_𝑛−1−𝑈_{𝑛−1,𝑛}𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑀^𝑛−1_𝑖=0𝜆_𝑖, (188)
for some constant 𝑀>0. Taking into account 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, we may assume, without loss of generality, that {𝛽_𝑛}⊂[𝑐,̂𝑑]. Utilizing (186)–(188), we have 𝑧_𝑛−𝑧_𝑛−1≤11−𝛽_𝑛𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−1≤11−𝛽_𝑛𝛼_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝛾_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀^𝑛−1_𝑖=0𝜆_𝑖+𝛿_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−1=11−𝛽_𝑛1−𝛽_𝑛−𝛼_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝛾_𝑛𝑀^𝑛−1_𝑖=0𝜆_𝑖+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−1=𝛼1−^𝑛1−𝜌1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛾_𝑛𝑀1−𝛽_𝑛^𝑛−1_𝑖=0𝜆_𝑖+𝛿_𝑛1−𝛽_𝑛𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+11−𝛽_𝑛×𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑥_𝑛−𝛽_𝑛−1𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑀^𝑛−1_𝑖=0𝜆_𝑖+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+11−𝛽_𝑛𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1×𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛×𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑀₁^𝑛−1_𝑖=0𝜆_𝑖+𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1, (189)
where sup_𝑛≥0{(1/(1−𝑑)²)(‖𝑓(𝑥_𝑛)‖+‖𝐵_𝑛𝑥_𝑛‖+‖𝑆_𝑛𝐺𝑥_𝑛‖+𝑀)}≤𝑀₁ for some 𝑀₁>0. Thus, from (189), conditions (i), (iii) and the assumption on {𝑆_𝑛}, it follows that (noting that 0<𝜆_𝑖≤𝑏<1, for all 𝑖≥0) lim_𝑛→∞𝑧_𝑛−𝑧_𝑛−1−𝑥_𝑛−𝑥_𝑛−1≤0. (190)
Since 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, by Lemma 20 we get lim_𝑛→∞𝑥_𝑛−𝑧_𝑛=0. (191)
Consequently, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛=0. (192)
Next we show that ‖𝑥_𝑛−𝐺𝑥_𝑛‖→0 as 𝑛→∞.
Indeed, 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=𝐺𝑥_𝑛 for all 𝑛≥0. From Lemma 26 we have 𝑢_𝑛−𝑞²=Π_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−Π_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²≤𝑥_𝑛−𝑝²−2𝜇₂𝛽₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²,𝑣_𝑛−𝑝²=Π_𝐶𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛−Π_𝐶𝑞−𝜇₁𝐵₁𝑞²≤𝑢_𝑛−𝑞−𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²≤𝑢_𝑛−𝑞²−2𝜇₁𝛽₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞². (193)(194)
Substituting (193) for (194), we obtain 𝑣_𝑛−𝑝²≤𝑥_𝑛−𝑝²−2𝜇₂𝛽₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−2𝜇₁𝛽₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞². (195)
By Lemma 8, we have from (178) and (195) 𝑥_𝑛+1−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝≤𝛼_𝑛𝜌²𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑣_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝≤𝛼_𝑛𝜌𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑥_𝑛−𝑝²−2𝜇₂𝛽₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−2𝜇₁𝛽₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1=−𝑝1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑝²−2𝛿_𝑛𝜇₂𝛽₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝜇₁𝛽₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1≤𝑥−𝑝_𝑛−𝑝²−2𝛿_𝑛𝜇₂𝛽₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝜇₁𝛽₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1−𝑝, (196)
which hence implies that 2𝛿_𝑛𝜇₂𝛽₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝜇₁𝛽₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1≤𝑥−𝑝_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1−𝑝. (197)
Since ‖𝑥_𝑛−𝑥_𝑛+1‖→0, 0<𝜇_𝑖<𝛽_𝑖/𝜅² for 𝑖=1,2, and {𝑥_𝑛} is bounded, we obtain from conditions (i), (ii) that lim_𝑛→∞𝐵₂𝑥_𝑛−𝐵₂𝑝=0,lim_𝑛→∞𝐵₁𝑢_𝑛−𝐵₁𝑞=0. (198)
Utilizing Proposition 6 and Lemma 9, 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, (199)
which implies that 𝑢_𝑛−𝑞²≤𝑥_𝑛−𝑝²−𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞. (200)
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, (201)
which implies that 𝑣_𝑛−𝑝²≤𝑢_𝑛−𝑞²−𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝. (202)
Substituting (200) for (202), 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. (203)
By Lemma 8, we have from (196) and (203) 𝑥_𝑛+1−𝑝²≤𝛼_𝑛𝜌²𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑣_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝≤𝛼_𝑛𝜌𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑥_𝑛−𝑝²−𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞−𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛×𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1≤−𝑝1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑝²−𝛿_𝑛𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛×𝑣_𝑛−𝑝+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1≤𝑥−𝑝_𝑛−𝑝²−𝛿_𝑛𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛×𝑣_𝑛−𝑝+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1−𝑝, (204)
which hence leads to 𝛿_𝑛𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝑔₂𝑢_𝑛−𝑣_𝑛+≤𝑥𝑝−𝑞_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1≤𝑥−𝑝_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1−𝑝. (205)
From (198), (205), conditions (i), (ii) and the boundedness of {𝑥_𝑛}, {𝑢_𝑛}, and {𝑣_𝑛}, we deduce that lim_𝑛→∞𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞=0,lim_𝑛→∞𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞=0. (206)
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. (207)
From (207), we get 𝑥_𝑛−𝑣_𝑛≤𝑥_𝑛−𝑢_𝑛−+𝑢𝑝−𝑞_𝑛−𝑣_𝑛+𝑝−𝑞⟶0as𝑛→∞. (208)
That is, lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0. (209)
Next, let us show that lim_𝑛→∞𝑆_𝑛𝐺𝑥_𝑛−𝑥_𝑛=0,lim_𝑛→∞𝐵_𝑛𝑥_𝑛−𝑥_𝑛=0. (210)
Indeed, observe that 𝑥_𝑛+1 can be rewritten as follows: 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛+𝛿_𝑛×𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝑒_𝑛𝑧_𝑛, (211)
where 𝑒_𝑛=𝛾_𝑛+𝛿_𝑛 and 𝑧_𝑛=(𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛)/(𝛾_𝑛+𝛿_𝑛). Utilizing Lemma 11 and (211), we have 𝑥_𝑛+1−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝑒_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝑒_𝑛𝑧_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₃𝑧_𝑛−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₃𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₃𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝛿−𝑝^𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₃𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₃𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛+𝛿−𝑝^𝑛𝛾_𝑛+𝛿_𝑛𝐺𝑥_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₃𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛+𝛿−𝑝^𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₃𝑧_𝑛−𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₃𝑧_𝑛−𝑥_𝑛, (212)
which hence implies that 𝛽_𝑛𝑒_𝑛𝑔₃𝑧_𝑛−𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝+𝑥_𝑛+1×𝑥−𝑝_𝑛−𝑥_𝑛+1. (213)
Utilizing (184), conditions (i), (ii), (iv), and the boundedness of {𝑥_𝑛} and {𝑓(𝑥_𝑛)}, we get lim_𝑛→∞𝑔₃𝑧_𝑛−𝑥_𝑛=0. (214)
From the properties of 𝑔₃, we have lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0. (215)
Utilizing Lemma 15 and the definition of 𝑧_𝑛, we have 𝑧_𝑛−𝑝²=𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛−𝑝²=𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝛿−𝑝^𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²≤𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₄𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝐺𝑥_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₄𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₄𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝑥_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₄𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛, (216)
which hence yields 𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₄𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝑥_𝑛−𝑝²−𝑧_𝑛−𝑝²≤𝑥_𝑛−𝑝+𝑧_𝑛−𝑝𝑥_𝑛−𝑧_𝑛. (217)
Since {𝑥_𝑛} and {𝑧_𝑛} are bounded and ‖𝑧_𝑛−𝑥_𝑛‖→0 as 𝑛→∞, we deduce from condition (ii) that lim_𝑛→∞𝑔₄‖𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛‖=0. (218)
From the properties of 𝑔₄, we have lim_𝑛→∞𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛=0. (219)
On the other hand, 𝑥_𝑛+1 can also be rewritten as follows: 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛼_𝑛+𝛿_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛼_𝑛+𝛿_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝑑_𝑛𝑧_𝑛, (220)
where 𝑑_𝑛=𝛼_𝑛+𝛿_𝑛 and 𝑧_𝑛=(𝛼_𝑛𝑓(𝑥_𝑛)+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛)/(𝛼_𝑛+𝛿_𝑛). Utilizing Lemma 11 and the convexity of ‖⋅‖², we have 𝑥_𝑛+1−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝+𝑑_𝑛𝑧_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝑧_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₅𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛼_𝑛+𝛿_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₅𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝛼_𝑛+𝛿_𝑛𝑓𝑥_𝑛+𝛿−𝑝^𝑛𝛼_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₅𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝛼_𝑛+𝛿_𝑛𝑓𝑥_𝑛−𝑝²+𝛿_𝑛𝛼_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₅𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝛼_𝑛+𝛿_𝑛𝑓𝑥_𝑛−𝑝²+𝛿_𝑛𝛼_𝑛+𝛿_𝑛𝐺𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₅𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝛼_𝑛+𝛿_𝑛𝑓𝑥_𝑛−𝑝²+𝛿_𝑛𝛼_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₅𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₅𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₅𝑥_𝑛−𝐵_𝑛𝑥_𝑛, (221)
which hence implies that 𝛽_𝑛𝛾_𝑛𝑔₅𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝+𝑥_𝑛+1×𝑥−𝑝_𝑛−𝑥_𝑛+1. (222)
From (184), conditions (i), (ii), (iv), and the boundedness of {𝑥_𝑛} and {𝑓(𝑥_𝑛)}, we have lim_𝑛→∞𝑔₅𝑥_𝑛−𝐵_𝑛𝑥_𝑛=0. (223)
Utilizing the properties of 𝑔₅, we have lim_𝑛→∞𝑥_𝑛−𝐵_𝑛𝑥_𝑛=0, (224)
which, together with (219), implies that 𝑆_𝑛𝐺𝑥_𝑛−𝑥_𝑛≤𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛+𝐵_𝑛𝑥_𝑛−𝑥_𝑛⟶0as𝑛→∞. (225)
That is, lim_𝑛→∞𝑆_𝑛𝐺𝑥_𝑛−𝑥_𝑛=0. (226)
We note that 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛𝑥_𝑛+𝑆_𝑛𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝐺𝑥_𝑛−𝑥_𝑛+𝑆_𝑛𝑥_𝑛−𝑆𝑥_𝑛. (227)
So, in terms of (209), (226), and Lemma 12, we have lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (228)
Suppose that 𝛽_𝑛≡𝛽 for some fixed 𝛽∈(0,1) such that 𝛼_𝑛+𝛽+𝛾_𝑛+𝛿_𝑛=1 for all 𝑛≥0. Define a mapping 𝑉𝑥=(1−𝜃₁−𝜃₂)𝑆𝑥+𝜃₁𝐵𝑥+𝜃₂𝐺𝑥, where 𝜃₁,𝜃₂∈(0,1) are two constants with 𝜃₁+𝜃₂<1. Then by Lemmas 14 and 17, we have that Fix(𝑉)=Fix(𝑆)∩Fix(𝐵)∩Fix(𝐺)=𝐹. For each 𝑘≥1, let {𝑝_𝑘} be a unique element of 𝐶 such that 𝑝_𝑘=1𝑘𝑓𝑝_𝑘+11−𝑘𝑉𝑝_𝑘. (229)
From Lemma 13, we conclude that 𝑝_𝑘→𝑞∈Fix(𝑉)=𝐹 as 𝑘→∞. Observe that for every 𝑛,𝑘𝑥_𝑛+1−𝐵𝑝_𝑘=𝛼_𝑛𝑓𝑥_𝑛−𝐵𝑝_𝑘+𝛽𝑥_𝑛−𝐵𝑝_𝑘+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵𝑝_𝑘≤𝛼_𝑛𝑓𝑥_𝑛−𝐵𝑝_𝑘+𝛽𝑥_𝑛−𝐵𝑝_𝑘+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛+𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘=𝛼_𝑛𝑓𝑥_𝑛−𝐵𝑝_𝑘+𝛽𝑥_𝑛−𝐵𝑝_𝑘+𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝐵𝑝_𝑘+𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛−𝛽𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝐵𝑝_𝑘+𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛×𝐵−𝛽_𝑛𝑥_𝑛−𝐵_𝑛𝑝_𝑘+𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝐵𝑝_𝑘+𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛼_𝑛−𝛽𝑥_𝑛−𝑝_𝑘+𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝐵𝑝_𝑘+𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝑥_𝑛−𝑝_𝑘+𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝜃_𝑛+𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝑥_𝑛−𝑝_𝑘, (230)
where 𝜃_𝑛=𝛼_𝑛‖𝑓(𝑥_𝑛)−𝐵𝑝_𝑘‖+(1−𝛽)‖𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘‖+𝛿_𝑛‖𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛‖. Since lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞‖𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘‖=lim_𝑛→∞‖𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛‖=0, we know that 𝜃_𝑛→0 as 𝑛→∞.
From (230), we obtain 𝑥_𝑛+1−𝐵𝑝_𝑘²≤𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝑥_𝑛−𝑝_𝑘²+𝜃_𝑛2𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝑥_𝑛−𝑝_𝑘+𝜃_𝑛=𝛽²𝑥_𝑛−𝐵𝑝_𝑘²+1−𝛽²𝑥_𝑛−𝑝_𝑘²+2𝛽1−𝛽𝑥_𝑛−𝐵𝑝_𝑘𝑥_𝑛−𝑝_𝑘+𝜏_𝑛≤𝛽²𝑥_𝑛−𝐵𝑝_𝑘²+1−𝛽²𝑥_𝑛−𝑝_𝑘²+𝛽1−𝛽𝑥_𝑛−𝐵𝑝_𝑘²+𝑥_𝑛−𝑝_𝑘²+𝜏_𝑛=𝛽𝑥_𝑛−𝐵𝑝_𝑘²+1−𝛽𝑥_𝑛−𝑝_𝑘²+𝜏_𝑛, (231)
where 𝜏_𝑛=𝜃_𝑛[2(𝛽‖𝑥_𝑛−𝐵𝑝_𝑘‖+(1−𝛽)‖𝑥_𝑛−𝑝_𝑘‖)+𝜃_𝑛]→0 as 𝑛→∞.
For any Banach limit 𝜇, from (231) we derive 𝜇_𝑛𝑥_𝑛−𝐵𝑝_𝑘²=𝜇_𝑛𝑥_𝑛+1−𝐵𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘². (232)
In addition, note that 𝑥_𝑛−𝐺𝑝_𝑘²≤𝑥_𝑛−𝐺𝑥_𝑛+𝐺𝑥_𝑛−𝐺𝑝_𝑘²≤𝑥_𝑛−𝐺𝑥_𝑛+𝑥_𝑛−𝑝_𝑘²=𝑥_𝑛−𝑝_𝑘²+𝑥_𝑛−𝐺𝑥_𝑛×2𝑥_𝑛−𝑝_𝑘+𝑥_𝑛−𝐺𝑥_𝑛,𝑥_𝑛−𝑆𝑝_𝑘²≤𝑥_𝑛−𝑆𝑥_𝑛+𝑆𝑥_𝑛−𝑆𝑝_𝑘²≤𝑥_𝑛−𝑆𝑥_𝑛+𝑥_𝑛−𝑝_𝑘²=𝑥_𝑛−𝑝_𝑘²+𝑥_𝑛−𝑆𝑥_𝑛×2𝑥_𝑛−𝑝_𝑘+𝑥_𝑛−𝑆𝑥_𝑛. (233)
It is easy to see from (209) and (228) that 𝜇_𝑛𝑥_𝑛−𝐺𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘²,𝜇_𝑛𝑥_𝑛−𝑆𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘². (234)
Utilizing (232) and (234), we deduce that 𝜇_𝑛𝑥_𝑛−𝑉𝑝_𝑘²=𝜇_𝑛1−𝜃₁−𝜃₂𝑥_𝑛−𝑆𝑝_𝑘+𝜃₁𝑥_𝑛−𝐵𝑝_𝑘+𝜃₂𝑥_𝑛−𝐺𝑝_𝑘²≤1−𝜃₁−𝜃₂𝜇_𝑛𝑥_𝑛−𝑆𝑝_𝑘²+𝜃₁𝜇_𝑛𝑥_𝑛−𝐵𝑝_𝑘²+𝜃₂𝜇_𝑛𝑥_𝑛−𝐺𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘². (235)
Also, observe that 𝑥_𝑛−𝑝_𝑘=1𝑘𝑥_𝑛−𝑓𝑝_𝑘+11−𝑘𝑥_𝑛−𝑉𝑝_𝑘; (236)
that is, 11−𝑘𝑥_𝑛−𝑉𝑝_𝑘=𝑥_𝑛−𝑝_𝑘−1𝑘𝑥_𝑛−𝑓𝑝_𝑘. (237)
It follows from Lemma 8(ii) and (237) that 11−𝑘²𝑥_𝑛−𝑉𝑝_𝑘²≥𝑥_𝑛−𝑝_𝑘²−2𝑘𝑥_𝑛−𝑝_𝑘+𝑝_𝑘−𝑓𝑝_𝑘,𝐽𝑥_𝑛−𝑝_𝑘=21−𝑘𝑥_𝑛−𝑝_𝑘²+2𝑘𝑓𝑝_𝑘−𝑝_𝑘,𝐽𝑥_𝑛−𝑝_𝑘. (238)
So by (235) and (238), we have 11−𝑘²𝜇_𝑛𝑥_𝑛−𝑝_𝑘²≥21−𝑘𝜇_𝑛𝑥_𝑛−𝑝_𝑘²+2𝑘𝜇_𝑛𝑓𝑝_𝑘−𝑝_𝑘,𝐽𝑥_𝑛−𝑝_𝑘, (239)
and hence 1𝑘²𝜇_𝑛𝑥_𝑛−𝑝_𝑘²≥2𝑘𝜇_𝑛𝑓𝑝_𝑘−𝑝_𝑘,𝐽𝑥_𝑛−𝑝_𝑘. (240)
This implies that 1𝜇2𝑘^𝑛𝑥_𝑛−𝑝_𝑘²≥𝜇_𝑛𝑓𝑝_𝑘−𝑝_𝑘,𝐽𝑥_𝑛−𝑝_𝑘. (241)
Since 𝑝_𝑘→𝑞∈Fix(𝑉)=𝐹 as 𝑘→∞, by the uniform Frechet differentiability of the norm of 𝑋, we have 𝜇_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0. (242)
On the other hand, from (184) and the norm-to-norm uniform continuity of 𝐽 on bounded subsets of 𝑋, it follows that lim_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞−𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞=0. (243)
So, utilizing Lemma 18 we deduce from (242) and (243) that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0, (244)
which, together with (184) and the norm-to-norm uniform continuity of 𝐽 on bounded subsets of 𝑋, implies that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞≤0. (245)
Finally, let us show that 𝑥_𝑛→𝑞 as 𝑛→∞. Utilizing Lemma 8 (i), from (178) and the convexity of ‖⋅‖², we get 𝑥_𝑛+1−𝑞²=𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑞+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑞+𝛼_𝑛𝑓𝑞−𝑞²≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑞+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑞²+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝑥_𝑛−𝑞²+𝛿_𝑛𝐺𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞≤𝛼_𝑛𝜌𝑥_𝑛−𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝑥_𝑛−𝑞²+𝛿_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1=−𝑞1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛+1=−𝑞1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+𝛼_𝑛2𝑓𝑞𝑥1−𝜌−𝑞,𝐽_𝑛+1−𝑞.1−𝜌 (246)
Applying Lemma 7 to (246), we obtain that 𝑥_𝑛→𝑞 as 𝑛→∞. This completes the proof.

Corollary 32. Let 𝐶 be a nonempty closed 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 {𝜌_𝑛}^∞_𝑛=0 be a sequence of positive numbers in (0,𝑏] for some 𝑏∈(0,1) and 𝐴_𝑖∶𝐶→𝑋 an 𝛼_𝑖-inverse strongly accretive mapping for each 𝑖=0,1,…. Define a mapping 𝐺_𝑖∶𝐶→𝐶 by Π_𝐶(𝐼−𝜆_𝑖𝐴_𝑖)𝑥=𝐺_𝑖𝑥 for all 𝑥∈𝐶 and 𝑖=0,1,…, where 𝜆_𝑖∈(0,𝛼_𝑖/𝜅²] and 𝜅 is the 2-uniformly smooth constant of 𝑋. Let 𝐵_𝑛∶𝐶→𝐶 be the 𝑊-mapping generated by 𝐺_𝑛,𝐺_𝑛−1,…,𝐺₀, and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₀. Let 𝑉∶𝐶→𝐶 be an 𝛼-strictly pseudocontractive mapping. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let {𝑆_𝑖}^∞_𝑖=0 be a countable family of nonexpansive mappings of 𝐶 into itself such that 𝐹=(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Fix(𝑉)∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖))≠∅. For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛((1−𝑙)𝐼+𝑙𝑉)𝑥_𝑛,∀𝑛≥0, (247)
where 0<𝑙<𝛼/𝜅² and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛} are the sequences in (0,1) such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 for all 𝑛≥0. Suppose that the following conditions hold:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(ii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iii)lim_𝑛→∞(|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|)=0;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_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 and suppose that Fix(𝑆)=⋂^∞_𝑖=0Fix(𝑆_𝑖). Then, there hold the following:(I)lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0;(II)the sequence {𝑥_𝑛}^∞_𝑛=0 converges strongly to some 𝑞∈𝐹 which is the unique solution of the variational inequality problem (VIP) 𝐼−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹, (248) provided 𝛽_𝑛≡𝛽 for some fixed 𝛽∈(0,1).

Proof. In Theorem 31, we put 𝐵₁=𝐼−𝑉, 𝐵₂=0 and 𝜇₁=𝑙 where 0<𝑙<𝛼/𝜅². Then GSVI (13) is equivalent to the VIP of finding 𝑥^∗∈𝐶 such that 𝐵₁𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (249)
In this case, 𝐵₁∶𝐶→𝑋 is 𝛼-inverse strongly accretive. Repeating the same arguments as those in the proof of Corollary 25, we can infer that Fix(𝑉)=VI(𝐶,𝐵₁). Accordingly, we know that 𝐹=⋂^∞_𝑖=0Fix(𝑇_𝑖)∩Ω∩𝐴⁻¹0=⋂^∞_𝑖=0Fix(𝑇_𝑖)∩Fix(𝑉)∩𝐴⁻¹0, and Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑥_𝑛=Π_𝐶𝐼−𝜇₁𝐵₁𝑥_𝑛=Π_𝐶(1−𝑙)𝑥_𝑛+𝑙𝑉𝑥_𝑛=((1−𝑙)𝐼+𝑙𝑉)𝑥_𝑛. (250)
So, scheme (178) reduces to (247). Therefore, the desired result follows from Theorem 31.

Theorem 33. Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let {𝜌_𝑛}^∞_𝑛=0 be a sequence of positive numbers in (0,𝑏] for some 𝑏∈(0,1) and 𝐴_𝑖∶𝐶→𝑋 𝜉_𝑖-strictly pseudocontractive and 𝛼_𝑖-strongly accretive with 𝜉_𝑖+𝛼_𝑖≥1 for each 𝑖=0,1,…. Define a mapping 𝐺_𝑖∶𝐶→𝐶 by Π_𝐶(𝐼−𝜆_𝑖𝐴_𝑖)𝑥=𝐺_𝑖𝑥 for all 𝑥∈𝐶 and 𝑖=0,1,…, where 1−(𝜉_𝑖/(1+𝜉_𝑖))(1−(1−𝛼_𝑖)/𝜉_𝑖)≤𝜆_𝑖≤1 for all 𝑖=0,1,…. Let 𝐵_𝑛∶𝐶→𝐶 be the 𝑊-mapping generated by 𝐺_𝑛,𝐺_𝑛−1,…,𝐺₀ and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₀. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝜁_𝑖-strictly pseudocontractive and 𝛽_𝑖-strongly accretive with 𝜁_𝑖+𝛽_𝑖≥1 for 𝑖=1,2. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let {𝑆_𝑖}^∞_𝑖=0 be a countable family of nonexpansive mappings of 𝐶 into itself such that 𝐹=(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Ω∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖))≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) with 1−(𝜁_𝑖/(1+𝜁_𝑖))(1−(1−𝛽_𝑖)/𝜁_𝑖)≤𝜇_𝑖≤1 for 𝑖=1,2. For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑥_𝑛+1=𝜎_𝑛𝐺𝑥_𝑛+1−𝜎_𝑛×𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛,∀𝑛≥0, (251)
where {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛿_𝑛}, and {𝜎_𝑛} are the sequences in (0,1) such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 for all 𝑛≥0. Suppose that the following conditions hold:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(ii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iii)∑^∞_𝑛=1(|𝜎_𝑛−𝜎_𝑛−1|+|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|)<∞;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and 0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_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 and suppose that Fix(𝑆)=⋂^∞_𝑖=0Fix(𝑆_𝑖). Then, there hold the following:(I)lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0;(II)the sequence {𝑥_𝑛}^∞_𝑛=0 converges strongly to some 𝑞∈𝐹 which is the unique solution of the variational inequality problem (VIP) 𝐼−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹, (252)

Proof. First of all, it is easy to see that (251) can be rewritten as follows: 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛,𝑥_𝑥+1=𝜎_𝑛𝐺𝑥_𝑛+1−𝜎_𝑛𝑦_𝑛,∀𝑛≥0. (253)
Take a fixed 𝑝∈𝐹 arbitrarily. Then, we obtain 𝑝=𝐺𝑝, 𝑝=𝐵_𝑛𝑝 and 𝑆_𝑛𝑝=𝑝 for all 𝑛≥0. Thus, we get from (253) 𝑦_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝≤𝛼_𝑛𝜌𝑥_𝑛−𝑝+𝑓𝑝−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑥_𝑛=−𝑝1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝, (254)
and hence 𝑥_𝑛+1−𝑝≤𝜎_𝑛𝐺𝑥_𝑛−𝑝+1−𝜎_𝑛𝑦_𝑛−𝑝≤𝜎_𝑛𝑥_𝑛−𝑝+1−𝜎_𝑛×1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝=−𝑝1−1−𝜎_𝑛𝛼_𝑛1−𝜌𝑥_𝑛+−𝑝1−𝜎_𝑛𝛼_𝑛𝑓𝑝=−𝑝1−1−𝜎_𝑛𝛼_𝑛1−𝜌𝑥_𝑛+−𝑝1−𝜎_𝑛𝛼_𝑛𝑓𝑝1−𝜌−𝑝𝑥1−𝜌≤max_𝑛,𝑓𝑝−𝑝−𝑝.1−𝜌 (255)
By induction, we have 𝑥_𝑛−𝑝≤max𝑥₀,𝑓𝑝−𝑝−𝑝1−𝜌,∀𝑛≥0. (256)
which implies that {𝑥_𝑛} is bounded and so are the sequences {𝑦_𝑛}, {𝐺𝑥_𝑛} and {𝑓(𝑥_𝑛)}.
Let us show that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (257)
As a matter of fact, observe that 𝑦_𝑛 can be rewritten as follows: 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛, (258)
where 𝑧_𝑛=(𝛼_𝑛𝑓(𝑥_𝑛)+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛)/(1−𝛽_𝑛). Observe that 𝑧_𝑛−𝑧_𝑛−1=𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛1−𝛽_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−11−𝛽_𝑛−1=𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1=𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛+𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1≤𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛+𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1=11−𝛽_𝑛𝑦_𝑛−𝛽_𝑛𝑥_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1+11−𝛽_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1=11−𝛽_𝑛𝑦_𝑛−𝛽_𝑛𝑥_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1=11−𝛽_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1−𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1−𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1≤11−𝛽_𝑛𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1. (259)
On the other hand, repeating the same arguments as those of (52) and (54) in the proof of Theorem 24, we can deduce that for all 𝑛≥1𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛−1𝐺𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1,𝐵_𝑛𝑥_𝑛−𝐵_𝑛−1𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑀^𝑛−1_𝑖=0𝜆_𝑖, (260)
for some constant 𝑀>0. Taking into account 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, we may assume, without loss of generality, that {𝛽_𝑛}⊂[𝑐,̂𝑑]. Utilizing (259)-(260) we have 𝑧_𝑛−𝑧_𝑛−1≤11−𝛽_𝑛×𝛼_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝛾_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀^𝑛−1_𝑖=0𝜆_𝑖+𝛿_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1=11−𝛽_𝑛1−𝛽_𝑛−𝛼_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝛾_𝑛𝑀^𝑛−1_𝑖=0𝜆_𝑖+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1=𝛼1−^𝑛1−𝜌1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛾_𝑛𝑀1−𝛽_𝑛^𝑛−1_𝑖=0𝜆_𝑖+𝛿_𝑛1−𝛽_𝑛𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+11−𝛽_𝑛𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1×𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1≤𝛼1−^𝑛1−𝜌1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀^𝑛−1_𝑖=0𝜆_𝑖+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1+11−𝛽_𝑛×𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛×𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−1𝐵_𝑛−1𝑥_𝑛−1+𝛿_𝑛−1𝑆_𝑛−1𝐺𝑥_𝑛−1≤𝛼1−^𝑛1−𝜌1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₁^𝑛−1_𝑖=0𝜆_𝑖+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1, (261)
where sup_𝑛≥0{(1/(1−𝑑)²)(‖𝑓(𝑥_𝑛)‖+‖𝐵_𝑛𝑥_𝑛‖+‖𝑆_𝑛𝐺𝑥_𝑛‖+𝑀)}≤𝑀₁ for some 𝑀₁>0. In the meantime, observe that 𝑥_𝑛+1−𝑥_𝑛=𝜎_𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝐺𝑥_𝑛−1−𝑧_𝑛−1+1−𝜎_𝑛𝑧_𝑛−𝑧_𝑛−1. (262)
This together with (261), implies that 𝑥_𝑛+1−𝑥_𝑛≤𝜎_𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝐺𝑥_𝑛−1−𝑧_𝑛−1+1−𝜎_𝑛𝑧_𝑛−𝑧_𝑛−1≤𝜎_𝑛𝑥_𝑛−𝑥_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝐺𝑥_𝑛−1−𝑧_𝑛−1+1−𝜎_𝑛𝛼1−^𝑛1−𝜌1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₁^𝑛−1_𝑖=0𝜆_𝑖+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1≤1−1−𝜎_𝑛𝛼_𝑛1−𝜌1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝐺𝑥_𝑛−1−𝑧_𝑛−1+𝑀₁^𝑛−1_𝑖=0𝜆_𝑖+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1≤1−1−𝜎_𝑛𝛼_𝑛1−𝜌1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₂^𝑛−1_𝑖=0𝜆_𝑖+𝜎_𝑛−𝜎_𝑛−1+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑆_𝑛𝐺𝑥_𝑛−1−𝑆_𝑛−1𝐺𝑥_𝑛−1, (263)
where sup_𝑛≥0{𝑀₁+‖𝐺𝑥_𝑛−𝑧_𝑛‖}≤𝑀₂ for some 𝑀₂>0. Since ∑^∞_𝑛=0𝛼_𝑛=∞ and (1−𝜎_𝑛)𝛼_𝑛(1−𝜌)/(1−𝛽_𝑛)≥(1−𝜎_𝑛)𝛼_𝑛(1−𝜌), we obtain from conditions (i) and (iv) that ∑^∞_𝑛=0((1−𝜎_𝑛)𝛼_𝑛(1−𝜌)/(1−𝛽_𝑛))=∞. Thus, applying Lemma 7 to (263), we deduce from condition (iii) and the assumption on {𝑆_𝑛} that (noting that 0<𝜆_𝑖≤𝑏<1, for all 𝑖≥0) lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (264)
Next, we show that ‖𝑥_𝑛−𝐺𝑥_𝑛‖→0 as 𝑛→∞.
Indeed, according to Lemma 8 we have from (253) 𝑦_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝≤𝛼_𝑛𝜌𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1=−𝑝1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1≤𝑥−𝑝_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1−𝑝. (265)
Utilizing Lemma 15 we get from (253) and (265) 𝑥_𝑛+1−𝑝²=𝜎_𝑛𝐺𝑥_𝑛−𝑝+1−𝜎_𝑛𝑦_𝑛−𝑝²≤𝜎_𝑛𝐺𝑥_𝑛−𝑝²+1−𝜎_𝑛𝑦_𝑛−𝑝²−𝜎_𝑛1−𝜎_𝑛𝑔𝐺𝑥_𝑛−𝑦_𝑛≤𝜎_𝑛𝑥_𝑛−𝑝²+1−𝜎_𝑛×𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1−𝑝−𝜎_𝑛1−𝜎_𝑛𝑔𝐺𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1−𝑝−𝜎_𝑛1−𝜎_𝑛𝑔𝐺𝑥_𝑛−𝑦_𝑛, (266)
which hence yields 𝜎_𝑛1−𝜎_𝑛𝑔𝐺𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1≤𝑥−𝑝_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+2𝛼_𝑛𝑓𝑝−𝑝𝑥_𝑛+1−𝑝. (267)
Since 𝛼_𝑛→0 and ‖𝑥_𝑛+1−𝑥_𝑛‖→0, from condition (iv) and the boundedness of {𝑥_𝑛}, it follows that lim_𝑛→∞𝑔𝐺𝑥_𝑛−𝑦_𝑛=0. (268)
Utilizing the properties of 𝑔, we have lim_𝑛→∞𝐺𝑥_𝑛−𝑦_𝑛=0, (269)
which, together with (253) and (257), implies that 𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑦_𝑛=𝑥_𝑛−𝑥_𝑛+1+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛⟶0as𝑛⟶∞. (270)
That is, lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (271)
Since 𝑥_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝐺𝑥_𝑛, (272)
it immediately follows from (269) and (271) that lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0. (273)
On the other hand, observe that 𝑦_𝑛 can be rewritten as follows: 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝑒_𝑛𝑧_𝑛, (274)
where 𝑒_𝑛=𝛾_𝑛+𝛿_𝑛 and 𝑧_𝑛=(𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛)/(𝛾_𝑛+𝛿_𝑛). Utilizing Lemma 11, we have 𝑦_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝑒_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝑒_𝑛𝑧_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₁𝑧_𝑛−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₁𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₁𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝛿−𝑝^𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₁𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₁𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₁𝑧_𝑛−𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₁𝑧_𝑛−𝑥_𝑛, (275)
which hence implies that 𝛽_𝑛𝑒_𝑛𝑔₁𝑧_𝑛−𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛. (276)
Utilizing (271), conditions (i), (ii), (iv), and the boundedness of {𝑥_𝑛}, {𝑦_𝑛} and {𝑓(𝑥_𝑛)}, we get lim_𝑛→∞𝑔₁𝑧_𝑛−𝑥_𝑛=0. (277)
From the properties of 𝑔₁, we have lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0. (278)
Utilizing Lemma 15 and the definition of 𝑧_𝑛, we have 𝑧_𝑛−𝑝²=𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛾_𝑛+𝛿_𝑛−𝑝²=𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛+𝛿−𝑝^𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²≤𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₂𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝑥_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₂𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛, (279)
which leads to 𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₂𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝑥_𝑛−𝑝²−𝑧_𝑛−𝑝²≤𝑥_𝑛−𝑝+𝑧_𝑛−𝑝𝑥_𝑛−𝑧_𝑛. (280)
Since {𝑥_𝑛} and {𝑧_𝑛} are bounded, we deduce from (278) and condition (ii) that lim_𝑛→∞𝑔₂𝐵_𝑛𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛=0. (281)
From the properties of 𝑔₂, we have lim_𝑛→∞𝐵_𝑛𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛=0. (282)
Furthermore, 𝑦_𝑛 can also be rewritten as follows: 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛼_𝑛+𝛿_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛼_𝑛+𝛿_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝑑_𝑛𝑧_𝑛, (283)
where 𝑑_𝑛=𝛼_𝑛+𝛿_𝑛 and 𝑧_𝑛=(𝛼_𝑛𝑓(𝑥_𝑛)+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛)/(𝛼_𝑛+𝛿_𝑛). Utilizing Lemma 11 and the convexity of ‖⋅‖², we have 𝑦_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝+𝑑_𝑛𝑧_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝑧_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛𝛼_𝑛+𝛿_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝛼_𝑛+𝛿_𝑛𝑓𝑥_𝑛+𝛿−𝑝^𝑛𝛼_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝛼_𝑛+𝛿_𝑛𝑓𝑥_𝑛−𝑝²+𝛿_𝑛𝛼_𝑛+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛, (284)
which hence implies that 𝛽_𝑛𝛾_𝑛𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛×𝑥−𝑝_𝑛−𝑦_𝑛. (285)
Utilizing (271), conditions (i), (ii), (iv), and the boundedness of {𝑥_𝑛}, {𝑦_𝑛} and {𝑓(𝑥_𝑛)}, we get lim_𝑛→∞𝑔₃𝑥_𝑛−𝐵_𝑛𝑥_𝑛=0. (286)
From the properties of 𝑔₃, we have lim_𝑛→∞𝑥_𝑛−𝐵_𝑛𝑥_𝑛=0. (287)
Thus, from (282) and (287), we get 𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛≤𝑥_𝑛−𝐵_𝑛𝑥_𝑛+𝐵_𝑛𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛⟶0as𝑛⟶∞. (288)
That is, lim_𝑛→∞𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛=0. (289)
Therefore, from Lemma 12, (273), and (289), it follows that 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝑆_𝑛𝐺𝑥_𝑛−𝑆_𝑛𝑥_𝑛+𝑆_𝑛𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆_𝑛𝐺𝑥_𝑛+𝐺𝑥_𝑛−𝑥_𝑛+𝑆_𝑛𝑥_𝑛−𝑆𝑥_𝑛⟶0as𝑛⟶∞. (290)
That is, lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (291)
Suppose that 𝛽_𝑛≡𝛽 for some fixed 𝛽∈(0,1) such that 𝛼_𝑛+𝛽+𝛾_𝑛+𝛿_𝑛=1 for all 𝑛≥0. Define a mapping 𝑉𝑥=(1−𝜃₁−𝜃₂)𝑆𝑥+𝜃₁𝐵𝑥+𝜃₂𝐺𝑥, where 𝜃₁,𝜃₂∈(0,1) are two constants with 𝜃₁+𝜃₂<1. Then, by Lemmas 14 and 17, we have that Fix(𝑉)=Fix(𝑆)∩Fix(𝐵)∩Fix(𝐺)=𝐹. For each 𝑘≥1, let {𝑝_𝑘} be a unique element of 𝐶 such that 𝑝_𝑘=1𝑘𝑓𝑝_𝑘+11−𝑘𝑉𝑝_𝑘. (292)
From Lemma 13, we conclude that 𝑝_𝑘→𝑞∈Fix(𝑉)=𝐹 as 𝑘→∞. Repeating the same arguments as those of (81) in the proof of Theorem 24, we can conclude that for every 𝑛,𝑘𝑦_𝑛−𝐵𝑝_𝑘≤𝛼_𝑛𝑓𝑥_𝑛−𝐵𝑝_𝑘+𝛽𝑥_𝑛−𝐵𝑝_𝑘+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝐵𝑝_𝑘+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝐵𝑝_𝑘≤𝜃_𝑛+𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝑥_𝑛−𝑝_𝑘, (293)
where 𝜃_𝑛=𝛼_𝑛‖𝑓(𝑥_𝑛)−𝐵𝑝_𝑘‖+(1−𝛽)‖𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘‖+𝛿_𝑛‖𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛‖. Since lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞‖𝐵_𝑛𝑝_𝑘−𝐵𝑝_𝑘‖=lim_𝑛→∞‖𝑆_𝑛𝐺𝑥_𝑛−𝐵_𝑛𝑥_𝑛‖=0, we know that 𝜃_𝑛→0 as 𝑛→∞. So, it immediately follows that 𝑥_𝑛+1−𝐵𝑝_𝑘²=𝑦_𝑛−𝐵𝑝_𝑘²+𝑥_𝑛+1−𝑦_𝑛×2𝑦_𝑛−𝐵𝑝_𝑘+𝑥_𝑛+1−𝑦_𝑛≤𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝑥_𝑛−𝑝_𝑘²+𝜃_𝑛2𝛽𝑥_𝑛−𝐵𝑝_𝑘+1−𝛽𝑥_𝑛−𝑝_𝑘+𝜃_𝑛+𝑥_𝑛+1−𝑦_𝑛2𝑦_𝑛−𝐵𝑝_𝑘+𝑥_𝑛+1−𝑦_𝑛=𝛽²𝑥_𝑛−𝐵𝑝_𝑘²+1−𝛽²𝑥_𝑛−𝑝_𝑘²+2𝛽1−𝛽𝑥_𝑛−𝐵𝑝_𝑘𝑥_𝑛−𝑝_𝑘+𝜏_𝑛≤𝛽²𝑥_𝑛−𝐵𝑝_𝑘²+1−𝛽²𝑥_𝑛−𝑝_𝑘²+𝛽1−𝛽𝑥_𝑛−𝐵𝑝_𝑘²+𝑥_𝑛−𝑝_𝑘²+𝜏_𝑛=𝛽𝑥_𝑛−𝐵𝑝_𝑘²+1−𝛽𝑥_𝑛−𝑝_𝑘²+𝜏_𝑛, (294)
where 𝜏_𝑛=𝜃_𝑛[2(𝛽‖𝑥_𝑛−𝐵𝑝_𝑘‖+(1−𝛽)‖𝑥_𝑛−𝑝_𝑘‖)+𝜃_𝑛]+‖𝑥_𝑛+1−𝑦_𝑛‖[2‖𝑦_𝑛−𝐵𝑝_𝑘‖+‖𝑥_𝑛+1−𝑦_𝑛‖]→0 as 𝑛→∞.
For any Banach limit 𝜇, from (294), we derive 𝜇_𝑛𝑥_𝑛−𝐵𝑝_𝑘²=𝜇_𝑛𝑥_𝑛+1−𝐵𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘². (295)
In addition, note that 𝑥_𝑛−𝐺𝑝_𝑘²≤𝑥_𝑛−𝐺𝑥_𝑛+𝐺𝑥_𝑛−𝐺𝑝_𝑘²≤𝑥_𝑛−𝐺𝑥_𝑛+𝑥_𝑛−𝑝_𝑘²=𝑥_𝑛−𝑝_𝑘²+𝑥_𝑛−𝐺𝑥_𝑛×2𝑥_𝑛−𝑝_𝑘+𝑥_𝑛−𝐺𝑥_𝑛,𝑥_𝑛−𝑆𝑝_𝑘²≤𝑥_𝑛−𝑆𝑥_𝑛+𝑆𝑥_𝑛−𝑆𝑝_𝑘²≤𝑥_𝑛−𝑆𝑥_𝑛+𝑥_𝑛−𝑝_𝑘²=𝑥_𝑛−𝑝_𝑘²+𝑥_𝑛−𝑆𝑥_𝑛×2𝑥_𝑛−𝑝_𝑘+𝑥_𝑛−𝑆𝑥_𝑛. (296)
It is easy to see from (273) and (291) that 𝜇_𝑛𝑥_𝑛−𝐺𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘²,𝜇_𝑛𝑥_𝑛−𝑆𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘². (297)
Utilizing (295) and (297), we deduce that 𝜇_𝑛𝑥_𝑛−𝑉𝑝_𝑘²=𝜇_𝑛1−𝜃₁−𝜃₂𝑥_𝑛−𝑆𝑝_𝑘+𝜃₁𝑥_𝑛−𝐵𝑝_𝑘+𝜃₂𝑥_𝑛−𝐺𝑝_𝑘²≤1−𝜃₁−𝜃₂𝜇_𝑛𝑥_𝑛−𝑆𝑝_𝑘²+𝜃₁𝜇_𝑛𝑥_𝑛−𝐵𝑝_𝑘²+𝜃₂𝜇_𝑛𝑥_𝑛−𝐺𝑝_𝑘²≤𝜇_𝑛𝑥_𝑛−𝑝_𝑘². (298)
Repeating the same arguments as those of (99) in the proof of Theorem 24, we can obtain that 1𝜇2𝑘^𝑛𝑥_𝑛−𝑝_𝑘²≥𝜇_𝑛𝑓𝑝_𝑘−𝑝_𝑘,𝐽𝑥_𝑛−𝑝_𝑘. (299)
Since 𝑝_𝑘→𝑞∈Fix(𝑉)=𝐹 as 𝑘→∞, by the uniform Gateaux differentiability of the norm of 𝑋 we have 𝜇_𝑛𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0. (300)
On the other hand, from (257) and the norm-to-weak* uniform continuity of 𝐽 on bounded subsets of 𝑋, it follows that lim_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛+1−𝑞−𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞=0. (301)
So, utilizing Lemma 18, we deduce from (300) and (301) that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0, (302)
which together with (271) and the norm-to-weak* uniform continuity of 𝐽 on bounded subsets of 𝑋, implies that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞≤0. (303)
Finally, let us show that 𝑥_𝑛→𝑞 as 𝑛→∞. Utilizing Lemma 8 (i), from (253) and the convexity of ‖⋅‖, we get 𝑦_𝑛−𝑞²≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑞+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝐵_𝑛𝑥_𝑛−𝑞²+𝛿_𝑛𝑆_𝑛𝐺𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞≤𝛼_𝑛𝜌𝑥_𝑛−𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝑥_𝑛−𝑞²+𝛿_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛=−𝑞1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞, (304)
and hence 𝑥_𝑛+1−𝑞²≤𝜎_𝑛𝐺𝑥_𝑛−𝑞²+1−𝜎_𝑛𝑦_𝑛−𝑞²≤𝜎_𝑛𝑥_𝑛−𝑞²+1−𝜎_𝑛×1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛=−𝑞1−1−𝜎_𝑛𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+21−𝜎_𝑛𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛=−𝑞1−1−𝜎_𝑛𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+1−𝜎_𝑛𝛼_𝑛2𝑓𝑞𝑦1−𝜌−𝑞,𝐽_𝑛−𝑞.1−𝜌 (305)
From conditions (i) and (iv), it is easy to see that ∑^∞_𝑛=0(1−𝜎_𝑛)𝛼_𝑛(1−𝜌)=∞. Applying Lemma 7 to (305), we infer that 𝑥_𝑛→𝑞 as 𝑛→∞. This completes the proof.

Corollary 34. Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let {𝜌_𝑛}^∞_𝑛=0 be a sequence of positive numbers in (0,𝑏] for some 𝑏∈(0,1) and 𝐴_𝑖∶𝐶→𝑋 𝜉_𝑖-strictly pseudocontractive and 𝛼_𝑖-strongly accretive with 𝜉_𝑖+𝛼_𝑖≥1 for each 𝑖=0,1,…. Define a mapping 𝐺_𝑖∶𝐶→𝐶 by Π_𝐶(𝐼−𝜆_𝑖𝐴_𝑖)𝑥=𝐺_𝑖𝑥 for all 𝑥∈𝐶 and 𝑖=0,1,…, where 1−(𝜉_𝑖/(1+𝜉_𝑖))(1−(1−𝛼_𝑖)/𝜉_𝑖)≤𝜆_𝑖≤1 for all 𝑖=0,1,…. Let 𝐵_𝑛∶𝐶→𝐶 be the 𝑊-mapping generated by 𝐺_𝑛,𝐺_𝑛−1,…,𝐺₀ and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₀. Let 𝑉∶𝐶→𝐶 be a self-mapping such that 𝐼−𝑉∶𝐶→𝑋 is 𝜁-strictly pseudocontractive and 𝜃-strongly accretive with 𝜃+𝜁≥1. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let {𝑆_𝑖}^∞_𝑖=0 be a countable family of nonexpansive mappings of 𝐶 into itself such that 𝐹=(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Fix(𝑉)∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖))≠∅. For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑥_𝑛+1=𝜎_𝑛((1−𝑙)𝐼+𝑙𝑉)𝑥_𝑛+1−𝜎_𝑛×𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵_𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛((1−𝑙)𝐼+𝑙𝑉)𝑥_𝑛,∀𝑛≥0, (306)
where 1−(𝜁/(1+𝜁))(1−(1−𝜃)/𝜁)≤𝑙≤1 and {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛿_𝑛}, and {𝜎_𝑛} are the sequences in (0,1) such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 for all 𝑛≥0. Suppose that the following conditions hold:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(ii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iii)∑^∞_𝑛=1(|𝜎_𝑛−𝜎_𝑛−1|+|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|)<∞;(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and 0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_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 and suppose that Fix(𝑆)=⋂^∞_𝑖=0Fix(𝑆_𝑖). Then, there hold the following:(I)lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0;(II)the sequence {𝑥_𝑛}^∞_𝑛=0 converges strongly to some 𝑞∈𝐹 which is the unique solution of the variational inequality problem (VIP) 𝐼−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹, (307) provided 𝛽_𝑛≡𝛽 for some fixed 𝛽∈(0,1).

Proof. In Theorem 33, we put 𝐵₁=𝐼−𝑉, 𝐵₂=0 and 𝜇₁=𝑙 where 1−(𝜁/(1+𝜁))(1−(1−𝜃)/𝜁)≤𝑙≤1. Then, GSVI (13) is equivalent to the VIP of finding 𝑥^∗∈𝐶 such that 𝐵₁𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (308)
In this case, 𝐵₁∶𝐶→𝑋 is 𝜁-strictly pseudocontractive and 𝜃-strongly accretive. Repeating the same arguments as those in the proof of Corollary 25, we can infer that Fix(𝑉)=VI(𝐶,𝐵₁). Accordingly, 𝐹=(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Ω∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖))=(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Fix(𝑉)∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖)), 𝐺𝑥_𝑛=((1−𝑙)𝐼+𝑙𝑉)𝑥_𝑛, (309)
So, the scheme (251) reduces to (306). Therefore, the desired result follows from Theorem 33.

Remark 35. Our Theorems 31 and 33 improve, extend, supplement and develop Ceng and Yao’s [10, Theorem  3.2], Cai and Bu’s [11, Theorem  3.1], Kangtunyakarn’s [38, Theorem  3.1], and Ceng and Yao’s [8, Theorem  3.1], in the following aspects.(i)The problem of finding a point 𝑞∈(⋂^∞_𝑖=0Fix(𝑆_𝑖))∩Ω∩(⋂^∞_𝑖=0VI(𝐶,𝐴_𝑖)) in our Theorems 31 and 33 is more general and more subtle than every one of the problem of finding a point 𝑞∈⋂^∞_𝑖=0Fix(𝑇_𝑖) in [10, Theorem  3.2], the problem of finding a point 𝑞∈⋂^∞_𝑖=1Fix(𝑇_𝑖)∩Ω in [11, Theorem  3.1], the problem of finding a point 𝑞∈Fix(𝑆)∩Fix(𝑉)∩(⋂^𝑁_𝑖=1VI(𝐶,𝐴_𝑖)) in [38, Theorem  3.1], and the problem of finding a point 𝑞∈Fix(𝑇) in [8, Theorem  3.1].(ii)The iterative scheme in [38, Theorem  3.1] is extended to develop the iterative scheme (178) of our Theorem 31, and the iterative scheme in [11, Theorem  3.1] is extended to develop the iterative scheme (251) of our Theorem 33. Iterative schemes (178) and (181) in our Theorems 31 and 33 are more advantageous and more flexible than the iterative scheme of [11, Theorem  3.1] because they both are one-step iteration schemes and involve several parameter sequences {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛿_𝑛}, (and {𝜎_𝑛}).(iii)Our Theorems 31 and 33 extend and generalize Ceng and Yao’s [8, Theorem  3.1] from a nonexpansive mapping to a countable family of nonexpansive mappings, and Ceng and Yao’s [10, Theorems  3.2] to the setting of the GSVI (13) and infinitely many VIPs, Kangtunyakarn’s [38, Theorem  3.1] from finitely many VIPs to infinitely many VIPs, from a nonexpansive mapping to a countable family of nonexpansive mappings and from a strict pseudocontraction to the GSVI (13). In the meantime, our Theorems 31 and 33 extend and generalize Cai and Bu’s [11, Theorem  3.1] to the setting of infinitely many VIPs.(iv)The iterative schemes (178) and (251) in our Theorems 31 and 33 are very different from every one in [10, Theorem  3.2], [11, Theorem  3.1], [38, Theorem  3.1], and [8, Theorem  3.1] because the mappings 𝐺 and 𝑇_𝑛 in [11, Theorem  3.1] and the mapping 𝑇 in [8, Theorem  3.1] are replaced with the same composite mapping 𝑆_𝑛𝐺 in the iterative schemes (42) and (130) and the mapping 𝑊_𝑛 in [10, Theorem  3.2] is replaced by 𝐵_𝑛.(v)Cai and Bu’s proof in [11, Theorem  3.1] depends on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), and the inequality in smooth and uniform convex Banach spaces (see Proposition 6). Because the composite mapping 𝑆_𝑛𝐺 appears in the iterative scheme (178) of our Theorem 31, the proof of our Theorem 31 depends on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), the inequality in smooth and uniform convex Banach spaces (see Proposition 6), the inequalities in uniform convex Banach spaces (see Lemmas 11 and 15 in Section 2 of this paper), and the properties of the 𝑊-mapping and the Banach limit (see Lemmas 16, 17, and 18 in Section 2 of this paper). However, the proof of our Theorem 33 does not depend on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), and the inequality in smooth and uniform convex Banach spaces (see Proposition 6). It depends on only the inequalities in uniform convex Banach spaces (see Lemmas 11 and 15 in Section 2 of this paper) and the properties of the 𝑊-mapping and the Banach limit (see Lemmas 16–18 in Section 2 of this paper).(vi)The assumption of the uniformly convex and 2-uniformly smooth Banach space 𝑋 in [11, Theorem  3.1] is weakened to the one of the uniformly convex Banach space 𝑋 having a uniformly Gateaux differentiable norm in our Theorem 33. Moreover, the assumption of the uniformly smooth Banach space 𝑋 in [8, Theorem  3.1] is replaced with the one of the uniformly convex Banach space 𝑋 having a uniformly Gateaux differentiable norm in our Theorem 33.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under Grant no. HiCi/15-130-1433. The authors, therefore, acknowledge technical and financial support of KAU. The authors would like to thank Professor J. C. Yao for motivation and many fruitful discussions regarding this work.