Mann-Type Viscosity Approximation Methods for Multivalued Variational Inclusions with Finitely Many Variational Inequality Constraints in Banach Spaces

Abstract

We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative 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 Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. 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 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. (2)

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 (3) exists for all 𝑥,𝑦∈𝑈; in this case, 𝑋 is also said to have a Gáteaux differentiable norm. 𝑋 is said to have a uniformly Gáteaux differentiable norm if, for each 𝑦∈𝑈, the limit is attained uniformly for 𝑥∈𝑈. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for 𝑥,𝑦∈𝑈. The norm of 𝑋 is said to be the Fréchet differential if, for each 𝑥∈𝑈, this limit is attained uniformly for 𝑦∈𝑈. In addition, we define a function 𝜌∶[0,∞)→[0,∞) called the modulus of smoothness of 𝑋 as follows: 𝜌(𝜏)1=sup2+‖𝑥‖𝑦.𝑥+𝑦𝑥−𝑦−1∶𝑥,𝑦∈𝑋,=1,=𝜏 (4)

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. It is well-known that 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 Gáteaux differentiable norm then the duality mapping 𝐽 is norm-to-weak* uniformly continuous on bounded subsets of 𝑋.

Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝑋. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (5)

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.

Definition 1. Let 𝐴∶𝐶→𝑋 be a mapping of 𝐶 into 𝑋. Then 𝐴 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, (6) where 𝐽 is the normalized duality mapping;(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, (7) 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, (8) for some 𝛽>0;(iv)𝜆-strictly pseudocontractive if for each 𝑥,𝑦∈𝐶 there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≤𝑥−𝑦²−𝜆𝑥−𝑦−𝐴𝑥−𝐴𝑦² (9) for some 𝜆∈(0,1).

Let 𝑋 be a real smooth Banach space. Let 𝐶 be a nonempty closed convex subset of 𝑋 and let 𝐴∶𝐶→𝑋 be a nonlinear mapping. The so-called variational inequality problem (VIP) is the problem of finding 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (10) which was considered by Aoyama et al. [1]. Note that VIP (10) is connected with the fixed point problem for nonlinear mapping (see e.g., [2]), the problem of finding a zero point of a nonlinear operator (see e.g., [3]), and so on. In particular, whenever 𝑋=𝐻 a Hilbert space, the VIP (10) reduces to the classical VIP of finding 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (11)

whose solution set is denoted by VI(𝐶,𝐴). Recently, in order to find a solution of VIP (10), Aoyama et al. [1] introduced Mann-type iterative scheme for an accretive operator 𝐴 as follows: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛Π_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,∀𝑛≥1, (12) where Π_𝐶 is a sunny nonexpansive retraction from 𝑋 onto 𝐶. Then they proved a weak convergence theorem.

Definition 2. 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. (13) Such a mapping 𝐾 is called the 𝐾-mapping generated by 𝑇₁,…,𝑇_𝑁 and 𝜆₁,…,𝜆_𝑁.

Lemma 3 (see [4]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space. Let {𝑇_𝑖}^𝑁_𝑖=1 be a finite family of nonexpansive mappings of 𝐶 into itself with ∩^𝑁_𝑖=1Fix(𝑇_𝑖)≠∅ and let 𝜆₁,…,𝜆_𝑁 be real numbers such that 0<𝜆_𝑖<1 for every 𝑖=1,…,𝑁−1 and 0<𝜆_𝑁≤1. Let 𝐾 be the 𝐾-mapping generated by 𝑇₁,…,𝑇_𝑁 and 𝜆₁,…,𝜆_𝑁. Then Fix(𝐾)=∩^𝑁_𝑖=1Fix(𝑇_𝑖).

From Lemma 3, it is easy to see that the 𝐾-mapping is a nonexpansive mapping.

On the other hand, let 𝐶𝐵(𝑋) be the family of all nonempty, closed, and bounded subsets of a real smooth Banach space 𝑋. Also, we denote by 𝐻(⋅,⋅) the Hausdorff metric on 𝐶𝐵(𝑋) defined by 𝐻(𝐴,𝐵)∶=maxsup_𝑥∈𝐵inf_𝑦∈𝐴𝑑𝑥,𝑦,sup_𝑥∈𝐴inf_𝑦∈𝐵𝑑,(𝑋).𝑥,𝑦∀𝐴,𝐵∈𝐶𝐵 (14)

Let 𝑇,𝐹∶𝑋→𝐶𝐵(𝑋) be two multivalued mappings, let 𝐴∶𝐷(𝐴)⊂𝑋→2^𝑋 be an 𝑚-accretive mapping, let 𝑔∶𝑋→𝐷(𝐴) be a single-valued mapping, and let 𝑁(⋅,⋅)∶𝑋×𝑋→𝑋 be a nonlinear mapping. Then for any given 𝑣∈𝑋, 𝜆>0, Chidume et al. [5] introduced and studied the multivalued variational inclusion (MVVI) of finding 𝑥∈𝐷(𝐴) such that (𝑥,𝑤,𝑘) is a solution of the following: 𝑣∈𝑁(𝑤,𝑘)+𝜆𝐴𝑔(𝑥),∀𝑤∈𝑇𝑥,𝑘∈𝐹𝑥. (15)

If 𝑣=0 and 𝜆=1, then the MVVI (15) reduces to the problem of finding 𝑥∈𝐷(𝐴) such that (𝑥,𝑤,𝑘) is a solution of the following: 0∈𝑁(𝑤,𝑘)+𝐴𝑔(𝑥),∀𝑤∈𝑇𝑥,𝑘∈𝐹𝑥. (16)

We denote by Γ the set of such solutions 𝑥 for MVVI (16).

The authors [5] established an existence theorem for MVVI (15) in a smooth Banach space 𝑋 and then proved that the sequence generated by their iterative algorithm converges strongly to a solution of MVVI (16).

Theorem 4 (see [5, Theorem 3.2]). Let 𝑋 be a real smooth Banach space. Let 𝑇,𝐹∶𝑋→𝐶𝐵(𝑋), and 𝐴∶𝐷(𝐴)⊂𝑋→2^𝑋 be three multivalued mappings, let 𝑔∶𝑋→𝐷(𝐴) be a single-valued mapping, and let 𝑁(⋅,⋅)∶𝑋×𝑋→𝑋 be a single-valued continuous mapping satisfying the following conditions:(C1)𝐴∘𝑔∶𝑋→2^𝑋 is 𝑚-accretive and 𝐻-uniformly continuous;(C2)𝑇∶𝑋→𝐶𝐵(𝑋) is 𝐻-uniformly continuous;(C3)𝐹∶𝑋→𝐶𝐵(𝑋) is 𝐻-uniformly continuous;(C4)the mapping 𝑥↦𝑁(𝑥,𝑦) is 𝜙-strongly accretive and 𝜇-𝐻-Lipschitz with respect to the mapping 𝑇, where 𝜙∶[0,∞)→[0,∞) is a strictly increasing function with 𝜙(0)=0;(C5)the mapping 𝑦↦𝑁(𝑥,𝑦) is accretive and 𝜉-𝐻-Lipschitz with respect to the mapping 𝐹.
For arbitrary 𝑥₀∈𝐷(𝐴), define the sequence {𝑥_𝑛} iteratively by 𝑥_𝑛+1=𝑥_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝑢_𝑛∈𝐴𝑔𝑥_𝑛, (17) where {𝑢_𝑛} is defined by 𝑢_𝑛−𝑢_𝑛+1≤(1+𝜀)𝐻𝐴𝑔𝑥_𝑛+1,𝐴𝑔𝑥_𝑛,∀𝑛≥0, (18) for any 𝑤_𝑛∈𝑇𝑥_𝑛, 𝑘_𝑛∈𝐹𝑥_𝑛, and some 𝜀>0, where {𝜎_𝑛} is a positive real sequence such that lim_𝑛→∞𝜎_𝑛=0,∑^∞_𝑛=0𝜎_𝑛=∞. Then, there exists 𝑑>0 such that, for 0<𝜎_𝑛≤𝑑 and for all 𝑛≥0, {𝑥_𝑛} converges strongly to 𝑥∈Γ, and, for any 𝑤∈𝑇𝑥 and 𝑘∈𝐹𝑥, (𝑥,𝑤,𝑘) is a solution of the MVVI (16).

Let 𝐶 be a nonempty closed convex subset of a real smooth Banach space 𝑋 and let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Motivated and inspired by the research going on this area, we introduce Mann-type viscosity approximation methods for finding solutions of the MVVI (16) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings. Here, the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative 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 Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of 𝜂_𝑖-strictly pseudocontractive mappings (𝑖=1,…,𝑁) and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. 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, [6–11].

2. Preliminaries

Let 𝑋 be a real Banach space with dual 𝑋^∗. We denote by 𝐽 the normalized duality mapping from 𝑋 to 2^𝑋∗ defined by 𝐽(𝑥)=𝑥^∗∈𝑋^∗∶𝑥,𝑥^∗=‖𝑥‖²=𝑥^∗2, (19) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. Throughout this paper, the single-valued normalized duality map is still denoted by 𝐽. Unless otherwise stated, we assume that 𝑋 is a smooth Banach space with dual 𝑋^∗.

A multivalued mapping 𝐴∶𝐷(𝐴)⊆𝑋→2^𝑋 is said to be(i)accretive, if 𝑢−𝑣,𝐽𝑥−𝑦≥0,∀𝑢∈𝐴𝑥,𝑣∈𝐴𝑦; (20)(ii)𝑚-accretive, if 𝐴 is accretive and (𝐼+𝑟𝐴)(𝐷(𝐴))=𝑋, for all 𝑟>0, where 𝐼 is the identity mapping;(iii)𝜁-inverse strongly accretive, if there exists a constant 𝜁>0 such that 𝑢−𝑣,𝐽𝑥−𝑦≥𝜁‖𝑢−𝑣‖²,∀𝑢∈𝐴𝑥,𝑣∈𝐴𝑦; (21)(iv)𝜙-strongly accretive, if there exists a strictly increasing continuous function 𝜙∶[0,∞)→[0,∞) with 𝜙(0)=0 such that 𝑢−𝑣,𝐽𝑥−𝑦,≥𝜙𝑥−𝑦𝑥−𝑦,∀𝑢∈𝐴𝑥,𝑣∈𝐴𝑦; (22)(v)𝜙-expansive, if ‖𝑢−𝑣‖≥𝜙𝑥−𝑦,∀𝑢∈𝐴𝑥,𝑣∈𝐴𝑦. (23)

It is easy to see that if 𝐴 is 𝜙-strongly accretive, then 𝐴 is 𝜙-expansive.

A mapping 𝑇∶𝑋→𝐶𝐵(𝑋) is said to be 𝐻-uniformly continuous, if for any given 𝜀>0, there exists a 𝛿>0 such that whenever ‖𝑥−𝑦‖<𝛿 then 𝐻(𝑇𝑥,𝑇𝑦)<𝜀.

A mapping 𝑁∶𝑋×𝑋→𝑋 is 𝜙-strongly accretive, with respect to 𝑇∶𝑋→𝐶𝐵(𝑋), in the first argument if 𝑁(𝑢,𝑧)−𝑁(𝑣,𝑧),𝐽𝑥−𝑦≥𝜙𝑥−𝑦𝑥−𝑦,∀𝑢∈𝑇𝑥,𝑣∈𝑇𝑦. (24)

A mapping 𝑆∶𝑋→2^𝑋 is called lower semicontinuous, if 𝑆⁻¹(𝑂)∶={𝑥∈𝑋∶𝑆𝑥∩𝑂≠∅} is open in 𝑋 whenever 𝑂⊂𝑌 is open.

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

Proposition 5 (see [12]). Let {𝜆_𝑛} and {𝑏_𝑛} be sequences of nonnegative numbers and {𝛼_𝑛}⊂(0,1) a sequence satisfying the conditions that {𝜆_𝑛} is bounded, ∑^∞_𝑛=0𝛼_𝑛=∞, and 𝑏_𝑛→0, as 𝑛→∞. Let the recursive inequality 𝜆²_𝑛+1≤𝜆²_𝑛−2𝛼_𝑛𝜓𝜆_𝑛+1+2𝛼_𝑛𝑏_𝑛𝜆_𝑛+1,∀𝑛≥0, (25) be given where 𝜓∶[0,∞)→[0,∞) is a strictly increasing function such that it is positive on (0,∞) and 𝜓(0)=0. Then 𝜆_𝑛→0, as 𝑛→∞.

Proposition 6 (see [13]). Let 𝑋 be a real smooth Banach space. Let 𝑇, and 𝐹∶𝑋→2^𝑋 be two multivalued mappings, and let 𝑁(⋅,⋅)∶𝑋×𝑋→𝑋 be a nonlinear mapping satisfying the following conditions:(i)the mapping  𝑥↦𝑁(𝑥,𝑦)is  𝜙-strongly accretive with respect to the mapping  𝑇;(ii)the mapping  𝑦↦𝑁(𝑥,𝑦)2009  is accretive with respect to the mapping  𝐹.

Then the mapping 𝑆∶𝑋→2^𝑋 defined by 𝑆𝑥=𝑁(𝑇𝑥,𝐹𝑥) is 𝜙-strongly accretive.

Proposition 7 (see [14]). Let 𝑋 be a real Banach space and let 𝑆∶𝑋→2^𝑋⧵{∅} be a lower semicontinuous and 𝜙-strongly accretive mapping; then, for any 𝑥∈𝑋,𝑆𝑥 is a one-point set; that is, 𝑆 is a single-valued mapping.

Lemma 8 can be found in [15]. Lemma 9 is an immediate consequence of the subdifferential inequality of the function (1/2)‖⋅‖².

Lemma 8. 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, (26) 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.

Lemma 9. In a smooth Banach space 𝑋, there holds the inequality 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽𝑥+𝑦,∀𝑥,𝑦∈𝑋. (27)

Lemma 10 (see [1]). 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 𝐴 be an accretive operator of 𝐶 into 𝑋. Then, for all 𝜆>0, VI(𝐶,𝐴)=FixΠ_𝐶(𝐼−𝜆𝐴). (28)

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), (29) 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 11 (see [16]). 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 12 (see [17]). Let 𝑋 be a uniformly convex Banach space and 𝐵_𝑟(0)∶={𝑥∈𝑋∶‖𝑥‖≤𝑟}, 𝑟>0. Then there exists a continuous, strictly increasing, and convex function 𝜑∶[0,∞]→[0,∞], 𝜑(0)=0 such that 𝛼𝑥+𝛽𝑦+𝛾𝑧²≤𝛼‖𝑥‖²+𝛽𝑦²+𝛾‖𝑧‖²−𝛼𝛽𝜑𝑥−𝑦, (30) for all 𝑥,𝑦,𝑧∈𝐵_𝑟(0) and all 𝛼,𝛽,𝛾∈[0,1] with 𝛼+𝛽+𝛾=1.

Lemma 13 (see [18]). 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 let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix(𝑇)≠∅. As previous, let Ξ_𝐶 be the set of all contractions on 𝐶. For 𝑡∈(0,1) and 𝑓∈Ξ_𝐶, let 𝑥_𝑡∈𝐶 be the unique fixed point of the contraction 𝑥↦𝑡𝑓(𝑥)+(1−𝑡)𝑇𝑥 on 𝐶; that is, 𝑥_𝑡=𝑡𝑓𝑥_𝑡+(1−𝑡)𝑇𝑥_𝑡. (31)

Lemma 14 (see [19]). Let 𝑋 be a uniformly smooth Banach space or a reflexive and strictly convex Banach space with a uniformly Gáteaux differentiable norm. Let 𝐶 be a nonempty closed convex subset of 𝑋, let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix(𝑇)≠∅, and let 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 one defines 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 as follows: 𝐼−𝑓𝑄𝑓,𝐽𝑄𝑓−𝑝≤0,∀𝑓∈Ξ_𝐶,𝑝∈Fix(𝑇). (32)

Lemma 15 (see [20]). 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 ⋂^∞_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 and nonexpansive, and Fix(𝑆)=⋂^∞_𝑛=0Fix(𝑇_𝑛) holds.

Lemma 16 (see [21]). Given a number 𝑟>0. A real Banach space 𝑋 is uniformly convex if and only if there exists a continuous strictly increasing function 𝜑∶[0,∞)→[0,∞),𝜑(0)=0, such that 𝜆𝑥+(1−𝜆)𝑦²≤𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝜆(1−𝜆)𝜑𝑥−𝑦, (33) for all 𝜆∈[0,1] and 𝑥,𝑦∈𝑋 such that ‖𝑥‖≤𝑟 and ‖𝑦‖≤𝑟.

3. Mann-Type Viscosity Algorithms in Uniformly Convex and 2-Uniformly Smooth Banach Spaces

In this section, we introduce Mann-type viscosity iterative algorithms in uniformly convex and 2-uniformly smooth Banach spaces and show strong convergence theorems. We will use the following useful lemma.

Lemma 17. Let 𝐶 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝑋. Let 𝐴∶𝐶→𝑋 be an 𝛼-inverse strongly accretive mapping. Then, one has (𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦²≤𝑥−𝑦²+2𝜆𝜆𝜅²−𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶, (34) where 𝜆>0. In particular, if 0<𝜆≤𝛼/𝜅², then 𝐼−𝜆𝐴 is nonexpansive.

Theorem 18. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space and let 𝐶 be a nonempty closed convex subset of 𝑋 such that 𝐶±𝐶⊂𝐶. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝑇, 𝐹∶𝑋→𝐶𝐵(𝑋), and 𝐴∶𝐶→2^𝐶 be three multivalued mappings, let 𝑔∶𝑋→𝐶 be a single-valued mapping, and let 𝑁(⋅,⋅)∶𝑋×𝑋→𝐶 be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4. Consider that
(C6) 𝑁(𝑇𝑥,𝐹𝑥)+𝐴(𝑔(𝑥))∶𝑋→2^𝐶⧵{∅} is 𝜁-inverse strongly accretive with 𝜁≥𝜅².

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 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 Δ∶=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Γ∩(∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖))≠∅. Suppose that {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛},{𝜎_𝑛}, and {𝜖_𝑛} are the sequences in [0,1],𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and satisfy the following conditions:(i)∑^∞_𝑛=1(|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|+|𝜎_𝑛−𝜎_𝑛−1|+|𝜖_𝑛−𝜖_𝑛−1|)<∞;(ii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(iii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;(v)0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1;(vi)0<liminf_𝑛→∞𝜖_𝑛≤limsup_𝑛→∞𝜖_𝑛<1.

For arbitrary 𝑥₀∈𝐶, define the sequence {𝑥_𝑛} iteratively by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛,𝑥_𝑛+1=𝜖_𝑛𝑥_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝜖_𝑛𝑦_𝑛,𝑢_𝑛∈𝐴𝑔𝑥_𝑛,∀𝑛≥0, (35) where {𝑢_𝑛} is defined by 𝑢_𝑛−𝑢_𝑛+1≤(1+𝜀)𝐻𝐴𝑔𝑥_𝑛+1,𝐴𝑔𝑥_𝑛,∀𝑛≥0, (36) for any 𝑤_𝑛∈𝑇𝑥_𝑛, 𝑘_𝑛∈𝐹𝑥_𝑛, and some 𝜀>0. 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 {𝑥_𝑛} converges strongly to 𝑞∈Δ, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈Δ, (37) and, for any 𝑤∈𝑇𝑞 and 𝑘∈𝐹𝑞, (𝑞,𝑤,𝑘) is a solution of the MVVI (16).

Proof. First of all, by Lemma 17 we know that 𝐼−𝜆_𝑖𝐴_𝑖 is a nonexpansive mapping, where 𝜆_𝑖∈(0,𝛼_𝑖/𝜅²) for each 𝑖=1,…,𝑁. Hence, from the nonexpansivity of Π_𝐶, it follows that 𝐺_𝑖 is a nonexpansive mapping for each 𝑖=1,…,𝑁. Since 𝐵∶𝐶→𝐶 is the 𝐾-mapping generated by 𝐺₁,…,𝐺_𝑁 and 𝜌₁,…,𝜌_𝑁, by Lemma 3, we deduce that Fix(𝐵)=∩^𝑁_𝑖=1Fix(𝐺_𝑖). Utilizing Lemma 10, and the definition of 𝐺_𝑖, we get Fix(𝐺_𝑖)=VI(𝐶,𝐴_𝑖) for each 𝑖=1,…,𝑁. Thus, we have Fix(𝐵)=^𝑁_𝑖=1Fix𝐺_𝑖=^𝑁_𝑖=1VI𝐶,𝐴_𝑖. (38)
Now, let us show that for any 𝑣∈𝐶, 𝜆>0, there exists a point 𝑥∈𝐶 such that (𝑥,𝑤,𝑘) is a solution of the MVVI (15), for any 𝑤∈𝑇𝑥 and 𝑘∈𝐹𝑥. Indeed, following the argument idea in the proof of Chidume et al. [5, Theorem 3.1], we put 𝑉𝑥∶=𝑁(𝑇𝑥,𝐹𝑥) for all 𝑥∈𝑋. Then by Proposition 6, 𝑉 is 𝜙-strongly accretive. Since 𝑇 and 𝐹 are 𝐻-uniformly continuous and 𝑁(⋅,⋅) is continuous, 𝑉𝑥 is continuous and hence lower semicontinuous. Thus, by Proposition 7,  𝑉𝑥 is single-valued. Moreover, since 𝑉 is 𝜙-strongly accretive and by assumption 𝐴∘𝑔∶𝑋→2^𝐶 is 𝑚-accretive, we have that 𝑉+𝜆𝐴∘𝑔 is an 𝑚-accretive and 𝜙-strongly accretive mapping, and hence by Cioranescu [22, page 184], for any 𝑥∈𝑋, we have that (𝑉+𝜆𝐴∘𝑔)(𝑥) is closed and bounded. Therefore, by Morales [23], 𝑉+𝜆𝐴∘𝑔 is surjective. Hence, for any 𝑣∈𝑋 and 𝜆>0, there exists 𝑥∈𝐷(𝐴)=𝐶 such that 𝑣∈𝑉𝑥+𝜆𝐴(𝑔(𝑥))=𝑁(𝑤,𝑘)+𝜆𝐴(𝑔(𝑥)), where 𝑤∈𝑇𝑥 and 𝑘∈𝐹𝑥. In addition, in terms of Proposition 7, we know that 𝑉+𝜆𝐴∘𝑔 is a single-valued mapping. Assume that 𝑁(𝑇𝑥,𝐹𝑥)+𝜆𝐴(𝑔(𝑥))∶𝑋→𝐶 is 𝜁-inverse strongly accretive with 𝜁≥𝜅². Then by Lemma 17, we conclude that the mapping 𝑥↦𝑥−(𝑁(𝑇𝑥,𝐹𝑥)+𝜆𝐴(𝑔(𝑥))) is nonexpansive.
Without loss of generality, we may assume that 𝑣=0 and 𝜆=1. Let 𝑝∈Δ and let 𝑟(≥‖𝑓(𝑝)−𝑝‖/(1−𝜌)) be sufficiently large such that 𝑥₀∈𝐵_𝑟(𝑝)=∶𝐵. Then 𝑝∈𝐷(𝐴)=𝐶 such that 0∈𝑁(𝑤,𝑘)+𝐴∘𝑔(𝑝) for any 𝑤∈𝑇𝑝 and 𝑘∈𝐹𝑝. Let 𝑀∶=sup{‖𝑢‖∶𝑢∈𝑁(𝑤,𝑘)+𝐴(𝑔(𝑥)), 𝑥∈𝐵, 𝑤∈𝑇𝑥, 𝑘∈𝐹𝑥}. Then as 𝐴∘𝑔,𝑇, and 𝐹 are 𝐻-uniformly continuous on 𝑋, for 𝜀₁∶=𝜙(𝑟)/8(1+𝜀), 𝜀₂∶=𝜙(𝑟)/8𝜇(1+𝜀), and 𝜀₃∶=𝜙(𝑟)/8𝜉(1+𝜀), there exist 𝛿₁, 𝛿₂, 𝛿₃>0 such that for any 𝑥,𝑦∈𝑋,‖𝑥−𝑦‖<𝛿₁,‖𝑥−𝑦‖<𝛿₂ and ‖𝑥−𝑦‖<𝛿₃ imply 𝐻(𝐴∘𝑔(𝑥),𝐴∘𝑔(𝑦))<𝜀₁,𝐻(𝑇𝑥,𝑇𝑦)<𝜀₂ and 𝐻(𝐹𝑥,𝐹𝑦)<𝜀₃, respectively.
Let us show that 𝑥_𝑛∈𝐵 for all 𝑛≥0. We show this by induction. First, 𝑥₀∈𝐵 by construction. Assume that 𝑥_𝑛∈𝐵. We show that 𝑥_𝑛+1∈𝐵. If possible we assume that 𝑥_𝑛+1∉𝐵, then ‖𝑥_𝑛+1−𝑝‖>𝑟. Further from (35) it follows that 𝑦_𝑛=𝛼−𝑝_𝑛𝑓𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝑓𝑝−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝑥_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=−𝑝1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝1−𝜌−𝑝𝑥1−𝜌≤max_𝑛,𝑓𝑝−𝑝−𝑝,1−𝜌 (39) and hence 𝑥_𝑛+1−𝑝²=𝜖_𝑛𝑥_𝑛−𝑝−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝜖_𝑛𝑦_𝑛−𝑝,𝐽𝑥_𝑛+1=𝜖−𝑝_𝑛𝑥_𝑛−𝑝+1−𝜖_𝑛𝑦_𝑛−𝑝,𝐽𝑥_𝑛+1−𝑝−𝜖_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝐽𝑥_𝑛+1≤𝛼−𝑝_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑦_𝑛−𝑝𝑥_𝑛+1−𝑝−𝜖_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝐽𝑥_𝑛+1≤𝜖−𝑝_𝑛𝑥_𝑛−𝑝+1−𝜖_𝑛𝑦_𝑛−𝑝𝑥_𝑛+1−𝑝−𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝐽𝑥_𝑛+1≤𝜖−𝑝_𝑛𝑥_𝑛−𝑝+1−𝜖_𝑛max𝑥_𝑛,𝑓𝑝−𝑝−𝑝𝑥1−𝜌^𝑛+1−𝑝−𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝐽𝑥_𝑛+1𝑥−𝑝≤max_𝑛,𝑓𝑝−𝑝−𝑝𝑥1−𝜌^𝑛+1−𝑝−𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝐽𝑥_𝑛+1≤1−𝑝2max𝑥_𝑛−𝑝²,𝑓𝑝−𝑝1−𝜌²+𝑥_𝑛+1−𝑝²−𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝐽𝑥_𝑛+1−𝑝, (40)
which immediately yields 𝑥_𝑛+1−𝑝²≤max𝑥_𝑛−𝑝²,𝑓𝑝−𝑝1−𝜌²−2𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝐽𝑥_𝑛+1𝑥−𝑝=max_𝑛−𝑝²,𝑓𝑝−𝑝1−𝜌²−2𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛+1,𝑘_𝑛+1+𝑢_𝑛+1,𝐽𝑥_𝑛+1−𝑝−2𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛−𝑁𝑤_𝑛+1,𝑘_𝑛+1+𝑢_𝑛+1,𝐽𝑥_𝑛+1−𝑝. (41)
Since 𝑁(⋅,⋅) is 𝜙-strongly accretive with respect to 𝑇 and 𝐴(𝑔(⋅)) is accretive, we deduce from (41) that 𝑥_𝑛+1−𝑝²≤max𝑥_𝑛−𝑝²,𝑓𝑝−𝑝1−𝜌²−2𝛼_𝑛𝜎_𝑛𝜙𝑥_𝑛+1−𝑝𝑥_𝑛+1−𝑝+2𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛+1,𝑘_𝑛+1−𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝑢_𝑛𝑥_𝑛+1𝑥−𝑝≤max_𝑛−𝑝²,𝑓𝑝−𝑝1−𝜌²−2𝛼_𝑛𝜎_𝑛𝜙𝑥_𝑛+1−𝑝𝑥_𝑛+1−𝑝+2𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛+1,𝑘_𝑛+1−𝑁𝑤_𝑛+1,𝑘_𝑛+𝑁𝑤_𝑛+1,𝑘_𝑛−𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝑢_𝑛𝑥_𝑛+1−𝑝. (42)
Again from (35), we have that 𝑥_𝑛+1−𝑝≤𝜖_𝑛𝑥_𝑛−𝑝−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝜖_𝑛𝑦_𝑛−𝑝≤𝜖_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝜖_𝑛max𝑥_𝑛,𝑓𝑝−𝑝−𝑝1−𝜌≤𝜖_𝑛𝑟+𝜎_𝑛𝑀+1−𝜖_𝑛𝑟≤2𝑟. (43)
Also, from Proposition 7, 𝑉𝑥=𝑁(𝑇𝑥,𝐹𝑥) is a single-valued mapping; that is, for any 𝑘,𝑘^′∈𝐹𝑥 and 𝑤,𝑤^′∈𝑇𝑥, we have 𝑁(𝑤,𝑘)=𝑁(𝑤,𝑘^′) and 𝑁(𝑤,𝑘)=𝑁(𝑤^′,𝑘). On the other hand, it follows from Nadler [24] that, for 𝑘_𝑛+1∈𝐹𝑥_𝑛+1 and 𝑤_𝑛+1∈𝑇𝑥_𝑛+1, there exist 𝑘^′_𝑛∈𝐹𝑥_𝑛 and 𝑤^′_𝑛∈𝑇𝑥_𝑛 such that 𝑘_𝑛+1−𝑘^′_𝑛≤(1+𝜀)𝐻𝐹𝑥_𝑛+1,𝐹𝑥_𝑛𝑤_𝑛+1,−𝑤^′_𝑛≤(1+𝜀)𝐻𝑇𝑥_𝑛+1,𝑇𝑥_𝑛, (44)(45) respectively. Therefore, from (42) and (36), we have 𝑥_𝑛+1−𝑝²≤max𝑥_𝑛−𝑝²,𝑓𝑝−𝑝1−𝜌²−2𝛼_𝑛𝜎_𝑛𝜙(𝑟)𝑟+2𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛+1,𝑘_𝑛+1−𝑁𝑤_𝑛+1,𝑘^′_𝑛+𝑁𝑤_𝑛+1,𝑘_𝑛−𝑁𝑤^′_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝑢_𝑛𝑥2𝑟≤max_𝑛−𝑝²,𝑓𝑝−𝑝1−𝜌²−2𝛼_𝑛𝜎_𝑛𝜙(𝑟)𝑟+2𝛼_𝑛𝜎_𝑛𝜉(1+𝜀)𝐻𝐹𝑥_𝑛+1,𝐹𝑥_𝑛+𝜇(1+𝜀)𝐻𝑇𝑥_𝑛+1,𝑇𝑥_𝑛+(1+𝜀)𝐻𝐴𝑔𝑥_𝑛+1,𝐴𝑔𝑥_𝑛𝑥2𝑟≤max_𝑛−𝑝²,𝑓𝑝−𝑝1−𝜌²−2𝛼_𝑛𝜎_𝑛𝜙(𝑟)𝑟+2𝛼_𝑛𝜎_𝑛𝜙(𝑟)8+𝜙(𝑟)8+𝜙(𝑟)8𝑥2𝑟=max_𝑛−𝑝²,𝑓𝑝−𝑝1−𝜌²−2𝛼_𝑛𝜎_𝑛𝜙(𝑟)𝑟+𝛼_𝑛𝜎_𝑛32𝜙(𝑟)𝑟≤max𝑥_𝑛−𝑝²,𝑓𝑝−𝑝1−𝜌². (46)
So, we get ‖𝑥_𝑛+1−𝑝‖≤𝑟, a contradiction. Therefore, {𝑥_𝑛} is bounded.
Let us show that lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.
Indeed, we define 𝐺∶𝐶→𝐶 by 𝐺𝑥∶=𝑥−(𝑁(𝑇𝑥,𝐹𝑥)+𝐴(𝑔(𝑥))) for all 𝑥∈𝐶. Then, 𝐺 is a nonexpansive mapping and the iterative scheme (35) can be rewritten as follows: 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛,𝑥_𝑛+1=𝜖_𝑛1−𝜎_𝑛𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛+1−𝜖_𝑛𝑦_𝑛,∀𝑛≥0. (47)
Taking into account condition (iv), we may assume that {𝛽_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1). From (47), we can rewrite 𝑦_𝑛 by 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛, (48) where 𝑧_𝑛=(𝛼_𝑛𝑓(𝑥_𝑛)+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛)/(1−𝛽_𝑛). Now, we have 𝑧_𝑛+1−𝑧_𝑛=𝛼_𝑛+1𝑓𝑥_𝑛+1+𝛾_𝑛+1𝐵𝑥_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝑥_𝑛+11−𝛽_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_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+1−𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛+1+𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−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+1+𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛+1−𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛=11−𝛽_𝑛+1𝑦_𝑛+1−𝛽_𝑛+1𝑥_𝑛+1−𝑦_𝑛−𝛽_𝑛𝑥_𝑛+11−𝛽_𝑛+1−11−𝛽_𝑛𝑦_𝑛−𝛽_𝑛𝑥_𝑛=11−𝛽_𝑛+1𝑦_𝑛+1−𝛽_𝑛+1𝑥_𝑛+1−𝑦_𝑛−𝛽_𝑛𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛1−𝛽_𝑛1−𝛽_𝑛+1𝑦_𝑛−𝛽_𝑛𝑥_𝑛=11−𝛽_𝑛+1𝛼_𝑛+1𝑓𝑥_𝑛+1+𝛾_𝑛+1𝐵𝑥_𝑛+1+𝛿_𝑛+1𝑆_𝑛+1𝑥_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛1−𝛽_𝑛1−𝛽_𝑛+1𝑦_𝑛−𝛽_𝑛𝑥_𝑛≤11−𝛽_𝑛+1𝛼_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝛾_𝑛+1𝐵𝑥_𝑛+1−𝐵𝑥_𝑛+𝛿_𝑛+1𝑆_𝑛+1𝑥_𝑛+1−𝑆_𝑛𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛+1−𝛿_𝑛𝑆_𝑛𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛1−𝛽_𝑛1−𝛽_𝑛+1𝑦_𝑛−𝛽_𝑛𝑥_𝑛≤11−𝛽_𝑛+1𝛼_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝛾_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛿_𝑛+1𝑆_𝑛+1𝑥_𝑛+1−𝑆_𝑛+1𝑥_𝑛+𝑆_𝑛+1𝑥_𝑛−𝑆_𝑛𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛×𝐵𝑥_𝑛+𝛿_𝑛+1−𝛿_𝑛𝑆_𝑛𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛1−𝛽_𝑛1−𝛽_𝑛+1𝑦_𝑛−𝛽_𝑛𝑥_𝑛≤11−𝛽_𝑛+1𝛼_𝑛+1𝜌𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛿_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝑆_𝑛+1𝑥_𝑛−𝑆_𝑛𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛+1−𝛿_𝑛𝑆_𝑛𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛1−𝛽_𝑛1−𝛽_𝑛+1𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛=1−𝛽_𝑛+1−𝛼_𝑛+11−𝜌1−𝛽_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛿_𝑛+11−𝛽_𝑛+1𝑆_𝑛+1𝑥_𝑛−𝑆_𝑛𝑥_𝑛+11−𝛽_𝑛+1𝛼_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛+1−𝛾_𝑛×𝐵𝑥_𝑛+𝛿_𝑛+1−𝛿_𝑛𝑆_𝑛𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛1−𝛽_𝑛1−𝛽_𝑛+1𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛≤𝛼1−^𝑛+11−𝜌1−𝛽_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝑆_𝑛+1𝑥_𝑛−𝑆_𝑛𝑥_𝑛+𝑀₀𝛼_𝑛+1−𝛼_𝑛+𝛽_𝑛+1−𝛽_𝑛+𝛾_𝑛+1−𝛾_𝑛+𝛿_𝑛+1−𝛿_𝑛, (49) where 1/(1−𝑏)²sup_𝑛≥0{‖𝑓(𝑥_𝑛)‖+‖𝐵𝑥_𝑛‖+‖𝑆_𝑛𝑥_𝑛‖}≤𝑀₀ for some 𝑀₀>0. By simple calculation, we have 𝑦_𝑛−𝑦_𝑛−1=𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1×𝑥_𝑛−1−𝑧_𝑛−1+1−𝛽_𝑛𝑧_𝑛−𝑧_𝑛−1. (50)
So, from (49), we get 𝑦_𝑛−𝑦_𝑛−1≤𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1−𝑧_𝑛−1+1−𝛽_𝑛𝑧_𝑛−𝑧_𝑛−1≤𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1−𝑧_𝑛−1+1−𝛽_𝑛𝛼1−^𝑛1−𝜌1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑆_𝑛𝑥_𝑛−1−𝑆_𝑛−1𝑥_𝑛−1+𝑀₀𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1≤1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1−𝑧_𝑛−1+𝑆_𝑛𝑥_𝑛−1−𝑆_𝑛−1𝑥_𝑛−1+𝑀₀𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1. (51)
Also, for convenience, we write 𝑥_𝑛+1=𝜖_𝑛𝑧_𝑛+1−𝜖_𝑛𝑦_𝑛,𝑧_𝑛=𝜎_𝑛𝐺𝑥_𝑛+1−𝜎_𝑛𝑥_𝑛. (52)
By simple calculation, we get 𝑥_𝑛+1−𝑥_𝑛=𝜖_𝑛𝑧_𝑛−𝑧_𝑛−1+𝜖_𝑛−𝜖_𝑛−1×𝑧_𝑛−1−𝑦_𝑛−1+1−𝜖_𝑛𝑦_𝑛−𝑦_𝑛−1,𝑧_𝑛−𝑧_𝑛−1=𝜎_𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1+𝜎_𝑛−𝜎_𝑛−1×𝐺𝑥_𝑛−1−𝑥_𝑛−1+1−𝜎_𝑛𝑥_𝑛−𝑥_𝑛−1. (53)
From (51) and (53), we deduce that 𝑧_𝑛−𝑧_𝑛−1≤𝜎_𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1+𝜎_𝑛−𝜎_𝑛−1×𝐺𝑥_𝑛−1−𝑥_𝑛−1+1−𝜎_𝑛𝑥_𝑛−𝑥_𝑛−1≤𝜎_𝑛𝑥_𝑛−𝑥_𝑛−1+𝜎_𝑛−𝜎_𝑛−1×𝐺𝑥_𝑛−1−𝑥_𝑛−1+1−𝜎_𝑛𝑥_𝑛−𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝐺𝑥_𝑛−1−𝑥_𝑛−1, (54) and hence 𝑥_𝑛+1−𝑥_𝑛≤𝜖_𝑛𝑧_𝑛−𝑧_𝑛−1+𝜖_𝑛−𝜖_𝑛−1×𝑧_𝑛−1−𝑦_𝑛−1+1−𝜖_𝑛𝑦_𝑛−𝑦_𝑛−1≤𝜖_𝑛𝑥_𝑛−𝑥_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝐺𝑥_𝑛−1−𝑥_𝑛−1+𝜖_𝑛−𝜖_𝑛−1𝑧_𝑛−1−𝑦_𝑛−1+1−𝜖_𝑛1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1−𝑧_𝑛−1+𝑆_𝑛𝑥_𝑛−1−𝑆_𝑛−1𝑥_𝑛−1+𝑀₀𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1≤1−1−𝜖_𝑛𝛼_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝐺𝑥_𝑛−1−𝑥_𝑛−1+𝜖_𝑛−𝜖_𝑛−1𝑧_𝑛−1−𝑦_𝑛−1+𝛽_𝑛−𝛽_𝑛−1×𝑥_𝑛−1−𝑧_𝑛−1+𝑆_𝑛𝑥_𝑛−1−𝑆_𝑛−1𝑥_𝑛−1+𝑀₀𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1≤1−1−𝜖_𝑛𝛼_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝑆_𝑛𝑥_𝑛−1−𝑆_𝑛−1𝑥_𝑛−1+𝑀₁𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝜎_𝑛−𝜎_𝑛−1+𝜖_𝑛−𝜖_𝑛−1, (55) where sup_𝑛≥1{‖𝐺𝑥_𝑛−1−𝑥_𝑛−1‖+‖𝑧_𝑛−1−𝑦_𝑛−1‖+‖𝑥_𝑛−1−𝑧_𝑛−1‖+𝑀₀}≤𝑀₁ for some 𝑀₁>0. Utilizing Lemma 17, we conclude from (55), conditions (i), (ii), and (vi), and the assumption on {𝑆_𝑛} that lim_𝑛→∞𝑥_𝑛−𝑥_𝑛+1=0. (56)
Furthermore, utilizing Lemma 16, we obtain from (39) and (47) that 𝑥_𝑛+1−𝑝²=𝜖_𝑛1−𝜎_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝐺𝑥_𝑛+−𝑝1−𝜖_𝑛𝑦_𝑛−𝑝²≤𝜖_𝑛1−𝜎_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝐺𝑥_𝑛−𝑝²+1−𝜖_𝑛𝑦_𝑛−𝑝²−𝜖_𝑛1−𝜖_𝑛×𝜑1−𝜎_𝑛𝑥_𝑛−𝑦_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛≤𝜖_𝑛1−𝜎_𝑛𝑥_𝑛−𝑝²+𝜎_𝑛𝐺𝑥_𝑛−𝑝²−𝜎_𝑛1−𝜎_𝑛𝜑₁𝑥_𝑛−𝐺𝑥_𝑛+1−𝜖_𝑛𝑦_𝑛−𝑝²−𝜖_𝑛1−𝜖_𝑛×𝜑1−𝜎_𝑛𝑥_𝑛−𝑦_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛≤𝜖_𝑛1−𝜎_𝑛𝑥_𝑛−𝑝²+𝜎_𝑛𝑥_𝑛−𝑝²−𝜎_𝑛1−𝜎_𝑛𝜑₁𝑥_𝑛−𝐺𝑥_𝑛+1−𝜖_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝²−𝜖_𝑛1−𝜖_𝑛𝜑1−𝜎_𝑛𝑥_𝑛−𝑦_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛=𝜖_𝑛𝑥_𝑛−𝑝²−𝜎_𝑛1−𝜎_𝑛𝜑₁𝑥_𝑛−𝐺𝑥_𝑛+1−𝜖_𝑛𝑥_𝑛−𝑝²+𝛼_𝑛𝑓𝑝×2𝑥−𝑝_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝−𝜖_𝑛1−𝜖_𝑛𝜑1−𝜎_𝑛𝑥_𝑛−𝑦_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑝²−𝜖_𝑛𝜎_𝑛1−𝜎_𝑛𝜑₁𝑥_𝑛−𝐺𝑥_𝑛+𝛼_𝑛𝑓𝑝−𝑝2𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝−𝜖_𝑛1−𝜖_𝑛𝜑1−𝜎_𝑛𝑥_𝑛−𝑦_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛, (57) which immediately yields 𝜖_𝑛𝜎_𝑛1−𝜎_𝑛𝜑₁𝑥_𝑛−𝐺𝑥_𝑛+𝜖_𝑛1−𝜖_𝑛×𝜑1−𝜎_𝑛𝑥_𝑛−𝑦_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝛼_𝑛𝑓𝑝×2𝑥−𝑝_𝑛−𝑝+𝛼_𝑛𝑓𝑝≤𝑥−𝑝_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑓𝑝−𝑝2𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝. (58) So, from (56) and conditions (ii), (v), and (vi), we get lim_𝑛→∞𝜑₁𝑥_𝑛−𝐺𝑥_𝑛=0,lim_𝑛→∞𝜑1−𝜎_𝑛𝑥_𝑛−𝑦_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛=0, (59) which together with the properties of 𝜑 and 𝜑₁ implies that lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0,lim_𝑛→∞1−𝜎_𝑛𝑥_𝑛−𝑦_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛=0. (60)
Note that 𝑥_𝑛−𝑦_𝑛=1−𝜎_𝑛𝑥_𝑛−𝑦_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛+𝜎_𝑛𝑥_𝑛−𝐺𝑥_𝑛≤1−𝜎_𝑛𝑥_𝑛−𝑦_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛+𝜎_𝑛𝑥_𝑛−𝐺𝑥_𝑛≤1−𝜎_𝑛𝑥_𝑛−𝑦_𝑛+𝜎_𝑛𝐺𝑥_𝑛−𝑦_𝑛+𝑥_𝑛−𝐺𝑥_𝑛. (61)
Hence, from (60), it follows that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (62)
Let us show that lim_𝑛→∞‖𝑥_𝑛−𝐵𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑆𝑥_𝑛‖=0.
Indeed, from the definition of 𝑦_𝑛, we can rewrite 𝑦_𝑛 by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛𝛾_𝑛+𝛿_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝑒_𝑛𝑧^′_𝑛, (63) where 𝑒_𝑛=𝛾_𝑛+𝛿_𝑛 and 𝑧^′_𝑛=(𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛)/(𝛾_𝑛+𝛿_𝑛).
Utilizing Lemma 12, from (63) 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_𝑛𝜑₂𝑧^′_𝑛−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝜑₂𝑧^′_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛𝛾_𝑛+𝛿_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝜑₂𝑧^′_𝑛−𝑥_𝑛+𝑒_𝑛𝛿1−^𝑛𝛾_𝑛+𝛿_𝑛𝐵𝑥_𝑛+𝛿−𝑝^𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝜑₂𝑧^′_𝑛−𝑥_𝑛+𝑒_𝑛𝛿1−^𝑛𝛾_𝑛+𝛿_𝑛𝐵𝑥_𝑛+𝛿−𝑝_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝜑₂𝑧^′_𝑛−𝑥_𝑛+𝑒_𝑛𝑥_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝜑₂𝑧^′_𝑛−𝑥_𝑛, (64) which implies that 𝛽_𝑛𝑒_𝑛𝜑₂𝑧^′_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. (65) From (62) and conditions (ii), (iii), and (iv), we have lim_𝑛→∞𝜑₂𝑧^′_𝑛−𝑥_𝑛=0. (66) From the properties of 𝜑₂, we have lim_𝑛→∞𝑧^′_𝑛−𝑥_𝑛=0. (67) By Lemma 16, we deduce from the definition of 𝑧^′_𝑛 the following 𝑧^′_𝑛−𝑝²=𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛𝛾_𝑛+𝛿_𝑛−𝑝²=𝛿1−^𝑛𝛿_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿−𝑝^𝑛𝛿_𝑛+𝛾_𝑛𝑆_𝑛𝑥_𝑛−𝑝²≤𝛿1−^𝑛𝛿_𝑛+𝛾_𝑛𝐵𝑥_𝑛−𝑝²+𝛿_𝑛𝛿_𝑛+𝛾_𝑛𝑆_𝑛𝑥_𝑛−𝑝²−𝛿_𝑛𝛿_𝑛+𝛾_𝑛𝛿1−^𝑛𝛿_𝑛+𝛾_𝑛𝜑₃𝐵𝑥_𝑛−𝑆_𝑛𝑥_𝑛≤𝑥_𝑛−𝑝²−𝛿_𝑛𝛿_𝑛+𝛾_𝑛𝛿1−^𝑛𝛿_𝑛+𝛾_𝑛𝜑₃𝐵𝑥_𝑛−𝑆_𝑛𝑥_𝑛, (68) which implies that 𝛿_𝑛𝛿_𝑛+𝛾_𝑛𝛿1−^𝑛𝛿_𝑛+𝛾_𝑛𝜑₃𝐵𝑥_𝑛−𝑆_𝑛𝑥_𝑛≤𝑥_𝑛−𝑝²−𝑧^′_𝑛−𝑝²≤𝑥_𝑛−𝑝+𝑧^′_𝑛−𝑝𝑥_𝑛−𝑧^′_𝑛. (69) From (67) and condition (iii), we have lim_𝑛→∞𝜑₃𝐵𝑥_𝑛−𝑆_𝑛𝑥_𝑛=0. (70) From the properties of 𝜑₃, we have lim_𝑛→∞𝐵𝑥_𝑛−𝑆_𝑛𝑥_𝑛=0. (71) From the definition of 𝑦_𝑛, we can rewrite 𝑦_𝑛 by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛼_𝑛+𝛿_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛𝛼_𝑛+𝛿_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝑑_𝑛𝑧_𝑛^′′, (72) where 𝑑_𝑛=𝛼_𝑛+𝛿_𝑛 and 𝑧_𝑛^′′=(𝛼_𝑛𝑓(𝑥_𝑛)+𝛿_𝑛𝑆_𝑛𝑥_𝑛)/(𝛼_𝑛+𝛿_𝑛).
Utilizing Lemma 12, from (72) and the convexity of ‖⋅‖², we have 𝑦_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵𝑥_𝑛−𝑝+𝑑_𝑛𝑧_𝑛^′′−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵𝑥_𝑛−𝑝²+𝑑_𝑛𝑧_𝑛^′′−𝑝²−𝛽_𝑛𝛾_𝑛𝜑₄𝑥_𝑛−𝐵𝑥_𝑛=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛𝛼_𝑛+𝛿_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝜑₄𝑥_𝑛−𝐵𝑥_𝑛=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝛼_𝑛+𝛿_𝑛𝑓𝑥_𝑛+𝛼−𝑝1−^𝑛𝛼_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝜑₄𝑥_𝑛−𝐵𝑥_𝑛≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝛼_𝑛+𝛿_𝑛𝑓𝑥_𝑛−𝑝²+𝛼1−^𝑛𝛼_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝜑₄𝑥_𝑛−𝐵𝑥_𝑛=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐵𝑥_𝑛−𝑝²+𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝜑₄𝑥_𝑛−𝐵𝑥_𝑛≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛿_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝜑₄𝑥_𝑛−𝐵𝑥_𝑛≤𝑥_𝑛−𝑝²+𝛼_𝑛𝑓𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝜑₄𝑥_𝑛−𝐵𝑥_𝑛, (73) which implies that 𝛽_𝑛𝛾_𝑛𝜑₄𝑥_𝑛−𝐵𝑥_𝑛≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²+𝛼_𝑛𝑓𝑥_𝑛−𝑝²≤𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛+𝛼_𝑛𝑓𝑥_𝑛−𝑝². (74) From (62), (74), and conditions (ii), (iii), and (iv), we have lim_𝑛→∞𝜑₄𝑥_𝑛−𝐵𝑥_𝑛=0. (75)
By the properties of 𝜑₄, we have lim_𝑛→∞𝑥_𝑛−𝐵𝑥_𝑛=0. (76)
From (71), (76), and 𝑥_𝑛−𝑆_𝑛𝑥_𝑛≤𝑥_𝑛−𝐵𝑥_𝑛+𝐵𝑥_𝑛−𝑆_𝑛𝑥_𝑛, (77) we have lim_𝑛→∞𝑥_𝑛−𝑆_𝑛𝑥_𝑛=0. (78)
Observe that 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆_𝑛𝑥_𝑛+𝑆_𝑛𝑥_𝑛−𝑆𝑥_𝑛. (79) Utilizing Lemma 13, we conclude from (78) that lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (80)
Define a mapping 𝑊𝑥=(1−𝜃₁−𝜃₂)𝐵𝑥+𝜃₁𝑆𝑥+𝜃₂𝐺𝑥, where 𝜃₁,𝜃₂∈(0,1) are two constants with 𝜃₁+𝜃₂<1. Then by Lemma 15, we have Fix(𝑊)=Fix(𝐵)∩Fix(𝑆)∩Fix(𝐺)=Δ. We observe that 𝑥_𝑛−𝑊𝑥_𝑛=1−𝜃₁−𝜃₂𝑥_𝑛−𝐵𝑥_𝑛+𝜃₁𝑥_𝑛−𝑆𝑥_𝑛+𝜃₂𝑥_𝑛−𝐺𝑥_𝑛≤1−𝜃₁−𝜃₂𝑥_𝑛−𝐵𝑥_𝑛+𝜃₁𝑥_𝑛−𝑆𝑥_𝑛+𝜃₂𝑥_𝑛−𝐺𝑥_𝑛. (81) From (60), (76), and (80), we obtain lim_𝑛→∞𝑥_𝑛−𝑊𝑥_𝑛=0. (82)
Now, we claim that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0, (83) where 𝑞=𝑠−lim_𝑡→0𝑥_𝑡 with 𝑥_𝑡 being the fixed point of the contraction 𝑥⟼𝑡𝑓(𝑥)+(1−𝑡)𝑊𝑥. (84) Then 𝑥_𝑡 solves the fixed point equation 𝑥_𝑡=𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑊𝑥_𝑡. Thus we have 𝑥_𝑡−𝑥_𝑛=(1−𝑡)𝑊𝑥_𝑡−𝑥_𝑛+𝑡𝑓𝑥_𝑡−𝑥_𝑛. (85) By Lemma 9, we conclude that 𝑥_𝑡−𝑥_𝑛²=(1−𝑡)𝑊𝑥_𝑡−𝑥_𝑛+𝑡𝑓𝑥_𝑡−𝑥_𝑛²≤(1−𝑡)²𝑊𝑥_𝑡−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤(1−𝑡)²𝑊𝑥_𝑡−𝑊𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤(1−𝑡)²𝑥_𝑡−𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛=(1−𝑡)²𝑥_𝑡−𝑥_𝑛²+2𝑥_𝑡−𝑥_𝑛×𝑊𝑥_𝑛−𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛+2𝑡𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛=1−2𝑡+𝑡²𝑥_𝑡−𝑥_𝑛²+𝑓_𝑛(𝑡)+2𝑡𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛+2𝑡𝑥_𝑡−𝑥_𝑛², (86) where 𝑓_𝑛(𝑡)=(1−𝑡)²2𝑥_𝑡−𝑥_𝑛+𝑥_𝑛−𝑊𝑥_𝑛×𝑥_𝑛−𝑊𝑥_𝑛⟶0,as𝑛⟶∞. (87) It follows from (86) that 𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤𝑡2𝑥_𝑡−𝑥_𝑛²+1𝑓2𝑡^𝑛(𝑡). (88)
Letting 𝑛→∞ in (88) and noticing (87), we derive limsup_𝑛→∞𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤𝑡2𝑀₂, (89) where 𝑀₂>0 is a constant such that ‖𝑥_𝑡−𝑥_𝑛‖²≤𝑀₂ for all 𝑡∈(0,1) and 𝑛≥0. Taking 𝑡→0 in (89), we have limsup_𝑡→0limsup_𝑛→∞𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤0. (90)
On the other hand, we have 𝑓𝑞−𝑞,𝐽𝑥_𝑛=𝑓𝑞𝑥−𝑞−𝑞,𝐽_𝑛−𝑞−𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑥_𝑡−𝑓𝑞−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑞−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡−𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡=𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞−𝐽𝑥_𝑛−𝑥_𝑡+𝑥_𝑡−𝑞,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑞−𝑓𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡. (91)
It follows that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞−𝐽𝑥_𝑛−𝑥_𝑡+𝑥_𝑡−𝑞limsup_𝑛→∞𝑥_𝑛−𝑥_𝑡+𝜌𝑞−𝑥_𝑡limsup_𝑛→∞𝑥_𝑛−𝑥_𝑡+limsup_𝑛→∞𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡. (92)
Taking into account that 𝑥_𝑡→𝑞 as 𝑡→0, we have limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞=limsup_𝑡→0limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤limsup_𝑡→0limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞−𝐽𝑥_𝑛−𝑥_𝑡. (93)
Since 𝑋 has a uniformly Fréchet differentiable norm, the duality mapping 𝐽 is norm-to-norm uniformly continuous on bounded subsets of 𝑋. Consequently, the two limits are interchangeable and hence (83) holds. Noticing that 𝐽 is norm-to-norm uniformly continuous on bounded subsets of 𝑋, we deduce from (62) that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞=limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛+𝑓𝑞𝑦−𝑞−𝑞,𝐽_𝑛−𝑞−𝐽𝑥_𝑛−𝑞=limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0. (94)
Finally, let us show that 𝑥_𝑛→𝑞 as 𝑛→∞. Indeed, utilizing Lemma 9, we obtain from (47) that 𝑦_𝑛−𝑞²=𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝐵𝑥_𝑛−𝑞+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑞+𝛼_𝑛𝑓𝑞−𝑞²≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝐵𝑥_𝑛−𝑞+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝐵𝑥_𝑛−𝑞²+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞≤𝛼_𝑛𝜌𝑥_𝑛−𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝑥_𝑛−𝑞+𝛿_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛=−𝑞1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞, (95) and hence 𝑥_𝑛+1𝑞²=𝜖_𝑛1−𝜎_𝑛𝑥_𝑛−𝑞+𝜎_𝑛𝐺𝑥_𝑛+−𝑞1−𝜖_𝑛𝑦_𝑛−𝑞²≤𝜖_𝑛1−𝜎_𝑛𝑥_𝑛−𝑞+𝜎_𝑛𝐺𝑥_𝑛−𝑞²+1−𝜖_𝑛𝑦_𝑛−𝑞²≤𝜖_𝑛1−𝜎_𝑛𝑥_𝑛−𝑞²+𝜎_𝑛𝐺𝑥_𝑛−𝑞²+1−𝜖_𝑛𝑦_𝑛−𝑞²≤𝜖_𝑛1−𝜎_𝑛𝑥_𝑛−𝑞²+𝜎_𝑛𝑥_𝑛−𝑞²+1−𝜖_𝑛𝑦_𝑛−𝑞²=𝜖_𝑛𝑥_𝑛−𝑞²+1−𝜖_𝑛𝑦_𝑛−𝑞²≤𝜖_𝑛𝑥_𝑛−𝑞²+1−𝜖_𝑛×1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛=−𝑞1−1−𝜖_𝑛𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+21−𝜖_𝑛𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛=−𝑞1−1−𝜖_𝑛𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+1−𝜖_𝑛𝛼_𝑛2𝑓𝑞𝑦1−𝜌−𝑞,𝐽_𝑛−𝑞.1−𝜌 (96)
Applying Lemma 8 to (96), we conclude from conditions (ii) and (vi) and (94) that 𝑥_𝑛→𝑞 as 𝑛→∞. This completes the proof.

Corollary 19. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space and let 𝐶 be a nonempty closed convex subset of 𝑋 such that 𝐶±𝐶⊂𝐶. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝑇,𝐹∶𝑋→𝐶𝐵(𝑋), and 𝐴∶𝐶→2^𝐶 be three multivalued mappings, let 𝑔∶𝑋→𝐶 be a single-valued mapping, and let 𝑁(⋅,⋅)∶𝑋×𝑋→𝐶 be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4 and (C6) 𝑁(𝑇𝑥,𝐹𝑥)+𝐴(𝑔(𝑥))∶𝑋→2^𝐶⧵{∅} is 𝜁-inverse strongly accretive with 𝜁≥𝜅².

Let 𝑇_𝑖∶𝐶→𝑋 be a 𝜂_𝑖-strictly pseudocontractive mapping for each 𝑖=1,…,𝑁. Define the mapping 𝐺_𝑖∶𝐶→𝐶 by 𝐺_𝑖=Π_𝐶(𝐼−𝜆_𝑖(𝐼−𝑇_𝑖)) for 𝑖=1,…,𝑁, where 𝜆_𝑖∈(0,𝜂_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 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 Δ∶=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Γ∩(∩^𝑁_𝑖=1Fix(𝑇_𝑖))≠∅. Suppose that {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛},{𝜎_𝑛}, and {𝜖_𝑛} are the sequences in [0,1],𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and satisfy the following conditions:(i)∑^∞_𝑛=1(|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|+|𝜎_𝑛−𝜎_𝑛−1|+|𝜖_𝑛−𝜖_𝑛−1|)<∞;(ii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(iii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;(v)0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1;(vi)0<liminf_𝑛→∞𝜖_𝑛≤limsup_𝑛→∞𝜖_𝑛<1.

For arbitrary 𝑥₀∈𝐶, define the sequence {𝑥_𝑛} iteratively by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛,𝑥_𝑛+1=𝜖_𝑛𝑥_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝜖_𝑛𝑦_𝑛,𝑢_𝑛∈𝐴𝑔𝑥_𝑛,∀𝑛≥0, (97) where {𝑢_𝑛} is defined by 𝑢_𝑛−𝑢_𝑛+1≤(1+𝜀)𝐻𝐴𝑔𝑥_𝑛+1,𝐴𝑔𝑥_𝑛,∀𝑛≥0, (98) for any 𝑤_𝑛∈𝑇𝑥_𝑛, 𝑘_𝑛∈𝐹𝑥_𝑛 and some 𝜀>0. 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 {𝑥_𝑛} converges strongly to 𝑞∈Δ, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈Δ, (99) and, for any 𝑤∈𝑇𝑞 and 𝑘∈𝐹𝑞,(𝑞,𝑤,𝑘) is a solution of the MVVI (16).

Proof. Since 𝑇_𝑖 is a 𝜂_𝑖-strictly pseudocontractive mapping for each 𝑖=1,…,𝑁, it is known that 𝐴_𝑖∶=𝐼−𝑇_𝑖 is 𝜂_𝑖-inverse strongly accretive for each 𝑖=1,…,𝑁. In Theorem 18, we put 𝐺_𝑖=Π_𝐶(𝐼−𝜆_i𝐴_𝑖) for 𝑖=1,…,𝑁, where 𝜆_𝑖∈(0,𝜂_𝑖/𝜅²). It is not hard to see that Fix(𝑇_𝑖)=VI(𝐶,𝐴_𝑖). As a matter of fact, we have, for 𝜆_𝑖>0, 𝑢∈VI𝐶,𝐴_𝑖⟺𝐴_𝑖⟺𝑢,𝐽𝑦−𝑢≥0∀𝑦∈𝐶𝑢−𝜆_𝑖𝐴_𝑖𝑢−𝑢,𝐽𝑢−𝑦≥0∀𝑦∈𝐶⟺𝑢=Π_𝐶𝑢−𝜆_𝑖𝐴_𝑖𝑢⟺𝑢=Π_𝐶𝑢−𝜆_𝑖𝑢+𝜆_𝑖𝑇_𝑖𝑢⟺𝑢−𝜆_𝑖𝑢+𝜆_𝑖𝑇_𝑖⟺𝑢−𝑢,𝐽𝑢−𝑦≥0∀𝑦∈𝐶𝑢−𝑇_𝑖𝑢,𝐽𝑢−𝑦≤0∀𝑦∈𝐶⟺𝑢=𝑇_𝑖𝑢⟺𝑢∈Fix𝑇_𝑖. (100) Accordingly, we conclude that Δ∶=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Γ∩(∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖))=  ⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Γ∩(∩^𝑁_𝑖=1Fix(𝑇_𝑖)). Therefore, the desired result follows from Theorem 18.

Remark 20. Theorem 18 improves, extends, supplements, and develops [5, Theorem 3.2] and [25, Theorem 3.1] in the following aspects.

(i) Kangtunyakarn's problem of finding a point of Fix(𝑆)∩Fix(𝑉)∩(∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖)) (see [25, Theorem 1.1]) is extended to develop our problem of finding a point of ⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Γ∩(∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖)) in Theorem 18 because 𝐵_𝐴∶=𝑆((1−𝛼)𝐼+𝛼𝑉) is nonexpansive with 𝛼∈(0,𝜂/𝜅²) and Fix(𝐵_𝐴)=Fix(𝑆)∩Fix(𝑉) (see [25, Lemma2.12]). It is clear that the problem of finding a point of ⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Γ∩(∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖)) in Theorem 18 is more general and more subtle than the problem of finding a point of Γ in [5, Theorem 3.2].

(ii) The iterative scheme in [25, Theorem 3.1] is extended to develop the iterative scheme (35) of Theorem 18 by virtue of the iterative schemes of [5, Theorem 3.2]. The iterative scheme (35) of Theorem 18 is more advantageous and more flexible than the iterative scheme of [10, Theorem 3.2] because it can be applied to solving three problems (i.e., MVVI (16), a finite family of VIPs, and the fixed point problem of {𝑆_𝑛}) and involves several parameter sequences {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, {𝛿_𝑛}, {𝜎_𝑛}, and {𝜖_𝑛}.

(iii) Theorem 18 extends and generalizes [5, Theorems 3.2] to the setting of a countable family of nonexpansive mappings and a finite family of VIPs. In the meantime, Theorem 18 extends and generalizes Kangtunyakarn [25, Theorem 3.1] to the setting of the MVVI (16).

(iv) The iterative scheme (35) in Theorem 18 is very different from every one in [5, Theorem 3.2] and [25, Theorem 3.1] because every iterative scheme in [25, Theorem 3.1] and [5, Theorem 3.2] is one-step iterative scheme and the iterative scheme (35) in Theorem 18 is the combination of two iterative schemes in [25, Theorem 3.1] and [5, Theorem 3.2].

(v) No boundedness condition on the ranges 𝑅(𝐼−𝑁(𝑇(⋅),𝐹(⋅))) and 𝑅(𝐴(𝑔(⋅))) is imposed in Theorems 18.

4. Mann-Type Viscosity Algorithms in a Uniformly Convex Banach Space Having a Uniformly Gáteaux Differentiable Norm

In this section, we introduce Mann-type viscosity iterative algorithms in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm and show strong convergence theorems. First, we give the following useful lemma.

Lemma 21. Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝑋 and let 𝐴∶𝐶→𝑋 be a 𝜉-strictly pseudocontractive and 𝜈-strongly accretive mapping with 𝜉+𝜈≥1. Then, for 𝜆∈(0,1], one has )()𝑦≤(𝐼−𝜆𝐴𝑥−𝐼−𝜆𝐴1−𝜈𝜉+()11−𝜆1+𝜉𝑥−𝑦,∀𝑥,𝑦∈𝐶. (101) In particular, if 1−(𝜉/(1+𝜉))(1−(1−𝜈)/𝜉)≤𝜆≤1, then 𝐼−𝜆𝐴 is nonexpansive.

Theorem 22. Let 𝑋 be a nonempty closed convex subset of a uniformly convex Banach space which has a uniformly Gáteaux differentiable norm and let 𝐶 be a nonempty closed convex subset of 𝑋 such that 𝐶±𝐶⊂𝐶. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝑇,𝐹∶𝑋→𝐶𝐵(𝑋), and 𝐴∶𝐶→2^𝐶 be three multivalued mappings, let 𝑔∶𝑋→𝐶 be a single-valued mapping, and let 𝑁(⋅,⋅)∶𝑋×𝑋→𝐶 be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4. Consider that
(H6) 𝑁(𝑇𝑥,𝐹𝑥)+𝐴(𝑔(𝑥))∶𝑋→𝐶 is 𝜉₀-strictly pseudocontractive and 𝜈₀-strongly accretive with 𝜉₀+𝜈₀≥1.

Let 𝐴_𝑖∶𝐶→𝑋 be 𝜉_𝑖-strictly pseudocontractive and 𝜈_𝑖-strongly accretive with 𝜉_𝑖+𝜈_𝑖≥1 for each 𝑖=1,…,𝑁. Define the mapping 𝐺_𝑖∶𝐶→𝐶 by 𝐺_𝑖=Π_𝐶(𝐼−𝜆_𝑖𝐴_𝑖) where 1−(𝜉_𝑖/(1+𝜉_𝑖))(1−(1−𝜈_𝑖)/𝜉_𝑖)≤𝜆_𝑖≤1 for each 𝑖=1,…,𝑁. 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 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 Δ∶=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Γ∩(∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖))≠∅. Suppose that {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛},{𝜎_𝑛}, and {𝜖_𝑛} are the sequences in [0,1],𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and satisfy the following conditions:(i)∑^∞_𝑛=1(|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|+|𝜎_𝑛−𝜎_𝑛−1|+|𝜖_𝑛−𝜖_𝑛−1|)<∞;(ii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(iii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;(v)0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1;(vi)0<liminf_𝑛→∞𝜖_𝑛≤limsup_𝑛→∞𝜖_𝑛<1.

For arbitrary 𝑥₀∈𝐶, define the sequence {𝑥_𝑛} iteratively by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛,𝑥_𝑛+1=𝜖_𝑛𝑥_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝜖_𝑛𝑦_𝑛,𝑢_𝑛∈𝐴𝑔𝑥_𝑛,∀𝑛≥0, (102) where {𝑢_𝑛} is defined by 𝑢_𝑛−𝑢_𝑛+1≤(1+𝜀)𝐻𝐴𝑔𝑥_𝑛+1,𝐴𝑔𝑥_𝑛,∀𝑛≥0, (103) for any 𝑤_𝑛∈𝑇𝑥_𝑛, 𝑘_𝑛∈𝐹𝑥_𝑛, and some 𝜀>0. 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 {𝑥_𝑛} converges strongly to 𝑞∈Δ, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈Δ, (104) and, for any 𝑤∈𝑇𝑞 and 𝑘∈𝐹𝑞, (𝑞,𝑤,𝑘) is a solution of the MVVI (16).

Proof. First of all, by Lemma 21, we know that 𝐼−𝜆_𝑖𝐴_𝑖 is a nonexpansive mapping, where 1−(𝜉_𝑖/(1+𝜉_𝑖))(1−(1−𝜈_𝑖)/𝜉_𝑖)≤𝜆_𝑖≤1 for each 𝑖=1,…,𝑁. Hence, from the nonexpansivity of Π_𝐶, it follows that 𝐺_𝑖 is a nonexpansive mapping for each 𝑖=1,…,𝑁. Since 𝐵∶𝐶→𝐶 is the 𝐾-mapping generated by 𝐺₁,…,𝐺_𝑁 and 𝜌₁,…,𝜌_𝑁, by Lemma 3, we deduce that Fix(𝐵)=∩^𝑁_𝑖=1Fix(𝐺_𝑖). Utilizing Lemma 10 and the definition of 𝐺_𝑖, we get Fix(𝐺_𝑖)=VI(𝐶,𝐴_𝑖) for each 𝑖=1,…,𝑁. Thus, we have Fix(𝐵)=^𝑁_𝑖=1Fix𝐺_𝑖=^𝑁_𝑖=1VI𝐶,𝐴_𝑖. (105)
Repeating the same arguments as those in the proof of Theorem 18, we can prove that for any 𝑣∈𝐶, 𝜆>0, there exists a point 𝑥∈𝐶 such that (𝑥,𝑤,𝑘) is a solution of the MVVI (15), for any 𝑤∈𝑇𝑥 and 𝑘∈𝐹𝑥. In addition, in terms of Proposition 7, we know that 𝑉+𝜆𝐴∘𝑔 is a single-valued mapping due to the fact that 𝑉+𝜆𝐴∘𝑔 is 𝜙-strongly accretive. Assume that 𝑁(𝑇𝑥,𝐹𝑥)+𝐴(𝑔(𝑥))∶𝑋→𝐶 is 𝜉₀-strictly pseudocontractive and 𝜈₀-strongly accretive with 𝜉₀+𝜈₀≥1. Then by Lemma 21, we conclude that the mapping 𝑥↦𝑥−(𝑁(𝑇𝑥,𝐹𝑥)+𝜆𝐴(𝑔(𝑥))) is nonexpansive.
Without loss of generality, we may assume that 𝑣=0 and 𝜆=1. Let 𝑝∈Δ and let 𝑟(≥‖𝑓(𝑝)−𝑝‖/(1−𝜌)) be sufficiently large such that 𝑥₀∈𝐵_𝑟(𝑝)=∶𝐵. Observe that 𝑦_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐵𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝑥_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_𝑛𝑓𝑝𝑥−𝑝≤max_𝑛,𝑓𝑝−𝑝−𝑝.1−𝜌 (106) Utilizing (106) and repeating the same arguments as those in the proof of Theorem 18, we can derive 𝑥_𝑛∈𝐵 for all 𝑛≥0. Hence {𝑥_𝑛} is bounded.
Let us show that lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.
Indeed, we define 𝐺∶𝐶→𝐶 by 𝐺𝑥∶=𝑥−(𝑁(𝑇𝑥,𝐹𝑥)+𝐴(𝑔(𝑥))) for all 𝑥∈𝐶. Then, 𝐺 is a nonexpansive mapping and the iterative scheme (102) can be rewritten as follows: 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛,𝑥_𝑛+1=𝜖_𝑛1−𝜎_𝑛𝑥_𝑛+𝜎_𝑛𝐺𝑥_𝑛+1−𝜖_𝑛𝑦_𝑛,∀𝑛≥0. (107) Repeating the same arguments as those of (56), (60), (62), (76), and (80) in the proof of Theorem 18, we can obtain that lim_𝑛→∞𝑥_𝑛−𝑥_𝑛+1=0,lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0,lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0,lim_𝑛→∞𝑥_𝑛−𝐵𝑥_𝑛=0,lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (108)(109)
Define a mapping 𝑊𝑥=(1−𝜃₁−𝜃₂)𝐵𝑥+𝜃₁𝑆𝑥+𝜃₂𝐺𝑥, where 𝜃₁,𝜃₂∈(0,1) are two constants with 𝜃₁+𝜃₂<1. Then by Lemma 15, we have that Fix(𝑊)=Fix(𝐵)∩Fix(𝑆)∩Fix(𝐺)=Δ. We observe that 𝑥_𝑛−𝑊𝑥_𝑛=1−𝜃₁−𝜃₂𝑥_𝑛−𝐵𝑥_𝑛+𝜃₁𝑥_𝑛−𝑆𝑥_𝑛+𝜃₂𝑥_𝑛−𝐺𝑥_𝑛≤1−𝜃₁−𝜃₂𝑥_𝑛−𝐵𝑥_𝑛+𝜃₁𝑥_𝑛−𝑆𝑥_𝑛+𝜃₂𝑥_𝑛−𝐺𝑥_𝑛. (110) From (109), we obtain lim_𝑛→∞𝑥_𝑛−𝑊𝑥_𝑛=0. (111)
Now, we claim that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0, (112) where 𝑞=𝑠−lim_𝑡→0𝑥_𝑡 with 𝑥_𝑡 being the fixed point of the contraction 𝑥⟼𝑡𝑓(𝑥)+(1−𝑡)𝑊𝑥. (113) Then 𝑥_𝑡 solves the fixed point equation 𝑥_𝑡=𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑊𝑥_𝑡. Repeating the same arguments as those of (93) in the proof of Theorem 18, we can deduce that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞=limsup_𝑡→0limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤limsup_𝑡→0limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞−𝐽𝑥_𝑛−𝑥_𝑡. (114) Since 𝑋 has a uniformly Gáteaux differentiable norm, the duality mapping 𝐽 is norm-to-weak∗ uniformly continuous on bounded subsets of 𝑋. Consequently, the two limits are interchangeable and hence (112) holds. Noticing that 𝐽 is norm-to-weak∗ uniformly continuous on bounded subsets of 𝑋, we conclude from (108) that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞=limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛+𝑓𝑞𝑦−𝑞−𝑞,𝐽_𝑛−𝑞−𝐽𝑥_𝑛−𝑞=limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0. (115)
Finally, let us show that 𝑥_𝑛→𝑞 as 𝑛→∞. Indeed, repeating the same arguments as those (96) in the proof of Theorem 18, we can deduce from (107) that 𝑥_𝑛+1−𝑞²≤1−1−𝜖_𝑛𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+1−𝜖_𝑛𝛼_𝑛2𝑓𝑞𝑦1−𝜌−𝑞,𝐽_𝑛−𝑞.1−𝜌 (116) Applying Lemma 8 to (116), we infer from conditions (ii) and (vi) and (115) that 𝑥_𝑛→𝑞 as 𝑛→∞. This completes the proof.

Corollary 23. Let 𝑋 be a uniformly convex Banach space which has a uniformly Gáteaux differentiable norm and let 𝐶 be a nonempty closed convex subset of 𝑋 such that 𝐶±𝐶⊂𝐶. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝑇,𝐹∶𝑋→𝐶𝐵(𝑋), and 𝐴∶𝐶→2^𝐶 be three multivalued mappings, let 𝑔∶𝑋→𝐶 be a single-valued mapping, and let 𝑁(⋅,⋅)∶𝑋×𝑋→𝐶 be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4 and (H6) 𝑁(𝑇𝑥,𝐹𝑥)+𝐴(𝑔(𝑥))∶𝑋→𝐶 is 𝜉₀-strictly pseudocontractive and 𝜈₀-strongly accretive with 𝜉₀+𝜈₀≥1.

For each 𝑖=1,…,𝑁, let 𝑇_𝑖∶𝐶→𝐶 be a self-mapping such that 𝐼−𝑇_𝑖∶𝐶→𝑋 is 𝜉_𝑖-strictly pseudocontractive and 𝜈_𝑖-strongly accretive with 𝜉_𝑖+𝜈_𝑖≥1. Define the mapping 𝐺_𝑖∶𝐶→𝐶 by 𝐺_𝑖=(1−𝜆_𝑖)𝐼+𝜆_𝑖𝑇_𝑖 where 1−(𝜉_𝑖/(1+𝜉_𝑖))(1−(1−𝜈_𝑖)/𝜉_𝑖)≤𝜆_𝑖≤1 for each 𝑖=1,…,𝑁. 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 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 Δ∶=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Γ∩(∩^𝑁_𝑖=1Fix(𝑇_𝑖))≠∅. Suppose that {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛},{𝜎_𝑛}, and {𝜖_𝑛} are the sequences in [0,1],𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and satisfy the following conditions:(i)∑^∞_𝑛=1(|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|+|𝜎_𝑛−𝜎_𝑛−1|+|𝜖_𝑛−𝜖_𝑛−1|)<∞;(ii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(iii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iv)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;(v)0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1;(vi)0<liminf_𝑛→∞𝜖_𝑛≤limsup_𝑛→∞𝜖_𝑛<1.For arbitrary 𝑥₀∈𝐶, define the sequence {𝑥_𝑛} iteratively by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛,𝑥_𝑛+1=𝜖_𝑛𝑥_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝜖_𝑛𝑦_𝑛,𝑢_𝑛∈𝐴𝑔𝑥_𝑛,∀𝑛≥0, (117) where {𝑢_𝑛} is defined by 𝑢_𝑛−𝑢_𝑛+1≤(1+𝜀)𝐻𝐴𝑔𝑥_𝑛+1,𝐴𝑔𝑥_𝑛,∀𝑛≥0, (118)

for any 𝑤_𝑛∈𝑇𝑥_𝑛, 𝑘_𝑛∈𝐹𝑥_𝑛, and some 𝜀>0. 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 {𝑥_𝑛} converges strongly to 𝑞∈Δ, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈Δ (119) and, for any 𝑤∈𝑇𝑞 and 𝑘∈𝐹𝑞, (𝑞,𝑤,𝑘) is a solution of the MVVI (16).

Conflict of Interests

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

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for the technical and financial support. This research was partially supported (to Lu-Chuan Ceng) by the National Science Foundation of China (11071169), the Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and the Ph.D. Program Foundation of Ministry of Education of China (20123127110002). Finally, the authors thank the referees for their valuable comments and appreciation.