Mann-Type Extragradient Methods for General Systems of Variational Inequalities with Multivalued Variational Inclusion Constraints in Banach Spaces

Abstract

We introduce Mann-type extragradient methods for a general system of variational inequalities with solutions of a multivalued variational inclusion and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type extragradient methods are based on Korpelevich’s extragradient method and Mann iteration method. We first consider and analyze a Mann-type extragradient algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another Mann-type extragradient algorithm in a smooth and uniformly convex Banach space. Under suitable assumptions, we derive some weak and strong convergence theorems. 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 as follows:(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 if for each 𝑥,𝑦∈𝐶 there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≤𝑥−𝑦²−𝜆𝑥−𝑦−𝐴𝑥−𝐴𝑦², (5) for some 𝜆∈(0,1).

Let 𝑈={𝑥∈𝑋∶‖𝑥‖=1} denote the unite sphere of 𝑋. A Banach space 𝑋 is said to be uniformly convex if for each 𝜖∈(0,2], there exists 𝛿>0 such that for all 𝑥,𝑦∈𝑈𝑥−𝑦≥𝜖⟹𝑥+𝑦2≤1−𝛿. (6) It is known that a uniformly convex Banach space is reflexive and strictly 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 Gáteaux differentiable norm. 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 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 [1], no Banach space is 𝑞-uniformly smooth for 𝑞>2. In addition, it is also known that 𝐽 is single-valued if and only if 𝑋 is smooth, whereas if 𝑋 is uniformly smooth, then 𝐽 is norm-to-norm uniformly continuous on bounded subsets of 𝑋.

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

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

Lemma 1 (see [3]). For given 𝑥,𝑦∈𝐶,(𝑥,𝑦) is a solution of problem (10) if and only if 𝑥 is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺(𝑥)=𝑃_𝐶𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶, (11) where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥) and 𝑃_𝐶 is the the projection of 𝐻 onto 𝐶.

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

Define the mapping 𝐺∶𝐶→𝐶 as follows: 𝐺(𝑥)∶=Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶. (12) The fixed point set of 𝐺 is denoted by Ω.

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 (13) Let 𝑇 and 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. [4] 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. (14)

If 𝑣=0 and 𝜆=1, then the MVVI (14) 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. (15) We denote by Γ the set of such solutions 𝑥 for MVVI (15).

The authors [4] first established an existence theorem for MVVI (14) in smooth Banach space 𝑋 and then proved that the sequence generated by their iterative algorithm converges strongly to a solution of MVVI (15).

Theorem 2 (see [4, Theorem 3.2]). Let 𝑋 be a real smooth Banach space. Let 𝑇 and 𝐹∶𝑋→𝐶𝐵(𝑋), let 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, (16) where {𝑢_𝑛} is defined by 𝑢_𝑛−𝑢_𝑛+1≤(1+𝜀)𝐻𝐴𝑔𝑥_𝑛+1,𝐴𝑔𝑥_𝑛,∀𝑛≥0, (17) 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 and ∑^∞_𝑛=0𝜎_𝑛=∞. Then, there exists 𝑑>0 such that for 0<𝜎_𝑛≤𝑑, for all 𝑛≥0, {𝑥_𝑛} converges strongly to 𝑥∈Γ; and for any 𝑤∈𝑇𝑥, 𝑘∈𝐹𝑥, (𝑥,𝑤,𝑘) is a solution of the MVVI (15).

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. Motivated and inspired by the research going on this area, we introduce Mann-type extragradient methods for finding solutions of the GSVI (9) which are also ones of the MVVI (15) and common fixed points of a countable family of nonexpansive mappings. Here the Mann-type extragradient methods are based on Korpelevich’s extragradient method and Mann iteration method. We first consider and analyze a Mann-type extragradient algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space, and then another Mann-type extragradient algorithm in a smooth and uniformly convex Banach space. Under suitable assumptions, we derive some weak and strong convergence theorems. 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, [2–10].

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, (18) 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 as follows:(i)accretive, if 𝑢−𝑣,𝐽𝑥−𝑦≥0,∀𝑢∈𝐴𝑥,𝑣∈𝐴𝑦; (19)(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; (20)(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; (21)(v)𝜙-expansive, if ‖𝑢−𝑣‖≥𝜙𝑥−𝑦,∀𝑢∈𝐴𝑥,𝑣∈𝐴𝑦. (22)

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. (23)

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

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

Proposition 3 (see [11]). 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, (24) 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 4 (see [12]). 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) 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 5 (see [13]). 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.

Recall that a Banach space 𝑋 is said to satisfy Opial’s condition, if whenever {𝑥_𝑛} is a sequence in 𝑋 which converges weakly to 𝑥 as 𝑛→∞, then limsup_𝑛→∞𝑥_𝑛−𝑥<limsup_𝑛→∞𝑥_𝑛−𝑦,∀𝑦∈𝑋with𝑥≠𝑦. (25)

Lemma 6 (Demiclosedness principle; see [14, Lemma 2]). Let 𝐶 be a nonempty closed convex subset of a reflexive Banach space 𝑋 that satisfies Opial’s condition and suppose that 𝑇∶𝐶→𝑋 is nonexpansive. Then the mapping 𝐼−𝑇 is demiclosed at zero, that is, 𝑥_𝑛⇀𝑥 and 𝑥_𝑛−𝑇𝑥_𝑛→0 imply 𝑥=𝑇𝑥; that is, 𝑥∈Fix(𝑇).

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

Lemma 7. In a real smooth Banach space 𝑋, there holds the inequality 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽𝑥+𝑦,∀𝑥,𝑦∈𝑋. (26)
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), (27) 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 8 (see [15]). 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. The following result is an easy consequence of Lemma 8.

Lemma 9. 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 (9) if and only if 𝑥^∗=Π_𝐶(𝑦^∗−𝜇₁𝐵₁𝑦^∗), where 𝑦^∗=Π_𝐶(𝑥^∗−𝜇₂𝐵₂𝑥^∗).

In terms of Lemma 9, we observe that 𝑥^∗=Π_𝐶Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗−𝜇₁𝐵₁Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗, (28) 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 10 (see [16]). 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, (29) for all 𝜆∈[0,1], and 𝑥,𝑦∈𝑋 such that ‖𝑥‖≤𝑟 and ‖𝑦‖≤𝑟.

Lemma 11 (see [17]). 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.

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

In this section, we introduce Mann-type extragradient algorithms in uniformly convex and 2-uniformly smooth Banach spaces and show weak and strong convergence theorems. We will use some useful lemmas in the sequel.

Lemma 12 (see [2, Lemma 2.8]). 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 𝛼_𝑖-inverse-strongly accretive. Then, we have 𝐼−𝜇_𝑖𝐵_𝑖𝑥−𝐼−𝜇_𝑖𝐵_𝑖𝑦²≤𝑥−𝑦²+2𝜇_𝑖𝜇_𝑖𝜅²−𝛼_𝑖𝐵_𝑖𝑥−𝐵_𝑖𝑦²,∀𝑥,𝑦∈𝐶, (30) for 𝑖=1,2, where 𝜇_𝑖>0. In particular, if 0<𝜇_𝑖≤𝛼_𝑖/𝜅², then 𝐼−𝜇_𝑖𝐵_𝑖 is nonexpansive for 𝑖=1,2.

Lemma 13 (see [2, Lemma 2.9]). 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 𝛼_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. (31) If 0<𝜇_𝑖≤𝛼_𝑖/𝜅² for 𝑖=1,2, then 𝐺∶𝐶→𝐶 is nonexpansive.

Theorem 14. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space satisfying Opial’s condition 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 and 𝐹∶𝑋→𝐶𝐵(𝑋) and let 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 2 and(C6)𝑁(𝑇𝑥,𝐹𝑥)+𝐴(𝑔(𝑥))∶𝑋→2^𝐶⧵{∅} is 𝜁-inverse strongly accretive with 𝜁≥𝜅². Let 𝐵_𝑖∶𝐶→𝑋 be 𝛼_𝑖-inverse strongly accretive for 𝑖=1,2. Let {𝑆_𝑖}^∞_𝑖=0 be a countable family of nonexpansive mappings of 𝐶 into itself such that Δ∶=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Ω∩Γ≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) with 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2. Assume that {𝛼_𝑛}, {𝛽_𝑛}, and {𝜎_𝑛} are sequences in [0,1] such that(i)0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1;(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;(iii)0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1. For arbitrary 𝑥₀∈𝐶 define the sequence {𝑥_𝑛} iteratively by 𝑦_𝑛=𝛽_𝑛𝑆_𝑛𝑥_𝑛+1−𝛽_𝑛Π_𝐶𝐼−𝜇₁𝐵₁×Π_𝐶𝐼−𝜇₂𝐵₂𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝛼_𝑛𝑦_𝑛,𝑢_𝑛∈𝐴𝑔𝑥_𝑛,∀𝑛≥0, (32) where {𝑢_𝑛} is defined by 𝑢_𝑛−𝑢_𝑛+1≤(1+𝜀)𝐻𝐴𝑔𝑥_𝑛+1,𝐴𝑔𝑥_𝑛,∀𝑛≥0, (33) 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 weakly to some 𝑥∈Δ, and for any 𝑤∈𝑇𝑥, 𝑘∈𝐹𝑥,(𝑥,𝑤,𝑘) is a solution of the MVVI (15).

Proof. First of all, 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 (14), for any 𝑤∈𝑇𝑥 and 𝑘∈𝐹𝑥. Indeed, following the argument idea in the proof of Chidume et al. [4, Theorem 3.1], we put 𝑉𝑥∶=𝑁(𝑇𝑥,𝐹𝑥) for all 𝑥∈𝑋. Then by Proposition 4, 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 5  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 [18, page 184] for any 𝑥∈𝑋 we have (𝑉+𝜆𝐴∘𝑔)(𝑥) is closed and bounded. Therefore, by Morales [19], 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 5 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 12, we conclude that the mapping 𝑥↦𝑥−(𝑁(𝑇𝑥,𝐹𝑥)+𝜆𝐴(𝑔(𝑥))) is nonexpansive. Meantime, by Lemma 13 we know that 𝐺∶𝐶→𝐶 is also nonexpansive.
Without loss of generality we may assume that 𝑣=0 and 𝜆=1. Let 𝑝∈Δ and let 𝑟>0 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), and 𝜀₂∶=𝜙(𝑟)/8𝜇(1+𝜀), 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 (32) it follows that 𝑦_𝑛−𝑝=𝛽_𝑛𝑆_𝑛𝑥_𝑛+1−𝛽_𝑛Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶×𝐼−𝜇₂𝐵₂𝑥_𝑛−𝑝≤𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝐺𝑥_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝑥_𝑛=𝑥−𝑝_𝑛−𝑝, (34) and hence 𝑥_𝑛+1−𝑝²=𝛼_𝑛𝑥_𝑛−𝑝−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝛼_𝑛𝑦_𝑛−𝑝,𝐽𝑥_𝑛+1=𝛼−𝑝_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛×𝑦_𝑛−𝑝,𝐽𝑥_𝑛+1−𝑝−𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝐽𝑥_𝑛+1≤𝛼−𝑝_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑦_𝑛×𝑥−𝑝_𝑛+1−𝑝−𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝐽𝑥_𝑛+1≤𝛼−𝑝_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑥_𝑛×𝑥−𝑝_𝑛+1−𝑝−𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝐽𝑥_𝑛+1=𝑥−𝑝_𝑛−𝑝𝑥_𝑛+1−𝑝−𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝐽𝑥_𝑛+1≤1−𝑝2𝑥_𝑛−𝑝²+𝑥_𝑛+1−𝑝²−𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝐽𝑥_𝑛+1−𝑝, (35) which immediately yields 𝑥_𝑛+1−𝑝²≤𝑥_𝑛−𝑝²−2𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛,𝐽𝑥_𝑛+1=𝑥−𝑝_𝑛−𝑝²−2𝛼_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. (36) Since 𝑁(⋅,⋅) is 𝜙-strongly accretive with respect to 𝑇 and 𝐴(𝑔(⋅)) is accretive, we deduce from (36) that 𝑥_𝑛+1−𝑝²≤𝑥_𝑛−𝑝²−2𝛼_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+1≤𝑥−𝑝_𝑛−𝑝²−2𝛼_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. (37) Again from (32) we have that 𝑥_𝑛+1−𝑝≤𝛼_𝑛𝑥_𝑛−𝑝−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝛼_𝑛𝑦_𝑛−𝑝≤𝛼_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝛼_𝑛𝑥_𝑛−𝑝≤𝛼_𝑛𝑟+𝜎_𝑛𝑀+1−𝛼_𝑛𝑟≤2𝑟. (38) Also, from Proposition 5, 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 [20] 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_𝑛,𝑤_𝑛+1−𝑤^′_𝑛≤(1+𝜀)𝐻𝑇𝑥_𝑛+1,𝑇𝑥_𝑛, (39) respectively. Therefore, from (37) and (33), we have that 𝑥_𝑛+1−𝑝²≤𝑥_𝑛−𝑝²−2𝛼_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_𝑛≤𝑥2𝑟_𝑛−𝑝²−2𝛼_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_𝑛≤𝑥2𝑟_𝑛−𝑝²−2𝛼_𝑛𝜎_𝑛𝜙(𝑟)𝑟+2𝛼_𝑛𝜎_𝑛𝜙(𝑟)8+𝜙(𝑟)8+𝜙(𝑟)8=𝑥2𝑟_𝑛−𝑝²−2𝛼_𝑛𝜎_𝑛𝜙(𝑟)𝑟+𝛼_𝑛𝜎_𝑛32𝜙(𝑟)𝑟≤𝑥_𝑛−𝑝². (40) So, we get ‖𝑥_𝑛+1−𝑝‖≤𝑟, a contradiction. Therefore, {𝑥_𝑛} is bounded.
Let us show that lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0.
Indeed, utilizing Lemma 10 and the nonexpansivity of the mapping 𝑥↦𝑥−(𝑁(𝑇𝑥,𝐹𝑥)+𝐴(𝑔(𝑥))), we obtain from (32) that for all 𝑛≥0𝑥_𝑛+1−𝑝²=𝛼_𝑛𝑥_𝑛−𝑝−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝛼_𝑛𝑦_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛²+1−𝛼_𝑛𝑦_𝑛−𝑝²−𝛼_𝑛1−𝛼_𝑛𝜑×𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛=𝛼_𝑛1−𝜎_𝑛𝑥_𝑛−𝑝+𝜎_𝑛𝑥_𝑛−𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛−𝑝²+1−𝛼_𝑛𝑦_𝑛−𝑝²−𝛼_𝑛1−𝛼_𝑛𝜑×𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛≤𝛼_𝑛1−𝜎_𝑛𝑥_𝑛−𝑝²+𝜎_𝑛𝑥_𝑛−𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛−𝑝²−𝜎_𝑛1−𝜎_𝑛𝜑₁𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝛼_𝑛𝑥_𝑛−𝑝²−𝛼_𝑛1−𝛼_𝑛𝜑×𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛≤𝛼_𝑛1−𝜎_𝑛𝑥_𝑛−𝑝²+𝜎_𝑛𝑥_𝑛−𝑝²−𝜎_𝑛1−𝜎_𝑛𝜑₁𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝛼_𝑛𝑥_𝑛−𝑝²−𝛼_𝑛1−𝛼_𝑛𝜑×𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛=𝛼_𝑛𝑥_𝑛−𝑝²−𝛼_𝑛𝜎_𝑛1−𝜎_𝑛𝜑₁×𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝛼_𝑛𝑥_𝑛−𝑝²−𝛼_𝑛1−𝛼_𝑛𝜑×𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛=𝑥_𝑛−𝑝²−𝛼_𝑛𝜎_𝑛1−𝜎_𝑛𝜑₁𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛−𝛼_𝑛1−𝛼_𝑛𝜑×𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛≤𝑥_𝑛−𝑝². (41) It is easy to see that the limit lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists. Meantime, it can be readily seen from (41) that 𝛼_𝑛𝜎_𝑛1−𝜎_𝑛𝜑₁𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+𝛼_𝑛1−𝛼_𝑛𝜑×𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝², (42) which together with conditions (i) and (iii) and the existence of lim_𝑛→∞‖𝑥_𝑛−𝑝‖, implies that lim_𝑛→∞𝜑₁𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛=0,lim_𝑛→∞𝜑𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛=0. (43) Utilizing the properties of 𝜑 and 𝜑₁, we get lim_𝑛→∞𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛=0,lim_𝑛→∞𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛=0. (44) Note that 𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛. (45) So, we have lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (46) Also, observe that 𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝛼_𝑛𝑥_𝑛−𝑦_𝑛≤𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+𝑥_𝑛−𝑦_𝑛. (47) Thus, from (44) and (46) it follows that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (48)
Let us show that lim_𝑛→∞‖𝑥_𝑛−𝐺𝑥_𝑛‖=lim_𝑛→∞‖𝑥_𝑛−𝑆𝑥_𝑛‖=0.
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 12 we have 𝑢_𝑛−𝑞²=Π_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−Π_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²≤𝑥_𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂×𝐵₂𝑥_𝑛−𝐵₂𝑝²,𝑣_𝑛−𝑝²=Π_𝐶𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛−Π_𝐶𝑞−𝜇₁𝐵₁𝑞²≤𝑢_𝑛−𝑞−𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²≤𝑢_𝑛−𝑞²−2𝜇₁𝛼₁−𝜅²𝜇₁×𝐵₁𝑢_𝑛−𝐵₁𝑞². (49)(50) Substituting (49) for (50), we obtain 𝑣_𝑛−𝑝²≤𝑥_𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞². (51) Utilizing [21, Proposition 1] and Lemma 10, from (32) and (51) we have 𝑦_𝑛−𝑝²≤𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝐺𝑥_𝑛−𝑝²−𝛽_𝑛1−𝛽_𝑛𝜑₂𝑆_𝑛𝑥_𝑛−𝐺𝑥_𝑛≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑣_𝑛−𝑝²−𝛽_𝑛1−𝛽_𝑛𝜑₂𝑆_𝑛𝑥_𝑛−𝐺𝑥_𝑛≤𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛1−𝛽_𝑛𝜑₂𝑆_𝑛𝑥_𝑛−𝐺𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²=𝑥_𝑛−𝑝²−𝛽_𝑛1−𝛽_𝑛𝜑₂𝑆_𝑛𝑥_𝑛−𝐺𝑥_𝑛−21−𝛽_𝑛𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞², (52) which hence implies that 𝛽_𝑛1−𝛽_𝑛𝜑₂𝑆_𝑛𝑥_𝑛−𝐺𝑥_𝑛+21−𝛽_𝑛𝜇₂𝛼₂−𝜅²𝜇₂×𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝜇₁𝛼₁−𝜅²𝜇₁×𝐵₁𝑢_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛. (53) Since 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2 and {𝑥_𝑛}, {𝑦_𝑛} are bounded, we obtain from (46), (53), condition (ii) and the properties of 𝜑₂ that lim_𝑛→∞𝐵₂𝑥_𝑛−𝐵₂𝑝=lim_𝑛→∞𝐵₁𝑢_𝑛−𝐵₁𝑞=0,lim_𝑛→∞𝑆_𝑛𝑥_𝑛−𝐺𝑥_𝑛=0. (54) Utilizing Proposition 3 and Lemma 8, 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−𝜓₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞, (55) which implies that 𝑢_𝑛−𝑞²≤𝑥_𝑛−𝑝²−𝜓₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞. (56) 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−𝜓₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝, (57) which implies that 𝑣_𝑛−𝑝²≤𝑢_𝑛−𝑞²−𝜓₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝. (58) Substituting (56) for (58), we get 𝑣_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝜓₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞−𝜓₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝. (59) By Lemma 7, we have from (52) and (59) 𝑦_𝑛−𝑝²≤𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑣_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛×𝑥_𝑛−𝑝²−𝜓₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞−𝜓₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛≤𝑥−𝑝_𝑛−𝑝²−1−𝛽_𝑛×𝜓₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝜓₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝, (60) which hence leads to 1−𝛽_𝑛𝜓₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝜓₂𝑢_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. (61) From (46), (54), (61), condition (ii), and the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, {𝑢_𝑛}, and {𝑣_𝑛}, we deduce that lim_𝑛→∞𝜓₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞=0,lim_𝑛→∞𝜓₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞=0. (62) Utilizing the properties of 𝜓₁ and 𝜓₂, we deduce that lim_𝑛→∞𝑥_𝑛−𝑢_𝑛−𝑝−𝑞=0,lim_𝑛→∞𝑢_𝑛−𝑣_𝑛+𝑝−𝑞=0. (63) From (63) we get 𝑥_𝑛−𝑣_𝑛≤𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝑢_𝑛−𝑣_𝑛+𝑝−𝑞⟶0as𝑛⟶∞, (64) which together with (54), leads to lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0,lim_𝑛→∞𝑥_𝑛−𝑆_𝑛𝑥_𝑛=0. (65) Since 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆_𝑛𝑥_𝑛+𝑆_𝑛𝑥_𝑛−𝑆𝑥_𝑛. (66) Utilizing the assumption on {𝑆_𝑛} and Lemma 11, from (65) we get lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (67)
Next, let us show that {𝑥_𝑛} converges weakly to some 𝑥∈Δ.
Indeed, since 𝑋 is reflexive and {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑥∈𝐶. Then by Lemma 6, we obtain from (44), (65), and (67) that 𝑥∈Γ, 𝑥∈Fix(𝐺)=Ω, and 𝑥∈Fix(𝑆)=⋂^∞_𝑖=0Fix(𝑆_𝑖). Thus, 𝑥∈Δ. In addition, if {𝑥_𝑚𝑗} is another subsequence of {𝑥_𝑛} such that 𝑥_𝑚𝑗⇀𝑥, then by Lemma 6 we also deduce from (44), (65), and (67) that 𝑥∈Δ. Thus, the limits lim_𝑛→∞‖𝑥_𝑛−𝑥‖ and lim_𝑛→∞‖𝑥_𝑛−𝑥‖ exist. Now we claim that 𝑥=𝑥. Assume that 𝑥≠𝑥. Then in terms of Opial’s condition, we get lim_𝑛→∞𝑥_𝑛−𝑥=limsup_𝑖→∞𝑥_𝑛𝑖−𝑥<limsup_𝑖→∞𝑥_𝑛𝑖−𝑥=limsup_𝑛→∞𝑥_𝑛−𝑥=limsup_𝑗→∞𝑥_𝑚𝑗−𝑥<limsup_𝑗→∞𝑥_𝑚𝑗−𝑥=lim_𝑛→∞𝑥_𝑛−𝑥, (68) which leads to a contradiction. So, we must have 𝑥=𝑥. Therefore, 𝑥_𝑛⇀𝑥∈Δ. This completes the proof.

Theorem 15. 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 and 𝐹∶𝑋→𝐶𝐵(𝑋) and let 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 2. Let 𝐵_𝑖∶𝐶→𝑋 be 𝛼_𝑖-inverse strongly accretive for 𝑖=1,2. Let {𝑆_𝑖}^∞_𝑖=0 be a countable family of nonexpansive mappings of 𝐶 into itself such that Δ∶=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Ω∩Γ≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) with 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2. Let {𝛼_𝑛},{𝛽_𝑛}⊂[0,1], and {𝜎_𝑛}⊂(0,∞) such that lim_𝑛→∞𝛼_𝑛𝜎_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛𝜎_𝑛=∞. For arbitrary 𝑥₀∈𝐶 define the sequence {𝑥_𝑛} iteratively by (32), where {𝑢_𝑛} is defined by (33) for any 𝑤_𝑛∈𝑇𝑥_𝑛, 𝑘_𝑛∈𝐹𝑥_𝑛 and some 𝜀>0. Then, there exists 𝑑>0 such that for 0<𝛼_𝑛𝜎_𝑛≤𝑑, for all 𝑛≥0, {𝑥_𝑛} converges strongly to 𝑝∈Δ provided lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0; in this case, for any 𝑤∈𝑇𝑥, 𝑘∈𝐹𝑥, (𝑥,𝑤,𝑘) is a solution of the MVVI (15).

Proof. First of all, repeating the same arguments as those in the proof of Theorem 14, we can prove that for any 𝑣∈𝐶, 𝜆>0, there exists a point 𝑥∈𝐷(𝐴)=𝐶 such that (𝑥,𝑤,𝑘) is a solution of the MVVI (14), for any 𝑤∈𝑇𝑥 and 𝑘∈𝐹𝑥. In addition, in terms of Proposition 5 we know that 𝑉+𝜆𝐴∘𝑔 is a single-valued mapping due to the fact that 𝑉+𝜆𝐴∘𝑔 is 𝜙-strongly accretive. Meantime, by Lemma 13 we know that 𝐺∶𝐶→𝐶 is nonexpansive.
Without loss of generality we may assume that 𝑣=0 and 𝜆=1. Let 𝑝∈Δ and let 𝑟>0 be sufficiently large such that 𝑥₀∈𝐵_𝑟(𝑝)=∶𝐵. 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) and 𝜀₂∶=𝜙(𝑟)/8𝜇(1+𝜀), 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 𝑑∶=(1/2)min{𝛿₂/𝑀,𝛿₃/𝑀,𝛿₁/𝑀,𝑟/𝑀}. Taking into account lim_𝑛→∞𝛼_𝑛𝜎_𝑛=0 we may assume that 0<𝛼_𝑛𝜎_𝑛≤𝑑, for all 𝑛≥0.
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 (32) it follows that 𝑦_𝑛−𝑝≤𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝐺𝑥_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝑥_𝑛=𝑥−𝑝_𝑛−𝑝. (69) Repeating the same arguments as those of (37) in the proof of Theorem 14, we can get 𝑥_𝑛+1−𝑝²≤𝑥_𝑛−𝑝²−2𝛼_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+1≤𝑥−𝑝_𝑛−𝑝²−2𝛼_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. (70) Utilizing (33) and (70) and repeating the same arguments as those of (40) in the proof of Theorem 14, we can derive ‖𝑥_𝑛+1−𝑝‖²≤‖𝑥_𝑛−𝑝‖². So, we have ‖𝑥_𝑛+1−𝑝‖≤𝑟, a contradiction. Therefore, {𝑥_𝑛} is bounded.
Next let us show that 𝑥_𝑛→𝑥 as 𝑛→∞.
Indeed, since ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞, we have that 𝐻(𝐴(𝑔(𝑥_𝑛+1)),𝐴(𝑔(𝑥_𝑛)))→0 and ‖𝑁(𝑤_𝑛+1,𝑘_𝑛+1)−𝑁(𝑤_𝑛,𝑘_𝑛)‖→0 as 𝑛→∞. The conclusion now follows from inequality (70) with the use of Proposition 3 and hence (𝑥,𝑤,𝑘) is a solution of the MVVI (15) for any 𝑤∈𝑇𝑥, 𝑘∈𝐹𝑥. This completes the proof.

Remark 16. Theorems 14 and 15 improve, extend, supplement, and develop [4, Theorem 3.2] and [2, Theorem 3.1] in the following aspects.(i)The problem of finding a point of ⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Ω∩Γ in Theorems 14 and 15 is more general and more subtle than every one of the problem of finding a point of Γ in [4, Theorem 3.2] and the problem of finding a point of ⋂^∞_𝑖=1Fix(𝑆_𝑖)∩Ω in [2, Theorem 3.1].(ii)The iterative scheme in [2, Theorem 3.1] is extended to develop the iterative scheme (32) of Theorems 14 and 15 by virtue of the iterative schemes of [4, Theorems 3.2]. The iterative scheme (32) of Theorems 14 and 15 is more advantageous and more flexible than the iterative scheme of [4, Theorem 3.2] because it can be applied to solving three problems (i.e., GSVI (9), MVVI (15), and the fixed point problem of {𝑆_𝑛}) and involves several parameter sequences {𝛼_𝑛}, {𝛽_𝑛}, and {𝜎_𝑛}.(iii)Theorems 14 and 15 extend and generalize [4, Theorems 3.2] to the setting of a countable family of nonexpansive mappings and the GSVI (9) for two inverse-strongly accretive mappings. In the meantime, Theorems 14 and 15 extend and generalize [2, Theorem 3.1] to the setting of the MVVI (15).(iv)The iterative scheme (32) in Theorems 14 and 15 is very different from everyone in [4, Theorem 3.2] and [2, Theorem 3.1] because the iterative scheme in [2, Theorem 3.1] involves the viscosity approximation method and the iterative scheme in [4, Theorem 3.2] is a one-step iterative scheme.(v)No boundedness condition on the ranges 𝑅(𝐼−𝑁(𝑇(⋅),𝐹(⋅))) and 𝑅(𝐴(𝑔(⋅))) is imposed in Theorems 14 and 15.

4. Mann-Type Extragradient Algorithms in Smooth and Uniformly Convex Banach Spaces

In this section, we introduce Mann-type extragradient algorithms in smooth and uniformly convex Banach spaces and show weak and strong convergence theorems. First, we give some useful lemmas whose proofs will be omitted.

Lemma 17. 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] we have 𝐼−𝜇_𝑖𝐵_𝑖𝑥−𝐼−𝜇_𝑖𝐵_𝑖𝑦≤1−𝛼_𝑖𝜆_𝑖+1−𝜇_𝑖11+𝜆_𝑖×𝑥−𝑦,∀𝑥,𝑦∈𝐶, (71) 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.

Lemma 18. 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. (72) If 1−(𝜆_𝑖/(1+𝜆_𝑖))(1−(1−𝛼_𝑖)/𝜆_𝑖)≤𝜇_𝑖≤1, then 𝐺∶𝐶→𝐶 is nonexpansive.

Theorem 19. Let 𝑋 be a smooth and uniformly convex Banach space satisfying Opial’s condition and 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 and 𝐹∶𝑋→𝐶𝐵(𝑋) and let 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 2 and(H6)𝑁(𝑇𝑥,𝐹𝑥)+𝐴(𝑔(𝑥))∶𝑋→𝐶 is 𝜆₀-strictly pseudocontractive and 𝛼₀-strongly accretive with 𝜆₀+𝛼₀≥1. Let 𝐵_𝑖∶𝐶→𝑋 be 𝜆_𝑖-strictly pseudocontractive and 𝛼_𝑖-strongly accretive with 𝜆_𝑖+𝛼_𝑖≥1 for 𝑖=1,2. Let {𝑆_𝑖}^∞_𝑖=0 be a countable family of nonexpansive mappings of 𝐶 into itself such that Δ∶=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Ω∩Γ≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) with 1−(𝜆_𝑖/(1+𝜆_𝑖))(1−(1−𝛼_𝑖)/𝜆_𝑖)≤𝜇_𝑖≤1 for 𝑖=1,2. Assume that {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝜎_𝑛} are sequences in [0,1] such that(i)0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1;(ii){𝛽_𝑛},{𝛾_𝑛},{𝛽_𝑛+𝛾_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iii)0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1. For arbitrary 𝑥₀∈𝐶 define the sequence {𝑥_𝑛} iteratively by 𝑦_𝑛=𝛽_𝑛𝑆_𝑛𝑥_𝑛+𝛾_𝑛𝑥_𝑛+1−𝛽_𝑛−𝛾_𝑛Π_𝐶𝐼−𝜇₁𝐵₁×Π_𝐶𝐼−𝜇₂𝐵₂𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝛼_𝑛𝑦_𝑛,𝑢_𝑛∈𝐴𝑔𝑥_𝑛,∀𝑛≥0, (73) where {𝑢_𝑛} is defined by 𝑢_𝑛−𝑢_𝑛+1≤(1+𝜀)𝐻𝐴𝑔𝑥_𝑛+1,𝐴𝑔𝑥_𝑛,∀𝑛≥0, (74) 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 weakly to some 𝑥∈Δ, and for any 𝑤∈𝑇𝑥, 𝑘∈𝐹𝑥, (𝑥,𝑤,𝑘) is a solution of the MVVI (15).

Proof. First of all, repeating the same arguments as those in the proof of Theorem 14, 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 (14), for any 𝑤∈𝑇𝑥 and 𝑘∈𝐹𝑥. In addition, in terms of Proposition 5 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)+𝜆𝐴(𝑔(𝑥))∶𝑋→𝐶 is 𝜆₀ strictly pseudocontractive and 𝛼₀ strongly accretive with 𝜆₀+𝛼₀≥1. Then by Lemma 17 we conclude that the mapping 𝑥↦𝑥−(𝑁(𝑇𝑥,𝐹𝑥)+𝜆𝐴(𝑔(𝑥))) is nonexpansive. Meantime, by Lemma 18 we know that 𝐺∶𝐶→𝐶 is also nonexpansive.
Without loss of generality we may assume that 𝑣=0 and 𝜆=1. Let 𝑝∈Δ and let 𝑟>0 be sufficiently large such that 𝑥₀∈𝐵_𝑟(𝑝)=∶𝐵. Observe that 𝑦_𝑛−𝑝≤𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛+−𝑝1−𝛽_𝑛−𝛾_𝑛𝐺𝑥_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛+−𝑝1−𝛽_𝑛−𝛾_𝑛𝑥_𝑛=𝑥−𝑝_𝑛−𝑝. (75) Utilizing (75) and repeating the same arguments as those in the proof of Theorem 14, we can derive 𝑥_𝑛∈𝐵 for all 𝑛≥0. Hence {𝑥_𝑛} is bounded.
Let us show that lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0.
Indeed, repeating the same arguments as those of (41) in the proof of Theorem 14, we obtain from (73) and (75) that for all 𝑛≥0𝑥_𝑛+1−𝑝²≤𝑥_𝑛−𝑝²−𝛼_𝑛𝜎_𝑛1−𝜎_𝑛𝜑₁×𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛−𝛼_𝑛1−𝛼_𝑛𝜑×𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛≤𝑥_𝑛−𝑝². (76) It is easy to see that the limit lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists. Meantime, it can be readily seen from (76) that 𝛼_𝑛𝜎_𝑛1−𝜎_𝑛𝜑₁𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+𝛼_𝑛1−𝛼_𝑛𝜑𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝², (77) which together with conditions (i), (iii), and the existence of lim_𝑛→∞‖𝑥_𝑛−𝑝‖, implies that lim_𝑛→∞𝜑₁𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛=0,lim_𝑛→∞𝜑𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛=0. (78) Utilizing the properties of 𝜑 and 𝜑₁, we get Lim_𝑛→∞𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛=0,lim_𝑛→∞𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛=0. (79) Note that 𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑦_𝑛−𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛. (80) So, we have lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (81) Also, observe that 𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛𝜎_𝑛𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+1−𝛼_𝑛𝑥_𝑛−𝑦_𝑛≤𝑁𝑤_𝑛,𝑘_𝑛+𝑢_𝑛+𝑥_𝑛−𝑦_𝑛. (82) Thus from (79) and (81) it follows that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (83)
Let us show that lim_𝑛→∞‖𝑥_𝑛−𝐺𝑥_𝑛‖=lim_𝑛→∞‖𝑥_𝑛−𝑆𝑥_𝑛‖=0.
Indeed, from (73) we obtain that 𝛽_𝑛𝛽_𝑛+𝛾_𝑛𝑆_𝑛𝑥_𝑛−𝐺𝑥_𝑛+𝛾_𝑛𝛽_𝑛+𝛾_𝑛𝑥_𝑛−𝐺𝑥_𝑛=𝛽_𝑛𝑆_𝑛𝑥_𝑛+𝛾_𝑛𝑥_𝑛𝛽_𝑛+𝛾_𝑛−𝐺𝑥_𝑛=𝑦_𝑛−1−𝛽_𝑛−𝛾_𝑛𝐺𝑥_𝑛𝛽_𝑛+𝛾_𝑛−𝐺𝑥_𝑛=𝑦_𝑛−𝐺𝑥_𝑛𝛽_𝑛+𝛾_𝑛. (84) Utilizing Lemma 10 we deduce from (73) and (84) that 𝑦_𝑛−𝑝²=𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛+−𝑝1−𝛽_𝑛−𝛾_𝑛𝐺𝑥_𝑛−𝑝²=𝛽_𝑛+𝛾_𝑛𝛽_𝑛𝛽_𝑛+𝛾_𝑛𝑆_𝑛𝑥_𝑛+𝛾−𝑝_𝑛𝛽_𝑛+𝛾_𝑛𝑥_𝑛+−𝑝1−𝛽_𝑛−𝛾_𝑛𝐺𝑥_𝑛−𝑝²≤𝛽_𝑛+𝛾_𝑛𝛽_𝑛𝛽_𝑛+𝛾_𝑛𝑆_𝑛𝑥_𝑛+𝛾−𝑝_𝑛𝛽_𝑛+𝛾_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛−𝛾_𝑛𝐺𝑥_𝑛−𝑝²−𝛽_𝑛+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛𝜑₂×𝛽_𝑛𝛽_𝑛+𝛾_𝑛𝑆_𝑛𝑥_𝑛−𝐺𝑥_𝑛+𝛾_𝑛𝛽_𝑛+𝛾_𝑛𝑥_𝑛−𝐺𝑥_𝑛=𝛽_𝑛+𝛾_𝑛𝛽_𝑛𝛽_𝑛+𝛾_𝑛𝑆_𝑛𝑥_𝑛+𝛾−𝑝_𝑛𝛽_𝑛+𝛾_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛−𝛾_𝑛𝐺𝑥_𝑛−𝑝²−𝛽_𝑛+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛𝜑₂×1𝛽_𝑛+𝛾_𝑛𝑦_𝑛−𝐺𝑥_𝑛≤𝛽_𝑛+𝛾_𝑛𝛽_𝑛𝛽_𝑛+𝛾_𝑛𝑆_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝛽_𝑛+𝛾_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝛽_𝑛+𝛾_𝑛²𝜑₃𝑆_𝑛𝑥_𝑛−𝑥_𝑛+1−𝛽_𝑛−𝛾_𝑛𝐺𝑥_𝑛−𝑝²−𝛽_𝑛+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛𝜑₂×1𝛽_𝑛+𝛾_𝑛𝑦_𝑛−𝐺𝑥_𝑛≤𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝜑₃𝑆_𝑛𝑥_𝑛−𝑥_𝑛+1−𝛽_𝑛−𝛾_𝑛𝐺𝑥_𝑛−𝑝²−𝛽_𝑛+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛𝜑₂×1𝛽_𝑛+𝛾_𝑛𝑦_𝑛−𝐺𝑥_𝑛≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝜑₃𝑆_𝑛𝑥_𝑛−𝑥_𝑛+1−𝛽_𝑛−𝛾_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛𝜑₂×1𝛽_𝑛+𝛾_𝑛𝑦_𝑛−𝐺𝑥_𝑛=𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝜑₃𝑆_𝑛𝑥_𝑛−𝑥_𝑛−𝛽_𝑛+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛𝜑₂×1𝛽_𝑛+𝛾_𝑛𝑦_𝑛−𝐺𝑥_𝑛, (85) which hence implies that 𝛽_𝑛𝛾_𝑛𝜑₃𝑆_𝑛𝑥_𝑛−𝑥_𝑛+𝛽_𝑛+𝛾_𝑛1−𝛽_𝑛−𝛾_𝑛𝜑₂×1𝛽_𝑛+𝛾_𝑛𝑦_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛. (86) Utilizing condition (ii) we conclude from (81) and (86) that lim_𝑛→∞𝜑₃𝑆_𝑛𝑥_𝑛−𝑥_𝑛=0,lim_𝑛→∞𝜑₂1𝛽_𝑛+𝛾_𝑛𝑦_𝑛−𝐺𝑥_𝑛=0. (87) From the properties of 𝜑₂ and 𝜑₃ and condition (ii), it immediately follows that lim_𝑛→∞𝑆_𝑛𝑥_𝑛−𝑥_𝑛=0,lim_𝑛→∞𝑦_𝑛−𝐺𝑥_𝑛=0. (88) Observe that 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆_𝑛𝑥_𝑛+𝑆_𝑛𝑥_𝑛−𝑆𝑥_𝑛,𝑥_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝐺𝑥_𝑛. (89) Thus, utilizing Lemma 11 and the assumption on {𝑆_𝑛}, from (81) and (88) we conclude that lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0,lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0. (90)
Next, let us show that {𝑥_𝑛} converges weakly to some 𝑥∈Δ.
Indeed, repeating the same arguments as those in the proof of Theorem 14, we can prove that {𝑥_𝑛} converges weakly to some 𝑥∈Δ. This completes the proof.

Theorem 20. 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 and 𝐹∶𝑋→𝐶𝐵(𝑋) and let 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 2. Let 𝐵_𝑖∶𝐶→𝑋 be 𝜆_𝑖 strictly pseudocontractive and 𝛼_𝑖 strongly accretive with 𝜆_𝑖+𝛼_𝑖≥1 for 𝑖=1,2. Let {𝑆_𝑖}^∞_𝑖=0 be a countable family of nonexpansive mappings of 𝐶 into itself such that Δ∶=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Ω∩Γ≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) with 1−(𝜆_𝑖/(1+𝜆_𝑖))(1−(1−𝛼_𝑖)/𝜆_𝑖)≤𝜇_𝑖≤1 for 𝑖=1,2. Let {𝛼_𝑛},{𝛽_𝑛}⊂[0,1] and {𝜎_𝑛}⊂(0,∞) such that lim_𝑛→∞𝛼_𝑛𝜎_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛𝜎_𝑛=∞. For arbitrary 𝑥₀∈𝐶 define the sequence {𝑥_𝑛} iteratively by (73), where {𝑢_𝑛} is defined by (74) for any 𝑤_𝑛∈𝑇𝑥_𝑛,𝑘_𝑛∈𝐹𝑥_𝑛 and some 𝜀>0. Then, there exists 𝑑>0 such that for 0<𝛼_𝑛𝜎_𝑛≤𝑑, for all 𝑛≥0, {𝑥_𝑛} converges strongly to 𝑝∈Δ provided lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0; in this case, for any 𝑤∈𝑇𝑥, 𝑘∈𝐹𝑥, (𝑥,𝑤,𝑘) is a solution of the MVVI (15).

Proof. First of all, repeating the same arguments as those in the proof of Theorem 14, we can prove that for any 𝑣∈𝐶,𝜆>0, there exists a point 𝑥∈𝐷(𝐴)=𝐶 such that (𝑥,𝑤,𝑘) is a solution of the MVVI (14), for any 𝑤∈𝑇𝑥 and 𝑘∈𝐹𝑥. In addition, in terms of Proposition 5 we know that 𝑉+𝜆𝐴∘𝑔 is a single-valued mapping due to the fact that 𝑉+𝜆𝐴∘𝑔 is 𝜙-strongly accretive. Meantime, by Lemma 18 we know that 𝐺∶𝐶→𝐶 is nonexpansive.
Without loss of generality we may assume that 𝑣=0 and 𝜆=1. Let 𝑝∈Δ. Then from (73) we deduce that for all 𝑛≥0𝑦_𝑛−𝑝≤𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛+−𝑝1−𝛽_𝑛−𝛾_𝑛𝐺𝑥_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛+−𝑝1−𝛽_𝑛−𝛾_𝑛𝑥_𝑛=𝑥−𝑝_𝑛−𝑝. (91) Repeating the same arguments as those of the remainder in the proof of Theorem 15, we can derive the desired result.

Conflict of Interests

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

Acknowledgments

This research was partially supported to the first author by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second and third authors, therefore, acknowledge with thanks DSR for technical and financial support. The authors are also thankful to the referees for their valuable suggestion/comments and appreciation.