Strong Convergence Theorems for a Generalized Mixed Equilibrium Problem and a Family of Total Quasi-𝝓-Asymptotically Nonexpansive Multivalued Mappings in Banach Spaces

Abstract

The main purpose of this paper is by using a hybrid algorithm to find a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of common fixed points for a infinite family of total quasi-𝜙-asymptotically nonexpansive multivalued mapping in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results announced by some authors.

1. Introduction

Throughout this paper, we always assume that 𝑋 is a real Banach space with the dual 𝑋^∗, 𝐶 is a nonempty closed convex subset of 𝑋, and 𝐽∶𝑋→2^𝑋 is the normalized duality mapping defined by𝐽(𝑥)=𝑓^∗∈𝑋^∗∶𝑥,𝑓^∗=‖𝑥‖²=𝑓^∗2,∀𝑥∈𝐸. (1.1) In the sequel, we use 𝐹(𝑇) to denote the set of fixed points of a mapping 𝑇 and use 𝓡 and 𝓡⁺ to denote the set of all real numbers and the set of all nonnegative real numbers, respectively. We denote by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥 the strong convergence and weak convergence of a sequence {𝑥_𝑛}, respectively.

Let Θ∶𝐶×𝐶→𝓡 be a bifunction, 𝜓∶𝐶→𝓡 a real valued function, and 𝐴∶𝐶→𝑋^∗ a nonlinear mapping. The so-called generalized mixed equilibrium problem is to find 𝑢∈𝐶 such thatΘ𝑢,𝑦+𝐴𝑢,𝑦−𝑢+𝜓𝑦−𝜓(𝑢)≥0,∀𝑦∈𝐶. (1.2) The set of solutions to (1.2) is denoted by Ω, that is,Ω=𝑢∈𝐶∶Θ𝑢,𝑦+𝐴𝑢,𝑦−𝑢+𝜓𝑦−𝜓(𝑢)≥0,∀𝑦∈𝐶. (1.3) Special examples are follows.(i)If 𝐴≡0, the problem (1.2) is equivalent to finding 𝑢∈𝐶 such that Θ𝑢,𝑦+𝜓𝑦−𝜓(𝑢)≥0,∀𝑦∈𝐶, (1.4) which is called the mixed equilibrium problem (MEP) [1].(ii)If Θ≡0, the problem (1.2) is equivalent to finding 𝑢∈𝐶 such that 𝐴𝑢,𝑦−𝑢+𝜓𝑦−𝜓(𝑢)≥0,∀𝑦∈𝐶, (1.5) which is called the mixed variational inequality of Browder type (VI) [2].

A Banach space 𝑋 is said to be strictly convex if ‖𝑥+𝑦‖/2<1 for all 𝑥,𝑦∈𝑈={𝑧∈𝑋∶‖𝑧‖=1} with 𝑥≠𝑦. 𝑋 is said to be uniformly convex if, for each 𝜖∈(0,2], there exists 𝛿>0 such that ‖𝑥+𝑦‖/2<1−𝛿 for all 𝑥,𝑦∈𝑈 with ‖𝑥−𝑦‖≥𝜖. 𝑋 is said to be smooth if the limitlim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (1.6) exists for all 𝑥,𝑦∈𝑈. 𝑋 is said to be uniformly smooth if the above limit is attained uniformly in 𝑥,𝑦∈𝑈.

Remark 1.1. The following basic properties of a Banach space 𝑋 can be found in Cioranescu [3].(i)If 𝑋 is uniformly smooth, then 𝐽 is uniformly continuous on each bounded subset of 𝑋.(ii)If 𝑋 is a reflexive and strictly convex Banach space, then 𝐽⁻¹ is norm-weak-continuous.(iii)If 𝑋 is a smooth, strictly convex, and reflexive Banach space, then 𝐽 is single-valued, one-to-one and onto.(iv)A Banach space 𝑋 is uniformly smooth if and only if 𝑋^∗ is uniformly convex.(v)Each uniformly convex Banach space 𝑋 has the Kadec-Klee property, that is, for any sequence {𝑥_𝑛}⊂𝑋, if 𝑥_𝑛⇀𝑥∈𝑋 and ‖𝑥_𝑛‖→‖𝑥‖, then 𝑥_𝑛→𝑥.

Let 𝑋 be a smooth Banach space. We always use 𝜙∶𝑋×𝑋→ℛ⁺ to denote the Lyapunov functional defined by𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝑋. (1.7) It is obvious from the definition of the function 𝜙 that‖𝑥‖−𝑦²≤𝜙𝑥,𝑦≤‖𝑥‖+𝑦²,∀𝑥,𝑦∈𝑋. (1.8) Following Alber [4], the generalized projection Π_𝐶∶𝑋→𝐶 is defined byΠ_𝐶(𝑥)=arginf_𝑦∈𝐶𝜙𝑦,𝑥,∀𝑥∈𝑋. (1.9)

Lemma 1.2 (see [4]). Let 𝑋 be a smooth, strictly convex, and reflexive Banach space and 𝐶 a nonempty closed convex subset of 𝑋. Then, the following conclusions hold:(a)𝜙(𝑥,Π_𝐶𝑦)+𝜙(Π_𝐶𝑦,𝑦)≤𝜙(𝑥,𝑦) for all 𝑥∈𝐶 and 𝑦∈𝑋,(b)if 𝑥∈𝑋 and 𝑧∈𝐶, then 𝑧=Π_𝐶𝑥if𝑓𝑧−𝑦,𝐽𝑥−𝐽𝑧≥0,∀𝑦∈𝐶, (1.10)(c)for 𝑥,𝑦∈𝑋,𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦.
Let 𝑋 be a smooth, strictly convex, and reflexive Banach space, 𝐶 a nonempty closed convex subset of 𝑋, and 𝑇∶𝐶→𝐶 a mapping. A point 𝑝∈𝐶 is said to be an asymptotic fixed point of T if there exists a sequence {𝑥_𝑛}⊂𝐶 such that 𝑥_𝑛⇀𝑝 and ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0. We denoted the set of all asymptotic fixed points of 𝑇 by 𝐹(𝑇).

Definition 1.3. (1) A mapping 𝑇∶𝐶→𝐶 is said to be relatively nonexpansive [5] if 𝐹(𝑇)≠∅,𝐹(𝑇)=𝐹(𝑇) and 𝜙𝑝,𝑇𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑇). (1.11)
(2) A mapping 𝑇∶𝐶→𝐶 is said to be closed if, for any sequence {𝑥_𝑛}⊂𝐶 with 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛→𝑦, 𝑇𝑥=𝑦.
Let 𝐶 be a nonempty closed convex subset of a Banach space 𝑋. Let 𝑁(𝐶) be the family of nonempty subsets of 𝐶.

Definition 1.4. (1) Let 𝑇∶𝐶→𝑁(𝐶) be a multivalued mapping and 𝑞 a point in 𝐶. The definitions of 𝑇𝑞,𝑇²𝑞,𝑇³𝑞,…,𝑇^𝑛𝑞,𝑛≥1 are as follows: 𝑇𝑞∶=𝑞₁∶𝑞₁𝑞,𝑇∈𝑇₂𝑞=𝑇𝑇𝑞∶=_{𝑞1∈𝑇(𝑞)}𝑇𝑞₁,𝑇³𝑞=𝑇𝑇²𝑞∶=_{𝑞2∈𝑇2(𝑞)}𝑇𝑞₂,⋮𝑇^𝑛𝑞=𝑇𝑇^𝑛−1𝑞∶=_{𝑞𝑛−1∈𝑇𝑛−1(𝑞)}𝑇𝑞^𝑛−1,𝑛≥1. (1.12)
(2) Let 𝑇∶𝐶→𝑁(𝐶) be a multivalued mapping. A point 𝑝∈𝐶 is said to be an asymptotic fixed point of 𝑇 if there exists a sequence {𝑥_𝑛}⊂𝐶 such that 𝑥_𝑛⇀𝑝 and lim_𝑛→∞𝑑(𝑥_𝑛,𝑇(𝑥_𝑛))=0. We denoted the set of all asymptotic fixed points of 𝑇 by 𝐹(𝑇).
(3) A multivalued mapping 𝑇∶𝐶→𝑁(𝐶) is said to be relatively nonexpansive [5] if 𝐹(𝑇)≠∅,𝐹(𝑇)=𝐹(𝑇) and 𝜙𝑝,𝑤≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑤∈𝑇𝑥,𝑝∈𝐹(𝑇). (1.13)
(4) A multivalued mapping 𝑇∶𝐶→𝑁(𝐶) is said to be closed if, for any sequence {𝑥_𝑛}⊂𝐶 with 𝑥_𝑛→𝑥 and 𝑤_𝑛∈𝑇(𝑥_𝑛) with 𝑤_𝑛→𝑦, then 𝑦∈𝑇𝑥.

Definition 1.5. (1) A multivalued mapping 𝑇∶𝐶→𝑁(𝐶) is said to be quasi-ϕ-nonexpansive if 𝐹(𝑇)≠∅ and 𝜙𝑝,𝑤≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑤∈𝑇𝑥,𝑝∈𝐹(𝑇). (1.14)
(2) A multivalued mapping 𝑇∶𝐶→𝑁(𝐶) is said to be quasi-ϕ-asymptotically nonexpansive if 𝐹(𝑇)≠∅ and there exists a real sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that 𝜙𝑝,𝑤_𝑛≤𝑘_𝑛𝜙𝑝,𝑥,∀𝑛≥1,𝑥∈𝐶,𝑤_𝑛∈𝑇^𝑛𝑥,𝑝∈𝐹(𝑇). (1.15)
(3) A multivalued mapping 𝑇∶𝐶→𝑁(𝐶) is said to be total quasi-ϕ-asymptotically nonexpansive if 𝐹(𝑇)≠∅ and there exist nonnegative real sequences {𝜈_𝑛},{𝜇_𝑛} with 𝜈_𝑛→0,𝜇_𝑛→0 (as 𝑛→∞) and a strictly increasing continuous function 𝜁∶ℛ⁺→ℛ⁺ with 𝜁(0)=0 such that for all 𝑥∈𝐶,𝑝∈𝐹(𝑇)𝜙𝑝,𝑤_𝑛≤𝜙𝑝,𝑥+𝜈_𝑛𝜁𝜙𝑝,𝑥+𝜇_𝑛,∀𝑛≥1,𝑤_𝑛∈𝑇^𝑛𝑥. (1.16)

Definition 1.6. (1) Let {𝑇_𝑖}^∞_𝑖=1∶𝐶→𝑁(𝐶) be a sequence of mappings. {𝑇_𝑖}^∞_𝑖=1 is said to be a family of uniformly total quasi-ϕ-asymptotically nonexpansive multivalued mappings if ∩^∞_𝑖=1𝐹(𝑇_𝑖)≠∅ and there exist nonnegative real sequences {𝜈_𝑛},{𝜇_𝑛} with 𝜈_𝑛→0,𝜇_𝑛→0 (as 𝑛→∞) and a strictly increasing continuous function 𝜁∶ℛ⁺→ℛ⁺ with 𝜁(0)=0 such that for all 𝑖≥1,𝑥∈𝐶,𝑝∈∩^∞_𝑖=1𝐹(𝑇_𝑖)𝜙𝑝,𝑤_𝑛,𝑖≤𝜙𝑝,𝑥+𝜈_𝑛𝜁𝜙𝑝,𝑥+𝜇_𝑛,∀𝑤_𝑛,𝑖∈𝑇^𝑛_𝑖𝑥,∀𝑛≥1. (1.17)
(2) A total quasi-𝜙-asymptotically nonexpansive multivalued mapping 𝑇∶𝐶→𝑁(𝐶) is said to be uniformly L-Lipschitz continuous if there exists a constant 𝐿>0 such that 𝑤_𝑛−𝑠_𝑛≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐶,𝑤_𝑛∈𝑇^𝑛𝑥,𝑠_𝑛∈𝑇^𝑛𝑦,𝑛≥1. (1.18)

In 2005, Matsushita and Takahashi [5] proved weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping in a uniformly convex and uniformly smooth Banach space 𝑋. In 2008, Plubtieng and Ungchittrakool [6] proved the strong convergence theorems to approximate a fixed point of two relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space 𝑋. In 2010, Chang et al. [7] obtained the strong convergence theorem for an infinite family of quasi-𝜙-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space 𝑋 with Kadec-Klee property. In 2011, Chang et al. [8] proved some approximation theorems of common fixed points for countable families of total quasi-𝜙-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space 𝑋 with Kadec-Klee property. In 2011, Homaeipour and Razani [9] proved weak and strong convergence theorems for a single relatively nonexpansive multivalued mapping in a uniformly convex and uniformly smooth Banach space 𝑋. On the other hand, In 2009, Zhang [10] proved the strong convergence theorem for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of fixed points of a finite family of quasi-𝜙-asymptotically nonexpansive mappings in a uniformly smooth and uniformly convex Banach space. Recently, Tang [11], Cho et al. [12–21], and Noor et al. [22–26] extended the finite family of quasi-𝜙-asymptotically nonexpansive mappings to infinite family of quasi-𝜙-asymptotically nonexpansive mappings.

Motivated and inspired by the researches going on in this direction, the purpose of this paper is by using the hybrid iterative algorithm to find a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of fixed points of a infinite family of total quasi-𝜙-asymptotically nonexpansive multivalued mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. In order to get the strong convergence theorems, the hybrid algorithms are presented and used to approximate the fixed point. The results presented in the paper improve and extend some recent results announced by some authors.

2. Preliminaries

Lemma 2.1 (see [8]). Let 𝑋 be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and 𝐶 a nonempty closed convex set of 𝑋. Let {𝑥_𝑛} and {𝑦_𝑛} be two sequences in 𝐶 such that 𝑥_𝑛→𝑝 and 𝜙(𝑥_𝑛,𝑦_𝑛)→0, where 𝜙 is the function defined by (1.7), and then 𝑦_𝑛→𝑝.

Lemma 2.2. Let 𝑋 and 𝐶 be as in Lemma 2.1. Let 𝑇∶𝐶→𝑁(𝐶) be a closed and total quasi-𝜙-asymptotically nonexpansive multivalued mapping with nonnegative real sequences {𝜈_𝑛},{𝜇_𝑛} and a strictly increasing continuous function 𝜁∶ℛ⁺→ℛ⁺ such that 𝜈_𝑛→0,𝜇_𝑛→0 (as 𝑛→∞), and 𝜁(0)=0. If 𝜇₁=0, then the fixed point set 𝐹(𝑇) is a closed and convex subset of 𝐶.

Proof. Letting {𝑥_𝑛} be a sequence in 𝐹(𝑇) with 𝑥_𝑛→𝑝 (as 𝑛→∞), we prove that 𝑝∈𝐹(𝑇). In fact, by the assumption that 𝑇 is a total quasi-𝜙-asymptotically nonexpansive multivalued mapping and 𝜇₁=0, we have 𝜙𝑥_𝑛,𝑢≤𝜙𝑥_𝑛,𝑝+𝜈₁𝜁𝜙𝑥_𝑛,𝑝,∀𝑢∈𝑇𝑝. (2.1) Furthermore, we have 𝜙𝑝,𝑢=lim_𝑛→∞𝜙𝑥_𝑛,𝑢≤lim_𝑛→∞𝜙𝑥_𝑛,𝑝+𝜈₁𝜁𝜙𝑥_𝑛,𝑝=0,∀𝑢∈𝑇𝑝. (2.2) By Lemma 1.2(c), 𝑝=𝑢. Hence, 𝑝∈𝑇𝑝. This implies that 𝑝∈𝐹(𝑇), that is, 𝐹(𝑇) is closed.
Next, we prove that 𝐹(𝑇) is convex. For any 𝑥,𝑦∈𝐹(𝑇),𝑡∈(0,1), putting 𝑞=𝑡𝑥+(1−𝑡)𝑦, we prove that 𝑞∈𝐹(𝑇). Indeed, let {𝑢_𝑛} be a sequence generated by 𝑢₁𝑢∈𝑇𝑞,₂∈𝑇𝑢₁⊂𝑇²𝑢𝑞,₃∈𝑇𝑢₂⊂𝑇³⋮𝑢𝑞,_𝑛∈𝑇𝑢_𝑛−1⊂𝑇^𝑛⋮𝑞, (2.3) In view of the definition of 𝜙(𝑥,𝑦), for all 𝑢_𝑛∈𝑇𝑢_𝑛−1⊂𝑇^𝑛𝑞, we have 𝜙𝑞,𝑢_𝑛=𝑞²−2𝑞,𝐽𝑢_𝑛+𝑢_𝑛²=𝑞²−2𝑡𝑥,𝐽𝑢_𝑛−2(1−𝑡)𝑦,𝐽𝑢_𝑛+𝑢_𝑛²=𝑞²+𝑡𝜙𝑥,𝑢_𝑛+(1−𝑡)𝜙𝑦,𝑢_𝑛−𝑡‖𝑥‖²−(1−𝑡)𝑦² (2.4) since 𝑡𝜙𝑥,𝑢_𝑛+(1−𝑡)𝜙𝑦,𝑢_𝑛≤𝑡𝜙𝑥,𝑞+𝜈_𝑛𝜁𝜙𝑥,𝑞+𝜇_𝑛+(1−𝑡)𝜙𝑦,𝑞+𝜈_𝑛𝜁𝜙𝑦,𝑞+𝜇_𝑛=𝑡‖𝑥‖²−2𝑥,𝐽𝑞+𝑞²+𝜈_𝑛𝜁𝜙𝑥,𝑞+𝜇_𝑛+(1−𝑡)𝑦²−2𝑦,𝐽𝑞+𝑞²+𝜈_𝑛𝜁𝜙𝑦,𝑞+𝜇_𝑛=𝑡‖𝑥‖²+(1−𝑡)𝑦²−𝑞²+𝑡𝜈_𝑛𝜁𝜙𝑥,𝑞+(1−𝑡)𝜈_𝑛𝜁𝜙𝑦,𝑞+𝜇_𝑛. (2.5) Substituting (2.5) into (2.4) and simplifying it, we have 𝜙𝑞,𝑢_𝑛≤𝑡𝜈_𝑛𝜁𝜙𝑥,𝑞+(1−𝑡)𝜈_𝑛𝜁𝜙𝑦,𝑞+𝜇_𝑛⟶0(𝑛⟶∞). (2.6) By Lemma 2.1, we have 𝑢_𝑛→𝑞 (as 𝑛→∞). This implies that 𝑢_𝑛+1→𝑞 (as 𝑛→∞). Since 𝑇 is closed, we have 𝑞∈𝑇𝑞, that is, 𝑞∈𝐹(𝑇).
This completes the proof of Lemma 2.2.

Lemma 2.3 (see [7]). Let 𝑋 be a uniformly convex Banach space, 𝑟>0, a positive number, and 𝐵_𝑟(0) a closed ball of 𝑋. Then, for any given sequence {𝑥_𝑛}^∞_𝑛=1⊂𝐵_𝑟(0) and for any given sequence {𝜆_𝑛}^∞_𝑛=1 of positive numbers with Σ^∞_𝑛=1𝜆_𝑛=1, there exists a continuous, strictly increasing, and convex function 𝑔∶[0,2𝑟)→[0,∞) with 𝑔(0)=0 such that for any positive integers 𝑖,𝑗 with 𝑖<𝑗, ^∞_𝑛=1𝜆_𝑛𝑥_𝑛²≤^∞_𝑛=1𝜆_𝑛𝑥_𝑛²−𝜆_𝑖𝜆_𝑗𝑔𝑥_𝑖−𝑥_𝑗. (2.7)
For solving the generalized mixed equilibrium problem, let us assume that the function 𝜓∶𝐶→ℛ is convex and lower semicontinuous, the nonlinear mapping 𝐴∶𝐶→𝑋^∗ is continuous and monotone, and the bifunction Θ∶𝐶×𝐶→ℛ satisfies the following conditions:(A₁)Θ(𝑥,𝑥)=0,forall𝑥∈𝐶, (A₂)Θ is monotone, that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)≤0,forall𝑥,𝑦∈𝐶,(A₃)limsup_𝑡↓0Θ(𝑥+𝑡(𝑧−𝑥),𝑦)≤Θ(𝑥,𝑦),forall𝑥,𝑦,𝑧∈𝐶, (A₄) the function 𝑦↦Θ(𝑥,𝑦) is convex and lower semicontinuous.

Lemma 2.4. Let 𝑋 be a smooth, strictly convex and reflexive Banach space and 𝐶 a nonempty closed convex subset of 𝑋. Let Θ∶𝐶×𝐶→ℛ be a bifunction satisfying conditions (A₁)–(A₄). Let 𝑟>0 and 𝑥∈𝑋. Then, the following hold.(i) [27] There exists 𝑧∈𝐶 such that Θ+1𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶. (2.8)(ii) [28] Define a mapping 𝑇_𝑟∶𝑋→𝐶 by 𝑇_𝑟+1𝑥=𝑧∈𝐶∶Θ𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶,𝑥∈𝑋. (2.9) Then, the following conclusions hold:(a)𝑇_𝑟 is single-valued,(b)𝑇_𝑟 is a firmly nonexpansive-type mapping, that is, forall𝑧,𝑦∈𝑋, 𝑇_𝑟(𝑧)−𝑇_𝑟𝑦,𝐽𝑇_𝑟(𝑧)−𝐽𝑇_𝑟𝑦≤𝑇_𝑟(𝑧)−𝑇_𝑟𝑦,𝐽𝑧−𝐽𝑦, (2.10)(c)𝐹(𝑇_𝑟)=𝐸𝑃(Θ)=𝐹(𝑇_𝑟),(d)𝐸𝑃(Θ) is closed and convex,(e)𝜙(𝑞,𝑇_𝑟(𝑥))+𝜙(𝑇_𝑟(𝑥),𝑥)≤𝜙(𝑞,𝑥),forall𝑞∈𝐹(𝑇_𝑟).

Lemma 2.5 (see [10]). Let 𝑋 be a smooth, strictly convex, and reflexive Banach space and 𝐶 a nonempty closed convex subset of 𝑋. Let 𝐴∶𝐶→𝑋^∗ be a continuous and monotone mapping, 𝜓∶𝐶→ℛ a lower semicontinuous and convex function, and Θ∶𝐶×𝐶→ℛ a bifunction satisfying conditions (A₁)–(A₄). Let 𝑟>0 be any given number and 𝑥∈𝑋 any given point. Then, the following hold.
(i) There exists 𝑢∈𝐶 such that forall𝑦∈𝐶Θ+𝑦(𝑢)+1𝑢,𝑦𝐴𝑢,𝑦−𝑢+𝜓−𝜓𝑟𝑦−𝑢,𝐽𝑢−𝐽𝑥≥0. (2.11)
(ii) If one defines a mapping 𝐾_𝑟∶𝐶→𝐶 by 𝐾_𝑟(𝑥)=+𝑦(𝑢)+1𝑢∈𝐶∶Θ𝑢,𝑦𝐴𝑢,𝑦−𝑢+𝜓−𝜓𝑟𝑦−𝑢,𝐽𝑢−𝐽𝑥≥0,∀𝑦∈𝐶,𝑥∈𝐶, (2.12) then, the mapping 𝐾_𝑟 has the following properties:(a)𝐾_𝑟 is single-valued,(b)𝐾_𝑟 is a firmly nonexpansive-type mapping, that is, forall𝑧,𝑦∈𝑋𝐾_𝑟(𝑧)−𝐾_𝑟𝑦,𝐽𝐾_𝑟(𝑧)−𝐽𝐾_𝑟𝑦≤𝐾_𝑟(𝑧)−𝐾_𝑟𝑦,𝐽𝑧−𝐽𝑦, (2.13)(c)𝐹(𝐾_𝑟)=Ω=𝐹(𝐾_𝑟),(d)Ω is a closed convex set of 𝐶,(e)𝜙(𝑝,𝐾_𝑟(𝑧))+𝜙(𝐾_𝑟(𝑧),𝑧)≤𝜙(𝑝,𝑧),forall𝑝∈𝐹(𝐾_𝑟),𝑧∈𝑋.

Remark 2.6. It follows from Lemma 2.4 that the mapping 𝐾_𝑟∶𝐶→𝐶 defined by (2.12) is a relatively nonexpansive mapping. Thus, it is quasi-𝜙-nonexpansive.

3. Main Results

In this section, we will use the hybrid iterative algorithm to find a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of fixed points of a infinite family of total quasi-𝜙-asymptotically nonexpansive multivalued mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property.

Theorem 3.1. Let 𝑋 be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and 𝐶 a nonempty closed and convex subset of 𝑋. Let Θ∶𝐶×𝐶→ℛ be a bifunction satisfying conditions (A₁)–(A₄), 𝐴∶𝐶→𝑋^∗ a continuous and monotone mapping, and 𝜓∶𝐶→ℛ a lower semicontinuous and convex function. Let {𝑇_𝑖}^∞_𝑖=1∶𝐶→𝑁(𝐶) be an infinite family of closed and uniformly total quasi-𝜙-asymptotically nonexpansive multivalued mappings with nonnegative real sequences {𝜈_𝑛},{𝜇_𝑛} and a strictly increasing continuous function 𝜁∶ℛ⁺→ℛ⁺ such that 𝜇₁=0, 𝜈_𝑛→0,𝜇_𝑛→0 (as 𝑛→∞) and 𝜁(0)=0 and for each 𝑖≥1, 𝑇_𝑖 is uniformly 𝐿_𝑖-Lipschitz continuous. Let 𝑥₀∈𝐶, 𝐶₀=𝐶, and let {𝑥_𝑛} be a sequence generated by 𝑥_𝑛+1=_𝐶𝑛+1𝑥₀,𝐶_𝑛+1=𝜈∈𝐶_𝑛∶𝜙𝜈,𝑢_𝑛≤𝜙𝜈,𝑥_𝑛+𝜉_𝑛𝑦,∀𝑛≥0,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛=𝐽⁻¹𝛽_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝐽𝑤_𝑛,𝑖,𝑢_𝑛∈𝐶suchthatΘ𝑢,∀𝑦∈𝐶,_𝑛,𝑦+𝐴𝑢_𝑛,𝑦−𝑢_𝑛+𝜓𝑦−𝜓𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛≥0, (3.1) where 𝑤_𝑛,𝑖∈𝑇^𝑛_𝑖𝑥_𝑛,forall𝑛≥1,𝑖≥1,𝜉_𝑛=𝜈_𝑛sup_𝑝∈ℱ𝜁(𝜙(𝑝,𝑥_𝑛))+𝜇_𝑛, Π_𝐶𝑛+1 is the generalized projection of 𝑋 onto 𝐶_𝑛+1, and {𝛼_𝑛} and {𝛽_𝑛,0,𝛽_𝑛,𝑖} are sequences in [0,1] satisfying the following conditions:(a)for each 𝑛≥0, 𝛽_𝑛,0+Σ^∞_𝑖=1𝛽_𝑛,𝑖=1,(b)liminf_𝑛→∞𝛽_𝑛,0𝛽_𝑛𝑖>0 for any 𝑖≥1,(c)0≤𝛼_𝑛≤𝛼<1 for some 𝛼∈(0,1).
If 𝐺∶=ℱ∩Ω=∩^∞_𝑖=1𝐹(𝑇_𝑖)∩Ω is a nonempty and bounded subset of 𝐶, then the sequence {𝑥_𝑛} converges strongly to Π_𝐺𝑥₀.

Proof. First, we define two functions 𝐻∶𝐶×𝐶→ℛ and 𝐾_𝑟∶𝐶→𝐶 by 𝐻+𝑦(𝑥)𝐾𝑥,𝑦=Θ𝑥,𝑦𝐴𝑥,𝑦−𝑥+𝜓−𝜓,∀𝑥,𝑦∈𝐶,_𝑟(𝑥)=+1𝑢∈𝐶∶𝐻𝑢,𝑦𝑟𝑦−𝑢,𝐽𝑢−𝐽𝑥≥0,∀𝑦∈𝐶,𝑥∈𝐶. (3.2)
By Lemma 2.5, we know that the function 𝐻 satisfies conditions (A₁)–(A₄) and 𝐾_𝑟 has properties (a)–(e). Therefore, (3.1) is equivalent to 𝑥_𝑛+1=_𝐶𝑛+1𝑥₀,𝐶_𝑛+1=𝜈∈𝐶_𝑛∶𝜙𝜈,𝑢_𝑛≤𝜙𝜈,𝑥_𝑛+𝜉_𝑛𝑦,∀𝑛≥0,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛=𝐽⁻¹𝛽_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝐽𝑤_𝑛,𝑖,𝑢_𝑛∈𝐶suchthat𝐻𝑢,∀𝑦∈𝐶,_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛≥0. (3.3)
Now we divide the proof of Theorem 3.1 into six steps.
(i)  ℱ and 𝐶_𝑛 are closed and convex for each 𝑛≥0.
In fact, it follows from Lemma 2.2 that 𝐹(𝑇_𝑖),𝑖≥1, is a closed and convex subset of 𝐶. Therefore, ℱ is a closed and convex subset 𝐶.
Again by the assumption, 𝐶₀=𝐶 is closed and convex. Suppose that 𝐶_𝑛 is closed and convex for some 𝑛≥1. Since the condition 𝜙(𝜈,𝑦_𝑛)≤𝜙(𝜈,𝑥_𝑛)+𝜉_𝑛 is equivalent to 2𝜈,𝐽𝑥_𝑛−𝐽𝑦_𝑛≤𝑥_𝑛²−𝑦_𝑛²+𝜉_𝑛,𝑛=1,2,…, (3.4) the set 𝐶_𝑛+1=𝜈∈𝐶_𝑛∶2𝜈,𝐽𝑥_𝑛−𝐽𝑦_𝑛≤𝑥_𝑛²−𝑦_𝑛²+𝜉_𝑛 (3.5) is closed and convex. Therefore, 𝐶_𝑛 is closed and convex for each 𝑛≥0.
(ii)  {𝑥_𝑛} is bounded and {𝜙(𝑥_𝑛,𝑥₀)} is a convergent sequence.
Indeed, it follows from (3.1) and Lemma 1.2(a) that for all 𝑛≥0,𝑢∈𝐹(𝑇)𝜙𝑥_𝑛,𝑥₀=𝜙_𝐶𝑛𝑥₀,𝑥₀≤𝜙𝑢,𝑥₀−𝜙𝑢,_𝐶𝑛𝑥₀≤𝜙𝑢,𝑥₀. (3.6) This implies that {𝜙(𝑥_𝑛,𝑥₀)} is bounded. By virtue of (1.3), we know that {𝑥_𝑛} is bounded.
In view of the structure of {𝐶_𝑛}, we have 𝐶_𝑛+1⊂𝐶_𝑛,𝑥_𝑛=Π_𝐶𝑛𝑥₀ and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀. This implies that 𝑥_𝑛+1∈𝐶_𝑛 and 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀,∀𝑛≥0. (3.7) Therefore, {𝜙(𝑥_𝑛,𝑥₀)} is a convergent sequence.
(iii)  𝐺∶=ℱ∩Ω⊂𝐶_𝑛 for all 𝑛≥0.
Indeed, it is obvious that 𝐺⊂𝐶₀=𝐶. Suppose that 𝐺⊂𝐶_𝑛 for some 𝑛∈𝒩. Since 𝑢_𝑛=𝐾_𝑟𝑛𝑦_𝑛, by Lemma 2.5 and Remark 2.6, 𝐾_𝑟𝑛 is quasi-𝜙-nonexpansive. Hence, for any given 𝑢∈𝐺⊂𝐶_𝑛 and 𝑛≥1 we have 𝜙𝑢,𝑢_𝑛=𝜙𝑢,𝐾_𝑟𝑛𝑦_𝑛≤𝜙𝑢,𝑦_𝑛=𝜙𝑢,𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛=‖𝑢‖²−2𝑢,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛+𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛²≤‖𝑢‖²−2𝛼_𝑛𝑢,𝐽𝑥_𝑛−21−𝛼_𝑛𝑢,𝐽𝑧_𝑛+𝛼_𝑛𝑥_𝑛²+1−𝛼_𝑛𝑧_𝑛²=𝛼_𝑛𝜙𝑢,𝑥_𝑛+1−𝛼_𝑛𝜙𝑢,𝑧_𝑛. (3.8) Furthermore, it follows from Lemma 2.3 that for any 𝑢∈𝐺⊂𝐶_𝑛,𝑤_𝑛,𝑖∈𝑇^𝑛_𝑖𝑥_𝑛, and 𝑖≥1 we have 𝜙𝑢,𝑧_𝑛=𝜙𝑢,𝐽⁻¹𝛽_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝐽𝑤_𝑛,𝑖=‖𝑢‖²−2𝑢,𝛽_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝐽𝑤_𝑛,𝑖+𝛽_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝐽𝑤_𝑛,𝑖²≤‖𝑢‖²−2𝛽_𝑛,0𝑢,𝐽𝑥_𝑛−2^∞_𝑖=1𝛽_𝑛,𝑖𝑢,𝐽𝑤_𝑛,𝑖+𝛽_𝑛,0𝑥_𝑛²+^∞_𝑖=1𝛽_𝑛,𝑖𝑤_𝑛,𝑖²−𝛽_𝑛,0𝛽_𝑛,𝑙𝑔𝐽𝑥_𝑛−𝐽𝑤_𝑛,𝑙=𝛽_𝑛,0𝜙𝑢,𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝜙𝑢,𝑤_𝑛,𝑖−𝛽_𝑛,0𝛽_𝑛,𝑙𝑔𝐽𝑥_𝑛−𝐽𝑤_𝑛,𝑙≤𝛽_𝑛,0𝜙𝑢,𝑥_𝑛+^∞_𝑖=1𝛽_𝑛,𝑖𝜙𝑢,𝑥_𝑛+𝜈_𝑛𝜁𝜙𝑢,𝑥_𝑛+𝜇_𝑛−𝛽_𝑛,0𝛽_𝑛,𝑙𝑔𝐽𝑥_𝑛−𝐽𝑤_𝑛,𝑙≤𝜙𝑢,𝑥_𝑛+𝜈_𝑛sup_𝑝∈ℱ𝜁𝜙𝑝,𝑥_𝑛+𝜇_𝑛−𝛽_𝑛,0𝛽_𝑛,𝑙𝑔𝐽𝑥_𝑛−𝐽𝑤_𝑛,𝑙=𝜙𝑢,𝑥_𝑛+𝜉_𝑛−𝛽_𝑛,0𝛽_𝑛,𝑙𝑔𝐽𝑥_𝑛−𝐽𝑤_𝑛,𝑙. (3.9) Substituting (3.9) into (3.8) and simplifying it, we have for all 𝑢∈𝐺𝜙𝑢,𝑢_𝑛≤𝜙𝑢,𝑦_𝑛≤𝜙𝑢,𝑥_𝑛+1−𝛼_𝑛𝜉_𝑛−1−𝛼_𝑛𝛽_𝑛,0𝛽_𝑛,𝑙𝑔𝐽𝑥_𝑛−𝐽𝑤_𝑛,𝑙≤𝜙𝑢,𝑥_𝑛+𝜉_𝑛−1−𝛼_𝑛𝛽_𝑛,0𝛽_𝑛,𝑙𝑔𝐽𝑥_𝑛−𝐽𝑤_𝑛,𝑙≤𝜙𝑢,𝑥_𝑛+𝜉_𝑛, (3.10) that is, 𝑢∈𝐶_𝑛+1 and so 𝐺⊂𝐶_𝑛+1 for all 𝑛≥0.
By the way, in view of the assumption on {𝜈_𝑛},{𝜇_𝑛} we have 𝜉_𝑛=𝜈_𝑛sup_𝑝∈ℱ𝜁𝜙𝑝,𝑥_𝑛+𝜇_𝑛⟶0(𝑛⟶∞). (3.11)
(iv) {𝑥_𝑛} converges strongly to some point 𝑝^∗∈𝐶.
In fact, since {𝑥_𝑛} is bounded and 𝑋 is reflexive, there exists a subsequence {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑝^∗ (some point in 𝐶). Since 𝐶_𝑛 is closed and convex and 𝐶_𝑛+1⊂𝐶_𝑛, this implies that 𝐶_𝑛 is weakly closed and 𝑝^∗∈𝐶_𝑛 for each 𝑛≥0. In view of 𝑥_𝑛𝑖=Π_𝐶𝑛𝑖𝑥₀, we have 𝜙𝑥_𝑛𝑖,𝑥₀≤𝜙𝑝^∗,𝑥₀,∀𝑛_𝑖≥0. (3.12) Since the norm ‖⋅‖ is weakly lower semicontinuous, we have liminf_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀=liminf_{𝑛𝑖→∞}𝑥_𝑛𝑖²−2𝑥_𝑛𝑖,𝐽𝑥₀+𝑥₀²≥𝑝^∗2−2𝑝^∗,𝐽𝑥₀+𝑥₀²=𝜙𝑝^∗,𝑥₀, (3.13) and so 𝜙𝑝^∗,𝑥₀≤liminf_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀≤limsup_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀≤𝜙𝑝^∗,𝑥₀. (3.14) This implies that lim_{𝑛𝑖→∞}𝜙(𝑥_𝑛𝑖,𝑥₀)=𝜙(𝑝^∗,𝑥₀), and so ‖𝑥_𝑛𝑖‖→‖𝑝^∗‖. Since 𝑥_𝑛𝑖⇀𝑝^∗, by virtue of Kadec-Klee property of 𝑋, we obtain that lim_{𝑛𝑖→∞}𝑥_𝑛𝑖=𝑝^∗. (3.15) Since {𝜙(𝑥_𝑛,𝑥₀)} is convergent, this together with lim_{𝑛𝑖→∞}𝜙(𝑥_𝑛𝑖,𝑥₀)=𝜙(𝑝^∗,𝑥₀), shows that lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀)=𝜙(𝑝^∗,𝑥₀). If there exists some sequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑗→𝑞, then from Lemma 1.2(a) we have that 𝜙𝑝^∗,𝑞=lim_{𝑛𝑖,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥_𝑛𝑗=lim_{𝑛𝑖,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,_𝐶𝑛𝑗𝑥₀≤lim_{𝑛𝑖,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥₀−𝜙_𝐶𝑛𝑗𝑥₀,𝑥₀=lim_{𝑛𝑖,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥₀−𝜙𝑥_𝑛𝑗,𝑥₀=𝜙𝑝^∗,𝑥₀−𝜙𝑝^∗,𝑥₀=0. (3.16) This implies that 𝑝^∗=𝑞 and lim_𝑛→∞𝑥_𝑛=𝑝^∗. (3.17)
(v) Now we prove that 𝑝^∗∈𝐺=ℱ∩Ω.
First, we prove that 𝑝^∗∈ℱ. In fact, since 𝑥_𝑛+1∈𝐶_𝑛+1⊂𝐶_𝑛, it follows from (3.1) and (3.17) that 𝜙𝑥_𝑛+1,𝑦_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+𝜉_𝑛⟶0(𝑛⟶∞). (3.18) By the virtue of Lemma 2.1, we have lim_𝑛→∞𝑦_𝑛=𝑝^∗. (3.19) From (3.10), for any 𝑢∈ℱ and 𝑤_𝑛,𝑖∈𝑇^𝑛_𝑖𝑥_𝑛, we have 𝜙𝑢,𝑦_𝑛≤𝜙𝑢,𝑥_𝑛+𝜉_𝑛−1−𝛼_𝑛𝛽_𝑛,0𝛽_𝑛,𝑙𝑔𝐽𝑥_𝑛−𝐽𝑤_𝑛,𝑙, (3.20) that is, 1−𝛼_𝑛𝛽_𝑛,0𝛽_𝑛,𝑙𝑔𝐽𝑥_𝑛−𝐽𝑤_𝑛,𝑙≤𝜙𝑢,𝑥_𝑛+𝜉_𝑛−𝜙𝑢,𝑦_𝑛⟶0(𝑛⟶∞). (3.21) By conditions (b) and (c) it is shown that lim_𝑛→∞𝑔(‖𝐽𝑥_𝑛−𝐽𝑤_𝑛,𝑙‖)=0. In view of property of 𝑔, we have 𝐽𝑥_𝑛−𝐽𝑤_𝑛,𝑙⟶0(𝑛⟶∞). (3.22) Since 𝐽𝑥_𝑛→𝐽𝑝^∗, this implies that 𝐽𝑤_𝑛,𝑙→𝐽𝑝^∗. From Remark 1.1(ii) it yields 𝑤_𝑛,𝑙⇀𝑝^∗(𝑛⟶∞),∀𝑙≥1. (3.23) Again since 𝑤_𝑛,𝑙−𝑝^∗=𝐽𝑤_𝑛,𝑙−𝐽𝑝^∗≤𝐽𝑤_𝑛,𝑙−𝐽𝑝^∗⟶0(𝑛⟶∞), (3.24) this together with (3.23) and the Kadec-Klee property of 𝑋 shows that lim_𝑛→∞𝑤_𝑛,𝑙=𝑝^∗,∀𝑙≥1. (3.25) Let {𝑠_𝑛,𝑙} be a sequence generated by 𝑠_2,𝑙∈𝑇_𝑙𝑤_1,𝑙⊂𝑇²_𝑙𝑥₁,𝑠_3,𝑙∈𝑇_𝑙𝑤_2,𝑙⊂𝑇³_𝑙𝑥₂,⋮𝑠_𝑛+1,𝑙∈𝑇_𝑙𝑤_𝑛,𝑙⊂𝑇_𝑙^𝑛+1𝑥_𝑛,⋮ (3.26) By the assumption that each 𝑇_𝑖 is uniformly 𝐿_𝑖-Lipschitz continuous, for any 𝑤_𝑛,𝑙∈𝑇^𝑛_𝑙𝑥_𝑛 and 𝑠_𝑛+1,𝑙∈𝑇_𝑙𝑤_𝑛⊂𝑇_𝑙^𝑛+1𝑥_𝑛 we have 𝑠_𝑛+1,𝑙−𝑤_𝑛,𝑙≤𝑠_𝑛+1,𝑙−𝑤_𝑛+1,𝑙+𝑤_𝑛+1,𝑙−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑤_𝑛,𝑙≤𝐿_𝑙+1𝑥_𝑛+1−𝑥_𝑛+𝑤_𝑛+1,𝑙−𝑥_𝑛+1+𝑥_𝑛−𝑤_𝑛,𝑙. (3.27) This together with (3.17) and (3.27) shows that lim_𝑛→∞‖𝑠_𝑛+1,𝑙−𝑤_𝑛,𝑙‖=0 and lim_𝑛→∞𝑠_𝑛+1,𝑙=𝑝^∗. In view of the closeness of 𝑇_𝑙, it yields that 𝑝^∗∈𝑇𝑝^∗, that is, 𝑝^∗∈𝐹(𝑇_𝑙). By the arbitrariness of 𝑙≥1, we have 𝑝^∗∈ℱ=^∞_𝑖=1𝐹𝑇_𝑖. (3.28)
Next, we prove that 𝑝^∗∈Ω. Since 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛, it follows from (3.1) and (3.17) that 𝜙𝑥_𝑛+1,𝑢_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+𝜉_𝑛⟶0(𝑛⟶∞). (3.29) Since 𝑥_𝑛→𝑝^∗, by virtue of Lemma 2.1 we have lim_𝑛→∞𝑢_𝑛=𝑝^∗. (3.30) This together with (3.19) shows that ‖𝑢_𝑛−𝑦_𝑛‖→0 and lim_𝑛→∞‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖→0. By the assumption that 𝑟_𝑛≥𝑎,forall𝑛≥0, we have lim_𝑛→∞𝐽𝑢_𝑛−𝐽𝑦_𝑛𝑟_𝑛=0. (3.31) Since 𝐻(𝑢_𝑛,𝑦)+(1/𝑟_𝑛)⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,forall𝑦∈𝐶, by condition (A₁), we have 1𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛≥−𝐻𝑢_𝑛,𝑦≥𝐻𝑦,𝑢_𝑛,∀𝑦∈𝐶. (3.32) By the assumption that 𝑦↦𝐻(𝑥,𝑦) is convex and lower semicontinuous, letting 𝑛→∞ in (3.32), from (3.30) and (3.31), we have 𝐻(𝑦,𝑝^∗)≤0,forall𝑦∈𝐶.
For 𝑡∈(0,1] and 𝑦∈𝐶, letting 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑝^∗, there are 𝑦_𝑡∈𝐶 and 𝐻(𝑦_𝑡,𝑝^∗)≤0. By conditions (A₁) and (A₄), we have 0=𝐻𝑦_𝑡,𝑦_𝑡≤𝑡𝐻𝑦_𝑡,𝑦+(1−𝑡)𝐻𝑦_𝑡,𝑝^∗≤𝑡𝐻𝑦_𝑡,𝑦. (3.33) Dividing both sides of the above equation by 𝑡, we have 𝐻(𝑦_𝑡,𝑦)≤0,forall𝑦∈𝐶. Letting 𝑡↓0, from condition (A₃), we have 𝐻(𝑝^∗,𝑦)≤0,forall𝑦∈𝐶, that is, 𝑝^∗∈Ω, and 𝑝^∗∈𝐺=ℱ∩Ω.
(vi) We prove that 𝑥_𝑛→𝑝^∗=Π_𝐺𝑥₀.
Let 𝑞=Π_𝐺𝑥₀. Since 𝑞∈𝐺⊂𝐶_𝑛 and 𝑥_𝑛=Π_𝐶𝑛𝑥₀, we have 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑞,𝑥₀,∀𝑛≥0. (3.34) This implies that 𝜙𝑝^∗,𝑥₀=lim_𝑛→∞𝜙𝑥_𝑛,𝑥₀≤𝜙𝑞,𝑥₀. (3.35) In view of the definition of Π_𝐺𝑥₀, from (3.35) we have 𝑝^∗=𝑞. Therefore, 𝑥_𝑛→𝑝^∗=Π_𝐺𝑥₀.
This completes the proof of Theorem 3.1.

4. Conclusions

Recently the extended general variational inequalities have been introduced and studied in Noor [24, 25]. We would like to point out that the results and the methods presented in this paper will be used to study this kind of extended general variational inequalities and its multivalued version.

Acknowledgment

The authors would like to express their thanks to the referees for their helpful suggestions and comments.