A New Iterative Method for Solving a System of Generalized Mixed Equilibrium Problems for a Countable Family of Generalized Quasi-ϕ-Asymptotically Nonexpansive Mappings

Abstract

By using a specific way of choosing the indexes, we introduce an up-to-date iterative algorithm for approximating common fixed points of a countable family of generalized quasi-ϕ-asymptotically nonexpansive mappings and obtain a strong convergence theorem under some suitable conditions. As application, an iterative solution to a system of generalized mixed equilibrium problems is studied. The results extend those of other authors, in which the involved mappings consist of just finite families.

1. Introduction

Throughout this paper, we assume that 𝐸 is a real Banach space with its dual 𝐸^∗, 𝐶 is a nonempty closed convex subset of 𝐸, and 𝐽∶𝐸→2^𝐸∗ is the normalized duality mapping defined by 𝐽𝑥=𝑓∈𝐸^∗∶𝑥,𝑓=‖𝑥‖²=𝑓²,∀𝑥∈𝐸. (1) In the sequel, we use 𝐹(𝑇) to denote the set of fixed points of a mapping 𝑇.

Definition 1. (1) [1] A mapping 𝑇∶𝐶→𝐶 is said to be generalized quasi-ϕ-asymptotically nonexpansive in the light of [1], if 𝐹(𝑇)≠∅, and there exist nonnegative real sequences {𝜈_𝑛} and {𝜇_𝑛} with 𝜈_𝑛,𝜇_𝑛→0 (as 𝑛→∞) such that 𝜙𝑝,𝑇^𝑛𝑥≤1+𝜈_𝑛𝜙𝑝,𝑥+𝜇_𝑛,∀𝑛≥1,𝑥∈𝐶,𝑝∈𝐹(𝑇), (2) where 𝜙∶𝐸×𝐸→ℝ denotes the Lyapunov functional defined by 𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝐸. (3) It is obvious from the definition of 𝜙 that ‖𝑥‖−𝑦²≤𝜙𝑥,𝑦≤‖𝑥‖+𝑦². (4)
(2) A mapping 𝑇∶𝐶→𝐶 is said to be uniformly L-Lipschitz continuous, if there exists a constant 𝐿>0 such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝐿𝑥−𝑦,∀𝑛≥1,𝑥,𝑦∈𝐶. (5)

Example 2. Let 𝐶 be a unit ball in a real Hilbert space 𝑙², and let 𝑇∶𝐶→𝐶 be a mapping defined by 𝑇𝑥₁,𝑥₂,…=0,𝑥²₁,𝑎₂𝑥₂,𝑎₃𝑥₃,…, (6) where {𝑎_𝑖} is a sequence in (0,1) satisfying ∏^∞_𝑖=2𝑎_𝑖=1/2. It is shown by Goebel and Kirk [2] that 𝜙𝑝,𝑇^𝑛𝑦≤1+𝜈_𝑛𝜙𝑝,𝑦+𝜇_𝑛,∀𝑛≥1,𝑦∈𝐶,𝑝∈𝐹(𝑇), (7) where 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖², 𝜈_𝑛=(2∏^𝑛_𝑖=2𝑎_𝑖)²−1, for all 𝑛≥1, and {𝜇_𝑛} is a nonnegative real sequence with 𝜇_𝑛→0 as 𝑛→∞. This shows that the mapping 𝑇 defined earlier is a generalized quasi-𝜙-asymptotically nonexpansive mapping.
Let 𝜃∶𝐶×𝐶→ℝ be a bifunction, 𝜓∶𝐶→ℝ a real valued function, and 𝐵∶𝐶→𝐸^∗ a nonlinear mapping. The so-called generalized mixed equilibrium problem (GMEP) is to find a 𝑢∈𝐶 such that 𝜃𝑢,𝑦+𝑦−𝑢,𝐵𝑢+𝜓𝑦−𝜓(𝑢)≥0,∀𝑦∈𝐶, (8) whose set of solutions is denoted by Ω(𝜃,𝐵,𝜓).

The equilibrium problem is a unifying model for several problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games, and others. It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems. Many authors have proposed some useful methods to solve the (equilibrium problem) EP, (generalized equilibrium problem) GEP, (mixed equilibrium problem) MEP, and GMEP. Concerning the weak and strong convergence of iterative sequences to approximate a common element of the set of solutions for the GMEP, the set of solutions to variational inequality problems and the set of common fixed points for relatively nonexpansive mappings, quasi-𝜙-nonexpansive mappings, and quasi-𝜙-asymptotically nonexpansive mappings have been studied by many authors in the setting of Hilbert or Banach spaces (e.g., see [3–16] and the references therein).

Inspired and motivated by the study mentioned earlier, in this paper, by using a specific way of choosing the indexes, we propose an up-to-date iteration scheme for approximating common fixed points of a countable family of generalized quasi-𝜙-asymptotically nonexpansive mappings and obtain a strong convergence theorem for solving a system of generalized mixed equilibrium problems. The results extend those of the authors, in which the involved mappings consist of just finite families.

2. Preliminaries

A Banach space 𝐸 is strictly convex if the following implication holds for 𝑥,𝑦∈𝐸: ‖𝑥‖=𝑦=1,𝑥≠𝑦⟹𝑥+𝑦2<1. (9) It is also said to be uniformly convex if for any 𝜖>0, there exists a 𝛿>0 such that ‖𝑥‖=𝑦=1,𝑥−𝑦≥𝜖⟹𝑥+𝑦2≤1−𝛿. (10) It is known that if 𝐸 is uniformly convex Banach space, then 𝐸 is reflexive and strictly convex. A Banach space 𝐸 is said to be smooth if lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (11) exists for each 𝑥,𝑦∈𝑆(𝐸)∶={𝑥∈𝐸∶‖𝑥‖=1}. In this case, the norm of 𝐸 is said to be Gâteaux differentiable. The space 𝐸 is said to have uniformly Gâteaux differentiable norm if for each 𝑦∈𝑆(𝐸), the limit (11) is attained uniformly for 𝑥∈𝑆(𝐸). The norm of 𝐸 is said to be Fréchet differentiable if for each 𝑥∈𝑆(𝐸), the limit (11) is attained uniformly for 𝑦∈𝑆(𝐸). The norm of 𝐸 is said to be uniformly Fréchet differentiable (and 𝐸 is said to be uniformly smooth) if the limit (11) is attained uniformly for 𝑥,𝑦∈𝑆(𝐸). Note that 𝐸 (𝐸^∗, resp.) is uniformly convex ⇔𝐸^∗ (𝐸, resp.) is uniformly smooth.

Following Alber [17], the generalized projection Π_𝐶∶𝐸→𝐶 is defined by Π_𝐶(𝑥)=arginf_𝑦∈𝐶𝜙𝑦,𝑥,∀𝑥∈𝐸. (12)

Lemma 3 (see [17]). Let E be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Then, the following conclusions hold:(1)𝜙(𝑥,Π_𝐶𝑦)+𝜙(Π_𝐶𝑦,𝑦)≤𝜙(𝑥,𝑦) for all 𝑥∈𝐶 and 𝑦∈𝐸;(2)if 𝑥∈𝐸 and 𝑧∈𝐶, then 𝑧=Π_𝐶𝑥⇔⟨𝑧−𝑦,𝐽𝑥−𝐽𝑧⟩≥0, for all 𝑦∈𝐶;(3)for 𝑥,𝑦∈𝐸, 𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦.

Remark 4. The following basic properties for a Banach space 𝐸 can be found in Cioranescu [18].(i)If 𝐸 is uniformly smooth, then 𝐽 is uniformly continuous on each bounded subset of 𝐸.(ii)If 𝐸 is reflexive and strictly convex, then 𝐽⁻¹ is norm-weak continuous.(iii)If 𝐸 is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping 𝐽∶𝐸→2^𝐸∗ is single valued, one-to-one, and onto.(iv)A Banach space 𝐸 is uniformly smooth if and only if 𝐸^∗ is uniformly convex.(v)Each uniformly convex Banach space 𝐸 has the Kadec-Klee property; that is, for any sequence {𝑥_𝑛}⊂𝐸, if 𝑥_𝑛⇀𝑥∈𝐸 and ‖𝑥_𝑛‖→‖𝑥‖, then 𝑥_𝑛→𝑥, where 𝑥_𝑛⇀𝑥 denotes that {𝑥_𝑛} converges weakly to 𝑥.

Lemma 5 (see [19]). Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and let C be a nonempty closed convex subset of E. 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 (3); then, 𝑦_𝑛→𝑝.

Lemma 6 (see [1]). Let E and C be the same as those in Lemma 5. Let 𝑇∶𝐶→𝐶 be a closed and generalized quasi-𝜙-asymptotically nonexpansive mapping with nonnegative real sequences {𝜈_𝑛} and {𝜇_𝑛}; then, the fixed point set F(T) of T is a closed and convex subset of C.

Lemma 7 (see [20]). Let 𝐸 be a real uniformly convex Banach space, and let 𝐵_𝑟(0) be the closed ball of 𝐸 with center at the origin and radius 𝑟>0. Then, there exists a continuous strictly increasing convex function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that 𝛼𝑥+𝛽𝑦²≤𝛼‖𝑥‖²+𝛽𝑦²−𝛼𝛽𝑔𝑥−𝑦 (13) for all 𝑥,𝑦,∈𝐵_𝑟(0), and 𝛼,𝛽∈[0,1] with 𝛼+𝛽=1.

3. Main Results

Theorem 8. Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, C a nonempty closed convex subset of E, and 𝑇_𝑖∶𝐶→𝐶, 𝑖=1,2,… a countable family of closed and generalized quasi-𝜙-asymptotically nonexpansive mappings with nonnegative real sequences {𝜈_𝑛^(𝑖)} and {𝜇_𝑛^(𝑖)} satisfying 𝜈_𝑛^(𝑖)→0 and 𝜇_𝑛^(𝑖)→0 (as 𝑛→∞ and for each 𝑖≥1), and each 𝑇_𝑖 is uniformly 𝐿_𝑖-Lipschitz continuous. Let {𝛼_𝑛} be a sequence in [0,𝜖] for some 𝜖∈(0,1), and let {𝛽_𝑛} be a sequence in [0,1] satisfying 0<liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛). Let {𝑥_𝑛} be the sequence generated by 𝑥₁∈𝐶;𝐶₁𝑦=𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛,𝐶_𝑛+1=𝑣∈𝐶_𝑛∶𝜙𝑣,𝑦_𝑛≤𝜙𝑣,𝑥_𝑛+𝜉_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₁,∀𝑛≥1, (14) where 𝜉_𝑛∶=𝜈^(𝑖𝑛)_𝑚𝑛sup_𝑝∈𝐹𝜙(𝑝,𝑥_𝑛)+𝜇^(𝑖𝑛)_𝑚𝑛, Π_𝐶𝑛+1 is the generalized projection of 𝐸 onto 𝐶_𝑛+1, and 𝑖_𝑛 and 𝑚_𝑛 satisfy the positive integer equation: 𝑛=𝑖+(𝑚−1)𝑚/2, 𝑚≥𝑖 (𝑚≥𝑖, 𝑛=1,2,…); that is, for each 𝑛≥1, there exist unique 𝑖_𝑛 and 𝑚_𝑛 such that 𝑖₁=1,𝑖₂=1,𝑖₃=2,𝑖₄=1,𝑖₅𝑖=2,₆=3,𝑖₇=1,𝑖₈𝑚=2,…;₁=1,𝑚₂=2,𝑚₃=2,𝑚₄=3,𝑚₅𝑚=3,₆=3,𝑚₇=4,𝑚₈=4,…. (15) If 𝐹∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖) is nonempty and bounded, then {𝑥_𝑛} converges strongly to Π_𝐹𝑥₁.

Proof. We divide the proof into several steps.(I)𝐹 and 𝐶_𝑛 (for all 𝑛≥1) both are closed and convex subsets in 𝐶.
In fact, it follows from Lemma 6 that each 𝐹(𝑇_𝑖) is a closed and convex subset of 𝐶, so is 𝐹. In addition, with 𝐶₁ (=𝐶) being closed and convex, we may assume that 𝐶_𝑛 is closed and convex for some 𝑛≥2. In view of the definition of 𝜙, we have that 𝐶_𝑛+1=𝑣∈𝐶∶𝜑(𝑣)≤𝑎∩𝐶_𝑛, (16) where 𝜑(𝑣)=2⟨𝑣,𝐽𝑥_𝑛−𝐽𝑦_𝑛⟩, and 𝑎=‖𝑥_𝑛‖²−‖𝑦_𝑛‖²+𝜉_𝑛. This shows that 𝐶_𝑛+1 is closed and convex.(II)𝐹 is a subset of ⋂^∞_𝑛=1𝐶_𝑛.
It is obvious that 𝐹⊂𝐶₁. Suppose that 𝐹⊂𝐶_𝑛 for some 𝑛≥2. Since 𝐸 is uniformly smooth, 𝐸^∗ is uniformly convex. Then, for any 𝑝∈𝐹⊂𝐶_𝑛, we have that 𝜙𝑝,𝑦_𝑛=𝜙𝑝,𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛=𝑝²−2𝑝,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛+𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛²≤𝑝²−2𝛼_𝑛𝑝,𝐽𝑥_𝑛−21−𝛼_𝑛𝑝,𝐽𝑧_𝑛+𝛼_𝑛𝑥_𝑛²+1−𝛼_𝑛𝑧_𝑛²=𝛼_𝑛𝜙𝑝,𝑥_𝑛+1−𝛼_𝑛𝜙𝑝,𝑧_𝑛. (17) Furthermore, it follows from Lemma 7 that for any 𝑝∈𝐹, we have that 𝜙𝑝,𝑧_𝑛=𝜙𝑝,𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛=𝑝²−2𝑝,𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛+𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛²≤𝑝²−2𝛽_𝑛𝑝,𝐽𝑥_𝑛−21−𝛽_𝑛×𝑝,𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛+𝛽_𝑛𝑥_𝑛²+1−𝛽_𝑛𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛=𝛽_𝑛𝜙𝑝,𝑥_𝑛+1−𝛽_𝑛𝜙𝑝,𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛−𝛽_𝑛1−𝛽_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛≤𝛽_𝑛𝜙𝑝,𝑥_𝑛+1−𝛽_𝑛×𝜙𝑝,𝑥_𝑛+𝜈^(𝑖𝑛)_𝑚𝑛sup_𝑝∈𝐹𝜙𝑝,𝑥_𝑛+𝜇^(𝑖𝑛)_𝑚𝑛−𝛽_𝑛1−𝛽_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛≤𝜙𝑝,𝑥_𝑛+𝜈^(𝑖𝑛)_𝑚𝑛sup_𝑝∈𝐹𝜙𝑝,𝑥_𝑛+𝜇^(𝑖𝑛)_𝑚𝑛−𝛽_𝑛1−𝛽_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛=𝜙𝑝,𝑥_𝑛+𝜉_𝑛−𝛽_𝑛1−𝛽_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛. (18) Substituting (18) into (17) and simplifying it, we have that 𝜙𝑝,𝑦_𝑛≤𝜙𝑝,𝑥_𝑛+1−𝛼_𝑛𝜉_𝑛≤𝜙𝑝,𝑥_𝑛+𝜉_𝑛. (19) This implies that 𝑝∈𝐶_𝑛+1, and so 𝐹⊂𝐶_𝑛+1.(III)𝑥_𝑛→𝑥^∗∈𝐶 as 𝑛→∞.
In fact, since 𝑥_𝑛=Π_𝐶𝑛𝑥₁, from Lemma 3 (2), we have that ⟨𝑥_𝑛−𝑦,𝐽𝑥₁−𝐽𝑥_𝑛⟩≥0, for all 𝑦∈𝐶_𝑛. Again, since 𝐹⊂⋂^∞_𝑛=1𝐶_𝑛, we have that ⟨𝑥_𝑛−𝑝,𝐽𝑥₁−𝐽𝑥_𝑛⟩≥0, for all 𝑝∈𝐹. It follows from Lemma 3 (1) that for each 𝑝∈𝐹 and for each 𝑛≥1, 𝜙𝑥_𝑛,𝑥₁=𝜙Π_𝐶𝑛𝑥₁,𝑥₁≤𝜙𝑝,𝑥₁−𝜙𝑝,𝑥_𝑛≤𝜙𝑝,𝑥₁, (20) which implies that {𝜙(𝑥_𝑛,𝑥₁)} is bounded, so is {𝑥_𝑛}. Since for all 𝑛≥1, 𝑥_𝑛=Π_𝐶𝑛𝑥₁ and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₁∈𝐶_𝑛+1⊂𝐶_𝑛, we have 𝜙(𝑥_𝑛,𝑥₁)≤𝜙(𝑥_𝑛+1,𝑥₁). This implies that {𝜙(𝑥_𝑛,𝑥₁)} is nondecreasing; hence, the limit lim_𝑛→∞𝜙𝑥_𝑛,𝑥₁exists. (21) Since 𝐸 is reflexive, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑥^∗∈𝐶 as 𝑖→∞. Since 𝐶_𝑛 is closed and convex and 𝐶_𝑛+1⊂𝐶_𝑛, this implies that 𝐶_𝑛 is weakly, closed and 𝑥^∗∈𝐶_𝑛 for each 𝑛≥1. In view of 𝑥_𝑛𝑖=Π_𝐶𝑛𝑖𝑥₁, we have that 𝜙𝑥_𝑛𝑖,𝑥₁≤𝜙𝑥^∗,𝑥₁,∀𝑖≥1. (22) Since the norm ‖⋅‖ is weakly lower semicontinuous, we have that liminf_𝑖→∞𝜙𝑥_𝑛𝑖,𝑥₁=liminf_𝑖→∞𝑥_𝑛𝑖²−2𝑥_𝑛𝑖,𝐽𝑥₁+𝑥₁²≥𝑥^∗2−2𝑥^∗,𝐽𝑥₁+𝑥₁²=𝜙𝑥^∗,𝑥₁, (23) and so 𝜙𝑥^∗,𝑥₁≤liminf_𝑖→∞𝜙𝑥_𝑛𝑖,𝑥₁≤limsup_𝑖→∞𝜙𝑥_𝑛𝑖,𝑥₁≤𝜙𝑥^∗,𝑥₁. (24) This implies that lim_𝑖→∞𝜙(𝑥_𝑛𝑖,𝑥₁)=𝜙(𝑥^∗,𝑥₁), and so ‖𝑥_𝑛𝑖‖→‖𝑥^∗‖ as 𝑖→∞. Since 𝑥_𝑛𝑖⇀𝑥^∗, by virtue of Kadec-Klee property of 𝐸, we obtain that lim_𝑖→∞𝑥_𝑛𝑖=𝑥^∗. (25) Since {𝜙(𝑥_𝑛,𝑥₁)} is convergent, this, together with lim_𝑖→∞𝜙(𝑥_𝑛𝑖,𝑥₁)=𝜙(𝑥^∗,𝑥₁), shows that lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₁)=𝜙(𝑥^∗,𝑥₁). If there exists some subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that 𝑥_𝑛𝑗→𝑦 as 𝑗→∞, then, from Lemma 3 (1), we have that 𝜙𝑥^∗,𝑦=lim_{𝑖,𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥_𝑛𝑗=lim_{𝑖,𝑗→∞}𝜙𝑥_𝑛𝑖,Π_𝐶𝑛𝑗𝑥₁≤lim_{𝑖,𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥₁−𝜙Π_𝐶𝑛𝑗𝑥₁,𝑥₁=lim_{𝑖,𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥₁−𝜙𝑥_𝑛𝑗,𝑥₁=𝜙𝑥^∗,𝑥₁−𝜙𝑥^∗,𝑥₁=0; (26) that is, 𝑥^∗=𝑦, and so lim_𝑛→∞𝑥_𝑛=𝑥^∗. (27)(IV)𝑥^∗ is some member of 𝐹.
Set 𝒦_𝑖={𝑘≥1∶𝑘=𝑖+(𝑚−1)𝑚/2,𝑚≥𝑖,𝑚∈ℕ} for each 𝑖≥1. Note that 𝜈^(𝑖𝑘)_𝑚𝑛=𝜈_𝑚^(𝑖)_𝑘 and 𝜇^(𝑖𝑘)_𝑚𝑛=𝜇_𝑚^(𝑖)_𝑘 whenever 𝑘∈𝒦_𝑖 for each 𝑖≥1. For example, by the definition of 𝒦₁, we have that 𝒦₁={1,2,4,7,11,16,…}, and 𝑖₁=𝑖₂=𝑖₄=𝑖₇=𝑖₁₁=𝑖₁₆=⋅⋅⋅=1. Then, we have that 𝜉_𝑘=𝜈_𝑚^(𝑖)_𝑘sup_𝑝∈𝐹𝜙𝑝,𝑥_𝑘+𝜇_𝑚^(𝑖)_𝑘,∀𝑘∈𝒦_𝑖. (28) Note that {𝑚_𝑘}_{𝑘∈𝒦𝑖}={𝑖,𝑖+1,𝑖+2,…}; that is, 𝑚_𝑘↑∞ as 𝒦_𝑖∋𝑘→∞. It follows from (27) and (28) that lim_{𝒦𝑖∋𝑘→∞}𝜉_𝑘=0. (29) Since 𝑥_𝑛+1∈𝐶_𝑛+1, it follows from (14), (27), and (29) that 𝜙𝑥_𝑘+1,𝑦_𝑘≤𝜙𝑥_𝑘+1,𝑥_𝑘+𝜉_𝑘⟶0 (30) as 𝒦_𝑖∋𝑘→∞. Since 𝑥_𝑘→𝑥^∗ as 𝒦_𝑖∋𝑘→∞, it follows from (30) and Lemma 5 that lim_{𝒦𝑖∋𝑘→∞}𝑦_𝑘=𝑥^∗. (31) Note that 𝑇^𝑚𝑘_𝑖𝑘=𝑇^𝑚𝑘_𝑖 whenever 𝑘∈𝒦_𝑖 for each 𝑖≥1. From (17) and (18), for any 𝑝∈𝐹, we have that 𝜙𝑝,𝑦_𝑘≤𝜙𝑝,𝑥_𝑘+𝜉_𝑘−1−𝛼_𝑘𝛽_𝑘1−𝛽_𝑘𝑔×𝐽𝑥_𝑘−𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘; (32) that is, 1−𝛼_𝑘𝛽_𝑘1−𝛽_𝑘𝑔𝐽𝑥_𝑘−𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘≤𝜙𝑝,𝑥_𝑘+𝜉_𝑘−𝜙𝑝,𝑦_𝑘⟶0𝒦_𝑖∋𝑘⟶∞. (33) This, together with assumption conditions imposed on the sequences {𝛼_𝑛} and {𝛽_𝑛}, shows that lim_{𝒦𝑖∋𝑘→∞}𝑔(‖𝐽𝑥_𝑘−𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘‖)=0. In view of property of 𝑔, we have that lim_{𝒦𝑖∋𝑘→∞}𝐽𝑥_𝑘−𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘=0. (34) In addition, 𝐽𝑥_𝑘→𝐽𝑥^∗ implies that lim_{𝒦𝑖∋𝑘→∞}𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘=𝐽𝑥^∗. From Remark 4 (ii), it yields that, as 𝒦_𝑖∋𝑘→∞, 𝑇^𝑚𝑘_𝑖𝑥_𝑘⇀𝑥^∗,∀𝑖≥1. (35) Again, since for each 𝑖≥1, as 𝒦_𝑖∋𝑘→∞, 𝑇^𝑚𝑘_𝑖𝑥_𝑘−𝑥^∗=𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘−𝐽𝑥^∗≤𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘−𝐽𝑥^∗⟶0. (36) This, together with (35) and the Kadec-Klee property of 𝐸, shows that lim_{𝒦𝑖∋𝑘→∞}𝑇^𝑚𝑘_𝑖𝑥_𝑘=𝑥^∗,∀𝑖≥1. (37) On the other hand, by the assumptions that for each 𝑖≥1, 𝑇_𝑖 is uniformly 𝐿_𝑖-Lipschitz continuous, and noting again that {𝑚_𝑘}_{𝑘∈𝒦𝑖}={𝑖,𝑖+1,𝑖+2,…}, that is, 𝑚_𝑘+1−1=𝑚_𝑘 for all 𝑘∈𝒦_𝑖, we then have 𝑇^𝑚𝑘+1_𝑖𝑥_𝑘−𝑇^𝑚𝑘_𝑖𝑥_𝑘≤𝑇^𝑚𝑘+1_𝑖𝑥_𝑘−𝑇^𝑚𝑘+1_𝑖𝑥_𝑘+1+𝑇^𝑚𝑘+1_𝑖𝑥_𝑘+1−𝑥_𝑘+1+𝑥_𝑘+1−𝑥_𝑘+𝑥_𝑘−𝑇^𝑚𝑘_𝑖𝑥_𝑘≤𝐿_𝑖+1𝑥_𝑘+1−𝑥_𝑘+𝑇^𝑚𝑘+1_𝑖𝑥_𝑘+1−𝑥_𝑘+1+𝑥_𝑘−𝑇^𝑚𝑘_𝑖𝑥_𝑘. (38) From (37) and 𝑥_𝑘→𝑥^∗, we have that lim_{𝒦𝑖∋𝑘→∞}‖𝑇^𝑚𝑘+1_𝑖𝑥_𝑘−𝑇^𝑚𝑘_𝑖𝑥_𝑘‖=0, and lim_{𝒦𝑖∋𝑘→∞}𝑇^𝑚𝑘+1_𝑖𝑥_𝑘=𝑥^∗; that is, lim_{𝒦𝑖∋𝑘→∞}𝑇_𝑖(𝑇^𝑚𝑘+1_𝑖⁻¹𝑥_𝑘)=𝑥^∗. It then follows that, for each 𝑖≥1, lim_{𝒦𝑖∋𝑘→∞}𝑇_𝑖𝑇^𝑚𝑘_𝑖𝑥_𝑘=𝑥^∗. (39) In view of the closeness of 𝑇_𝑖, it follows from (37) that 𝑇_𝑖𝑥^∗=𝑥^∗, namely, for each 𝑖≥1, 𝑥^∗∈𝐹(𝑇_𝑖), and, hence, 𝑥^∗∈𝐹. (V)𝑥^∗=Π_𝐹𝑥₁, and so 𝑥_𝑛→Π_𝐹𝑥₁ as 𝑛→∞.
Put 𝑢=Π_𝐹𝑥₁. Since 𝑢∈𝐹⊂𝐶_𝑛 and 𝑥_𝑛=Π_𝐶𝑛𝑥₁, we have that 𝜙(𝑥_𝑛,𝑥₁)≤𝜙(𝑢,𝑥₁), for all 𝑛≥1. Then, 𝜙𝑥^∗,𝑥₁=lim_𝑛→∞𝜙𝑥_𝑛,𝑥₁≤𝜙𝑢,𝑥₁, (40) which implies that 𝑥^∗=𝑢 since 𝑢=Π_𝐹𝑥₁, and, hence, 𝑥_𝑛→𝑥^∗=Π_𝐹𝑥₁.
This completes the proof.

4. Applications

Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸. Let {𝐵_𝑖}^∞_𝑖=1∶𝐶→𝐸^∗ be a sequence of 𝛽_𝑖-inverse strongly monotone mappings, {𝜓}^∞_𝑖=1∶𝐶→ℝ a sequence of lower semicontinuous and convex functions, and {𝜃_𝑖}^∞_𝑖=1∶𝐶×𝐶→ℝ a sequence of bifunctions satisfying the following conditions:(A₁)𝜃(𝑥,𝑥)=0;(A₂)𝜃 is monotone; that is, 𝜃(𝑥,𝑦)+𝜃(𝑦,𝑥)≤0;(A₃)limsup_𝑡↓0𝜃(𝑥+𝑡(𝑧−𝑥),𝑦)≤𝜃(𝑥,𝑦);(A₄) the mapping 𝑦↦𝜃(𝑥,𝑦) is convex and lower semicontinuous.

A system of generalized mixed equilibrium problems (GMEPs), for {𝜃_𝑖}^∞_𝑖=1, {𝐵_𝑖}^∞_𝑖=1, and {𝜓_𝑖}^∞_𝑖=1 is to find an 𝑥^∗∈𝐶 such that 𝜃_𝑖𝑥^∗,𝑦+𝑦−𝑥^∗,𝐵_𝑖𝑥^∗+𝜓_𝑖𝑦−𝜓_𝑖𝑥^∗≥0,∀𝑦∈𝐶,𝑖≥1, (41) whose set of common solutions is denoted by Ω∶=∩^∞_𝑖=1Ω_𝑖, where Ω_𝑖 denotes the set of solutions to generalized mixed equilibrium problem for 𝜃_𝑖, 𝐵_𝑖, and 𝜓_𝑖.

Define a countable family of mappings {𝑇_𝑟,𝑖}^∞_𝑖=1∶𝐸→𝐶 with 𝑟>0 as follows: 𝑇_𝑟,𝑖(𝑥)=𝑧∈𝐶∶𝜏_𝑖+1𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶,∀𝑖≥1, (42) where 𝜏_𝑖(𝑥,𝑦)=𝜃_𝑖(𝑥,𝑦)+⟨𝑦−𝑥,𝐵_𝑖𝑥⟩+𝜓_𝑖(𝑦)−𝜓_𝑖(𝑥), for all 𝑥,𝑦∈𝐶, 𝑖≥1. It has been shown by Zhang [15] that (1){𝑇_𝑟,𝑖}^∞_𝑖=1 is a sequence of single-valued mappings;(2){𝑇_𝑟,𝑖}^∞_𝑖=1 is a sequence of closed quasi-𝜙-nonexpansive mappings;(3)⋂^∞_𝑖=1𝐹(𝑇_𝑟,𝑖)=Ω.

Now, we have the following result.

Theorem 9. Let 𝐸 be the same as that in Theorem 8, and let C be a nonempty closed convex subset of E. Let {𝑇_𝑟,𝑖}^∞_𝑖=1∶𝐶→𝐶 be a sequence of mappings defined by (42) with 𝐹∶=⋂^∞_𝑖=1𝐹(𝑇_𝑟,𝑖)≠∅. Let {𝛼_𝑛} be a sequence in [0,𝜖] for some 𝜖∈(0,1), and let {𝛽_𝑛} be a sequence in [0,1] satisfying 0<liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛). Let {𝑥_𝑛} be the sequence generated by 𝑥₁∈𝐶;𝐶₁𝑦=𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇_{𝑟,𝑖𝑛}𝑥_𝑛,𝐶_𝑛+1=𝑣∈𝐶_𝑛∶𝜙𝑣,𝑦_𝑛≤𝜙𝑣,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₁,∀𝑛≥1, (43) where 𝑖_𝑛 satisfies the positive integer equation: 𝑛=𝑖+(𝑚−1)𝑚/2, and 𝑚≥𝑖 (𝑚≥𝑖, 𝑛=1,2,…). Then, {𝑥_𝑛} converges strongly to Π_𝐹𝑥₁ which is some solution to the system of generalized mixed equilibrium problems for {𝑇_𝑟,𝑖}^∞_𝑖=1.

Proof. Note that {𝑇_𝑟,𝑖}^∞_𝑖=1 are quasi-𝜙-nonexpansive mappings; so, they are obviously generalized quasi-𝜙-asymptotically nonexpansive. Therefore, this conclusion can be obtained immediately from Theorem 8.

Acknowledgments

The author is greatly grateful to the referees for their useful suggestions by which the contents of this paper are improved. This work was supported by the National Natural Science Foundation of China (Grant no. 11061037).