Strong Convergence to Common Fixed Points of a Countable Family of Asymptotically Strictly Quasi-𝜙-Pseudocontractions

Abstract

Based on an original idea, namely, a specific way of choosing the indexes of the involved mappings, we propose a new hybrid shrinking iteration scheme for approximating some common fixed points of a countable family of asymptotically strictly quasi-𝜙-pseudocontractions and obtain a strong convergence theorem in the framework of Banach space. Our result extends other authors, related results existing in the current literature. As application, an iterative solution to a system of equilibrium problems is provided.

1. Introduction

Fixed point theory as an important branch of nonlinear analysis theory has been applied in many disciplines. From the viewpoint of applications in real world, it is not only to show the existence of fixed points of nonlinear mappings but also to construct iterative algorithms to approximate their fixed points, which explains the popularity of iteration methods for finding the fixed points of various nonlinear mappings. For instance, iteration approximation of common fixed points of asymptotically nonexpansive mappings, an important class of nonlinear mappings introduced by Goebel and Kirk [1], has been studied by many authors (see, e.g., [2–7]).

In this paper, we discuss the approximation of common fixed points of a family of asymptotically strictly quasi-𝜙-pseudocontractions.

Throughout this paper we assume that 𝐸 is a real Banach space with 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 (see [8]). A mapping 𝑇∶𝐶→𝐶 is said to be an asymptotically strictly quasi- 𝜙-pseudocontraction, if 𝐹(𝑇)≠∅, and there exists a nonnegative real sequence {𝜇_𝑛} with 𝜇_𝑛→0 (as 𝑛→∞) and a constant 𝜅∈[0,1) such that 𝜙𝑝,𝑇^𝑛𝑥≤1+𝜇_𝑛𝜙𝑝,𝑥+𝜅𝜙𝑥,𝑇^𝑛𝑥,∀𝑛≥1,𝑥∈𝐶,𝑝∈𝐹(𝑇), (2) where 𝜙∶𝐸×𝐸→ℝ⁺ denotes the Lyapunov functional defined by 𝜙𝑥,𝑦=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+𝑦²,∀𝑥,𝑦∈𝐸, (3) where ⟨⋅,⋅⟩ stands for the duality product. It is obvious from the definition of 𝜙 that 𝜙𝑥,𝑦=𝜙(𝑥,𝑧)+𝜙𝑧,𝑦+2𝑥−𝑧,𝐽𝑧−𝐽𝑦,∀𝑥,𝑦,𝑧∈𝐸,‖𝑥‖−𝑦²≤𝜙𝑥,𝑦≤‖𝑥‖+𝑦²,∀𝑥,𝑦∈𝐸. (4)(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 in [8] that 𝜙𝑝,𝑇^𝑛𝑦≤1+𝜇_𝑛𝜙,(𝑇),𝑝,𝑦∀𝑛≥1,𝑦∈𝐶,𝑝∈𝐹 (7) where 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖²,𝜇_𝑛=(2∏^𝑛_𝑖=2𝑎_𝑖)²−1 for all 𝑛≥1. This shows that the mapping 𝑇 defined above is an asymptotically strictly quasi-𝜙-pseudocontraction.

Definition 3 (see [9]). A mapping 𝑇∶𝐶→𝐶 is said to be asymptotically regular on 𝐶 if, for any bounded subset 𝐾 of 𝐶, lim_𝑛→∞sup_𝑥∈𝐾𝑇^𝑛+1𝑥−𝑇^𝑛𝑥=0. (8)

In 2012, Zhang [9] used the following hybrid projection algorithm for a single asymptotically strictly quasi-𝜙-pseudocontraction to obtain strong convergence under a limit condition only in the framework of reflexive, strictly convex, and smooth Banach spaces such that both 𝐸 and 𝐸^∗ have the Kadec-Klee property 𝑥₀∈𝐸chosenarbitrarily;𝐶₁𝑥=𝐶,₁=Π_𝐶1𝑥₀,𝐶_𝑛+1=𝑢∈𝐶_𝑛∶𝜙𝑥_𝑛,𝑇^𝑛𝑥_𝑛≤2𝑥1−𝜅^𝑛−𝑢,𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑥_𝑛+𝜇_𝑛𝑀_𝑛,𝑥1−𝜅_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥1, (9) where 𝑀_𝑛=sup{𝜙(𝑝,𝑥_𝑛)∶𝑝∈𝐹(𝑇)} and Π_𝐶𝑛+1 is the generalized projection (see (10)) of 𝐸 onto 𝐶_𝑛+1. Their results improved the corresponding results of Zhou and Gao [10] and Qin et al. [8].

Inspired and motivated by the studies mentioned above, we introduce an up-to-date iteration scheme for approximating common fixed points of a countable family of asymptotically strictly quasi-𝜙-pseudocontractions and obtain a strong convergence theorem. The result extends the corresponding one for one map in [9].

2. Preliminaries

Following Alber [11], the generalized projection Π_𝐶∶𝐸→𝐶 is defined by Π_𝐶=arginf_𝑦∈𝐶𝜙𝑦,𝑥,∀𝑥∈𝐸. (10)

Lemma 4 (see [11]). Let 𝐸 be a smooth, strictly convex, and reflexive Banach space and C 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 5. The following basic properties for a Banach space 𝐸 can be found in [12]:(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 as 𝑛→∞.

Lemma 6 (see [13]). Let 𝐸 be a strictly convex and smooth Banach space, let 𝐶 be a nonempty and closed convex subset of 𝐸, and let 𝑇 be a quasi-𝜙-nonexpansive mapping from 𝐶 into 𝐸. Then 𝐹(𝑇) is closed and convex.

Lemma 7 (see [14]). The unique solutions to the positive integer equation ()𝑚𝑛=𝑖+𝑚−12,𝑚≥𝑖,𝑛=1,2,3,… (11) are ()𝑚𝑖=𝑛−𝑚−12,1𝑚=−2−12𝑛+4,𝑛=1,2,3,…, (12) where [𝑥] denotes the maximal integer that is not larger than 𝑥.

3. Main Results

Recall that an operator 𝑇 in a Banach space is closed if 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛→𝑦 as  𝑛→∞; then 𝑇𝑥=𝑦.

Theorem 8. Let 𝐸 be a reflexive, strictly convex, and smooth Banach space such that both 𝐸 and 𝐸^∗ have Kadec-Klee property, let C be a nonempty closed and convex subset of 𝐸, and {𝑇_𝑖}∶𝐶→𝐶 be a countable family of closed and asymptotically strictly quasi-𝜙-pseudocontractions with a nonnegative real sequence {𝜇_𝑛^(𝑖)} satisfying 𝜇_𝑛^(𝑖)→0(as 𝑛→∞ for each 𝑖≥1) and a sequence {𝜅_𝑖}⊂[0,1). Let {𝑥_𝑛} be the sequence generated by 𝑥₁∈𝐸;𝐶₁𝐶=𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑥_𝑛,𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛≤21−𝜅_𝑖𝑛⟨𝑥_𝑛−𝑧,𝐽𝑥_𝑛−𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛⟩+𝜇^(𝑖𝑛)_𝑚𝑛𝜈_𝑛1−𝜅_𝑖𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₁,∀𝑛≥1, (13) where 𝜈_𝑛∶=sup_𝑝∈𝐹𝜙(𝑝,𝑥_𝑛); Π_𝐶𝑛+1 is the generalized projection of 𝐸 onto 𝐶_𝑛+1; 𝑖_𝑛 and 𝑚_𝑛 are the solutions to the positive integer equation 𝑛=𝑖+((𝑚−1)𝑚/2)(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,𝑚₄𝑚=3,₅=3,𝑚₆=3,𝑚₇=4,𝑚₈=4,…. (14) If {𝑇_𝑖} are asymptotically regular on C and 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) Both 𝐹 and 𝐶_𝑛 (for all 𝑛≥1) are closed and convex subsets in 𝐶.
For each 𝑖≥1, the closedness of 𝐹(𝑇_𝑖) follows from that of 𝑇_𝑖, and hence 𝐹 is closed. In addition, using the same argument as presented in the proof of Theorem 2.1 in [9], we have that each 𝐹_𝑖 is convex and so is 𝐹.
Next, we show that 𝐶_𝑛 is closed and convex for all 𝑛≥1. 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 𝐶_𝑛+1 we have that 𝐶_𝑛+1=𝑧∈𝐶∶𝜑(𝑧)≤𝑎∩𝐶_𝑛, (15) where 𝜑(𝑧)=(2/𝜅_𝑖𝑛−1)⟨𝑥_𝑛−𝑧,𝐽𝑥_𝑛−𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛⟩ and 𝑎=𝜇^(𝑖𝑛)_𝑚𝑛(𝜈_𝑛/1−𝜅_𝑖𝑛)−𝜙(𝑥_𝑛,𝑇^𝑚𝑛_𝑖𝑛𝑥_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. It follows from (4) that 𝜙𝑝,𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛=𝜙𝑝,𝑥_𝑛+𝜙𝑥_𝑛,𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛+2⟨𝑝−𝑥_𝑛,𝐽𝑥_𝑛−𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛⟩ (16) for any 𝑝∈𝐹⊂𝐶_𝑛. Then, from (16) we have 𝜙𝑥_𝑛,𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛≤𝜇^(𝑖𝑛)_𝑚𝑛1−𝜅_𝑖𝑛𝜙𝑝,𝑥_𝑛+21−𝜅_𝑖𝑛𝑥_𝑛−𝑝,𝐽𝑥_𝑛−𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛≤𝜇^(𝑖𝑛)_𝑚𝑛𝜈_𝑛1−𝜅_𝑖𝑛𝜙𝑝,𝑥_𝑛+21−𝜅_𝑖𝑛𝑥_𝑛−𝑝,𝐽𝑥_𝑛−𝐽𝑇^𝑚𝑛_𝑖𝑛𝑥_𝑛. (17) This implies that 𝑝∈𝐶_𝑛+1, and so 𝐹⊂𝐶_𝑛+1.
(iii) Consider 𝑥_𝑛→𝑥^∗∈𝐶 as 𝑛→∞.
In fact, since 𝑥_𝑛=Π_𝐶𝑛𝑥₁, from Lemma 4 (2) we have ⟨𝑥_𝑛−𝑦,𝐽𝑥₁−𝐽𝑥_𝑛⟩≥0, for all 𝑦∈𝐶_𝑛. Again since 𝐹⊂∩^∞_𝑛=1𝐶_𝑛, we have ⟨𝑥_𝑛−𝑝,𝐽𝑥₁−𝐽𝑥_𝑛⟩≥0, for all 𝑝∈𝐹. It follows from Lemma 4 (1) that for each 𝑝∈𝐹 and for each 𝑛≥1, 𝜙𝑥_𝑛,𝑥₁=𝜙Π_𝐶𝑛𝑥₁,𝑥₁≤𝜙𝑝,𝑥₁−𝜙𝑝,𝑥_𝑛≤𝜙𝑝,𝑥₁, (18) which implies that {𝜙(𝑥_𝑛,𝑥₁)} is bounded and hence 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. (19) 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 𝜙𝑥_𝑛𝑖,𝑥₁≤𝜙𝑥^∗,𝑥₁,∀𝑖≥1. (20) Since the norm ‖⋅‖ is weakly lower semicontinuous, we have liminf_𝑖→∞𝜙𝑥_𝑛𝑖,𝑥₁=liminf_𝑖→∞𝑥_𝑛𝑖²−2𝑥_𝑛𝑖,𝐽𝑥₁+𝑥₁²≥𝑥^∗2−2⟨𝑥^∗,𝐽𝑥₁⟩+𝑥₁²=𝜙𝑥^∗,𝑥₁, (21) and so 𝜙𝑥^∗,𝑥₁≤liminf_𝑖→∞𝜙𝑥_𝑛𝑖,𝑥₁≤limsup_𝑖→∞𝜙𝑥_𝑛𝑖,𝑥₁≤𝜙𝑥^∗,𝑥₁. (22) 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^∗. (23) 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 4 (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. (24) That is, 𝑥^∗=𝑦 and so lim_𝑛→∞𝑥_𝑛=𝑥^∗. (25)
(iv) 𝑥^∗ is a member of 𝐹.
Set ℕ_𝑖={𝑘∈ℕ∶𝑘=𝑖+((𝑚−1)𝑚/2),𝑚≥𝑖,𝑚∈ℕ} for each 𝑖≥1. Note that 𝑇^𝑚𝑘_𝑖𝑘=𝑇^𝑚𝑘_𝑖,𝜅_𝑖𝑘=𝜅_𝑖, and 𝜇^(𝑖𝑘)_𝑚𝑘=𝜇_𝑚^(𝑖)_𝑘 whenever 𝑘∈ℕ_𝑖 for each 𝑖≥1. For example, by Lemma 7 and the definition of ℕ₁, we have ℕ₁={1,2,4,7,11,16,…} and 𝑖₁=𝑖₂=𝑖₄=𝑖₇=𝑖₁₁=𝑖₁₆=⋅⋅⋅=1. By virtue of 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₁∈𝐶_𝑛+1, we have 𝜙𝑥_𝑘,𝑇^𝑚𝑘_𝑖𝑥_𝑘≤21−𝜅_𝑖⟨𝑥_𝑘−𝑥_𝑘+1,𝐽𝑥_𝑘−𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘⟩+𝜇_𝑚^(𝑖)_𝑘𝜈_𝑘1−𝜅_𝑖 (26) for all 𝑘∈ℕ_𝑖. Note that {𝑚_𝑘}_{𝑘∈ℕ𝑖}={𝑖,𝑖+1,𝑖+2,…}; that is, 𝑚_𝑘↑∞ as ℕ_𝑖∋𝑘→∞. It then follows from (25) that lim_{ℕ𝑖∋𝑘→∞}𝜙𝑥_𝑘,𝑇^𝑚𝑘_𝑖𝑥_𝑘=0,∀𝑖≥1. (27) In view of (5), we have lim_{ℕ𝑖∋𝑘→∞}𝑥_𝑘−𝑇^𝑚𝑘_𝑖𝑥_𝑘=0,∀𝑖≥1. (28) This, in view of (25), implies that lim_{ℕ𝑖∋𝑘→∞}𝑇^𝑚𝑘_𝑖𝑥_𝑘=𝑥^∗,∀𝑖≥1, (29) which yields that lim_{ℕ𝑖∋𝑘→∞}𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘=𝐽𝑥^∗,∀𝑖≥1. (30) This shows that {𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘}_{𝑘∈ℕ𝑖} is bounded for each 𝑖≥1. Note that both 𝐸 and 𝐸^∗ are reflexive. We may assume that 𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘⇀𝑓∈𝐸^∗ as ℕ_𝑖∋𝑘→∞. In view of the reflexivity of 𝐸, we know that there exists an element 𝑦∈𝐸 such that 𝐽𝑦=𝑓. Then we have 𝜙𝑥_𝑘,𝑇^𝑚𝑘_𝑖𝑥_𝑘=𝑥_𝑘²−2𝑥_𝑘,𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘+𝑇^𝑚𝑘_𝑖𝑥_𝑘²=𝑥_𝑘²−2⟨𝑥_𝑘,𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘⟩+𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘². (31) Taking liminf_{ℕ𝑖∋𝑘→∞} on both sides of the above equation yields that 0≥𝑥^∗2−2𝑥^∗,𝑓+𝑓²=𝑥^∗2−2⟨𝑥^∗,𝐽𝑦⟩+𝐽𝑦²=𝑥^∗2−2⟨𝑥^∗,𝐽𝑦⟩+𝑦²=𝜙𝑥^∗,𝑦. (32) That is, 𝑥^∗=𝑦, which in turn implies that 𝑓=𝐽𝑥^∗. It follows that 𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘⇀𝐽𝑥^∗∈𝐸^∗ as ℕ_𝑖∋𝑘→∞. In the light of the Kadec-Klee property of 𝐸^∗, we obtain from (30) that lim_{ℕ𝑖∋𝑘→∞}𝐽𝑇^𝑚𝑘_𝑖𝑥_𝑘=𝐽𝑥^∗. Since 𝐽⁻¹∶𝐸^∗→𝐸 is norm-weak-continuous, 𝑇^𝑚𝑘_𝑖𝑥_𝑘⇀𝑥^∗ as ℕ_𝑖∋𝑘→∞. This implies, from (29) and the Kadec-Klee property of 𝐸, that lim_{ℕ𝑖∋𝑘→∞}𝑇^𝑚𝑘_𝑖𝑥_𝑘=𝑥^∗,∀𝑖≥1. (33) Again note that {𝑚_𝑘}_{𝑘∈ℕ𝑖}={𝑖,𝑖+1,𝑖+2,…}; that is, 𝑚_𝑘+1−1=𝑚_𝑘 whenever 𝑘∈ℕ_𝑖 for each 𝑖≥1. We then have 𝑇^𝑚𝑘+1_𝑖𝑥_𝑘−𝑥^∗≤𝑇^𝑚𝑘+1_𝑖𝑥_𝑘−𝑇^𝑚𝑘_𝑖𝑥_𝑘+𝑇^𝑚𝑘_𝑖𝑥_𝑘−𝑥_𝑘=𝑇^𝑚𝑘+1_𝑖𝑥_𝑘−𝑇^𝑚𝑘+1_𝑖⁻¹𝑥_𝑘+𝑇^𝑚𝑘_𝑖𝑥_𝑘−𝑥_𝑘. (34) It follows from (33) and the asymptotic regularity of each 𝑇_𝑖 that lim_{ℕ𝑖∋𝑘→∞}𝑇^𝑚𝑘+1_𝑖𝑥_𝑘−𝑥^∗=0,∀𝑖≥1. (35) That is, lim_{ℕ𝑖∋𝑘→∞}𝑇_𝑖𝑇^𝑚𝑘_𝑖𝑥_𝑘=𝑥^∗,∀𝑖≥1. (36) In view of the closedness of 𝑇_𝑖, it follows from (33) that 𝑇_𝑖𝑥^∗=𝑥^∗, namely, for each 𝑖≥1, 𝑥^∗∈𝐹(𝑇_𝑖) and hence 𝑥^∗∈𝐹.
(v) Consider 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 𝜙(𝑥_𝑛,𝑥₁)≤𝜙(𝑢,𝑥₁), for all 𝑛≥1. Then 𝜙𝑥^∗,𝑥₁=lim_𝑛→∞𝜙𝑥_𝑛,𝑥₁≤𝜙𝑢,𝑥₁, (37) which implies that 𝑥^∗=𝑢 since 𝑢=Π_𝐹𝑥₁, and hence 𝑥_𝑛→𝑥^∗=Π_𝐹𝑥₁. This completes the proof.

4. Applications to a System of Equilibrium Problems

The kind of equilibrium problems 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) and generalized equilibrium problem (GEP) (see, e.g., [15–19]).

Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, 𝐶 a nonempty closed and convex subset of 𝐸, and {𝑓_𝑖}^∞_𝑖=1∶𝐶×𝐶→ℝ a countable family of bifunctions satisfying the conditions: for each 𝑖≥1,(A₁)𝑓_𝑖(𝑥,𝑥)=0;(A₂)𝑓_𝑖 is monotone; that is, 𝑓_𝑖(𝑥,𝑦)+𝑓_𝑖(𝑦,𝑥)≤0;(A₃)limsup_𝑡↓0𝑓_𝑖(𝑥+𝑡(𝑧−𝑥),𝑦)≤𝑓_𝑖(𝑥,𝑦);(A₄) the mapping 𝑦↦𝑓_𝑖(𝑥,𝑦) is convex and lower semicontinuous. A system of equilibrium problems for {𝑓_𝑖}^∞_𝑖=1 is to find an 𝑥^∗∈𝐶 such that 𝑓_𝑖𝑥^∗,𝑦≥0,∀𝑦∈𝐶,𝑖≥1, (38) whose set of common solutions is denoted by EP∶=∩^∞_𝑖=1EP(𝑓_𝑖), where EP(𝑓_𝑖) denotes the set of solutions to the equilibrium problem for 𝑓_𝑖(𝑖=1,2,…). It will be shown in Theorem 9 that such a system of problems can be reduced to approximation of some fixed points of a countable family of nonlinear mappings.

Let 𝑟>0. Define a countable family of mappings {𝑇_𝑖}^∞_𝑖=1∶𝐶→𝐶 as follows: 𝑇_𝑖(𝑥)=𝑧∈𝐶∶𝑓_𝑖+1𝑧,𝑦𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶,∀𝑖≥1. (39) It is shown in [15] that (1){𝑇_𝑖}^∞_𝑖=1 is a sequence of single-valued mappings;(2){𝑇_𝑖}^∞_𝑖=1 is a sequence of closed quasi-𝜙-nonexpansive mappings; that is, 𝜙𝑝,𝑇_𝑖𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹𝑇_𝑖; (40)(3)𝐹∶=∩^∞_𝑖=1𝐹(𝑇_𝑖)=EP.

Now, we have the following result.

Theorem 9. Let 𝐸 be a reflexive, strictly convex, and smooth Banach space such that both 𝐸 and 𝐸^∗ have Kadec-Klee property, C a nonempty closed and convex subset of 𝐸, and {𝑇_𝑖}^∞_𝑖=1∶𝐶→𝐶 be a countable family of mappings defined by (39). Let {𝑥_𝑛} be the sequence generated by𝑥₁∈𝐸;𝐶₁𝐶=𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑥_𝑛,𝑇_𝑖𝑛𝑥_𝑛≤2𝑥_𝑛−𝑧,𝐽𝑥_𝑛−𝐽𝑇_𝑖𝑛𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₁,∀𝑛≥1, (41) where Π_𝐶𝑛+1 is the generalized projection of 𝐸 onto 𝐶_𝑛+1; 𝑖_𝑛 is the solution to the positive integer equation 𝑛=𝑖+((𝑚−1)𝑚/2)(𝑚≥𝑖,𝑛=1,2,…). If 𝐹∶=∩^∞_𝑖=1𝐹(𝑇_𝑖)≠∅, then {𝑥_𝑛} strongly converges to Π_𝐹𝑥₁ which is a common solution of the system of equilibrium problems for {𝑓_𝑖}^∞_𝑖=1.

Proof. Firstly, it follows from Lemma 6 that 𝐹 is closed and convex. Secondly, {𝑇_𝑖}^∞_𝑖=1 are obviously asymptotically strictly quasi-𝜙-pseudocontractions with 𝜇₁^(𝑖)=0 and 𝜅_𝑖=0 for all 𝑖≥1, since they are quasi-𝜙-nonexpansive. Finally, (13) clearly reduces to (41) with 𝑚_𝑛=1 for all 𝑛≥1. Therefore conclusion can be obtained immediately from Theorem 8.

Acknowledgments

The author is very grateful to the referees for their useful suggestions, by which the content of this paper has been improved. This work is supported by the General Project of Scientific Research Foundation of Yunnan University of Finance and Economics (YC2013A02).