A New Approach to the Approximation of Common Fixed Points of an Infinite Family of Relatively Quasinonexpansive Mappings with Applications

Abstract

By using a specific way of choosing the indexes, we propose an iteration algorithm generated by the monotone CQ method for approximating common fixed points of an infinite family of relatively quasinonexpansive mappings. A strong convergence theorem without the stronger assumptions of the AKTT condition and the ^∗AKTT condition imposed on the involved mappings is established in the framework of Banach space. As application, an iterative solution to a system of equilibrium problems is studied. The result is more applicable than those of other authors with related interest.

1. Introduction

Let 𝐶 be a nonempty and closed convex subset of a real Banach space 𝐸. A mapping 𝑇∶𝐶→𝐸 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.1) A mapping 𝑇 is said to be quasi-nonexpansive if 𝐹(𝑇)∶={𝑥∈𝐶∶𝑥=𝑇𝑥}≠∅ and ‖𝑇𝑥−𝑝‖≤‖𝑥−𝑝‖,∀𝑥∈𝐶,𝑝∈𝐹(𝑇).(1.2) It is easy to see that if 𝑇 is nonexpansive with 𝐹(𝑇)≠∅, then it is quasi-nonexpansive. There are many methods for approximating fixed points of quasi-nonexpansive mappings. In 1953, Mann [1] introduced the iteration as follows: a sequence {𝑥_𝑛} is defined by 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,(1.3) where the initial element 𝑥₀∈𝐶 is arbitrary and {𝛼_𝑛} is a sequence of real numbers in [0,1]. Approximation of fixed points of nonexpansive mappings via Mann's algorithm has extensively been investigated. One of the fundamental convergence results was proved by Reich [2]. In infinite-dimensional Hilbert spaces, Mann iteration can yield only weak convergence (see [3, 4]).

Attempts to modify the Mann iteration method (1.3) for strong convergence have recently been made. Nakajo and Takahashi [5] proposed the following modification of Mann iteration method (1.3) for a nonexpansive mapping 𝑇 from 𝐶 into itself in a Hilbert space: from an arbitrary 𝑥₀∈𝐶, 𝐶_𝑛𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑄=𝑧∈𝐶∶‖‖𝑦_𝑛−𝑧‖‖≤‖‖𝑥_𝑛−𝑧_𝑛‖‖,=𝑥𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0,(1.4) where 𝑃_𝐾 denotes the metric projection from a Hilbert space 𝐻 onto a closed convex subset 𝐾 of 𝐻. They proved that the sequence {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)𝑥₀.

Recently, Su and Qin [6] introduced a monotone CQ method for nonexpansive mapping, defined as follows: from an arbitrary 𝑥₀∈𝐶, 𝐶₀𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶=𝑧∈𝐶∶‖‖𝑦₀−𝑧‖‖≤‖‖𝑥₀−𝑧‖‖,𝑄₀=𝐶,_𝑛𝑄=𝑧∈𝐶_𝑛−1∩𝑄_𝑛−1∶‖‖𝑦_𝑛−𝑧‖‖≤‖‖𝑥_𝑛−𝑧_𝑛‖‖,=𝑧∈𝑥𝐶_𝑛−1∩𝑄_𝑛−1∶⟨𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0,(1.5) and it proved that the sequence {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)𝑥₀.

We now recall some definitions concerning relatively quasi-nonexpansive mappings. Let 𝐸 be a real smooth Banach space with norm ‖⋅‖ and let 𝐸^∗ be the dual of 𝐸. The normalized duality mapping 𝐽 from 𝐸 to 𝐸^∗ is defined by 𝐽𝑥=𝑓∈𝐸^∗∶⟨𝑥,𝑓⟩=‖𝑥‖²=‖𝑓‖²,∀𝑥∈𝐸,(1.6) where ⟨⋅,⋅⟩ denotes the pairing between 𝐸 and 𝐸^∗. Readers are directed to [7] (and its review [8]), where the properties of the duality mapping and several related topics are presented. The function 𝜙∶𝐸×𝐸→R⁺ is defined by 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,∀𝑥,𝑦∈𝐸.(1.7)

Let 𝑇 be a mapping from 𝐶 into 𝐸. A point 𝑝 in 𝐶 is said to be an asymptotic fixed point [9] of 𝑇 if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 and lim_𝑛→∞(𝑥_𝑛−𝑇𝑥_𝑛)=0. The set of asymptotic fixed points of 𝑇 is denoted by 𝐹(𝑇).

We say that the mapping 𝑇 is relatively nonexpansive (see [10]) if the following conditions are satisfied:(R1)𝐹(𝑇)≠∅; (R2)𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥), ∀𝑝∈𝐹(𝑇);(R3)𝐹(𝑇)=𝐹(𝑇).If 𝑇 satisfies (R1) and (R2), then 𝑇 is called relatively quasi-nonexpansive.

Several articles have provided methods for approximating fixed points of relatively quasi-nonexpansive mappings [11–16]. Employing the ideas of Su and Qin [6], and of Aoyama et al. [17], in 2008, Nilsrakoo and Saejung [18] used the following iterations to obtain strong convergence theorems for common fixed points of a countable family of relatively quasi-nonexpansive mappings in a Banach space 𝑦𝑥₀∈𝐶,𝐶₋₁=𝑄₋₁=𝐶;_𝑛𝐶_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇_𝑛𝑥_𝑛,𝑄=𝑧∈𝐶_𝑛−1∩𝑄_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛,=𝑧∈𝐶𝑥_𝑛−1∩𝑄_𝑛−1∶⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0_𝑛+1,=Π_𝑛𝑥₀,∀𝑛≥0.(1.8) However, the results were obtained under two stronger assumption conditions, namely, the 𝐴𝐾𝑇𝑇-condition and the ^∗𝐴𝐾𝑇𝑇-condition imposed on the involved mappings.

Inspired and motivated by those studies mentioned above, in this paper, we use a modified type of the iteration scheme (1.8) for approximating common fixed points of an infinite family of relatively quasi-nonexpansive mappings; without stronger assumptions imposed on the involved mappings, a strong convergence theorem in Banach spaces is obtained for solving a system of equilibrium problems. The results improve those of other authors with related interest.

2. Preliminaries

Throughout the paper, let 𝐸 be a real Banach space. We say that 𝐸 is strictly convex if the following implication holds for 𝑥,𝑦∈𝐸: ‖𝑥‖=‖𝑦‖=1,𝑥≠𝑦⟹‖‖‖𝑥+𝑦2‖‖‖<1.(2.1) 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.(2.2) 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→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.3) 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 (2.3) 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 (2.3) 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 (2.3) is attained uniformly for 𝑥,𝑦∈𝑆(𝐸).

We also know the following properties (see, e.g., [19] for details). (1)𝐸 (𝐸^∗, resp.) is uniformly convex ⇔𝐸^∗ (𝐸, resp.) is uniformly smooth.(2)𝐽𝑥≠∅ for each 𝑥∈𝐸.(3)If 𝐸 is reflexive, then 𝐽 is a mapping from 𝐸 onto 𝐸^∗.(4)If 𝐸 is strictly convex, then 𝐽𝑥∩𝐽𝑦=∅ as 𝑥≠𝑦.(5)If 𝐸 is smooth, then 𝐽 is single-valued.(6)If 𝐸 has a Fréchet differentiable norm, then 𝐽 is norm-to-norm continuous.(7)If 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸.(8)If 𝐸 is a Hilbert space, then 𝐽 is the identity operator.

Let 𝐸 be a smooth Banach space. The function 𝜙∶𝐸×𝐸→R⁺ is defined by 𝜙(𝑥,𝑦)∶=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖².(2.4) It is obvious from the definition of the function 𝜙 that (‖𝑥‖−‖𝑦‖)²≤𝜙(𝑥,𝑦)≤(‖𝑥‖+‖𝑦‖)².(2.5) Moreover, we know the following results.

Lemma 2.1 (see [13]). Let 𝐸 be a strictly convex and smooth Banach space, then 𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦.

Lemma 2.2 (see [11]). Let 𝐸 be a uniformly convex and smooth Banach space and let 𝑟>0. Then there exists a continuous, strictly increasing, and convex function 𝑔∶[0,2𝑟]→[0,∞) such that 𝑔(0)=0 and 𝑔(‖𝑥‖−‖𝑦‖)≤𝜙(𝑥,𝑦)(2.6) for all 𝑥,𝑦∈𝐵_𝑟∶={𝑧∈𝐸∶P𝑧P≤𝑟}.

Let 𝐶 be a nonempty and closed convex subset of 𝐸. Suppose that 𝐸 is reflexive, strictly convex, and smooth. It is known in [20] that for any 𝑥∈𝐸, there exists a unique point 𝑥^∗∈𝐶 such that 𝜙𝑥^∗,𝑥=min_𝑦∈𝐶𝜙(𝑦,𝑥).(2.7) Following Alber [21], we denote such an 𝑥^∗ by Π_𝐶𝑥. The mapping Π_𝐶 is called the generalized projection from 𝐸 onto 𝐶. It is easy to see that in a Hilbert space, the mapping Π_𝐶 coincides with the metric projection 𝑃_𝐶. What follows are the well-known facts concerning the generalized projection.

Lemma 2.3 (see [20]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸 and let 𝑥∈𝐸. Then 𝑥^∗=Π_𝐶𝑥⟺⟨𝑥^∗−𝑦,𝐽𝑥−𝐽𝑥^∗⟩≥0,∀𝑦∈𝐶.(2.8)

Lemma 2.4 (see [20]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑥∈𝐸. Then 𝜙𝑦,Π_𝐶𝑥+𝜙Π_𝐶𝑥,𝑥≤𝜙(𝑦,𝑥),∀𝑦∈𝐶.(2.9)

Dealing with the generalized projection from 𝐸 onto the fixed point set of a relatively quasi-nonexpansive mapping, we have the following result.

Lemma 2.5 (see [18]). 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 relatively quasi-nonexpansive mapping from 𝐶 into 𝐸. Then 𝐹(𝑇) is closed and convex.

Let 𝐶 be a subset of a Banach space 𝐸 and let {𝑇_𝑛} be a family of mappings from 𝐶 into 𝐸. For a subset 𝐵 of 𝐶, we say that (i)({𝑇_𝑛},𝐵) satisfies AKTT-condition if ^∞_𝑛=1sup‖‖𝑇_𝑛+1𝑧−𝑇_𝑛𝑧‖‖∶𝑧∈𝐵<∞;(2.10)(ii)({𝑇_𝑛},𝐵) satisfies ∗AKTT-condition if ^∞_𝑛=1sup‖‖𝐽𝑇_𝑛+1𝑧−𝐽𝑇_𝑛𝑧‖‖∶𝑧∈𝐵<∞.(2.11)

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 3.1. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, 𝐶 a nonempty and closed convex subset of 𝐸. Let {𝑇_𝑖}^∞𝑖=1∶𝐶→𝐸 be a sequence of closed and relatively quasi-nonexpansive mappings with 𝐹∶=⋂^∞𝑖=1𝐹(𝑇_𝑖)≠∅. Starting from an arbitrary 𝑥₁∈𝐶, the sequence {𝑥_𝑛} is define by 𝑦𝑥₁∈𝐶,𝐶₀=𝑄₀=𝐶;_𝑛𝐶_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇_𝑖𝑛𝑥_𝑛,𝑄=𝑧∈𝐶_𝑛−1∩𝑄_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛,=𝑧∈𝐶𝑥_𝑛−1∩𝑄_𝑛−1∶⟨𝑥_𝑛−𝑧,𝐽𝑥₁−𝐽𝑥_𝑛⟩≥0_𝑛+1,=Π_𝑛𝑥₁,∀𝑛≥1,(3.1) where Π_𝑛∶=Π_{𝐶𝑛⋂𝑄𝑛} and {𝛼_𝑛} is a sequence in [0,1) with limsup_𝑛→∞𝛼_𝑛<1; 𝑖_𝑛 is the solution to the positive integer equation: 𝑛=𝑖+(𝑚−1)𝑚/2(𝑚≥𝑖,𝑛=1,2,…), that is, for each 𝑛≥1, there exists a unique 𝑖_𝑛 such that 𝑖₁=1,𝑖₂=1,𝑖₃=𝑖2,𝑖₄=1,𝑖₅=2,𝑖₆=3,𝑖₇=1,𝑖₈=2,₉=3,𝑖₁₀=4,𝑖₁₁=1,….(3.2) Then {𝑥_𝑛} converges strongly to Π_𝐹𝑥₁.

Proof. We first claim that both 𝐶_𝑛 and 𝑄_𝑛 are closed and convex. This follows from the fact that 𝜙(𝑧,𝑦_𝑛)≤𝜙(𝑧,𝑥_𝑛) is equivalent to the following: 2⟨𝑧,𝐽𝑥_𝑛−𝐽𝑦_𝑛⟩≤‖‖𝑥_𝑛‖‖²−‖‖𝑦_𝑛‖‖².(3.3)
It is clear that 𝐹⊂𝐶=𝐶₀∩𝑄₀. Next, we show that 𝐹⊂𝐶_𝑛∩𝑄_𝑛,∀𝑛≥1.(3.4) Suppose that 𝐹⊂𝐶_𝑘−1∩𝑄_𝑘−1 for some 𝑘≥2. Letting 𝑝∈𝐹, we then have 𝜙𝑝,𝑦_𝑘==𝜙‖𝑝‖𝑝₂,𝐽⁻¹𝛼_𝑘𝐽𝑥_𝑘+1−𝛼_𝑘𝐽𝑇_𝑖𝑘−𝑥_𝑘−2⟨𝑝,𝛼_𝑘𝐽𝑥_𝑘+1−𝛼_𝑘𝐽𝑇_𝑖𝑘𝑥_𝑘⟩+‖‖𝛼_𝑘𝐽𝑥_𝑘+(1−𝛼_𝑘)𝐽𝑇_𝑖𝑘𝑥_𝑘‖‖²≤‖𝑝‖²−2𝛼_𝑘⟨𝑝,𝐽𝑥_𝑘⟩−21−𝛼_𝑘⟨𝑝,𝐽𝑇_𝑖𝑘𝑥_𝑘⟩+𝛼_𝑘‖‖𝑥_𝑘‖‖²+1−𝛼_𝑘‖‖𝑇_𝑖𝑘𝑥_𝑘‖‖²==𝛼𝛼_𝑘𝑘‖𝑝‖²−2⟨𝑝,𝐽𝑥_𝑘⟩+‖‖𝑥_𝑘‖‖²+1−𝛼^𝑘‖𝑝‖²−2⟨𝑝,𝐽𝑇_𝑖𝑘𝑥_𝑘⟩+‖‖𝑇_𝑖𝑘𝑥_𝑘‖‖²≤𝛼_𝑘𝜙𝑝,𝑥_𝑘+1−𝛼_𝑘𝜙𝑝,𝑇_𝑖𝑘𝑥_𝑘=𝜙𝜙𝑝,𝑥𝑝,_𝑘𝑥_𝑘+1−𝛼_𝑘𝜙𝑝,𝑥_𝑘.(3.5) This implies that 𝐹⊂𝐶_𝑘. It follows from 𝑥_𝑘=Π_𝑘−1𝑥₁ and Lemma 2.3 that ⟨𝑥_𝑘−𝑧,𝐽𝑥₁−𝐽𝑥_𝑘⟩≥0,∀𝑧∈𝐶_𝑘−1∩𝑄_𝑘−1.(3.6) Particularly, ⟨𝑥_𝑘−𝑧,𝐽𝑥₁−𝐽𝑥_𝑘⟩≥0,∀𝑝∈𝐹(3.7) and hence 𝐹⊂𝑄_𝑘, which yields that 𝐹⊂𝐶_𝑘∩𝑄_𝑘.(3.8) By induction, (3.4) holds. This implies that {𝑥_𝑛} is well defined. It follows from the definition of 𝑄_𝑛 and Lemma 2.3 that 𝑥_𝑛=Π_𝑄𝑛𝑥₁. Since 𝑥_𝑛+1=Π_𝑛𝑥₁∈𝑄_𝑛, we have 𝜙𝑥_𝑛,𝑥₁≤𝜙𝑥_𝑛+1,𝑥₁,∀𝑛≥1.(3.9) Therefore, {𝜙(𝑥_𝑛,𝑥₁)} is nondecreasing. Using 𝑥_𝑛=Π_𝑄𝑛𝑥₁ and Lemma 2.4, we have 𝜙𝑥_𝑛,𝑥₁=𝜙Π_𝑄𝑛𝑥₁,𝑥₁≤𝜙𝑝,𝑥₁−𝜙𝑝,𝑥_𝑛≤𝜙𝑝,𝑥₁(3.10) for all 𝑝∈𝐹 and for all 𝑛≥1, that is, {𝜙(𝑥_𝑛,𝑥₁)} is bounded. Then lim_𝑛→∞𝜙𝑥_𝑛,𝑥₁exists.(3.11) In particular, by (2.5), the sequence {(‖𝑥_𝑛‖−‖𝑥₁‖)²} is bounded. This implies that {𝑥_𝑛} is bounded. Note again that 𝑥_𝑛=Π_𝑄𝑛𝑥₁ and for any positive integer 𝑘, 𝑥_𝑛+𝑘∈𝑄_{𝑛+𝑘−1}⊂𝑄_𝑛. By Lemma 2.4, 𝜙𝑥_𝑛+𝑘,𝑥_𝑛≤=𝜙𝜙𝑥𝑥_{𝑛+𝑘𝑛+𝑘},Π_𝑄𝑛𝑥₁=𝜙𝑥_𝑛+𝑘,𝑥₁−𝜙Π_𝑄𝑛𝑥₁,𝑥₁,𝑥₁−𝜙𝑥_𝑛,𝑥₁.(3.12) By Lemma 2.2, we have, for any positive integers 𝑚,𝑛 with 𝑚>𝑛, 𝑔‖‖𝑥_𝑚−𝑥_𝑛‖‖≤𝜙𝑥_𝑚,𝑥_𝑛≤𝜙𝑥_𝑚,𝑥₁−𝜙𝑥_𝑛,𝑥₁,(3.13) where 𝑔∶[0,∞)→[0,∞) is a continuous, strictly increasing, and convex function with 𝑔(0)=0. Then the properties of the function 𝑔 yield that {𝑥_𝑛} is a Cauchy sequence in 𝐶, so there exists an 𝑥^∗∈𝐶 such that 𝑥_𝑛⟶𝑥^∗(𝑛⟶∞).(3.14) In view of 𝑥_𝑛+1=Π_𝑛𝑥₁∈𝐶_𝑛 and the definition of 𝐶_𝑛, we also have 𝜙𝑥_𝑛+1,𝑦_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛,∀𝑛≥1.(3.15) This implies that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑦_𝑛=lim_𝑛→∞𝜙𝑥_𝑛+1,𝑥_𝑛=0.(3.16) It follows from Lemma 2.2 that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑦_𝑛‖‖=lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=0.(3.17) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we have lim_𝑛→∞‖‖𝐽𝑥_𝑛+1−𝐽𝑦_𝑛‖‖=lim_𝑛→∞‖‖𝐽𝑥_𝑛+1−𝐽𝑥_𝑛‖‖=0.(3.18) On the other hand, we have, for each 𝑛≥1, ‖‖𝐽𝑥_𝑛+1−𝐽𝑦_𝑛‖‖==‖‖‖‖𝐽𝑥1−_𝑛+1𝛼−_𝑛𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇_𝑖𝑛𝑥_𝑛‖‖≥1−𝛼_𝑛𝐽𝑥_𝑛+1−𝐽𝑇_𝑖𝑛𝑥_𝑛−𝛼_𝑛𝐽𝑥_𝑛−𝐽𝑥_𝑛+1‖‖‖‖𝐽𝑥_𝑛+1−𝐽𝑇_𝑖𝑛𝑥_𝑛‖‖−𝛼_𝑛‖‖𝐽𝑥_𝑛−𝐽𝑥_𝑛+1‖‖,(3.19) and hence ‖‖𝐽𝑥_𝑛+1−𝐽𝑇_𝑖𝑛𝑥_𝑛‖‖≤11−𝛼_𝑛‖‖𝐽𝑥_𝑛+1−𝐽𝑦_𝑛‖‖+𝛼^𝑛1−𝛼_𝑛‖‖𝐽𝑥_𝑛−𝐽𝑥_𝑛+1‖‖.(3.20) From (3.18) and limsup_𝑛→∞𝛼_𝑛<1, we obtain that lim_𝑛→∞‖‖𝐽𝑥_𝑛+1−𝐽𝑇_𝑖𝑛𝑥_𝑛‖‖=0.(3.21) Since 𝐽⁻¹ is uniformly norm-to-norm continuous on bounded sets, we have lim_𝑛→∞‖‖𝑥_𝑛+1−𝑇_𝑖𝑛𝑥_𝑛‖‖=lim_𝑛→∞‖‖𝐽⁻¹𝐽𝑥_𝑛+1−𝐽⁻¹𝐽𝑇_𝑖𝑛𝑥_𝑛‖‖=0.(3.22) It follows from (3.17) that, as 𝑛→∞, ‖‖𝑥_𝑛−𝑇_𝑖𝑛𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑇_𝑖𝑛𝑥_𝑛‖‖⟶0.(3.23) Now, set 𝒦_𝑖={𝑘≥1∶𝑘=𝑖+(𝑚−1)𝑚/2,𝑚≥𝑖,𝑚∈ℤ⁺} for each 𝑖≥1. Note that 𝑇_𝑖𝑘=𝑇_𝑖 whenever 𝑘∈𝒦_𝑖. For example, by the definition of 𝒦₁, we have 𝒦₁={1,2,4,7,11,16,…} and 𝑖₁=𝑖₂=𝑖₄=𝑖₇=𝑖₁₁=𝑖₁₆=⋯=1. Then it follows from (3.23) that lim_{𝒦𝑖∋𝑘→∞}‖‖𝑇_𝑖𝑥_𝑘−𝑥_𝑘‖‖=0,∀𝑖≥1.(3.24) Since {𝑥_𝑘}_{𝑘∈𝒦𝑖} is a subsequence of {𝑥_𝑛}, (3.14) implies that 𝑥_𝑘→𝑥^∗ as 𝒦_𝑖∋𝑘→∞. It immediately follows from (3.24) and the closedness of 𝑇_𝑖 that 𝑥^∗∈𝐹(𝑇_𝑖) for each 𝑖≥1, and hence 𝑥^∗∈𝐹. Furthermore, by (3.10), 𝜙𝑥_∗,𝑥₁=lim_𝑛→∞𝜙𝑥_𝑛,𝑥₁≤𝜙𝑝,𝑥₁,∀𝑝∈𝐹.(3.25) This implies that 𝑥^∗=Π_𝐹𝑥₁. The proof is completed.

Remark 3.2. Note that the algorithm (3.1) is based on the projection onto an intersection of two closed and convex sets. An example [22] of how to compute such a projection is given as follows.

Dykstra's Algorithm
Let Ω₁,Ω₂,…,Ω_𝑝 be closed and convex subsets of ℝ^𝑛. For any 𝑖=1,2,…,𝑝 and 𝑥⁰∈ℝ^𝑛, the sequences {𝑥^𝑘𝑖} are defined by the following recursive formulae: 𝑥_𝑘𝑖𝑥^𝑘0=𝑥^𝑝𝑘−1,𝑦=𝑃_Ω𝑖𝑥^𝑘𝑖−1−𝑦^𝑖𝑘−1,𝑖=1,2,…,𝑝,_𝑘𝑖=𝑥^𝑘𝑖−𝑥^𝑘𝑖−1−𝑦^𝑖𝑘−1,𝑖=1,2,…,𝑝,(3.26) for 𝑘=1,2,… with initial values 𝑥^0𝑝=𝑥⁰ and 𝑦^0𝑖=0 for 𝑖=1,2,…,𝑝. If Ω∶=⋂^𝑝𝑖=1Ω_𝑖≠∅, then {𝑥^𝑘𝑖} converges to 𝑥^∗=𝑃_Ω(𝑥⁰), where 𝑃_Ω(𝑥)∶=arginf_𝑦∈Ω‖𝑦−𝑥‖²,forall𝑥∈ℝ^𝑛.

4. Applications

The so-called convex feasibility problem for a family of mappings {𝑇_𝑖}^∞𝑖=1 is to find a point in the nonempty intersection ⋂^∞𝑖=1𝐹(𝑇_𝑖), which exactly illustrates the importance of finding fixed points of infinite families. The following example also clarifies the same thing.

Example 4.1. Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, 𝐶 a nonempty and closed convex subset of 𝐸, and {𝑓_𝑖}^∞𝑖=1∶𝐶→𝐶 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_𝑖𝑥^∗,𝑦≥0,∀𝑦∈𝐶,𝑖≥1,(4.1) 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 4.3 that such a system of problems can be reduced to approximation of some fixed points of a countable family of nonexpansive mappings.

Example 4.2 (see [23]). Let 𝑟>0. Define a countable family of mappings {𝑇_𝑟,𝑖}^∞𝑖=1∶𝐸→𝐶 as follows: 𝑇_𝑟,𝑖(𝑥)=𝑧∈𝐶∶𝑓_𝑖(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶,∀𝑖≥1.(4.2) Then we have 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 relatively quasi-nonexpansive mappings;(3)𝐹∶=⋂^∞𝑖=1𝐹(𝑇_𝑟,𝑖)=𝐸𝑃.

Now, we have the following result.

Theorem 4.3. Let 𝐶,𝐸, and {𝛼_𝑛} be the same as those in Theorem 3.1. Let {𝑓_𝑖}^∞𝑖=1∶𝐶→𝐶 be a countable family of bifunctions satisfying the conditions (𝐴₁)–(𝐴₄). Let {𝑇_𝑟,𝑖}^∞𝑖=1∶𝐸→𝐶 be a countable family of mappings defined by (4.2). Let {𝑥_𝑛} be the sequence generated by 𝑓𝑥₁∈𝐶,𝐶₀=𝑄₀=𝐶;_𝑖𝑛𝑢_𝑛,𝑦𝑦+1𝑟⟨𝑦−𝑢^𝑛,𝐽𝑢_𝑛−𝐽𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝐶_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑢_𝑛,𝑄=𝑧∈𝐶_𝑛−1∩𝑄_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛,=𝑧∈𝐶𝑥_𝑛−1∩𝑄_𝑛−1∶⟨𝑥_𝑛−𝑧,𝐽𝑥₁−𝐽𝑥_𝑛⟩≥0_𝑛+1,=Π_𝑛𝑥₁,∀𝑛≥1,(4.3) where 𝑖_𝑛 satisfies the positive integer equation: 𝑛=𝑖+(𝑚−1)𝑚/2(𝑚≥𝑖,𝑛=1,2,…). If 𝐹∶=⋂^∞𝑖=1𝐹(𝑇_𝑟,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. Since each 𝑇_𝑟,𝑖 is single-valued, 𝑢_𝑛=𝑇_{𝑟,𝑖𝑛}𝑥_𝑛 for all 𝑛≥1. In addition, we have pointed out in Example 4.2 that 𝐹=𝐸𝑃 and {𝑇_𝑟,𝑖}^∞𝑖=1 is a sequence of closed relatively quasi-nonexpansive mappings. Hence, (4.3) can be rewritten as follows: 𝑦𝑥₁∈𝐶,𝐶₀=𝑄₀=𝐶;_𝑛𝐶_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇_{𝑟,𝑖𝑛}𝑥_𝑛,𝑄=𝑧∈𝐶_𝑛−1∩𝑄_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛,=𝑧∈𝐶𝑥_𝑛−1∩𝑄_𝑛−1∶⟨𝑥_𝑛−𝑧,𝐽𝑥₁−𝐽𝑥_𝑛⟩≥0_𝑛+1,=Π_𝑛𝑥₁,∀𝑛≥1.(4.4) Therefore, this conclusion can be obtained immediately from Theorem 3.1.

Acknowledgments

The author is greatly grateful to the referees for their useful suggestions by which the contents of this article are improved. This study was supported by the General Project of Scientific Research Foundation of Yunnan University of Finance and Economics (YC2012A02).