Strong Convergence Theorems for a Common Fixed Point of Two Countable Families of Relatively Quasi Nonexpansive Mappings and Applications

Abstract

The purpose of this paper is to prove strong convergence theorems for common fixed points of two countable families of relatively quasi nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalized 𝑓-projection operator. In order to get the strong convergence theorems, a new iterative scheme by monotone hybrid method is presented and is used to approximate the common fixed points. Then, two examples of countable families of uniformly closed nonlinear mappings are given. The results of this paper modify and improve the results of Li et al. (2010), the results of Takahashi and Zembayashi (2008), and many others.

1. Introduction

Let 𝐸 be a real Banach space with the dual 𝐸^∗. We denote by 𝐽 the normalized duality mapping from 𝐸 to 2^𝐸∗ defined by 𝐽𝑥=𝑓∈𝐸^∗∶𝑥,𝑓=‖𝑥‖²=𝑓², (1.1) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. The duality mapping 𝐽 has the following properties:(i)if 𝐸 is smooth, then 𝐽 is single-valued;(ii)if 𝐸 is strictly convex, then 𝐽 is one-to-one;(iii)if 𝐸 is reflexive, then 𝐽 is surjective;(iv)if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸;(v)if 𝐸^∗ is uniformly convex, then 𝐽 is uniformly continuous on bounded subsets of 𝐸 and 𝐽 is single valued and also one-to-one (see [1–4]).

Let 𝐸 be a smooth Banach space with the dual 𝐸^∗. The functional 𝜙∶𝐸×𝐸→𝑅 is defined by 𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦², (1.2) for all 𝑥,𝑦∈𝐸.

Let 𝐶 be a closed convex subset of 𝐸, and let 𝑇 be a mapping from 𝐶 into itself. We denote by 𝐹(𝑇) the set of fixed points of 𝑇. A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 [5] if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that the strong lim_𝑛→∞(𝑥_𝑛−𝑇𝑥_𝑛)=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇 from 𝐶 into itself is called nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶 and relatively nonexpansive [1, 6–8] if 𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for all 𝑥∈𝐶 and 𝑝∈𝐹(𝑇). The asymptotic behavior of relatively nonexpansive mapping was studied in [1, 6–8].

Three classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping. The first one is introduced in 1953 by Mann [9] which is well known as Mann’s iteration process and is defined as follows: 𝑥₀𝑥chosenarbitrarily,_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0, (1.3) where the sequence {𝛼_𝑛} is chosen in [0, 1]. Fourteen years later, Halpern [10] proposed the new innovation iteration process which resembles Manns iteration (1.3), it is defined by 𝑥₀𝑥chosenarbitrarily,_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0, (1.4) where the element 𝑢∈𝐶 is fixed. Seven years later, Ishikawa [2] enlarged and improved Mann's iteration (1.3) to the new iteration method, it is often cited as Ishikawa's iteration process which is defined recursively by 𝑥₀𝑦chosenarbitrarily,_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑦_𝑛,𝑛≥0, (1.5) where {𝛼_𝑛} and {𝛽_𝑛} are sequences in the interval [0, 1].

In both Hilbert space [10–12] and uniformly smooth Banach space [13–15] the iteration process (1.4) has been proved to be strongly convergent if the sequence {𝛼_𝑛} satisfies the following conditions:(i)𝛼_𝑛→0;(ii)∑^∞_𝑛=0𝛼_𝑛=∞;(iii)∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞ or lim_𝑛→∞(𝛼_𝑛/𝛼_𝑛+1)=1. By the restriction of condition (ii), it is widely believed that Halpern’s iteration process (1.4) has slow convergence though the rate of convergence has not been determined. Halpern [10] proved that conditions (i) and (ii) are necessary in the strong convergence of (1.4) for a nonexpansive mapping 𝑇 on a closed convex subset 𝐶 of a Hilbert space 𝐻. Moreover, Wittmann [12] showed that (1.4) converges strongly to 𝑃_𝐹(𝑇)𝑢 when {𝛼_𝑛} satisfies (i), (ii), and (iii), where 𝑃_𝐹(𝑇)(⋅) is the metric projection onto 𝐹(𝑇).

Both iterations processes (1.3) and (1.5) have only weak convergence, in general Banach space (see [16] for more details). As a matter of fact, process (1.3) may fail to converge while process (1.5) can still converge for a Lipschitz pseudo contractive mapping in a Hilbert space [17]. For example, Reich [18] proved that if 𝐸 is a uniformly convex Banach space with Frechet differentiable norm and if {𝛼_𝑛} is chosen such that ∑^∞_𝑛=0𝛼_𝑛(1−𝛼_𝑛)=∞, then the sequence {𝑥_𝑛} defined by (1.3) converges weakly to a fixed point of 𝑇. However, we note that Manns iteration process (1.3) has only weak convergence even in a Hilbert space [16].

Some attempts to modify the Mann’s iteration method so that strong convergence guaranteed has recently been made. Nakajo and Takahashi [19] proposed the following modification of the Mann iteration method for a single nonexpansive mapping 𝑇 in a Hilbert space 𝐻: 𝑥₀𝑦∈𝐶chosenarbitrarily,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀, (1.6) where 𝐶 is a closed convex subset of 𝐻, 𝑃_𝐾 denotes the metric projection from 𝐻 onto a closed convex subset 𝐾 of 𝐻. They proved that if the sequence {𝛼_𝑛} is bounded above from one then the sequence {𝑥_𝑛} generated by (1.6) converges strongly to 𝑃_𝐹(𝑇)(𝑥₀), where 𝐹(𝑇) denotes the fixed points set of 𝑇.

The ideas to generalize the process (1.6) from Hilbert space to Banach space have recently been made. By using available properties on uniformly convex and uniformly smooth Banach space, Matsushita and Takahashi [8] presented their ideas as the following method for a single relatively nonexpansive mapping 𝑇 in a Banach space 𝐸: 𝑥₀𝑦∈𝐶chosenarbitrarily,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀. (1.7)

They proved the following convergence theorem.

Theorem MT. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑇 be a relatively nonexpansive mapping from 𝐶 into itself, and let {𝛼_𝑛} be a sequence of real numbers such that 0≤𝛼_𝑛<1 and limsup_𝑛→∞𝛼_𝑛<1. Suppose that {𝑥_𝑛} is given by (1.7), where 𝐽 is the duality mapping on 𝐸. If 𝐹(𝑇) is nonempty, then {𝑥_𝑛} converges strongly to Π_𝐹(𝑇)𝑥₀, where Π_𝐹(𝑇)(⋅) is the generalized projection from 𝐶 onto 𝐹(𝑇).

In 2007, Plubtieng and Ungchittrakool [20] proposed the following hybrid algorithms for two relatively nonexpansive mappings in a Banach space and proved the following convergence theorems.

Theorem SK 1. Let 𝐸 be a uniformly convex and uniformly smooth real Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑇,𝑆 be two relatively nonexpansive mappings from 𝐶 into itself with 𝐹∶=𝐹(𝑇)∩𝐹(𝑆) is nonempty. Let a sequence {𝑥_𝑛} be defined by 𝑥₀𝑦∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆𝑥_𝑛,𝐻_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑊_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐻𝑛∩𝑊𝑛}𝑥₀, (1.8) with the following restrictions:(i)0≤𝛼_𝑛<1,limsup_𝑛→∞𝛼_𝑛<1;(ii)0≤𝛽_𝑛⁽¹⁾,𝛽_𝑛⁽¹⁾,𝛽_𝑛⁽³⁾≤1,lim_𝑛→∞𝛽_𝑛⁽¹⁾=0,liminf_𝑛→∞𝛽_𝑛⁽²⁾𝛽_𝑛⁽³⁾>0.
Then the {𝑥_𝑛} converges strongly to Π_𝐹𝑥₀, where Π_𝐹 is the generalized projection from 𝐶 onto 𝐹.

Theorem SK 2. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑇,𝑆 be two relatively nonexpansive mappings from 𝐶 into itself with 𝐹∶=𝐹(𝑇)∩𝐹(𝑆) is nonempty. Let a sequence {𝑥_𝑛} be defined by 𝑥₀𝑦∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆𝑥_𝑛,𝐻_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+𝛼_𝑛𝑥₀²+2𝑧,𝐽𝑥_𝑛−𝐽𝑥₀,𝑊_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐻𝑛∩𝑊𝑛}𝑥₀, (1.9) with the following restrictions:(i)0<𝛼_𝑛<1,limsup_𝑛→∞𝛼_𝑛<1;(ii)0≤𝛽_𝑛⁽¹⁾,𝛽_𝑛⁽¹⁾,𝛽_𝑛⁽³⁾≤1,lim_𝑛→∞𝛽_𝑛⁽¹⁾=0,liminf_𝑛→∞𝛽_𝑛⁽²⁾𝛽_𝑛⁽³⁾>0.
Then the {𝑥_𝑛} converges strongly to Π_𝐹𝑥₀, where Π_𝐹 is the generalized projection from 𝐶 onto 𝐹.

In 2010, Su et al. [21] proposed the following hybrid algorithms for two countable families of weak relatively nonexpansive mappings in a Banach space and proved the following convergence theorems.

Theorem SXZ 1. Let 𝐸 be a uniformly convex and uniformly smooth real Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let {𝑇_𝑛},{𝑆_𝑛} be two countable families of weak relatively nonexpansive mappings from 𝐶 into itself such that 𝐹∶=(∩^∞_𝑛=0𝐹(𝑇_𝑛))⋂(∩^∞_𝑛=0𝐹(𝑆_𝑛))≠∅. Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥₀𝑧∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝐶₀=𝑧∈𝐶∶𝜙𝑧,𝑦₀≤𝜙𝑧,𝑥₀,𝑄_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑄≥0₀𝑥=𝐶,_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀ (1.10) with the conditions:(i)𝛽_𝑛⁽¹⁾+𝛽_𝑛⁽²⁾+𝛽_𝑛⁽³⁾=1;(ii)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾>0; (iii)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽³⁾>0;(iv)0≤𝛼_𝑛≤𝛼<1 for some 𝛼∈(0,1).Then {𝑥_𝑛} converges strongly to Π_𝐹𝑥₀, where Π_𝐹 is the generalized projection from 𝐶 onto 𝐹.

Theorem SXZ 2. Let 𝐸 be a uniformly convex and uniformly smooth real Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let {𝑇_𝑛},{𝑆_𝑛} be two countable families of weak relatively nonexpansive mappings from 𝐶 into itself such that 𝐹∶=(∩^∞_𝑛=0𝐹(𝑇_𝑛))⋂(∩^∞_𝑛=0𝐹(𝑆_𝑛))≠∅. Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥₀𝑧∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝜙𝑧,𝑦_𝑛≤1−𝛼_𝑛𝛽_𝑛⁽¹⁾𝜙𝑧,𝑥_𝑛+𝛼_𝑛𝛽_𝑛⁽¹⁾𝜙𝑧,𝑥₀,𝐶₀=𝑧∈𝐶∶𝜙𝑧,𝑦₀≤𝜙𝑧,𝑥₀,𝑄_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑄≥0₀𝑥=𝐶,_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀, (1.11) with the conditions:(i)𝛽_𝑛⁽¹⁾+𝛽_𝑛⁽²⁾+𝛽_𝑛⁽³⁾=1; (ii)lim_𝑛→∞𝛽_𝑛⁽¹⁾=0;(iii)limsup_𝑛→∞𝛽_𝑛⁽²⁾𝛽_𝑛⁽³⁾>0;(iv)0≤𝛼_𝑛≤𝛼<1𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝛼∈(0,1). Then {𝑥_𝑛} converges strongly to Π_𝐹𝑥₀.

Recently, Li et al. [22] introduced the following hybrid iterative scheme for approximation of fixed points of a relatively nonexpansive mapping using the properties of generalized 𝑓-projection operator in a uniformly smooth real Banach space which is also uniformly convex: 𝑥₀∈𝐶, 𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝐶_𝑛=𝑤∈𝐶_𝑛∶𝐺𝑤,𝐽𝑦_𝑛≤𝐺𝑤,𝐽𝑥_𝑛,𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀,𝑛≥1. (1.12) They proved a strong convergence theorem for finding an element in the fixed points set of 𝑇. We remark here that the results of Li et al. [22] extended and improved on the results of Matsushita and Takahashi [8].

Motivated by the above-mentioned results and the ongoing research, it is our purpose in this paper to prove a strong convergence theorem for two countable families of relatively quasi nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalized 𝑓-projection operator. Our results extend the results of Li et al. [22], Takahashi and Zembayashi [23], and many other recent known results in the literature.

2. Preliminaries

Let 𝐸 be a smooth Banach space with the dual 𝐸^∗. The functional 𝜙∶𝐸×𝐸→𝑅 is defined by 𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦², (2.1) for all 𝑥,𝑦∈𝐸. Observe that, in a Hilbert space 𝐻, (2.1) reduces to 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖²,𝑥,𝑦∈𝐻.

Recall that if 𝐶 is a nonempty, closed, and convex subset of a Hilbert space 𝐻 and 𝑃_𝐶∶𝐻→𝐶 is the metric projection of 𝐻 onto 𝐶, then 𝑃_𝐶 is nonexpansive. This is true only when 𝐻 is a real Hilbert space. In this connection, Alber [24] has recently introduced a generalized projection operator Π_𝐶 in a Banach space 𝐸 which is an analogue of the metric projection in Hilbert spaces. The generalized projection Π_𝐶∶𝐸→𝐶 is a map that assigns to an arbitrary point 𝑥∈𝐸, the minimum point of the functional 𝜙(𝑦,𝑥), that is, Π_𝐶𝑥=𝑥, where 𝑥 is the solution to the minimization problem 𝜙(𝑥,𝑥)=min_𝑦∈𝐶𝜙𝑦,𝑥, (2.2) existence and uniqueness of the operator Π_𝐶 follow from the properties of the functional 𝜙(𝑦,𝑥) and strict monotonicity of the mapping 𝐽. In Hilbert space, Π_𝐶=𝑃_𝐶. It is obvious from the definition of the functional 𝜙 that ‖𝑥‖−𝑦²≤𝜙𝑦,𝑥≤𝑦²+‖𝑥‖²,𝜙𝑥,𝑦=𝜙(𝑥,𝑧)+𝜙𝑧,𝑦−2𝑥−𝑧,𝐽𝑧−𝐽𝑦 (2.3)(2.4) for all 𝑥,𝑦∈𝐸. See [25] for more details.

This section collects some definitions and lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive.

Remark 2.1. If 𝐸 is a reflexive strictly convex and smooth Banach space, then for 𝑥,𝑦∈𝐸, 𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦. It is sufficient to show that if 𝜙(𝑥,𝑦)=0, then 𝑥=𝑦. From (2.3), we have ‖𝑥‖=‖𝑦‖. This implies ⟨𝑥,𝐽𝑦⟩=‖𝑥‖²=‖𝐽𝑦‖². From the definition of 𝐽, we have 𝐽𝑥=𝐽𝑦. Since 𝐽 is one-to-one, then we have 𝑥=𝑦; see [12, 15, 26] for more details.
Let 𝐶 be a closed convex subset of 𝐸, and let {𝑇_𝑛}^∞_𝑛=0 be a countable family of mappings from 𝐶 into itself. We denote by 𝐹 the set of common fixed points of {𝑇_𝑛}^∞_𝑛=0, that is, 𝐹=⋂^∞_𝑛=0F(𝑇_𝑛), where 𝐹(𝑇_𝑛) denote the set of fixed points of 𝑇_𝑛, for all 𝑛≥0. A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of {𝑇_𝑛}^∞_𝑛=0 if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑇_𝑛𝑥_𝑛−𝑥_𝑛‖=0. The set of asymptotic fixed point of {𝑇_𝑛}^∞_𝑛=0 will be denoted by 𝐹({𝑇_𝑛}^∞_𝑛=0). A point 𝑝 in 𝐶 is said to be a strong asymptotic fixed point of {𝑇_𝑛}^∞_𝑛=0 if 𝐶 contains a sequence {𝑥_𝑛} which converges strongly to 𝑝 such that lim_𝑛→∞‖𝑇_𝑛𝑥_𝑛−𝑥_𝑛‖=0. The set of strong asymptotic fixed point of {𝑇_𝑛}^∞_𝑛=0 will be denoted by 𝐹({𝑇_𝑛}^∞_𝑛=0) [21].

Definition 2.2. Countable family of mappings {𝑇_𝑛}^∞_𝑛=0 is said to be countable family of relatively nonexpansive mappings if the following conditions are satisfied:(i)𝐹({𝑇_𝑛}^∞_𝑛=0) is nonempty;(ii)𝜙(𝑢,𝑇_𝑛𝑥)≤𝜙(𝑢,𝑥), forall𝑢∈𝐹(𝑇_𝑛),𝑥∈𝐶,𝑛≥0;(iii)𝐹({𝑇_𝑛}^∞_𝑛=0)=⋂^∞_𝑛=0𝐹(𝑇_𝑛).

Definition 2.3. Countable family of mappings {𝑇_𝑛}^∞_𝑛=0 is said to be countable family of weak relatively nonexpansive mappings if the following conditions are satisfied:(i)𝐹({𝑇_𝑛}^∞_𝑛=0) is nonempty;(ii)𝜙(𝑢,𝑇_𝑛𝑥)≤𝜙(𝑢,𝑥), forall𝑢∈𝐹(𝑇_𝑛),𝑥∈𝐶,𝑛≥0;(iii)𝐹({𝑇_𝑛}^∞_𝑛=0)=⋂^∞_𝑛=0𝐹(𝑇_𝑛).

Definition 2.4. Countable family of mappings {𝑇_𝑛}^∞_𝑛=0 is said to be countable family of relatively quasi nonexpansive mappings if the following conditions are satisfied:(i)𝐹({𝑇_𝑛}^∞_𝑛=0) is nonempty;(ii)𝜙(𝑢,𝑇_𝑛𝑥)≤𝜙(𝑢,𝑥), forall𝑢∈𝐹(𝑇_𝑛),𝑥∈𝐶,𝑛≥0.

Definition 2.5. A mapping 𝑇 is said to be relatively nonexpansive mappings if the following conditions are satisfied:(i)𝐹(𝑇)isnonempty; (ii)𝜙(𝑢,𝑇𝑥)≤𝜙(𝑢,𝑥),forall𝑢∈𝐹(𝑇),𝑥∈𝐶;(iii)𝐹(𝑇)=𝐹(𝑇).

Definition 2.6. A mapping 𝑇 is said to be weak relatively nonexpansive mappings if the following conditions are satisfied:(i)𝐹(𝑇)isnonempty; (ii)𝜙(𝑢,𝑇𝑥)≤𝜙(𝑢,𝑥),forall𝑢∈𝐹(𝑇),𝑥∈𝐶;(iii)𝐹(𝑇)=𝐹(𝑇).

Definition 2.7. A mapping 𝑇 is said to be relatively quasi nonexpansive mappings if the following conditions are satisfied:(i)𝐹(𝑇) is nonempty;(ii)𝜙(𝑢,𝑇𝑥)≤𝜙(𝑢,𝑥),  forall𝑢∈𝐹(𝑇),𝑥∈𝐶.

The Definition 2.5 (Definitions 2.6 and 2.7) is a special form of the Definition 2.2 (Definitions 2.3 and 2.4) as 𝑇_𝑛≡𝑇 for all 𝑛≥0. The following conclusions are obvious: (1) relatively nonexpansive mapping must be weak relatively nonexpansive mapping; (2) weak relatively nonexpansive mapping must be relatively quasi nonexpansive mapping.

The hybrid algorithms for fixed point of relatively nonexpansive mappings and applications have been studied by many authors, for example, [1, 6, 7, 17, 27, 28]. In recent years, the definition of relatively quasi nonexpansive mapping has been presented and studied by many authors [7, 17, 26, 28]. Now we give an example which is a countable family of relatively quasi nonexpansive mappings but not a countable family of relatively nonexpansive mappings.

Example 2.8. Let 𝐸=𝑙², where 𝑙²=𝜉=𝜉₁,𝜉₂,𝜉₃,...,𝜉_𝑛,...∶^∞_𝑛=1𝑥_𝑛²,𝜉=<∞_∞𝑛=1𝜉_𝑛^21/2,∀𝜉∈𝑙²,𝜉,𝜂=^∞_𝑛=1𝜉_𝑛𝜂_𝑛,∀𝜉=𝜉₁,𝜉₂,𝜉₃,...,𝜉_𝑛,...,𝜂=𝜂₁,𝜂₂,𝜂₃,...,𝜂_𝑛....∈𝑙². (2.5) It is well known that 𝑙² is a Hilbert space, so that (𝑙²)^∗=𝑙². Let {𝑥_𝑛}⊂𝐸 be a sequence defined by 𝑥₀=(),𝑥1,0,0,0,…₁=(),𝑥1,1,0,0,…₂=(),𝑥1,0,1,0,0,…₃=(),⋮𝑥1,0,0,1,0,0,…_𝑛=𝜉_𝑛,1,𝜉_𝑛,2,𝜉_𝑛,3,…,𝜉_𝑛,𝑘,⋮,… (2.6) where 𝜉_𝑛,𝑘=1if𝑘=1,𝑛+1,0if𝑘≠1,𝑘≠𝑛+1, (2.7) for all 𝑛≥1.
Define a countable family of mappings 𝑇_𝑛∶𝐸→𝐸 as follows: 𝑇_𝑛(𝑥)=𝑛𝑥𝑛+1^𝑛if𝑥=𝑥_𝑛,−𝑥if𝑥≠𝑥_𝑛, (2.8) for all 𝑛≥0.

Conclusion 2.9. {𝑥_𝑛} converges weakly to 𝑥₀.

Proof. For any 𝑓=(𝜁₁,𝜁₂,𝜁₃,...,𝜁_𝑘,...)∈𝑙²=(𝑙²)^∗, we have 𝑓𝑥_𝑛−𝑥₀=𝑓,𝑥_𝑛−𝑥₀=^∞_𝑘=2𝜁_𝑘𝜉_𝑛,𝑘=𝜁_𝑛+1→0, (2.9) as 𝑛→∞. That is, {𝑥_𝑛} converges weakly to 𝑥₀

Conclusion 2.10. {𝑥_𝑛} is not a Cauchy sequence, so that, it does not converge strongly to any element of 𝑙².

Proof. In fact, we have ‖𝑥_𝑛−𝑥_𝑚‖=√2 for any 𝑛≠𝑚. Then {𝑥_𝑛} is not a Cauchy sequence.

Conclusion 2.11. 𝑇_𝑛 has a unique fixed point 0, that is, 𝐹(𝑇_𝑛)={0}, for all 𝑛≥0.

Proof. The conclusion is obvious.

Conclusion 2.12. 𝑥₀ is an asymptotic fixed point of {𝑇_𝑛}^∞_𝑛=0.

Proof. Since {𝑥_𝑛} converges weakly to 𝑥₀ and 𝑇_𝑛𝑥_𝑛−𝑥_𝑛=𝑛𝑥𝑛+1^𝑛−𝑥_𝑛=1𝑥𝑛+1^𝑛→0 (2.10) as 𝑛→∞, so that, 𝑥₀ is an asymptotic fixed point of {𝑇_𝑛}^∞_𝑛=0.

Conclusion 2.13. {𝑇_𝑛}^∞_𝑛=0 is a countable family of relatively quasi nonexpansive mappings.

Proof. Since 𝐸=𝐿² is a Hilbert space, for any 𝑛≥0 we have 𝜙0,𝑇_𝑛𝑥=0−𝑇_𝑛𝑥²=𝑇_𝑛𝑥²≤‖𝑥‖²=‖𝑥−0‖²=𝜙(0,𝑥),∀𝑥∈𝐸. (2.11) then {𝑇_𝑛}^∞_𝑛=0 is a countable family of relatively quasi nonexpansive mappings

Conclusion 2.14. {𝑇_𝑛}^∞_𝑛=0 is not a countable family of relatively nonexpansive mappings.

Proof. From Conclusions 2.11 and 2.12, we have ⋂^∞_𝑛=0𝐹(𝑇_𝑛)≠𝐹({𝑇_𝑛}^∞_𝑛=0), so that, {𝑇_𝑛}^∞_𝑛=0 is not a countable family of relatively nonexpansive mapping.

Next, we recall the concept of generalized 𝑓-projector operator, together with its properties. Let 𝐺∶𝐶×𝐸^∗→𝑅∪{+∞} be a functional defined as follows: 𝐺𝜉,𝜑=𝜉²−2𝜉,𝜑+𝜑²+2𝜌𝑓(𝜉), (2.12) where 𝜉∈𝐶,𝜑∈𝐸,𝜌 is a positive number, and 𝑓∶𝐶→𝑅∪{+∞} is proper, convex, and lower semicontinuous. From the definitions of 𝐺 and 𝑓, it is easy to see the following properties:(i)𝐺(𝜉,𝜑) is convex and continuous with respect to 𝜑 when 𝜉 is fixed;(ii)𝐺(𝜉,𝜑) is convex and lower semi-continuous with respect to 𝜉 when 𝜑 is fixed.

Definition 2.15 (see [29]). Let 𝐸 be a real Banach space with its dual 𝐸^∗. Let 𝐶 be a nonempty, closed and convex subset of 𝐸. We say that Π^𝑓_𝐶∶𝐸^∗→2^𝐶 is a generalized 𝑓-projection operator if Π^𝑓_𝐶𝜑=𝑢∈𝐶∶𝐺𝑢,𝜑=inf_𝜉∈𝐶𝐺𝜉,𝜑,∀𝜑∈𝐸^∗. (2.13)

For the generalized 𝑓-projection operator, Wu and Huang [42] proved the following theorem basic properties

Lemma 2.16 (see [29]). Let 𝐸 be a real reflexive Banach space with its dual 𝐸^∗. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Then the following statements hold:(i)Π^𝑓_𝐶 is a nonempty closed convex subset of 𝐶 for all 𝜑∈𝐸^∗;(ii)if 𝐸 is smooth, then for all 𝜑∈𝐸^∗, 𝑥∈Π^𝑓_𝐶 if and only if 𝑥−𝑦,𝜑−𝐽𝑥+𝜌𝑓𝑦−𝜌𝑓(𝑥)≥0,∀𝑦∈𝐶; (2.14)(iii)if 𝐸 is strictly convex and 𝑓∶𝐶→𝑅∪{+∞} is positive homogeneous (𝑖.𝑒.,𝑓(𝑡𝑥)=𝑡𝑓(𝑥)𝑓𝑜𝑟𝑎𝑙𝑙𝑡>0such that 𝑡𝑥∈𝐶,𝑤ℎ𝑒𝑟𝑒𝑥∈𝐶), then Π^𝑓_𝐶 is a single valued mapping.

Fan et al. [30] showed that the condition 𝑓 is a positive homogeneous which appeared in Lemma 2.13 can be removed.

Lemma 2.17 (see [30]). Let 𝐸 be a real reflexive Banach space with its dual 𝐸^∗ and 𝐶 a nonempty, closed, and convex subset of 𝐸. Then if 𝐸 is strictly convex, then Π^𝑓_𝐶 is a single-valued mapping.

Recall that 𝐽 is a single-valued mapping when 𝐸 is a smooth Banach space. There exists a unique element 𝜑∈𝐸. such that 𝜑=𝐽𝑥 for each 𝑥∈𝐸. This substitution in (2.15) gives 𝐺(𝜉,𝐽𝑥)=𝜉²−2𝜉,𝐽𝑥+‖𝑥‖²+2𝜌𝑓(𝜉). (2.15) Now, we consider the second generalized 𝑓-projection operator in a Banach space.

Definition 2.18. Let 𝐸 be a real Banach space and 𝐶 a nonempty, closed, and convex subset of 𝐸. We say that Π^𝑓_𝐶∶𝐸→2^𝐶 is a generalized 𝑓-projection operator if Π^𝑓_𝐶(𝑥)=𝑢∈𝐶∶𝐺(𝑢,𝐽𝑥)=inf_𝜉∈𝐶𝐺(𝜉,𝐽𝑥),∀∈𝐸. (2.16) Obviously, the definition of relatively quasi nonexpansive mapping 𝑇 is equivalent to(R1)𝐹(𝑇)≠∅;(R2)𝐺(𝑝,𝐽𝑇𝑥)≤𝐺(𝑝,𝐽𝑥),forall𝑥∈𝐶,𝑝∈𝐹(𝑇).

Lemma 2.19 (see [31]). Let 𝐸 be a Banach space and let 𝑓∶𝐸→𝑅∪{+∞} be a lower semi-continuous convex functional. Then there exists 𝑥^∗∈𝐸 and 𝛼∈𝑅 such that 𝑓(𝑥)≥𝑥,𝑥^∗+𝛼,∀𝑥∈𝐸. (2.17)

We know that the following lemmas hold for operator Π^𝑓_𝐶.

Lemma 2.20 (see [22]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth and reflexive Banach space 𝐸. Then the following statements hold:(i)Π^𝑓_𝐶𝑥 is a nonempty closed and convex subset of 𝐶 for all 𝑥∈𝐸;(ii)for all 𝑥∈𝐸, 𝑥∈Π^𝑓_𝐶𝑥 if and only if 𝑥−𝑦,𝐽𝑥−𝐽𝑥+𝜌𝑓𝑦−𝜌𝑓(𝑥)≥0,∀𝑦∈𝐶; (2.18)(iii)if 𝐸 is strictly convex, then Π^𝑓_𝐶𝑥 is a single valued mapping.

Lemma 2.21 (see [22]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth and reflexive Banach space 𝐸. let 𝑥∈𝐸 and 𝑥∈Π^𝑓_𝐶, then 𝜙𝑦,𝑥+𝐺(𝑥,𝐽𝑥)≤𝐺𝑦,𝐽𝑥,∀𝑦∈𝐶. (2.19)

The fixed points set 𝐹(𝑇) of a relatively quasi nonexpansive mapping is closed convex as given in the following lemma.

Lemma 2.22 (see [32, 33]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth and reflexive Banach space 𝐸. let 𝑇 be a closed relatively quasi nonexpansive mapping of 𝐶 into itself. Then 𝐹(𝑇) is closed and convex.

Also, this following lemma will be used in the sequel.

Lemma 2.23 (see [25]). Let 𝐸 be a uniformly convex and smooth real Banach space and let {𝑥_𝑛},{𝑦_𝑛} be two sequences of 𝐸. If 𝜙(𝑥_𝑛,𝑦_𝑛)→0 and either {𝑥_𝑛} or {𝑦_𝑛} is bounded, then ‖𝑥_𝑛−𝑦_𝑛‖→0.

Lemma 2.24 (see [24, 25, 27]). Let 𝐶 be a nonempty closed convex subset of a smooth real Banach space 𝐸 and 𝑥∈𝐸. Then, 𝑥₀=Π_𝐶𝑥 if and only if 𝑥₀−𝑦,𝐽𝑥−𝐽𝑥₀≥0,∀𝑦∈𝐶. (2.20)

Lemma 2.25 (see [28]). Let 𝐸 be a uniformly convex Banach space and let 𝐵_𝑟(0)={𝑥∈𝐸∶‖𝑥‖≤𝑟} be a closed ball of 𝐸. Then there exists a continuous strictly increasing convex function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that 𝜆𝑥+𝜇𝑦+𝛾𝑧²≤𝜆‖𝑥‖²+𝜇𝑦²+𝛾‖𝑧‖²−𝜆𝜇𝑔𝑥−𝑦 (2.21) for all 𝑥,𝑦,and𝑧∈𝐵_𝑟(0) and 𝜆,𝜇,and𝛾∈[0,1] with 𝜆+𝜇+𝛾=1.

It is easy to prove the following result.

Lemma 2.26. Let 𝐸 be a strictly convex and smooth real Banach space, let 𝐶 be a closed convex subset of 𝐸, and let 𝑇 be a relatively quasi nonexpansive mapping from 𝐶 into itself. Then 𝐹(𝑇) is closed and convex.

Lemma 2.27. Let 𝐸 be a 𝑝-uniformly convex Banach space with 𝑝≥2. Then, for all 𝑥,𝑦∈𝐸, 𝑗(𝑥)∈𝐽_𝑝(𝑥) and 𝑗(𝑦)∈𝐽_𝑝(𝑦), (𝑥)𝑦≥𝑐𝑥−𝑦,𝑗−𝑗^𝑝𝑐^𝑝−2𝑝𝑥−𝑦^𝑝, (2.22) where 𝐽_𝑝 is the generalized duality mapping from 𝐸 into 𝐸^∗ and 1/𝑐 is the 𝑝-uniformly convexity constant of 𝐸.

Observe that an infinite family of operators {𝑇_𝑛}^∞_𝑛=1 in a Banach space is said to be uniformly closed, if 𝑥_𝑛→𝑥, 𝑥_𝑛−𝑇_𝑛𝑥_𝑛→0 then 𝑇_𝑛𝑥=𝑥 (i.e., 𝑥∈∩^∞_𝑛=1𝐹(𝑇_𝑛)). Obviously, a countable family of uniformly closed of relatively quasi nonexpansive mappings is a countable family of weak relatively nonexpansive mappings.

3. Main Results

Theorem 3.1. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let {𝑇_𝑛}^∞_𝑛=1,{𝑆_𝑛}^∞_𝑛=1 be two countable families of uniformly closed of relatively quasi nonexpansive mappings of 𝐶 into itself such that 𝐹∶=(∩^∞_𝑛=1𝐹(𝑇_𝑛))∩(∩^∞_𝑛=1𝐹(𝑆_𝑛))≠∅. Let 𝑓∶𝐸→𝑅 be a convex and lower semicontinuous mapping with 𝐶⊂int(𝐷(𝑓)). For any given gauss 𝑥₀∈𝐶, define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝐺𝑧,𝐽𝑦_𝑛≤𝐺𝑧,𝐽𝑥_𝑛,𝐶₀=𝑧∈𝐶∶𝐺𝑧,𝐽𝑦₀≤𝐺𝑧,𝐽𝑥₀,𝑄_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑄≥0₀𝑥=𝐶,_𝑛+1=Π^𝑓_{𝐶𝑛⋂𝑄𝑛}𝑥₀, (3.1) with the conditions(i)𝛽_𝑛⁽¹⁾+𝛽_𝑛⁽²⁾+𝛽_𝑛⁽³⁾=1;(ii)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾>0;(iii)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽³⁾>0;(iv)0≤𝛼_𝑛≤𝛼<1𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝛼∈(0,1). Then {𝑥_𝑛} converges strongly to 𝑥^∗, where 𝑥^∗=Π^𝑓_𝐹𝑥₀.

Proof. Step 1. We show that 𝐶_𝑛 and 𝑄_𝑛 are closed and convex for each 𝑛≥0.
From the definitions of 𝐶_𝑛 and 𝑄_𝑛, it is obvious that 𝑄_𝑛 is closed and convex and 𝐶_𝑛 is closed for each 𝑛≥0. Moreover, since 𝐺(𝑧,𝐽𝑦_𝑛)≤𝐺(𝑧,𝐽𝑥_𝑛) is equivalent to 𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛, (3.2) and is equivalent to 2𝑧,𝐽𝑥_𝑛−𝐽𝑦_𝑛≤𝑥_𝑛²+𝑦_𝑛², (3.3) it follows that 𝐶_𝑛 is convex for each 𝑛≥0. So, 𝐶_𝑛⋂𝑄_𝑛 is a closed convex subset of 𝐸 for all 𝑛∈𝑁∪{0}.
Step 2. We show that 𝐹⊂𝐶_𝑛⋂𝑄_𝑛 for all 𝑛≥0. Observe that 𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛. (3.4) Hence from the definition of 𝐺(𝑥,𝐽𝑦) and the convexity of ‖⋅‖²we have, for all 𝑝∈𝐹, that 𝐺𝑝,𝐽𝑧_𝑛=𝐺𝑝,𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛=𝑝²−2𝑝,𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛+𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛²𝑝≤𝑝+2𝜌𝑓₂−2𝑝,𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛+𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛²+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛²+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛²+2𝜌𝑓𝑝=𝛽_𝑛⁽¹⁾𝐺𝑝,𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐺𝑝,𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐺𝑝,𝐽𝑆_𝑛𝑥_𝑛≤𝛽_𝑛⁽¹⁾𝐺𝑝,𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐺𝑝,𝐽𝑥_𝑛+𝛽_𝑛⁽³⁾𝐺𝑝,𝐽𝑥_𝑛=𝐺𝑝,𝐽𝑥_𝑛. (3.5)
By the similar reason we have, for all 𝑝∈𝐹, that 𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝐺𝑝,𝐽𝑦_𝑛=𝐺𝑝,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛=𝑝²−2𝑝,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛+𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛²𝑝≤𝑝+2𝜌𝑓₂−2𝑝,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛+𝛼_𝑛𝐽𝑥_𝑛²+1−𝛼_𝑛𝐽𝑧_𝑛²+2𝜌𝑓𝑝=𝛼_𝑛𝐺𝑝,𝐽𝑥_𝑛+1−𝛼_𝑛𝐺𝑝,𝐽𝑧_𝑛≤𝛼_𝑛𝐺𝑝,𝐽𝑥_𝑛+1−𝛼_𝑛𝐺𝑝,𝐽𝑥_𝑛=𝐺𝑝,𝐽𝑥_𝑛. (3.6)(3.7) That is, 𝑝∈𝐶_𝑛 for all 𝑛≥0.
Next, we show that 𝐹⊂𝑄_𝑛 for all 𝑛≥0, we prove this by induction. For 𝑛=0, we have 𝐹⊂𝐶=𝑄₀. Assume that 𝐹⊂𝑄_𝑛. Since 𝑥_𝑛+1=Π^𝑓_{𝐶𝑛⋂𝑄𝑛}(𝑥₀), by Definition 2.15 we have 𝐺𝑥_𝑛+1,𝐽𝑥₀≤𝐺𝑧,𝐽𝑥₀,∀𝑧∈𝐶_𝑛𝑄_𝑛. (3.8) It is equivalent to 𝜙𝑥_𝑛+1,𝐽𝑥₀≤𝜙𝑧,𝐽𝑥₀, (3.9) and is equivalent to 𝑥_𝑛+1−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛+1≥0,∀𝑧∈𝐶_𝑛𝑄_𝑛. (3.10) As 𝐹⊂𝐶_𝑛⋂𝑄_𝑛, by the induction assumptions, the last inequality holds, in particular, for all 𝑧∈𝐹. This together with the definition of 𝑄_𝑛+1 implies that 𝐹⊂𝑄_𝑛+1.
Step 3. We show that 𝑥_𝑛→𝑥^∗ as 𝑛→∞, and 𝑥^∗∈𝐹.
We now show that lim_𝑛→∞𝐺(𝑥_𝑛,𝐽𝑥₀) exists. Since 𝑓∶𝐸→𝑅 is a convex and lower semi-continuous, applying Lemma 2.19, we see that there exists 𝑢^∗∈𝐸^∗ and 𝛼∈𝑅 such that 𝑓𝑦≥𝑦,𝑢^∗+𝛼,∀𝑦∈𝐸. (3.11) It follows that 𝐺𝑥_𝑛+1,𝐽𝑥₀=𝑥_𝑛+1²−2𝑥_𝑛+1,𝐽𝑥₀+𝑥₀²+2𝜌𝑓𝑥_𝑛+1≥𝑥_𝑛+1²−2𝑥_𝑛+1,𝐽𝑥₀+𝑥₀²+2𝜌𝑥_𝑛+1,𝑢^∗=𝑥+2𝜌𝛼_𝑛+12−2𝑥_𝑛+1,𝐽𝑥₀−𝜌𝑢^∗+𝑥₀²≥𝑥+2𝜌𝛼_𝑛+12−2𝑥_𝑛+1𝐽𝑥₀−𝜌𝑢^∗+𝑥₀²=𝑥+2𝜌𝛼_𝑛+1−𝐽𝑥₀−𝜌𝑢^∗2−𝐽𝑥₀−𝜌𝑢^∗2+𝑥₀²+2𝜌𝛼. (3.12)
Since 𝑥_𝑛+1=Π^𝑓_{𝐶𝑛⋂𝑄𝑛}(𝑥₀), for each 𝑝∈𝐹, it follows from (3.12) that 𝐺𝑝,𝐽𝑥₀≥𝐺𝑥_𝑛+1,𝐽𝑥₀≥𝑥_𝑛+1−𝐽𝑥₀−𝜌𝑢^∗2−𝐽𝑥₀−𝜌𝑢^∗2+𝑥₀²+2𝜌𝛼. (3.13) This implies that {𝑥_𝑛}^∞_𝑛=1 is bounded and so is {𝐺(𝑥_𝑛,𝐽𝑥₀)}^∞_𝑛=0.
Since 𝑥_𝑛+1=Π^𝑓_{𝐶𝑛⋂𝑄𝑛}(𝑥₀)∈𝐶_𝑛⋂𝑄_𝑛⊂𝐶_𝑛−1⋂𝑄_𝑛−1, by Lemma 2.21 we have 𝜙𝑥_𝑛+1,𝑥_𝑛+𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑥_𝑛+1,𝐽𝑥₀. (3.14) It is obvious that 𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑥_𝑛+1,𝐽𝑥₀, (3.15) and so {𝐺(𝑥_𝑛,𝐽𝑥₀)}^∞_𝑛=0 is nondecreasing. It follows that the limit of {𝐺(𝑥_𝑛,𝐽𝑥₀)}^∞_𝑛=0 exists.
By the fact that 𝐶_𝑛+𝑚⋂𝑄_𝑛+𝑚⊂𝐶_𝑛−1⋂𝑄_𝑛−1 and 𝑥_𝑛+1=Π^𝑓_{𝐶𝑛⋂𝑄𝑛}(𝑥₀)∈𝐶_𝑛−1⋂𝑄_𝑛−1, by Lemma 2.21 we obtain 𝜙𝑥_𝑛+𝑚,𝑥_𝑛+𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑥_𝑛+𝑚,𝐽𝑥₀. (3.16) Taking the limit as 𝑚,𝑛→∞ in (3.16), we obtain lim_𝑛→∞𝜙𝑥_𝑛+𝑚,𝑥_𝑛=0, (3.17) which holds uniformly for all 𝑚. By using Lemma 2.21, we get that lim_𝑛→∞𝑥_𝑛+𝑚−𝑥_𝑛=0, (3.18) which holds uniformly for all 𝑚. Then {𝑥_𝑛} is a Cauchy sequence, therefore there exists a point 𝑥^∗∈𝐶 such that 𝑥_𝑛→𝑥^∗. In particular, we have lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.19) Since 𝑥_𝑛+1=Π^𝑓_{𝐶𝑛⋂𝑄𝑛}(𝑥₀)∈𝐶_𝑛, from the definition of 𝐶_𝑛, we know that 𝐺𝑥_𝑛+1,𝐽𝑦_𝑛≤𝐺𝑥_𝑛+1,𝐽𝑥_𝑛 (3.20) is equivalent to 𝜙𝑥_𝑛+1,𝑦_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛. (3.21) So, 𝜙(𝑥_𝑛+1,𝑦_𝑛)→0.
By using Lemma 2.23, we get that 𝑥_𝑛+1−𝑦_𝑛→0, (3.22) as 𝑛→∞.
Hence 𝑦_𝑛→𝑥^∗ as 𝑛→∞. Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we have lim_𝑛→∞1−𝛼_𝑛𝐽𝑧_𝑛−𝐽𝑥_𝑛=𝐽𝑦_𝑛−𝐽𝑥_𝑛=0. (3.23) Since 0≤𝛼_𝑛≤𝛼<1, then lim_𝑛→∞𝐽𝑧_𝑛−𝐽𝑥_𝑛=0. (3.24) Since 𝐽⁻¹ is also uniformly norm-to-norm continuous on bounded sets, we have lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0 (3.25) so that 𝑧_𝑛→𝑥^∗ as 𝑛→∞.
Since {𝑥_𝑛} is convergent, then {𝑥_𝑛} is bounded, so are {𝑧_𝑛}, {𝐽𝑇_𝑛𝑥_𝑛}, and {𝐽𝑆_𝑛𝑥_𝑛}. From the definition of 𝜙(𝑥,𝑦) and 𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛, (3.26) we have, for all 𝑝∈𝐹 that 𝜙𝑝,𝑧_𝑛=𝜙𝑝,𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛=𝑝²−2𝑝,𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛+𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛². (3.27) Therefore by using Lemma 2.25 (inequality (2.21)), for all 𝑝∈𝐹, we have 𝜙𝑝,𝑧_𝑛≤𝑝²−2𝑝,𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛+𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛²+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛²+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛²−𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛=𝛽_𝑛⁽¹⁾𝜙𝑝,𝑥_𝑛+𝛽_𝑛⁽²⁾𝜙𝑝,𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝜙𝑝,𝑆_𝑛𝑥_𝑛−𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛≤𝛽_𝑛⁽¹⁾𝜙𝑝,𝑥_𝑛+𝛽_𝑛⁽²⁾𝜙𝑝,𝑥_𝑛+𝛽_𝑛⁽³⁾𝜙𝑝,𝑥_𝑛−𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛=𝜙𝑝,𝑥_𝑛−𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛, (3.28) and hence 𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑧_𝑛→0, (3.29) as 𝑛→∞. By using the same way, we can prove that 𝛽_𝑛⁽¹⁾𝛽_𝑛⁽³⁾𝑔𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑧_𝑛→0, (3.30) as 𝑛→∞. From the properties of the mapping 𝑔, we have 𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛→0, (3.31) as 𝑛→∞, and 𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛→0, (3.32) as 𝑛→∞. Since 𝐽⁻¹ is also uniformly norm-to-norm continuous on any bounded sets, we have 𝑥_𝑛−𝑇_𝑛𝑥_𝑛→0, (3.33) as 𝑛→∞, and 𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛→0, (3.34) as 𝑛→∞. Since 𝑥_𝑛→𝑥^∗ and 𝑇_𝑛, 𝑆_𝑛 are uniformly closed, 𝑥^∗=𝑇_𝑛𝑥^∗, and 𝑥^∗=𝑆_𝑛𝑥^∗.
Step 4. we show that 𝑥^∗=Π^𝑓_𝐹𝑥₀.
Since 𝐹∶=(∩^∞_𝑛=1𝐹(𝑇_𝑛))∩(∩^∞_𝑛=1𝐹(𝑆_𝑛)) is a closed and convex set, from Lemma 2.19, we know that Π^𝑓_𝐹𝑥₀ is single valued and denote 𝑤=Π^𝑓_𝐹𝑥₀. Since 𝑥_𝑛=Π^𝑓_{𝐶𝑛−1⋂𝑄𝑛−1}(𝑥₀) and 𝑤∈𝐹⊂𝐶_𝑛−1⋂𝑄_𝑛−1, we have 𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑤,𝐽𝑥₀,∀𝑛≥0. (3.35) We know that 𝐺(𝜉,𝐽𝜑) is convex and lower semicontinuous with respect to 𝜉 when 𝜑 is fixed. This implies that 𝐺𝑥^∗,𝐽𝑥₀≤liminf_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀≤limsup_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑤,𝐽𝑥₀. (3.36) From the definition of Π^𝑓_𝐹𝑥₀ and 𝑥^∗∈𝐹, we see that 𝑥^∗=𝑤. This completes the proof.

Based on Theorem 3.1, we have the following.

Corollary 3.2. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let {𝑇_𝑛}^∞_𝑛=1,{𝑆_𝑛}^∞_𝑛=1 be two countable families of weak relatively nonexpansive mappings of 𝐶 into itself such that 𝐹∶=(∩^∞_𝑛=1𝐹(𝑇_𝑛))∩(∩^∞_𝑛=1𝐹(𝑆_𝑛))≠∅. For any given gauss 𝑥₀∈𝐶, define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙z,𝑥_𝑛,𝐶₀=𝑧∈𝐶∶𝜙𝑧,𝑦₀≤𝜙𝑧,𝑥₀,𝑄_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑄≥0₀𝑥=𝐶,_𝑛+1=Π_{𝐶𝑛⋂𝑄𝑛}𝑥₀, (3.37) with the conditions:(i)𝛽_𝑛⁽¹⁾+𝛽_𝑛⁽²⁾+𝛽_𝑛⁽³⁾=1;(ii)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾>0;(iii)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽³⁾>0;(iv)0≤𝛼_𝑛≤𝛼<1for𝑠ome𝛼∈(0,1). Then {𝑥_𝑛} converges strongly to 𝑥^∗, where 𝑥^∗=Π_𝐹𝑥₀.

Proof. Putting 𝑓(𝑥)≡0, we can conclude from Theorem 3.1 the desired conclusion immediately.

Theorem 3.3. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, let {𝑇_𝑛}^∞_𝑛=1,{𝑆_𝑛}^∞_𝑛=1 be two countable families of uniformly closed of relatively quasi nonexpansive mappings of 𝐶 into itself such that 𝐹∶=(∩^∞_𝑛=1𝐹(𝑇_𝑛))∩(∩^∞_𝑛=1𝐹(𝑆_𝑛))≠∅. Let 𝑓∶𝐸→𝑅 be a convex and lower semicontinuous mapping with 𝐶⊂𝑖𝑛𝑡(𝐷(𝑓)). For any given gauss 𝑥₀∈𝐶, define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝐺𝑧,𝐽𝑦_𝑛≤𝛽_𝑛⁽¹⁾−𝛼_𝑛𝛽_𝑛⁽¹⁾𝐺𝑧,𝐽𝑥₀+1−𝛽_𝑛⁽¹⁾+𝛼_𝑛𝛽_𝑛⁽¹⁾𝐺𝑧,𝐽𝑥_𝑛,𝐶₀=𝑧∈𝐶∶𝐺𝑧,𝐽𝑦₀≤𝐺𝑧,𝐽𝑥₀,𝑄_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑄≥0₀𝑥=𝐶,_𝑛+1=Π^𝑓_{𝐶𝑛⋂𝑄𝑛}𝑥₀, (3.38) with the conditions:(i)𝛽_𝑛⁽¹⁾+𝛽_𝑛⁽²⁾+𝛽_𝑛⁽³⁾=1;(ii)lim_𝑛→∞𝛽_𝑛⁽¹⁾=0; (iii)limsup_𝑛→∞𝛽_𝑛⁽²⁾𝛽_𝑛⁽³⁾>0; (iv)0≤𝛼_𝑛≤𝛼<1forsome𝛼∈(0,1). Then {𝑥_𝑛} converges strongly to 𝑥^∗, where 𝑥^∗=Π^𝑓_𝐹𝑥₀.

Proof. Step 1. We show that 𝐶_𝑛 and 𝑄_𝑛 are closed and convex for each 𝑛≥0.
From the definitions of 𝐶_𝑛 and 𝑄_𝑛, it is obvious that 𝑄_𝑛 is closed and convex and 𝐶_𝑛 is closed for each 𝑛≥0. Moreover, since 𝐺(𝑧,𝐽𝑦_𝑛)≤(𝛽_𝑛⁽¹⁾−𝛼_𝑛𝛽_𝑛⁽¹⁾)𝐺(𝑧,𝐽𝑥₀)+(1−𝛽_𝑛⁽¹⁾+𝛼_𝑛𝛽_𝑛⁽¹⁾)𝐺(𝑧,𝐽𝑥_𝑛) is equivalent to 2𝑧,𝛽_𝑛⁽¹⁾−𝛼_𝑛𝛽_𝑛⁽¹⁾𝐽𝑥₀+1−𝛽_𝑛⁽¹⁾+𝛼_𝑛𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛−𝐽𝑦_𝑛≤𝛽_𝑛⁽¹⁾−𝛼_𝑛𝛽_𝑛⁽¹⁾𝑥₀²+1−𝛽_𝑛⁽¹⁾+𝛼_𝑛𝛽_𝑛⁽¹⁾𝑥_𝑛²−𝑦_𝑛². (3.39) it follows that 𝐶_𝑛 is convex for each 𝑛≥0. So, 𝐶_𝑛⋂𝑄_𝑛 is a closed convex subset of 𝐸 for all 𝑛∈𝑁∪{0}.
Step 2. We show that 𝐹⊂𝐶_𝑛⋂𝑄_𝑛 for all 𝑛≥0. Observe that 𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛. (3.40) Hence from the definition of 𝐺(𝑥,𝐽𝑦) and the convexity of ‖⋅‖²we have, for all 𝑝∈𝐹, that 𝐺𝑝,𝐽𝑧_𝑛=𝐺𝑝,𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛=𝑝²−2𝑝,𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛+𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛²𝑝≤𝑝+2𝜌𝑓₂−2𝑝,𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛+𝛽_𝑛⁽¹⁾𝐽𝑥₀²+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛²+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛²+2𝜌𝑓𝑝=𝛽_𝑛⁽¹⁾𝐺𝑝,𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐺𝑝,𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐺𝑝,𝐽𝑆_𝑛𝑥_𝑛≤𝛽_𝑛⁽¹⁾𝐺𝑝,𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐺𝑝,𝐽𝑥_n+𝛽_𝑛⁽³⁾𝐺𝑝,𝐽𝑥_𝑛=𝛽_𝑛⁽¹⁾𝐺𝑝,𝐽𝑥₀+1−𝛽_𝑛⁽¹⁾𝐺𝑝,𝐽𝑥_𝑛. (3.41) By the similar reason we have, for all 𝑝∈𝐹, that 𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝐺𝑝,𝐽𝑦_𝑛=𝐺𝑝,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛=𝑝²−2𝑝,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛+𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛²𝑝≤𝑝+2𝜌𝑓₂−2𝑝,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛+𝛼_𝑛𝐽𝑥_𝑛²+1−𝛼_𝑛𝐽𝑧_𝑛²+2𝜌𝑓𝑝=𝛼_𝑛𝐺𝑝,𝐽𝑥_𝑛+1−𝛼_𝑛𝐺𝑝,𝐽𝑧_𝑛≤𝛼_𝑛𝐺𝑝,𝐽𝑥_𝑛+1−𝛼_𝑛𝛽_𝑛⁽¹⁾𝐺𝑝,𝐽𝑥₀+1−𝛽_𝑛⁽¹⁾𝐺𝑝,𝐽𝑥_𝑛=1−𝛼_𝑛𝛽_𝑛⁽¹⁾𝐺𝑝,𝐽𝑥₀+1−𝛽_𝑛⁽¹⁾+𝛼_𝑛𝛽_𝑛⁽¹⁾𝐺𝑝,𝐽𝑥_𝑛. (3.42) That is, 𝑝∈𝐶_𝑛 for all 𝑛≥0.
Next, we show that 𝐹⊂𝑄_𝑛 for all 𝑛≥0, we prove this by induction. For 𝑛=0, we have 𝐹⊂𝐶=𝑄₀. Assume that 𝐹⊂𝑄_𝑛. Since 𝑥_𝑛+1=Π^𝑓_{𝐶𝑛⋂𝑄𝑛}(𝑥₀),  by Definition 2.15 we have 𝐺𝑥_𝑛+1,𝐽𝑥₀≤𝐺𝑧,𝐽𝑥₀,∀𝑧∈𝐶_𝑛𝑄_𝑛. (3.43) It is equivalent to 𝜙𝑥_𝑛+1,𝐽𝑥₀≤𝜙𝑧,𝐽𝑥₀, (3.44) and is equivalent to 𝑥_𝑛+1−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛+1≥0,∀𝑧∈𝐶_𝑛𝑄_𝑛. (3.45) As 𝐹⊂𝐶_𝑛⋂𝑄_𝑛, by the induction assumptions, the last inequality holds, in particular, for all 𝑧∈𝐹. This together with the definition of 𝑄_𝑛+1 implies that 𝐹⊂𝑄_𝑛+1.
Step 3. We show that 𝑥_𝑛→𝑥^∗ as 𝑛→∞, and 𝑥^∗∈𝐹.
We now show that lim_𝑛→∞𝐺(𝑥_𝑛,𝐽𝑥₀) exists. Since 𝑓∶𝐸→𝑅 is a convex and lower semi-continuous, applying Lemma 2.19, we see that there exists 𝑢^∗∈𝐸^∗ and 𝛼∈𝑅 such that 𝑓𝑦≥𝑦,𝑢^∗+𝛼,∀𝑦∈𝐸. (3.46)
It follows that 𝐺𝑥_𝑛+1,𝐽𝑥₀=𝑥_𝑛+1²−2𝑥_𝑛+1,𝐽𝑥₀+𝑥₀²+2𝜌𝑓𝑥_𝑛+1≥𝑥_𝑛+1²−2𝑥_𝑛+1,𝐽𝑥₀+𝑥₀²+2𝜌𝑥_𝑛+1,𝑢^∗=𝑥+2𝜌𝛼_𝑛+12−2𝑥_𝑛+1,𝐽𝑥₀−𝜌𝑢^∗+𝑥₀²≥𝑥+2𝜌𝛼_𝑛+12−2𝑥_𝑛+1𝐽𝑥₀−𝜌𝑢^∗+𝑥₀²=𝑥+2𝜌𝛼_𝑛+1−𝐽𝑥₀−𝜌𝑢^∗2−𝐽𝑥₀−𝜌𝑢^∗2+𝑥₀²+2𝜌𝛼 (3.47)
Since 𝑥_𝑛+1=Π^𝑓_{𝐶𝑛⋂𝑄𝑛}(𝑥₀), for each 𝑝∈𝐹, it follows from (3.47) that 𝐺𝑝,𝐽𝑥₀≥𝐺𝑥_𝑛+1,𝐽𝑥₀≥𝑥_𝑛+1−𝐽𝑥₀−𝜌𝑢^∗2−𝐽𝑥₀−𝜌𝑢^∗2+𝑥₀²+2𝜌𝛼. (3.48)
This implies that {𝑥_𝑛}^∞_𝑛=1 is bounded and so is {𝐺(𝑥_𝑛,𝐽𝑥₀)}^∞_𝑛=0.
Since 𝑥_𝑛+1=Π^𝑓_{𝐶𝑛⋂𝑄𝑛}(𝑥₀)∈𝐶_𝑛⋂𝑄_𝑛⊂𝐶_𝑛−1⋂𝑄_𝑛−1, by Lemma 2.21, we have 𝜙𝑥_𝑛+1,𝑥_𝑛+𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑥_𝑛+1,𝐽𝑥₀. (3.49) It is obvious that 𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑥_𝑛+1,𝐽𝑥₀, (3.50) and so {𝐺(𝑥_𝑛,𝐽𝑥₀)}^∞_𝑛=0 is nondecreasing. It follows that the limit of {𝐺(𝑥_𝑛,𝐽𝑥₀)}^∞_𝑛=0 exists.
By the fact that 𝐶_𝑛+𝑚⋂𝑄_𝑛+𝑚⊂𝐶_𝑛−1⋂𝑄_𝑛−1 and 𝑥_𝑛+1=Π^𝑓_{𝐶𝑛⋂𝑄𝑛}(𝑥₀)∈𝐶_𝑛−1⋂𝑄_𝑛−1, by Lemma 2.21 we obtain 𝜙𝑥_𝑛+𝑚,𝑥_𝑛+𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑥_𝑛+𝑚,𝐽𝑥₀. (3.51)
Taking the limit as 𝑚,𝑛→∞ in (3.51), we obtain lim_𝑛→∞𝜙𝑥_𝑛+𝑚,𝑥_𝑛=0, (3.52) which holds uniformly for all 𝑚. By using Lemma 2.23, we get that lim_𝑛→∞𝑥_𝑛+𝑚−𝑥_𝑛=0, (3.53) which holds uniformly for all 𝑚. Then {𝑥_𝑛} is a Cauchy sequence, therefore there exists a point 𝑥^∗∈𝐶 such that 𝑥_𝑛→𝑥^∗. In particular, we have lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.54) Since 𝑥_𝑛+1=Π^𝑓_{𝐶𝑛⋂𝑄𝑛}(𝑥₀)∈𝐶_𝑛, from the definition of 𝐶_𝑛, we know that 𝐺𝑥_𝑛+1,𝐽𝑦_𝑛≤𝐺𝑧,𝐽𝑦_𝑛≤𝛽_𝑛⁽¹⁾−𝛼_𝑛𝛽_𝑛⁽¹⁾𝐺𝑥_𝑛+1,𝐽𝑥₀+1−𝛽_𝑛⁽¹⁾+𝛼_𝑛𝛽_𝑛⁽¹⁾𝐺𝑥_𝑛+1,𝐽𝑥_𝑛, (3.55) and is equivalent to 𝜙𝑥_𝑛+1,𝑦_𝑛+2𝜌𝑓𝑥_𝑛+1≤𝛽_𝑛⁽¹⁾−𝛼_𝑛𝛽_𝑛⁽¹⁾𝜙𝑥_𝑛+1,𝑥₀+2𝜌𝑓𝑥_𝑛+1+1−𝛽_𝑛⁽¹⁾+𝛼_𝑛𝛽_𝑛⁽¹⁾𝜙𝑥_𝑛+1,𝑥_𝑛+2𝜌𝑓𝑥_𝑛+1, (3.56) and is equivalent to 𝜙𝑥_𝑛+1,𝑦_𝑛≤𝛽_𝑛⁽¹⁾−𝛼_𝑛𝛽_𝑛⁽¹⁾𝜙𝑥_𝑛+1,𝑥₀+1−𝛽_𝑛⁽¹⁾+𝛼_𝑛𝛽_𝑛⁽¹⁾𝜙𝑥_𝑛+1,𝑥_𝑛, (3.57)
So, 𝜙(𝑥_𝑛+1,𝑦_𝑛)→0.
By using Lemma 2.23, we get that 𝑥_𝑛+1−𝑦_𝑛→0, (3.58) as 𝑛→∞,
and hence 𝑦_𝑛→𝑥^∗ as 𝑛→∞. Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we have lim_𝑛→∞1−𝛼_𝑛𝐽𝑧_𝑛−𝐽𝑥_𝑛=𝐽𝑦_𝑛−𝐽𝑥_𝑛=0. (3.59)
Since 0≤𝛼_𝑛≤𝛼<1, then lim_𝑛→∞𝐽𝑧_𝑛−𝐽𝑥_𝑛=0. (3.60)
Since 𝐽⁻¹ is also uniformly norm-to-norm continuous on bounded sets, we have lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0, (3.61) so that 𝑧_𝑛→𝑥^∗ as 𝑛→∞.
Since {𝑥_𝑛} is convergent, then {𝑥_𝑛} is bounded, so are {𝑧_𝑛}, {𝐽𝑇_𝑛𝑥_𝑛}, and {𝐽𝑆_𝑛𝑥_𝑛}. From the definition of 𝜙(𝑥,𝑦) and 𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛, (3.62) we have, for all 𝑝∈𝐹 that 𝜙𝑝,𝑧_𝑛=𝜙𝑝,𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛x_𝑛=𝑝²−2𝑝,𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛+𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛². (3.63) Therefore by using Lemma 2.25 (inequality (2.21)), for all 𝑝∈𝐹, we have 𝜙𝑝,𝑧_𝑛≤𝑝²−2𝑝,𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛+𝛽_𝑛⁽¹⁾𝐽𝑥₀²+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛²+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛²−𝛽_𝑛⁽²⁾𝛽_𝑛⁽³⁾𝑔𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛=𝛽_𝑛⁽¹⁾𝜙𝑝,𝑥₀+𝛽_𝑛⁽²⁾𝜙𝑝,𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝜙𝑝,𝑆_𝑛𝑥_𝑛−𝛽_𝑛⁽²⁾𝛽_𝑛⁽³⁾𝑔𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛≤𝛽_𝑛⁽¹⁾𝜙𝑝,𝑥₀+𝛽_𝑛⁽²⁾𝜙𝑝,𝑥_𝑛+𝛽_𝑛⁽³⁾𝜙𝑝,𝑥_𝑛−𝛽_𝑛⁽²⁾𝛽_𝑛⁽³⁾𝑔𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛=𝛽_𝑛⁽¹⁾𝜙𝑝,𝑥₀+1−𝛽_𝑛⁽¹⁾𝜙𝑝,𝑥_𝑛−𝛽_𝑛⁽²⁾𝛽_𝑛⁽³⁾𝑔𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛, (3.64) and hence 𝛽_𝑛⁽²⁾𝛽_𝑛⁽³⁾𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛≤𝛽_𝑛⁽¹⁾𝜙𝑝,𝑥₀+1−𝛽_𝑛⁽¹⁾𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑧_𝑛. (3.65)
From lim_𝑛→∞𝛽_𝑛⁽¹⁾=0 and 𝑥_𝑛→𝑥^∗, 𝑧_𝑛→𝑥^∗,we have 𝛽_𝑛⁽²⁾𝛽_𝑛⁽³⁾𝑔𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛→0, (3.66) as 𝑛→∞. From the properties of the mapping 𝑔, we have 𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛→0, (3.67) as 𝑛→∞.
Since 𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛, (3.68) then we have 𝐽𝑧_𝑛=𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛. (3.69) Therefore 𝐽𝑥_𝑛−𝐽𝑧_𝑛=𝐽𝑥_𝑛−𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛=𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛−𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛≥𝛽_𝑛⁽²⁾𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛−𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛−𝐽𝑥₀, (3.70) which leads to 𝛽_𝑛⁽²⁾𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛≤𝐽𝑥_𝑛−𝐽𝑧_𝑛+𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛−𝐽𝑥₀. (3.71) Since 𝑥_𝑛→𝑥^∗, 𝑧_𝑛→𝑥^∗ and lim_𝑛→∞𝛽_𝑛⁽¹⁾=0. Then from above inequality we obtain 𝛽_𝑛⁽²⁾𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛→0. (3.72) On the other hand, by using the property of norm ‖.‖, we have 𝛽_𝑛⁽²⁾𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛=𝛽_𝑛⁽²⁾𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛−𝛽_𝑛⁽³⁾𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛=𝛽_𝑛⁽²⁾+𝛽_𝑛⁽³⁾𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑇𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛≥𝛽_𝑛⁽²⁾+𝛽_𝑛⁽³⁾𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛−𝛽_𝑛⁽³⁾𝐽𝑇𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛, (3.73) which leads to the following inequality 𝛽_𝑛⁽²⁾+𝛽_𝑛⁽³⁾𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛≤𝛽_𝑛⁽²⁾𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑇𝑥_𝑛−𝐽𝑆_𝑛𝑥_𝑛. (3.74)
Therefore, by using (3.67) and (3.72) we have 𝛽_𝑛⁽²⁾+𝛽_𝑛⁽³⁾𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛→0. (3.75)
This together with condition (ii) of Theorem 3.3 implies that 𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛→0. (3.76) Since 𝐽⁻¹ is uniformly norm-to-norm continuous on bounded sets, then we have 𝑥_𝑛−𝑇_𝑛𝑥_𝑛→0, (3.77) as 𝑛→∞.  By using the same way, we can prove that 𝑥_𝑛−𝑆_𝑛𝑥_𝑛→0 (3.78) as 𝑛→∞. Since 𝑥_𝑛→𝑥^∗ and 𝑇_𝑛, 𝑆_𝑛 are uniformly closed, 𝑥^∗=𝑇_𝑛𝑥^∗, and 𝑥^∗=𝑆_𝑛𝑥^∗, so that 𝑥^∗∈𝐹.
Step 4. we show that 𝑥^∗=Π^𝑓_𝐹𝑥₀.
Since 𝐹∶=(∩^∞_𝑛=1𝐹(𝑇_𝑛))∩(∩^∞_𝑛=1𝐹(𝑆_𝑛)) is a closed and convex set, from Lemma 2.19, we know that Π^𝑓_𝐹𝑥₀ is single valued and denote 𝑤=Π^𝑓_𝐹𝑥₀. Since 𝑥_𝑛=Π^𝑓_{𝐶𝑛−1⋂𝑄𝑛−1}(𝑥₀) and 𝑤∈𝐹⊂𝐶_𝑛−1⋂𝑄_𝑛−1 we have 𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑤,𝐽𝑥₀,∀𝑛≥0. (3.79) We know that 𝐺(𝜉,𝐽𝜑) is convex and lower semicontinuous with respect to 𝜉 when 𝜑 is fixed. This implies that 𝐺𝑥^∗,𝐽𝑥₀≤liminf_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀≤limsup_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑤,𝐽𝑥₀. (3.80) From the definition of Π^𝑓_𝐹𝑥₀ and 𝑥^∗∈𝐹, we see that 𝑥^∗=𝑤. This completes the proof.

Based on Theorem 3.3, we have the following.

Corollary 3.4. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let {𝑇_𝑛}^∞_𝑛=1,{𝑆_𝑛}^∞_𝑛=1 be two countable families of weak relatively nonexpansive mappings of 𝐶 into itself such that 𝐹∶=(∩^∞_𝑛=1𝐹(𝑇_𝑛))∩(∩^∞_𝑛=1𝐹(𝑆_𝑛))≠∅. For any given gauss 𝑥₀∈𝐶, define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥₀+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝛽_𝑛⁽¹⁾−𝛼_𝑛𝛽_𝑛⁽¹⁾𝜙𝑧,𝑥₀+1−𝛽_𝑛⁽¹⁾+𝛼_𝑛𝛽_𝑛⁽¹⁾𝜙𝑧,𝑥_𝑛,𝐶₀=𝑧∈𝐶∶𝜙𝑧,𝑦₀≤𝜙𝑧,𝑥₀,𝑄_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑄≥0₀𝑥=𝐶,_𝑛+1=Π_{𝐶𝑛⋂𝑄𝑛}𝑥₀, (3.81) with the conditions:(i)𝛽_𝑛⁽¹⁾+𝛽_𝑛⁽²⁾+𝛽_𝑛⁽³⁾=1; (ii)lim_𝑛→∞𝛽_𝑛⁽¹⁾=0; (iii)limsup_𝑛→∞𝛽_𝑛⁽²⁾𝛽_𝑛⁽³⁾>0; (iv)0≤𝛼_𝑛≤𝛼<1𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝛼∈(0,1). Then {𝑥_𝑛} converges strongly to 𝑥^∗, where 𝑥^∗=Π_𝐹𝑥₀.

Proof. Putting 𝑓(𝑥)≡0, we can conclude from Theorem 3.3 the desired conclusion immediately.

4. Applications

Let 𝐸 be a real Banach space and let 𝐸^∗ be the dual space of 𝐸. Let 𝐶 be a closed convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅=(−∞,+∞). The equilibrium problem is to find 𝑥∈𝐶 such that 𝑓𝑥,𝑦≥0,∀𝑦∈𝐶. (4.1) The set of solutions of (4.1) is denoted by 𝐸𝑃(𝑓). Given a mapping 𝑇∶𝐶→𝐸^∗ let 𝑓(𝑥,𝑦)=⟨𝑇𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶. Then, 𝑝∈𝐸𝑃(𝑓) if and only if ⟨𝑇𝑝,𝑦−𝑝⟩≥0 for all 𝑦∈𝐶, that is, 𝑝 is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (4.1). Some methods have been proposed to solve the equilibrium problem in Hilbert spaces.

For solving the equilibrium problem, let us assume that a bifunction 𝑓 satisfies the following conditions:(A1)𝑓(𝑥,𝑥)=0,forall𝑥∈𝐸;(A2)𝑓 is monotone, that is, 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0,forall𝑥,𝑦∈𝐸;(A3) for all 𝑥,𝑦,𝑧∈𝐸, limsup_𝑡↓0𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦);(A4) for all 𝑥∈𝐶, 𝑓(𝑥,⋅) is convex and lower semi-continuous.

Lemma 4.1 (see [34]). Let 𝐶 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅=(−∞,+∞) satisfying (A1)–(A4), and let 𝑟>0 and 𝑥∈𝐸. Then, there exists 𝑧∈𝐶 such that 𝑓+1𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶. (4.2)

Lemma 4.2 (see [34]). Let 𝐶 be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space 𝐸, and let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅=(−∞,+∞) satisfying (A1)–(A4). For 𝑟>0, define a mapping 𝑇_𝑟∶𝐸→𝐶 as follows: 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶𝑓𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶, (4.3) for all 𝑥∈𝐸. Then, the following hold(1)𝑇_𝑟 is single valued;(2)𝑇_𝑟 is a firmly nonexpansive-type mapping, that is, for all 𝑥,𝑦∈𝐸, 𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑇_𝑟𝑥−𝐽𝑇_𝑟𝑦≤𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑥−𝐽𝑦; (4.4)(3)𝐹(𝑇_𝑟)=𝐸𝑃(𝑓);(4)𝐸𝑃(𝑓) is closed and convex;(5)𝑇_𝑟 is also a relatively nonexpansive mapping.

Lemma 4.3 (see [34]). Let 𝐶 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅=(−∞,+∞) satisfying (A1)–(A4), and let 𝑟>0 and 𝑥∈𝐸, 𝑞∈𝐹(𝑇_𝑟), then the following holds 𝜙𝑞,𝑇_𝑟𝑥+𝜙𝑇_𝑟𝑥,𝑥≤𝜙𝑞,𝑥. (4.5)

Lemma 4.4. Let 𝐸 be a 𝑝-uniformly convex with 𝑝≥2 and uniformly smooth Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to 𝑅=(−∞,+∞) satisfying (A1)–(A4). Let {𝑟_𝑛} be a positive real sequence such that lim_𝑛→∞𝑟_𝑛=𝑟>0. Then the sequence of mappings {𝑇_𝑟𝑛} is uniformly closed.

Proof. (1) Let {𝑥_𝑛} be a convergent sequence in 𝐶. Let 𝑧_𝑛=𝑇_𝑟𝑛𝑥_𝑛 for all 𝑛, then 𝑓𝑧_𝑛+1,𝑦𝑟_𝑛𝑦−𝑧_𝑛,𝐽𝑧_n−𝐽𝑥_𝑛𝑓𝑧≥0,∀𝑦∈𝐶,_𝑛+𝑚+1,𝑦𝑟_𝑛+𝑚𝑦−𝑧_𝑛+𝑚,𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛+𝑚≥0,∀𝑦∈𝐶. (4.6)(4.7)
Putting 𝑦=𝑧_𝑛+𝑚 in (4.6) and 𝑦=𝑧_𝑛 in (4.7), we have 𝑓𝑧_𝑛,𝑧_𝑛+𝑚+1𝑟_𝑛𝑧_𝑛+𝑚−𝑧_𝑛,𝐽𝑧_𝑛−𝐽𝑥_𝑛𝑓𝑧≥0,∀𝑦∈𝐶,_𝑛+𝑚,𝑧_𝑛+1𝑟_𝑛+𝑚𝑧_𝑛−𝑧_𝑛+𝑚,𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛+𝑚≥0,∀𝑦∈𝐶. (4.8)
So, from (A2) we have 𝑧_𝑛+𝑚−𝑧_𝑛,𝐽𝑧_𝑛−𝐽𝑥_𝑛𝑟_𝑛−𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛+𝑚𝑟_𝑛+𝑚≥0, (4.9) and hence 𝑧_𝑛+𝑚−𝑧_𝑛,𝐽𝑧_𝑛−𝐽𝑥_𝑛−𝑟_𝑛𝑟_𝑛+𝑚𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛+𝑚≥0. (4.10) Thus, we have 𝑧_𝑛+𝑚−𝑧_𝑛,𝐽𝑧_𝑛−𝐽𝑧_𝑛+𝑚+𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛−𝑟_𝑛𝑟_𝑛+𝑚𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛+𝑚≥0, (4.11) which implies that 𝑧_𝑛+𝑚−𝑧_𝑛,𝐽𝑧_𝑛+𝑚−𝐽𝑧_𝑛≤𝑧_𝑛+𝑚−𝑧_𝑛,𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛−𝑟_𝑛𝑟_𝑛+𝑚𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛+𝑚≥0. (4.12) By using Lemma 2.25, we obtain 𝑐^𝑝𝑐^𝑝−2𝑝𝑧_𝑛+𝑚−𝑧_𝑛^𝑝≤𝑧_𝑛+𝑚−𝑧_𝑛,𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛−𝑟_𝑛𝑟_𝑛+𝑚𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛+𝑚=𝑧_𝑛+𝑚−𝑧_𝑛,𝑟1−^𝑛𝑟_𝑛+𝑚𝐽𝑧_𝑛+𝑚+𝑟_𝑛𝑟_𝑛+𝑚𝐽𝑥_𝑛+𝑚−𝐽𝑥_𝑛. (4.13)
Therefore, we get 𝑐^𝑝𝑐^𝑝−2𝑝𝑧_𝑛+𝑚−𝑧_𝑛^𝑝−1≤𝑟1−^𝑛𝑟_𝑛+𝑚𝐽𝑧_𝑛+𝑚+𝑟_𝑛𝑟_𝑛+𝑚𝐽𝑥_𝑛+𝑚−𝐽𝑥_𝑛. (4.14) On the other hand, for any 𝑝∈𝐸𝑃(𝑓), from 𝑧_𝑛=𝑇_𝑟𝑛𝑥_𝑛, we have 𝑧_𝑛−𝑝=𝑇_𝑟𝑛𝑥_𝑛−𝑝≤𝑥_𝑛−𝑝, (4.15) so that {𝑧_𝑛} is bounded. Since lim_𝑛→∞𝑟_𝑛=𝑟>0, this together with (4.14) implies that {𝑧_𝑛} is a Cauchy sequence. Hence 𝑇_𝑟𝑛𝑥_𝑛=𝑧_𝑛 is convergent.
(2) By using the Lemma 4.2, we know that ^∞_𝑛=1𝐹𝑇_𝑟𝑛=𝐸𝑃𝑓≠∅. (4.16)
(3) From (1) we know that, lim_𝑛→∞𝑇_𝑟𝑛𝑥 exists for all 𝑥∈𝐶. So, we can define a mapping 𝑇 from 𝐶 into itself by 𝑇𝑥=lim_𝑛→∞𝑇_𝑟𝑛𝑥,∀𝑥∈𝐶. (4.17) It is obvious that, 𝑇 is nonexpansive. It is easy to see that 𝐸𝑃𝑓=^∞_𝑛=1𝐹𝑇_𝑟𝑛⊂𝐹(𝑇). (4.18) On the other hand, let 𝑤∈𝐹(𝑇),𝑤_𝑛=𝑇_𝑟𝑛𝑤, we have 𝑓𝑤_𝑛+1,𝑦𝑟_𝑛𝑦−𝑤_𝑛,𝐽𝑤_𝑛−𝐽𝑤≥0,∀𝑦∈𝐶. (4.19)
By (A2) we know 1𝑟_𝑛𝑦−𝑤_𝑛,𝐽𝑤_𝑛−𝐽𝑤≥𝑓𝑦,𝑤_𝑛,∀𝑦∈𝐶. (4.20) Since 𝑤_𝑛→𝑇𝑤=𝑤 and from (A4), we have 𝑓(𝑦,𝑤)≤0, for all 𝑦∈𝐶. Then, for 𝑡∈(0,1] and 𝑦∈𝐶, ()()𝑤()+()𝑓()().0=𝑓𝑡𝑦+1−𝑡𝑤,𝑡𝑦+1−𝑡≤𝑡𝑓𝑡𝑦+1−𝑡𝑤,𝑦1−𝑡𝑡𝑦+1−𝑡𝑤,𝑤≤𝑡𝑓𝑡𝑦+1−𝑡𝑤,𝑦 (4.21) Therefore, we have 𝑓𝑡𝑦+(1−𝑡)𝑤,𝑦≥0. (4.22) Letting 𝑡↓0 and using (A3), we get 𝑓𝑤,𝑦≥0,∀𝑦∈𝐶 (4.23) and hence 𝑤∈𝐸𝑃(𝑓). From the above two respects, we know that 𝐹(𝑇)=⋂^∞_𝑛=0𝐹(𝑇_𝑟𝑛).
Next we show {𝑇_𝑟𝑛} is uniformly closed. Assume 𝑥_𝑛→𝑥 and ‖𝑥_𝑛−𝑇_𝑟𝑛𝑥_𝑛‖→0, from the above results we know that 𝑇𝑥=lim_𝑛→∞𝑇_𝑟𝑛𝑥. On the other hand, from ‖𝑥_𝑛−𝑇_𝑟𝑛𝑥_𝑛‖→0, we also get lim_𝑛→∞𝑇_𝑟𝑛𝑥=𝑥, so that 𝑥∈𝐹(𝑇)=⋂^∞_𝑛=1𝐹(𝑇_𝑟𝑛). That is, the sequence of mappings {𝑇_𝑟𝑛} is uniformly closed. This completes the proof.

Let 𝐴 be a multi-valued operator from 𝐸 to 𝐸^∗ with domain 𝐷(𝐴)={𝑧∈𝐸∶𝐴𝑧≠∅} and range 𝑅(𝐴)={𝑧∈𝐸∶𝑧∈𝐷(𝐴)}. An operator 𝐴 is said to be monotone if 𝑥₁−𝑥₂,𝑦₁−𝑦₂≥0, (4.24) for each 𝑥₁,𝑥₂∈𝐷(𝐴) and 𝑦₁∈𝐴𝑥₁,𝑦₂∈𝐴𝑥₂. A monotone operator A is said to be maximal if it is graph 𝐺(𝐴)={(𝑥,𝑦)∶𝑦∈𝐴𝑥} is not properly contained in the graph of any other monotone operator. We know that if 𝐴 is a maximal monotone operator, then 𝐴⁻¹0 is closed and convex. The following result is also well known.

Theorem 4.5. Let 𝐸 be a reflexive, strictly convex, and smooth Banach space and let 𝐴 be a monotone operator from 𝐸 to 𝐸^∗. Then 𝐴 is maximal if and only if 𝑅(𝐽+𝑟𝐴)=𝐸^∗, for all 𝑟>0.

Let 𝐸 be a reflexive, strictly convex, and smooth Banach space, and let 𝐴 be a maximal monotone operator from 𝐸 to 𝐸^∗. Using Theorem 4.5 and strict convexity of 𝐸, one obtains that for every 𝑟>0 and 𝑥∈𝐸, there exists a unique 𝑥_𝑟 such that 𝐽𝑥∈𝐽𝑥_𝑟+𝑟𝐴𝑥_𝑟. (4.25) Then one can defines a single-valued mapping 𝐽_𝑟∶𝐸→𝐷(𝐴) by 𝐽_𝑟=(𝐽+𝑟𝐴)⁻¹𝐽 and such a 𝐽_𝑟 is called the resolvent of 𝐴, one knows that 𝐴⁻¹0=𝐹(𝐽_𝑟) for all 𝑟>0.

Theorem 4.6. Let 𝐸 be a uniformly convex and a uniformly smooth Banach space, and let 𝐴 be a maximal monotone operator from 𝐸 to 𝐸^∗, let 𝐽_𝑟 be a resolvent of 𝐴 for 𝑟>0. Then for any sequence {𝑟_𝑛}^∞_𝑛=1 such that liminf_𝑛→∞𝑟_𝑛>0, {𝐽_𝑟𝑛}^∞_𝑛=1 is a uniformly closed sequence of relatively quasi nonexpansive mappings.

Proof. Firstly, we show that {𝐽_𝑟𝑛}^∞_𝑛=1 is uniformly closed. Let {𝑧_𝑛}⊂𝐸 be a sequence such that 𝑧_𝑛→𝑝 and lim_𝑛→∞‖𝑧_𝑛−𝐽_𝑟𝑛𝑧_𝑛‖=0. Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we obtain 1𝑟_𝑛𝐽𝑧_𝑛−𝐽𝐽_𝑟𝑛𝑧_𝑛→0. (4.26) It follows from 1𝑟_𝑛𝐽𝑧_𝑛−𝐽𝐽_𝑟𝑛𝑧_𝑛∈𝐴𝐽_𝑟𝑛𝑧_𝑛 (4.27)
and the monotonicity of 𝐴 that 𝑤−𝐽_𝑟𝑛𝑧_𝑛,𝑤^∗−1𝑟_𝑛𝐽𝑧_𝑛−𝐽𝐽_𝑟𝑛𝑧_𝑛≥0 (4.28) for all 𝑤∈𝐷(𝐴) and 𝑤^∗∈𝐴𝑤. Letting 𝑛→∞, we have ⟨𝑤−𝑝,𝑤^∗⟩≥0 for all 𝑤∈𝐷(𝐴) and 𝑤^∗∈𝐴𝑤. Therefore from the maximality of 𝐴, we obtain 𝑝∈𝐴⁻¹0=𝐹(𝐽_𝑟𝑛) for all 𝑛≥1, that is, 𝑝∈⋂^∞_𝑛=1𝐹(𝐽_𝑟𝑛).
Next we show 𝐽_𝑟𝑛 is a relatively quasi nonexpansive mapping for all 𝑛≥1. For any 𝑤∈𝐸 and 𝑝∈𝐹(𝐽_𝑟𝑛)=𝐴⁻¹0, from the monotonicity of 𝐴, we have 𝜙𝑝,𝐽_𝑟𝑛𝑤=𝑝²−2𝑝,𝐽𝐽_𝑟𝑛𝑤+𝐽_𝑟𝑛𝑤²=𝑝²+2𝑝,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤−𝐽𝑤+𝐽_𝑟𝑛𝑤²=𝑝²+2𝑝,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤−2𝑝,𝐽𝑤+𝐽_𝑟𝑛𝑤²=𝑝²−2𝐽_𝑟𝑤−𝑝−𝐽_𝑟𝑛𝑤,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤−2𝑝,𝐽𝑤+𝐽_𝑟𝑛𝑤²=𝑝²−2𝐽_𝑟𝑛𝑤−𝑝,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤+2𝐽_𝑟𝑛𝑤,𝐽𝑤−𝐽𝐽_𝑟n𝑤−2𝑝,𝐽𝑤+𝐽_𝑟𝑛𝑤²≤𝑝²+2𝐽_𝑟𝑛𝑤,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤−2𝑝,𝐽𝑤+𝐽_𝑟𝑛𝑤²=𝑝²−2𝑝,𝐽𝑤+‖𝑤‖²−𝐽_𝑟𝑛𝑤²+2𝐽_𝑟𝑛𝑤,𝐽𝑤−‖𝑤‖²=𝜙𝑝,𝑤−𝜙𝐽_𝑟𝑛.𝑤,𝑤≤𝜙𝑝,𝑤 (4.29) This implies that 𝐽_𝑟𝑛 is a relatively quasi nonexpansive mapping for all 𝑛≥1. This completes the proof.