Iterative Schemes for Fixed Points of Relatively Nonexpansive Mappings and Their Applications

Abstract

We present two iterative schemes with errors which are proved to be strongly convergent to a common element of the set of fixed points of a countable family of relatively nonexpansive mappings and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly smooth and uniformly convex Banach space. Using the result we consider strong convergence theorems for variational inequalities and equilibrium problems in a real Hilbert space and strong convergence theorems for maximal monotone operators in a real uniformly smooth and uniformly convex Banach space.

1. Introduction

Let 𝐸 be a real Banach space, and 𝐸^∗ the dual space of 𝐸. The function 𝜙∶𝐸→𝐸^∗ is denoted by 𝜙𝑦,𝑥=𝑦²−2𝑦,𝐽𝑥+‖𝑥‖² (1.1) for all 𝑥,𝑦∈𝐸, where 𝐽 is the normalized duality mapping from 𝐸 to 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 [1] if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that the strong lim_𝑛→∞(𝑥_𝑛−𝑇𝑥_𝑛) equals 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 nonexpansive with respect to the Lyapunov functional [2] if 𝜙(𝑇𝑥,𝑇𝑦)≤𝜙(𝑥,𝑦) for all 𝑥,𝑦∈𝐶 and it is called relatively nonexpansive [3–6] 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 [3–6].

There are many methods for approximating fixed points of a nonexpansive mapping. In 1953, Mann [7] introduced the iteration as follows: a sequence {𝑥_𝑛} is defined by𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛, (1.2) where the initial guess element 𝑥₀∈𝐶 is arbitrary and {𝛼_𝑛} is a real sequence in [0,1]. Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results was proved by Reich [1]. In an infinite-dimensional Hilbert space, Mann iteration can yield only weak convergence (see [8, 9]). Attempts to modify the Mann iteration method (1.2) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [10] proposed the following modification of Mann iteration method (1.2) for nonexpansive mapping 𝑇 in a Hilbert space: in particular, they studied the strong convergence of the sequence {𝑥_𝑛} generated by𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑧−𝑦_𝑛≤𝑧−𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥,𝑛=0,1,2,…, (1.3) where {𝛼_𝑛}⊂[0,1] and 𝑃_{𝐶𝑛∩𝑄𝑛} is the metric projection from 𝐶 onto 𝐶_𝑛∩𝑄_𝑛.

Recently, Takahashi et al. [11] extended iteration (1.6) to obtain strong convergence to a common fixed point of a countable family of nonexpansive mappings; let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. Let {𝑇_𝑛} and 𝒯 be families of nonexpansive mappings of 𝐶 into itself such that ∅≠𝐹(𝒯)=⋂^∞_𝑛=1𝐹(𝑇_𝑛) and let 𝑥₀∈𝐻. Suppose that {𝑇_𝑛} satisfies the NST-condition (I) with 𝒯; that is, for each bounded sequence {𝑧_𝑛}⊂𝐶, lim_𝑛→∞‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖=0 implies that lim_𝑛→∞‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖=0 for all 𝑇∈𝒯. For 𝑥₁=𝑃_𝐶𝑥₀, define a sequence {𝑥_𝑛} of 𝐶 as follows:𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑧−𝑦_𝑛≤𝑧−𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑢≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛∈ℕ, (1.4) where 0≤𝛼_𝑛≤𝛼<1. Then, {𝑢_𝑛} converges strongly to 𝑧₀=𝑃_𝐹(𝑇)𝑥₀.

On the other hand, Halpern [12] introduced the following iterative scheme for approximating a fixed point of 𝑇:𝑥_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑇𝑥_𝑛 (1.5) for all 𝑛∈ℕ, where 𝑥₁=𝑥∈𝐶 and {𝛼_𝑛} is a sequence of [0,1]. Strong convergence of this type of iterative sequence has been widely studied: for instance, see [13, 14] and the references therein. In 2006, Martinez-Yanes and Xu [15] have adapted Nakajo and Takahashi's [10] idea to modify the process (1.5) for a nonexpansive mapping 𝑇 in a Hilbert space:𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛𝑥₀²+2𝑥_𝑛−𝑥₀,𝑄,𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛=0,1,2,…, (1.6) where 𝑃_{𝐶𝑛∩𝑄𝑛} is the metric projection from 𝐶 onto 𝐶_𝑛∩𝑄_𝑛. They proved that if {𝛼_𝑛}⊂(0,1) and lim_𝑛→∞𝛼_𝑛=0, then the sequence {𝑥_𝑛} generated by (1.6) converges strongly to 𝑃_𝐹(𝑇)𝑥.

The ideas to generalize the process (1.3) and (1.6) from Hilbert space to Banach space have been studied by many authors. Matsushita and Takahashi [6], Qin and Su [16], and Plubtieng and Ungchittrakool [17] generalized the process (1.3) and (1.6) and proved the strong convergence theorems for relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach spacey; see, for instance, [2, 6, 16, 18–22] and the references therein.

Recently, Nakajo et al. [18] introduced the following condition. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, let {𝑇_𝑛} be a family of mappings of 𝐶 into itself with 𝐹∶=⋂^∞_𝑛=1𝐹(𝑇_𝑛)≠∅, and let 𝜔_𝑤(𝑧_𝑛) denote the set of all weak subsequential limits of a bounded sequence {𝑧_𝑛} in 𝐶. {𝑇_𝑛} is said to satisfy the NST-condition (II) if, for every bounded sequence {𝑧_𝑛} in 𝐶, lim_𝑛→∞𝑧_𝑛−𝑇_𝑛𝑧_𝑛=0implies𝜔_𝑤𝑧_𝑛⊂𝐹. (1.7) Very recently, Nakajo et al. [19] introduced the more general condition so-called the NST∗-condition, {𝑇_𝑛} is said to satisfy the NST^∗-condition if, for every bounded sequence {𝑧_𝑛} in 𝐶, lim_𝑛→∞𝑧_𝑛−𝑇_𝑛𝑧_𝑛=lim_𝑛→∞𝑧_𝑛−𝑧_𝑛+1=0implies𝜔_𝑤𝑧_𝑛⊂𝐹. (1.8) It follows directly from the definitions above that if {𝑇_𝑛} satisfies the NST-condition (I), then {𝑇_𝑛} satisfies the NST^∗-condition.

Motivated and inspired by Wei and Cho [2], in this paper, we introduce two iterative schemes (3.1) and (3.14) and use the NST^∗-condition for a countable family of relatively nonexpansive mappings to obtain the strong convergence theorems for finding a common element of the set of fixed points of a countable family of relatively nonexpansive mappings and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly smooth and uniformly convex Banach space. Using this result, we also discuss the problem of strong convergence concerning variational inequality, equilibrium, and nonexpansive mappings in Hilbert spaces. Moreover, we also apply our convergence theorems to the maximal monotone operators in Banach spaces. The results obtained in this paper improve and extend the corresponding result of Matsushita and Takahashi [6], Qin and Su [16], Wei and Cho [2], Wei and Zhou [22], and many others.

2. Preliminaries

Let 𝐸 be a real Banach space with norm ‖⋅‖ and let 𝐸^∗ be the dual of 𝐸. For all 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗, we denote the value of 𝑥^∗ at 𝑥 by ⟨𝑥,𝑥^∗⟩. We denote the strong convergence and the weak convergence of a sequence {𝑥_𝑛} to 𝑥 in 𝐸 by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥, respectively. We also denote the weak^∗ convergence of a sequence {𝑥^∗_𝑛} to 𝑥^∗ in 𝐸^∗ by 𝑥^∗_𝑛⇀^∗𝑥^∗. An operator 𝑇⊂𝐸×𝐸^∗ is said to be monotone if ⟨𝑥−𝑦,𝑥^∗−𝑦^∗⟩≥0 whenever (𝑥,𝑥^∗),(𝑦,𝑦^∗)∈𝑇. We denote the set {𝑥∈𝐸∶0∈𝑇𝑥} by 𝑇⁻¹0. A monotone 𝑇 is said to be maximal if its graph 𝐺(𝑇)={(𝑥,𝑦)∶𝑦∈𝑇𝑥} is not properly contained in the graph of any other monotone operator. If 𝑇 is maximal monotone, then the solution set 𝑇⁻¹0 is closed and convex.

The normalized duality mapping 𝐽 from 𝐸 to 𝐸^∗ is defined by 𝐽𝑥=𝑥^∗∈𝐸^∗∶𝑥,𝑥^∗=‖𝑥‖²=𝑥^∗2 (2.1) for 𝑥∈𝐸. By Hahn-Banach theorem, 𝐽(𝑥) is nonempty; see [23] for more details. A Banach space 𝐸 is said to be strictly convex if ‖(𝑥+𝑦)/2‖<1 for all 𝑥,𝑦∈𝐸 with ‖𝑥‖=‖𝑦‖=1 and 𝑥≠𝑦. It is also said to be uniformly convex if, for each 𝜀∈(0,2], there exists 𝛿>0 such that ‖𝑥+𝑦‖/2≤𝛿 for 𝑥,𝑦∈𝐸 with ‖𝑥‖=‖𝑦‖=1 and ‖𝑥−𝑦‖≥𝜀. Let 𝑆(𝐸)={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸. Then the Banach space 𝐸 is said to be smooth provided thatlim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (2.2) exists for each 𝑥,𝑦∈𝑆(𝐸). It is also said to be uniformly smooth if the limit is attained uniformly for 𝑥,𝑦∈𝑆(𝐸). It is well known that if 𝐸 is smooth, strictly convex, and reflexive, then the duality mapping 𝐽 is single valued, one-to-one, and onto.

Let 𝐸 be a smooth, strictly convex, and reflexive Banach space and let 𝐶 be a nonempty closed convex subset of 𝐸. Throughout this paper, we denote by 𝜙 the function defined by 𝜙𝑦,𝑥=𝑦²−2𝑦,𝐽𝑥+‖𝑥‖²,∀𝑥,𝑦∈𝐶. (2.3) It is obvious from the definition of the function 𝜙 that (‖𝑥‖−‖𝑦‖)²⩽𝜙(𝑦,𝑥)⩽(‖𝑦‖²+‖𝑥‖²) for all 𝑥,𝑦∈𝐸.

Following Alber [24], the generalized projection 𝑃_𝐶 from 𝐸 onto 𝐶 is defined by 𝑃_𝐶(𝑥)=argmin_𝑦∈𝐶𝜙𝑦,𝑥,∀𝑥∈𝐸. (2.4) If 𝐸 is a Hilbert space, then 𝜙(𝑦,𝑥)=‖𝑦−𝑥‖² and 𝑃_𝐶 is the metric projection of 𝐻 onto 𝐶.

We need the following lemmas for the proof of our main results.

Lemma 2.1 (Kamimura and Takahashi [25]). Let 𝐸 be a uniformly convex and smooth Banach space and let {𝑥_𝑛} and {𝑦_𝑛} be sequences in 𝐸 such that either {𝑥_𝑛} or {𝑦_𝑛} is bounded. If lim_𝑛→∞𝜙(𝑥_𝑛,𝑦_𝑛)=0, then lim_𝑛→∞‖x_𝑛−𝑦_𝑛‖=0.

Lemma 2.2 (Alber [24], Kamimura and Takahashi [25]). Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Then 𝜙𝑥,𝑃_𝐶𝑦+𝜙𝑃_𝐶𝑦,𝑦≤𝜙𝑥,𝑦,∀𝑥∈𝐶,𝑦∈𝐸. (2.5)

Lemma 2.3 (Alber [24], Kamimura and Takahashi [25]). Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝑥∈𝐸, and let 𝑥₀∈𝐶. Then 𝑥₀=𝑃_𝐶𝑥⟺⟨𝑧−𝑥₀,𝐽𝑥₀−𝐽𝑥⟩≥0,∀𝑧∈𝐶. (2.6)

Lemma 2.4 (Matsushita and Takahashi [6]). Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, and let 𝑇 be a relatively nonexpansive mapping from 𝐶 into itself. Then 𝐹(𝑇) is closed and convex.

Lemma 2.5 (see [2]). Let 𝐸 be a real smooth and uniformly convex Banach space. If 𝑆∶𝐸→𝐸 is a mapping which is nonexpansive with respect to the Lyapunov functional, then 𝐹(𝑆) is convex and closed subset of 𝐸.

3. Main Results

In this section, by using the NST^∗-condition, we proved two strong convergence theorems for finding a common element of the set of fixed points of a countable family of relatively nonexpansive mappings and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly smooth and uniformly convex Banach space.

Theorem 3.1. Let 𝐸 be a real uniformly smooth and uniformly convex Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑆∶𝐶→𝐶 be nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous. Let {𝑇_𝑛} be a family of relatively nonexpansive mappings of 𝐶 into itself such that 𝐹∶=⋂^∞_𝑛=1𝐹(𝑇_𝑛)∩𝐹(𝑆)≠∅ and satisfy the NST^∗-condition. Then the sequence {𝑥_𝑛} generated by 𝑥₀𝑦∈𝐶,_𝑛=𝑇_𝑛𝑥_𝑛+𝑒_𝑛,𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑦_𝑛,𝑢_𝑛=𝑆𝑧_𝑛,𝐻_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑢_𝑛⩽𝜙𝑧,𝑧_𝑛⩽𝛼_𝑛𝜙𝑧,𝑥_𝑛+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝑒_𝑛,𝑊_𝑛=𝑧∈𝐶∶𝑧−𝑥_𝑛,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≤0_𝑛+1=𝑃_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛=0,1,2,… (3.1) converges strongly to 𝑃_𝐹𝑥₀ provided that(i){𝛼_𝑛}⊂[0,1] with 𝛼_𝑛≤1−𝛽 for some 𝛽∈(0,1);(ii)the error sequence is {𝑒_𝑛}⊂𝐶 such that ‖𝑒_𝑛‖→0 as 𝑛→∞.

Proof. We split the proof into five steps.Step 1 (Both 𝐻_𝑛 and 𝑊_𝑛 are closed and convex subset of 𝐶). Noting the facts that 𝜙𝑧,𝑧_𝑛⩽𝛼_𝑛𝜙𝑧,𝑥_𝑛+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝑒_𝑛⟺𝑧_𝑛²−𝛼_𝑛𝑥_𝑛²−1−𝛼_𝑛𝑥_𝑛+𝑒_𝑛²≤2𝑧,𝐽𝑧_𝑛−𝛼_𝑛𝐽𝑥_𝑛−1−𝛼_𝑛𝐽𝑥_𝑛+𝑒_𝑛,𝜙𝑧,𝑢_𝑛⩽𝜙𝑧,𝑧_𝑛⟺𝑧_𝑛²−𝑢_𝑛²≥2𝑧,𝐽𝑧_𝑛−𝐽𝑢_𝑛, (3.2) we can easily know that 𝐻_𝑛 is closed and convex subset of 𝐸. It is obvious that 𝑊_𝑛 is also a closed and convex subset of 𝐸.Step 2 (𝐹∶=(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩𝐹(𝑆)⊆𝐻_𝑛∩𝑊_𝑛 for all 𝑛∈ℕ∪{0}). To observe this, take 𝑝∈𝐹. Then it follows from the convexity of ‖⋅‖² that 𝜙𝑝,𝑢_𝑛=𝜙𝑆𝑝,𝑆𝑧_𝑛≤𝜙𝑝,𝑧_𝑛=𝑝²−2𝑝,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑦_𝑛+𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑦_𝑛²≤𝑝²−2𝛼_𝑛𝑝,𝐽𝑥_𝑛−21−𝛼_𝑛𝑝,𝐽𝑦_𝑛+𝛼_𝑛𝑥_𝑛²+1−𝛼_𝑛𝑦_𝑛²=𝛼_𝑛𝜙𝑝,𝑥_𝑛+1−𝛼_𝑛𝜙𝑝,𝑦_𝑛=𝛼_𝑛𝜙𝑝,𝑥_𝑛+1−𝛼_𝑛𝜙𝑝,𝑇_𝑛𝑥_𝑛+𝑒_𝑛≤𝛼_𝑛𝜙𝑝,𝑥_𝑛+1−𝛼_𝑛𝜙𝑝,𝑥_𝑛+𝑒_𝑛 (3.3) for all 𝑛∈ℕ∪{0}. Hence 𝐹⊂𝐻_𝑛 for all 𝑛∈ℕ∪{0}. On the other hand, it is clear that 𝑝∈𝑊₀=𝐶. Then 𝑝∈𝐻₀∩𝑊₀ and 𝑥₁=𝑃_{𝐻0∩𝑊0}𝑥₀ is well defined. Suppose that 𝑝∈𝑊_𝑛−1 for some 𝑛≥1. Then 𝑝∈𝐻_𝑛−1∩𝑊_𝑛−1 and 𝑥_𝑛=𝑃_{𝐻𝑛−1∩𝑊𝑛−1}𝑥₀ is well defined. It follows from Lemma 2.3 that 𝑝−𝑥_𝑛,𝐽𝑥₀−𝐽𝑥_𝑛=𝑝−𝑃_{𝐻𝑛−1∩𝑊𝑛−1}𝑥₀,𝐽𝑥₀−𝐽𝑃_{𝐻𝑛−1∩𝑊𝑛−1}𝑥₀≤0, (3.4) which implies that 𝑝∈𝑊_𝑛. Therefore 𝑝∈𝐻_𝑛∩𝑊_𝑛, and hence 𝑥_𝑛+1=𝑃_{𝐻𝑛∩𝑊𝑛}𝑥₀ is well defined. Then by induction, the sequence {𝑥_𝑛} generated by (3.1) is well defined, for each 𝑛≥0. Moreover 𝐹⊂𝐻_𝑛∩𝑊_𝑛 for each nonnegative integer 𝑛.Step 3 ({𝑥_𝑛} is bounded sequence of 𝐶). In fact, for all 𝑝∈𝐹⊂𝐻_𝑛∩𝑊_𝑛⊂𝑊_𝑛, it follows from Lemma 2.2 that 𝜙𝑝,𝑃_𝑊𝑛𝑥₀+𝜙𝑃_𝑊𝑛𝑥₀,𝑥₀≤𝜙𝑝,𝑥₀. (3.5) By the definition of 𝑊_𝑛 and Lemma 2.2, we note that 𝑥_𝑛=𝑃_𝑊𝑛𝑥₀ and hence 𝜙𝑝,𝑥_𝑛+𝜙𝑥_𝑛,𝑥₀≤𝜙𝑝,𝑥₀. (3.6) Therefore {𝑥_𝑛} is bounded.Step 4 (𝜔_𝑤(𝑥_𝑛)⊂𝐹∶=(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩𝐹(𝑆)). From the facts 𝑥_𝑛=𝑃_𝑊𝑛𝑥₀, 𝑥_𝑛+1∈𝑊_𝑛, and Lemma 2.2, we have 𝜙𝑥_𝑛+1,𝑥_𝑛+𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀. (3.7) Therefore, lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists. Then 𝜙(𝑥_𝑛+1,𝑥_𝑛)→0, which implies from Lemma 2.1 that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞. Since 𝑥_𝑛+1∈𝐻_𝑛, we have 𝜙𝑥_𝑛+1,𝑢_𝑛⩽𝜙𝑥_𝑛+1,𝑧_𝑛⩽𝛼_𝑛𝜙𝑥_𝑛+1,𝑥_𝑛+1−𝛼_𝑛𝜙𝑥_𝑛+1,𝑥_𝑛+𝑒_𝑛. (3.8) Note that 𝜙𝑥_𝑛+1,𝑥_𝑛+𝑒_𝑛−𝜙𝑥_𝑛+1,𝑥_𝑛=𝑥_𝑛+𝑒_𝑛²−𝑥_𝑛²+2𝑥_𝑛+1,𝐽𝑥_𝑛−𝐽𝑥_𝑛+𝑒_𝑛. (3.9) Since 𝐽 is uniform continuous on each bounded subset of 𝐸 and ‖𝑒_𝑛‖→0, we have 𝜙(𝑥_𝑛+1,𝑥_𝑛+𝑒_𝑛)→0 as 𝑛→∞. This implies that 𝜙(𝑥_𝑛+1,𝑢_𝑛)→0 and 𝜙(𝑥_𝑛+1,𝑧_𝑛)→0 as 𝑛→∞. Using Lemma 2.1, we obtain lim_𝑛→∞‖𝑥_𝑛+1−(𝑥_𝑛+𝑒_𝑛)‖=0, lim_𝑛→∞‖𝑥_𝑛+1−𝑧_𝑛‖=0, and lim_𝑛→∞‖𝑥_𝑛+1−𝑢_𝑛‖=0. Since both 𝐽∶𝐸→𝐸^∗ and 𝐽⁻¹∶𝐸^∗→𝐸 are uniformly continuous on bounded subsets, it follows that lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.
Put 𝑣_𝑛∶=𝑥_𝑛+𝑒_𝑛 for all 𝑛∈ℕ∪{0}. Since {𝑥_𝑛} is bounded and ‖𝑒_𝑛‖→0, we have {𝑣_𝑛} is bounded. Note that 𝑣_𝑛+1−𝑣_𝑛=𝑥_𝑛+1+𝑒_𝑛+1−𝑥_𝑛+𝑒_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑒_𝑛+𝑒_𝑛+1,𝑣_𝑛−𝑇_𝑛𝑣_𝑛=𝑥_𝑛+𝑒_𝑛−𝑇_𝑛𝑥_𝑛+𝑒_𝑛=𝑥_𝑛+𝑒_𝑛−𝑦_𝑛≤𝑥_𝑛+𝑒_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛 (3.10) for all 𝑛∈ℕ∪{0}. It implies that lim_𝑛→∞‖𝑣_𝑛+1−𝑣_𝑛‖=lim_𝑛→∞‖𝑣_𝑛−𝑇_𝑛𝑣_𝑛‖=0. Therefore, we have 𝜔_𝑤(𝑣_𝑛)⊂⋂^∞_𝑛=1𝐹(𝑇_𝑛) since {𝑇_𝑛} satisfies NST^∗-condition. Note that ‖𝑣_𝑛−𝑥_𝑛‖=‖(𝑥_𝑛+𝑒_𝑛)−𝑥_𝑛‖=‖𝑒_𝑛‖→0 as 𝑛→∞. Hence, we also have 𝜔_𝑤(𝑥_𝑛)⊂⋂^∞_𝑛=1𝐹(𝑇_𝑛).
On the other hand, from Step 3, we know that 𝜔_𝑤(𝑥_𝑛)≠∅. Then, for all 𝑞∈𝜔_𝑤(𝑥_𝑛), there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑞 as 𝑖→∞. Therefore, 𝑢_𝑛𝑖⇀𝑞 and 𝑧_𝑛𝑖⇀𝑞 as 𝑖→∞. Since 𝑆∶𝐶→𝐶 is weakly continuous and 𝑢_𝑛𝑖=𝑆𝑧_𝑛𝑖, we have 𝑞∈𝐹(𝑆). Hence 𝜔_𝑤(𝑥_𝑛)⊂𝐹∶=(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩𝐹(𝑆).Step 5 (𝑥_𝑛→𝑞^∗=𝑃_𝐹𝑥₀, as 𝑛→∞). Let {𝑥_𝑛𝑗} be any subsequence of {𝑥_𝑛} which weakly converges to 𝑞∈𝐹. Since 𝑥_𝑛+1=𝑃_{𝐻𝑛∩𝑊𝑛}𝑥₀ and 𝑞^∗∈𝐹⊂𝐻_𝑛∩𝑊_𝑛, we have 𝜙(𝑥_𝑛+1,𝑥₀)≤𝜙(𝑞^∗,𝑥₀). Then, 𝜙𝑥_𝑛,𝑞^∗=𝜙𝑥_𝑛,𝑥₀+𝜙𝑥₀,𝑞^∗−2𝑥_𝑛−𝑥₀,𝐽𝑞^∗−𝐽𝑥₀≤𝜙𝑞^∗,𝑥₀+𝜙𝑥₀,𝑞^∗−2𝑥_𝑛−𝑥₀,𝐽𝑞^∗−𝐽𝑥₀, (3.11) which yields limsup_𝑗→∞𝜙𝑥_𝑛𝑗,𝑞^∗≤𝜙𝑞^∗,𝑥₀+𝜙𝑥₀,𝑞^∗−2𝑞−𝑥₀,𝐽𝑞^∗−𝐽𝑥₀=−2𝑞^∗−𝑞,𝐽𝑞^∗−𝐽𝑥₀≤0. (3.12) Hence 𝜙(𝑥_𝑛𝑗,𝑞^∗)→0 as 𝑗→∞. It follows from Lemma 2.1 that 𝑥_𝑛𝑗→𝑞^∗ as 𝑗→∞. Therefore, 𝑥_𝑛→𝑞^∗=𝑃_𝐹𝑥₀ as 𝑛→∞. This completes the proof.

Corollary 3.2 (see Matsushita and Takahashi [6]). Let 𝐸 be a real uniformly smooth and uniformly convex Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, let 𝑇 be a relatively nonexpansive mappings of 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 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_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,𝑛=0,1,2,…, (3.13) where 𝐽 is the duality mapping on 𝐸. If 𝐹(𝑇) is nonempty, then {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)𝑥, where 𝑃_𝐹(𝑇) is the generalized projection from 𝐶 onto 𝐹(𝑇).

Proof. Suppose that 𝐹(𝑇)≠∅ and put 𝑇_𝑛≡𝑇, 𝑆≡𝐼, and 𝑒_𝑛=0 for all 𝑛∈ℕ∪{0}. Let {𝑧_𝑛} be a bounded sequence in 𝐶 with lim_𝑛→∞‖𝑧_𝑛−𝑇𝑧_𝑛‖=0 and let 𝑧∈𝜔_𝑤(𝑧_𝑛). Then there exists subsequence {𝑧_𝑛𝑘} of {𝑧_𝑛} such that 𝑧_𝑛𝑘⇀𝑧. It follows directly from the definition of 𝑇 that 𝑧∈𝐹(𝑇)=𝐹(𝑇). Hence 𝑇 satisfies NST-condition; by Theorem 3.1, {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)𝑥.

Theorem 3.3. Let 𝐸 be a real uniformly smooth and uniformly convex Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑆∶𝐶→𝐶 be nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous. Let {𝑇_𝑛} be a family of relatively nonexpansive mappings of 𝐶 into itself such that 𝐹∶=⋂^∞_𝑛=1𝐹(𝑇_𝑛)∩𝐹(𝑆)≠∅ and satisfy the NST^∗-condition. Then the sequence {𝑥_𝑛} generated by 𝑥₀𝑦∈𝐶,_𝑛=𝑇_𝑛𝑥_𝑛+𝑒_𝑛,𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝐽𝑦_𝑛,𝑢_𝑛=𝑆𝑧_𝑛,𝐻_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑢_𝑛⩽𝜙𝑧,𝑧_𝑛⩽𝛼_𝑛𝜙𝑧,𝑥₀+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝑒_𝑛,𝑊_𝑛=𝑧∈𝐶∶𝑧−𝑥_𝑛,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≤0_𝑛+1=𝑃_{𝐻𝑛∩𝑊𝑛}𝑥₀ (3.14) converges strongly to 𝑃_𝐹𝑥₀ provided that(i){𝛼_𝑛}⊂[0,1] is a sequence such that 𝛼_𝑛→0 as 𝑛→∞;(ii)the error sequence is {𝑒_𝑛}⊂𝐸 such that ‖𝑒_𝑛‖→0 as 𝑛→∞.

Proof. By slightly modifying the corresponding proof in Theorem 3.1, we can easily obtain the result.

Setting 𝑇_𝑛≡𝑇, 𝑆≡𝐼, and 𝑒_𝑛=0 for all 𝑛∈ℕ∪{0} in Theorem 3.3, we obtain the following corollary.

Corollary 3.4 (see Qin and Su [16]). Let 𝐸 be a real uniformly smooth and uniformly convex Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑇 be a relatively nonexpansive mappings of 𝐶 into itself, and let {𝛼_𝑛} be a sequence in (0,1) and lim_𝑛→∞𝛼_𝑛=0. Define a sequence {𝑥_𝑛} in 𝐶 by 𝑥₀𝑦∈𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛⩽𝛼_𝑛𝜙𝑧,𝑥₀+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛=0,1,2,…, (3.15) where 𝐽 is the duality mapping on 𝐸. If 𝐹(𝑇) is nonempty, then {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)𝑥, where 𝑃_𝐹(𝑇) is the generalized projection from 𝐶 onto 𝐹(𝑇).

4. Applications to the Variational Inequality Problem, Equilibrium Problem, and Fixed Points Problem of Nonexpansive Mappings in a Real Hilbert Space

In this section, using Theorems 3.1 and 3.3, we prove the strong convergence theorems for finding a common element of the set of fixed points of nonexpansive mapping, the solution set of the variational inequality problems, and the solution set of an equilibrium problems in a Hilbert space.

4.1. Common Solutions of a Fixed Point Problem and a Variational Inequality Problem

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝐴 be a mapping of 𝐶 into 𝐻. A classical variational inequality problem, denoted by VI(𝐶,𝐴), is to find an element 𝑢^∗∈𝐶 such that ⟨𝑢−𝑢^∗,𝐴𝑢^∗⟩≥0 for all 𝑢∈𝐶. It is known that for 𝜆>0, 𝑥∈𝐶 is a solution of the variational inequality of 𝐴 if and only if 𝑥=𝑃_𝐶(𝐼−𝜆𝐴)𝑥, where 𝑃_𝐶 is the metric projection from 𝐻 onto 𝐶. We denote by 𝑁_𝐶(𝑣) the normal cone for 𝐶 at a point 𝑣∈𝐶, that is, 𝑁_𝐶(𝑣)={𝑤∈𝐻∶⟨𝑤,𝑣−𝑢⟩≥0 for all 𝑢∈𝐶}. Define 𝑓∶𝐻→𝐻 by𝑓(𝑣)=𝐴𝑣+𝑁_𝐶(𝑣),∀𝑣∈𝐶,∅,otherwise. (4.1) Then 𝑓 is maximal monotone and 𝑓⁻¹0=VI(𝐶,𝐴) (see [26]). A mapping 𝐴∶𝐶→𝐻 is called α-inverse strongly monotone if there exists an 𝛼>0 such that ⟨𝐴𝑢−𝐴𝑣,𝑢−𝑣⟩≥𝛼‖𝐴𝑢−𝐴𝑣‖²,∀𝑢,𝑣∈𝐶. (4.2) It is well known that if 0<𝜆<2𝛼, then 𝑃_𝐶(𝐼−𝜆𝐴) is nonexpansive of 𝐶 into itself.

Lemma 4.1 (see [27]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝐴 be α-inverse strongly monotone mapping of 𝐶 into 𝐻. Let {𝜆_𝑛}⊂(0,2𝛼) and let 𝑃_𝐶 be a metric projection from 𝐻 onto 𝐶. Let 𝑇_𝑛=𝑃_𝐶(𝐼−𝜆_𝑛𝐴), for all 𝑛∈ℕ. Then {𝑇_𝑛} is a sequence of nonexpansive mappings and satisfies NST-condition.

By using Lemma 4.1 and Theorem 3.1 with 𝑒_𝑛=0, for all 𝑛∈ℕ∪{0}, we obtain the following theorem.

Theorem 4.2. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝐴 be α-inverse strongly monotone mapping of 𝐶 into 𝐻. Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that 𝐹∶=𝐹(𝑆)∩𝑉𝐼(𝐶,𝐴)≠∅. Let 𝑃_𝐶 be a metric projection from 𝐻 onto 𝐶. Then the sequence {𝑥_𝑛} generated by 𝑥₀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑧_𝑛=𝑆𝑦_𝑛,𝐻_𝑛=𝑧∈𝐶∶𝑧−𝑧_𝑛≤𝑧−𝑥_𝑛,𝑊_𝑛=𝑧∈𝐶∶𝑧−𝑥_𝑛,𝑥₀−𝑥_𝑛,𝑥≤0_𝑛+1=𝑃_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛=0,1,2,… (4.3) converges strongly to 𝑃_𝐹𝑥₀ provided that(i){𝛼_𝑛}⊂[0,1] with 𝛼_𝑛≤1−𝛽 for some 𝛽∈(0,1);(ii){𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,2𝛼) with 𝑎≤𝑏.

Next, by using Lemma 4.1 and Theorem 3.3 with 𝑒_𝑛=0, for all 𝑛∈ℕ∪{0}, we obtain the following theorem.

Theorem 4.3. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝐴 be α-inverse strongly monotone mapping of 𝐶 into 𝐻. Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that 𝐹∶=𝐹(𝑆)∩𝑉𝐼(𝐶,𝐴)≠∅. Let 𝑃_𝐶 be a metric projection from 𝐻 onto 𝐶. Then the sequence {𝑥_𝑛} generated by 𝑥₀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_n,𝑧_𝑛=𝑆𝑦_𝑛,𝐻_𝑛=𝑧∈𝐶∶𝑧−𝑧_𝑛²≤𝑧−𝑥_𝑛²+𝛼_𝑛𝑥₀²+2𝑥_𝑛−𝑥₀,𝑊,𝑣_𝑛=𝑧∈𝐶∶𝑧−𝑥_𝑛,𝑥₀−𝑥_𝑛,𝑥≤0_𝑛+1=𝑃_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛=0,1,2,… (4.4) converges strongly to 𝑃_𝐹𝑥₀ provided that(i){𝛼_𝑛}⊂[0,1] is a sequence such that 𝛼_𝑛→0 as 𝑛→∞;(ii){𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,2𝛼) with 𝑎≤𝑏.

4.2. Common Solutions of a Fixed Point Problem and an Equilibrium Problem

Let 𝐶 be a nonempty closed convex subset of 𝐻 and let 𝐹 be a bifunction of 𝐶×𝐶 into ℝ, where ℝ is the set of real numbers. The equilibrium problem for 𝐹∶𝐶×𝐶→ℝ is to find 𝑥∈𝐶 such that𝐹𝑥,𝑦≥0,∀𝑦∈𝐶. (4.5) The set of solutions of (4.5) is denoted by EP(𝐹). Numerous problems in physics, optimization, and economics reduce to find a solution of (4.5). In 1997, Combettes and Hirstoaga [15] introduced an iterative scheme of finding the best approximation to the initial data when EP(𝐹) is nonempty and proved a strong convergence theorem.

For solving the equilibrium problem for a bifunction 𝐹∶𝐶×𝐶→ℝ, let us assume that 𝐹 satisfies the following conditions:(A1)𝐹(𝑥,𝑥)=0 for all 𝑥∈𝐶;(A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 forall 𝑥,𝑦∈𝐶;(A3) for each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡→0𝐹(𝑡𝑧+(1−𝑡)𝑥),𝑦≤𝐹𝑥,𝑦; (4.6)(A4) for each 𝑥∈𝐶, 𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous.

The following lemma appears implicitly in [15, 28].

Lemma 4.4 (see [15, 28]). Let 𝐶 be a nonempty closed convex subset of 𝐻 and let 𝐹 be a bifunction of 𝐶×𝐶 in to ℝ satisfying (A1)–(A4). Let 𝑟>0 and 𝑥∈𝐻. Then, there exists 𝑧∈𝐶 such that 𝐹+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (4.7) Define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶𝐹𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶 (4.8) for all 𝑧∈𝐻. Then, the following holds:(1)𝑇_𝑟 is single-valued; (2)𝑇_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻, 𝑇_𝑟𝑥−𝑇_𝑟𝑦²≤𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦; (4.9)(3)𝐹(𝑇_𝑟)=𝐸𝑃(𝐹); (4)𝐸𝑃(𝐹) is closed and convex.

Lemma 4.5 (see [27]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑓 be a bifunction from 𝐶×𝐶 into ℝ satisfying (A1)–(A4) and 𝐸𝑃(𝑓)≠∅. If {𝑟_𝑛} is a sequence in (0,∞) satisfying liminf_𝑛→∞𝑟_𝑛>0, then {𝑇_𝑟𝑛} is a family of firmly nonexpansive mappings of 𝐻 into 𝐶 and satisfies NST-condition.

The following theorem follows directly from Lemma 4.5 and Theorem 3.1.

Theorem 4.6. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝑓 be a bifunction from 𝐶×𝐶 into ℝ satisfying (A1)–(A4). Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that 𝐹∶=𝐹(𝑆)∩𝐸𝑃(𝑓)≠∅. Then the sequence {𝑥_𝑛} generated by 𝑥₀𝑢∈𝐶,_𝑛∈𝐶suchthat𝑓𝑢_𝑛+1,𝑦𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦⟩≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑢_𝑛,𝑧_𝑛=𝑆𝑦_𝑛,𝐻_𝑛=𝑧∈𝐶∶𝑧−𝑧_𝑛≤𝑧−𝑥_𝑛,𝑊_𝑛=𝑧∈𝐶∶𝑧−𝑥_𝑛,𝑥₀−𝑥_𝑛,𝑥≤0_𝑛+1=𝑃_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛=0,1,2,… (4.10) converges strongly to 𝑃_𝐹𝑥₀ provided that(i){𝛼_𝑛}⊂[0,1] with 𝛼_𝑛≤1−𝛽 for some 𝛽∈(0,1);(ii){𝑟_𝑛}⊂(0,∞) such that liminf_𝑛→∞𝑟_𝑛>0,

Proof. Put 𝑇_𝑛≡𝑇_𝑟𝑛 and 𝑒_𝑛=0 for all 𝑛∈ℕ∪{0}. Then, it follows from Lemma 4.5 and Theorem 3.1 that {𝑥_𝑛} converges strongly to 𝑃_𝐹𝑥₀.

Next, by using Lemma 4.5 and Theorem 3.3 with 𝑒_𝑛=0, for all 𝑛∈ℕ∪{0}, we obtain the following result.

Theorem 4.7. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝑓 be a bifunction from 𝐶×𝐶 into ℝ satisfying (A1)–(A4). Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that 𝐹∶=𝐹(𝑆)∩𝐸𝑃(𝑓)≠∅. Then the sequence {𝑥_𝑛} generated by 𝑥₀𝑢∈𝐶,_𝑛∈𝐶suchthat𝑓𝑢_𝑛+1,𝑦𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦⟩≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑢_𝑛,𝑧_𝑛=𝑆𝑦_𝑛,𝐻_𝑛=𝑧∈𝐶∶𝑧−𝑧_𝑛²≤𝑧−𝑥_𝑛²+𝛼_𝑛𝑥₀²+2𝑥_𝑛−𝑥₀,𝑊,𝑣_𝑛=𝑧∈𝐶∶𝑧−𝑥_𝑛,𝑥₀−𝑥_𝑛,𝑥≤0_𝑛+1=𝑃_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛=0,1,2,… (4.11)(1) converges strongly to 𝑃_𝐹𝑥₀ provided that(i){𝛼_𝑛}⊂[0,1] is a sequence such that 𝛼_𝑛→0 as 𝑛→∞;(ii){𝑟_𝑛}⊂(0,∞) such that liminf_𝑛→∞𝑟_𝑛>0,

5. Applications to Maximal Monotone Operators in Banach Space

In this section, we discuss the problem of strong convergence concerning maximal monotone operators in a real uniformly smooth and uniformly convex Banach space.

Let 𝐸 be a smooth, strictly convex, and reflexive Banach space and let 𝑇⊂𝐸×𝐸^∗ be a maximal monotone operator. Then for each 𝑟>0 and 𝑥∈𝐸, there corresponds a unique element 𝑥_𝑟∈𝐷(𝑇) satisfying 𝐽(𝑥)∈𝐽𝑥_𝑟+𝑟𝑇𝑥_𝑟; (5.1) see [23]. We define the resolvent of 𝑇 by 𝐽_𝑟𝑥=𝑥_𝑟. In other words, 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽 for all 𝑟>0. We know that 𝐽_𝑟 is relatively nonexpansive and 𝑇⁻¹0=𝐹(𝐽_𝑟) for all 𝑟>0 (see [6, 23]), where 𝐹(𝐽_𝑟) denotes the set of all fixed points of 𝐽_𝑟. We can also define, for each 𝑟>0, the Yosida approximation of 𝑇 by 𝐴_𝑟=𝜆⁻¹(𝐽−𝐽𝐽_𝑟). We know that (𝐽_𝑟𝑥,𝐴_𝑟𝑥)∈𝑇 for all 𝑟>0.

Lemma 5.1. Let 𝐸 be a real uniformly smooth and uniformly convex Banach space and let 𝑇∶𝐸→𝐸^∗ be a maximal monotone operator with 𝐹∶=𝑇⁻¹0≠∅ and 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽 for all 𝑟>0. Let {𝑇_𝑛} be a sequence of relatively nonexpansive mappings of 𝐸 into itself defined by 𝑇_𝑛≡𝐽_𝑟𝑛 for all 𝑛∈ℕ∪{0}, where {𝑟_𝑛} is a sequence in (0,∞) such that liminf_𝑛≥0𝑟_𝑛>0. Then, {𝑇_𝑛} satisfies the NST-condition.

Proof. It easy to see that ⋂^∞_𝑛=0𝐹(𝑇_𝑛)=⋂^∞_𝑛=0𝐹(𝐽_𝑟𝑛)=𝑇⁻¹0. Let {𝑣_𝑛} be a bounded sequence in 𝐸 such that lim_𝑛→∞‖𝑣_𝑛−𝐽_𝑟𝑛𝑣_𝑛‖=0 and let 𝑣₀∈𝜔_𝑤(𝑣_𝑛). Then, there exists a subsequence {𝑣_𝑛𝑘} of {𝑣_𝑛} such that 𝑣_𝑛𝑘⇀𝑣₀. By the uniform smoothness of 𝐸, we have lim_𝑛→∞‖𝐽𝑣_𝑛−𝐽𝐽_𝑟𝑛𝑣_𝑛‖=0. Since liminf_𝑛→∞𝑟_𝑛>0, we have lim_𝑛→∞𝐴_𝑟𝑛𝑣_𝑛=lim_𝑛→∞1𝑟_𝑛𝐽𝑣_𝑛−𝐽𝐽_𝑟𝑛𝑣_𝑛=0. (5.2) Let (𝑢,𝑢^∗)∈𝑇. Then it holds from the monotonicity of 𝑇 that ⟨𝑢−𝐽_𝑟𝑛𝑘𝑣_𝑛𝑘,𝑢^∗−𝐴_𝑟𝑛𝑘𝑣_𝑛𝑘⟩≥0 (5.3) for all 𝑘∈ℕ. Letting 𝑘→∞, we get ⟨𝑢−𝑣₀,𝑢^∗⟩≥0. Then, the maximality of 𝑇 implies 𝑣₀∈𝑇⁻¹0∶=⋂^∞_𝑛=1𝐹(𝐽_𝑟𝑛)=⋂^∞_𝑛=0𝐹(𝑇_𝑛). Hence 𝜔_𝑤(𝑣_𝑛)⊂⋂^∞_𝑛=0𝐹(𝑇_𝑛). Therefore, {𝑇_𝑛} satisfies the NST-condition.

Using Theorem 3.1 and Lemma 5.1, we first obtain the result of [2].

Theorem 5.2 (see [2]). Let 𝐸 be a real uniformly smooth and uniformly convex Banach space and let 𝑆∶𝐸→𝐸 be nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous. Let 𝑇∶𝐸→𝐸^∗ be a maximal monotone operator with 𝐹∶=𝑇⁻¹0∩𝐹(𝑆)≠∅ and 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽 for all 𝑟>0. Then, sequence {𝑥_𝑛} generated by the following scheme 𝑥₀∈𝐸,𝑟₀𝑦>0,_𝑛=𝐽_𝑟𝑛𝑥_𝑛+𝑒_𝑛,𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑦_𝑛,𝑢_𝑛=𝑆𝑧_𝑛,𝐻_𝑛=𝑧∈𝐸∶𝜙𝑧,𝑢_𝑛⩽𝜙𝑧,𝑧_𝑛⩽𝛼_𝑛𝜙𝑧,𝑥_𝑛+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝑒_𝑛,𝑊_𝑛=𝑧∈𝐸∶𝑧−𝑥_𝑛,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≤0_𝑛+1=𝑃_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛=0,1,2,… (5.4) converges strongly to 𝑃_𝐹𝑥₀ provided that(i){𝛼_𝑛}⊂[0,1] with 𝛼_𝑛≤1−𝛽 for some 𝛽∈(0,1);(ii){𝑟_𝑛}⊂(0,+∞) is a sequence such that liminf_𝑛≥0𝑟_𝑛>0;(iii)the error sequence is {𝑒_𝑛}⊂𝐸 such that ‖𝑒_𝑛‖→0 as 𝑛→∞.

Proof. Put 𝑇_𝑛≡𝐽_𝑟𝑛 for all 𝑛∈ℕ∪{0}. Hence by using Lemma 5.1 and Theorem 3.1, we obtain the result.

Putting 𝑆≡𝐼 in Theorem 5.2, we obtain the following corollary.

Corollary 5.3 (see [22]). Let 𝐸 be a real uniformly smooth and uniformly convex Banach space and let 𝑇∶𝐸→𝐸^∗ be a maximal monotone operator with 𝐹∶=𝑇⁻¹0∩𝐹(𝑆)≠∅. and 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽 for all 𝑟>0. Then, sequence {𝑥_𝑛} generated by the following scheme 𝑥₀∈𝐸,𝑟₀𝑦>0,_𝑛=𝐽_𝑟𝑛𝑥_𝑛+𝑒_𝑛,𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑦_𝑛,𝐻_𝑛=𝑧∈𝐸∶𝜙𝑧,𝑧_𝑛⩽𝛼_𝑛𝜙𝑧,𝑥_𝑛+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝑒_𝑛,𝑊_𝑛=𝑧∈𝐸∶𝑧−𝑥_𝑛,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≤0_𝑛+1=𝑃_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛=0,1,2,… (5.5) converges strongly to 𝑃_𝐹𝑥₀ provided that(i){𝛼_𝑛}⊂[0,1] with 𝛼_𝑛≤1−𝛽 for some 𝛽∈(0,1);(ii){𝑟_𝑛}⊂(0,+∞) is a sequence such that liminf_𝑛≥0𝑟_𝑛>0;(iii)the error sequence is {𝑒_𝑛}⊂𝐸 such that ‖𝑒_𝑛‖→0 as 𝑛→∞.

The following theorem follows directly from Lemma 5.1 and Theorem 3.3

Theorem 5.4 (see [2]). Let 𝐸 be a real uniformly smooth and uniformly convex Banach space and let 𝑆∶𝐸→𝐸 be nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous. Let 𝑇∶𝐸→𝐸^∗ be a maximal monotone operator with 𝐹∶=𝑇⁻¹0∩𝐹(𝑆)≠∅. and 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽 for all 𝑟>0. Then, sequence {𝑥_𝑛} generated by the following scheme 𝑥₀∈𝐸,𝑟₀𝑦>0,_𝑛=𝐽_𝑟𝑛𝑥_𝑛+𝑒_𝑛,𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝐽𝑦_𝑛,𝑢_𝑛=𝑆𝑧_𝑛,𝐻_𝑛=𝑧∈𝐸∶𝜙𝑧,𝑢_𝑛⩽𝜙𝑧,𝑧_𝑛⩽𝛼_𝑛𝜙𝑧,𝑥₀+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝑒_𝑛,𝑊_𝑛=𝑧∈𝐸∶𝑧−𝑥_𝑛,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≤0_𝑛+1=𝑃_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛=0,1,2,… (5.6) converges strongly to 𝑃_𝐹𝑥₀ provided that(i){𝛼_𝑛}⊂[0,1] is a sequence such that 𝛼_𝑛→0 as 𝑛→∞;(ii){𝑟_𝑛}⊂(0,+∞) is a sequence such that liminf_𝑛≥0𝑟_𝑛>0;(iii)the error sequence is {𝑒_𝑛}⊂𝐸 such that ‖𝑒_𝑛‖→0 as 𝑛→∞.

Putting 𝑆≡𝐼 in Theorem 5.4, we obtain the following corollary.

Corollary 5.5 (see [22]). Let 𝐸 be a real uniformly smooth and uniformly convex Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑇∶𝐸→𝐸^∗ be a maximal monotone operator with 𝐹∶=𝑇⁻¹0∩𝐹(𝑆)≠∅ and 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽 for all 𝑟>0. Then, sequence {𝑥_𝑛} generated by the following scheme 𝑥₀∈𝐸,𝑟₀𝑦>0,_𝑛=𝐽_𝑟𝑛𝑥_𝑛+𝑒_𝑛,𝑧_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝐽𝑦_𝑛,𝐻_𝑛=𝑧∈𝐸∶𝜙𝑧,𝑧_𝑛⩽𝛼_𝑛𝜙𝑧,𝑥₀+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝑒_𝑛,𝑊_𝑛=𝑧∈𝐸∶𝑧−𝑥_𝑛,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≤0_𝑛+1=𝑃_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛=0,1,2,… (5.7) converges strongly to 𝑃_𝐹𝑥₀ provided that(i){𝛼_𝑛}⊂[0,1] is a sequence such that 𝛼_𝑛→0 as 𝑛→∞;(ii){𝑟_𝑛}⊂(0,+∞) is a sequence such that liminf_𝑛≥0𝑟_𝑛>0;(iii)the error sequence is {𝑒_𝑛}⊂𝐸 such that ‖𝑒_𝑛‖→0 as 𝑛→∞.

Acknowledgment

The authors thank the National Centre of Excellence in Mathematics, PERDO, for the Financial support. Moreover, the second author would like to thank the National Centre of Excellence in Mathematics, PERDO, under the Commission on Higher Education, Ministry of Education, Thailand