Hybrid Algorithm for Common Fixed Points of Uniformly Closed Countable Families of Hemirelatively Nonexpansive Mappings and Applications

Abstract

The authors have obtained the following results: (1) the definition of uniformly closed countable family of nonlinear mappings, (2) strong convergence theorem by the monotone hybrid algorithm for two countable families of hemirelatively nonexpansive mappings in a Banach space with new method of proof, (3) two examples of uniformly closed countable families of nonlinear mappings and applications, (4) an example which is hemirelatively nonexpansive mapping but not weak relatively nonexpansive mapping, and (5) an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping. Therefore, the results of this paper improve and extend the results of Plubtieng and Ungchittrakool (2010) and many others.

1. Introduction and Preliminaries

Let 𝐸 be a 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=‖𝑓‖²,(1.1)where ⟨⋅,⋅⟩ denotes the generalized duality pairing. It is well known that the normalized duality 𝐽 has the following properties: (1) if 𝐸 is smooth, then 𝐽 is single valued; (2) if 𝐸 is strictly convex, then 𝐽 is one-to-one (i.e., 𝐽𝑥∩𝐽𝑦=∅ for all 𝑥≠𝑦); (3) if 𝐸 is reflexive, then 𝐽 is surjective; (4) if 𝐸 has Frchet differentiable norm, then 𝐽 is uniformly norm-to-norm continuous; (5) if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸; (7) if 𝐸 is a Hilbert space, then 𝐽 is the identity operator.

As we all know that if 𝐶 is a nonempty closed 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 fact actually characterizes Hilbert spaces, and, consequently, it is not available in more general Banach spaces. In this connection, Alber [1] 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.

Let 𝐸 be a smooth Banach space. Consider the function defined by 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²for𝑥,𝑦∈𝐸.(1.2) Observe that, in a Hilbert space 𝐻, (1.2) reduces to 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖²,𝑥,𝑦∈𝐻.

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(𝑥,𝑦), that is, Π_𝐶𝑥=𝑥, where 𝑥 is the solution to the minimization problem 𝜙𝑥,𝑥=min_𝑦∈𝐶𝜙(𝑦,𝑥),(1.3) existence and uniqueness of the operator Π_𝐶 follow from the properties of the functional 𝜙(𝑥,𝑦) and strict monotonicity of the mapping 𝐽 (see, e.g., [1, 2]). In Hilbert space, Π_𝐶=𝑃_𝐶. It is obvious from the definition of function 𝜙 that (‖𝑦‖−‖𝑥‖)²≤𝜙(𝑦,𝑥)≤(‖𝑦‖+‖𝑥‖)²∀𝑥,𝑦∈𝐸.(1.4)

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 (1.4), we have ‖𝑥‖=‖𝑦‖. This implies that ⟨𝑥,𝐽𝑦⟩=‖𝑥‖²=‖𝐽𝑦‖². From the definitions of 𝑗, we have 𝐽𝑥=𝐽𝑦. That is, 𝑥=𝑦; see [3, 4] for more details.

Let 𝐶 be a closed convex subset of 𝐸, and let 𝑇 be a mapping from 𝐶 into itself with nonempty set of fixed points. We denote by 𝐹(𝑇) the set of fixed points of 𝑇. 𝑇 is called hemi-relatively nonexpansive if 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for all 𝑥∈𝐶 and 𝑝∈𝐹(𝑇).

A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that the strong lim_𝑛→∞(𝑇𝑥_𝑛−𝑥_𝑛)=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A hemi-relatively nonexpansive mapping 𝑇 from 𝐶 into itself is called relatively nonexpansive if 𝐹(𝑇)=𝐹(𝑇) (see, [5]).

A point 𝑝 in 𝐶 is said to be a strong asymptotic fixed point of 𝑇 if 𝐶 contains a sequence {𝑥_𝑛} which converges strongly to 𝑝 such that the strong lim_𝑛→∞(𝑇𝑥_𝑛−𝑥_𝑛)=0. The set of strong asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A hemi-relatively nonexpansive mapping 𝑇 from 𝐶 into itself is called weak relatively nonexpansive if 𝐹(𝑇)=𝐹(𝑇) (see, [6]).

The following conclusions are obvious: (1) relatively nonexpansive mapping must be weak relatively nonexpansive mapping; (2) weak relatively nonexpansive mapping must be hemi-relatively nonexpansive mapping.

In this paper, we will give two examples to show that the inverses of above two conclusions are not hold.

In an infinite-dimensional Hilbert space, Mann's iterative algorithm has only weak convergence, in general, even for nonexpansive mappings. Hence in order to have strong convergence, in recent years, the hybrid iteration methods for approximating fixed points of nonlinear mappings has been introduced and studied by various authors.

In 2003, Nakajo and Takahashi [7] proposed the following modification of Mann iteration method for a single nonexpansive mapping 𝑇 in a Hilbert space 𝐻: 𝑥𝑦₀∈𝐶chosenonlyarbitrarily,_𝑛𝐶_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑄=𝑧∈𝐶∶‖‖𝑦_𝑛−𝑧‖‖≤‖‖𝑥_𝑛−𝑧_𝑛‖‖,𝑥=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,(1.5) where 𝐶 is a closed convex subset of 𝐻, and 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 (1.5) converges strongly to 𝑃_𝐹(𝑇)(𝑥₀), where 𝐹(𝑇) denote the fixed points set of 𝑇.

The ideas to generalize the process (1.5) 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: 𝑥𝑦₀∈𝐶chosenonlyarbitrarily,_𝑛𝐶_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝑄=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛,𝑥=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0_𝑛+1,=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,(1.6) where 𝐽 is the duality mapping on 𝐸, and Π_𝐾(⋅) is the generalized projection from 𝐸 onto a nonempty closed convex subset 𝐾. 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, 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 (1.6), 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).

Recently, Plubtieng and Ungchittrakool [9] here proposed the following hybrid iteration method for a countable family of relatively nonexpansive mappings in a Banach space and proved the convergence theorem.

Theorem PU. Let 𝐸 be a uniformly smooth and uniformly convex Banach space and let 𝐶 and 𝐶 be two nonempty closed convex subsets of 𝐸 such that 𝐶⊂𝐶. Let {𝑇_𝑛} be a sequence of relatively nonexpansive mappings from 𝐶 into 𝐸 such that ⋂^∞𝑛=1𝐹(𝑇_𝑛) is nonempty, and let {𝑥_𝑛} be a sequence defined as follows: 𝐶𝑥₀∈𝐶,₁𝑥=𝐶,₁𝑦_𝑛=Π_𝐶1𝑥₀,𝐶_𝑛+1=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇_𝑛𝑥_𝑛,𝑥=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑛≥1,_𝑛+1=Π_𝐶𝑛+1𝑥₀,(1.7) where 𝛼_𝑛∈[0,1] satisfies either(a)0≤𝛼_𝑛<1 for all 𝑛≥1 and limsup_𝑛→∞𝛼_𝑛<1 or(b)liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0. Suppose that for any bounded subset 𝐵 of 𝐶 there exists an increasing, continuous and convex function ℎ_𝐵 from 𝑅⁺ into 𝑅⁺ such that ℎ_𝐵(0)=0, and lim_{𝑙,𝑘→∞}supℎ_𝐵‖‖𝑇_𝑙𝑧−𝑇_𝑘𝑧‖‖∶𝑧∈𝐵=0.(1.8) Let 𝑇 be a mapping from 𝐶 into 𝐸 defined by 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥 for all 𝑥∈𝐶 and suppose that 𝐹(𝑇)=^∞_𝑛=1𝐹𝑇_𝑛=^∞_𝑛=1𝐹𝑇_𝑛=𝐹(𝑇).(1.9) Then {𝑥_𝑛}, {𝑇_𝑛𝑥_𝑛}, and {𝑦_𝑛} converge strongly to Π_𝐹(𝑇)𝑥₀.

In this paper, the authors have obtained the following results: (1) the definition of uniformly closed countable family of nonlinear mappings, (2) strong convergence theorem by the monotone hybrid algorithm for a countable family of hemi-relatively nonexpansive mappings in a Banach space with new method of proof, (3) two examples of uniformly closed countable families of nonlinear mappings and applications, (4) an example which is hemi-relatively nonexpansive mapping but not weak relatively nonexpansive mapping, and (5) an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping. Therefore, the results of this paper improve and extend the results of Plubtieng and Ungchittrakool [9] and many others.

We need the following definitions and lemmas.

Lemma 1.1 (Kamimura and Takahashi [10]). Let 𝐸 be a uniformly convex and smooth Banach space, and let {𝑥_𝑛} and {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 1.2 (Alber [1]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸 and 𝑥∈𝐸. Then, 𝑥₀=Π_𝐶𝑥 if and only if ⟨𝑥₀−𝑦,𝐽𝑥−𝐽𝑥₀⟩≥0for𝑦∈𝐶.(1.10)

Lemma 1.3 (Alber [1]). 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.(1.11)

The following lemma is not hard to prove.

Lemma 1.4. Let 𝐸 be a strictly convex and smooth Banach space, let 𝐶 be a closed convex subset of 𝐸, and let 𝑇 be a hemi-relatively nonexpansive mapping from 𝐶 into itself. Then 𝐹(𝑇) is closed and convex.

In this paper, we present the definition of uniformly closed for a sequence of mappings as follows.

Definition 1.5. Let 𝐸 be a Banach space, 𝐶 be a closed convex subset of 𝐸, let {𝑇_𝑛}^∞𝑛=1 be a sequence of mappings of 𝐶 into 𝐸 such that ⋂^∞𝑛=1𝐹(𝑇_𝑛) is nonempty. We say that {𝑇_𝑛}^∞𝑛=1 is uniformly closed if 𝑝∈⋂^∞𝑛=1𝐹(𝑇_𝑛) whenever {𝑥_𝑛}⊂𝐶 converges strongly to 𝑝 and ‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖→0 as 𝑛→∞.

Lemma 1.6 (see [11, 12]). 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,𝑗(𝑥)−𝑗(𝑦)⟩≥𝑐^𝑝𝑐^𝑝−2𝑝‖𝑥−𝑦‖^𝑝,(1.12) where 𝐽_𝑝 is the generalized duality mapping from 𝐸 into 𝐸^∗, and 1/𝑐 is the 𝑝-uniformly convexity constant of 𝐸.

2. Main Results

Definition 2.1. Let 𝐸 be a Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let {𝑇_𝑛}^∞𝑛=1∶𝐶→𝐸, {𝑆_𝑛}^∞𝑛=1∶𝐶→𝐸 be two sequences of mappings. If for any convergence sequence {𝑥_𝑛}⊂𝐶, the following holds: lim_𝑛→+∞‖‖𝑇_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛‖‖=0.(2.1) Then {𝑇_𝑛} and {𝑆_𝑛} are said to satisfy the only asymptotically condition.

Theorem 2.2. 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∶𝐶→𝐸, {𝑆_𝑛}^∞𝑛=1∶𝐶→𝐸 be two uniformly closed sequences of hemi-relatively nonexpansive mappings satisfying the asymptotically condition such that 𝐹₁=⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅, 𝐹₂=⋂^∞𝑛=1𝐹(𝑆_𝑛)≠∅, and 𝐹=𝐹₁⋂𝐹₂≠∅. Assume that {𝛼_𝑛}^∞𝑛=0, {𝛽_𝑛}^∞𝑛=0, {𝛾_𝑛}^∞𝑛=0 and {𝛿_𝑛}^∞𝑛=0 are four sequences in [0,1] such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1, lim_𝑛→∞𝛼_𝑛=0 and 0<𝛾≤𝛾_𝑛≤1, 𝛿_𝑛→0 for some constant 𝛾∈(0,1). Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥𝑦₀∈𝐶arbitrarily,_𝑛𝐶=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇_𝑛𝑥_𝑛+𝛿_𝑛𝐽𝑆_𝑛𝑥_𝑛,𝑛≥1,_𝑛𝐶=𝑧∈𝐶_𝑛−1∶𝜙𝑧,𝑦_𝑛≤1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝛼_𝑛𝜙𝑧,𝑥₀,𝑛≥1,₀𝑥=𝐶,_𝑛+1=Π_𝐶𝑛𝑥₀,𝑛≥0.(2.2) Then {𝑥_𝑛} converges strongly to 𝑞=Π_𝐹𝑥₀.

Proof . We first show that 𝐶_𝑛 is closed and convex for all 𝑛≥0. From the definitions of 𝐶_𝑛, it is obvious that 𝐶_𝑛 is closed for all 𝑛≥0. Next, we prove that 𝐶_𝑛 is convex for all 𝑛≥0. Since 𝜙𝑧,𝑦_𝑛≤1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝛼_𝑛𝜙𝑧,𝑥₀(2.3) is equivalent to 2𝑧,1−𝛼_𝑛𝐽𝑥_𝑛+𝛼_𝑛𝐽𝑥₀−𝐽𝑦_𝑛≤1−𝛼_𝑛‖‖𝑥_𝑛‖‖²+𝛼_𝑛‖‖𝑥₀‖‖²−‖‖𝑦_𝑛‖‖²,(2.4) it is easy to get that 𝐶_𝑛 is convex for all 𝑛≥0.
Next, we show that 𝐹⊂𝐶_𝑛 for all 𝑛≥1. Indeed, for each 𝑝∈𝐹, we have 𝜙𝑝,𝑦_𝑛==𝜙‖𝑝‖𝑝₂,𝐽⁻¹𝛼_𝑛𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇_𝑛𝑥_𝑛+𝛿_𝑛𝐽𝑆_𝑛𝑥_𝑛+‖‖𝛼−2𝑝,𝛼_𝑛𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇_𝑛𝑥_𝑛+𝛿_𝑛𝐽𝑆_𝑛𝑥_𝑛_𝑛𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇_𝑛𝑥_𝑛+𝛿_𝑛𝐽𝑆_𝑛𝑥_𝑛‖‖²≤+‖𝑝𝛼‖²_𝑛−2𝛼_𝑛⟨𝑝,𝐽𝑥₀⟩−2𝛽_𝑛⟨𝑝,𝐽𝑥_𝑛⟩−2𝛾_𝑛⟨𝑝,𝐽𝑇_𝑛𝑥_𝑛⟩−2𝛿_𝑛⟨𝑝,𝐽𝑆_𝑛𝑥_𝑛⟩‖‖𝑥₀‖‖²+𝛽_𝑛‖‖𝑥_𝑛‖‖+𝛾_𝑛‖‖𝑇_𝑛𝑥_𝑛‖‖²+𝛿_𝑛‖‖𝑆_𝑛𝑥_𝑛‖‖²≤≤𝛼𝛼_𝑛𝑛𝜙𝑝,𝑥₀+𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝜙𝑝,𝑇_𝑛𝑥_𝑛+𝛿_𝑛𝜙𝑝,𝑆_𝑛𝑥_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝜙𝑝,𝑥_𝑛.(2.5) So, 𝑝∈𝐶_𝑛, which implies that 𝐹⊂𝐶_𝑛 for all 𝑛≥1.
Since 𝑥_𝑛+1=Π_𝐶𝑛𝑥₀ and 𝐶_𝑛⊂𝐶_𝑛−1, then we get the following: 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀,∀𝑛≥0.(2.6) Therefore, {𝜙(𝑥_𝑛,𝑥₀)} is nondecreasing. On the other hand, by Lemma 1.3, we have 𝜙𝑥_𝑛,𝑥₀=𝜙Π_𝐶𝑛−1𝑥₀,𝑥₀≤𝜙𝑝,𝑥₀−𝜙𝑝,𝑥_𝑛≤𝜙𝑝,𝑥₀,(2.7) for all 𝑝∈𝐹(𝑇)⊂𝐶_𝑛−1 and for all 𝑛≥1. Therefore, 𝜙(𝑥_𝑛,𝑥₀) is also bounded. This together with (3.1) implies that the limit of {𝜙(𝑥_𝑛,𝑥₀)} exists. Put the following: lim_𝑛→∞𝜙𝑥_𝑛,𝑥₀=𝑑.(2.8) From Lemma 1.3, we have, for any positive integer 𝑚, that 𝜙𝑥_𝑛+𝑚,𝑥_𝑛+1==𝜙𝜙𝑥𝑥_{𝑛+𝑚𝑛+𝑚},Π_𝐶𝑛𝑥₀≤𝜙𝑥_𝑛+𝑚,𝑥₀−𝜙Π_𝐶𝑛𝑥₀,𝑥₀,𝑥₀−𝜙𝑥_𝑛+1,𝑥₀,(2.9) for all 𝑛≥0. This together with (3.6) implies that lim_𝑛→∞𝜙𝑥_𝑛+𝑚,𝑥_𝑛+1=0(2.10) is, uniformly for all 𝑚, holds. By using Lemma 1.1, we get that lim_𝑛→∞‖‖𝑥_𝑛+𝑚−𝑥_𝑛+1‖‖=0(2.11) is, uniformly for all 𝑚, holds. Then {𝑥_𝑛} is a Cauchy sequence; therefore, there exists a point 𝑝∈𝐶 such that 𝑥_𝑛→𝑝.
Since 𝑥_𝑛+1=Π_𝐶𝑛𝑥₀∈𝐶_𝑛, from the definition of 𝐶_𝑛, we have the following: 𝜙𝑥_𝑛+1,𝑦_𝑛≤1−𝛼_𝑛𝜙𝑥_𝑛+1,𝑥_𝑛+𝛼_𝑛𝜙𝑥_𝑛+1,𝑥₀.(2.12) This together with (2.10) and lim_𝑛→∞𝛼_𝑛=0 implies that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑦_𝑛=0.(2.13) Therefore, by using Lemma 1.1, we obtain the following: lim_𝑛→∞‖‖𝑥_𝑛+1−𝑦_𝑛‖‖=0.(2.14) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, then we have the following: lim_𝑛→∞‖‖𝐽𝑥_𝑛+1−𝐽𝑦_𝑛‖‖=lim_𝑛→∞‖‖𝐽𝑥_𝑛+1−𝐽𝑥_𝑛‖‖=0.(2.15) Noticing that ‖‖𝐽𝑥_𝑛+1−𝐽𝑦_𝑛‖‖==‖‖‖‖𝛼𝐽_𝑛𝑥_𝑛+1−𝛼_𝑛𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇_𝑛𝑥_𝑛+𝛿_𝑛𝐽𝑆_𝑛𝑥_𝑛‖‖≥𝛾_𝑛𝐽𝑥_𝑛+1−𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+1−𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑥_𝑛+1−𝐽𝑇_𝑛𝑥_𝑛+𝛿_𝑛𝐽𝑥_𝑛+1−𝐽𝑇_𝑛𝑥_𝑛‖‖−𝛼‖‖_𝑛𝐽𝑥_𝑛+1−𝐽𝑇_𝑛𝑥_𝑛‖‖−𝛿_𝑛‖‖𝐽𝑥_𝑛+1−𝐽𝑆_𝑛𝑥_𝑛‖‖‖‖𝐽𝑥_𝑛+1−𝐽𝑥₀‖‖−𝛽_𝑛‖‖𝐽𝑥_𝑛+1−𝐽𝑥_𝑛‖‖,(2.16) which leads to 𝛾_𝑛‖‖𝐽𝑥_𝑛+1−𝐽𝑇_𝑛𝑥_𝑛‖‖≤+‖‖𝛽𝐽_𝑛𝑥_𝑛+1−𝐽𝑦_𝑛‖‖+𝛼_𝑛‖‖𝐽𝑥₀−𝐽𝑥_𝑛+1‖‖‖‖𝐽𝑥_𝑛+1−𝐽𝑥_𝑛‖‖+𝛿_𝑛‖‖𝐽𝑥_𝑛+1−𝐽𝑆_𝑛𝑥_𝑛‖‖.(2.17) From (2.15) andlim_𝑛→∞𝛼_𝑛=0, lim_𝑛→∞𝛿_𝑛=0, 0<𝛾≤𝛾_𝑛≤1 we obtain that lim_𝑛→∞‖‖𝐽𝑥_𝑛+1−𝐽𝑇_𝑛𝑥_𝑛‖‖=0.(2.18) Since 𝐽⁻¹ is also uniformly norm-to-norm continuous on bounded sets, then we obtain that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑇_𝑛𝑥_𝑛‖‖=0.(2.19) This together with (2.11) implies that lim_𝑛→∞‖‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖‖=0.(2.20) Sine {𝑇_𝑛} and {𝑆_𝑛} satisfy the asymptotically condition, we also have lim_𝑛→∞‖‖𝑥_𝑛−𝑆_𝑛𝑥_𝑛‖‖=0.(2.21) Since 𝑥_𝑛→𝑝 and {𝑇_𝑛}^∞𝑛=1, {𝑆_𝑛}^∞𝑛=1 are uniformly closed, we have 𝑝∈𝐹=^∞_𝑛=1𝐹𝑇_𝑛^∞_𝑛=1𝐹𝑆_𝑛.(2.22)
Finally, we prove that 𝑝=Π_𝐹𝑥₀, from Lemma 1.3, we have 𝜙𝑝,Π_𝐹𝑥₀+𝜙Π_𝐹𝑥₀,𝑥₀≤𝜙𝑝,𝑥₀.(2.23) On the other hand, 𝑥_𝑛+1=Π_𝐶𝑛𝑥₀ and 𝐹⊂𝐶_𝑛, for all 𝑛. Also from Lemma 1.3, we have 𝜙Π_𝐹𝑥₀,𝑥_𝑛+1+𝜙𝑥_𝑛+1,𝑥₀≤𝜙Π_𝐹𝑥₀,𝑥₀.(2.24) By the definition of 𝜙(𝑥,𝑦), we know that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑥₀=𝜙𝑝,𝑥₀.(2.25) Combining (2.24) and (2.25), we know that 𝜙(𝑝,𝑥₀)=𝜙(Π_𝐹𝑥₀,𝑥₀). Therefore, it follows from the uniqueness of Π_𝐹𝑥₀ that 𝑝=Π_𝐹𝑥₀. This completes the proof.

When 𝛼_𝑛≡0, 𝛿_𝑛≡0 in Theorem 2.2, we obtain the following result.

Theorem 2.3. 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∶𝐶→𝐸 be a uniformly closed sequence of hemi-relatively nonexpansive mappings such that 𝐹=⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅. Assume that {𝛼_𝑛}^∞𝑛=0 is a sequences in [0,1] such that 0≤𝛼_𝑛≤𝛼<1 for some constant 𝛼∈(0,1). Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥𝑦₀∈𝐶arbitrarily,_𝑛𝐶=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇_𝑛𝑥_𝑛,𝑛≥1,_𝑛𝐶=𝑧∈𝐶_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑛≥1,₀𝑥=𝐶,_𝑛+1=Π_𝐶𝑛𝑥₀,𝑛≥0.(2.26) Then {𝑥_𝑛} converges strongly to 𝑞=Π_𝐹𝑥₀.

3. Applications for Equilibrium Problem

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 𝑓(𝑥,𝑦)≥0,∀𝑦∈𝐶.(3.1) The set of solutions of (1.2) 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 (1.2). Some methods have been proposed to solve the equilibrium problem in Hilbert spaces, see, for instance, [13–15].

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, only limsup_𝑡↓0𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦),(A4) for all 𝑥∈𝐶, 𝑓(𝑥,⋅) is convex and lower semicontinuous.

Lemma 3.1 (Blum and Oettli [13]). 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(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(3.2)

Lemma 3.2 (Takahashi and Zembayashi [15]). 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)=𝑧∈𝐶∶𝑓(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶(3.3) 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⟩,(3.4)(3)𝐹(𝑇_𝑟)=EP(𝑓);(4)EP(𝑓) is closed and convex;(5)𝑇_𝑟 is also a relatively nonexpansive mapping.

Lemma 3.3 (Takahashi and Zembayashi [15]). 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), let 𝑟>0, and let 𝑥∈𝐸, 𝑞∈𝐹(𝑇_𝑟), then the following holds: 𝜙𝑞,𝑇_𝑟𝑥+𝜙𝑇_𝑟𝑥,𝑥≤𝜙(𝑞,𝑥).(3.5)

Lemma 3.4. 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 𝑓𝑧_𝑛,𝑦+1𝑟_𝑛𝑓𝑧⟨𝑦−𝑧_𝑛,𝐽𝑧_𝑛−𝐽𝑥_𝑛⟩≥0,∀𝑦∈𝐶,(3.6)_𝑛+𝑚,𝑦+1𝑟_𝑛+𝑚𝑦−𝑧_𝑛+𝑚,𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛+𝑚≥0,∀𝑦∈𝐶.(3.7) Putting 𝑦=𝑧_𝑛+𝑚 in (3.6) and 𝑦=𝑧_𝑛 in (3.7), we have 𝑓𝑧_𝑛,𝑧_𝑛+𝑚+1𝑟_𝑛𝑧_𝑛+𝑚𝑓𝑧−𝑧_𝑛,𝐽𝑧_𝑛−𝐽𝑥_𝑛≥0,∀𝑦∈𝐶,_𝑛+𝑚,𝑧_𝑛+1𝑟_𝑛+𝑚𝑧_𝑛−𝑧_𝑛+𝑚,𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛+𝑚≥0,∀𝑦∈𝐶.(3.8) So, from (A2), we have 𝑧^𝑛+𝑚−𝑧_𝑛,𝐽𝑧^𝑛−𝐽𝑥_𝑛𝑟_𝑛−𝐽𝑧^𝑛+𝑚−𝐽𝑥_𝑛+𝑚𝑟_𝑛+𝑚≥0,(3.9) and hence 𝑧^𝑛+𝑚−𝑧_𝑛,𝐽𝑧_𝑛−𝐽𝑥_𝑛−𝑟^𝑛𝑟_𝑛+𝑚𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛+𝑚≥0.(3.10) Thus, we have 𝑧^𝑛+𝑚−𝑧_𝑛,𝐽𝑧_𝑛−𝐽𝑧_𝑛+𝑚+𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛−𝑟^𝑛𝑟_𝑛+𝑚𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛+𝑚≥0,(3.11) which implies that 𝑧_𝑛+𝑚−𝑧_𝑛,𝐽𝑧_𝑛+𝑚−𝐽𝑧_𝑛≤𝑧_𝑛+𝑚−𝑧_𝑛,𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛−𝑟𝑟^𝑛_𝑛+𝑚𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛+𝑚≥0.(3.12) By using Lemma 1.6, we obtain the following: 𝑐^𝑝𝑐^𝑝−2𝑝‖‖𝑧^𝑛+𝑚−𝑧_𝑛‖‖^𝑝≤𝑧_𝑛+𝑚−𝑧_𝑛,𝐽𝑧_𝑛+𝑚−𝐽𝑥_𝑛−𝑟^𝑛𝑟_𝑛+𝑚𝐽𝑧_𝑛+𝑚=𝑧−𝐽𝑥_𝑛+𝑚≥0_𝑛+𝑚−𝑧_𝑛,1−𝑟^𝑛𝑟_𝑛+𝑚𝐽𝑧^𝑛+𝑚+𝑟^𝑛𝑟_𝑛+𝑚𝐽𝑥_𝑛+𝑚−𝐽𝑥_𝑛.(3.13) Therefore, we get the following: 𝑐^𝑝𝑐^𝑝−2𝑝‖‖𝑧^𝑛+𝑚−𝑧_𝑛‖‖^𝑝−1≤||||1−𝑟^𝑛𝑟_𝑛+𝑚||||‖‖𝐽𝑧_𝑛+𝑚‖‖+‖‖‖𝑟^𝑛𝑟_𝑛+𝑚𝐽𝑥_𝑛+𝑚−𝐽𝑥_𝑛‖‖‖.(3.14) 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‖‖,(3.15) so that {𝑧_𝑛} is bounded. Since lim_𝑛→∞𝑟_𝑛=𝑟>0, this together with (3.14) implies that {𝑧_𝑛} is a Cauchy sequence. Hence 𝑇_𝑟𝑛𝑥_𝑛=𝑧_𝑛 is convergent.
(2) By using Lemma 3.2, we know that ^∞_𝑛=1𝐹𝑇_𝑟𝑛=EP(𝑓)≠∅.(3.16)
(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.(3.17) It is obvious that 𝑇 is nonexpansive. It is easy to see that EP(𝑓)=^∞_𝑛=1𝐹𝑇_𝑟𝑛⊂𝐹(𝑇).(3.18) On the other hand, let 𝑤∈𝐹(𝑇) and 𝑤_𝑛=𝑇_𝑟𝑛𝑤, we have 𝑓𝑤_𝑛,𝑦+1𝑟_𝑛⟨𝑦−𝑤_𝑛,𝐽𝑤_𝑛−𝐽𝑤⟩≥0,∀𝑦∈𝐶.(3.19) By (A2) we know that 1𝑟_𝑛⟨𝑦−𝑤_𝑛,𝐽𝑤_𝑛−𝐽𝑤⟩≥𝑓𝑦,𝑤_𝑛,∀𝑦∈𝐶.(3.20) 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, 0=≤≤𝑓𝑡𝑡𝑓𝑓(((𝑡𝑡𝑦𝑡𝑦𝑦+++(((111−−−𝑡)𝑡𝑡))𝑤𝑤𝑤,,,𝑡𝑦𝑦𝑦))+.+(1(1−−𝑡)𝑡)𝑤𝑓)(𝑡𝑦+(1−𝑡)𝑤,𝑤)(3.21) Therefore, we have 𝑓(𝑡𝑦+(1−𝑡)𝑤,𝑦)≥0.(3.22) Letting 𝑡↓0 and using (A3), we get the following: 𝑓(𝑤,𝑦)≥0,∀𝑦∈𝐶,(3.23) and hence 𝑤∈EP(𝑓). From 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 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.

Theorem 3.5. 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 and 𝑔 be two bifunctions from 𝐶×𝐶 to 𝑅=(−∞,+∞) satisfying (A1)–(A4). Assume that {𝛼_𝑛}^∞𝑛=0, {𝛽_𝑛}^∞𝑛=0, {𝛾_𝑛}^∞𝑛=0, and {𝛿_𝑛}^∞𝑛=1 are four sequences in [0,1] such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1, lim_𝑛→∞𝛼_𝑛=0,lim_𝑛→∞𝛿_𝑛=0, and 0<𝛾≤𝛾_𝑛≤1 for some constant 𝛾∈(0,1). Let {𝑥_𝑛} be a sequence generated by 𝑥𝑦₀∈𝐶arbitrarily,_𝑛𝐶=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝐽𝑆_𝑟𝑛𝑥_𝑛,𝑛≥1,_𝑛𝐶=𝑧∈𝐶_𝑛−1∶𝜙z,𝑦_𝑛≤1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝛼_𝑛𝜙𝑧,𝑥₀,𝑛≥1,₀𝑥=𝐶,_𝑛+1=Π_𝐶𝑛𝑥₀,𝑛≥0,(3.24) where 𝑇_𝑟(𝑥)=𝑧∈𝐶∶𝑓(𝑧,𝑦)+1𝑆𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶,∀𝑥∈𝐸,_𝑟(𝑥)=𝑧∈𝐶∶𝑔(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶,∀𝑥∈𝐸,(3.25) and lim_𝑛→∞𝑟_𝑛=𝑟>0. Assume that the mappings 𝑇_𝑛 and 𝑆_𝑛 satisfy the asymptotically condition. Then {𝑥_𝑛} converges strongly to 𝑞=Π_{EP(𝑓)∩∩EP(𝑔)}𝑥₀.

Proof. By Lemma 3.4, {𝑇_𝑟𝑛}^∞𝑛=1, {𝑆_𝑟𝑛}^∞𝑛=1 are uniformly closed; therefore, by using Theorem 2.2 and Lemma 3.2, we can obtain the conclusion of Theorem 3.5. This completes the proof.

When 𝛼_𝑛≡0, 𝛿_𝑛≡0 in the Theorem 3.5, we obtain the following result.

Theorem 3.6. 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). Assume that {𝛼_𝑛}^∞𝑛=0 is a sequences in [0,1] such that 0≤𝛼_𝑛≤𝛼<1 for some constant 𝛼∈(0,1). Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥𝑦₀∈𝐶arbitrarily,_𝑛𝐶=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇_𝑟𝑛𝑥_𝑛,𝑛≥1,_𝑛𝐶=𝑧∈𝐶_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑛≥1,₀𝑥=𝐶,_𝑛+1=Π_𝐶𝑛𝑥₀,𝑛≥0,(3.26) where lim_𝑛→∞𝑟_𝑛=𝑟>0. Then {𝑥_𝑛} converges strongly to 𝑞=Π_EP(𝑓)𝑥₀.

4. Applications for Maximal Monotone Operators

In this section, we apply our above results to prove some strong convergence theorem concerning maximal monotone operators in a Banach space 𝐸.

Let 𝐴 be a multivalued 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(4.1) for each 𝑥₁,𝑥₂∈𝐷(𝐴) and 𝑦₁∈𝐴𝑥₁, 𝑦₂∈𝐴𝑥₂. A monotone operator 𝐴 is said to be maximal if it's 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 wellknown.

Theorem 4.1. 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 4.1 and strict convexity of 𝐸, we obtain that for every 𝑟>0 and 𝑥∈𝐸, there exists a unique 𝑥_𝑟 such that 𝐽𝑥∈𝐽𝑥_𝑟+𝑟𝐴𝑥_𝑟.(4.2)

Then we can define 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, we know that 𝐴⁻¹0=𝐹(𝐽_𝑟) for all 𝑟>0.

Theorem 4.2. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐴 be a maximal monotone operator from 𝐸 to 𝐸^∗, and 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 hemi-relatively nonespansive 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 that 1𝑟_𝑛𝐽𝑧_𝑛−𝐽𝐽_𝑟𝑛𝑧_𝑛⟶0.(4.3) It follows from 1𝑟_𝑛𝐽𝑧_𝑛−𝐽𝐽_𝑟𝑛𝑧_𝑛∈𝐴𝐽_𝑟𝑛𝑧_𝑛(4.4) and the monotonicity of 𝐴 that 𝑤−𝐽^𝑟𝑛𝑧_𝑛,𝑤^∗−1𝑟_𝑛𝐽𝑧_𝑛−𝐽𝐽_𝑟𝑛𝑧_𝑛≥0(4.5) 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 that 𝑝∈𝐴⁻¹0=𝐹(𝐽_𝑟𝑛) for all 𝑛≥1, that is, 𝑝∈⋂^∞𝑛=1𝐹(𝐽_𝑟𝑛).
Next we show that 𝐽_𝑟𝑛 is a hemi-relatively nonexpansive mapping for all 𝑛≥1. For any 𝑤∈𝐸 and 𝑝∈𝐹(𝐽_𝑟𝑛)=𝐴⁻¹0, from the monotonicity of 𝐴, we have 𝜙𝑝,𝐽_𝑟𝑛𝑤=‖𝑝‖²−2𝑝,𝐽𝐽_𝑟𝑛𝑤+‖‖𝐽_𝑟𝑛𝑤‖‖²=‖𝑝‖²+2𝑝,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤−𝐽𝑤+‖‖𝐽_𝑟𝑛𝑤‖‖²=‖𝑝‖²+2𝑝,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤−2⟨𝑝,𝐽𝑤⟩+‖‖𝐽_𝑟𝑛𝑤‖‖²=‖𝑝‖²−2𝐽_𝑟𝑤−𝑝−𝐽_𝑟𝑛𝑤,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤−2⟨𝑝,𝐽𝑤⟩+‖‖𝐽_𝑟𝑛𝑤‖‖²=+‖𝑝2‖²𝐽−_𝑟𝑛2𝐽_𝑟𝑛𝑤−𝑝,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤𝑤,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤−2⟨𝑝,𝐽𝑤⟩+‖‖𝐽_𝑟𝑛𝑤‖‖²≤‖𝑝‖²+2𝐽_𝑟𝑛𝑤,𝐽𝑤−𝐽𝐽_𝑟𝑛𝑤−2⟨𝑝,𝐽𝑤⟩+‖‖𝐽_𝑟𝑛𝑤‖‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑤⟩+‖𝑤‖²−‖‖𝐽_𝑟𝑛𝑤‖‖²+2𝐽_𝑟𝑛𝑤,𝐽𝑤−‖𝑤‖²≤=𝜙𝜙((𝑝𝑝,,𝑤𝑤)).−𝜙𝐽_𝑟𝑛𝑤,𝑤(4.6) This implies that 𝐽_𝑟𝑛 is a hemi-relatively nonexpansive mapping for all 𝑛≥1. This completes the proof.

Theorem 4.3. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐴 and 𝐵 be two maximal monotone operators from 𝐸 to 𝐸^∗ with nonempty common zero point set 𝐴⁻¹(0)∩𝐵⁻¹(0), let 𝐽^𝐴𝑟 be a resolvent of 𝐴 and 𝐽^𝐵𝑟 a resolvent of 𝐵 for 𝑟>0. Assume that {𝛼_𝑛}^∞𝑛=0, {𝛽_𝑛}^∞𝑛=0, {𝛾_𝑛}^∞𝑛=0, and {𝛿_𝑛}^∞𝑛=1 are four sequences in [0,1] such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1, lim_𝑛→∞𝛼_𝑛=0, lim_𝑛→∞𝛿_𝑛=0, and 0<𝛾≤𝛾_𝑛≤1 for some constant 𝛾∈(0,1). Let {𝑥_𝑛} be a sequence generated by 𝑥𝑦₀∈𝐶arbitrarily,_𝑛𝐶=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝐽^𝐴𝑟_𝑛𝑥_𝑛+𝛿_𝑛𝐽𝐽^𝐵𝑟_𝑛𝑥_𝑛,𝑛≥1,_𝑛𝐶=𝑧∈𝐶_𝑛−1∶𝜙𝑧,𝑦_𝑛≤1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝛼_𝑛𝜙𝑧,𝑥₀,𝑛≥1,₀𝑥=𝐶,_𝑛+1=Π_𝐶𝑛𝑥₀,𝑛≥0,(4.7) where liminf_𝑛→∞𝑟_𝑛>0. Assume that the mappings 𝐽^𝐴𝑟_𝑛 and 𝐽^𝐵𝑟_𝑛 satisfy the asymptotically condition. Then {𝑥_𝑛} converges strongly to 𝑞=Π_𝐴⁻¹_(0)∩𝐵⁻¹₍₀₎𝑥₀.

Proof. From Theorem 4.2, {𝐽_𝑟𝑛}^∞𝑛=1 is uniformly closed countable family of hemi-relatively nonexpansive mappings, on the other hand, 𝐴⁻¹(0)=⋂^∞𝑛=1𝐹(𝐽_𝑟𝑛), by using Theorem 2.2, we can obtain the conclusion of Theorem 4.3.

When 𝛼_𝑛≡0,𝛿_𝑛≡0 in Theorem 4.3, we obtain the following result.

Theorem 4.4. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐴 be a maximal monotone operator from 𝐸 to 𝐸^∗ with nonempty zero point set 𝐴⁻¹(0), and let 𝐽_𝑟 be a resolvent of 𝐴 for 𝑟>0. Assume that {𝛼_𝑛}^∞𝑛=0 is a sequence in [0,1] such that 0≤𝛼_𝑛≤𝛼<1 for some constant 𝛼∈(0,1). Define a sequence {𝑥_𝑛} in 𝐶 by the following algorithm: 𝑥𝑦₀∈𝐶arbitrarily,_𝑛𝐶=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝐽_𝑟𝑛𝑥_𝑛,𝑛≥1,_𝑛𝐶=𝑧∈𝐶_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑛≥1,₀𝑥=𝐶,_𝑛+1=Π_𝐶𝑛𝑥₀,𝑛≥0,(4.8) where liminf_𝑛→∞𝑟_𝑛>0. Then {𝑥_𝑛} converges strongly to 𝑞=Π_𝐴⁻¹₍₀₎𝑥₀.

5. Examples

Firstly, we give an example which is hemi-relatively nonexpansive mapping but not weak relatively nonexpansive mapping.

Example 5.1. Let 𝐸=𝑅^𝑛 and 𝑥₀≠0 be a any element of 𝐸. We define a mapping 𝑇∶𝐸→𝐸 as follows: 𝑇(𝑥)=⎧⎪⎨⎪⎩12+21^𝑛+1𝑥⁰if𝑥=12+21^𝑛𝑥⁰−𝑥if𝑥≠1,2+21^𝑛𝑥⁰,(5.1) for 𝑛=1,2,3,…. Next we show that 𝑇 is a hemi-relatively nonexpansive mapping but no weak relatively nonexpansive mapping. First, it is obvious that 𝐹(𝑇)={0}. In addition, it is easy to see that ‖𝑇𝑥‖≤‖𝑥‖,∀𝑥∈𝐸.(5.2) This implies that ‖𝑇𝑥‖²−‖𝑥‖²≤2⟨0,𝐽𝑇𝑥−𝐽𝑥⟩=2⟨𝑝,𝐽𝑇𝑥−𝐽𝑥⟩(5.3) for all 𝑝∈𝐹(𝑇). It follows from above inequality that ‖𝑝‖²−2⟨𝑝,𝐽𝑇𝑥⟩+‖𝑇𝑥‖²≤‖𝑝‖²−2⟨𝑝,𝐽𝑥⟩+‖𝑥‖²,(5.4) for all 𝑝∈𝐹(𝑇) and 𝑥∈𝐸. That is 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥),(5.5) for all 𝑝∈𝐹(𝑇) and 𝑥∈𝐸; hence 𝑇 is a hemi-relatively nonexpansive mapping. Finally, we show that 𝑇 is not weak relatively nonexpansive mapping. In fact that, letting 𝑥_𝑛=12+21^𝑛𝑥⁰,𝑛=1,2,3,…(5.6) from the definition of 𝑇, we have 𝑇𝑥_𝑛=12+21^𝑛+1𝑥⁰,𝑛=1,2,3,…(5.7) which implies that ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0 and 𝑥_𝑛→𝑥₀ (𝑥_𝑛⇀𝑥₀) as 𝑛→∞. That is 𝑥₀∈𝐹(𝑇) but 𝑥₀∈𝐹(𝑇).
Next, we give an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping.

Example 5.2. Let 𝐸=𝑙², where 𝑙²=𝜉=𝜉₁,𝜉₂,𝜉₃,…,𝜉_𝑛,…∶^∞_𝑛=1‖𝜉‖=_∞||𝜉_𝑛||²<∞,_𝑛=1||𝜉_𝑛||²^1/2⟨𝜉,𝜂⟩=_∞,∀𝜉∈𝑙²,_𝑛=1𝜉_𝑛𝜂_𝑛,∀𝜉=𝜉₁,𝜉₂,𝜉₃,…,𝜉_𝑛,…,𝜂=𝜂₁,𝜂₂,𝜂₃,…,𝜂_𝑛,….∈𝑙².(5.8) It is well known that 𝑙² is a Hilbert space, so that (𝑙²)^∗=𝑙². Let {𝑥_𝑛}⊂𝐸 be a sequence defined by 𝑥𝑥₀=(1,0,0,0,…),₁𝑥=(1,1,0,0,…),₂𝑥=(1,0,1,0,0,…),₃𝑥⋮=(1,0,0,1,0,0,…),_𝑛=𝜉_𝑛,1,𝜉_𝑛,2,𝜉_𝑛,3,…,𝜉_𝑛,𝑘,…,(5.9) where 𝜉_𝑛,𝑘=10iiff𝑘𝑘≠=11,,𝑘𝑛≠+𝑛1+,1,(5.10) for all 𝑛≥1. Define a mapping 𝑇∶𝐸→𝐸 as follows: 𝑇(𝑥)=𝑛+𝑛1𝑥^𝑛−𝑥ifif𝑥𝑥=≠𝑥𝑥_𝑛𝑛(∃𝑛≥1),(∀𝑛≥1).(5.11)

Conclusion 1. {𝑥_𝑛} converges weakly to 𝑥₀.

Proof. For any 𝑓=(𝜁₁,𝜁₂,𝜁₃,…,𝜁_𝑘,…)∈𝑙²=(𝑙²)^∗, we have 𝑓𝑥_𝑛−𝑥₀=⟨𝑓,𝑥_𝑛−𝑥₀⟩=^∞_𝑘=2𝜁_𝑘𝜉_𝑛,𝑘=𝜁_𝑛+1⟶0,(5.12) as 𝑛→∞. That is, {𝑥_𝑛} converges weakly to 𝑥₀.

Conclusion 2. {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 3. 𝑇 has a unique fixed point 0, that is, 𝐹(𝑇)={0}.

Proof . The conclusion is obvious.

Conclusion 4. 𝑥₀ is an asymptotic fixed point of 𝑇.

Proof. Since {𝑥_𝑛} converges weakly to 𝑥₀ and ‖‖𝑇𝑥_𝑛−𝑥_𝑛‖‖=‖‖‖𝑛𝑛+1𝑥^𝑛−𝑥_𝑛‖‖‖=1𝑛+1‖‖𝑥^𝑛‖‖⟶0(5.13) as 𝑛→∞, so that, 𝑥₀ is an asymptotic fixed point of 𝑇.

Conclusion 5. 𝑇 has a unique strong asymptotic fixed point 0, so that 𝐹(𝑇)=𝐹(𝑇).

Proof . In fact that, for any strong convergent sequence {𝑧_𝑛}⊂𝐸 such that 𝑧_𝑛→𝑧₀ and ‖𝑧_𝑛−𝑇𝑧_𝑛‖→0 as 𝑛→∞, from Conclusion 2, there exists sufficiently large nature number 𝑁 such that 𝑧_𝑛≠𝑥_𝑚, for any 𝑛,𝑚>𝑁. Then 𝑇𝑧_𝑛=−𝑧_𝑛 for 𝑛>𝑁, and it follows from ‖𝑧_𝑛−𝑇𝑧_𝑛‖→0 that 2𝑧_𝑛→0 and hence 𝑧_𝑛→𝑧₀=0.

Conclusion 6. 𝑇 is a weak relatively nonexpansive mapping.

Proof. Since 𝐸=𝑙² is a Hilbert space, we have 𝜙(0,𝑇𝑥)=‖0−𝑇𝑥‖²=‖𝑇𝑥‖²≤‖𝑥‖²=‖𝑥−0‖²=𝜙(0,𝑥),∀𝑥∈𝐸.(5.14) From Conclusion 2, we have 𝐹(𝑇)=𝐹(𝑇), then 𝑇 is a weak relatively nonexpansive mapping.

Conclusion 7. 𝑇 is not a relatively nonexpansive mapping.

Proof. From Conclusions 3 and 4, we have 𝐹(𝑇)≠𝐹(𝑇), so that 𝑇 is not a relatively nonexpansive mapping.

Acknowledgment

This project is supported by the National Natural Science Foundation of China under Grant (11071279).