A viscosity approximation method for weakly relatively nonexpansive mappings by the sunny nonexpansive retractions in Banach spaces
- Chin-Tzong Pang^{1},
- Eskandar Naraghirad^{2}Email author and
- Ching-Feng Wen^{3}
https://doi.org/10.1186/s13660-014-0546-7
© Pang et al.; licensee Springer. 2015
Received: 29 August 2014
Accepted: 27 December 2014
Published: 3 February 2015
Abstract
In this paper, we introduce a new viscosity approximation method by using the shrinking projection algorithm to approximate a common fixed point of a countable family of nonlinear mappings in a Banach space. Under quite mild assumptions, we establish the strong convergence of the sequence generated by the proposed algorithm and provide an affirmative answer to an open problem posed by Maingé (Comput. Math. Appl. 59:74-79, 2010) for quasi-nonexpansive mappings. In contrast with related processes, our method does not require any demiclosedness principle condition imposed on the involved operators belonging to the wide class of quasi-nonexpansive operators. As an application, we also introduce an iterative algorithm for finding a common element of the set of common fixed points of an infinite family of quasi-nonexpansive mappings and the set of solutions of a mixed equilibrium problem in a real Banach space. We prove a strong convergence theorem by using the proposed algorithm under some suitable conditions. Our results improve and generalize many known results in the current literature.
Keywords
MSC
1 Introduction
Let C be a nonempty, closed, and convex subset of a Banach space E and \(x\in E\). Then there exists a unique nearest point \(z\in C\) such that \(\|x-z\|=\inf_{y\in C}\|x-y\|\). We denote such a correspondence by \(z=P_{C}x\). The mapping \(P_{C}\) is called metric projection of E onto C.
- (1)
\(F(T)\) is nonempty;
- (2)
\(\|p-Tv\|\leq\|p-v\|\), \(\forall p\in F(T)\), \(v\in C\);
- (3)
\(\hat{F}(T)=F(T)\).
- (1)
\(F(T)\) is nonempty;
- (2)
\(\|p-Tv\|\leq\|p-v\|\), \(\forall p\in F(T)\), \(v\in C\);
- (3)
\(\tilde{F}(T)=F(T)\).
Example 1.1
Remark 1.1
It is worth mentioning that the class of weakly relatively nonexpansive mappings introduced in the present paper is different from the class of Bregman weak relatively nonexpansive mappings introduced in [5]. It is well known that in a Banach space E the Bregman projection operator \(\operatorname{proj}^{g}_{C}\) is a Bregman weak relatively nonexpansisive mapping but it is not a quasi-nonexpansive mapping with respect to the norm of the space; see, for example, [5, 9].
Definition 1.1
Let E be a real Banach space and D be a closed subset of E. A mapping \(T:D\to D\) is said to be demiclosed at the origin if, for any sequence \(\{x_{n}\}_{n\in \Bbb{N}}\) in D, the conditions \(x_{n}\rightharpoonup x_{0}\) and \(Tx_{n}\to0\) imply \(Tx_{0}=0\).
Recently, Zegeye and Shahzad [14] proved the following fixed point theorem for quasi-nonexpansive mappings in a Banach space.
Theorem 1.1
- (i)
\(\lim_{n\to\infty}\alpha_{n} = 0\);
- (ii)
\(\sum_{n=1}^{\infty}\alpha_{n}=\infty\).
Very recently, Maingé [15] studied strong convergence theorems for quasi-nonexpansive mappings and posed the following open problem in his final remark.
Question 1.1
Is there any strong convergence theorem of Moudafi’s type for quasi-nonexpansive mappings without using the demiclosedness principle in a Banach space E?
Many problems in nonlinear analysis can be formulated as a problem of finding a fixed point of a nonexpansive-type mapping. There exists an extensive literature regarding the convergence analysis of iterative methods for approximation fixed points of several types of mappings T, in the settings of Hilbert and Banach spaces (see, e.g., [16–21]). However, to the best of our knowledge, there is no strong convergence result regarding the viscosity approximation of a weakly relatively nonexpansive mapping in a Banach space. In this paper, we first introduce a new viscosity approximation method based on the shrinking projection algorithm to approximate a common fixed point of a countable family of nonlinear mappings in a Banach space. Under quite mild assumptions, we establish the strong convergence of the sequence generated by the proposed algorithm. In contrast with other related processes, our method does not require any demiclosedness principle condition imposed on the involved operators belonging to the vide class of quasi-nonexpansive operators. As an application, we also introduce an iterative algorithm for finding a common element of the set of common fixed points of an infinite family of quasi-nonexpansive mappings and the set of solutions of a mixed equilibrium problem in a real Banach space. We prove a strong convergence theorem by using the proposed algorithm under some suitable conditions. Our results improve and generalize many known results in the current literature; see, for example, [14, 15, 22].
2 Preliminaries
In this section, we collect some lemmas which will be used in the proofs for the main results in next sections.
Lemma 2.1
[21]
Lemma 2.2
[21]
Theorem 2.1
A Meir-Keeler contraction defined on a complete metric space has a unique fixed point. We have the following result, given by Suzuki [13], for Meir-Keeler contractions defined on a Banach space.
Lemma 2.3
(Suzuki [13])
Let C be a nonempty convex subset of a Banach space E and \(f:C\to E\) be a Meir-Keeler contraction. Then, for every \(\epsilon> 0\), there exists \(r\in(0, 1)\) such that \(\|x-y\|\geq\epsilon\) implies that \(\|f(x)-f(y)\|\leq r\|x - y\|\) for \(x, y \in C\).
Let \(\{C_{n}\}_{n\in\Bbb{N}}\) be a sequence of nonempty, closed, and convex subsets of a reflexive Banach space E. We define a subset \(\mbox{s-Li}_{n}C_{n}\) of E as follows: \(x\in\mbox{s-Li}_{n}C_{n}\) if and only if there exists \(\{x_{n}\}_{n\in\Bbb{N}}\subset E\) such that \(\{x_{n}\}_{n\in\Bbb{N}}\) converges strongly to x and such that \(x_{n}\in C_{n}\) for all \(n\in\Bbb{N}\). Similarly, a subset \(\mbox{w-Ls}_{n}C_{n}\) of E is defined by the following: \(y\in \mbox{w-Ls}_{n}C_{n}\) if and only if there exist a subsequence \(\{C_{n_{i}}\}_{i\in\Bbb{N}}\) of \(\{C_{n}\}_{n\in\Bbb{N}}\) and a sequence \(\{y_{i}\}_{i\in\Bbb{N}}\subset E\) such that \(\{y_{i}\}_{i\in\Bbb{N}}\) converges weakly to y and such that \(y_{i}\in C_{n_{i}}\) for all \(i\in\Bbb{N}\). If \(C_{0}\subset E\) satisfies \(C_{0} =\mbox{s-Li}_{n} C_{n} = \mbox{w-Ls}_{n} C_{n}\), it is said that \(\{C_{n}\}_{n\in\Bbb{N}}\) converges to \(C_{0}\) in the sense of Mosco [23], and we write \(C_{0}=\mbox{M-}\!\lim_{n}C_{n}\). One of the simplest examples of Mosco convergence is a decreasing sequence \(\{C_{n}\}_{n\in\Bbb{N}}\) with respect to inclusion. The Mosco limit of such a sequence is \(\bigcap_{n=1}^{\infty}C_{n}\). For more details, see [24]. Tsukada [25] proved the following theorem for the metric projection in a Banach space.
Theorem 2.2
(Tsukada [25])
Let \(\{C_{n}\}_{n\in\Bbb{N}}\) be a sequence of nonempty, closed, and convex subsets of a Banach space E. If \(C_{0} =\mathrm{M}\mbox{-}\!\lim_{n}C_{n}\) exists and is nonempty, then, for each \(x\in E\), \(\{P_{C_{n}}x\}_{n\in\Bbb{N}}\) converges strongly to \(P_{C_{0}}x\).
3 Strong convergence theorems
In this section, we prove a strong convergence theorem for approximating common fixed points of weakly relatively nonexpansive mappings in a Banach space.
Theorem 3.1
Proof
- (i)
There exists \(n_{1} \geq n_{0}\) such that \(\|x_{n_{1}}-v \|< \delta+\epsilon\).
- (ii)
\(\|x-y\|\geq\delta+\epsilon\) for every \(n\geq n_{0}\).
Remark 3.1
(1) Theorem 3.1 extends and improves Theorem 1.1. We did not use the demiclosedness principle (the condition (A) of the sequence \(\{x_{n}\}_{n\in\Bbb{N}}\)) in our discussion. Our Theorem 3.1 is also valid in a wide class of general Banach spaces while Theorem 1.1 is valid in Banach spaces having weakly sequentially duality mappings.
(2) We note also that the main result of the paper provides a positive answer to open Question 1.1. So, our Theorem 3.1 improves the main result of [15] from a Hilbert space to a Banach space.
4 Application to equilibrium problems
The equilibrium problem was first introduced by Fan in [26] (see, also [27]). It is well known that the equilibrium problem includes many important problems in nonlinear analysis and optimization such as the Nash equilibrium problem, variational inequalities, complementarity problems, vector optimization problems, fixed point problems, saddle point problems, and game theory; see for example [28] and its references. Existence results for solutions to equilibrium problems have been extensively studied, as can be seen in [27, 28].
- (A1)
\(h(x,x)=0\) for all \(x\in C\);
- (A2)
h is monotone, i.e., \(h(x,y)+h(y,x)\leq0\) for all \(x,y\in C\);
- (A3)
for all \(y\in C\), \(h(\cdot,y)\) is weakly upper semicontinuous;
- (A4)
for all \(x\in C\), \(h(x,\cdot)\) is convex.
Lemma 4.1
[30]
- (1)
\(T_{r}(x)\) is nonempty for every \(x\in E\);
- (2)
\(T_{r}\) is single-valued;
- (3)
\(\langle T_{r}x-T_{r}y,J(T_{r}x-x)\rangle\leq \langle T_{r}x-T_{r}y,J(T_{r}y-y)\rangle\), for all \(x, y\in E\);
- (4)
\(F(T_{r})=MEP(h,\phi)\);
- (5)
\(MEP(h,\phi)\) is nonempty, closed, and convex.
Theorem 4.1
Proof
Remark 4.1
The main result of [30] gave a strong convergence theorem to approximate fixed point of an infinite family of nonexpansive mappings, while the main result of the present paper gives a strong convergence theorem to approximate common fixed points of an infinite family of weakly relatively nonexpansive mappings in a uniformly convex Banach space. We note that the proof of Theorem 3.1 (lines 24-25, where the authors used the nonexpansivity of the mapping T) in [30] is not valid in our discussion. So our result extends and improves the corresponding results of [30].
Remark 4.2
(1) In Theorem 4.1, we present a strong convergence result for a system of equilibrium problems with new algorithms and new control conditions. This is complementary to the main results of [31–34]. In addition, our scheme in Theorem 4.1 has an advantage that it does not require any demiclosedness principle condition imposed on the involved operators belonging to the wide class of quasi-nonexpansive operators. Indeed, we propose different approaches, based on shrinking projection algorithms, to solve the equilibrium problem in a Banach space. So, our Theorem 4.1 improves the main results of [31–34].
Declarations
Acknowledgements
This research was partially supported by a grant from NSC.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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