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Strong convergence theorems for equilibrium problems and asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense
- Ren-Xing Ni^{1, 2}View ORCID ID profile,
- Jian-Shuai Jin^{1, 2} and
- Ching-Feng Wen^{3}Email author
https://doi.org/10.1186/s13663-015-0404-4
© Ni et al. 2015
Received: 7 April 2015
Accepted: 12 August 2015
Published: 4 September 2015
Abstract
In this paper, we investigate common solutions to a family of mixed equilibrium problems with a relaxed η-α-monotone mapping and a nonlinear operator equation involving an infinite family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense. Strong convergence theorems of common solutions are established in a strictly convex and uniformly smooth Banach space. These results extend many important recent ones in the literature.
Keywords
- asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense
- generalized projection
- equilibrium problem
- relaxed η-α-monotone
- fixed point
MSC
- 46B20
- 46T99
- 47H05
- 47H10
- 47J05
- 47J25
- 54C20
1 Introduction
It is well known that equilibrium problems and mixed equilibrium problems have been important tools for solving problems arising in the fields of linear or nonlinear programming, complementary problems, optimization problems, variational inequalities, fixed point problems and in certain applications to economics, physics, mechanics and engineering sciences, etc. One of the most significant topics in the theory of equilibria is to develop effective and implementable algorithms for solving equilibrium problems and mixed equilibrium problems (see, e.g., [1–13] and the references therein).
The aim of this paper is to present an iterative method for solving solutions of a family of mixed equilibrium problems with a relaxed η-α-monotone mapping and a nonlinear operator equation involving an infinite family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense.
The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, an iterative algorithm is presented. A strong convergence theorem is established in a reflexive Banach space. Some results in Hilbert spaces are also discussed.
2 Preliminaries
In this paper, without other specifications, let \(N^{+}\) and \(\mathbb{R}\) be the sets of positive integers and real numbers, respectively, C be a nonempty, closed, and convex subset of a real reflexive Banach space E with the dual space \(E^{*}\). The norm and the dual pair between \(E^{*}\) and E are denoted by \(\|\cdot\|\) and \(\langle\cdot,\cdot\rangle\), respectively. Recall that the normalized duality mapping J from E to \(2^{E^{*}}\) is defined by \(Jx=\{x^{*}\in E^{*}:\langle x,x^{*}\rangle =\|x\|^{2}=\|x^{*}\|^{2}\}\).
Recall that E is said to be strictly convex if \(\|\frac{x+y}{2}\|<1\) for all \(x,y\in E\) with \(\|x\|=\|y\|=1\) and \(x\neq y\). It is said to be uniformly convex if \(\lim_{n\to\infty}\|x_{n}-y_{n}\|=0\) for any two sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) in E such that \(\|x_{n}\|=\|y_{n}\|=1\) and \(\lim_{n\to\infty}\|\frac{x_{n}+y_{n}}{2}\|=1\). Let \(U=\{x\in E:\|x\|=1\}\) be the unit sphere of E. A Banach space E is said to be smooth provided \(\lim_{t\to0}\frac{\|x+ty\|-\|x\|}{t}\) exists for each \(x,y\in U\). It is said to be uniformly smooth if the limit is attained uniformly for \(x,y\in E\). It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E and E is uniformly smooth if and only if \(E^{*}\) is uniformly convex.
In the paper, we use → and ⇀ to denote the strong convergence and weak convergence, respectively. Recall that a Banach space E enjoys the Kadec-Klee property if for any sequence \(\{x_{n}\} \subset E\), and \(x\in E\) with \(x_{n}\rightharpoonup x\), and \(\|x_{n}\|\to \|x\|\), then \(\|x_{n}-x\|\to0\) as \(n\to\infty\) (see, e.g., [14] and the references therein). It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property.
As we all know, if C is a nonempty closed convex subset of a Hilbert space H and \(P_{C}:H\to C\) 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, recently, Alber [15] introduced a generalized projection operator \(\Pi_{C}\) in a Banach space E which is an analog of the metric projection in Hilbert spaces.
Remark 2.1
If E is a smooth, strictly convex, and reflexive Banach space, then \(\phi(x,y)=0\) if and only if \(x=y\) (see [14, 15] and the references therein).
(III) If \(\Theta=0\), then the problem (2.7) is equivalent to the variational inequality (2.5) and (2.6). Denote the set of solutions of (2.5) and (2.6) by Ω.
- (C1)
\(\Theta(x,x)=0\) for all \(x\in C\);
- (C2)
Θ is monotone; that is, \(\Theta(x,y)+\Theta(y,x)\leq0\) for all \(x,y\in C\);
- (C3)
for all \(x,y,z\in C\), \(\limsup_{t\downarrow0}\Theta (tz+(1-t)x,y)\leq\Theta(x,y)\);
- (C4)
for all \(x\in C\), \(\Theta(x,\cdot)\) is convex and lower semicontinuous.
Let C be a nonempty subset of E and let \(T:C\to C\) be a mapping. In this paper, we use \(F(T)\) to denote the fixed point set of T. T is said to be asymptotically regular on C if for any bounded subset K of C, \(\lim_{n\to+\infty}\sup_{x\in K}\|T^{n+1}x-T^{n}x\|=0\). T is said to be closed if for any sequence \(\{x_{n}\}\subset C\) such that \(\lim_{n\to\infty}x_{n}=x_{0}\) and \(\lim_{n\to\infty}Tx_{n}=y_{0}\), then \(Tx_{0}=y_{0}\).
Recall that a point p in C is said to be an asymptotic fixed point of T [20] if C contains a sequence \(\{x_{n}\}\) which converges weakly to p such that \(\lim_{n\to\infty}\|x_{n}-Tx_{n}\|=0\). The set of asymptotic fixed points of T will be denoted by \(\widetilde{F}(T)\).
Remark 2.2
The class of relatively asymptotically nonexpansive mappings was first considered in [21] (see also, [22] and the reference therein).
Remark 2.3
The class of quasi-ϕ-nonexpansive mappings was first considered in [23]. The class of asymptotically quasi-ϕ-nonexpansive mappings that was studied in [24] and [25] includes the class of quasi-ϕ-nonexpansive mappings as a special cases.
Remark 2.4
The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings which require the strong restriction that \(\widetilde{F}(T)=F(T)\).
Remark 2.5
The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces.
Remark 2.6
The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense was first considered by Qin and Wang in [26].
The following Example 2.1 and Example 2.2 show that there is an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense with the nonempty fixed point set which is not ϕ-nonexpansive mapping.
Example 2.1
Example 2.2
Let \(E=l^{2}\) and \(C=\{x\in l^{2} \mid \|x\|\leq1\}\), where \(l^{2}= \{\sigma=(\sigma_{1},\sigma_{2},\ldots,\sigma _{n},\ldots) \mid \sum_{n=1}^{+\infty}|\sigma_{n}|^{2}<+\infty \}\). \(\|\sigma\|= (\sum_{n=1}^{+\infty}|\sigma_{n}|^{2} )^{\frac{1}{2}}\), \(\forall\sigma=(\sigma _{1},\sigma_{2},\ldots,\sigma_{n},\ldots)\in l^{2}\); \(\langle\sigma,\eta\rangle =\sum_{n=1}^{+\infty}\sigma_{n}\eta_{n}\), \(\forall\sigma=(\sigma_{1},\sigma _{2},\ldots,\sigma_{n},\ldots)\), \(\eta=(\eta_{1},\eta_{2},\ldots,\eta_{n},\ldots )\in l^{2}\).
- (i)
\(\|Tx-Ty\|\leq2\|x-y\|\), \(\forall x,y\in C\);
- (ii)
\(\|T^{n}x-T^{n}y\|\leq(2\prod_{j=2}^{n}a_{j})\|x-y\|\), \(\forall x,y\in C\), \(\forall n\geq2\).
Remark 2.7
The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasi-nonexpansive mappings in the intermediate sense, which was considered by Kirk [28], in the framework of Banach spaces.
The following lemmas are needed for the proof of our main results in next section.
Lemma 2.1
[15]
Lemma 2.2
[15]
Lemma 2.3
[5]
- (i)
\(\eta(x,y)+\eta(y,x)=0\), for all \(x,y\in C\);
- (ii)
for any fixed \(u,v\in C\), the mapping \(x\mapsto\langle Av,\eta (x,u)\rangle\) is convex and lower semicontinuous;
- (iii)
\(\alpha:E\to\mathbb{R}\) is weakly lower semicontinuous; that is, for any net \(\{x_{\beta}\}\), \(x_{\beta}\) converges to x in \(\sigma (E,E^{*})\) implying that \(\alpha(x)\leq\liminf\alpha(x_{\beta})\);
- (iv)
for any \(x,y\in C\), \(\alpha(x-y)+\alpha(y-x)\geq0\);
- (v)
\(\langle A(tz_{1}+(1-t)z_{2}),\eta(y,tz_{1}+(1-t)z_{2})\rangle\geq t\langle Az_{1},\eta(y,z_{1})\rangle+(1-t)\langle Az_{2},\eta(y,z_{2})\rangle\), for any \(z_{1},z_{2},y\in C\) and \(t\in[0,1]\).
- (1)
\(T_{r}\) is single-valued;
- (2)\(T_{r}\) is a firmly nonexpansive-type mapping; that is, for all \(x,y\in E\),$$\langle T_{r}x-T_{r}y,JT_{r}x-JT_{r}y \rangle\leq\langle T_{r}x-T_{r}y,Jx-Jy\rangle; $$
- (3)
\(F(T_{r})=EP(\Theta,A)\);
- (4)
\(T_{r}\) is quasi-ϕ-nonexpansive satisfying \(\phi(w,T_{r}x)+\phi (T_{r}x,x)\leq\phi(w,x)\) for all \(w\in F(T_{r})\) and \(x\in E\);
- (5)
\(EP(\Theta,A)\) is closed and convex.
Lemma 2.4
[29]
3 Main results
Theorem 3.1
- (vi)for all \(x,y,z,w\in C\),$$\limsup_{t\downarrow0}\bigl\langle Az,\eta\bigl(x,ty+(1-t)w\bigr)\bigr\rangle \leq\bigl\langle Az,\eta(x,w)\bigr\rangle . $$
Proof
The proof is split into the following six steps.
Step 1. We first show that F is closed and convex.
From Theorem 3.1 in [26], we see that \(\bigcap_{i=1}^{\infty}F(T_{i})\) is closed and convex, which combines with Lemma 2.3 shows that the common element set F is closed and convex.
Step 2. Next, we show that \(C_{n}\) is closed and convex for each \(n\geq1\).
Step 3. We prove that \(F\subset C_{n}\) for each \(n\geq1\).
Step 4. Next, we prove that the sequence \(\{x_{n}\}\) is bounded.
Step 5. Now we show that \(x_{n}\to x^{*}\), where \(x^{*}\in F\) as \(n\to\infty\).
(a) First we prove that \(x^{*}\in\bigcap_{i=1}^{\infty}F(T_{i})\).
(b) Next, we show that \(x^{*}\in\bigcap_{j\in\triangle}EP(\Theta_{j},A)\).
Step 6. Finally, we prove \(x^{*}=\Pi_{F}x_{1}\).
Remark 3.1
- (1)
From a quasi-ϕ-nonexpansive mapping to an infinite family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense.
- (2)
From a mixed equilibrium problem to a finite family of mixed equilibrium problems.
- (3)
From a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space such that the space has the Kadec-Klee property.
The space in Theorem 3.1 can be applicable to \(L^{P}\), \(P>1\). Now, we give Example 3.1 in order to support Theorem 3.1.
Example 3.1
For any sequence \(\{y_{n}\}\subseteq C\) such that \(\lim_{n\to+\infty }y_{n}=x^{0}\) and \(\lim_{n\to+\infty}T_{i}y_{n}=y^{0}\), we consider the following two cases:
In summary, we can see that \(T_{i}\) is closed for every \(i\in N^{+}\).
Finally, it is obvious that the family \(\{T_{i}\}_{i\in N^{+}}\) satisfies all the aspects of the hypothesis of Theorem 3.1.
For a single mapping and bifunction in Theorem 3.1, we have Corollary 3.1.
Corollary 3.1
- (vi)
for all \(x,y,z,w\in C\), \(\limsup_{t\downarrow0}\langle Az,\eta(x,ty+(1-t)w)\rangle\leq\langle Az,\eta(x,w)\rangle\).
Remark 3.2
- (1)
From a quasi-ϕ-nonexpansive mapping to an asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense.
- (2)
From a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space such that the space has the Kadec-Klee property.
Setting \(A\equiv0\) in Theorem 3.1, we have Corollary 3.2.
Corollary 3.2
Setting \(A\equiv0\), \(f\equiv0\) in Theorem 3.1, we have Corollary 3.3.
Corollary 3.3
Remark 3.3
Corollary 3.3 improves the main theorem in Huang and Ma [8] from an equilibrium problem to a family of equilibrium problems.
Setting \(\Theta\equiv0\) in Theorem 3.1, we have Corollary 3.4.
Corollary 3.4
- (vi)
for all \(x,y,z,w\in C\), \(\limsup_{t\downarrow0}\langle Az,\eta(x,ty+(1-t)w)\rangle\leq\langle Az,\eta(x,w)\rangle\).
Remark 3.4
Corollary 3.4 improves Corollary 15 in Chen et al. [5] from a quasi-ϕ-nonexpansive mapping to an infinite family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense, from a mixed equilibrium problem to a family of mixed equilibrium problems, and from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space such that the space has the Kadec-Klee property.
Setting E to be a Hilbert space in Corollary 3.3, we have Corollary 3.5.
Corollary 3.5
Proof
Note that \(\phi(x,y)=\|x-y\|^{2}\), \(J=I\), the identity mapping, and the generalized projection is reduced to the metric projection. In the framework of Hilbert spaces, the class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense is reduced to the class of asymptotically quasi-nonexpansive mappings in the intermediate sense. By Corollary 3.3, we draw the desired conclusion immediately. □
Remark 3.5
Corollary 3.5 improves Theorem 4.1 in Zhang and Wu [9] from asymptotically quasi-nonexpansive mappings to asymptotically quasi-nonexpansive mappings in the intermediate sense.
Setting \(T_{i}=I\) in Corollary 3.5, we have Corollary 3.6.
Corollary 3.6
Setting \(\Theta\equiv0\) in Corollary 3.5, we have Corollary 3.7.
Corollary 3.7
Remark 3.6
Corollary 3.7 improves Corollary 4.3 in Zhang and Wu [9] from asymptotically quasi-nonexpansive mappings to asymptotically quasi-nonexpansive mappings in the intermediate sense.
From Corollary 3.7, we can obtain Corollary 3.8 easily.
Corollary 3.8
Declarations
Acknowledgements
The research of the first author was partially supported by the National Natural Science Foundation of China (Grant No. 10971194). The research of the third author was partially supported by the grant MOST 104-2115-M-037-001.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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