Weak convergence theorems for inverse-strongly skew-monotone operators and generalized mixed equilibrium problems in Banach spaces
- Junmin Chen^{1}Email author and
- Tiegang Fan^{1, 2}
https://doi.org/10.1186/s13663-015-0261-1
© Chen and Fan; licensee Springer. 2015
Received: 3 May 2014
Accepted: 6 January 2015
Published: 1 February 2015
Abstract
In this paper, we consider an iterative algorithm for finding the common element of the set of solutions for the generalized mixed equilibrium problems, the common fixed points set of two generalized nonexpansive type mappings, and the set of solutions of the variational inequality for an inverse-strongly skew-monotone operator in Banach spaces. Under mild conditions, the weak convergence theorem is established by using the sunny generalized nonexpansive retraction in Banach spaces. Our results refine, supplement, and extend the corresponding results in (Saewan et al. in Optim. Lett. 8:501-518, 2014), and other results announced by many other authors.
Keywords
generalized nonexpansive type mapping generalized mixed equilibrium problem maximal monotone operator inverse-strongly skew-monotone operatorMSC
47H05 47H09 47J251 Introduction
Let E be a real Banach space with dual space \(E^{*}\), whose inner product and norm are denoted by \(\langle\cdot,\cdot\rangle\) and \(\|\cdot\|\), respectively. Let C be a nonempty, closed, and convex set in E and J be the duality mapping from E to \(E^{*}\) such that JC is a closed and convex subset of \(E^{*}\).
- (A1)
\(F(x^{*},x^{*})=0\), \(\forall x^{*}\in JC\);
- (A2)
F is monotone, i.e., \(F(x^{*},y^{*})+F(y^{*},x^{*})\leq0\), \(\forall x^{*}, y^{*}\in J(C)\);
- (A3)
\(\lim_{t\downarrow0} F(tz^{*}+(1-t)x^{*}, y^{*})\leq F(x^{*}, y^{*})\), \(\forall x^{*}, y^{*}, z^{*} \in J(C)\);
- (A4)
for each \(x^{*}\in J(C)\), \(y^{*}\mapsto F(x^{*},y^{*})\) is convex and lower semicontinuous.
There are many authors who studied the problem of finding a common element of the fixed point of nonlinear mappings and the set of solutions of the equilibrium problem in a Hilbert space or in a Banach space, for instance, [6–24]. In [13], Saewan et al. introduced a new iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problem and the set of fixed points for a closed ϕ-nonexpansive mapping by using the sunny generalized nonexpansive retraction in Banach spaces. In [18], using the hybrid method, Takahashi and Yao proved a strong convergence theorem for generalized nonexpansive type mappings with equilibrium problems in Banach spaces.
Motivated by [3, 18], and [13], in this paper, using the projection algorithm method with the sunny generalized nonexpansive retraction \(R_{C}\), we introduce an iterative scheme to find a common element of the set of solutions for the generalized mixed equilibrium problem, the common fixed points for two generalized nonexpansive type mappings and the set of solutions of the variational inequality in Banach spaces. Our results refine, supplement, and extend the corresponding results in [13], and other results announced by many others.
2 Preliminaries
In this paper, we denote the strong convergence, weak convergence, and weak ^{∗} convergence of a sequence \(\{x_{n}\}\) by \(x_{n}\rightarrow x\), \(x_{n}\rightharpoonup x\), and \(x_{n}\rightharpoonup^{*} x\), respectively.
The duality mapping J is said to be weakly sequentially continuous if the weak convergence of a sequence \(\{x_{n}\}\) to x implies the weak^{∗} convergence of \(\{Jx_{n}\}\) to Jx in \(E^{*}\).
Definition 2.1
- (1)
\(T: C\rightarrow E\) is said to be ϕ-nonexpansive [16] if \(F(T)\neq\emptyset\) and \(\phi(Tx, Ty)\leq\phi(x,y)\) for all \(x,y \in C\);
- (2)
\(T: C\rightarrow E\) is said to be generalized nonexpansive if \(F(T)\neq\emptyset\) and \(\phi(Tx, y)\leq\phi(x,y)\) for all \(x\in C\), \(y\in F(T)\).
Definition 2.2
Definition 2.3
The asymptotic behavior of non-spreading mappings and generalized nonexpansive type mappings was studied in [18].
We need the following lemmas and theorems for the proofs of our main results.
Lemma 2.1
[27]
Lemma 2.2
[28]
Let E be a uniformly convex and smooth Banach space, and let \(\{x_{n}\}\), \(\{y_{n}\}\) be sequences in E. If \(\{x_{n}\}\) or \(\{y_{n}\}\) is bounded and \(\lim_{n\rightarrow\infty} \phi (x_{n},y_{n})=0\), then \(\lim_{n\rightarrow\infty}\|x_{n}-y_{n}\|=0\).
Lemma 2.3
The following theorems are in Ibaraki and Takahashi [4].
Theorem 2.4
[4]
- (1)
R is sunny generalized nonexpansive;
- (2)
\(\langle x-Rx, Jy-JRx\rangle\leq0\) for all \(x\in E\) and \(y\in C\).
Theorem 2.5
[4]
Let C be a nonempty closed subset and a sunny generalized nonexpansive retraction of a smooth and strictly convex Banach space E. Then the sunny generalized nonexpansive retraction from E onto C is uniquely determined.
Theorem 2.6
[4]
- (1)
\(z=Rx\) if and only if \(\langle x-z, Jy-Jz\rangle\leq0\) for all \(y\in C\);
- (2)
\(\phi(x, Rx)+\phi(Rx, z)\leq\phi(x,z)\).
Plubtieng and Sriprad [3] proved the following theorems.
Theorem 2.7
[3]
Theorem 2.8
[3]
An operator \(A: D(A)\subset E^{*}\rightarrow E\) is said to be hemicontinuous if for all \(x^{*}, y^{*}\in D(A)\), the mapping f of \([0,1]\) into E defined by \(f(t)=A(tx^{*}+(1-t)y^{*})\) is continuous.
Theorem 2.9
[3]
Let E be a smooth Banach space and let C be a nonempty subset of E. Let \(T: C\rightarrow C\) be a mapping. Then \(p\in C\) is called a generalized asymptotically fixed point of T if there exists \(\{ x_{n}\}\subset C\) such that \(Jx_{n}\rightharpoonup Jp\) and \(\lim_{n\rightarrow\infty}\|Jx_{n}-JTx_{n}\|=0\). We denote the set of generalized asymptotically fixed points of T by \(\check{F}(T)\).
Lemma 2.10
[18]
- (1)
\(\check{F}(T)=F(T)\);
- (2)
\(JF(T)\) is closed and convex;
- (3)
\(F(T)\) is closed.
Lemma 2.11
[28]
Let E be a smooth and uniformly convex Banach space and \(r>0\). Then there exists a strictly increasing, continuous and convex function \(g: [0, 2r]\rightarrow\mathcal{R}\) such that \(g(0)=0\) and \(g(\|x-y\|)\leq\phi(x,y)\) for all \(x, y\in B_{r}(0)\), where \(B_{r}(0)=\{z\in E: \|z\|\leq r\}\).
Lemma 2.12
[31]
Lemma 2.13
[32]
3 Weak convergence theorems
In this section, we prove a weak convergence theorem for an inverse-strongly skew-monotone operator and two generalized nonexpansive type mappings applying the sunny generalized nonexpansive retraction in Banach spaces.
Lemma 3.1
Lemma 3.2
- (1)
\(K_{r}\) is single valued;
- (2)\(K_{r}\) is firmly generalized nonexpansive, i.e.,$$\langle K_{r} x-K_{r} y, JK_{r}x-JK_{r} y\rangle\leq\langle x-y, JK_{r}x-JK_{r} y\rangle,\quad \forall x, y\in E; $$
- (3)
\(F(K_{r})= \operatorname{GMEP}(F,A,\psi)\);
- (4)
\(J(\operatorname{GMEP}(F,A,\psi))\) is convex and closed;
- (5)
\(\phi(x, K_{r} x)+\phi(K_{r} x, p)\leq\phi(x, p)\), \(\forall x\in E\), \(p\in F(K_{r})\),
Remark 3.1
The proof of Lemmas 3.1 and 3.2 is similar to the proof of Lemma 24 and Theorem 25 in [33]; for details please refer to [33].
Theorem 3.3
- (i)
\(\beta_{n}\in(0,1)\), \(\sum_{n=1}^{\infty}\beta_{n}<+\infty\);
- (ii)
\(\alpha_{n}^{(1)}+\alpha_{n}^{(2)}+\alpha_{n}^{(3)}=1\), \(\limsup_{n\rightarrow\infty}\alpha_{n}^{(1)}<1\), \(\liminf_{n\rightarrow\infty }\alpha_{n}^{(1)}\alpha_{n}^{(2)}> 0\), and \(\liminf_{n\rightarrow\infty }\alpha_{n}^{(1)}\alpha_{n}^{(3)}> 0\);
- (iii)
\(\liminf_{n\rightarrow\infty}r_{n}=\eta>0\).
Then \(x_{n}\) converges weakly to \(u\in\Gamma\), where \(u=\lim_{n\rightarrow\infty}R_{\Gamma}x_{n}\).
Proof
First, we show that \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{z_{n}\}\), and \(\{u_{n}\}\) are bounded.
From \(\{x_{n}\}\) being bounded, one finds that \(\{Jx_{n}\}\) is bounded, so there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that \(Jx_{n_{i}}\rightharpoonup u^{*}\), and we get \(Jz_{n_{i}}\rightharpoonup u^{*}\). From (3.18), we have \(J^{-1}u^{*}\in\check{F}(S)\), and we can also obtain \(J^{-1}u^{*}\in\check{F}(T)\). From Lemma 2.10, putting \(u=J^{-1}u^{*}\), we have \(u\in F(S)\) and \(u\in F(T)\), i.e., \(u\in F(S)\cap F(T)\).
Hence, \(u\in\Gamma:=\operatorname{VI}(JC,B)\cap \operatorname{GMEP}(F, A, \psi)\cap F(S)\cap F(T)\).
Replacing \(x_{n}\) by x in (3.25), we see that \(\{\phi(x,\xi_{n})\}\) is bounded.
If we put \(A\equiv0\) and \(\psi\equiv0\) in Theorem 3.3, then Theorem 3.3 reduces to the following result.
Corollary 3.4
- (i)
\(\sum_{n=1}^{\infty}\beta_{n}<\infty\);
- (ii)
\(\alpha_{n}^{(1)}+\alpha_{n}^{(2)}+\alpha_{n}^{(3)}=1\), \(\limsup_{n\rightarrow\infty}\alpha_{n}^{(1)}<1\), \(\liminf_{n\rightarrow\infty }\alpha_{n}^{(1)}\alpha_{n}^{(2)}> 0\), and \(\liminf_{n\rightarrow\infty}\alpha_{n}^{(1)}\alpha_{n}^{(3)}> 0\);
- (iii)
\(\liminf_{n\rightarrow\infty}r_{n}=\eta>0\).
If J is weakly sequentially continuous, then \(x_{n}\) converges weakly to \(u\in\Gamma\), where \(u=\lim_{n\rightarrow\infty}R_{\Gamma}x_{n}\).
If we put \(A\equiv0\), \(\psi\equiv0\), and \(T=S\) in Theorem 3.3, then Theorem 3.3 reduces to the following corollary.
Corollary 3.5
- (i)
\(\sum_{n=1}^{\infty}\beta_{n}<\infty\);
- (ii)
\(0<\liminf_{n\rightarrow\infty}\alpha_{n}^{(1)}\leq\limsup_{n\rightarrow\infty}\alpha_{n}^{(1)}<1\);
- (iii)
\(\liminf_{n\rightarrow\infty}r_{n}=\eta>0\).
If J is weakly sequentially continuous, then \(x_{n}\) converges weakly to \(u\in\Gamma\), where \(u=\lim_{n\rightarrow\infty}R_{\Gamma}x_{n}\).
Remark 3.2
- (1)
Theorem 3.3 and Corollary 3.4 extend the result [18] on the iterative construction of the fixed point of a single generalized nonexpansive type mapping to the case of common fixed points of two generalized nonexpansive type mappings.
- (2)
Phuangphoo and Kumam [33] considered the fixed point problem of one closed ϕ-nonexpansive mapping; in this paper, we discuss fixed points of two generalized nonexpansive type mappings, and the closeness of the mapping is omitted.
- (3)
In this paper, the iterative scheme which we introduced is more general because it can be applied to find a common element of the set of solutions for the generalized mixed equilibrium problem, the common fixed points of two generalized nonexpansive type mappings, and the set of solutions of the variational inequality in Banach spaces.
4 Example
In this paper, we consider the convergence of the iteration which we suggest in the general Banach space. In order to make the theoretical results more intuitive, we give an example in the real number field. In a Hilbert space, the duality mapping J is the identity operator and we have the function \(\phi(x,y)=\|x-y\|^{2}\), \(R_{C}\) is the metric projection \(P_{C}\). The generalized nonexpansive type mapping should be a non-spreading mapping; the inverse-strongly skew-monotone operator should be an inverse-strongly monotone operator.
Example 4.1
It is easy to verify A, B are inverse-strongly monotone operators with coefficients \(\alpha=1\) and \(\beta=1\), respectively. T and S are non-spreading mappings and \(F(T)\cap F(S)=\{1\}\).
- (1)
\(F(x,x)=0\);
- (2)
\(F(x,y)\) is monotone, i.e., \(F(x,y)+F(y,x)=0\);
- (3)
for each \(x, y, w\in[1,100]\), \(\lim_{t\rightarrow0}F(tw+(1-t)x, y)\leq F(x,y)\);
- (4)
for each \(x\in[1,100]\), \(y\rightarrow F(x,y)\) is convex and lower semicontinuous.
Declarations
Acknowledgements
This project is supported by the NSF of Hebei province (A2014201033) and the NNSF of China (11101115, 11271106).
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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