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.
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1 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 \(E^{*}\).
Let \(A: C\rightarrow E^{*}\) be a monotone operator. The variational inequality problem is to find a point \(x\in C\) such that
The set of solutions of the variational inequality problem (1.1) is denoted by \(\operatorname{VI}(C,A)\). Such a problem is connected with the convex minimization problem, the complementarity problem. A well-known method for solving the variational inequality (1.1) is the projection method. Many researchers have studied this algorithm in a Hilbert space and in a Banach space, for instance, [1, 2].
Recall that a mapping \(A: C^{*}\subset E^{*}\rightarrow E\) is said to be skew-monotone, if for each \(x^{*}, y^{*} \in C^{*}\),
Recall that a mapping \(A: C^{*}\subset E^{*}\rightarrow E\) is said to be α-inverse-strongly skew-monotone, if there exists a positive number α such that
for all \(x^{*}, y^{*}\in C^{*}\). In this case, A is Lipschitz continuous, that is,
for all \(x^{*}, y^{*}\in D(A)\).
Recently, Plubtieng and Sriprad [3] considered the variational inequality problem to find \(u\in C\) such that
The set of solutions of the variational inequality problem (1.2) is denoted by \(\operatorname{VI}(JC,A)\). By using the projection algorithm method with a sunny generalized nonexpansive retraction which was introduced by Ibaraki and Takahashi [4], Plubtieng and Sriprad [3] introduced the following iterative scheme: \(x_{1}=x\in C\) and
for every \(n=1,2,\ldots\) , where \(R_{C}\) is the sunny generalized nonexpansive retraction of E onto C, \(A: D(A)\subset E^{*}\rightarrow E\) is an α-inverse-strongly skew-monotone operator, and A satisfies the condition: \(\|AJy\|\leq\|AJy-AJu\|\) for all \(y\in C\) and \(u\in \operatorname{VI}(JC,A)\). They proved that the sequence \(\{x_{n}\}\) generated by (1.3) converges weakly to some element of \(\operatorname{VI}(JC,A)\).
The equilibrium problem represents an important area of mathematical sciences such as game theory, financial mathematics, optimization, and so on. Let C be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E such that JC is closed and convex. In order to solve the equilibrium problem, we define a bifunction \(F: JC\times JC\rightarrow\mathcal{R}\) satisfying the following conditions:
-
(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.
The generalized mixed equilibrium problem is to find \(p\in C\) such that
where F is a bifunction satisfying suitable conditions, \(A: JC\rightarrow E\) is a skew-monotone operator, ψ is a real-valued function. The set of solutions of (1.4) is denoted by \(\operatorname{GMEP}(F, A, \psi )\), i.e.,
If \(A=0\), the problem (1.4) reduces to the mixed equilibrium problem, which is to find \(p\in C\) such that
The set of solutions of (1.5) is denoted by \(\operatorname{MEP}(F,\psi)\), i.e.,
If \(A=0\), \(\psi=0\), the problem (1.4) reduces to the equilibrium problem which is to find \(p\in C\) such that
The set of solutions of (1.6) is denoted by \(\operatorname{EP}(F)\). The above formulation (1.6) was considered by Takahashi and Zembayashi [5] and they proved a strong convergence theorem for finding a solution of the equilibrium problem (1.6) in Banach spaces.
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
Let E be a real Banach space with dual space \(E^{*}\), C be a nonempty, closed, and convex subset of E, \(\langle\cdot,\cdot\rangle\) denotes the generalized duality pairing. The normalized duality mapping J from E into \(2^{E^{*}}\) is given by \(J(x)=\{f\in E^{*}: \langle x, f\rangle= \|x\|^{2}=\|f\|^{2}\}\), \(x\in E\). Let \(U=\{x\in E: \|x\|=1\}\). The norm of E is said to be Gâteaux differentiable if, for each \(x,y\in U\), the limit \(\lim_{t\rightarrow0}\frac{\|x+ty\|-\| x\|}{t}\) exists. The modulus of convexity of E is the function \(\delta _{E}: (0, 2]\rightarrow[0,1]\) defined by
for every ϵ with \(0\leq\epsilon\leq2\). E is said to be strictly convex if \(\|\frac{x+y}{2}\| < 1\) for all \(x\neq y\in E\) with \(\|x\|=\|y\|= 1\). It is said to be uniformly convex if \(\delta_{E}(\epsilon)>0\) for every \(\epsilon\in(0,2]\). It is well known that E is reflexive if and only if J is surjective; E is smooth if and only if J is single valued; E is strictly convex if and only if J is one-to-one. When E is a reflexive, strictly convex, and smooth space, \(J^{-1}\) is also single valued, one-to-one, surjective, and it is also the duality mapping from \(E^{*}\) to E.
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 \(E^{*}\).
The function \(\phi: E\times E\rightarrow(-\infty,+\infty)\) is defined by
for all \(x,y\in E\); refer to [25]. From \((\|x\|^{2}-\|y\| ^{2})^{2}\leq\phi(x,y)\), for all \(x,y\in E\), it is easy to see that \(\phi (x,y)\geq0\). From the definition of ϕ, we obtain
If C is a nonempty, closed, and convex subset of E, then for all \(x\in E\) there exists a unique \(z\in C\) such that
We denote z by \(\Pi_{C} x\), the mapping \(\Pi_{C}\) is called a generalized projection from E to C; see [25].
Definition 2.1
Let C be a nonempty closed subset of a real Banach space E,
-
(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
Let C be a nonempty closed subset of a real Banach space E. A mapping \(T: C\rightarrow C\) is said to be
-
(1)
non-spreading [18], if \(\phi(Tx, Ty)+\phi(Ty,Tx)\leq \phi(Tx, y)+\phi(Ty,x)\) for all \(x, y\in E\);
-
(2)
quasi-non-spreading or hemi-relatively nonexpansive [18, 26] if \(F(T)\neq\emptyset\) and \(\phi(y,Tx)\leq\phi(y,x)\) for all \(x\in E\), \(y\in F(T)\).
Definition 2.3
Let C be a nonempty closed subset of a real Banach space E. A mapping \(T: C\rightarrow C\) is said to be generalized nonexpansive type if
for all \(x, y\in E\).
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]
Let E be a 2-uniformly smooth Banach space. Then for all \(x,y\in E\) there exists a constant \(c>0\) such that
where J is the duality mapping of E.
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 \(\lim_{n\rightarrow\infty}\|x_{n}-y_{n}\|=0\).
We make use of the following mapping V:
for all \(x\in E\) and \(x^{*}\in E^{*}\). It is obvious that \(V(x,x^{*})=\phi (x,J^{-1}x^{*})\) and \(V(x,Jy)=\phi(x,y)\).
Lemma 2.3
Let E be a strictly convex, smooth, and reflexive Banach space. Then
for all \(x\in E\) and \(x^{*}, y^{*}\in E^{*}\).
A mapping \(R: E\rightarrow C\) is called sunny if
A mapping \(R: E\rightarrow C\) is said to be a retraction if \(R^{2}x=Rx\), \(\forall x\in E\). If E is smooth and strictly convex, then a sunny generalized nonexpansive retraction of E to C is uniquely determined if it exists; refer to [4]. We also know that if E is reflexive, smooth and strictly convex, C is a nonempty closed subset of E, then there exists a sunny generalized nonexpansive retraction \(R_{C}\) from E onto C if and only if \(J(C)\) is closed and convex. In this case, \(R_{C}\) is given by \(R_{C}=J^{-1}\Pi_{J(C)}J\); refer to [29].
The following theorems are in Ibaraki and Takahashi [4].
Theorem 2.4
[4]
Let C be a nonempty closed and a sunny generalized nonexpansive retraction of a smooth, strictly convex Banach space E, and let R be a retraction from E to C. Then the following are equivalent:
-
(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]
Let C be a nonempty closed subset of a smooth and strictly convex Banach space E such that there exists a sunny generalized nonexpansive retraction R from E onto C. Let \(x\in E\) and \(z\in C\). Then the following hold:
-
(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]
Let E be a Banach space with the dual space \(E^{*}\). Let \(C^{*}\) be a nonempty, compact, and convex subset of \(E^{*}\) and let A be a skew-monotone and hemicontinuous operator of \(C^{*}\) into E. Then there exists \(x^{*}_{0}\in C^{*}\) such that
Theorem 2.8
[3]
Let C be a nonempty, compact, and convex subset of a smooth, strictly convex, and reflexive Banach space E such that JC is a closed and convex set. Let A be a skew-monotone operator of JC into E. Then
where \(R_{C}\) is the generalized nonexpansive retraction of E onto C.
Let E be a Banach space with the dual space \(E^{*}\) and let C be a nonempty, compact, and convex subset such that JC is a closed and convex set. Let \(i_{JC}\) be indicator of JC. Since \(i_{JC}: E^{*}\rightarrow(-\infty, +\infty)\) is proper, lower semicontinuous, and convex, the subdifferential \(\partial i_{JC}\) of \(i_{JC}\) defined by
is a maximal skew-monotone operator [30]. We denote by \(N_{JC}(x^{*})\) the skew normal cone of JC at a point \(x^{*}\in JC\), that is,
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 C be a nonempty, compact, and convex subset of a smooth, strictly convex, and reflexive Banach space E such that JC is closed and convex and let A be a skew-monotone and hemicontinuous operator of JC into E. Let \(B\subset E^{*}\times E\) be an operator defined as follows:
Then B is maximal skew-monotone and \((BJ)^{-1}0=\operatorname{VI}(JC,A)\).
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]
Let E be a reflexive smooth Banach space and \(E^{*}\) has a uniformly Gâteaux differentiable norm. Let C be a nonempty closed subset of E such that JC is closed and convex. Let T be a generalized nonexpansive type mapping of C into itself, i.e.,
for all \(x,y\in C\). Then the following hold:
-
(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]
Let E be a uniformly convex Banach space and \(r>0\). Then there exists a strictly increasing, continuous and convex function \(h: [0, \infty)\rightarrow[0, \infty)\) such that \(h(0)=0\) and
for all \(x, y, z\in B_{r}(0)\), where \(B_{r}(0)=\{z\in E: \|z\|\leq r\}\), and \(\lambda+\mu+\gamma=1\).
Lemma 2.13
[32]
Let \(\{a_{n}\}\) and \(\{b_{n}\}\) be two sequences of nonnegative real numbers satisfying the inequality
If \(\sum_{n=0}^{\infty}b_{n}<\infty\), then \(\lim_{n\rightarrow\infty}a_{n}\) exists.
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
Let E be a reflexive, strictly convex, and uniformly smooth Banach space with dual space \(E^{*}\), and let C be a nonempty, compact, and convex subset of E such that JC is closed and convex, \(C^{*}\) be a nonempty, closed, and convex subset of \(E^{*}\). \(F: JC\times JC\rightarrow(-\infty, +\infty)\) that satisfies the conditions (A1)-(A4). Let \(A: C^{*}\rightarrow E^{*}\) be a β-inverse-strongly skew-monotone operator and \(\psi: JC\rightarrow (-\infty,+\infty)\) be a convex and lower semicontinuous, \(r>0\) be a given real number, and \(x\in E\) be any point. Then there exists \(z\in C\) such that
where J is the duality mapping from E to \(E^{*}\).
Lemma 3.2
Let C be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex real Banach space E such that JC is closed and convex, let \(C^{*}\) be a nonempty, closed, and convex subset of \(E^{*}\). \(F: JC\times JC\rightarrow(-\infty, +\infty)\) that satisfies the conditions (A1)-(A4). Let \(A: C^{*}\rightarrow E^{*}\) be a β-inverse-strongly skew-monotone operator and \(\psi: JC\rightarrow(-\infty,+\infty)\) be a convex and lower semicontinuous function, \(r>0\) be a given real number, and \(x\in E\) be any point. Define a mapping \(K_{r}: E\rightarrow C\) as follows:
Then the following hold:
-
(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)
\(\phi(x, K_{r} x)+\phi(K_{r} x, p)\leq\phi(x, p)\), \(\forall x\in E\), \(p\in F(K_{r})\),
where J is the duality mapping from E to \(E^{*}\).
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
Let E be a uniformly convex and 2-uniformly smooth Banach space whose duality mapping J is weakly sequentially continuous, and let C be a closed convex subset of E such that JC is closed and convex. Let \(F: JC\times JC\rightarrow(-\infty, +\infty )\) be a bifunction satisfying (A1)-(A4), \(A: JC\rightarrow E\) be a β-inverse-strongly skew-monotone operator. Let \(B: JC\rightarrow E\) be an α-inverse-strongly skew-monotone operator such that \(\operatorname{VI}(JC,B)\neq\emptyset\) and \(\|BJy\|\leq\|BJy-BJu\|\) for all \(y\in C\) and \(u\in \operatorname{VI}(JC,B)\). Let S, T be two generalized nonexpansive type mappings of C into itself such that \(\Gamma:=\operatorname{VI}(JC,B)\cap \operatorname{GMEP}(F, A, \psi)\cap F(S)\cap F(T)\) is not empty. Let \(\{ x_{n}\}\) be a sequence generated by \(x_{1}=x\in C\) and
where \(R_{C}\) is the sunny generalized nonexpansive retraction of E onto C, J is the duality mapping on E. \(\{\lambda_{n}\}\subset[a,b]\) for some a, b with \(0< a<b<\frac{\alpha }{c}\), where c is the constant in Lemma 2.1. The following conditions are satisfied:
-
(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.
Let \(p\in\Gamma\),
Let \(v_{n}=u_{n}-\lambda_{n} BJu_{n}\), from Lemma 2.3 and Theorem 2.6, we have
Since B is α-inverse-strongly monotone and \(p\in \operatorname{VI}(JC,B)\), it follows that
By Lemma 2.1 and the assumption, we get
substituting (3.3) and (3.4) into (3.2), and from \(b<\frac{\alpha}{c}\), we can get
Since S, T are two generalized nonexpansive type mappings, we have
Using Theorem 2.6, from (3.5) and (3.6), we have
then, from Lemma 2.13, we see that \(\lim_{n\rightarrow\infty}\phi (x_{n},p)\) exists. Therefore, \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{z_{n}\}\), and \(\{u_{n}\} \) are bounded.
Applying (3.4)-(3.6) again, it follows that
So, we have
from \(\lim_{n\rightarrow\infty}\beta_{n}=0\), \(\limsup_{n\rightarrow\infty }\alpha_{n}^{(1)}<1\), and \(\lim_{n\rightarrow\infty}\phi(x_{n},p)\) exists; we have
From Theorem 2.6, Lemma 2.3, and (3.10), we have
then, from Lemma 2.2, we get
Since \(\{x_{n}\}\), \(\{z_{n}\}\) are bounded, by Lemma 2.12, we get
From the definition of \(R_{C}\) and (3.12), we obtain
this implies that
Notice that the limit of \(\{\phi(x_{n},p)\}\) exists and we have the conditions (i), (ii), so we get
From the property of g, we have
In a similar way, we can conclude
Noticing \(u_{n+1}=K_{r_{n+1}}x_{n+1}\), by Lemma 3.2, we get
therefore \(\lim_{n\rightarrow\infty}\phi(u_{n},p)\) exists. Let \(t_{1}=\sup_{n\geq1}\{\|x_{n}\|, \|u_{n}\|\}\). From Lemma 2.11, there exists a continuous strictly increasing and convex function \(g_{1}\) with \(g_{1}(0)=0\) such that \(g_{1}(\|x-y\|)\leq\phi(x,y)\), \(\forall x,y\in B_{t_{1}}(0)\). From the definition of \(u_{n}\) and Lemma 3.2, for \(p\in \Gamma\), we have
Let \(n\rightarrow\infty\), we obtain \(\lim_{n\rightarrow\infty}g_{1}(\| x_{n}- u_{n}\|)=0\). From the property of \(g_{1}\), we also get
Since
we get
Since
we have
From E is uniformly smooth, J is norm-to-norm continuous. So, we have
and
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., \(u\in F(S)\cap F(T)\).
Next we show that \(u\in \operatorname{VI}(JC,B)\). Let \(H\subset E^{*}\times E\) be an operator as follows:
By Theorem 2.9, H is maximal skew-monotone and \((HJ)^{-1}0=\operatorname{VI}(JC,B)\). Let \((v^{*}, w)\in G(H)\). Since \(w\in Hv^{*}=Bv^{*}+N_{JC}(v^{*})\), it follows that \(w-Bv^{*}\in N_{JC}(v^{*})\). From \(Jz_{n}\in JC\), we have
Since \(w\in Bv^{*}\), we get \(v^{*}\in JC\). This implies that there is \(v\in C\) such that \(Jv=v^{*}\). Thus it follows from (3.19) that
On the other hand, from \(z_{n}=R_{C}(u_{n}-\lambda_{n} BJu_{n})\), by Theorem 2.6, we have \(\langle(u_{n}-\lambda_{n} BJu_{n})-z_{n}, Jz_{n}-Jv\rangle\geq0\), and
It follows from (3.20) and (3.21) that
where \(M=\sup_{n\geq1}\{\|Jv-Jz_{n}\|\}\). Taking \(n=n_{i}\), from (3.11) and (3.20) and the weakly sequential continuity of J, we have \(\langle w, Jv-Ju\rangle\geq0\) as \(i\rightarrow\infty\). So, by the maximality of H, we obtain \(Ju\in H^{-1}0\), that is, \(u\in(HJ)^{-1}0=\operatorname{VI}(JC,B)\).
Now we show that \(u\in \operatorname{GMEP}(F, A, \psi)\). From \(u_{n}=K_{r_{n}}x_{n}\), we have
where \(\Theta(Ju_{n},Jy)=F(Ju_{n}, Jy) +\langle AJu_{n}, Jy-Ju_{n}\rangle+\phi (Jy)-\phi(Ju_{n})\). Replacing n by \(n_{i}\), we have
From \(0<\eta\leq r_{n}\) and (3.17), we have
So, we have
Put \(u^{*}_{t}=tJy+(1-t)u^{*}\) for \(t\in(0,1]\) and \(y\in C\). From JC is convex, we have \(u^{*}_{t}\in JC\). From (A1), (A4), and (3.24), we have
and then
letting \(t\rightarrow0\), from (A3), we have
This implies that \(u\in \operatorname{GMEP}(F,A,\psi)\).
Hence, \(u\in\Gamma:=\operatorname{VI}(JC,B)\cap \operatorname{GMEP}(F, A, \psi)\cap F(S)\cap F(T)\).
Let \(\xi_{n}=R_{\Gamma}x_{n}\) and \(p\in\Gamma\). From \(\xi_{n}\in\Gamma\) and Theorem 2.6, we have
since \(\{x_{n}\}\) is bounded, \(\{\xi_{n}\}\) is also bounded.
Replacing \(x_{n}\) by x in (3.25), we see that \(\{\phi(x,\xi_{n})\}\) is bounded.
By Theorem 2.6 and (3.6), (3.7), we get
From \(\{\phi(x,\xi_{n})\}\) being bounded, \(\sum_{n=1}^{\infty}\beta _{n}<+\infty\), and by Lemma 2.13, the limit of \(\{\phi(x_{n},\xi_{n})\}\) exists.
For any \(m\in\mathcal{N}\), from (3.25), we have \(\phi(x_{n+m},\xi_{n})\leq\phi(x_{n},\xi_{n})\). Noticing \(\xi_{n+m}=R_{\Gamma}x_{n+m}\), from Theorem 2.6, we obtain
so
Let \(t_{2}=\sup_{n\geq1}\{\xi_{n}\}\), from Lemma 2.11, there exists a continuous strictly increasing and convex function \(g_{2}\) with \(g_{2}(0)=0\) such that \(g_{2}(\|x-y\|)\leq\phi(x,y)\), \(\forall x,y\in B_{t_{2}}(0)\). Therefore, we have
Since \(\{\phi(x_{n}, \xi_{n})\}\) is a convergent sequence, by the property of \(g_{2}\), we see that \(\{ \xi_{n}\}\) is a Cauchy sequence. Suppose that \(\{\xi_{n}\}\) converges strongly to \(w\in\Gamma\). Noticing \(u\in\Gamma\), \(\xi_{n}=R_{\Gamma}x_{n}\), and from Theorem 2.6, we obtain
J is weakly sequentially continuous, and we have
On the other hand, J is monotone, and we have
So, we get
Therefore \(u=w\). The proof of Theorem 3.3 is completed. □
If we put \(A\equiv0\) and \(\psi\equiv0\) in Theorem 3.3, then Theorem 3.3 reduces to the following result.
Corollary 3.4
Let E be a uniformly convex and 2-uniformly smooth Banach space and let C be a closed subset of E such that JC is closed and convex. Let \(F: JC\times JC\rightarrow(-\infty, +\infty)\) be a bifunction satisfying (A1)-(A4). Let \(B: JC\rightarrow E\) be an α-inverse-strongly skew-monotone operator such that \(\operatorname{VI}(JC,B)\neq\emptyset\) and \(\|BJy\|\leq\|BJy-BJu\|\) for all \(y\in C\) and \(u\in \operatorname{VI}(JC,B)\). Let S, T be two generalized nonexpansive type mappings of C into itself such that \(\Gamma:=\operatorname{VI}(JC,B)\cap \operatorname{EP}(F)\cap F(S)\cap F(T)\) is not empty. Let \(\{ x_{n}\}\) be a sequence generated by \(x_{1}=x\in C\) and
where \(R_{C}\) is the sunny generalized nonexpansive retraction of E onto C, J is the duality mapping on E. \(\{\lambda_{n}\}\subset[a,b]\) for some a, b with \(0< a<b<\frac{\alpha }{c}\), where c is the constant in Lemma 2.1. The following conditions are satisfied:
-
(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
Let E be a uniformly convex and 2-uniformly smooth Banach space and let C be a closed subset of E such that JC is closed and convex. Let \(F: JC\times JC\rightarrow(-\infty, +\infty)\) be a bifunction satisfying (A1)-(A4). Let \(B: JC\rightarrow E\) be an α-inverse-strongly skew-monotone operator such that \(\operatorname{VI}(JC,B)\neq\emptyset\) and \(\|BJy\|\leq\|BJy-BJu\|\) for all \(y\in C\) and \(u\in \operatorname{VI}(JC,B)\) and T be a generalized nonexpansive type mapping of C into itself such that \(\Gamma:=\operatorname{VI}(JC,B)\cap \operatorname{EP}(F)\cap F(T)\) is not empty. Let \(\{ x_{n}\}\) be a sequence generated by \(x_{1}=x\in C\) and
where \(R_{C}\) is the sunny generalized nonexpansive retraction of E onto C, J is the duality mapping on E. \(\{\lambda_{n}\}\subset[a,b]\) for some a, b with \(0< a<b<\frac{\alpha }{c}\), where c is the constant in Lemma 2.1. The following conditions are satisfied:
-
(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
The results presented in this paper substantially improve and extend the results of others in the following aspects.
-
(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
Let \(A: R\rightarrow R\), \(B: R\rightarrow R\) be defined: for all \(x\in R\),
The classical variational inequality is as follows: finding a \(u\in C=[1,100]\) such that
The solution of (4.1), denoted by \(\operatorname{VI}([1,100], B)\), is \(\{1\}\), and B is satisfies the condition: \(\|By\|\leq\|By-Bp\|\), \(\forall y\in C\), \(p\in \operatorname{VI}(C,B)\).
Let \(T: [1,100]\rightarrow[1,100]\), \(S: [1,100]\rightarrow[1,100]\), \(F:[1,100]\times[1,100]\rightarrow R\) be defined as
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 \(F(T)\cap F(S)=\{1\}\).
It is obvious that \(F(x,y)\) satisfies the following conditions:
-
(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.
Then we have
So, we get the solution of the generalized mixed equilibrium problem \(\operatorname{GMEP}(F, A, \psi)\) is \(\{1\}\). At the same time, we obtain \(F(T)\cap F(S) \cap \operatorname{GMEP}(F, A, \psi)\cap \operatorname{VI}(C, B)=\{1\}\).
Observe that for all \(y\in[1,100]\),
Let \(a=r_{n}\), \(b=r_{n}u_{n}-3r_{n}+u_{n}-x_{n}\), \(c=3r_{n}u_{n}-2r_{n}u_{n}^{2}-u_{n}^{2}+x_{n}u_{n}\),
Let \(H(y)=r_{n}y^{2}+(r_{n}u_{n}-3r_{n}+u_{n}-x_{n})y+(3r_{n}u_{n}-2r_{n}u_{n}^{2}-u_{n}^{2}+x_{n}u_{n})\), and we have \(H(y)\geq0\). If \(H(y)\) has at most one solution in R, then we get \(\Delta\leq0\). This implies that
Put \(\alpha_{n}^{(1)}=\frac{2n}{4n+3}\), \(\alpha_{n}^{(2)}=\frac{n+2}{4n+3}\), \(\alpha_{n}^{(3)}=\frac{n+1}{4n+3}\), \(\beta_{n}=\frac{1}{n^{2}}\), \(r_{n}=\frac {n}{3n+2}\), \(\lambda_{n}=\frac{n+1}{3n+2}\). It is easy to see that the sequences \(\alpha_{n}^{(1)}\), \(\alpha_{n}^{(2)}\), \(\alpha_{n}^{(3)}\), \(\beta_{n}\), \(r_{n}\), and \(\lambda_{n}\) satisfy the conditions in Theorem 3.3. We can rewrite the sequence in Theorem 3.3: \(x_{1}=x\in[1,100]\),
Then from Theorem 3.3 we conclude that the sequence \(\{x_{n}\}\) converges weakly to 1.
References
Iiduka, H, Takahashi, W, Toyoda, M: Approximation of solutions of variational inequalities for monotone mappings. Panam. Math. J. 14, 417-429 (2004)
Iiduka, H, Takahashi, W: Weak convergence of a projection algorithm for variational inequalities in a Banach spaces. J. Math. Anal. Appl. 339, 668-679 (2008)
Plubtieng, S, Sriprad, W: Existence and approximation of solution of the variational inequality problem with a skew monotone operator defined on the dual spaces of Banach spaces. J. Nonlinear Anal. Optim. 1, 23-33 (2010)
Ibaraki, T, Takahashi, W: A new projection and convergence theorems for the projections in Banach spaces. J. Approx. Theory 149, 1-14 (2007)
Takahashi, W, Zembayashi, K: A strong convergence theorem for the equilibrium problem with a bifunction defined on the dual space of a Banach space. In: Proceedings of the 8th International Conference on Fixed Point Theory and Its Applications, pp. 197-209. Yokohama Publishers, Yokohama (2008)
Takahashi, S, Takahashi, W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69, 1025-1033 (2008)
Peng, JW, Yao, JC: Ishikawa iterative algorithms for a generalized equilibrium problem and fixed point problems of a pseudo-contraction mapping. J. Glob. Optim. 46, 331-345 (2010)
Zeng, LC, Wu, SY, Yao, JC: Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems. Taiwan. J. Math. 10, 1497-1514 (2006)
Peng, JW, Yao, JC: A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point problems and variational inequality problems. Taiwan. J. Math. 12, 1401-1433 (2008)
Peng, JW, Yao, JC: A new extragradient method for mixed equilibrium problems, fixed point problems and variational inequality problems. Math. Comput. Model. 49, 1816-1828 (2009)
Zeng, LC, Ansari, QH, Yao, JC: Viscosity approximation methods for generalized equilibrium problems and fixed point problems. J. Glob. Optim. 43, 487-502 (2009)
Qin, X, Cho, SY, Kang, SM: Strong convergence of shrinking projection methods for quasi-ϕ-nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 234, 750-760 (2010)
Saewan, S, Cho, YJ, Kumam, P: Weak and strong convergence theorems for mixed equilibrium problems in Banach spaces. Optim. Lett. 8, 501-518 (2014)
Cho, YJ, Argyros, IK, Petrot, N: Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems. Comput. Math. Appl. 50, 2292-2301 (2010)
Qin, X, Chang, SS, Cho, YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal., Real World Appl. 11, 2963-2972 (2010)
Qin, X, Cho, YJ, Kang, SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 225, 20-30 (2009)
Takahashi, W, Zembayashi, K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70, 45-57 (2009)
Takahashi, W, Yao, JC: Nonlinear operators of monotone type and convergence theorems with equilibrium problems in Banach spaces. Taiwan. J. Math. 15, 787-818 (2011)
Yao, Y, Cho, YJ, Chen, R: An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems. Nonlinear Anal. 71, 3363-3373 (2009)
Yao, Y, Cho, YJ, Liou, Y: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res. 212, 242-250 (2011)
Cai, G, Bu, S: Weak convergence theorems for general equilibrium problems and variational inequality problems and fixed point problems in Banach spaces. Acta Math. Sci. Ser. B 33, 303-320 (2013)
Ceng, LC, Petru, A, Yao, JC: Composite viscosity approximation methods for equilibrium problem, variational inequality and common fixed points. J. Nonlinear Convex Anal. 15, 219-240 (2014)
Al-Mazrooei, AE, Latif, A, Yao, JC: Solving generalized mixed equilibria, variational inequalities and constrained convex minimization. Abstr. Appl. Anal. 2014, Article ID 587865 (2014)
Ceng, LC, Yao, JC: On the triple hierarchical variational inequalities with constraints of mixed equilibria, variational inclusion and system of generalized equilibria. Tamkang J. Math. 45(3), 297-334 (2014)
Alber, YI: Metric and generalized projection in Banach space: properties and applications. In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15-50. Dekker, New York (1996)
Su, Y, Xu, HK, Wang, Z: Strong convergence theorem for a common fixed point of two hemi-relatively nonexpansive mappings. Nonlinear Anal. 71, 5616-5628 (2009)
Xu, ZB, Roach, GF: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157, 189-210 (1991)
Kamimura, S, Takahashi, W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938-945 (2002)
Kohsaha, F, Takahashi, W: Generalized nonexpansive retractions and a proximal-type algorithm in Banach spaces. J. Nonlinear Convex Anal. 8, 197-209 (2007)
Rockafellar, RT: Maximal monotone operators and proximal point algorithm. SIAM J. Control Optim. 14, 887-898 (1976)
Cho, YJ, Zhou, HY, Guo, G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 47, 707-717 (2004)
Tan, KK, Xu, HK: Approximating fixed points of nonexpansive mappings by Ishikawa iteration process. J. Math. Anal. Appl. 178, 301-308 (1993)
Phuangphoo, P, Kumam, P: Existence and modification of Halpern-Mann iterations for fixed point and generalized mixed equilibrium problems with a bifunction defined on the dual space. J. Appl. Math. 2013, Article ID 753096 (2013)
Acknowledgements
This project is supported by the NSF of Hebei province (A2014201033) and the NNSF of China (11101115, 11271106).
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Chen, J., Fan, T. Weak convergence theorems for inverse-strongly skew-monotone operators and generalized mixed equilibrium problems in Banach spaces. Fixed Point Theory Appl 2015, 13 (2015). https://doi.org/10.1186/s13663-015-0261-1
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DOI: https://doi.org/10.1186/s13663-015-0261-1