Weak convergence theorems for inverse-strongly skew-monotone operators and generalized mixed equilibrium problems in Banach spaces

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.

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

$$ \langle Ax, y-x\rangle \geq0, \quad \forall y\in C. $$

(1.1)

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 $x^{*}, y^{*} \in C^{*}$,

$$\bigl\langle Ax^{*}-Ay^{*}, x^{*}-y^{*}\bigr\rangle \geq0. $$

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

$$\bigl\langle Ax^{*}-Ay^{*}, x^{*}-y^{*}\bigr\rangle \geq\alpha\bigl\Vert Ax^{*}-Ay^{*}\bigr\Vert ^{2} $$

for all $x^{*}, y^{*}\in C^{*}$. In this case, A is Lipschitz continuous, that is,

$$\bigl\Vert Ax^{*}-Ay^{*}\bigr\Vert \leq\frac{1}{\alpha}\bigl\Vert x^{*}-y^{*} \bigr\Vert $$

for all $x^{*}, y^{*}\in D(A)$.

Recently, Plubtieng and Sriprad [3] considered the variational inequality problem to find $u\in C$ such that

$$ \langle AJu, Jv-Ju\rangle\geq0,\quad \forall v\in C. $$

(1.2)

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

$$ x_{n+1}=R_{C} (x_{n}-\lambda _{n} AJx_{n}) $$

(1.3)

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

$$ F(Jp,Jy)+\langle AJp, Jy-Jp\rangle+\psi(Jy)-\psi(Jp)\geq0,\quad \forall y\in C, $$

(1.4)

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.,

$$\begin{aligned} \operatorname{GMEP}(F,A,\psi) =&\bigl\{ p\in C: F(Jp,Jy)+\langle AJp, Jy-Jp \rangle \\ &{}+\psi(Jy)-\psi(Jp)\geq0, \forall y\in C\bigr\} . \end{aligned}$$

If $A=0$, the problem (1.4) reduces to the mixed equilibrium problem, which is to find $p\in C$ such that

$$ F(Jp,Jy)+\psi(Jy)-\psi (Jp)\geq0,\quad \forall y\in C. $$

(1.5)

The set of solutions of (1.5) is denoted by $\operatorname{MEP}(F,\psi)$, i.e.,

$$\operatorname{MEP}(F,\psi)=\bigl\{ p\in C: F(Jp,Jy)+\psi(Jy)-\psi(Jp)\geq0, \forall y\in C\bigr\} . $$

If $A=0$, $\psi=0$, the problem (1.4) reduces to the equilibrium problem which is to find $p\in C$ such that

$$ F(Jp,Jy)\geq0, \quad \forall y\in C. $$

(1.6)

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

$$\delta_{E}(\epsilon)=\inf \biggl\{ 1-\frac{\|x+y\|}{2}:\|x\|\leq1, \|y \| \leq1, \|x-y\|\leq\epsilon \biggr\} $$

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

$$\phi(x,y)=\|x\|^{2}-2\langle x,Jy\rangle+\|y\|^{2} $$

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

$$\phi(x,y)=\phi(x,z)+\phi(z,y)+2\langle x-z, Jz-Jy\rangle. $$

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

$$\phi(z,x)=\min_{y\in C}\phi(y,x). $$

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

$$\phi(Tx, Ty)+\phi(Ty,Tx)\leq\phi(x, Ty)+\phi(y,Tx) $$

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

$$\|Jx-Jy\|\leq c\|x-y\|, $$

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:

$$V\bigl(x, x^{*}\bigr)=\|x\|^{2}-2\bigl\langle x,x^{*}\bigr\rangle +\bigl\Vert x^{*}\bigr\Vert ^{2} $$

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

$$V\bigl(x,x^{*}\bigr)+2\bigl\langle J^{-1}x^{*} -x, y^{*}\bigr\rangle \leq V \bigl(x, x^{*}+y^{*}\bigr) $$

for all $x\in E$ and $x^{*}, y^{*}\in E^{*}$.

A mapping $R: E\rightarrow C$ is called sunny if

$$R\bigl(Rx+t(x-Rx)\bigr)=Rx, \quad \forall x\in E, \forall t\geq0. $$

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

$$\bigl\langle Ax_{0}^{*}, x^{*}-x_{0}^{*}\bigr\rangle \geq0,\quad \forall x^{*}\in C^{*}. $$

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

$$u\in \operatorname{VI}(JC,A)\quad \Leftrightarrow\quad u=R_{C}(u- \lambda Au),\quad \forall\lambda>0, $$

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

$$\partial i_{JC}=\bigl\{ x\in E: i_{JC}\bigl(y^{*}\bigr)\geq i_{JC}\bigl(x^{*}\bigr)+\bigl\langle x, y^{*}-x^{*}\bigr\rangle , \forall y^{*}\in E^{*}\bigr\} , \quad \forall x^{*}\in E^{*}, $$

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,

$$N_{JC}\bigl(x^{*}\bigr)=\bigl\{ x\in E: \bigl\langle x, x^{*}-y^{*}\bigr\rangle \geq0, \forall y^{*}\in JC\bigr\} . $$

From [3, 30], we know that

$$\partial i_{JC}=\left \{ \begin{array}{l@{\quad}l} N_{JC}(x^{*}), &x^{*}\in JC, \\ \emptyset,& x^{*}\notin JC. \end{array} \right . $$

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:

$$Bv^{*}=\left \{ \begin{array}{l@{\quad}l} Av^{*}+ N_{JC}(x^{*}),& v^{*}\in JC, \\ \emptyset,& v^{*}\notin JC. \end{array} \right . $$

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.,

$$\phi(Tx,Ty)+\phi(Ty,Tx)\leq\phi(x,Ty)+\phi(y,Tx) $$

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

$$\|\lambda x+\mu y+\gamma z\|^{2}\leq\lambda\|x\|^{2}+\mu\|y \|^{2}+\gamma\| z\|^{2}-\lambda\mu h\bigl(\Vert x-y\Vert \bigr) $$

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

$$a_{n+1}\leq a_{n}+b_{n},\quad \forall n\geq1. $$

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

$$F(Jz, Jy) +\langle AJz, Jy-Jz\rangle+\psi(Jy)-\psi(Jz)+\frac {1}{r}\langle z-x, Jy-Jz\rangle\geq0,\quad \forall y\in C, $$

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:

$$\begin{aligned} K_{r}(x) =&\biggl\{ z\in C: F(Jz, Jy) +\langle AJz, Jy-Jz\rangle \\ &{} +\psi(Jy)-\psi(Jz)+\frac{1}{r}\langle z-x, Jy-Jz\rangle\geq0, \forall y\in C\biggr\} . \end{aligned}$$

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

$$\left \{ \begin{array}{l} u_{n}=K_{r_{n}}x_{n}, \\ z_{n}= R_{C}(u_{n}-\lambda_{n}BJu_{n}), \\ y_{n}=\alpha_{n}^{(1)} u_{n} +\alpha_{n}^{(2)}Sz_{n}+\alpha_{n}^{(3)}Tz_{n}, \\ x_{n+1}=R_{C}(\beta_{n} x+(1-\beta_{n})y_{n}), \end{array} \right . $$

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 $p\in\Gamma$,

$$ \phi(u_{n}, p)=\phi(K_{r_{n}}x_{n}, p)\leq\phi(x_{n}, p). $$

(3.1)

Let $v_{n}=u_{n}-\lambda_{n} BJu_{n}$, from Lemma 2.3 and Theorem 2.6, we have

$$\begin{aligned} \phi(z_{n}, p) =&\phi\bigl(R_{C}(u_{n}- \lambda_{n}BJu_{n}),p\bigr) \\ \leq& \phi(v_{n},p)-\phi(v_{n},z_{n}) \\ \leq &\phi(u_{n}-\lambda_{n}BJu_{n}, p) \\ =& V(u_{n}-\lambda_{n}BJu_{n}, Jp) \\ \leq& V(u_{n}-\lambda_{n}BJu_{n}+ \lambda_{n}BJu_{n}, Jp)- 2\bigl\langle \lambda_{n} BJu_{n}, J(u_{n}-\lambda_{n} BJu_{n})-Jp\bigr\rangle \\ =& V(u_{n},Jp)-2 \lambda_{n}\bigl\langle BJu_{n}, J(u_{n}-\lambda_{n} BJu_{n})-Jp\bigr\rangle \\ =& \phi(u_{n}, p)-2 \lambda_{n}\langle BJu_{n}, Ju_{n}-Jp\rangle \\ & {}+2\bigl\langle -\lambda_{n} BJu_{n}, J(u_{n}-\lambda_{n} BJu_{n})-Ju_{n} \bigr\rangle . \end{aligned}$$

(3.2)

Since B is α-inverse-strongly monotone and $p\in \operatorname{VI}(JC,B)$, it follows that

$$\begin{aligned}& -2 \lambda_{n}\langle BJu_{n}, Ju_{n}-Jp\rangle \\& \quad = -2 \lambda_{n}\langle BJu_{n}-BJp, Ju_{n}-Jp\rangle-2 \lambda_{n}\langle BJp, Ju_{n}-Jp\rangle \\& \quad \leq -2\lambda_{n}\alpha\|BJu_{n}-BJp \|^{2}. \end{aligned}$$

(3.3)

By Lemma 2.1 and the assumption, we get

$$\begin{aligned}& 2\bigl\langle -\lambda_{n} BJu_{n}, J(u_{n}-\lambda_{n} BJu_{n})-Ju_{n} \bigr\rangle \\& \quad \leq 2\|\lambda_{n} BJu_{n}\|\bigl\Vert J(u_{n}-\lambda_{n} BJu_{n})-Ju_{n} \bigr\Vert \\& \quad \leq 2c\lambda_{n}\|BJu_{n}\|\|u_{n}- \lambda_{n} BJu_{n}-u_{n}\| \\& \quad = 2c\lambda_{n}^{2}\|BJu_{n} \|^{2}\leq2c\lambda_{n}^{2}\|BJu_{n}-Jp \|^{2}, \end{aligned}$$

(3.4)

substituting (3.3) and (3.4) into (3.2), and from $b<\frac{\alpha}{c}$, we can get

$$\begin{aligned} \phi(z_{n}, p) \leq& \phi(v_{n},p)- \phi(v_{n},z_{n}) \\ \leq& \phi(u_{n}, p)-2\lambda_{n}\alpha \|BJu_{n}-BJp\|^{2}+2c\lambda_{n}^{2}\| BJu_{n}-Jp\|^{2} \\ \leq& \phi(u_{n},p)+2a(bc-\alpha)\|BJu_{n}-BJp \|^{2} \\ \leq& \phi(u_{n},p)\leq\phi(x_{n},p). \end{aligned}$$

(3.5)

Since S, T are two generalized nonexpansive type mappings, we have

$$\begin{aligned} \phi(y_{n},p) =&\phi\bigl(\alpha_{n}^{(1)} u_{n} +\alpha_{n}^{(2)}Sz_{n}+\alpha _{n}^{(3)}Tz_{n},p\bigr) \\ \leq&\alpha_{n}^{(1)}\phi(u_{n}, p)+ \alpha_{n}^{(2)}\phi(Sz_{n},p)+\alpha _{n}^{(3)}\phi(Tz_{n},p) \\ \leq&\alpha_{n}^{(1)}\phi(u_{n},p)+ \alpha_{n}^{(2)}\phi(z_{n},p)+\alpha _{n}^{(3)}\phi(z_{n},p) \\ \leq&\phi(x_{n},p). \end{aligned}$$

(3.6)

Using Theorem 2.6, from (3.5) and (3.6), we have

$$\begin{aligned} \phi(x_{n+1}, p) =&\phi\bigl(R_{C}\bigl( \beta_{n} x+(1-\beta_{n})y_{n}\bigr),p\bigr) \\ \leq&\phi\bigl(\beta_{n} x+(1-\beta_{n})y_{n},p \bigr) \\ \leq&\beta_{n}\phi(x,p)+(1-\beta_{n}) \phi(y_{n},p) \\ \leq&\beta_{n}\phi(x,p)+(1-\beta_{n}) \phi(x_{n},p) \\ \leq&\beta_{n}\phi(x,p)+\phi(x_{n},p), \end{aligned}$$

(3.7)

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

$$\begin{aligned}& \phi(x_{n+1},p) \\& \quad \leq \beta_{n}\phi(x,p)+(1-\beta_{n})\bigl\{ \alpha_{n}^{(1)}\phi (x_{n},p) \\& \qquad {}+\bigl(1-\alpha_{n}^{(1)}\bigr) \bigl[\phi( x_{n},p)-2\lambda_{n}\alpha\| BJu_{n}-BJp \|^{2}+2c\lambda_{n}^{2}\|BJu_{n}-Jp \|^{2} \bigr]\bigr\} \\& \quad \leq \beta_{n}\phi(x,p)+(1-\beta_{n}) \phi(x_{n},p) \\& \qquad {} +(1-\beta_{n}) \bigl(1-\alpha_{n}^{(1)} \bigr)2a(cb-\alpha)\|BJu_{n}-BJp\|^{2}. \end{aligned}$$

(3.8)

So, we have

$$\begin{aligned}& (1-\beta_{n}) \bigl(1-\alpha_{n}^{(1)} \bigr)2a(\alpha-cb)\|BJu_{n}-BJp\|^{2} \\& \quad \leq \beta_{n}\phi(x,p)+(1-\beta_{n}) \phi(x_{n},p)-\phi(x_{n+1},p) \\& \quad \leq \beta_{n}\phi(x,p)+\phi(x_{n},p)- \phi(x_{n+1},p), \end{aligned}$$

(3.9)

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

$$ \lim_{n\rightarrow\infty}\|BJu_{n}-BJp\|=0. $$

(3.10)

From Theorem 2.6, Lemma 2.3, and (3.10), we have

$$\begin{aligned} \phi(z_{n}, u_{n}) =&\phi\bigl(R_{C}(u_{n}- \lambda_{n}BJu_{n}), u_{n}\bigr) \\ \leq&\phi(u_{n}-\lambda_{n}BJu_{n}, u_{n}) \\ =& V(u_{n}-\lambda_{n}BJu_{n}, Ju_{n}) \\ \leq& V(u_{n}-\lambda_{n}BJu_{n}+ \lambda_{n}BJu_{n}, Ju_{n}) \\ &{}- 2\bigl\langle \lambda_{n} BJu_{n}, J(u_{n}-\lambda_{n} BJu_{n})-Ju_{n} \bigr\rangle \\ =&V(u_{n},Ju_{n})-2 \lambda_{n}\bigl\langle BJu_{n}, J(u_{n}-\lambda_{n} BJu_{n})-Ju_{n}\bigr\rangle \\ =&\phi(u_{n},u_{n})-2 \lambda_{n}\bigl\langle BJu_{n}, J(u_{n}-\lambda_{n} BJu_{n})-Ju_{n}\bigr\rangle \\ =&-2 \lambda_{n}\bigl\langle BJu_{n}, J(u_{n}- \lambda_{n} BJu_{n})-Ju_{n}\bigr\rangle \\ \leq&2\lambda_{n}\|BJu_{n}\|\bigl\Vert J(u_{n}-\lambda_{n} BJu_{n})-Ju_{n} \bigr\Vert \\ \leq& 2c\lambda_{n}^{2}\|BJu_{n}\|\leq2c \lambda_{n}^{2}\|BJu_{n}-BJp\| ^{2} \rightarrow0, \end{aligned}$$

then, from Lemma 2.2, we get

$$ \lim_{n\rightarrow\infty}\|z_{n}-u_{n} \|=0. $$

(3.11)

Since $\{x_{n}\}$, $\{z_{n}\}$ are bounded, by Lemma 2.12, we get

$$\begin{aligned}& \phi(y_{n},p) \\& \quad = \phi\bigl(\alpha_{n}^{(1)} u_{n} + \alpha_{n}^{(1)}Sz_{n}+\alpha _{n}^{(3)}Tz_{n},p \bigr) \\& \quad = \bigl\Vert \alpha_{n}^{(1)} u_{n} + \alpha_{n}^{(1)}Sz_{n}+\alpha_{n}^{(3)}Tz_{n} \bigr\Vert ^{2}-2\bigl\langle \alpha_{n}^{(1)} x_{n} +\alpha_{n}^{(2)}Sz_{n}+ \alpha_{n}^{(3)}Tz_{n}, Jp\bigr\rangle +\Vert p \Vert ^{2} \\& \quad \leq \alpha_{n}^{(1)}\Vert u_{n}\Vert ^{2}+\alpha_{n}^{(2)}\Vert Sz_{n} \Vert ^{2}+\alpha_{n}^{(3)}\Vert Tz_{n}\Vert ^{2}-\alpha_{n}^{(1)} \alpha_{n}^{(2)}g\bigl(\Vert x_{n}-Sz_{n} \Vert \bigr) \\& \qquad {} -2 \alpha_{n}^{(1)} \langle u_{n},Jp \rangle+2\alpha_{n}^{(2)}\langle Sz_{n}, Jp\rangle+2 \alpha_{n}^{(3)}\langle Tz_{n}, Jp\rangle+\Vert p \Vert ^{2} \\& \quad = \alpha_{n}^{(1)} \phi(u_{n},p)+ \alpha_{n}^{(2)}\phi(Sz_{n},p)+ \alpha _{n}^{(3)}\phi(Tz_{n},p)- \alpha_{n}^{(1)} \alpha_{n}^{(2)}g\bigl(\Vert x_{n}-Sz_{n} \Vert \bigr) \\& \quad \leq \phi(x_{n},p)-\alpha_{n}^{(1)} \alpha_{n}^{(2)}g\bigl(\Vert x_{n}-Sz_{n} \Vert \bigr). \end{aligned}$$

(3.12)

From the definition of $R_{C}$ and (3.12), we obtain

$$\begin{aligned}& \phi(x_{n+1}, p) \\& \quad = \phi\bigl(R_{C}\bigl(\beta_{n} x+(1- \beta_{n})y_{n}\bigr),p\bigr) \\& \quad \leq \phi\bigl(\beta_{n} x+(1-\beta_{n})y_{n},p \bigr) \\& \quad \leq \beta_{n}\phi(x,p)+\phi(y_{n},p) \\& \quad = \beta_{n}\phi(x,p)+(1-\beta_{n})\bigl[ \phi(x_{n},p)-\alpha_{n}^{(1)}\alpha _{n}^{(2)}g\bigl(\Vert x_{n}-Sz_{n} \Vert \bigr)\bigr], \end{aligned}$$

(3.13)

this implies that

$$ (1-\beta_{n})\alpha_{n}^{(1)} \alpha_{n}^{(2)}g\bigl(\Vert x_{n}-Sz_{n} \Vert \bigr)\leq\beta_{n}\phi(x,p)+(1-\beta_{n}) \phi(x_{n},p)-\phi(x_{n+1},p). $$

(3.14)

Notice that the limit of $\{\phi(x_{n},p)\}$ exists and we have the conditions (i), (ii), so we get

$$\lim_{n\rightarrow\infty}g\bigl(\Vert x_{n}-Sz_{n} \Vert \bigr)=0. $$

From the property of g, we have

$$ \lim_{n\rightarrow\infty}\|x_{n}-Sz_{n} \|=0. $$

(3.15)

In a similar way, we can conclude

$$ \lim_{n\rightarrow\infty}\|x_{n}-Tz_{n} \|=0. $$

(3.16)

Noticing $u_{n+1}=K_{r_{n+1}}x_{n+1}$, by Lemma 3.2, we get

$$\begin{aligned} \phi(u_{n+1}, p) =&\phi(K_{r_{n+1}}x_{n+1},p) \\ \leq&\phi(x_{n+1},p) \\ \leq&\beta_{n}\phi(x,p)+(1-\beta_{n}) \phi(u_{n},p) \\ \leq&\beta_{n}\phi(x,p)+\phi(u_{n},p), \end{aligned}$$

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

$$\begin{aligned} g_{1}\bigl(\Vert x_{n}- u_{n}\Vert \bigr) \leq&\phi(x_{n}, u_{n}) \\ \leq&\phi(x_{n},K_{r_{n}}x_{n}) \\ \leq& \phi(x_{n},p)-\phi(u_{n},p) \\ \leq&\beta_{n-1}\phi(x,p)+\phi(u_{n-1},p)- \phi(u_{n},p). \end{aligned}$$

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

$$ \lim_{n\rightarrow\infty}\|x_{n}-u_{n} \|=0. $$

(3.17)

Since

$$\|z_{n}-x_{n}\|\leq\|z_{n}-u_{n}\|+ \|u_{n}-x_{n}\|, $$

we get

$$ \lim_{n\rightarrow\infty}\|z_{n}-x_{n}\|=0. $$

Since

$$\|z_{n}-Sz_{n}\|\leq\|z_{n}-x_{n}\|+ \|x_{n}-Sz_{n}\|, $$

we have

$$ \lim_{n\rightarrow\infty}\|z_{n}-Sz_{n} \|=0. $$

(3.18)

From E is uniformly smooth, J is norm-to-norm continuous. So, we have

$$\|Jx_{n}-Jz_{n}\|\rightarrow0 $$

and

$$\|Jz_{n}-JSz_{n}\|\rightarrow0. $$

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:

$$Hv^{*}=\left \{ \begin{array}{l@{\quad}l} Bv^{*}+N_{JC}(v^{*}),& v^{*}\in JC, \\ \emptyset,& v^{*}\notin JC. \end{array} \right . $$

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

$$ \bigl\langle w-Bv^{*}, v^{*}-Jz_{n}\bigr\rangle \geq0. $$

(3.19)

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

$$ \langle w-BJv, Jv-Jz_{n}\rangle\geq0. $$

(3.20)

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

$$ \biggl\langle \frac{u_{n}-z_{n}}{\lambda_{n}}-BJu_{n}, Jv-Jz_{n}\biggr\rangle \leq0. $$

(3.21)

It follows from (3.20) and (3.21) that

$$\begin{aligned} \langle w, Jv-Jz_{n}\rangle \geq& \langle BJv, Jv-Jz_{n} \rangle \\ \geq&\langle BJv, Jv-Jz_{n}\rangle+ \biggl\langle \frac{u_{n}-z_{n}}{\lambda _{n}}-BJu_{n}, Jv-Jz_{n}\biggr\rangle \\ =&\langle BJv-BJu_{n}, Jv-Jz_{n}\rangle+\biggl\langle \frac{u_{n}-z_{n}}{\lambda_{n}}, Jv-Jz_{n}\biggr\rangle \\ =&\langle BJv-BJz_{n}, Jv-Jz_{n}\rangle+\langle BJz_{n}-BJu_{n}, Jv-Jz_{n}\rangle \\ &{}+\biggl\langle \frac{u_{n}-z_{n}}{\lambda_{n}}, Jv-Jz_{n}\biggr\rangle \\ \geq&-\frac{1}{\alpha}\|Jz_{n}-Ju_{n}\| \|Jv-Jz_{n}\|-\frac{1}{a}\|z_{n}-u_{n}\|\| Jv-Jz_{n}\| \\ \geq&-M\biggl(\frac{1}{\alpha}\|Jz_{n}-Ju_{n}\|+ \frac{1}{a}\|z_{n}-u_{n}\|\biggr), \end{aligned}$$

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

$$ \Theta(Ju_{n},Jy) +\frac{1}{r_{n}}\langle u_{n}-x_{n}, Jy-Ju_{n}\rangle\geq0, $$

(3.22)

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

$$ \frac{1}{r_{n_{i}}}\langle u_{n_{i}}-x_{n_{i}}, Jy-Ju_{n_{i}}\rangle\geq-\Theta(Ju_{n_{i}},y)\geq\Theta (Jy,Ju_{n_{i}}),\quad \forall y\in C. $$

(3.23)

From $0<\eta\leq r_{n}$ and (3.17), we have

$$\lim_{n\rightarrow\infty}\frac{u_{n_{i}}-x_{n_{i}}}{r_{n_{i}}}=0. $$

So, we have

$$ \Theta\bigl( Jy, u^{*}\bigr)\leq0. $$

(3.24)

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

$$\begin{aligned} 0 =&\Theta\bigl(u^{*}_{t},u^{*}_{t}\bigr)\leq t\Theta \bigl(u^{*}_{t}, Jy\bigr)+(1-t)\Theta\bigl(u^{*}_{t},u^{*}\bigr) \\ \leq&t\Theta\bigl(u^{*}_{t}, Jy\bigr) \end{aligned}$$

and then

$$\Theta\bigl(u^{*}_{t}, Jy\bigr)\geq0, $$

letting $t\rightarrow0$, from (A3), we have

$$\Theta\bigl(u^{*}, Jy\bigr)\geq0,\quad \forall y\in C. $$

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

$$\begin{aligned} \phi(x_{n}, \xi_{n}) =&\phi(x_{n}, R_{\Gamma}x_{n}) \\ \leq&\phi(x_{n},p)-\phi(R_{\Gamma}x_{n}, p) \\ \leq&\phi(x_{n}, p), \end{aligned}$$

(3.25)

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

$$\begin{aligned} \phi(x_{n+1}, \xi_{n+1}) =&\phi(x_{n+1},R_{\Gamma}x_{n+1}) \\ \leq&\phi(x_{n+1}, \xi_{n})-\phi(R_{\Gamma}x_{n+1}, \xi_{n}) \\ \leq&\phi(x_{n+1}, \xi_{n}) \\ \leq&\beta_{n}\phi(x, \xi_{n})+\phi(y_{n}, \xi_{n}) \\ \leq&\beta_{n}\phi(x, \xi_{n})+\phi(x_{n}, \xi_{n}). \end{aligned}$$

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

$$\phi(\xi_{n+m}, \xi_{n})+\phi(x_{n+m}, \xi_{n+m})\leq\phi(x_{n+m},\xi _{n})\leq \phi(x_{n},\xi_{n}), $$

so

$$\phi(\xi_{n+m}, \xi_{n})\leq\phi(x_{n}, \xi_{n})-\phi(x_{n+m},\xi_{n+m}). $$

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

$$g_{2}\bigl(\Vert \xi_{n+m}-\xi_{n}\Vert \bigr) \leq\phi(\xi_{n+m}, \xi_{n})\leq\phi(x_{n},\xi _{n})-\phi(x_{n+m},\xi_{n+m}). $$

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

$$\langle x_{n}-\xi_{n}, Ju-J\xi_{n}\rangle\leq0. $$

J is weakly sequentially continuous, and we have

$$\langle u-w, Ju-Jw\rangle\leq0. $$

On the other hand, J is monotone, and we have

$$\langle u-w, Ju-Jw\rangle\geq0. $$

So, we get

$$\langle u-w, Ju-Jw\rangle= 0. $$

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

$$\left \{ \begin{array}{l} F(Ju_{n}, Jy)+\frac{1}{r_{n}}\langle u_{n}-x_{n}, Jy-Ju_{n}\rangle\geq0,\quad \forall y\in C, \\ z_{n}= R_{C}(u_{n}-\lambda_{n}BJu_{n}), \\ y_{n}=\alpha_{n}^{(1)} u_{n} +\alpha_{n}^{(2)}Sz_{n}+\alpha_{n}^{(3)}Tz_{n}, \\ x_{n+1}=R_{C}(\beta_{n} x+(1-\beta_{n})y_{n}), \end{array} \right . $$

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

$$\left \{ \begin{array}{l} F(Ju_{n}, Jy)+\frac{1}{r_{n}}\langle u_{n}-x_{n}, Jy-Ju_{n}\rangle\geq0,\quad \forall y\in C, \\ z_{n}= R_{C}(u_{n}-\lambda_{n}BJu_{n}), \\ y_{n}=\alpha_{n}^{(1)} u_{n} +(1-\alpha_{n}^{(1)})Tz_{n}, \\ x_{n+1}=R_{C}(\beta_{n} x+(1-\beta_{n})y_{n}), \end{array} \right . $$

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 $x\in R$,

$$Ax=\frac{x}{2},\qquad Bx=\frac{x-1}{2}, \qquad \psi(x)= \frac{x^{2}}{3}. $$

The classical variational inequality is as follows: finding a $u\in C=[1,100]$ such that

$$ \langle Bu, v-u\rangle\geq0, \quad \forall v\in[1,100]. $$

(4.1)

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

$$Tx=\frac{3x+4}{7},\qquad Sx=\frac{2x+5}{7},\qquad F(x,y)=- \frac{7}{6}y^{2}+\frac{xy}{2}+3x-3y+\frac{2}{3}y^{2} . $$

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

$$\begin{aligned}& F(x,y)+\langle Ax, y-x\rangle+\psi(y)-\psi(x) \\& \quad = -\frac{7}{6}y^{2}+\frac{xy}{2}+3x-3y+ \frac{2}{3}y^{2}+\frac {x}{2}(y-x)+\frac{y^{2}}{3}- \frac{x^{2}}{3} \\& \quad = -2x^{2}+(y+3)x+y^{2}-3y. \end{aligned}$$

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 $y\in[1,100]$,

$$\begin{aligned}& 0 \leq F(u_{n}, y)+\langle Au_{n}, y-u_{n} \rangle+\psi(y)-\psi(u_{n})+\frac {1}{r_{n}}\langle u_{n}-x_{n}, y-u_{n}\rangle \\& \hphantom{0} = \bigl[-2u_{n}^{2}+(y+3)u_{n}+y^{2}-3y \bigr]+\frac{1}{r_{n}}(u_{n}-x_{n}) (y-u_{n}) \\& \quad \Leftrightarrow\quad 0 \leq r_{n}\bigl[-2u_{n}^{2}+(y+3)u_{n}+y^{2}-3y \bigr]+\bigl(u_{n}y-u_{n}^{2}-x_{n}y+x_{n}u_{n} \bigr) \\& \hphantom{\quad \Leftrightarrow\quad0} = r_{n}y^{2}+(r_{n}u_{n}-3r_{n}+u_{n}-x_{n})y+ \bigl(3r_{n}u_{n}-2r_{n}u_{n}^{2}-u_{n}^{2}+x_{n}u_{n} \bigr). \end{aligned}$$

Let $a=r_{n}$, $b=r_{n}u_{n}-3r_{n}+u_{n}-x_{n}$, $c=3r_{n}u_{n}-2r_{n}u_{n}^{2}-u_{n}^{2}+x_{n}u_{n}$,

$$\begin{aligned} \Delta =&b^{2}-4ac \\ =&(r_{n}u_{n}-3r_{n}+u_{n}-x_{n})^{2}-4r_{n} \bigl(3r_{n}u_{n}-2r_{n}u_{n}^{2}-u_{n}^{2}+x_{n}u_{n} \bigr) \\ =&(u_{n}-3r_{n}+3r_{n}u_{n}-x_{n})^{2}. \end{aligned}$$

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

$$ u_{n}=\frac{x_{n}+3r_{n}}{1+3r_{n}}. $$

(4.2)

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: $x_{1}=x\in[1,100]$,

$$\left \{ \begin{array}{l} u_{n}=\frac{x_{n}+3r_{n}}{3r_{n}+1}, \\ z_{n}= P_{[1,100]}(u_{n}-\lambda_{n}\frac{1}{2}Bu_{n}), \\ y_{n}=\alpha_{n}^{(1)} u_{n} +\alpha_{n}^{(2)}Sz_{n}+\alpha_{n}^{(3)}Tz_{n}, \\ x_{n+1}=P_{[1,100]}(\beta_{n} x+(1-\beta_{n})y_{n}). \end{array} \right . $$

Then from Theorem 3.3 we conclude that the sequence $\{x_{n}\}$ converges weakly to 1.