Convergence theorems for split equality mixed equilibrium problems with applications

Abstract

In this paper, we introduce a new algorithm for solving split equality mixed equilibrium problems in the framework of infinite-dimensional real Hilbert spaces. The strong and weak convergence theorems are obtained. As application, we shall utilize our results to study the split equality mixed variational inequality problem and the split equality convex minimization problem. Our results presented in this paper improve and extend some recent corresponding results.

1 Introduction

Let H be a real Hilbert space with the inner product $\langle\cdot ,\cdot\rangle$ and the norm$\|\cdot\|$. Let D be a nonempty closed convex subset of H. Let $T:D\rightarrow D$ be a nonlinear mapping. The fixed point set of T is denoted by $F(T)$, that is, $F(T)=\{x\in D: Tx=x\}$. A mapping T is said to be nonexpansive if

$$ \|Tx-Ty\|\leq\|x-y\|,\quad\forall x,y\in D. $$

(1.1)

If D is a bounded nonempty closed convex subset of H and T is a nonexpansive mapping of D into itself, then $F(T)$ is nonempty [1].

A mapping T is said to be quasi-nonexpansive if $F(T)\neq\emptyset$, and

$$ \|Tx-p\|\leq\|x-p\| \quad\mbox{for each } x\in D \mbox{ and } p \in F(T). $$

(1.2)

For modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2], in 1994 Censor and Elfving [3] firstly introduced the following split feasibility problem (SFP) in finite-dimensional Hilbert spaces:

Let C and Q be nonempty closed convex subsets of the Hilbert spaces $H_{1}$ and $H_{2}$, respectively, let $A:H_{1}\rightarrow H_{2}$ be a bounded linear operator. The split feasibility problem (SFP) is formulated as finding a point $x^{*}$ with the property

$$ x^{*}\in C \quad\mbox{and}\quad Ax^{*}\in Q. $$

(1.3)

It has been found that the $SFP$ can be used in many areas such as image restoration, computer tomograph, and radiation therapy treatment planing [4–6]. Some methods have been proposed to solve split feasibility problems; see, for instance, [2, 7–14].

Assuming that the $SFP$ is consistent (i.e., (1.3) has a solution), it is not hard to see that

$$ x^{*}=P_{C}\bigl(I+\gamma A^{*}(P_{Q}-I)\bigr)Ax^{*},\quad\forall x\in C, $$

(1.4)

where $P_{C}$ and $P_{Q}$ are the (orthogonal) projections onto C and Q, respectively, $\gamma>0$, and $A^{*}$ denotes the adjoint of A. That is, $x^{*}$ solves $SFP$ (1.3) if and only if $x^{*}$ solves fixed point equation (1.4) (see [15]). This implies that $SFP$ can be solved by using fixed point algorithms.

Recently, Moudafi [16] introduced the following new split feasibility problem, which is also called general split equality problem:

Let $H_{1}$, $H_{2}$, $H_{3}$ be real Hilbert spaces, $C\subset H_{1}$, $Q\subset H_{2}$ be two nonempty closed convex sets, $A:H_{1}\rightarrow H_{3}$, $B:H_{2}\rightarrow H_{3}$ be two bounded linear operators. The new split feasibility problem is to

$$ \mbox{find } x^{*}\in C, y^{*}\in Q \mbox{ such that } Ax^{*}=By^{*}. $$

(1.5)

This allows asymmetric and partial relations between the variables x and y.

It is easy to see that problem (1.5) reduces to problem (1.3) as $H_{2}=H_{3}$ and $B=I$ (I stands for the identity mapping from $H_{2}$ to $H_{2}$) in (1.5). Therefore the new split feasibility problem (1.5) proposed by Moudafi is a generalization of split feasibility problem (1.3). The interest of this problem is to cover many situations, for instance, in decomposition methods for $PDE$’s, applications in game theory and in intensity-modulated radiation therapy.

Many authors have proposed some useful methods to solve some kinds of general split feasibility problems and general split equality problems in real Hilbert spaces, and under suitable conditions some strong convergence theorems have been proved; see, for instance, [17–19] and the references therein.

The equilibrium problem (for short, EP) is to find $x^{\ast}\in C$ such that

$$ F\bigl(x^{\ast},y\bigr)\ge0, \quad \forall y \in C. $$

(1.6)

The set of solutions of EP is denoted by $EP(F)$. Given a mapping $T:C\to C$, let $F(x,y)=\langle Tx,y-x\rangle$ for all $x,y \in C$. Then $x^{\ast}\in EP(F)$ if and only if $x^{\ast}\in C$ is a solution of the variational inequality $\langle Tx,y-x\rangle\geq0$ for all $y\in C$, i.e., $x^{*}$ is a solution of the variational inequality.

Let $\phi:C\rightarrow R\cup\{+\infty\}$ be a function. The mixed equilibrium problem (for short, $MEP$) is to find $x^{\ast}\in C$ such that

$$ F\bigl(x^{\ast},y\bigr)+\phi(y)-\phi\bigl(x^{*}\bigr)\geq0, \quad\forall y \in C. $$

(1.7)

The set of solutions of $MEP$ is denoted by $MEP (F,\phi)$.

If $\phi=0$, then the mixed equilibrium problem (1.7) reduces to (1.6).

If $F=0$, then the mixed equilibrium problem (1.7) reduces to the following convex minimization problem:

$$ \mbox{find } x^{\ast}\in C \mbox{ such that } \phi(y)\geq\phi\bigl(x^{*} \bigr),\quad \forall y \in C. $$

(1.8)

The set of solutions of (1.8) is denoted by $CMP (\phi)$.

The mixed equilibrium problem (MEP) includes serval important problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games and others. It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems (e.g., [20–23]).

Recently, Bnouhachem [24] introduced the following split equilibrium problems:

Let $F : C \times C \rightarrow R$ and $G : Q \times Q\rightarrow R$ be nonlinear bifunctions and $A : H_{1}\rightarrow H_{2}$ be a bounded linear operator, then the split equilibrium problem (SEP) is to find $x^{*}\in C$ such that

$$ F\bigl(x^{\ast},x\bigr)\ge0, \quad \forall x \in C, $$

(1.9)

and such that

$$ y^{*}=Ax^{*}\in Q \mbox{ solves } G\bigl(y^{\ast},y\bigr)\ge0, \quad \forall y \in Q. $$

(1.10)

In this paper, we consider the following pair of equilibrium problems called split equality equilibrium problems (SEEP).

Definition 1.1

Let $F : C \times C \rightarrow R$ and $G : Q \times Q\rightarrow R$ be nonlinear bifunctions, let $A : H_{1}\rightarrow H_{3}$ and $B : H_{2}\rightarrow H_{3}$ be two bounded linear operators, then the split equality equilibrium problem (SEEP) is to find $x^{*}\in C$ and $y^{*}\in Q$ such that

$$ F\bigl(x^{\ast},x\bigr)\ge0, \quad\forall x \in C, \qquad G \bigl(y^{\ast},y\bigr)\ge0, \quad \forall y \in Q \quad\mbox{and}\quad Ax^{*}=By^{*}. $$

(1.11)

The set of solutions of (1.11) is denoted by $SEEP (F, G)$.

The split equality mixed equilibrium problem (SEMEP) is defined as follows.

Definition 1.2

Let $F : C \times C \rightarrow R$ and $G : Q \times Q\rightarrow R$ be nonlinear bifunctions, let $\phi:C\rightarrow R\cup\{+\infty\}$ and $\varphi: Q\rightarrow R\cup\{+\infty\}$ be proper lower semi-continuous and convex functions such that $C\cap \operatorname{dom}\phi\neq\emptyset$ and $Q\cap \operatorname{dom}\varphi\neq\emptyset$, and let $A : H_{1}\rightarrow H_{3}$ and $B : H_{2}\rightarrow H_{3}$ be two bounded linear operators, then the split equality mixed equilibrium problem (SEMEP) is to find $x^{*}\in C$ and $y^{*}\in Q$ such that

$$\begin{aligned} &F\bigl(x^{\ast},x\bigr)+\phi(x)-\phi\bigl(x^{*}\bigr)\ge0,\quad \forall x \in C,\qquad G\bigl(y^{\ast},y\bigr)+\varphi(y)-\varphi\bigl(y^{*}\bigr) \ge0, \quad \forall y \in Q \\ &\quad\mbox{and}\quad Ax^{*}=By^{*}. \end{aligned}$$

(1.12)

The set of solutions of (1.12) is denoted by $SEMEP (F, G, \phi, \varphi)$.

Remark 1.3

(1) In (1.12), if $\phi=0$, then the split equality mixed equilibrium problem (1.12) reduces to (1.11).

(2) If $F=0$ and $G=0$, then the split equality mixed equilibrium problem (1.12) reduces to the following split equality convex minimization problem: find $x^{\ast}\in C$ and $y^{*}\in Q$ such that

$$ \phi(x)\geq\phi\bigl(x^{*}\bigr), \quad\forall x \in C, \qquad \varphi(y)\geq \varphi \bigl(y^{*}\bigr), \quad\forall y \in Q \quad\mbox{and}\quad Ax^{*}=By^{*}. $$

(1.13)

The set of solutions of (1.13) is denoted by $SECMP (\phi,\varphi )$.

(3) If $F=0$, $G=0$, $B=I$ and $y^{*}=Ax^{*}$, then the split equality mixed equilibrium problem (1.12) reduces to the following split convex minimization problem: find $x^{\ast}\in C$ such that

$$ \phi(x)\geq\phi\bigl(x^{*}\bigr), \quad\forall x \in C, \quad \mbox{and}\quad y^{*}=Ax^{*}\in Q,\quad \varphi(y)\geq\varphi\bigl(y^{*}\bigr), \quad \forall y \in Q. $$

(1.14)

The set of solutions of (1.14) is denoted by $SCMP (\phi,\varphi )$.

In order to solve the split equality problem (1.5), Moudafi and Al-Shemas [25] presented the following simultaneous iterative method and obtained weak convergence theorem:

$$ (SIM-FPP)\left \{ \begin{array}{@{}l} x_{k+1}= U(x_{k}-\gamma_{k} A^{*}(Ax_{k}-By_{k}));\\ y_{k+1}= T(y_{k}+\gamma_{k} B^{*}(Ax_{k}-By_{k})), \end{array} \right . $$

(1.15)

where $H_{1}$, $H_{2}$, $H_{3}$ are real Hilbert spaces, $U:H_{1}\rightarrow H_{1}$, $T:H_{2}\rightarrow H_{2}$ are two firmly quasi-nonexpansive mappings, $A:H_{1}\rightarrow H_{3}$, $B:H_{2}\rightarrow H_{3}$ are two bounded linear operators, $A^{*}$ and $B^{*}$ are the adjoint of A and B, respectively. Under some suitable conditions, they obtained some weak convergence theorems.

In this paper, motivated by the above works and related literature, we introduce a new algorithm for solving split equality mixed equilibrium problems in the framework of infinite-dimensional real Hilbert spaces. Under suitable conditions some strong and weak convergence theorems are obtained. As application, we shall utilize our results to study the split equality mixed variational inequality problem and the split equality convex minimization problem. Our results presented in this paper improve and extend some recent corresponding results.

2 Preliminaries

Throughout this paper, we denote the strong convergence and weak convergence of a sequence $\{x_{n}\}$ to a point $x\in X$ by $x_{n}\rightarrow x$ and $x_{n}\rightharpoonup x$, respectively.

Let H be a Hilbert space with the inner product $\langle\cdot,\cdot\rangle$ and the norm $\|\cdot\|$, C be a nonempty closed convex subset of H. For every point $x\in H$, there exists a unique nearest point of C, denoted by $P_{C}x$, such that $\|x-P_{C}x\|\leq\|x-y\|$ for all $y\in C$. The mapping $P_{C}$ is called the metric projection from H onto C. It is well known that $P_{C}$ is a firmly nonexpansive mapping from H to C, i.e.,

$$\|P_{C} x-P_{C} y\|^{2}\leq\langle P_{C} x-P_{C} y,x-y\rangle, \quad\forall x,y\in H. $$

Further, for any $x\in H$ and $z\in C$, $z=P_{C} x$ if and only if

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

(2.1)

For solving mixed equilibrium problems, we assume that the bifunction $F:C\times C\rightarrow R$ satisfies the following conditions:

• (A1) $F(x,x)=0$, $\forall x\in C$;

• (A2) $F(x,y)+F(y,x)\leq0$, $\forall x,y\in C$;

• (A3) For all $x,y,z\in C$, $\lim_{t\downarrow0}F(tz+(1-t)x,y)\leq F(x,y)$;

• (A4) For each $x\in C$, the function $y\longmapsto F(x,y)$ is convex and lower semi-continuous;

• (A5) For fixed $r> 0$ and $z \in C$, there exists a bounded subset K of $H_{1}$ and $x\in C\cap K$ such that

  $$F(z,x)+\frac{1}{r}\langle y-x,x-z\rangle \geq0, \quad\forall y \in C \backslash K. $$

Lemma 2.1

([26])

Let C be a nonempty closed convex subset of a Hilbert space H. Let F be a bifunction from $C\times C$ to R satisfying (A1)-(A5), and let $\phi:C\rightarrow R\cup\{ +\infty\}$ be a proper lower semi-continuous and convex function such that $C\cap \operatorname{dom}\phi\neq\emptyset$. For $r>0$ and $x\in H_{1}$, define a mapping $T_{r}^{F}:H_{1}\rightarrow C$ as follows:

$$ T_{r}^{F}(x)=\biggl\{ z\in C:F(z,y)+\phi(y)-\phi(z)+ \frac{1}{r}\langle y-z,z-x\rangle \geq0, \forall y\in C\biggr\} . $$

(2.2)

Then

• (1) For each $x\in H$, $T_{r}^{F}(x)\neq\emptyset$;

• (2) $T_{r}^{F}$ is single-valued;

• (3) $T_{r}^{F}$ is firmly nonexpansive, that is, $\forall x,y\in H_{1}$,

  $$\bigl\| T_{r}^{F}x-T_{r}^{F}y \bigr\| ^{2}\leq\bigl\langle T_{r}^{F}x-T_{r}^{F}y,x-y \bigr\rangle ; $$

• (4) $F(T_{r}^{F})=MEP(F,\phi)$;

• (5) $MEP(F, \phi)$ is closed and convex.

Assume that $G:Q\times Q$ to R satisfying (A1)-(A5), and let $\varphi:Q\rightarrow R\cup\{+\infty\}$ be a proper lower semi-continuous and convex function such that $Q\cap \operatorname{dom}\varphi\neq \emptyset$, and for $s>0$ and $\forall u\in H_{2}$, define a mapping $T_{s}^{G}:H_{2}\rightarrow Q$ as follows:

$$ T_{s}^{G}(u)=\biggl\{ \nu\in Q:G(\nu,w)+\varphi(w)-\varphi( \nu)+\frac{1}{s}\langle w-\nu ,\nu-u\rangle \geq0, \forall w\in Q\biggr\} . $$

(2.3)

Then it follows from Lemma 2.1 that $T_{s}^{G}$ satisfies (1)-(5) of Lemma 2.1, and $F(T_{s}^{G})=MEP(G,\varphi)$.

Definition 2.2

Let H be a Hilbert space.

• (1) A single-value mapping $T:H\rightarrow H$ is said to be demiclosed at origin if, for any sequence $\{x_{n}\}\subset H$ with $x_{n}\rightharpoonup x^{*}$ and $\|x_{n}-Tx_{n}\|\rightarrow0$, we have $x^{*}=Tx^{*}$.

• (2) A single-value mapping $T:H\rightarrow H$ is said to be semi-compact if, for any bounded sequence $\{x_{n}\}\subset H$ with $\| x_{n}-Tx_{n}\|\rightarrow0$, there exists a subsequence $\{x_{n_{i}}\} \subset\{x_{n}\}$ such that $\{x_{n_{i}}\}$ converges strongly to a point $x^{*}\in H$.

Lemma 2.3

([27])

Let C be a nonempty closed convex subset of a Hilbert space and T be a nonexpansive mapping from C into itself. If T has a fixed point, then $I-T$ is demiclosed at origin, where I is the identity mapping of H, that is, whenever $\{x_{n}\}$ is a sequence in C weakly converging to some $x\in C$ and the sequence $\{(I-T)x_{n}\}$ converges strongly to some 0, it follows that $Tx=x$.

Lemma 2.4

([25])

Let H be a Hilbert space and $\{\mu_{n}\}$ be a sequence in H such that there exists a nonempty set $W\subset H$ satisfying:

• (i) For every $\mu^{*} \in W$, $\lim_{n\rightarrow\infty}\|\mu _{n}-\mu^{*}\|$ exists.

• (ii) Any weak-cluster point of the sequence $\{\mu_{n}\}$ belongs to W.

Then there exists $\mu^{*}\in W$ such that $\{\mu_{n}\}$ weakly converges to $\mu^{*}$.

Lemma 2.5

([28])

Let H be a real Hilbert space, then for all $x, y\in H$, we have

$$ \|x-y\|^{2}=\|x\|^{2}-\|y\|^{2}-2\langle x-y,y \rangle . $$

(2.4)

3 Main results

Theorem 3.1

Let $H_{1}$, $H_{2}$, $H_{3}$ be real Hilbert spaces, $C\subseteq H_{1}$ and $Q\subseteq H_{2}$ be nonempty closed convex subsets of Hilbert spaces $H_{1}$ and $H_{2}$, respectively. Assume that $F:C\times C\rightarrow R$ and $G:Q\times Q\rightarrow R$ are bifunctions satisfying (A1)-(A5), and let $\phi:C\rightarrow R\cup\{ +\infty\}$ and $\varphi: Q\rightarrow R\cup\{+\infty\}$ be proper lower semi-continuous and convex functions such that $C\cap \operatorname{dom}\phi\neq \emptyset$ and $Q\cap \operatorname{dom}\varphi\neq\emptyset$. Let $T:H_{1}\rightarrow H_{1}$, $S:H_{2}\rightarrow H_{2}$ be two nonexpansive mappings, and $A:H_{1}\rightarrow H_{3}$, $B:H_{2}\rightarrow H_{3}$ be two bounded linear operators. Let $(x_{1},y_{1})\in C \times Q$ and the iteration scheme $\{ (x_{n}, y_{n})\}$ be defined as follows:

$$ \left \{ \begin{array}{@{}l} {F(u_{n},u)+\phi(u)-\phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n}\rangle \geq0,\quad \forall u\in C};\\ {G(v_{n},v)+\varphi(v)-\varphi(v_{n})+\frac{1}{r_{n}}\langle v-v_{n},v_{n}-y_{n}\rangle \geq0,\quad \forall v\in Q};\\ {x_{n+1}= \alpha_{n} u_{n}+ (1- \alpha_{n})T(u_{n}- \rho_{n} A^{*}(Au_{n}-Bv_{n}))};\\ {y_{n+1}= \alpha_{n}v_{n}+ ( 1- \alpha_{n})S(v_{n}+ \rho_{n} B^{*}(Au_{n}-Bv_{n})), \quad\forall n\geq1}; \end{array} \right . $$

(3.1)

where $\lambda_{A}$ and $\lambda_{B}$ stand for the spectral radii of $A^{*}A$ and $B^{*}B$ respectively, $\{\rho_{n}\}$ is a positive real sequence such that $\rho_{n}\in(\varepsilon,\frac{2}{\lambda_{A}+\lambda _{B}}-\varepsilon)$ (for ε small enough), $\{\alpha_{n}\}$ is a sequence in $(0, 1)$ and $\{r_{n}\}\subset(0,\infty)$ satisfies the following conditions:

• (1) $0<\alpha\leq\alpha_{n}\leq\beta<1$ (for some $\alpha, \beta\in (0,1)$);

• (2) $\liminf_{n\rightarrow\infty}r_{n}>0$ and $\lim_{n\rightarrow\infty }|r_{n+1}-r_{n}|=0$.

If $\Gamma:=F(T)\cap F(S)\cap SEMEP(F,G,\phi,\varphi)\neq\emptyset$, then

• (I) The sequence $\{(x_{n}, y_{n})\}$ converges weakly to a solution of problem (1.12).

• (II) In addition, if S, T are also semi-compact, then $\{(x_{n},y_{n})\}$ converges strongly to a solution of problem (1.12).

Proof

Now we prove conclusion (I).

Taking $(x,y)\in\Gamma$, it follows from Lemma 2.1 that $x=T_{r_{n}}^{F}x$ and $y=T_{r_{n}}^{G}y$, we have

$$\begin{aligned}& \|u_{n}-x\|=\bigl\| T_{r_{n}}^{F}x_{n}-T_{r_{n}}^{F}x \bigr\| \leq\|x_{n}-x\|, \end{aligned}$$

(3.2)

$$\begin{aligned}& \|v_{n}-y\|=\bigl\| T_{r_{n}}^{G}y_{n}-T_{r_{n}}^{G}y \bigr\| \leq\|y_{n}-y\|. \end{aligned}$$

(3.3)

Let $(x,y)\in\Gamma$. Since $\|\cdot\|^{2}$ is convex and S, T are nonexpansive mappings, we have

$$ \begin{aligned}[b] \|x_{n+1}-x\|^{2}={}&\bigl\| \alpha_{n}u_{n}+(1- \alpha_{n})T\bigl(u_{n}-\rho_{n} A^{*}(Au_{n}-Bv_{n})\bigr)-x\bigr\| ^{2} \\ ={}&\alpha_{n}^{2}\|u_{n}-x\|^{2}+(1- \alpha_{n})^{2}\bigl\| T\bigl(u_{n}-\rho_{n} A^{*}(Au_{n}-Bv_{n})\bigr)-x\bigr\| ^{2} \\ &{} +2\alpha_{n}(1-\alpha_{n})\bigl\langle u_{n}-x,T\bigl(u_{n}-\rho_{n} A^{*}(Au_{n}-Bv_{n})\bigr)-x\bigr\rangle \\ \leq{}&\alpha_{n}^{2}\|u_{n}-x \|^{2}+(1-\alpha_{n})^{2}\bigl\| u_{n}- \rho_{n} A^{*}(Au_{n}-Bv_{n})-x\bigr\| ^{2} \\ &{} +2\alpha_{n}(1-\alpha_{n})\bigl\langle u_{n}-x,T\bigl(u_{n}-\rho_{n} A^{*}(Au_{n}-Bv_{n})\bigr)-x\bigr\rangle \\ \leq{}&\alpha_{n}^{2}\|u_{n}-x \|^{2}+(1-\alpha_{n})^{2}\bigl\| u_{n}- \rho_{n} A^{*}(Au_{n}-Bv_{n})-x\bigr\| ^{2} \\ &{} +\alpha_{n}(1-\alpha_{n}) \bigl(\|u_{n}-x \|^{2}+\bigl\| u_{n}-\rho_{n} A^{*}(Au_{n}-Bv_{n})-x\bigr\| ^{2}\bigr) \\ ={}& \alpha_{n}\|u_{n}-x\|^{2}+(1- \alpha_{n})\bigl\| u_{n}-\rho_{n} A^{*}(Au_{n}-Bv_{n})-x \bigr\| ^{2} \\ \leq{}&\alpha_{n}\|u_{n}-x\|^{2}+(1- \alpha_{n}) \bigl(\|u_{n}-x\|^{2}+\bigl\| \rho _{n}A^{*}(Au_{n}-Bv_{n})\bigr\| ^{2} \\ &{} -2\rho_{n} \langle Au_{n}-Ax,Au_{n}-Bv_{n} \rangle \bigr) \\ \leq{}&\|x_{n}-x\|^{2}+(1-\alpha_{n})\bigl\| \rho_{n}A^{*}(Au_{n}-Bv_{n})\bigr\| ^{2} \\ &{} -2(1-\alpha_{n})\rho_{n} \langle Au_{n}-Ax,Au_{n}-Bv_{n} \rangle . \end{aligned} $$

(3.4)

Since

$$\begin{aligned} \bigl\| \rho_{n}A^{*}(Au_{n}-Bv_{n}) \bigr\| ^{2}&=\rho_{n}^{2}\bigl\langle A^{*}(Au_{n}-Bv_{n}),A^{*}(Au_{n}-Bv_{n}) \bigr\rangle \\ &=\rho_{n}^{2}\bigl\langle Au_{n}-Bv_{n},AA^{*}(Au_{n}-Bv_{n}) \bigr\rangle \\ &\leq\lambda_{A}\rho_{n}^{2}\langle Au_{n}-Bv_{n},Au_{n}-Bv_{n}\rangle \\ &=\lambda_{A}\rho_{n}^{2}\|Au_{n}-Bv_{n} \|^{2}. \end{aligned}$$

(3.5)

Combine (3.4) and (3.5), then we have

$$\begin{aligned} \|x_{n+1}-x\|^{2} \leq{}&\|x_{n}-x \|^{2}+(1-\alpha_{n})\lambda_{A} \rho_{n}^{2}\| Au_{n}-Bv_{n} \|^{2} \\ &{}-2(1-\alpha_{n})\rho_{n} \langle Au_{n}-Ax,Au_{n}-Bv_{n} \rangle . \end{aligned}$$

(3.6)

Similarly, from the fourth equality in (3.1), we can get

$$\begin{aligned} \|y_{n+1}-y\|^{2} ={}&\|y_{n}-x \|^{2}+(1-\alpha_{n})\lambda_{B} \rho_{n}^{2}\|Au_{n}-Bv_{n} \|^{2} \\ &{} +2(1-\alpha_{n})\rho_{n} \langle Bv_{n}-By,Au_{n}-Bv_{n} \rangle . \end{aligned}$$

(3.7)

Since $(x,y)\in\Gamma$, so we know that $Ax=By$, and finally we have

$$\begin{aligned} &\|x_{n+1}-x\|^{2}+\|y_{n+1}-y \|^{2} \\ &\quad\leq\|x_{n}-x\|^{2}+\|y_{n}-y \|^{2}-\rho_{n}(1-\alpha_{n}) \bigl(2- \rho_{n}(\lambda _{A}+\lambda_{B})\bigr) \|Au_{n}-Bv_{n}\|^{2}. \end{aligned}$$

(3.8)

Let $\Gamma_{n}(x,y):=\|x_{n}-x\|^{2}+\|y_{n}-y\|^{2}$, then we have

$$ \Gamma_{n+1}(x,y)\leq\Gamma_{n}(x,y)-\rho_{n}(1- \alpha_{n}) \bigl(2-\rho_{n}(\lambda _{A}+ \lambda_{B})\bigr)\|Au_{n}-Bv_{n} \|^{2}. $$

(3.9)

Obviously the sequence $\{\Gamma_{n}(x,y)\}$ is decreasing and is lower bounded by 0, so it converges to some finite limit, say $\omega(x,y)$. This means that the first condition of Lemma 2.4 (Opial’s lemma) is satisfied with $\mu_{n}=(x_{n},y_{n})$, $\mu^{*}=(x,y)$ and $W=\Gamma$. And by passing to limit in (3.9), we obtain that

$$ \lim_{n\rightarrow\infty}\|Au_{n}-Bv_{n} \|=0. $$

(3.10)

Since $\|x_{n}-x\|^{2}\leq\Gamma_{n}(x,y)$, $\|y_{n}-y\|^{2}\leq\Gamma_{n}(x,y)$ and $\lim_{n\rightarrow\infty}\Gamma_{n}(x,y)$ exists, we know that $\{ x_{n}\}$ and $\{y_{n}\}$ are bounded, and $\limsup_{n\rightarrow\infty}\| x_{n}-x\|$ and $\limsup_{n\rightarrow\infty}\|y_{n}-y\|$ exist. From (3.2) and (3.3), we have $\limsup_{n\rightarrow\infty}\|u_{n}-x\|$ and $\limsup_{n\rightarrow\infty}\|v_{n}-y\|$ also exist. Let $x^{*}$ and $y^{*}$ be respectively weak cluster points of the sequences $\{x_{n}\}$ and $\{y_{n}\}$. From Lemma 2.5, we have

$$\begin{aligned} \|x_{n+1}-x_{n}\|^{2}&=\|x_{n+1}-x-x_{n}+x \|^{2} \\ &=\|x_{n+1}-x\|^{2}-\|x_{n}-x\|^{2}-2 \langle x_{n+1}-x_{n},x_{n}-x\rangle \\ &=\|x_{n+1}-x\|^{2}-\|x_{n}-x\|^{2}-2 \bigl\langle x_{n+1}-x^{*},x_{n}-x\bigr\rangle +2\bigl\langle x_{n}-x^{*},x_{n}-x\bigr\rangle . \end{aligned}$$

So

$$ \limsup_{n\rightarrow\infty}\|x_{n+1}-x_{n} \|=0. $$

(3.11)

Similarly, we have

$$ \limsup_{n\rightarrow\infty}\|y_{n+1}-y_{n} \|=0. $$

(3.12)

This implies that

$$ \lim_{n\rightarrow\infty}\|x_{n+1}-x_{n} \|=0, $$

(3.13)

and

$$ \lim_{n\rightarrow\infty}\|y_{n+1}-y_{n} \|=0. $$

(3.14)

It follows from Lemma 2.1 that $u_{n}=T_{r_{n}}^{F}x_{n}$ and $u_{n+1}=T_{r_{n+1}}^{F}x_{n+1}$, we have

$$F(u_{n+1},u)+\phi(u)-\phi(u_{n+1})+\frac {1}{r_{n+1}}\langle u-u_{n+1},u_{n+1}-x_{n+1}\rangle \geq0, \quad \forall u\in C, $$

and

$$F(u_{n},u)+\phi(u)-\phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n}\rangle \geq0,\quad \forall u \in C. $$

Particularly, we have

$$ F(u_{n+1},u_{n})+\phi(u_{n})- \phi(u_{n+1})+\frac {1}{r_{n+1}}\langle u_{n}-u_{n+1},u_{n+1}-x_{n+1} \rangle \geq0 $$

(3.15)

and

$$ F(u_{n},u_{n+1})+\phi(u_{n+1})- \phi(u_{n})+\frac {1}{r_{n}}\langle u_{n+1}-u_{n},u_{n}-x_{n} \rangle \geq0. $$

(3.16)

Summing up (3.15) and (3.16) and using (A2), we obtain

$$\frac{1}{r_{n+1}}\langle u_{n}-u_{n+1},u_{n+1}-x_{n+1} \rangle +\frac{1}{r_{n}}\langle u_{n+1}-u_{n},u_{n}-x_{n} \rangle \geq0, $$

thus

$$\biggl\langle u_{n+1}-u_{n},\frac{u_{n}-x_{n}}{r_{n}}- \frac{u_{n}-x_{n+1}}{r_{n+1}}\biggr\rangle \geq0, $$

which implies that

$$\begin{aligned} 0&\leq\biggl\langle u_{n+1}-u_{n},u_{n}-x_{n}- \frac {r_{n}}{r_{n+1}}(u_{n+1}-x_{n+1})\biggr\rangle \\ &=\biggl\langle u_{n+1}-u_{n},u_{n}-u_{n+1}+u_{n+1}-x_{n}- \frac {r_{n}}{r_{n+1}}(u_{n+1}-x_{n+1})\biggr\rangle . \end{aligned}$$

Therefore,

$$\begin{aligned} \|u_{n+1}-u_{n}\|^{2}&\leq\biggl\langle u_{n+1}-u_{n},x_{n+1}-x_{n}+\biggl(1- \frac {r_{n}}{r_{n+1}}\biggr) (u_{n+1}-x_{n+1})\biggr\rangle \\ &\leq\|u_{n+1}-u_{n}\|\cdot\biggl[\|x_{n+1}-x_{n} \|+\biggl|1-\frac{r_{n}}{r_{n+1}}\biggr|\cdot\| u_{n+1}-x_{n+1}\|\biggr]. \end{aligned}$$

Thus, we have

$$ \|u_{n+1}-u_{n}\|\leq\|x_{n+1}-x_{n} \|+\biggl|1-\frac{r_{n}}{r_{n+1}}\biggr|\cdot\| u_{n+1}-x_{n+1} \|. $$

(3.17)

Since $\lim_{n\rightarrow\infty}|r_{n+1}-r_{n}|=0$, $\{u_{n}\}$ and $\{ x_{n}\}$ are bounded, from (3.13) we have

$$ \lim_{n\rightarrow\infty}\|u_{n+1}-u_{n} \|=0. $$

(3.18)

Using the same argument as the proof of the above, we have

$$ \lim_{n\rightarrow\infty}\|v_{n+1}-v_{n} \|=0. $$

(3.19)

It follows from (3.6) and (3.7) that

$$\begin{aligned} \|x_{n+1}-x\|^{2} \leq{}&\|u_{n}-x \|^{2}+(1-\alpha_{n})\lambda_{A} \rho_{n}^{2}\| Au_{n}-Bv_{n} \|^{2} \\ &{} -2(1-\alpha_{n})\rho_{n} \langle Au_{n}-Ax,Au_{n}-Bv_{n} \rangle \end{aligned}$$

(3.20)

and

$$ \begin{aligned}[b] \|y_{n+1}-y\|^{2} ={}&\|v_{n}-x \|^{2}+(1-\alpha_{n})\lambda_{B} \rho_{n}^{2}\|Au_{n}-Bv_{n} \|^{2}\\ &{} +2(1-\alpha_{n})\rho_{n} \langle Bv_{n}-By,Au_{n}-Bv_{n} \rangle . \end{aligned} $$

(3.21)

By adding the last two inequalities and by taking into account the fact that $Ax=By$, we have

$$\begin{aligned} &\|x_{n+1}-x\|^{2}+\|y_{n+1}-y \|^{2} \\ &\quad\leq\|u_{n}-x\|^{2}+\|v_{n}-y \|^{2}-\rho_{n}(1-\alpha_{n}) \bigl(2- \rho_{n}(\lambda _{A}+\lambda_{B})\bigr) \|Au_{n}-Bv_{n}\|^{2}, \end{aligned}$$

(3.22)

where

$$\begin{aligned} \|u_{n}-x\|^{2}&=\bigl\| T_{r_{n}}^{F}x_{n}-T_{r_{n}}^{F}x \bigr\| ^{2} \\ &\leq\langle x_{n}-x,u_{n}-x\rangle \\ &=\frac{1}{2}\bigl(\|x_{n}-x\|^{2}+ \|u_{n}-x\|^{2}-\|x_{n}-u_{n} \|^{2}\bigr), \end{aligned}$$

(3.23)

and

$$\begin{aligned} \|v_{n}-y\|^{2}&=\bigl\| T_{r_{n}}^{G}y_{n}-T_{r_{n}}^{G}y \bigr\| ^{2} \\ &\leq\langle y_{n}-y,v_{n}-y\rangle \\ &=\frac{1}{2}\bigl(\|y_{n}-y\|^{2}+ \|v_{n}-x\|^{2}-\|y_{n}-v_{n} \|^{2}\bigr). \end{aligned}$$

(3.24)

It follows from (3.22), (3.23) and (3.24) that

$$\begin{aligned} &\|x_{n}-u_{n}\|^{2}+ \|y_{n}-v_{n}\|^{2} \\ &\quad\leq\|x_{n}-x\|^{2}-\|x_{n+1}-x \|^{2}+\|y_{n}-y\|^{2}-\|y_{n+1}-y \|^{2} \\ &\qquad{} -\rho_{n}(1-\alpha_{n}) \bigl(2- \rho_{n}(\lambda_{A}+\lambda_{B})\bigr) \|Au_{n}-Bv_{n}\|^{2}. \end{aligned}$$

(3.25)

By (3.13) and (3.14), we obtain

$$\begin{aligned}& \lim_{n\rightarrow\infty}\|x_{n}-u_{n} \|=0, \end{aligned}$$

(3.26)

$$\begin{aligned}& \lim_{n\rightarrow\infty}\|y_{n}-v_{n} \|=0. \end{aligned}$$

(3.27)

It follows from (3.26) and (3.27) that $u_{n}\rightharpoonup x^{*}$ and $v_{n}\rightharpoonup y^{*}$, respectively.

Since T and S are nonexpansive mappings, so

$$\begin{aligned}[b] \|u_{n}-Tu_{n}\|={}&\|u_{n}-x_{n+1}+x_{n+1}-Tu_{n} \| \\ \leq{}&\|u_{n}-x_{n+1}\|+\|x_{n+1}-Tu_{n} \| \\ ={}&\|u_{n}-u_{n+1}-u_{n+1}-x_{n+1}\| \\ &{} +\bigl\| \alpha_{n} u_{n}+ (1- \alpha_{n})T \bigl(u_{n}- \rho_{n} A^{*}(Au_{n}-Bv_{n}) \bigr)-Tu_{n}\bigr\| \\ \leq{}&\|u_{n}-u_{n+1}\|+\|u_{n+1}-x_{n+1} \|+\alpha_{n} \|u_{n}-Tu_{n}\| \\ &{} + (1- \alpha_{n})\bigl\| T\bigl(u_{n}- \rho_{n} A^{*}(Au_{n}-Bv_{n})\bigr)-Tu_{n}\bigr\| \\ \leq{}&\|u_{n}-u_{n+1}\|+\|u_{n+1}-x_{n+1} \|+\alpha_{n} \|u_{n}-Tu_{n}\| \\ &{} + (1- \alpha_{n})\bigl\| -\rho_{n} A^{*}(Au_{n}-Bv_{n})\bigr\| . \end{aligned} $$

That is,

$$ (1-\alpha_{n})\|u_{n}-Tu_{n}\| \leq \|u_{n}-u_{n+1}\|+\|u_{n+1}-x_{n+1}\|+ (1- \alpha_{n})\bigl\| -\rho_{n} A^{*}(Au_{n}-Bv_{n}) \bigr\| . $$

(3.28)

By (3.10), (3.18) and (3.26), we get

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

(3.29)

Similarly,

$$ \lim_{n\rightarrow\infty}\|Sv_{n}-v_{n} \|=0. $$

(3.30)

Since

$$\begin{aligned} \|x_{n}-Tx_{n}\|&\leq\|x_{n}-u_{n}+u_{n}-Tu_{n}+Tu_{n}-Tx_{n} \| \\ &\leq\|x_{n}-u_{n}\|+\|u_{n}-Tu_{n} \|+\|Tu_{n}-Tx_{n}\| \\ &\leq2\|x_{n}-u_{n}\|+\|u_{n}-Tu_{n} \|. \end{aligned}$$

(3.31)

It follows from (3.26) and (3.29) that

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

(3.32)

In addition, since

$$\begin{aligned} \|y_{n}-Sy_{n}\|\leq\|y_{n}-v_{n} \|+\|v_{n}-Sv_{n}\|+\|Sv_{n}-Sy_{n}\| \leq2\|y_{n}-v_{n}\|+\|v_{n}-Sv_{n} \|, \end{aligned}$$

(3.33)

then, from (3.27) and (3.30), we have

$$ \lim_{n\rightarrow\infty}\|y_{n}-Sy_{n} \|=0. $$

(3.34)

Since $\{x_{n}\}$ and $\{y_{n}\}$ converge weakly to $x^{*}$ and $y^{*}$, respectively, then it follows from (3.32), (3.34) and Lemma 2.3 that $x^{*}\in F(T)$ and $y^{*}\in F(S)$. Since every Hilbert space satisfies Opial’s condition, Opial’s condition guarantees that the weakly subsequential limit of $\{(x_{n}, y_{n})\}$ is unique.

We now prove $x^{*}\in MEP(F,\phi)$ and $y^{*}\in MEP(G,\varphi)$.

Since $u_{n}=T_{r_{n}}^{F}x_{n}$, we have

$$ F(u_{n},u)+\phi(u)-\phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n}\rangle \geq0,\quad \forall u \in C. $$

(3.35)

From (A2) we obtain

$$ \phi(u)-\phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n} \rangle \geq-F(u_{n},u)\geq F(u,u_{n}), \quad\forall u\in C. $$

(3.36)

And hence

$$ \phi(u)-\phi(u_{n_{j}})+\frac{1}{r_{n_{j}}}\langle u-u_{n_{j}},u_{n_{j}}-x_{n_{j}} \rangle \geq F(u,u_{n_{j}}), \quad\forall u\in C. $$

(3.37)

From (3.26) we obtain $u_{n_{j}}\rightharpoonup x^{*}$. It follows from (A4) that $\lim_{j\rightarrow\infty}\frac{\|u_{n_{j}}-x_{n_{j}}\|}{r_{n_{j}}}=0$, and from the proper lower semicontinuity of ϕ that

$$ F\bigl(u,x^{*}\bigr)+\phi\bigl(x^{*}\bigr)-\phi(u)\leq0,\quad \forall u\in C. $$

(3.38)

Put $z_{t}=tu+(1-t)x^{*}$ for all $t\in(0,1]$ and $u\in C$. Consequently, we get $z_{t}\in C$ and hence $F(z_{t},x^{*})+\phi (x^{*})-\phi(z_{t})\leq0$. So from (A1) and (A4) we have

$$\begin{aligned} 0&=F(z_{t},z_{t})-\phi(z_{t})+ \phi(z_{t}) \\ &\leq t F(z_{t},u)+(1-t)G\bigl(z_{t},x^{*}\bigr)+t \phi(u)+(1-t)\phi\bigl(x^{*}\bigr)-\phi(z_{t}) \\ &\leq t\bigl[F(z_{t},x)+\phi(u)-\phi(z_{t})\bigr]. \end{aligned}$$

(3.39)

Hence, we have

$$ F(z_{t},u)+\phi(u)-\phi(z_{t})\geq0, \quad\forall u\in C. $$

(3.40)

Letting $t\rightarrow0$, from (A4) and the proper lower semicontinuity of ϕ, we have

$$ F\bigl(x^{*},u\bigr)+\phi(u)-\phi\bigl(x^{*}\bigr)\geq0,\quad \forall u\in C. $$

(3.41)

This implies that $x^{*}\in MEP(F,\phi)$.

Following a similar argument as the proof of the above, we have $y^{*}\in MEP(G,\varphi)$.

On the other hand, since the squared norm is weakly lower semicontinuous, we have

$$\bigl\| Ax^{*}-By^{*}\bigr\| ^{2}\leq \liminf_{n\rightarrow\infty}\|Au_{n}-Bv_{n} \|^{2}=0, $$

therefore $Ax^{*}=By^{*}$. This implies that $(x^{*},y^{*})\in SEMEP(F,G,\phi ,\varphi)$. Therefore, $(x^{*},y^{*})\in\Gamma$. Thus from Lemma 2.4 we know that $\{(x_{n},y_{n})\}$ converges weakly to $(x^{*},y^{*})$. The proof of conclusion (I) is completed.

Next, we prove conclusion (II).

Since T and S are semi-compact, $\{x_{n}\}$ and $\{y_{n}\}$ are bounded and $\lim_{n\rightarrow\infty}\|x_{n}-Tx_{n}\|=0$, $\lim_{n\rightarrow\infty }\|y_{n}-Sy_{n}\|=0$, then there exist subsequences $\{x_{n_{j}}\}$ and $\{ y_{n_{j}}\}$ of $\{x_{n}\}$ and $\{y_{n}\}$ such that $\{x_{n_{j}}\}$ and $\{ y_{n_{j}}\}$ converge strongly to $u^{*}$ and $v^{*}$ (some point in $H_{1}$ and $H_{2}$, respectively), respectively. Since $\{x_{n_{j}}\}$ and $\{ y_{n_{j}}\}$ converge weakly to $x^{*}$ and $y^{*}$, respectively, this implies that $x^{*}=u^{*}$ and $y^{*}=v^{*}$. From Lemma 2.3, we have $x^{*}\in F(T)$ and $y^{*}\in F(S)$. Using the same argument as in the proof in conclusion (I), we have $x^{*}\in MEP(F,\phi)$ and $y^{*}\in MEP(G,\varphi)$. Further, since the norm is weakly lower semicontinuous and $Au_{n_{j}}-Bv_{n_{j}}\rightarrow Ax^{*}-By^{*}$, we have

$$\bigl\| Ax^{*}-By^{*}\bigr\| ^{2}\leq \liminf_{j\rightarrow\infty}\|Au_{n_{j}}-Bv_{n_{j}} \|^{2}=0, $$

so $Ax^{*}=By^{*}$. This implies that $(x^{*},y^{*})\in\Gamma$.

On the other hand, since $\Gamma_{n}(x,y)=\|x_{n}-x\|^{2}+\|y_{n}-y\|^{2}$ for any $(x,y)\in\Gamma$, we know that $\lim_{j\rightarrow\infty}\Gamma _{n_{j}}(x^{*},y^{*})=0$. From conclusion (I), we have $\lim_{n\rightarrow\infty }\Gamma_{n}(x^{*},y^{*})$ exists, therefore $\lim_{n\rightarrow\infty}\Gamma _{n}(x^{*},y^{*})=0$. Further, we can obtain that $\lim_{n\rightarrow\infty}\| x_{n}-x^{*}\|=0$ and $\lim_{n\rightarrow\infty}\|y_{n}-y^{*}\|=0$. This completes the proof of conclusion (II). □

Taking $\phi= 0$ and $\varphi=0$ in Theorem 3.1, we also have the following result.

Corollary 3.2

Let $H_{1}$, $H_{2}$, $H_{3}$ be real Hilbert spaces, $C\subseteq H_{1}$ and $Q\subseteq H_{2}$ be nonempty closed convex subsets of Hilbert spaces $H_{1}$ and $H_{2}$, respectively. Assume that $F:C\times C\rightarrow R$ and $G:Q\times Q\rightarrow R$ are bifunctions satisfying (A1)-(A4). Let $T:H_{1}\rightarrow H_{1}$, $S:H_{2}\rightarrow H_{2}$ be two nonexpansive mappings, and $A:H_{1}\rightarrow H_{3}$, $B:H_{2}\rightarrow H_{3}$ be two bounded linear operators. Let $(x_{1},y_{1})\in C \times Q$ and the iteration scheme $\{ (x_{n}, y_{n})\}$ be defined as follows:

$$\left \{ \begin{array}{@{}l} F(u_{n},u)+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n}\rangle \geq0, \quad\forall u\in C;\\ G(v_{n},v)+\frac{1}{r_{n}}\langle v-v_{n},v_{n}-y_{n}\rangle \geq0, \quad \forall v\in Q;\\ {x_{n+1}= \alpha_{n} u_{n}+ (1- \alpha_{n})T(u_{n}- \rho_{n} A^{*}(Au_{n}-Bv_{n}))};\\ y_{n+1}= \alpha_{n}v_{n}+ ( 1- \alpha_{n})S(v_{n}+ \rho_{n} B^{*}(Au_{n}-Bv_{n})), \quad\forall n\geq1; \end{array} \right . $$

where $\lambda_{A}$ and $\lambda_{B}$ stand for the spectral radii of $A^{*}A$ and $B^{*}B$, respectively, $\{\rho_{n}\}$ is a positive real sequence such that $\rho_{n}\in(\varepsilon,\frac{2}{\lambda_{A}+\lambda _{B}}-\varepsilon)$ (for ε small enough), $\{\alpha_{n}\}$ is a sequence in $(0, 1)$ and $\{r_{n}\}\subset(0,\infty)$ satisfies the following conditions:

• (1) $0<\alpha\leq\alpha_{n}\leq\beta<1$ (for some $\alpha, \beta\in (0,1)$);

• (2) $\liminf_{n\rightarrow\infty}r_{n}>0$ and $\lim_{n\rightarrow\infty }|r_{n+1}-r_{n}|=0$.

If $\Gamma:=F(T)\cap F(S)\cap SEEP(F,G,\phi,\varphi)\neq\emptyset$, then

• (I) The sequence $\{(x_{n}, y_{n})\}$ converges weakly to a solution of problem (1.11).

• (II) In addition, if S, T are also semi-compact, then $\{ (x_{n},y_{n})\}$ converges strongly to a solution of problem (1.11).

In Theorem 3.1 taking $B=I$ and $H_{2}=H_{3}$, from Theorem 3.1 we can obtain the following convergence theorem for general split equilibrium problem (1.10)

Corollary 3.3

Let $H_{1}$ and $H_{2}$ be real Hilbert spaces, $C\subseteq H_{1}$ and $Q\subseteq H_{2}$ be nonempty closed convex subsets of Hilbert spaces $H_{1}$ and $H_{2}$, respectively. Assume that $F:C\times C\rightarrow R$ and $G:Q\times Q\rightarrow R$ are bifunctions satisfying (A1)-(A4), and let $\phi:C\rightarrow R\cup\{ +\infty\}$, $\varphi: Q\rightarrow R\cup\{+\infty\}$ be proper lower semi-continuous and convex functions such that $C\cap \operatorname{dom}\phi\neq \emptyset$ and $Q\cap \operatorname{dom}\varphi\neq\emptyset$. Let $T:H_{1}\rightarrow H_{1}$, $S:H_{2}\rightarrow H_{2}$ be two nonexpansive mappings, and $A:H_{1}\rightarrow H_{2}$ be a bounded linear operator. Let $(x_{1},y_{1})\in C \times Q$ and the iteration scheme $\{(x_{n}, y_{n})\}$ be defined as follows:

$$\left \{ \begin{array}{@{}l} F(u_{n},u)+\phi(u)-\phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n}\rangle \geq0, \quad\forall u\in C;\\ G(v_{n},v)+\varphi(v)-\varphi(v_{n})+\frac{1}{r_{n}}\langle v-v_{n},v_{n}-y_{n}\rangle \geq0,\quad \forall v\in Q;\\ {x_{n+1}= \alpha_{n} u_{n}+ (1- \alpha_{n})T(u_{n}- \rho_{n} A^{*}(Au_{n}-v_{n}))};\\ y_{n+1}= \alpha_{n}v_{n}+ ( 1- \alpha_{n})S(v_{n}+ \rho_{n} (Au_{n}-v_{n})),\quad \forall n\geq1; \end{array} \right . $$

where $\lambda_{A}$ stands for the spectral radius of $A^{*}A$, $\{\rho_{n}\}$ is a positive real sequence such that $\rho_{n}\in(\varepsilon,\frac {1}{\lambda_{A}}-\varepsilon)$ (for ε small enough), $\{\alpha _{n}\}$ is a sequence in $(0, 1)$ and $\{r_{n}\}\subset(0,\infty)$ satisfies the following conditions:

• (1) $0<\alpha\leq\alpha_{n}\leq\beta<1$ (for some $\alpha, \beta\in (0,1)$);

• (2) $\liminf_{n\rightarrow\infty}r_{n}>0$ and $\lim_{n\rightarrow\infty }|r_{n+1}-r_{n}|=0$.

If $\Gamma:=F(T)\cap F(S)\cap SMEP(F,G,\phi,\varphi)\neq\emptyset$, then

• (I) The sequence $\{(x_{n}, y_{n})\}$ converges weakly to a solution of problem (1.10).

• (II) In addition, if S, T are also semi-compact, then $\{ (x_{n},y_{n})\}$ converges strongly to a solution of problem (1.10).

4 Applications

4.1 Application to the split equality mixed variational inequality problem

The variational inequality problem (VIP) is formulated as the problem of finding a point $x^{*}$ with property $x^{*}\in C$, $\langle Ax^{*},z-x^{*}\rangle\geq0$, $\forall z\in C$. We will denote the solution set of VIP by $VI(A, C)$.

In [29], the mixed variational inequality of Browder type (VI) is shown to be equivalent to finding a point $u\in C$ such that

$$\langle Au,y-u \rangle +\varphi(y)-\varphi(u)\geq0, \quad \forall y\in C. $$

We will denote the solution set of a mixed variational inequality of Browder type by $VI(A, C, \varphi)$.

A mapping $A:C\rightarrow H$ is said to be an α-inverse-strongly monotone mapping if there exists a constant $\alpha>0$ such that $\langle Ax-Ay,x-y\rangle\geq\alpha\|Ax-Ay\|^{2}$ for any $x,y\in C$. Setting $F(x,y)=\langle Ax,y-x\rangle$, it is easy to show that F satisfies conditions (A1)-(A4) as A is an α-inverse-strongly monotone mapping.

In 2012, Censor et al. [30] introduced the split variational inequality problem (SVIP) which is formulated as follows:

$$\begin{aligned} & \mbox{find a point } x^{*}\in C \mbox{ such that }\bigl\langle f \bigl(x^{*}\bigr),x-x^{*}\bigr\rangle \geq0 \quad\mbox{for all } x\in C, \\ &\mbox{and such that} \\ &y^{*}=Ax^{*}\in Q \mbox{ solves } \bigl\langle g\bigl(y^{*}\bigr),y-y^{*}\bigr\rangle \geq0\quad \mbox{for all } y\in Q. \end{aligned}$$

(4.1)

The so-called split equality mixed variational inequality problem is shown to be equivalent to finding $x^{*}\in C$, $y^{*}\in Q$ such that

$$\begin{aligned} &\bigl\langle B_{1}\bigl(x^{*}\bigr),x-x^{*}\bigr\rangle + \phi(x)-\phi\bigl(x^{*}\bigr)\geq0 \quad \mbox{for all } x\in C,\quad\mbox{and} \\ &\bigl\langle B_{2}\bigl(y^{*}\bigr),y-y^{*}\bigr\rangle +\varphi(y)- \varphi\bigl(y^{*}\bigr)\geq0 \quad \mbox{for all } y\in Q, \\ &\mbox{and such that} \\ &Ax^{*}=By^{*}. \end{aligned}$$

(4.2)

We will denote the solution set of a split equality mixed variational inequality problem by $SEMVIP(\phi,\varphi)$.

The so-called split mixed variational inequality problem is shown to be equivalent to

$$\begin{aligned} & \mbox{finding a point } x^{*}\in C \mbox{ such that }\bigl\langle B_{1}\bigl(x^{*}\bigr),x-x^{*}\bigr\rangle +\phi (x)-\phi\bigl(x^{*}\bigr) \geq0 \quad \mbox{for all } x\in C, \\ &\mbox{and such that} \\ &y^{*}=Ax^{*}\in Q \mbox{ solves }\bigl\langle B_{2}\bigl(y^{*}\bigr),y-y^{*} \bigr\rangle +\varphi(y)-\varphi\bigl(y^{*}\bigr)\geq0\quad \mbox{for all } y\in Q. \end{aligned}$$

(4.3)

The set of solutions of a split mixed variational inequality problem is denoted by $SMVIP(\phi,\varphi)$.

Setting $F(x,y)=\langle B_{1}x,y-x\rangle$ and $G(x,y)=\langle B_{2}x,y-x\rangle$, it is easy to show that F and G satisfy conditions (A1)-(A5) as $B_{i}$ ($i=1,2$) is an $\eta_{i}$-inverse-strongly monotone mapping. Then it follows from Theorem 3.1 that the following result holds.

Theorem 4.1

Let $H_{1}$, $H_{2}$, $H_{3}$ be real Hilbert spaces, $C\subseteq H_{1}$ and $Q\subseteq H_{2}$ be nonempty closed convex subsets of Hilbert spaces $H_{1}$ and $H_{2}$, respectively. Let $B_{i}$ ($i=1,2$) be $\eta_{i}$-inverse strongly monotone mappings, and let $\phi :C\rightarrow R\cup\{+\infty\}$ and $\varphi: Q\rightarrow R\cup\{ +\infty\}$ be proper lower semi-continuous and convex functions such that $C\cap \operatorname{dom}\phi\neq\emptyset$ and $Q\cap \operatorname{dom}\varphi\neq\emptyset$. Let $T:H_{1}\rightarrow H_{1}$, $S:H_{2}\rightarrow H_{2}$ be two nonexpansive mappings, and $A:H_{1}\rightarrow H_{3}$, $B:H_{2}\rightarrow H_{3}$ be two bounded linear operators. Assume that $(x_{1},y_{1})\in C \times Q$ and the iteration scheme $\{(x_{n}, y_{n})\}$ is defined as follows:

$$\left \{ \begin{array}{@{}l} \langle B_{1}u_{n},u-u_{n}\rangle +\phi(u)-\phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n}\rangle \geq0, \quad\forall u\in C;\\ {\langle B_{2}(v_{n}),v-v_{n}\rangle +\varphi(v)-\varphi(v_{n})+\frac {1}{r_{n}}\langle v-v_{n},v_{n}-y_{n}\rangle \geq0, \quad\forall v\in Q};\\ {x_{n+1}= \alpha_{n} u_{n}+ (1- \alpha_{n})T(u_{n}- \rho_{n} A^{*}(Au_{n}-Bv_{n}))};\\ y_{n+1}= \alpha_{n}v_{n}+ ( 1- \alpha_{n})S(v_{n}+ \rho_{n} B^{*}(Au_{n}-Bv_{n})), \quad\forall n\geq1; \end{array} \right . $$

where $\lambda_{A}$ and $\lambda_{B}$ stand for the spectral radii of $A^{*}A$ and $B^{*}B$, respectively, $\{\rho_{n}\}$ is a positive real sequence such that $\rho_{n}\in(\varepsilon,\frac{2}{\lambda_{A}+\lambda _{B}}-\varepsilon)$ (for ε small enough), $\eta_{i}>0$ ($i=1,2$), $\{\alpha_{n}\}$ is a sequence in $(0, 1)$ and $\{r_{n}\}\subset(0,\infty)$ satisfies the following conditions:

• (1) $0<\alpha\leq\alpha_{n}\leq\beta<1$ (for some $\alpha, \beta\in (0,1)$);

• (2) $\liminf_{n\rightarrow\infty}r_{n}>0$ and $\lim_{n\rightarrow\infty }|r_{n+1}-r_{n}|=0$.

If $\Gamma:=F(T)\cap F(S)\cap SEMVIP(\phi,\varphi)\neq\emptyset$, then

• (I) The sequence $\{(x_{n}, y_{n})\}$ converges weakly to a solution of the split equality mixed variational inequality problem (4.2).

• (II) In addition, if S, T are also semi-compact, then $\{ (x_{n},y_{n})\}$ converges strongly to a solution of the split equality mixed variational inequality problem (4.2).

In Theorem 4.1 taking $B=I$ and $H_{2}=H_{3}$, from Theorem 4.1 we can obtain the following convergence theorem for split mixed variational inequality problem $SMVIP(\phi,\varphi)$.

Corollary 4.2

Let $H_{1}$ and $H_{2}$ be real Hilbert spaces, $C\subseteq H_{1}$ and $Q\subseteq H_{2}$ be nonempty closed convex subsets of Hilbert spaces $H_{1}$ and $H_{2}$, respectively. Let $B_{i}$ ($i=1,2$) be $\eta_{i}$-inverse strongly monotone mappings, and let $\phi :C\rightarrow R\cup\{+\infty\}$ and $\varphi: Q\rightarrow R\cup\{ +\infty\}$ be proper lower semi-continuous and convex functions such that $C\cap \operatorname{dom}\phi\neq\emptyset$ and $Q\cap \operatorname{dom}\varphi\neq\emptyset$. Let $T:H_{1}\rightarrow H_{1}$, $S:H_{2}\rightarrow H_{2}$ be two nonexpansive mappings, and $A:H_{1}\rightarrow H_{2}$ be a bounded linear operator. Assume that $(x_{1},y_{1})\in C \times Q$ and the iteration scheme $\{(x_{n}, y_{n})\}$ is defined as follows:

$$\left \{ \begin{array}{@{}l} \langle B_{1}u_{n},u-u_{n}\rangle +\phi(u)-\phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n}\rangle \geq0, \quad\forall u\in C;\\ {\langle B_{2}(v_{n}),v-v_{n}\rangle +\varphi(v)-\varphi(v_{n})+\frac {1}{r_{n}}\langle v-v_{n},v_{n}-y_{n}\rangle \geq0,\quad \forall v\in Q};\\ {x_{n+1}= \alpha_{n} u_{n}+ (1- \alpha_{n})T(u_{n}- \rho_{n} A^{*}(Au_{n}-v_{n}))};\\ y_{n+1}= \alpha_{n}v_{n}+ ( 1- \alpha_{n})S(v_{n}+ \rho_{n} (Au_{n}-v_{n})), \quad\forall n\geq1; \end{array} \right . $$

where $\lambda_{A}$ stands for the spectral radius of $A^{*}A$, $\{\rho_{n}\}$ is a positive real sequence such that $\rho_{n}\in(\varepsilon,\frac {2}{\lambda_{A}}-\varepsilon)$ (for ε small enough), $\eta _{i}>0$ ($i=1,2$), $\{\alpha_{n}\}$ is a sequence in $(0, 1)$ and $\{r_{n}\} \subset(0,\infty)$ satisfies the following conditions:

• (1) $0<\alpha\leq\alpha_{n}\leq\beta<1$ (for some $\alpha, \beta\in (0,1)$);

• (2) $\liminf_{n\rightarrow\infty}r_{n}>0$ and $\lim_{n\rightarrow\infty }|r_{n+1}-r_{n}|=0$.

If $\Gamma:=F(T)\cap F(S)\cap SMVIP(\phi,\varphi)\neq\emptyset$, then

• (I) The sequence $\{(x_{n}, y_{n})\}$ converges weakly to a solution of the split mixed variational inequality problem (4.3).

• (II) In addition, if S, T are also semi-compact, then $\{ (x_{n},y_{n})\}$ converges strongly to a solution of the split mixed variational inequality problem (4.3).

4.2 Application to the split equality convex minimization problem

It is easy to see that the split equality mixed equilibrium problem (1.12) reduces to the split equality convex minimization problem (1.13) as $F=0$ and $G=0$. Therefore, Theorem 3.1 can be used to solve split equality convex minimization problem (1.13), and the following result can be directly deduced from Theorem 3.1.

Theorem 4.3

Let $H_{1}$, $H_{2}$, $H_{3}$ be real Hilbert spaces, $C\subseteq H_{1}$ and $Q\subseteq H_{2}$ be nonempty closed convex subsets of Hilbert spaces $H_{1}$ and $H_{2}$, respectively. Let $\phi :C\rightarrow R\cup\{+\infty\}$ and $\varphi: Q\rightarrow R\cup\{ +\infty\}$ be proper lower semi-continuous and convex functions. Let $T:H_{1}\rightarrow H_{1}$, $S:H_{2}\rightarrow H_{2}$ be two nonexpansive mappings, and $A:H_{1}\rightarrow H_{3}$, $B:H_{2}\rightarrow H_{3}$ be two bounded linear operators. Assume that $(x_{1},y_{1})\in C \times Q$ and the iteration scheme $\{(x_{n}, y_{n})\}$ is defined as follows:

$$\left \{ \begin{array}{@{}l} {\phi(u)-\phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n}\rangle \geq0, \quad\forall u\in C};\\ \varphi(v)-\varphi(v_{n})+\frac{1}{r_{n}}\langle v-v_{n},v_{n}-y_{n}\rangle \geq0,\quad \forall v\in Q;\\ {x_{n+1}= \alpha_{n} u_{n}+ (1- \alpha_{n})T(u_{n}- \rho_{n} A^{*}(Au_{n}-Bv_{n}))};\\ y_{n+1}= \alpha_{n}v_{n}+ ( 1- \alpha_{n})S(v_{n}+ \rho_{n} B^{*}(Au_{n}-Bv_{n})),\quad \forall n\geq1; \end{array} \right . $$

where $\lambda_{A}$ and $\lambda_{B}$ stand for the spectral radii of $A^{*}A$ and $B^{*}B$, respectively, $\{\rho_{n}\}$ is a positive real sequence such that $\rho_{n}\in(\varepsilon,\frac{2}{\lambda_{A}+\lambda _{B}}-\varepsilon)$ (for ε small enough), $\{\alpha_{n}\}$ is a sequence in $(0, 1)$ and $\{r_{n}\}\subset(0,\infty)$ satisfies the following conditions:

• (1) $0<\alpha\leq\alpha_{n}\leq\beta<1$ (for some $\alpha, \beta\in (0,1)$);

• (2) $\liminf_{n\rightarrow\infty}r_{n}>0$ and $\lim_{n\rightarrow\infty }|r_{n+1}-r_{n}|=0$.

If $\Gamma:=F(T)\cap F(S)\cap SECMP(\phi,\varphi)\neq\emptyset$, then

• (I) The sequence $\{(x_{n}, y_{n})\}$ converges weakly to a solution of problem (1.13).

• (II) In addition, if S, T are also semi-compact, then $\{ (x_{n},y_{n})\}$ converges strongly to a solution of problem (1.13).

In Theorem 4.3 taking $B=I$ and $H_{2}=H_{3}$, from Theorem 4.3 we can obtain the following convergence theorem for split convex minimization problem (1.14) $SCMP(\phi,\varphi)$.

Corollary 4.4

Let $H_{1}$ and $H_{2}$ be real Hilbert spaces, $C\subseteq H_{1}$ and $Q\subseteq H_{2}$ be nonempty closed convex subsets of Hilbert spaces $H_{1}$ and $H_{2}$, respectively. Let $\phi :C\rightarrow R\cup\{+\infty\}$ and $\varphi: Q\rightarrow R\cup\{ +\infty\}$ be proper lower semi-continuous and convex functions. Let $T:H_{1}\rightarrow H_{1}$, $S:H_{2}\rightarrow H_{2}$ be two nonexpansive mappings, and $A:H_{1}\rightarrow H_{2}$ be a bounded linear operator. Assume that $(x_{1},y_{1})\in C \times Q$ and the iteration scheme $\{(x_{n}, y_{n})\}$ is defined as follows:

$$\left \{ \begin{array}{@{}l} \phi(u)-\phi(u_{n})+\frac{1}{r_{n}}\langle u-u_{n},u_{n}-x_{n}\rangle \geq0,\quad \forall u\in C;\\ \varphi(v)-\varphi(v_{n})+\frac{1}{r_{n}}\langle v-v_{n},v_{n}-y_{n}\rangle \geq0,\quad \forall v\in Q;\\ {x_{n+1}= \alpha_{n} u_{n}+ (1- \alpha_{n})T(u_{n}- \rho_{n} A^{*}(Au_{n}-v_{n}))};\\ y_{n+1}= \alpha_{n}v_{n}+ ( 1- \alpha_{n})S(v_{n}+ \rho_{n} (Au_{n}-v_{n})),\quad \forall n\geq1; \end{array} \right . $$

where $\lambda_{A}$ stands for the spectral radius of $A^{*}A$, $\{\rho_{n}\}$ is a positive real sequence such that $\rho_{n}\in(\varepsilon,\frac {2}{\lambda_{A}}-\varepsilon)$ (for ε small enough), $\{\alpha _{n}\}$ is a sequence in $(0, 1)$ and $\{r_{n}\}\subset(0,\infty)$ satisfies the following conditions:

• (1) $0<\alpha\leq\alpha_{n}\leq\beta<1$ (for some $\alpha, \beta\in (0,1)$);

• (2) $\liminf_{n\rightarrow\infty}r_{n}>0$ and $\lim_{n\rightarrow\infty }|r_{n+1}-r_{n}|=0$.

If $\Gamma:=F(T)\cap F(S)\cap SCMP(\phi,\varphi)\neq\emptyset$, then

• (I) The sequence $\{(x_{n}, y_{n})\}$ converges weakly to a solution of problem (1.14).

• (II) In addition, if S, T are also semi-compact, then $\{ (x_{n},y_{n})\}$ converges strongly to a solution of problem (1.14).