# The split equilibrium problem and its convergence algorithms

## Abstract

The purpose of this paper is to introduce a split equilibrium problem (SEP) and find a solution of the equilibrium problem such that its image under a given bounded linear operator is a solution of another equilibrium problem. By using the iterative method, we construct some iterative algorithms to solve such problem in real Hilbert spaces and obtain some strong and weak convergence theorems. Finally, we point out that there exist many SEPs which need the use of new methods to solve them. Some examples are given to illustrate our results.

MSC:47J25, 47H09, 65K10.

## 1 Introduction

Throughout this paper, the symbols $\mathbb{N}$ and $\mathbb{R}$ are used to denote the sets of positive integers and real numbers, respectively.

In this paper, we propose a new equilibrium problem, which is called a split equilibrium problem (SEP). Let ${E}_{1}$ and ${E}_{2}$ be two real Banach spaces. Let C be a closed convex subset of ${E}_{1}$, K a closed convex subset of ${E}_{2}$, and $A:{E}_{1}â†’{E}_{2}$ a bounded linear operator. f is a bi-function from $CÃ—C$ into $\mathbb{R}$ and g is a bi-function from $KÃ—K$ into $\mathbb{R}$. The SEP is

$\text{to find an element}pâˆˆC\phantom{\rule{1em}{0ex}}\text{such that}\phantom{\rule{1em}{0ex}}f\left(p,y\right)â‰¥0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}yâˆˆC,$
(1.1)

and

$\text{such that}u:=ApâˆˆK\phantom{\rule{1em}{0ex}}\text{solves}\phantom{\rule{1em}{0ex}}g\left(u,v\right)â‰¥0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}vâˆˆK.$
(1.2)

If we consider only the problem (1.1), then (1.1) is a classical equilibrium problem. From (1.1) and (1.2), we can see that the SEP contains two equilibrium problems, and the image of a solution of one equilibrium problem under a given bounded linear operator is a solution of another equilibrium problem. Since many problems coming from physics, optimization, and economics reduce to find a solution of the equilibrium problem (1.1) (see, for instance, [1, 2]), the equilibrium problem (1.1) is very important in the field of applied mathematics. Some authors have proposed some methods to find the solution of the equilibrium problem (1.1). As a generalization of the equilibrium problem (1.1), when finding a common solution for some equilibrium problems, it has been considered in the same subset of the same space; see [3â€“5]. However, in general, some equilibrium problems always belong to different subsets of spaces, so the SEP is important and quite general. The SEP should enable us to split the solution between two different subsets of spaces so that the image of a solution point of one problem, under a given bounded linear operator, is a solution point of another problem. A special case of the SEP is the split variational inequality problem (SVIP); see [6].

For convenience, in this paper let $EP\left(f\right)$, $EP\left(g\right)$ and $\mathrm{Î©}=\left\{pâˆˆEP\left(f\right):ApâˆˆEP\left(g\right)\right\}$ denote the solution set of (1.1), (1.2) and the SEP, respectively.

Example 1.1 Let ${E}_{1}={E}_{2}=\mathbb{R}$, $C:=\left[1,+\mathrm{âˆž}\right)$ and $K:=\left(âˆ’\mathrm{âˆž},âˆ’4\right]$. Let $A\left(x\right)=âˆ’4x$ for all $xâˆˆ\mathbb{R}$, then A is a bounded linear operator. Let $f:CÃ—Câ†’\mathbb{R}$, and $g:KÃ—Kâ†’\mathbb{R}$ be defined by $f\left(x,y\right)=yâˆ’x$, $g\left(u,v\right)=2\left(uâˆ’v\right)$, respectively. Clearly, $EP\left(f\right)=\left\{1\right\}$ and $A\left(1\right)=âˆ’4âˆˆEP\left(g\right)$. So .

Example 1.2 Let ${E}_{2}=\mathbb{R}$ with the standard norm $|â‹\dots |$ and ${E}_{1}={\mathbb{R}}^{2}$ with the norm $âˆ¥\mathrm{Î±}âˆ¥={\left({a}_{1}^{2}+{a}_{2}^{2}\right)}^{\frac{1}{2}}$ for some $\mathrm{Î±}=\left({a}_{1},{a}_{2}\right)âˆˆ{\mathbb{R}}^{2}$. $K:=\left[1,+\mathrm{âˆž}\right)$ and $C:=\left\{\mathrm{Î±}=\left({a}_{1},{a}_{2}\right)âˆˆ{\mathbb{R}}^{2}|{a}_{2}âˆ’{a}_{1}â‰¥1\right\}$. Define a bi-function $f\left(w,\mathrm{Î±}\right)={w}_{1}âˆ’{w}_{2}+{a}_{2}âˆ’{a}_{1}$, where $w=\left({w}_{1},{w}_{2}\right)$, $\mathrm{Î±}=\left({a}_{1},{a}_{2}\right)âˆˆC$, then f is a bi-function from $CÃ—C$ into $\mathbb{R}$ with $EP\left(f\right)=\left\{p=\left({p}_{1},{p}_{2}\right)|{p}_{2}âˆ’{p}_{1}=1\right\}$. For each $\mathrm{Î±}=\left({a}_{1},{a}_{2}\right)âˆˆ{E}_{1}$, let $A\mathrm{Î±}={a}_{2}âˆ’{a}_{1}$, then A is a bounded linear operator from ${E}_{1}$ into ${E}_{2}$. In fact, it is also easy to verify that $A\left(a{\mathrm{Î±}}_{1}+b{\mathrm{Î±}}_{2}\right)=aA\left({\mathrm{Î±}}_{1}\right)+bA\left({\mathrm{Î±}}_{2}\right)$ and $âˆ¥Aâˆ¥=\sqrt{2}$ for some ${\mathrm{Î±}}_{1},{\mathrm{Î±}}_{2}âˆˆ{E}_{1}$ and $a,bâˆˆ\mathbb{R}$. Now define another bi-function g as follows: $g\left(u,v\right)=vâˆ’u$ for all $u,vâˆˆK$. Then g is a bi-function from $KÃ—K$ into $\mathbb{R}$ with $EP\left(g\right)=\left\{1\right\}$.

Clearly, when $pâˆˆEP\left(f\right)$, we have $Ap=1âˆˆEP\left(g\right)$. So .

Remark 1.1 The SEP in Example 1.1 lies in two different subsets of the same space. While the SEP in Example 1.2 lies in two different subsets of the different space.

In this paper, we construct some iterative algorithms to solve the SEP. Some strong and weak convergence theorems are established. The results obtained in this paper can be reckoned as the new development of the equilibrium problem (1.1). Finally, we point out that there exist many SEPs which need the use of new methods to solve them. Some examples are given to illustrate our results.

## 2 Preliminaries

We assume that H is a real Hilbert space with zero vector Î¸ whose inner product and norm are denoted by $ã€ˆâ‹\dots ,â‹\dots ã€‰$ and $âˆ¥â‹\dots âˆ¥$, respectively; and we use symbols â†’ and â‡€ to denote strong and weak convergence, respectively.

Let ${H}_{1}$ and ${H}_{2}$ be two Hilbert spaces. The operator A from ${H}_{1}$ into ${H}_{2}$ and the operator B from ${H}_{2}$ into ${H}_{1}$ are two bounded linear operators. B is called the adjoint operator of A, if for all $zâˆˆ{H}_{1}$, $wâˆˆ{H}_{2}$, B satisfies $ã€ˆAz,wã€‰=ã€ˆz,Bwã€‰$. Especially, if ${H}_{1}={H}_{2}$, then B reduces to the well-known adjoint operator of A.

Remark 2.1 It is easy to verify that the operator B, an adjoint operator of A, has the following characters:

1. (i)

$âˆ¥Bâˆ¥=âˆ¥Aâˆ¥$; (ii) B is a unique adjoint operator of A.

Example 2.1 Let ${H}_{2}=\mathbb{R}$ with the standard norm $|â‹\dots |$ and ${H}_{1}={\mathbb{R}}^{2}$ with the norm $âˆ¥\mathrm{Î±}âˆ¥={\left({a}_{1}^{2}+{a}_{2}^{2}\right)}^{\frac{1}{2}}$ for some $\mathrm{Î±}=\left({a}_{1},{a}_{2}\right)âˆˆ{\mathbb{R}}^{2}$. $ã€ˆx,yã€‰=xy$ denotes the inner product of ${H}_{2}$ for some $x,yâˆˆ{H}_{2}$ and $ã€ˆ\mathrm{Î±},\mathrm{Î²}ã€‰={a}_{1}{b}_{1}+{a}_{2}{b}_{2}$ denotes the inner product of ${H}_{1}$ for some $\mathrm{Î±}=\left({a}_{1},{a}_{2}\right)$, $\mathrm{Î²}=\left({b}_{1},{b}_{2}\right)âˆˆ{H}_{1}$. Let $A\mathrm{Î±}={a}_{2}âˆ’{a}_{1}$, then A is a bounded linear operator from ${H}_{1}$ into ${H}_{2}$ with $âˆ¥Aâˆ¥=\sqrt{2}$. For $xâˆˆ{H}_{2}$, let $Bx=\left(âˆ’x,x\right)$, then B is a bounded linear operator from ${H}_{2}$ into ${H}_{1}$ with $âˆ¥Bâˆ¥=\sqrt{2}$. Moreover, for any $\mathrm{Î±}=\left({a}_{1},{a}_{2}\right)âˆˆ{H}_{1}$ and $xâˆˆ{H}_{2}$, $ã€ˆA\mathrm{Î±},xã€‰=ã€ˆ\mathrm{Î±},Bxã€‰$, so B is an adjoint operator of A.

Example 2.2 Let ${H}_{1}={\mathbb{R}}^{2}$ with the norm $âˆ¥\mathrm{Î±}âˆ¥={\left({a}_{1}^{2}+{a}_{2}^{2}\right)}^{\frac{1}{2}}$ for some $\mathrm{Î±}=\left({a}_{1},{a}_{2}\right)âˆˆ{\mathbb{R}}^{2}$ and ${H}_{2}={\mathbb{R}}^{3}$ with the norm $âˆ¥\mathrm{Î³}âˆ¥={\left({c}_{1}^{2}+{c}_{2}^{2}+{c}_{3}^{2}\right)}^{\frac{1}{2}}$ for some $\mathrm{Î³}=\left({c}_{1},{c}_{2},{c}_{3}\right)âˆˆ{\mathbb{R}}^{3}$. Let $ã€ˆ\mathrm{Î±},\mathrm{Î²}ã€‰={a}_{1}{b}_{1}+{a}_{2}{b}_{2}$ and $ã€ˆ\mathrm{Î³},\mathrm{Î·}ã€‰={c}_{1}{d}_{1}+{c}_{2}{d}_{2}+{c}_{3}{d}_{3}$ denote the inner product of ${H}_{1}$ and ${H}_{2}$, respectively, where $\mathrm{Î±}=\left({a}_{1},{a}_{2}\right)$, $\mathrm{Î²}=\left({b}_{1},{b}_{2}\right)âˆˆ{H}_{1}$, $\mathrm{Î³}=\left({c}_{1},{c}_{2},{c}_{3}\right)$, $\mathrm{Î·}=\left({d}_{1},{d}_{2},{d}_{3}\right)âˆˆ{\mathbb{R}}^{3}$. Let $A\mathrm{Î±}=\left({a}_{2},{a}_{1},{a}_{1}âˆ’{a}_{2}\right)$ for $\mathrm{Î±}=\left({a}_{1},{a}_{2}\right)âˆˆ{H}_{1}$, then A is a bounded linear operator from ${H}_{1}$ into ${H}_{2}$ with $âˆ¥Aâˆ¥=\sqrt{3}$ because $âˆ¥\left(\frac{\sqrt{2}}{2},âˆ’\frac{\sqrt{2}}{2},âˆ’\sqrt{2}\right)âˆ¥â‰¤{sup}_{âˆ¥\mathrm{Î±}âˆ¥=1}âˆ¥A\mathrm{Î±}âˆ¥â‰¤\sqrt{3}$. For $\mathrm{Î³}=\left({c}_{1},{c}_{2},{c}_{3}\right)âˆˆ{H}_{2}$, let $B\mathrm{Î³}=\left({c}_{2}+{c}_{3},{c}_{1}âˆ’{c}_{3}\right)$, then B is a bounded linear operator from ${H}_{2}$ into ${H}_{1}$ with $âˆ¥Bâˆ¥=\sqrt{3}$. Moreover, for any $\mathrm{Î±}=\left({a}_{1},{a}_{2}\right)âˆˆ{H}_{1}$ and $\mathrm{Î³}=\left({c}_{1},{c}_{2},{c}_{3}\right)âˆˆ{H}_{2}$, $ã€ˆA\mathrm{Î±},\mathrm{Î³}ã€‰=ã€ˆ\mathrm{Î±},B\mathrm{Î³}ã€‰$, so B is an adjoint operator of A.

Let K be a closed convex subset of a real Hilbert space H. For each point $xâˆˆH$, there exists a unique nearest point in K, denoted by ${P}_{K}x$, such that

$âˆ¥xâˆ’{P}_{K}xâˆ¥â‰¤âˆ¥xâˆ’yâˆ¥,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}yâˆˆK.$

The mapping ${P}_{K}$ is called the metric projection from H onto K. It is well known that ${P}_{K}$ has the following characters:

1. (i)

$ã€ˆxâˆ’y,{P}_{K}xâˆ’{P}_{K}yã€‰â‰¥{âˆ¥{P}_{K}xâˆ’{P}_{K}yâˆ¥}^{2}$ for every $x,yâˆˆH$.

2. (ii)

For $xâˆˆH$, and $zâˆˆK$, $z={P}_{K}\left(x\right)â‡”ã€ˆxâˆ’z,zâˆ’yã€‰â‰¥0$, $\mathrm{âˆ€}yâˆˆK$.

3. (iii)

For $xâˆˆH$ and $yâˆˆK$,

${âˆ¥yâˆ’{P}_{K}\left(x\right)âˆ¥}^{2}+{âˆ¥xâˆ’{P}_{K}\left(x\right)âˆ¥}^{2}â‰¤{âˆ¥xâˆ’yâˆ¥}^{2}.$
(2.1)

A Banach space $\left(X,âˆ¥â‹\dots âˆ¥\right)$ is said to satisfy Opialâ€™s condition if, for each sequence $\left\{{x}_{n}\right\}$ in X which converges weakly to a point $xâˆˆX$, we have

It is well known that each Hilbert space satisfies Opialâ€™s condition.

The following results are crucial to our main results.

Lemma 2.1 (see [1])

Let K be a nonempty closed convex subset of H and F be a bi-function of$KÃ—K$into R satisfying the following conditions:

(A1) $F\left(x,x\right)=0$for all$xâˆˆK$;

(A2) F is monotone, that is, $F\left(x,y\right)+F\left(y,x\right)â‰¤0$for all$x,yâˆˆK$;

(A3) for each$x,y,zâˆˆK$,

$\underset{tâ†“0}{limâ€‰sup}F\left(tz+\left(1âˆ’t\right)x,y\right)â‰¤F\left(x,y\right);$

(A4) for each$xâˆˆK$, $yâ†¦F\left(x,y\right)$is convex and lower semi-continuous.

Let$r>0$and$xâˆˆH$. Then, there exists$zâˆˆK$such that

$F\left(z,y\right)+\frac{1}{r}ã€ˆyâˆ’z,zâˆ’xã€‰â‰¥0,\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}yâˆˆK.$

Lemma 2.2 (see [7])

Let K be a nonempty closed convex subset of H and let F be a bi-function of$KÃ—K$into R satisfying (A 1)-(A 4). For$r>0$and$xâˆˆH$, define a mapping${T}_{r}^{F}:Hâ†’K$as follows:

${T}_{r}^{F}\left(x\right)=\left\{zâˆˆK:F\left(z,y\right)+\frac{1}{r}ã€ˆyâˆ’z,zâˆ’xã€‰â‰¥0,\mathrm{âˆ€}yâˆˆK\right\}$
(2.2)

for all$xâˆˆH$. Then the following hold:

1. (i)

${T}_{r}^{F}$is single-valued;

2. (ii)

${T}_{r}^{F}$is firmly non-expansive, that is, for any$x,yâˆˆH$,

${âˆ¥{T}_{r}^{F}xâˆ’{T}_{r}^{F}yâˆ¥}^{2}â‰¤ã€ˆ{T}_{r}^{F}xâˆ’{T}_{r}^{F}y,xâˆ’yã€‰;$
3. (iii)

$F\left({T}_{r}^{F}\right)=EP\left(F\right)$for$\mathrm{âˆ€}r>0$;

4. (iv)

$EP\left(F\right)$is closed and convex.

Lemma 2.3 (see [3])

Let H be a real Hilbert space. Then for any${x}_{1},{x}_{2},â€¦,{x}_{k}âˆˆH$and${a}_{1},{a}_{2},â€¦,{a}_{k}âˆˆ\left[0,1\right]$with${âˆ‘}_{i=1}^{k}{a}_{i}=1$, $kâˆˆ\mathbb{N}$, we have

${âˆ¥\underset{i=1}{\overset{k}{âˆ‘}}{a}_{i}{x}_{i}âˆ¥}^{2}=\underset{i=1}{\overset{k}{âˆ‘}}{a}_{i}{âˆ¥{x}_{i}âˆ¥}^{2}âˆ’\underset{i=1}{\overset{kâˆ’1}{âˆ‘}}\underset{j=i+1}{\overset{k}{âˆ‘}}{a}_{i}{a}_{j}{âˆ¥{x}_{i}âˆ’{x}_{j}âˆ¥}^{2}.$

In particular, we have

1. (1)

${âˆ¥\mathrm{Î±}x+\left(1âˆ’\mathrm{Î±}\right)yâˆ¥}^{2}=\mathrm{Î±}{âˆ¥xâˆ¥}^{2}+\left(1âˆ’\mathrm{Î±}\right){âˆ¥yâˆ¥}^{2}âˆ’\mathrm{Î±}\left(1âˆ’\mathrm{Î±}\right){âˆ¥xâˆ’yâˆ¥}^{2}$for all$x,yâˆˆH$and$\mathrm{Î±}âˆˆ\left[0,1\right]$;

2. (2)

the map$f:Hâ†’\mathbf{R}$defined by$f\left(x\right)={âˆ¥xâˆ¥}^{2}$is convex.

Lemma 2.4 (see, e.g., [8])

Let H be a real Hilbert space. Then the following hold:

1. (a)

${âˆ¥x+yâˆ¥}^{2}â‰¤{âˆ¥yâˆ¥}^{2}+2ã€ˆx,x+yã€‰$for all$x,yâˆˆH$;

2. (b)

${âˆ¥xâˆ’yâˆ¥}^{2}={âˆ¥xâˆ¥}^{2}+{âˆ¥yâˆ¥}^{2}âˆ’2ã€ˆx,yã€‰$for all$x,yâˆˆH$.

Lemma 2.5 Let K be a nonempty closed convex subset of H. For$xâˆˆH$, let the mapping${T}_{r}^{F}\left(x\right)$be the same as in Lemma 2.2. Then for$r,s>0$and$x,yâˆˆH$,

$âˆ¥{T}_{r}^{F}\left(x\right)âˆ’{T}_{s}^{F}\left(y\right)âˆ¥â‰¤âˆ¥yâˆ’xâˆ¥+\frac{|sâˆ’r|}{s}âˆ¥{T}_{s}^{F}\left(y\right)âˆ’yâˆ¥.$

Proof For $r,s>0$ and $x,yâˆˆH$, by (i) of Lemma 2.2, we can let ${z}_{1}={T}_{r}^{F}\left(x\right)$ and ${z}_{2}={T}_{s}^{F}\left(y\right)$. By the definition of ${T}_{r}^{F}$, we have

$F\left({z}_{1},u\right)+\frac{1}{r}ã€ˆuâˆ’{z}_{1},{z}_{1}âˆ’xã€‰â‰¥0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}uâˆˆK$
(2.3)

and

$F\left({z}_{2},u\right)+\frac{1}{s}ã€ˆuâˆ’{z}_{2},{z}_{2}âˆ’yã€‰â‰¥0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}uâˆˆK.$
(2.4)

Taking $u={z}_{2}$ in (2.3) and $u={z}_{1}$ in (2.4), we have

$F\left({z}_{1},{z}_{2}\right)+\frac{1}{r}ã€ˆ{z}_{2}âˆ’{z}_{1},{z}_{1}âˆ’xã€‰â‰¥0,$
(2.5)

and

$F\left({z}_{2},{z}_{1}\right)+\frac{1}{s}ã€ˆ{z}_{1}âˆ’{z}_{2},{z}_{2}âˆ’yã€‰â‰¥0.$
(2.6)

Since the bi-function F satisfies the condition (A2), from (2.5) and (2.6) we have

$\frac{1}{r}ã€ˆ{z}_{2}âˆ’{z}_{1},{z}_{1}âˆ’xã€‰+\frac{1}{s}ã€ˆ{z}_{1}âˆ’{z}_{2},{z}_{2}âˆ’yã€‰â‰¥0,$

which implies that

$ã€ˆ{z}_{2}âˆ’{z}_{1},{z}_{1}âˆ’xã€‰âˆ’ã€ˆ{z}_{2}âˆ’{z}_{1},r\frac{{z}_{2}âˆ’y}{s}ã€‰â‰¥0.$

Thus, we have

$ã€ˆ{z}_{2}âˆ’{z}_{1},{z}_{1}âˆ’{z}_{2}+{z}_{2}âˆ’xâˆ’\frac{r}{s}\left({z}_{2}âˆ’y\right)ã€‰â‰¥0,$

this implies that

${âˆ¥{z}_{2}âˆ’{z}_{1}âˆ¥}^{2}â‰¤ã€ˆ{z}_{2}âˆ’{z}_{1},{z}_{2}âˆ’xâˆ’\frac{r}{s}\left({z}_{2}âˆ’y\right)ã€‰â‰¤âˆ¥{z}_{2}âˆ’{z}_{1}âˆ¥âˆ¥{z}_{2}âˆ’xâˆ’\frac{r}{s}\left({z}_{2}âˆ’y\right)âˆ¥,$

so

$\begin{array}{rcl}âˆ¥{z}_{2}âˆ’{z}_{1}âˆ¥& â‰¤& âˆ¥{z}_{2}âˆ’xâˆ’\frac{r}{s}\left({z}_{2}âˆ’y\right)âˆ¥â‰¤âˆ¥yâˆ’xâˆ¥+âˆ¥\left(1âˆ’\frac{r}{s}\right)\left({z}_{2}âˆ’y\right)âˆ¥\\ =& âˆ¥yâˆ’xâˆ¥+\frac{|sâˆ’r|}{s}âˆ¥{T}_{s}^{F}\left(y\right)âˆ’yâˆ¥,\end{array}$

namely,

$âˆ¥{T}_{r}^{F}\left(x\right)âˆ’{T}_{s}^{F}\left(y\right)âˆ¥â‰¤âˆ¥yâˆ’xâˆ¥+\frac{|sâˆ’r|}{s}âˆ¥{T}_{s}^{F}\left(y\right)âˆ’yâˆ¥.$

The proof is completed.â€ƒâ–¡

## 3 Main results

In this section, we will solve the SEP which satisfies the conditions (A1)-(A4).

Theorem 3.1 (Weak convergence theorem)

Let C be a nonempty closed convex subset of${H}_{1}$and K a nonempty closed convex subset of${H}_{2}$, where${H}_{1}$and${H}_{2}$are two real Hilbert spaces. $âˆ§:=\left\{1,2,â€¦,k\right\}$denotes a finite index set. For any$iâˆˆâˆ§$, ${f}_{i}:CÃ—Câ†’\mathbf{R}$is a bi-function with. Let$A:{H}_{1}â†’{H}_{2}$be a bounded linear operator with the adjoint B and$g:KÃ—Kâ†’\mathbf{R}$a bi-function with. Let$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}^{i}\right\}$ ($iâˆˆâˆ§$) be sequences generated by

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ {f}_{i}\left({u}_{n}^{i},y\right)+\frac{1}{{r}_{n}}ã€ˆyâˆ’{u}_{n}^{i},{u}_{n}^{i}âˆ’{x}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}yâˆˆC,iâˆˆâˆ§,\hfill \\ {\mathrm{Ï„}}_{n}=\frac{{u}_{n}^{1}+â‹¯+{u}_{n}^{k}}{k},\hfill \\ g\left({w}_{n},z\right)+\frac{1}{{r}_{n}}ã€ˆzâˆ’{w}_{n},{w}_{n}âˆ’A{\mathrm{Ï„}}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}zâˆˆK,\hfill \\ {x}_{n+1}={P}_{C}\left({\mathrm{Ï„}}_{n}+\mathrm{Î¼}B\left({w}_{n}âˆ’A{\mathrm{Ï„}}_{n}\right)\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}nâˆˆ\mathbb{N},\hfill \end{array}$
(3.1)

where$\left\{{r}_{n}\right\}âŠ‚\left(0,+\mathrm{âˆž}\right)$with${limâ€‰inf}_{nâ†’\mathrm{âˆž}}{r}_{n}>0$, ${P}_{C}$is a projection operator from${H}_{1}$into C and$\mathrm{Î¼}âˆˆ\left(0,\frac{1}{{âˆ¥Bâˆ¥}^{2}}\right)$is a constant. Suppose that, then the sequences$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}^{i}\right\}$ ($iâˆˆâˆ§$) converge weakly to an element$pâˆˆ\mathrm{Î“}$, while$\left\{{w}_{n}\right\}$converges weakly to$ApâˆˆEP\left(g\right)$.

Proof For each $iâˆˆâˆ§$ and each $r>0$, let ${T}_{r}^{{f}_{i}}$: ${H}_{1}â†’C$ be defined by (2.2), then ${u}_{n}^{i}={T}_{{r}_{n}}^{{f}_{i}}{x}_{n}$ for all $nâˆˆ\mathbb{N}$ by Lemma 2.2. Again let ${T}_{r}^{g}$: ${H}_{2}â†’K$ be defined by (2.2), then ${w}_{n}={T}_{{r}_{n}}^{g}A{\mathrm{Ï„}}_{n}$ for all $nâˆˆ\mathbb{N}$. So (3.1) can be rewritten as follows:

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ {u}_{n}^{i}={T}_{{r}_{n}}^{{f}_{i}}{x}_{n},\phantom{\rule{1em}{0ex}}iâˆˆâˆ§,\hfill \\ {\mathrm{Ï„}}_{n}=\frac{{u}_{n}^{1}+â‹¯+{u}_{n}^{k}}{k},\hfill \\ {w}_{n}={T}_{{r}_{n}}^{g}A{\mathrm{Ï„}}_{n},\hfill \\ {x}_{n+1}={P}_{C}\left({\mathrm{Ï„}}_{n}+\mathrm{Î¼}B\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}nâˆˆ\mathbb{N}.\hfill \end{array}$
(3.2)

Let ${x}^{âˆ—}âˆˆC$ be a point such that ${x}^{âˆ—}âˆˆ{â‹‚}_{i=1}^{k}EP\left({f}_{i}\right)$ and $A{x}^{âˆ—}âˆˆEP\left(g\right)$, namely, ${x}^{âˆ—}âˆˆ\mathrm{Î©}$. By Lemma 2.2 and Lemma 2.4, it follows that

$\begin{array}{rcl}{âˆ¥{u}_{n}^{i}âˆ’{x}^{âˆ—}âˆ¥}^{2}& =& {âˆ¥{T}_{{r}_{n}}^{{f}_{i}}{x}_{n}âˆ’{T}_{{r}_{n}}^{{f}_{i}}{x}^{âˆ—}âˆ¥}^{2}â‰¤ã€ˆ{T}_{{r}_{n}}^{{f}_{i}}{x}_{n}âˆ’{T}_{{r}_{n}}^{{f}_{i}}{x}^{âˆ—},{x}_{n}âˆ’{x}^{âˆ—}ã€‰\\ =& \frac{1}{2}\left\{{âˆ¥{u}_{n}^{i}âˆ’{x}^{âˆ—}âˆ¥}^{2}+{âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}âˆ’{âˆ¥{u}_{n}^{i}âˆ’{x}_{n}âˆ¥}^{2}\right\},\end{array}$

hence

${âˆ¥{u}_{n}^{i}âˆ’{x}^{âˆ—}âˆ¥}^{2}â‰¤{âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}âˆ’{âˆ¥{u}_{n}^{i}âˆ’{x}_{n}âˆ¥}^{2}.$
(3.3)

Applying Lemma 2.3, we get

${âˆ¥{\mathrm{Ï„}}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}â‰¤\frac{1}{k}\underset{i=1}{\overset{k}{âˆ‘}}{âˆ¥{u}_{n}^{i}âˆ’{x}^{âˆ—}âˆ¥}^{2}.$
(3.4)

(3.3) and (3.4) imply that

${âˆ¥{\mathrm{Ï„}}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}â‰¤{âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}âˆ’\frac{1}{k}\underset{i=1}{\overset{k}{âˆ‘}}{âˆ¥{u}_{n}^{i}âˆ’{x}_{n}âˆ¥}^{2}.$
(3.5)

Again from Lemma 2.2, we have

$âˆ¥{w}_{n}âˆ’A{x}^{âˆ—}âˆ¥=âˆ¥{T}_{{r}_{n}}^{g}A{\mathrm{Ï„}}_{n}âˆ’A{x}^{âˆ—}âˆ¥â‰¤âˆ¥A{\mathrm{Ï„}}_{n}âˆ’A{x}^{âˆ—}âˆ¥\phantom{\rule{1em}{0ex}}\text{for each}nâˆˆ\mathbb{N}.$
(3.6)

By (b) of Lemma 2.4 and (3.6), for each $nâˆˆ\mathbb{N}$, we have

(3.7)

We also have

${âˆ¥B\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}â‰¤{âˆ¥Bâˆ¥}^{2}{âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}.$
(3.8)

From (3.2), (3.5)-(3.8), we have

$\begin{array}{rcl}{âˆ¥{x}_{n+1}âˆ’{x}^{âˆ—}âˆ¥}^{2}& =& {âˆ¥{P}_{C}\left({\mathrm{Ï„}}_{n}+\mathrm{Î¼}B\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}\right)âˆ’{P}_{C}{x}^{âˆ—}âˆ¥}^{2}\\ â‰¤& {âˆ¥{\mathrm{Ï„}}_{n}+\mathrm{Î¼}B\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}\right)âˆ’{x}^{âˆ—}âˆ¥}^{2}\\ =& {âˆ¥{\mathrm{Ï„}}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}+{âˆ¥\mathrm{Î¼}B\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}+2\mathrm{Î¼}ã€ˆ{\mathrm{Ï„}}_{n}âˆ’{x}^{âˆ—},B\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}ã€‰\\ â‰¤& {âˆ¥{\mathrm{Ï„}}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}+{\mathrm{Î¼}}^{2}{âˆ¥Bâˆ¥}^{2}{âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}âˆ’\mathrm{Î¼}{âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}\\ =& {âˆ¥{\mathrm{Ï„}}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}âˆ’\mathrm{Î¼}\left(1âˆ’\mathrm{Î¼}{âˆ¥Bâˆ¥}^{2}\right){âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}\\ â‰¤& {âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}âˆ’\mathrm{Î¼}\left(1âˆ’\mathrm{Î¼}{âˆ¥Bâˆ¥}^{2}\right){âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}.\end{array}$
(3.9)

Notice $\mathrm{Î¼}âˆˆ\left(0,\frac{1}{{âˆ¥Bâˆ¥}^{2}}\right)$, $\mathrm{Î¼}\left(1âˆ’\mathrm{Î¼}{âˆ¥Bâˆ¥}^{2}\right)>0$. It follows from (3.9) that

$âˆ¥{x}_{n+1}âˆ’{x}^{âˆ—}âˆ¥â‰¤âˆ¥{\mathrm{Ï„}}_{n}âˆ’{x}^{âˆ—}âˆ¥â‰¤âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥$
(3.10)

and

$\mathrm{Î¼}\left(1âˆ’\mathrm{Î¼}{âˆ¥Bâˆ¥}^{2}\right){âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}â‰¤{âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥}^{2}âˆ’{âˆ¥{x}_{n+1}âˆ’{x}^{âˆ—}âˆ¥}^{2}.$
(3.11)

(3.10) implies ${lim}_{nâ†’\mathrm{âˆž}}âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥$ exists. Further, from (3.10)-(3.11),

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥=\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{\mathrm{Ï„}}_{n}âˆ’{x}^{âˆ—}âˆ¥,\phantom{\rule{2em}{0ex}}\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥=0.$
(3.12)

Again from (3.5), we have

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{u}_{n}^{i}âˆ’{x}_{n}âˆ¥=0,\phantom{\rule{1em}{0ex}}iâˆˆâˆ§,$
(3.13)

which yields that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥\left({T}_{{r}_{n}}^{{f}_{i}}âˆ’I\right){x}_{n}âˆ¥=\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{T}_{{r}_{n}}^{{f}_{i}}{x}_{n}âˆ’{x}_{n}âˆ¥=\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{u}_{n}^{i}âˆ’{x}_{n}âˆ¥=0,\phantom{\rule{1em}{0ex}}iâˆˆâˆ§,$
(3.14)

and

$âˆ¥{\mathrm{Ï„}}_{n}âˆ’{x}_{n}âˆ¥â‰¤âˆ¥{u}_{n}^{1}âˆ’{x}_{n}âˆ¥+â‹¯+âˆ¥{u}_{n}^{k}âˆ’{x}_{n}âˆ¥â†’0\phantom{\rule{1em}{0ex}}\text{as}nâ†’\mathrm{âˆž}.$
(3.15)

Because ${lim}_{nâ†’\mathrm{âˆž}}âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥$ exists, which implies $\left\{{x}_{n}\right\}$ is bounded, hence $\left\{{x}_{n}\right\}$ has a weakly convergence subsequence $\left\{{x}_{{n}_{j}}\right\}$. Assume that ${x}_{{n}_{j}}â‡€p$ for some $pâˆˆC$. Then ${u}_{{n}_{j}}^{i}â‡€p$, ${\mathrm{Ï„}}_{{n}_{j}}â‡€p$ and $A{\mathrm{Ï„}}_{{n}_{j}}â‡€ApâˆˆK$ by (3.13) and (3.15).

Now we prove $pâˆˆ\mathrm{Î©}$ or, to be more precise, we prove $pâˆˆ{â‹‚}_{i=1}^{k}EP\left({f}_{i}\right)$ and $ApâˆˆEP\left(g\right)$. By Lemma 2.2, for any $r>0$, $EP\left({f}_{i}\right)=F\left({T}_{r}^{{f}_{i}}\right)$, $iâˆˆâˆ§$, and $EP\left(g\right)=F\left({T}_{r}^{g}\right)$. For $iâˆˆâˆ§$, since $\left(Iâˆ’{T}_{{r}_{n}}^{{f}_{i}}\right){x}_{n}â†’0$ by (3.14), we have ${T}_{r}^{{f}_{i}}p=p$ for $r>0$. Otherwise, if for all $iâˆˆâˆ§$, then by Opialâ€™s condition, we have

$\begin{array}{rcl}\underset{jâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{{n}_{j}}âˆ’pâˆ¥& <& \underset{jâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{{n}_{j}}âˆ’{T}_{r}^{{f}_{i}}pâˆ¥\\ =& \underset{jâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{{n}_{j}}âˆ’{T}_{{r}_{{n}_{j}}}^{{f}_{i}}{x}_{{n}_{j}}+{T}_{{r}_{{n}_{j}}}^{{f}_{i}}{x}_{{n}_{j}}âˆ’{T}_{r}^{{f}_{i}}pâˆ¥\\ â‰¤& \underset{jâ†’\mathrm{âˆž}}{limâ€‰inf}\left\{âˆ¥{x}_{{n}_{j}}âˆ’{T}_{{r}_{{n}_{j}}}^{{f}_{i}}{x}_{{n}_{j}}âˆ¥+âˆ¥{T}_{{r}_{{n}_{j}}}^{{f}_{i}}{x}_{{n}_{j}}âˆ’{T}_{r}^{{f}_{i}}pâˆ¥\right\}\\ =& \underset{jâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{T}_{{r}_{{n}_{j}}}^{{f}_{i}}{x}_{{n}_{j}}âˆ’{T}_{r}^{{f}_{i}}pâˆ¥\\ =& \underset{jâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{T}_{r}^{{f}_{i}}pâˆ’{T}_{{r}_{{n}_{j}}}^{{f}_{i}}{x}_{{n}_{j}}âˆ¥\\ â‰¤& \underset{jâ†’\mathrm{âˆž}}{limâ€‰inf}\left(âˆ¥{x}_{{n}_{j}}âˆ’pâˆ¥+\frac{|{r}_{{n}_{j}}âˆ’r|}{{r}_{{n}_{j}}}âˆ¥{T}_{{r}_{{n}_{j}}}^{{f}_{i}}{x}_{{n}_{j}}âˆ’{x}_{{n}_{j}}âˆ¥\right)\phantom{\rule{1em}{0ex}}\text{(by Lemma 2.5)}\\ =& \underset{jâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{{n}_{j}}âˆ’pâˆ¥,\end{array}$

this is a contradiction. So this shows that $pâˆˆ{â‹‚}_{i=1}^{k}F\left({T}_{r}^{{f}_{i}}\right)={â‹‚}_{i=1}^{k}EP\left({f}_{i}\right)$. Similarly, we can prove $ApâˆˆEP\left(g\right)$.

Finally, we prove $\left\{{x}_{n}\right\}$ and $\left\{{u}_{n}^{i}\right\}$ converge weakly to $pâˆˆ\mathrm{Î©}$, while $\left\{{w}_{n}\right\}$ converges weakly to $ApâˆˆEP\left(g\right)$. Firstly, if there exists other subsequence of $\left\{{x}_{n}\right\}$ which is denoted by $\left\{{x}_{{n}_{t}}\right\}$ such that ${x}_{{n}_{t}}â‡€qâˆˆ\mathrm{Î©}$ with , then, by Opialâ€™s condition,

$\underset{tâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{{n}_{t}}âˆ’qâˆ¥<\underset{tâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{{n}_{t}}âˆ’pâˆ¥<\underset{tâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{{n}_{t}}âˆ’qâˆ¥.$

This is a contradiction. Hence $\left\{{x}_{n}\right\}$ and $\left\{{u}_{n}^{i}\right\}$ converge weakly to $pâˆˆ\mathrm{Î©}$, respectively.

On the other hand, by (3.15) we also have ${\mathrm{Ï„}}_{n}â‡€p$. Notice that $âˆ¥{w}_{n}âˆ’A{\mathrm{Ï„}}_{n}âˆ¥=âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥â†’0$ by (3.12), so we have ${\mathrm{Ï„}}_{n}â‡€Ap$ and ${w}_{n}â‡€Ap$. We obtain the desired result.â€ƒâ–¡

Corollary 3.1 Let C be a nonempty closed convex subset of${H}_{1}$and K a nonempty closed convex subset of${H}_{2}$, where${H}_{1}$and${H}_{2}$are two real Hilbert spaces. $f:CÃ—Câ†’\mathbf{R}$is a bi-function with. Let$A:{H}_{1}â†’{H}_{2}$be a bounded linear operator with the adjoint B and$g:KÃ—Kâ†’\mathbf{R}$a bi-function with. Let$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}\right\}$be sequences generated by

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ f\left({u}_{n},y\right)+\frac{1}{{r}_{n}}ã€ˆyâˆ’{u}_{n},{u}_{n}âˆ’{x}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}yâˆˆC,\hfill \\ g\left({w}_{n},z\right)+\frac{1}{{r}_{n}}ã€ˆzâˆ’{w}_{n},{w}_{n}âˆ’A{u}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}zâˆˆK,\hfill \\ {x}_{n+1}={P}_{C}\left({u}_{n}+\mathrm{Î¼}B\left({w}_{n}âˆ’A{u}_{n}\right)\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}nâˆˆ\mathbb{N},\hfill \end{array}$
(3.16)

where$\left\{{r}_{n}\right\}âŠ‚\left(0,+\mathrm{âˆž}\right)$with${limâ€‰inf}_{nâ†’\mathrm{âˆž}}{r}_{n}>0$, ${P}_{C}$is a projection operator from${H}_{1}$into C and$\mathrm{Î¼}âˆˆ\left(0,\frac{1}{{âˆ¥Bâˆ¥}^{2}}\right)$is a constant. Suppose that, then the sequences$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}\right\}$converge weakly to an element$pâˆˆ\mathrm{Î©}$, while$\left\{{w}_{n}\right\}$converges weakly to$ApâˆˆEP\left(g\right)$.

Theorem 3.2 (Strong convergence theorem)

Let C be a nonempty closed convex subset of${H}_{1}$and K a nonempty closed convex subset of${H}_{2}$, where${H}_{1}$and${H}_{2}$are two real Hilbert spaces. $âˆ§:=\left\{1,2,â€¦,k\right\}$denotes a finite index set. For any$iâˆˆâˆ§$, ${f}_{i}:CÃ—Câ†’\mathbf{R}$is a bi-function with. Let$A:{H}_{1}â†’{H}_{2}$be a bounded linear operator with the adjoint B and$g:KÃ—Kâ†’\mathbf{R}$a bi-function with. Let${C}_{1}=C$and$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}^{i}\right\}$ ($iâˆˆâˆ§$) be sequences generated by

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ {f}_{i}\left({u}_{n}^{i},y\right)+\frac{1}{{r}_{n}}ã€ˆyâˆ’{u}_{n}^{i},{u}_{n}^{i}âˆ’{x}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}yâˆˆC,iâˆˆâˆ§,\hfill \\ {\mathrm{Ï„}}_{n}=\frac{{u}_{n}^{1}+â‹¯+{u}_{n}^{k}}{k},\hfill \\ g\left({w}_{n},z\right)+\frac{1}{{r}_{n}}ã€ˆzâˆ’{w}_{n},{w}_{n}âˆ’A{\mathrm{Ï„}}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}zâˆˆK,\hfill \\ {y}_{n}={P}_{C}\left({\mathrm{Ï„}}_{n}+\mathrm{Î¼}B\left({w}_{n}âˆ’A{\mathrm{Ï„}}_{n}\right)\right),\hfill \\ {C}_{n+1}=\left\{vâˆˆ{C}_{n}:âˆ¥{y}_{n}âˆ’vâˆ¥â‰¤âˆ¥{\mathrm{Ï„}}_{n}âˆ’vâˆ¥â‰¤âˆ¥{x}_{n}âˆ’vâˆ¥\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right),\phantom{\rule{1em}{0ex}}nâˆˆ\mathbb{N},\hfill \end{array}$
(3.17)

where$\left\{{r}_{n}\right\}âŠ‚\left(0,+\mathrm{âˆž}\right)$with${limâ€‰inf}_{nâ†’\mathrm{âˆž}}{r}_{n}>0$, ${P}_{C}$is a projection operator from${H}_{1}$into C and$\mathrm{Î¼}âˆˆ\left(0,\frac{1}{{âˆ¥Bâˆ¥}^{2}}\right)$is a constant. Suppose that, then the sequences$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}^{i}\right\}$ ($iâˆˆâˆ§$) converge strongly to an element${x}^{âˆ—}âˆˆ\mathrm{Î©}$, while$\left\{{w}_{n}\right\}$converges strongly to$A{x}^{âˆ—}âˆˆEP\left(g\right)$.

Proof By Lemma 2.2, ${u}_{n}^{i}={T}_{{r}_{n}}^{{f}_{i}}{x}_{n}$ for all $iâˆˆâˆ§$, $nâˆˆ\mathbb{N}$ and ${w}_{n}={T}_{{r}_{n}}^{g}A{\mathrm{Ï„}}_{n}$ for all $nâˆˆ\mathbb{N}$. We claim for $nâˆˆ\mathbb{N}$. In fact $\mathrm{Î©}âŠ‚{C}_{n}$ for $nâˆˆ\mathbb{N}$. Indeed, let $pâˆˆ\mathrm{Î©}$, it follows from (3.7) and (3.8) that

$2\mathrm{Î¼}ã€ˆ{\mathrm{Ï„}}_{n}âˆ’p,B\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}ã€‰â‰¤âˆ’\mathrm{Î¼}{âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2},$
(3.18)

and

${âˆ¥B\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}â‰¤{âˆ¥Bâˆ¥}^{2}{âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}.$
(3.19)

From (3.17)-(3.19) we have

$\begin{array}{rcl}{âˆ¥{y}_{n}âˆ’pâˆ¥}^{2}& â‰¤& {âˆ¥{\mathrm{Ï„}}_{n}+\mathrm{Î¼}B\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ’pâˆ¥}^{2}\\ =& {âˆ¥{\mathrm{Ï„}}_{n}âˆ’pâˆ¥}^{2}+{âˆ¥\mathrm{Î¼}B\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}+2\mathrm{Î¼}ã€ˆ{\mathrm{Ï„}}_{n}âˆ’p,B\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}ã€‰\\ â‰¤& {âˆ¥{\mathrm{Ï„}}_{n}âˆ’pâˆ¥}^{2}+{\mathrm{Î¼}}^{2}{âˆ¥Bâˆ¥}^{2}{âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}âˆ’\mathrm{Î¼}{âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}\\ =& {âˆ¥{\mathrm{Ï„}}_{n}âˆ’pâˆ¥}^{2}âˆ’\mathrm{Î¼}\left(1âˆ’\mathrm{Î¼}{âˆ¥Bâˆ¥}^{2}\right){âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}\\ â‰¤& {âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’\mathrm{Î¼}\left(1âˆ’\mathrm{Î¼}{âˆ¥Bâˆ¥}^{2}\right){âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}.\end{array}$
(3.20)

Notice $\mathrm{Î¼}âˆˆ\left(0,\frac{1}{{âˆ¥Bâˆ¥}^{2}}\right)$, $\mathrm{Î¼}\left(1âˆ’\mathrm{Î¼}{âˆ¥Bâˆ¥}^{2}\right)>0$. It follows from (3.20) that

$âˆ¥{y}_{n}âˆ’pâˆ¥â‰¤âˆ¥{\mathrm{Ï„}}_{n}âˆ’pâˆ¥â‰¤âˆ¥{x}_{n}âˆ’pâˆ¥\phantom{\rule{1em}{0ex}}\text{for all}nâˆˆ\mathbb{N},$
(3.21)

this shows $pâˆˆ{C}_{n}$ for all $nâˆˆ\mathbb{N}$, so $\mathrm{Î©}âŠ‚{C}_{n}$ and for $nâˆˆ\mathbb{N}$.

We want to prove ${C}_{n}$ is a closed convex set for $nâˆˆ\mathbb{N}$. It is easy to verify that ${C}_{n}$ is closed for $nâˆˆ\mathbb{N}$, so it suffices to verify ${C}_{n}$ is convex for $nâˆˆ\mathbb{N}$. In fact, let ${v}_{1},{v}_{2}âˆˆ{C}_{n+1}$, for each $\mathrm{Î»}âˆˆ\left(0,1\right)$, we have

$\begin{array}{rcl}{âˆ¥{y}_{n}âˆ’\left(\mathrm{Î»}{v}_{1}+\left(1âˆ’\mathrm{Î»}\right){v}_{2}\right)âˆ¥}^{2}& =& {âˆ¥\mathrm{Î»}\left({y}_{n}âˆ’{v}_{1}\right)+\left(1âˆ’\mathrm{Î»}\right)\left({y}_{n}âˆ’{v}_{2}\right)âˆ¥}^{2}\\ =& \mathrm{Î»}{âˆ¥{y}_{n}âˆ’{v}_{1}âˆ¥}^{2}+\left(1âˆ’\mathrm{Î»}\right){âˆ¥{y}_{n}âˆ’{v}_{2}âˆ¥}^{2}âˆ’\mathrm{Î»}\left(1âˆ’\mathrm{Î»}\right){âˆ¥{v}_{1}âˆ’{v}_{2}âˆ¥}^{2}\\ â‰¤& \mathrm{Î»}{âˆ¥{\mathrm{Ï„}}_{n}âˆ’{v}_{1}âˆ¥}^{2}+\left(1âˆ’\mathrm{Î»}\right){âˆ¥{\mathrm{Ï„}}_{n}âˆ’{v}_{2}âˆ¥}^{2}âˆ’\mathrm{Î»}\left(1âˆ’\mathrm{Î»}\right){âˆ¥{v}_{1}âˆ’{v}_{2}âˆ¥}^{2}\\ =& {âˆ¥{\mathrm{Ï„}}_{n}âˆ’\left(\mathrm{Î»}{v}_{1}+\left(1âˆ’\mathrm{Î»}\right){v}_{2}\right)âˆ¥}^{2},\end{array}$

namely, $âˆ¥{y}_{n}âˆ’\left(\mathrm{Î»}{v}_{1}+\left(1âˆ’\mathrm{Î»}\right){v}_{2}\right)âˆ¥â‰¤âˆ¥{\mathrm{Ï„}}_{n}âˆ’\left(\mathrm{Î»}{v}_{1}+\left(1âˆ’\mathrm{Î»}\right){v}_{2}\right)âˆ¥$. Similarly, we have $âˆ¥{\mathrm{Ï„}}_{n}âˆ’\left(\mathrm{Î»}{v}_{1}+\left(1âˆ’\mathrm{Î»}\right){v}_{2}\right)âˆ¥â‰¤âˆ¥{x}_{n}âˆ’\left(\mathrm{Î»}{v}_{1}+\left(1âˆ’\mathrm{Î»}\right){v}_{2}\right)âˆ¥$, this shows $\mathrm{Î»}{v}_{1}+\left(1âˆ’\mathrm{Î»}\right){v}_{2}âˆˆ{C}_{n+1}$ and ${C}_{n+1}$ is a convex set for $nâˆˆ\mathbb{N}$.

By (iv) of Lemma 2.2, Î© is a closed convex set, so there exists a unique element $q={P}_{\mathrm{Î©}}\left({x}_{1}\right)âˆˆ\mathrm{Î©}âŠ‚{C}_{n}$. Since ${x}_{n}={P}_{{C}_{n}}\left({x}_{1}\right)$, we have $âˆ¥{x}_{n}âˆ’{x}_{1}âˆ¥â‰¤âˆ¥qâˆ’{x}_{1}âˆ¥$, which shows that $\left\{{x}_{n}\right\}$ is bounded. So are $\left\{{\mathrm{Ï„}}_{n}\right\}$ and $\left\{{y}_{n}\right\}$. Notice that ${C}_{n+1}âŠ‚{C}_{n}$ and ${x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right)âŠ‚{C}_{n}$, then

$âˆ¥{x}_{n+1}âˆ’{x}_{1}âˆ¥â‰¤âˆ¥{x}_{n}âˆ’{x}_{1}âˆ¥,\phantom{\rule{1em}{0ex}}nâ‰¥2.$
(3.22)

It follows that ${lim}_{nâ†’\mathrm{âˆž}}âˆ¥{x}_{n}âˆ’{x}_{0}âˆ¥$ exists.

For some $m,nâˆˆ\mathbb{N}$ with $m>n$, from ${x}_{m}={P}_{{C}_{m}}\left({x}_{1}\right)âŠ‚{C}_{n}$ and (2.1), we have

${âˆ¥{x}_{n}âˆ’{x}_{m}âˆ¥}^{2}+{âˆ¥{x}_{1}âˆ’{x}_{m}âˆ¥}^{2}={âˆ¥{x}_{n}âˆ’{P}_{{C}_{m}}\left({x}_{1}\right)âˆ¥}^{2}+{âˆ¥{x}_{1}âˆ’{P}_{{C}_{m}}\left({x}_{0}\right)âˆ¥}^{2}â‰¤{âˆ¥{x}_{n}âˆ’{x}_{1}âˆ¥}^{2}.$
(3.23)

By (3.22)-(3.23) we have ${lim}_{nâ†’\mathrm{âˆž}}âˆ¥{x}_{n}âˆ’{x}_{m}âˆ¥=0$, so $\left\{{x}_{n}\right\}$ is a Cauchy sequence. Let ${x}_{n}â†’{x}^{âˆ—}$.

Next we prove ${x}^{âˆ—}âˆˆ\mathrm{Î©}$. Firstly, by ${x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right)âˆˆ{C}_{n+1}âŠ‚{C}_{n}$, from (3.17) we have

(3.24)

Again from (3.20), we have

$\begin{array}{rcl}{âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥}^{2}& â‰¤& \frac{1}{\mathrm{Î¼}\left(1âˆ’\mathrm{Î¼}{âˆ¥Bâˆ¥}^{2}\right)}\left\{{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{y}_{n}âˆ’pâˆ¥}^{2}\right\}\\ â‰¤& \frac{1}{\mathrm{Î¼}\left(1âˆ’\mathrm{Î¼}{âˆ¥Bâˆ¥}^{2}\right)}âˆ¥{x}_{n}âˆ’{y}_{n}âˆ¥\left\{âˆ¥{x}_{n}âˆ’pâˆ¥+âˆ¥{y}_{n}âˆ’pâˆ¥\right\}â†’0.\end{array}$
(3.25)

So

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥=0.$
(3.26)

Notice ${\mathrm{Ï„}}_{n}=\frac{{u}_{n}^{1}+â‹¯+{u}_{n}^{k}}{k}$, hence from (3.24) and (3.5), we obtain

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{T}_{{r}_{n}}^{{f}_{i}}{x}_{n}âˆ’{x}_{n}âˆ¥=\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{u}_{n}^{i}âˆ’{x}_{n}âˆ¥=0,\phantom{\rule{1em}{0ex}}iâˆˆâˆ§.$
(3.27)

Since ${x}_{n}â†’{x}^{âˆ—}$, (3.27) and Lemma 2.5 imply that for $r>0$,

$\begin{array}{rcl}âˆ¥{T}_{r}^{{f}_{i}}{x}^{âˆ—}âˆ’{x}^{âˆ—}âˆ¥& â‰¤& âˆ¥{T}_{r}^{{f}_{i}}{x}^{âˆ—}âˆ’{T}_{{r}_{n}}^{{f}_{i}}{x}_{n}âˆ¥+âˆ¥{T}_{{r}_{n}}^{{f}_{i}}{x}_{n}âˆ’{x}_{n}âˆ¥+âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥\\ â‰¤& âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥+\frac{|{r}_{n}âˆ’r|}{{r}_{n}}âˆ¥{T}_{{r}_{n}}^{{f}_{i}}{x}_{n}âˆ’{x}_{n}âˆ¥+âˆ¥{T}_{{r}_{n}}^{{f}_{i}}{x}_{n}âˆ’{x}_{n}âˆ¥+âˆ¥{x}_{n}âˆ’{x}^{âˆ—}âˆ¥â†’0,\end{array}$

which yields that ${x}^{âˆ—}âˆˆF\left({T}_{r}^{{f}_{i}}\right)$ for all $iâˆˆâˆ§$, further ${x}^{âˆ—}âˆˆ{â‹‚}_{i=1}^{k}EP\left({f}_{i}\right)$. Since A is a bounded linear operator, $âˆ¥A{x}_{n}âˆ’A{x}^{âˆ—}âˆ¥â†’0$ by ${x}_{n}â†’{x}^{âˆ—}$. Then for $r>0$, by (3.26) and Lemma 2.5 we have

$\begin{array}{rcl}âˆ¥{T}_{r}^{g}A{x}^{âˆ—}âˆ’A{x}^{âˆ—}âˆ¥& â‰¤& âˆ¥{T}_{r}^{g}A{x}^{âˆ—}âˆ’{T}_{{r}_{n}}^{g}A{x}_{n}âˆ¥+âˆ¥{T}_{{r}_{n}}^{g}A{x}_{n}âˆ’A{x}_{n}âˆ¥+âˆ¥A{x}_{n}âˆ’A{x}^{âˆ—}âˆ¥\\ â‰¤& âˆ¥A{x}_{n}âˆ’A{x}^{âˆ—}âˆ¥+\frac{|{r}_{n}âˆ’r|}{{r}_{n}}âˆ¥{T}_{{r}_{n}}^{g}A{x}_{n}âˆ’A{x}_{n}âˆ¥+âˆ¥{T}_{{r}_{n}}^{g}A{x}_{n}âˆ’A{x}_{n}âˆ¥\\ +âˆ¥A{x}_{n}âˆ’A{x}^{âˆ—}âˆ¥â†’0,\end{array}$

hence, $A{x}^{âˆ—}âˆˆF\left({T}_{r}^{g}\right)=EP\left(g\right)$ for $r>0$. Thus we have proved ${x}^{âˆ—}âˆˆ\mathrm{Î©}$, namely, $\left\{{x}_{n}\right\}$ converges strongly to ${x}^{âˆ—}âˆˆ\mathrm{Î©}$. Notice (3.27), we also have $\left\{{u}_{n}^{i}\right\}$ ($iâˆˆâˆ§$) converges strongly to ${x}^{âˆ—}âˆˆ\mathrm{Î©}$.

Since $âˆ¥{\mathrm{Ï„}}_{n}âˆ’{x}_{n}âˆ¥â†’0$ by (3.24), we have ${\mathrm{Ï„}}_{n}â†’{x}^{âˆ—}$ by ${x}_{n}â†’{x}^{âˆ—}$. Again from (3.26) we have

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{w}_{n}âˆ’A{\mathrm{Ï„}}_{n}âˆ¥=\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥\left({T}_{{r}_{n}}^{g}âˆ’I\right)A{\mathrm{Ï„}}_{n}âˆ¥=0,$

hence ${w}_{n}â†’A{x}^{âˆ—}âˆˆEP\left(g\right)$. The proof is completed.â€ƒâ–¡

Corollary 3.2 Let C be a nonempty closed convex subset of${H}_{1}$and K a nonempty closed convex subset of${H}_{2}$, where${H}_{1}$and${H}_{2}$are two real Hilbert spaces. $f:CÃ—Câ†’\mathbf{R}$is a bi-function with. Let$A:{H}_{1}â†’{H}_{2}$be a bounded linear operator with the adjoint B and$g:KÃ—Kâ†’\mathbf{R}$a bi-function with. Let${C}_{1}=C$and$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}\right\}$be sequences generated by

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ f\left({u}_{n},y\right)+\frac{1}{{r}_{n}}ã€ˆyâˆ’{u}_{n},{u}_{n}âˆ’{x}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}yâˆˆC,\hfill \\ g\left({w}_{n},z\right)+\frac{1}{{r}_{n}}ã€ˆzâˆ’{w}_{n},{w}_{n}âˆ’A{u}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}zâˆˆK,\hfill \\ {y}_{n}={P}_{C}\left({u}_{n}+\mathrm{Î¼}B\left({w}_{n}âˆ’A{u}_{n}\right)\right),\hfill \\ {C}_{n+1}=\left\{vâˆˆ{C}_{n}:âˆ¥{y}_{n}âˆ’vâˆ¥â‰¤âˆ¥{u}_{n}âˆ’vâˆ¥â‰¤âˆ¥{x}_{n}âˆ’vâˆ¥\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right),\phantom{\rule{1em}{0ex}}nâˆˆ\mathbb{N},\hfill \end{array}$
(3.28)

where$\left\{{r}_{n}\right\}âŠ‚\left(0,+\mathrm{âˆž}\right)$with${limâ€‰inf}_{nâ†’\mathrm{âˆž}}{r}_{n}>0$, $\mathrm{Î¼}âˆˆ\left(0,\frac{1}{{âˆ¥Bâˆ¥}^{2}}\right)$is a constant. Suppose that. Then the sequences$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}\right\}$converge strongly to an element${x}^{âˆ—}âˆˆ\mathrm{Î©}$, while$\left\{{w}_{n}\right\}$converges strongly to$A{x}^{âˆ—}âˆˆEP\left(g\right)$.

If $C={H}_{1}$ and $K={H}_{2}$ in Theorem 3.1 and Theorem 3.2, we have the following corollaries.

Corollary 3.3 Let${H}_{1}$and${H}_{2}$be two real Hilbert spaces. $âˆ§:=\left\{1,2,â€¦,k\right\}$denotes a finite index set. For any$iâˆˆâˆ§$, ${f}_{i}:{H}_{1}Ã—{H}_{1}â†’\mathbf{R}$is a bi-function with. Let$A:{H}_{1}â†’{H}_{2}$be a bounded linear operator with the adjoint B and$g:{H}_{2}Ã—{H}_{2}â†’\mathbf{R}$a bi-function with. Let$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}^{i}\right\}$be sequences generated by

$\left\{\begin{array}{c}{x}_{1}âˆˆ{H}_{1},\hfill \\ {f}_{i}\left({u}_{n}^{i},y\right)+\frac{1}{{r}_{n}}ã€ˆyâˆ’{u}_{n}^{i},{u}_{n}^{i}âˆ’{x}_{n}ã€‰â‰¥0,yâˆˆ{H}_{1},\phantom{\rule{1em}{0ex}}iâˆˆâˆ§,\hfill \\ {\mathrm{Ï„}}_{n}=\frac{{u}_{n}^{1}+â‹¯+{u}_{n}^{k}}{k},\hfill \\ g\left({w}_{n},z\right)+\frac{1}{{r}_{n}}ã€ˆzâˆ’{w}_{n},{w}_{n}âˆ’A{\mathrm{Ï„}}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}zâˆˆ{H}_{2},\hfill \\ {x}_{n+1}={\mathrm{Ï„}}_{n}+\mathrm{Î¼}B\left({w}_{n}âˆ’A{\mathrm{Ï„}}_{n}\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}nâˆˆ\mathbb{N},\hfill \end{array}$
(3.29)

where$\left\{{r}_{n}\right\}âŠ‚\left(0,+\mathrm{âˆž}\right)$with${limâ€‰inf}_{nâ†’\mathrm{âˆž}}{r}_{n}>0$, $\mathrm{Î¼}âˆˆ\left(0,\frac{1}{{âˆ¥Bâˆ¥}^{2}}\right)$is a constant. Suppose that. Then the sequences$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}^{i}\right\}$ ($iâˆˆâˆ§$) converge weakly to an element$pâˆˆ\mathrm{Î©}$, while$\left\{{w}_{n}\right\}$converges weakly to$ApâˆˆEP\left(g\right)$.

Corollary 3.4 Let${H}_{1}$and${H}_{2}$be two real Hilbert spaces. $f:{H}_{1}Ã—{H}_{1}â†’\mathbf{R}$is a bi-function with. Let$A:{H}_{1}â†’{H}_{2}$be a bounded linear operator with the adjoint B and$g:{H}_{2}Ã—{H}_{2}â†’\mathbf{R}$a bi-function with. Let$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}\right\}$be sequences generated by

$\left\{\begin{array}{c}{x}_{1}âˆˆ{H}_{1},\hfill \\ f\left({u}_{n},y\right)+\frac{1}{{r}_{n}}ã€ˆyâˆ’{u}_{n},{u}_{n}âˆ’{x}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}yâˆˆ{H}_{1},\hfill \\ g\left({w}_{n},z\right)+\frac{1}{{r}_{n}}ã€ˆzâˆ’{w}_{n},{w}_{n}âˆ’A{u}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}zâˆˆ{H}_{2},\hfill \\ {x}_{n+1}={u}_{n}+\mathrm{Î¼}B\left({w}_{n}âˆ’A{u}_{n}\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}nâˆˆ\mathbb{N},\hfill \end{array}$
(3.30)

where$\left\{{r}_{n}\right\}âŠ‚\left(0,+\mathrm{âˆž}\right)$with${limâ€‰inf}_{nâ†’\mathrm{âˆž}}{r}_{n}>0$, $\mathrm{Î¼}âˆˆ\left(0,\frac{1}{{âˆ¥Bâˆ¥}^{2}}\right)$is a constant. Suppose that. Then the sequences$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}\right\}$converge weakly to an element$pâˆˆ\mathrm{Î©}$, while$\left\{{w}_{n}\right\}$converges weakly to$ApâˆˆEP\left(g\right)$.

Corollary 3.5 Let${H}_{1}$and${H}_{2}$be two real Hilbert spaces. $âˆ§:=\left\{1,2,â€¦,k\right\}$denotes a finite index set. For any$iâˆˆâˆ§$, ${f}_{i}:{H}_{1}Ã—{H}_{1}â†’\mathbf{R}$is a bi-function with. Let$A:{H}_{1}â†’{H}_{2}$be a bounded linear operator with the adjoint B and$g:{H}_{2}Ã—{H}_{2}â†’\mathbf{R}$a bi-function with. Let${C}_{1}={H}_{1}$and$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}^{i}\right\}$ ($iâˆˆâˆ§$) be sequences generated by

$\left\{\begin{array}{c}{x}_{1}âˆˆ{H}_{1},\hfill \\ {f}_{i}\left({u}_{n}^{i},y\right)+\frac{1}{{r}_{n}}ã€ˆyâˆ’{u}_{n}^{i},{u}_{n}^{i}âˆ’{x}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}yâˆˆ{H}_{1},iâˆˆâˆ§,\hfill \\ {\mathrm{Ï„}}_{n}=\frac{{u}_{n}^{1}+â‹¯+{u}_{n}^{k}}{k},\hfill \\ g\left({w}_{n},z\right)+\frac{1}{{r}_{n}}ã€ˆzâˆ’{w}_{n},{w}_{n}âˆ’A{\mathrm{Ï„}}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}zâˆˆ{H}_{2},\hfill \\ {y}_{n}={\mathrm{Ï„}}_{n}+\mathrm{Î¼}B\left({w}_{n}âˆ’A{\mathrm{Ï„}}_{n}\right),\hfill \\ {C}_{n+1}=\left\{vâˆˆ{C}_{n}:âˆ¥{y}_{n}âˆ’vâˆ¥â‰¤âˆ¥{\mathrm{Ï„}}_{n}âˆ’vâˆ¥â‰¤âˆ¥{x}_{n}âˆ’vâˆ¥\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right),\phantom{\rule{1em}{0ex}}nâˆˆ\mathbb{N},\hfill \end{array}$
(3.31)

where$\left\{{r}_{n}\right\}âŠ‚\left(0,+\mathrm{âˆž}\right)$with${limâ€‰inf}_{nâ†’\mathrm{âˆž}}{r}_{n}>0$, $\mathrm{Î¼}âˆˆ\left(0,\frac{1}{{âˆ¥Bâˆ¥}^{2}}\right)$is a constant. Suppose that. Then the sequences$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}^{i}\right\}$ ($iâˆˆâˆ§$) converge strongly to an element$pâˆˆ\mathrm{Î©}$, while$\left\{{w}_{n}\right\}$converges strongly to$A{x}^{âˆ—}âˆˆEP\left(g\right)$.

Corollary 3.6 Let${H}_{1}$and${H}_{2}$be two real Hilbert spaces. $f:{H}_{1}Ã—{H}_{1}â†’\mathbf{R}$is a bi-function with. Let$A:{H}_{1}â†’{H}_{2}$be a bounded linear operator with the adjoint B and$g:{H}_{2}Ã—{H}_{2}â†’\mathbf{R}$a bi-function with. Let${C}_{1}={H}_{1}$and$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}\right\}$be sequences generated by

$\left\{\begin{array}{c}{x}_{1}âˆˆ{H}_{1},\hfill \\ f\left({u}_{n},y\right)+\frac{1}{{r}_{n}}ã€ˆyâˆ’{u}_{n},{u}_{n}âˆ’{x}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}yâˆˆ{H}_{1},\hfill \\ g\left({w}_{n},z\right)+\frac{1}{{r}_{n}}ã€ˆzâˆ’{w}_{n},{w}_{n}âˆ’A{u}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}zâˆˆ{H}_{2},\hfill \\ {y}_{n}={u}_{n}+\mathrm{Î¼}B\left({w}_{n}âˆ’A{u}_{n}\right),\hfill \\ {C}_{n+1}=\left\{vâˆˆ{C}_{n}:âˆ¥{y}_{n}âˆ’vâˆ¥â‰¤âˆ¥{u}_{n}âˆ’vâˆ¥â‰¤âˆ¥{x}_{n}âˆ’vâˆ¥\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right),\phantom{\rule{1em}{0ex}}nâˆˆ\mathbb{N},\hfill \end{array}$
(3.32)

where$\left\{{r}_{n}\right\}âŠ‚\left(0,+\mathrm{âˆž}\right)$with${limâ€‰inf}_{nâ†’\mathrm{âˆž}}{r}_{n}>0$, $\mathrm{Î¼}âˆˆ\left(0,\frac{1}{{âˆ¥Bâˆ¥}^{2}}\right)$is a constant. Suppose that. Then the sequences$\left\{{x}_{n}\right\}$and$\left\{{u}_{n}\right\}$converge strongly to an element${x}^{âˆ—}âˆˆ\mathrm{Î©}$, while$\left\{{w}_{n}\right\}$converges strongly to$A{x}^{âˆ—}âˆˆEP\left(g\right)$.

Remark 3.1 Since Example 1.1 and Example 1.2 satisfy the conditions of Corollary 3.1 and Corollary 3.2, the SEPs in Example 1.1 and Example 1.2 can be solved by the algorithm (3.16) and (3.28).

Remark 3.2 The results of this paper provide some solution algorithms for some SEPs; however, there are still some SEPs which cannot be solved by the results of this paper. The following examples belong to the case.

Example 3.1 Let ${H}_{2}=\mathbb{R}$ and ${H}_{1}={\mathbb{R}}^{2}$ with the norm $âˆ¥zâˆ¥={\left({x}^{2}+{y}^{2}\right)}^{\frac{1}{2}}$ for some $z=\left(x,y\right)âˆˆ{\mathbb{R}}^{2}$. $K:=\left[1,+\mathrm{âˆž}\right)$ and $C:=\left\{z=\left(x,y\right)âˆˆ{\mathbb{R}}^{2}|yâˆ’xâ‰¥1\right\}$. Define a bi-function $f\left(w,z\right)={x}_{1}+{y}_{1}+{y}_{2}âˆ’{x}_{2}$, where $w=\left({x}_{1},{y}_{1}\right)$, $z=\left({x}_{2},{y}_{2}\right)âˆˆC$, then f is a bi-function from $CÃ—C$ into $\mathbb{R}$ with $EP\left(f\right)=\left\{w=\left(x,y\right)|yâˆ’xâ‰¥1,x+yâ‰¥âˆ’1\right\}$. For each $z=\left(x,y\right)âˆˆ{H}_{1}$, let $Az=yâˆ’x$, then A is a bounded linear operator from ${H}_{1}$ into ${H}_{2}$. Now define another bi-function g as follows: $g\left(u,v\right)=vâˆ’u$ for all $u,vâˆˆK$. Then g is a bi-function from $KÃ—K$ into $\mathbb{R}$ with $EP\left(g\right)=\left\{1\right\}$.

Clearly, when $p=\left(x,y\right)âˆˆEP\left(f\right)$ with $yâˆ’x=1$ and $x+yâ‰¥âˆ’1$, we have $Ap=1âˆˆEP\left(g\right)$. So . However, because the bi-function f does not satisfy the conditions (A1)-(A4), the SEP in this example cannot be solved by Corollary 3.1 or Corollary 3.2.

Example 3.2 Let ${H}_{1}$, ${H}_{2}$, A and B be the same as Example 2.2. Let $C:=\left\{\mathrm{Î±}=\left({a}_{1},{a}_{2}\right)âˆˆ{H}_{1}|{a}_{2}âˆ’{a}_{1}â‰¥1\right\}$ and $K:=\left\{\mathrm{Î³}=\left({c}_{1},{c}_{2},{c}_{3}\right)âˆˆ{H}_{2}|âˆ¥\mathrm{Î³}âˆ¥â‰¤2\right\}$. Define a bi-function $f\left(\mathrm{Î±},\mathrm{Î²}\right)={\left({b}_{2}âˆ’{b}_{1}\right)}^{2}âˆ’{\left({a}_{1}+{a}_{2}\right)}^{2}$, where $\mathrm{Î±}=\left({a}_{1},{a}_{2}\right)$, $\mathrm{Î²}=\left({b}_{1},{b}_{2}\right)âˆˆC$, then f is a bi-function from $CÃ—C$ into $\mathbb{R}$ with $EP\left(f\right)=\left\{p=\left({p}_{1},{p}_{2}\right)|{p}_{2}âˆ’{p}_{1}â‰¥1â‰¥{p}_{1}+{p}_{2}â‰¥âˆ’1\right\}$. Define another bi-function $g\left(\mathrm{Î³},\mathrm{Î·}\right)={c}_{2}^{2}+{c}_{3}^{2}âˆ’\left({c}_{1}^{2}+{d}_{1}^{2}+{d}_{2}^{2}+{d}_{3}^{2}\right)$, where $\mathrm{Î³}=\left({c}_{1},{c}_{2},{c}_{3}\right)$, $\mathrm{Î·}=\left({d}_{1},{d}_{2},{d}_{3}\right)âˆˆK$, then $EP\left(g\right)=\left\{u=\left(0,{u}_{2},{u}_{3}\right)âˆˆK|{u}_{2}^{2}+{u}_{3}^{2}=2\right\}$.

Clearly, when $p=\left(âˆ’1,0\right)âˆˆEP\left(f\right)$, we have $Ap=\left(0,âˆ’1,âˆ’1\right)âˆˆEP\left(g\right)$. So . However, since all f and g do not satisfy the conditions (A1)-(A4), we cannot use the results obtained in this paper to solve the SEP in this example.

## 4 Conclusion

There are still many SEPs which do not satisfy the conditions (A1)-(A4), so we need to develop some new methods to solve these problems in the future.

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## Acknowledgement

The author was supported by the Natural Science Foundation of Yunnan Province (2010ZC152).

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Correspondence to Zhenhua He.

### Competing interests

The author declares that he has no competing interests.

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He, Z. The split equilibrium problem and its convergence algorithms. J Inequal Appl 2012, 162 (2012). https://doi.org/10.1186/1029-242X-2012-162