# New feasible iterative algorithms and strong convergence theorems for bilevel split equilibrium problems

## Abstract

In this paper, we first introduce and investigate a bilevel split equilibrium problem (BSEP) which can be regarded as a new development in the field of equilibrium problems. We provide some new feasible iterative algorithms for BSEP and establish strong convergence theorems for these iterative algorithms in different spaces.

MSC:47J25, 47H09, 65K10.

## 1 Introduction and preliminaries

Let K be a closed convex subset of a real Hilbert space H. Let $〈\cdot ,\cdot 〉$ and $\parallel \cdot \parallel$ denote the inner product of H and the norm of H, respectively. For each point $x\in H$, there exists a unique nearest point in K, denoted by ${P}_{K}x$, such that

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

1. (i)

$〈x-y,{P}_{K}x-{P}_{K}y〉\ge {\parallel {P}_{K}x-{P}_{K}y\parallel }^{2}$ for every $x,y\in H$.

2. (ii)

For $x\in H$ and $z\in K$, $z={P}_{K}\left(x\right)⇔〈x-z,z-y〉\ge 0$ for all $y\in K$.

3. (iii)

For $x\in H$ and $y\in K$,

${\parallel y-{P}_{K}\left(x\right)\parallel }^{2}+{\parallel x-{P}_{K}\left(x\right)\parallel }^{2}\le {\parallel x-y\parallel }^{2}.$
(1.1)

Let ${H}_{1}$ and ${H}_{2}$ be two Hilbert spaces. Let $A:{H}_{1}\to {H}_{2}$ and ${A}^{\ast }:{H}_{2}\to {H}_{1}$ be two bounded linear operators. ${A}^{\ast }$ is called the adjoint operator (or adjoint) of A if

It is known that the adjoint operator of a bounded linear operator on a Hilbert space always exists and is bounded linear and unique. Moreover, it is not hard to show that if ${A}^{\ast }$ is an adjoint operator of A, then $\parallel A\parallel =\parallel {A}^{\ast }\parallel$. The symbols and are used to denote the sets of positive integers and real numbers, respectively.

Example 1.1 ([1])

Let ${H}_{2}=\mathbb{R}$ with the standard norm $|\cdot |$ and ${H}_{1}={\mathbb{R}}^{2}$ with the norm $\parallel \alpha \parallel ={\left({a}_{1}^{2}+{a}_{2}^{2}\right)}^{\frac{1}{2}}$ for some $\alpha =\left({a}_{1},{a}_{2}\right)\in {\mathbb{R}}^{2}$. $〈x,y〉=xy$ denotes the inner product of ${H}_{2}$ for some $x,y\in {H}_{2}$, and $〈\alpha ,\beta 〉={\sum }_{i=1}^{2}{a}_{i}{b}_{i}$ denotes the inner product of ${H}_{1}$ for some $\alpha =\left({a}_{1},{a}_{2}\right),\beta =\left({b}_{1},{b}_{2}\right)\in {H}_{1}$. Let $A\alpha ={a}_{2}-{a}_{1}$ for $\alpha =\left({a}_{1},{a}_{2}\right)\in {H}_{1}$ and $Bx=\left(-x,x\right)$ for $x\in {H}_{2}$, then B is an adjoint operator of A.

Example 1.2 ([1])

Let ${H}_{1}={\mathbb{R}}^{2}$ with the norm $\parallel \alpha \parallel ={\left({a}_{1}^{2}+{a}_{2}^{2}\right)}^{\frac{1}{2}}$ for some $\alpha =\left({a}_{1},{a}_{2}\right)\in {\mathbb{R}}^{2}$ and ${H}_{2}={\mathbb{R}}^{3}$ with the norm $\parallel \gamma \parallel ={\left({c}_{1}^{2}+{c}_{2}^{2}+{c}_{3}^{2}\right)}^{\frac{1}{2}}$ for some $\gamma =\left({c}_{1},{c}_{2},{c}_{3}\right)\in {\mathbb{R}}^{3}$. Let $〈\alpha ,\beta 〉={\sum }_{i=1}^{2}{a}_{i}{b}_{i}$ and $〈\gamma ,\eta 〉={\sum }_{i=1}^{3}{c}_{i}{d}_{i}$ denote the inner product of ${H}_{1}$ and ${H}_{2}$, respectively, where $\alpha =\left({a}_{1},{a}_{2}\right),\beta =\left({b}_{1},{b}_{2}\right)\in {H}_{1}$, $\gamma =\left({c}_{1},{c}_{2},{c}_{3}\right),\eta =\left({d}_{1},{d}_{2},{d}_{3}\right)\in {H}_{2}$. Let $A\alpha =\left({a}_{2},{a}_{1},{a}_{1}-{a}_{2}\right)$ for $\alpha =\left({a}_{1},{a}_{2}\right)\in {H}_{1}$ and $B\gamma =\left({c}_{2}+{c}_{3},{c}_{1}-{c}_{3}\right)$ for $\gamma =\left({c}_{1},{c}_{2},{c}_{3}\right)\in {H}_{2}$. Obviously, B is an adjoint operator of A.

Let f be a bi-function from $C×C$ to . The classical equilibrium problem (EP, for short) is defined as follows.

(EP) Find $p\in C$ such that $f\left(p,y\right)\ge 0$, $\mathrm{\forall }y\in C$.

The set of such solutions is denoted by $EP\left(f\right)$, that is, $EP\left(f\right)=\left\{u\in C:f\left(u,v\right)\ge 0,\mathrm{\forall }v\in C\right\}$. In fact, equilibrium problem has an important relationship with variational inequality problem. For example, let $T:C\to H$ be a nonlinear mapping satisfying $〈Tx,y-x〉\ge 0$ for all $x,y\in C$. Then $x\in EP\left(f\right)$ if and only if $x\in C$ is a solution of the variational inequality $〈Tx,y-x〉\ge 0$ for all $y\in C$. It is known that the EP is an important mathematical model for nonlinear analysis and applied sciences which is generalized to many new mathematical models and includes many important problems arising in physics, engineering, science optimization, economics, network, game theory, complementary problems, variational inequalities problems, saddle point problems, fixed point problems and others; for details, one can refer to [28] and references therein. Many authors have proposed some useful methods to solve the EP; see, for instance, [25, 917] and references therein.

Recent investigations and developments in equilibrium theory as well as optimization theory have been applied to connect fundamental sciences with the real world. According to our experience, useful methods of real world problems often need to be used to solve several problems arising in different spaces. In view of this, recent studies focus on split problems which are more closed in the real world applications; see, for instance, [1, 1824] and the references therein. Recently, He [1] considered the following split equilibrium problem. Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces. Let C be a closed convex subset of ${H}_{1}$ and K be a closed convex subset of ${H}_{2}$. Let $f:C×C\to \mathbb{R}$ and $g:K×K\to \mathbb{R}$ be two bi-functions, and $A:{H}_{1}\to {H}_{2}$ be a bounded linear operator. The split equilibrium problem (SEP, in short) is defined as follows:

(SEP) Find $p\in C$ such that $f\left(p,y\right)\ge 0$, $\mathrm{\forall }y\in C$, and $u:=Ap$ satisfying $g\left(u,v\right)\ge 0$, $\mathrm{\forall }v\in K$.

In [1], the author established weak convergence algorithms and strong convergence algorithms for SEP (see [1] for more details).

Motivated and inspired by the works mentioned above, in this paper we shall introduce and investigate the following new problem. Let ${H}_{1}$, ${H}_{2}$ and ${H}_{3}$ be three real Hilbert spaces. Let C be a closed convex subset of ${H}_{1}$, Q be a closed convex subset of ${H}_{2}$ and K be a closed convex subset of ${H}_{3}$. Let $f:C×C\to \mathbb{R}$, $g:Q×Q\to \mathbb{R}$ and $h:K×K\to \mathbb{R}$ be three bi-functions. Let $A:{H}_{1}\to {H}_{3}$ and $B:{H}_{2}\to {H}_{3}$ be two bounded linear operators with theirs adjoint operators ${A}^{\ast }$ and ${B}^{\ast }$, respectively. The mathematical model about bilevel split equilibrium problem (BSEP, in short) is defined as follows:

(BSEP) Find $p\in C$ and $q\in Q$ such that

1. (i)

$f\left(p,x\right)\ge 0$ and $g\left(q,y\right)\ge 0$ for all $x\in C$ and $y\in Q$;

2. (ii)

$Ap=Bq:=u$;

3. (iii)

$h\left(u,z\right)\ge 0$ for all $z\in K$.

In fact, BSEP can be regarded as a new development in the field of equilibrium problems and contains several important problems as special cases. It was profoundly believed that BSEP will motivate and inspire further scientific activities in the fields of equilibrium problems, optimization problems, game problems, complementary problems, variational inequalities problems, fixed point problems and their applications.

Example A Let ${H}_{1}$, ${H}_{2}$ and ${H}_{3}$ be three real Hilbert spaces. Let $C\subset {H}_{1}$, $Q\subset {H}_{2}$ and $K\subset {H}_{3}$ be three closed convex sets. Let ${f}^{\ast }:C\to \mathbb{R}$, ${g}^{\ast }:Q\to \mathbb{R}$ and ${h}^{\ast }:K\to \mathbb{R}$ be three convex functions. Let $A:{H}_{1}\to {H}_{3}$ and $B:{H}_{2}\to {H}_{3}$ be two bounded linear operators with their adjoint operators ${A}^{\ast }$ and ${B}^{\ast }$, respectively. Let

and

Then BSEP reduces the bilevel convex optimization problem (BCOP):

(BCOP) Find $p\in C$ and $q\in Q$ such that $u:=Ap=Bq\in K$, ${f}^{\ast }\left(x\right)\ge {f}^{\ast }\left(p\right)$, ${g}^{\ast }\left(y\right)\ge {g}^{\ast }\left(q\right)$ and ${h}^{\ast }\left(z\right)\ge {h}^{\ast }\left(u\right)$ for all $x\in C$, $y\in Q$ and $z\in K$.

Example B Let ${H}_{1}$, ${H}_{2}$ and ${H}_{3}$ be three real Hilbert spaces. Let $C\subset {H}_{1}$, $Q\subset {H}_{2}$ and $K\subset {H}_{3}$ be three closed convex sets. Let $T:C\to {H}_{1}$, $S:Q\to {H}_{2}$ and $G:K\to {H}_{3}$ be three nonlinear operators. Let $A:{H}_{1}\to {H}_{3}$ and $B:{H}_{2}\to {H}_{3}$ be two bounded linear operators with their adjoint operators ${A}^{\ast }$ and ${B}^{\ast }$, respectively. If $f\left(p,x\right)=〈Tp,x-p〉$, $g\left(q,y\right)=〈Sq,y-q〉$ and $h\left(u,z\right)=〈Tu,z-u〉$, then BSEP reduces to the bilevel split variational inequality problem (BSVI):

(BSVI) Find $p\in C$ and $q\in Q$ such that $u:=Ap=Bq\in K$ satisfying $〈Tp,x-p〉\ge 0$, $〈Sq,y-q〉\ge 0$ and $〈Tu,z-u〉\ge 0$ for all $x\in C$, $y\in Q$ and $z\in K$.

Example C Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces and $B:{H}_{1}\to {H}_{2}$ be a bounded linear operator with its adjoint operator ${B}^{\ast }$. Let $C\subset {H}_{1}$, $Q\subset {H}_{1}$ and $K\subset {H}_{2}$ be three closed convex sets. If ${H}_{1}={H}_{2}$ and $A=B$, then BSEP reduces to the following split equilibrium problem (1) (SEP${\left(1\right)}^{}$):

(${\mathrm{SEP}}^{\left(1\right)}$) Find $p\in C$ and $q\in Q$ such that $u:=Bp=Bq\in K$ satisfying $f\left(p,x\right)\ge 0$, $g\left(q,y\right)\ge 0$ and $h\left(u,z\right)\ge 0$ for all $x\in C$, $y\in Q$ and $z\in K$.

Example D Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces and $A:{H}_{1}\to {H}_{2}$ be a bounded linear operator with its adjoint operator ${A}^{\ast }$. Let $C\subset {H}_{1}$, $Q\subset {H}_{2}$ and $K\subset {H}_{2}$ be three closed convex sets with $Q\cap K\ne \mathrm{\varnothing }$. If ${H}_{2}={H}_{3}$ and $B=I$ (identity operator), then BSEP reduces to the following split equilibrium problem (2) (SEP${\left(2\right)}^{}$):

(${\mathrm{SEP}}^{\left(2\right)}$) Find $p\in C$ such that $u:=Ap\in Q\cap K$ satisfying $f\left(p,x\right)\ge 0$, $g\left(u,y\right)\ge 0$ and $h\left(u,z\right)\ge 0$ for each $x\in C$, $y\in Q$ and $z\in K$.

Especially, if $g\left(p,y\right)\equiv 0$ for all $p,y\in Q$, then (SEP${\left(1\right)}^{}$) reduces to finding $p\in C$ such that $u:=Ap\in K$ satisfying $f\left(p,x\right)\ge 0$ and $h\left(u,z\right)\ge 0$ for all $x\in C$ and $z\in K$, which was studied in [1].

Example E Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces and $B:{H}_{1}\to {H}_{2}$ be a bounded linear operator with its adjoint operator ${A}^{\ast }$. Let $C\subset {H}_{1}$, $Q\subset {H}_{1}$ and $K\subset {H}_{2}$ be three closed convex sets with $C\cap Q\ne \mathrm{\varnothing }$. If ${H}_{1}={H}_{2}$ and $A=I$ (identity operator), then BSEP reduces to the following split equilibrium problems (3) (SEP${\left(3\right)}^{}$):

(${\mathrm{SEP}}^{\left(3\right)}$) Find $p\in C\cap Q$ such that $u:=Bp\in K$ satisfying $f\left(p,x\right)\ge 0$, $g\left(p,y\right)\ge 0$ and $h\left(u,z\right)\ge 0$ for all $x\in C$, $y\in Q$ and $z\in K$.

Example F In Example A, if ${H}_{1}={H}_{2}={H}_{3}:=H$, $C=Q=K\subset H$ and $A=B=I$ (identity operator), then BSEP reduces to the common solution of equilibrium problems (CEP):

(CEP) Find $p\in C$ such that $f\left(p,x\right)\ge 0$, $g\left(p,y\right)\ge 0$ and $h\left(p,z\right)\ge 0$ for each $x,y,z\in C$.

The paper is divided into four sections. In Sections 1 and 2, we first introduce and investigate a bilevel split equilibrium problem (BSEP) and then provide some new feasible iterative algorithms for BSEP and establish strong convergence theorems for these iterative algorithms in different spaces. In Section 3, we give the proof of the main result Theorem 2.1 in detail. Finally, an example illustrating Theorem 2.1 is given in Section 4.

## 2 Feasible iterative algorithms for BSEP and their strong convergence theorems

In 1994, Blum and Oettli [2] established the following important existence theorem which plays a key role in solving equilibrium problems, variational inequality problems and optimization problems.

Lemma 2.1 (Blum and Oettli [2])

Let K be a nonempty closed convex subset of H and F be a bi-function of $K×K$ into satisfying the following conditions.

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

(A2) F is monotone, that is, $F\left(x,y\right)+F\left(y,x\right)\le 0$ for all $x,y\in K$;

(A3) for each $x,y,z\in K$,

$\underset{t\to {0}^{+}}{lim sup}F\left(tz+\left(1-t\right)x,y\right)\le F\left(x,y\right);$

(A4) for each $x\in K$, $y↦F\left(x,y\right)$ is convex and lower semi-continuous.

Let $r>0$ and $x\in H$. Then there exists $z\in K$ such that

$F\left(z,y\right)+\frac{1}{r}〈y-z,z-x〉\ge 0\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}y\in K.$

In this paper, we first introduce a new iterative algorithm for BSEP and establish a strong convergence theorem for this iterative algorithm. Here, the space ${H}_{1}×{H}_{2}$ denotes the product space of two real Hilbert spaces ${H}_{1}$ and ${H}_{2}$, which is endowed with the usual linear operation and norm, namely, for $\left(x,y\right),\left(\overline{x},\overline{y}\right)\in {H}_{1}×{H}_{2}$ and $a,b\in \mathbb{R}$,

$a\left(x,y\right)+b\left(\overline{x},\overline{y}\right)=\left(ax+b\overline{x},ay+b\overline{y}\right)$

and

$\parallel \left(x,y\right)\parallel =\parallel x\parallel +\parallel y\parallel .$

Theorem 2.1 Let ${H}_{1}$, ${H}_{2}$ and ${H}_{3}$ be three real Hilbert spaces. Let C be a closed convex subset of ${H}_{1}$, Q be a closed convex subset of ${H}_{2}$ and K be a closed convex subset of ${H}_{3}$. Let $f:C×C\to \mathbb{R}$, $g:Q×Q\to \mathbb{R}$ and $h:K×K\to \mathbb{R}$ be three bi-functions. $A:{H}_{1}\to {H}_{3}$ and $B:{H}_{2}\to {H}_{3}$ are two bounded linear operators with their adjoint operators ${A}^{\ast }$ and ${B}^{\ast }$, respectively. Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let ${x}_{1}\in C$, ${y}_{1}\in Q$, ${C}_{1}=C$, ${Q}_{1}=Q$, $\left\{{x}_{n}\right\}$, $\left\{{y}_{n}\right\}$, $\left\{{u}_{n}\right\}$, $\left\{{v}_{n}\right\}$ and $\left\{{w}_{n}\right\}$ be sequences generated by

$\left\{\begin{array}{c}{u}_{n}={T}_{{r}_{n}}^{f}{x}_{n},\phantom{\rule{2em}{0ex}}{v}_{n}={T}_{{r}_{n}}^{g}{y}_{n},\phantom{\rule{2em}{0ex}}{w}_{n}={T}_{{r}_{n}}^{h}\left(\frac{1}{2}A{u}_{n}+\frac{1}{2}B{v}_{n}\right),\hfill \\ {l}_{n}={P}_{C}\left({u}_{n}-\xi {A}^{\ast }\left(A{u}_{n}-{w}_{n}\right)\right),\phantom{\rule{2em}{0ex}}{k}_{n}={P}_{Q}\left({v}_{n}-\xi {B}^{\ast }\left(B{v}_{n}-{w}_{n}\right)\right),\hfill \\ {C}_{n+1}×{Q}_{n+1}=\left\{\left(x,y\right)\in {C}_{n}×{Q}_{n}:{\parallel {l}_{n}-x\parallel }^{2}+{\parallel {k}_{n}-y\parallel }^{2}\hfill \\ \phantom{{C}_{n+1}×{Q}_{n+1}=\left\{}\le {\parallel {u}_{n}-x\parallel }^{2}+{\parallel {v}_{n}-y\parallel }^{2}\le {\parallel {x}_{n}-x\parallel }^{2}+{\parallel {y}_{n}-y\parallel }^{2}\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right),\hfill \\ {y}_{n+1}={P}_{{Q}_{n+1}}\left({y}_{1}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$
(2.1)

where $\xi \in \left(0,min\left(\frac{1}{{\parallel A\parallel }^{2}},\frac{1}{{\parallel B\parallel }^{2}}\right)\right)$ and $\left\{{r}_{n}\right\}\subset \left(0,+\mathrm{\infty }\right)$ with ${lim inf}_{n\to +\mathrm{\infty }}{r}_{n}>0$, ${P}_{C}$ and ${P}_{Q}$ are two projection operators from ${H}_{1}$ into C and from ${H}_{2}$ into Q, respectively. Suppose that

$\mathrm{\Omega }=\left\{\left(p,q\right)\in EP\left(f\right)×EP\left(g\right):Ap=Bq\in EP\left(h\right)\right\}\ne \mathrm{\varnothing }.$

Then there exists $\left(p,q\right)\in \mathrm{\Omega }$ such that

1. (a)

$\left({x}_{n},{y}_{n}\right)\to \left(p,q\right)$ as $n\to \mathrm{\infty }$;

2. (b)

$\left({u}_{n},{v}_{n}\right)\to \left(p,q\right)$ as $n\to \mathrm{\infty }$;

3. (c)

${w}_{n}\to {w}^{\ast }:=Ap=Bq\in EP\left(h\right)$ as $n\to \mathrm{\infty }$.

The following conclusion is immediate from Theorem 2.1 by putting $A=B$.

Corollary 2.1 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces. Let C and Q be two closed convex subsets of ${H}_{1}$ and K be a closed convex subset of ${H}_{2}$. Let $f:C×C\to \mathbb{R}$, $g:Q×Q\to \mathbb{R}$ and $h:K×K\to \mathbb{R}$ be three bi-functions. $B:{H}_{1}\to {H}_{2}$ is a bounded linear operator with its adjoint operator ${B}^{\ast }$. Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let ${x}_{1}\in C$, ${y}_{1}\in Q$, ${C}_{1}=C$, ${Q}_{1}=Q$, $\left\{{x}_{n}\right\}$, $\left\{{y}_{n}\right\}$, $\left\{{u}_{n}\right\}$, $\left\{{v}_{n}\right\}$ and $\left\{{w}_{n}\right\}$ be sequences generated by

$\left\{\begin{array}{c}{u}_{n}={T}_{{r}_{n}}^{f}{x}_{n},\phantom{\rule{2em}{0ex}}{v}_{n}={T}_{{r}_{n}}^{g}{y}_{n},\phantom{\rule{2em}{0ex}}{w}_{n}={T}_{{r}_{n}}^{h}\left(\frac{1}{2}B{u}_{n}+\frac{1}{2}B{v}_{n}\right),\hfill \\ {l}_{n}={P}_{C}\left({u}_{n}-\xi {B}^{\ast }\left(B{u}_{n}-{w}_{n}\right)\right),\phantom{\rule{2em}{0ex}}{k}_{n}={P}_{Q}\left({v}_{n}-\xi {B}^{\ast }\left(B{v}_{n}-{w}_{n}\right)\right),\hfill \\ {C}_{n+1}×{Q}_{n+1}=\left\{\left(x,y\right)\in {C}_{n}×{Q}_{n}:{\parallel {l}_{n}-x\parallel }^{2}+{\parallel {k}_{n}-y\parallel }^{2}\le {\parallel {u}_{n}-x\parallel }^{2}+{\parallel {v}_{n}-y\parallel }^{2}\hfill \\ \phantom{{C}_{n+1}×{Q}_{n+1}=\left\{}\le {\parallel {x}_{n}-x\parallel }^{2}+{\parallel {y}_{n}-y\parallel }^{2}\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right),\hfill \\ {y}_{n+1}={P}_{{Q}_{n+1}}\left({y}_{1}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$

where $\xi \in \left(0,\frac{1}{{\parallel B\parallel }^{2}}\right)$ and $\left\{{r}_{n}\right\}\subset \left(0,+\mathrm{\infty }\right)$ with ${lim inf}_{n\to +\mathrm{\infty }}{r}_{n}>0$, ${P}_{C}$ and ${P}_{Q}$ are two projection operators from ${H}_{1}$ into C and from ${H}_{1}$ into Q, respectively. Suppose that

$\mathrm{\Omega }=\left\{\left(p,q\right)\in EP\left(f\right)×EP\left(g\right):Bp=Bq\in EP\left(h\right)\right\}\ne \mathrm{\varnothing }.$

Then there exists $\left(p,q\right)\in \mathrm{\Omega }$ such that

1. (a)

$\left({x}_{n},{y}_{n}\right)\to \left(p,q\right)$ as $n\to \mathrm{\infty }$;

2. (b)

$\left({u}_{n},{v}_{n}\right)\to \left(p,q\right)$ as $n\to \mathrm{\infty }$;

3. (c)

${w}_{n}\to {w}^{\ast }:=Bp=Bq\in EP\left(h\right)$ as $n\to \mathrm{\infty }$.

If ${H}_{2}={H}_{3}$ and $B=I$, then Theorem 2.1 reduces to the following corollary.

Corollary 2.2 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces. Let $C\subset {H}_{1}$ and $Q,K\subset {H}_{2}$ be three closed convex sets. Let $f:C×C\to \mathbb{R}$, $g:Q×Q\to \mathbb{R}$ and $h:K×K\to \mathbb{R}$ be three bi-functions. $A:{H}_{1}\to {H}_{2}$ is a bounded linear operator with its adjoint operator ${A}^{\ast }$. Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let ${x}_{1}\in C$, ${y}_{1}\in Q$, ${C}_{1}=C$, ${Q}_{1}=Q$, $\left\{{x}_{n}\right\}$, $\left\{{y}_{n}\right\}$, $\left\{{u}_{n}\right\}$, $\left\{{v}_{n}\right\}$ and $\left\{{w}_{n}\right\}$ be sequences generated by

$\left\{\begin{array}{c}{u}_{n}={T}_{{r}_{n}}^{f}{x}_{n},\phantom{\rule{2em}{0ex}}{v}_{n}={T}_{{r}_{n}}^{g}{y}_{n},\phantom{\rule{2em}{0ex}}{w}_{n}={T}_{{r}_{n}}^{h}\left(\frac{1}{2}A{u}_{n}+\frac{1}{2}{v}_{n}\right),\hfill \\ {l}_{n}={P}_{C}\left({u}_{n}-\xi {A}^{\ast }\left(A{u}_{n}-{w}_{n}\right)\right),\phantom{\rule{2em}{0ex}}{k}_{n}={v}_{n}-\xi \left({v}_{n}-{w}_{n}\right),\hfill \\ {C}_{n+1}×{Q}_{n+1}=\left\{\left(x,y\right)\in {C}_{n}×{Q}_{n}:{\parallel {l}_{n}-x\parallel }^{2}+{\parallel {k}_{n}-y\parallel }^{2}\le {\parallel {u}_{n}-x\parallel }^{2}+{\parallel {v}_{n}-y\parallel }^{2}\hfill \\ \phantom{{C}_{n+1}×{Q}_{n+1}=\left\{}\le {\parallel {x}_{n}-x\parallel }^{2}+{\parallel {y}_{n}-y\parallel }^{2}\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right),\hfill \\ {y}_{n+1}={P}_{{Q}_{n+1}}\left({y}_{1}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$

where $\xi \in \left(0,min\left\{1,\frac{1}{{\parallel A\parallel }^{2}}\right\}\right)$ and $\left\{{r}_{n}\right\}\subset \left(0,+\mathrm{\infty }\right)$ with ${lim inf}_{n\to +\mathrm{\infty }}{r}_{n}>0$, ${P}_{C}$ and ${P}_{Q}$ are two projection operators from ${H}_{1}$ into C and from ${H}_{2}$ into Q, respectively. Suppose that

$\mathrm{\Omega }=\left\{p\in EP\left(f\right):Ap\in EP\left(g\right)\cap EP\left(h\right)\right\}\ne \mathrm{\varnothing }.$

Then there exists $p\in \mathrm{\Omega }$ such that

1. (a)

${x}_{n}\to p$ as $n\to \mathrm{\infty }$;

2. (b)

${u}_{n}\to p$ as $n\to \mathrm{\infty }$;

3. (c)

${v}_{n},{y}_{n},{w}_{n}\to {w}^{\ast }:=Ap$ as $n\to \mathrm{\infty }$.

If ${H}_{1}={H}_{2}$ and $A=I$, then Theorem 2.1 reduces to the following corollary.

Corollary 2.3 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces. Let $C,Q\subset {H}_{1}$ and $K\subset {H}_{2}$ be three closed convex sets. Let $f:C×C\to \mathbb{R}$, $g:Q×Q\to \mathbb{R}$ and $h:K×K\to \mathbb{R}$ be three bi-functions. $B:{H}_{1}\to {H}_{2}$ is a bounded linear operator with its adjoint operator ${B}^{\ast }$. Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let ${x}_{1}\in C$, ${y}_{1}\in Q$, ${C}_{1}=C$, ${Q}_{1}=Q$, $\left\{{x}_{n}\right\}$, $\left\{{y}_{n}\right\}$, $\left\{{u}_{n}\right\}$, $\left\{{v}_{n}\right\}$ and $\left\{{w}_{n}\right\}$ be sequences generated by

$\left\{\begin{array}{c}{u}_{n}={T}_{{r}_{n}}^{f}{x}_{n},\phantom{\rule{2em}{0ex}}{v}_{n}={T}_{{r}_{n}}^{g}{y}_{n},\phantom{\rule{2em}{0ex}}{w}_{n}={T}_{{r}_{n}}^{h}\left(\frac{1}{2}{u}_{n}+\frac{1}{2}B{v}_{n}\right),\hfill \\ {l}_{n}={u}_{n}-\xi \left({u}_{n}-{w}_{n}\right),\phantom{\rule{2em}{0ex}}{k}_{n}={P}_{q}\left({v}_{n}-\xi {B}^{\ast }\left(B{v}_{n}-{w}_{n}\right)\right),\hfill \\ {C}_{n+1}×{Q}_{n+1}=\left\{\left(x,y\right)\in {C}_{n}×{Q}_{n}:{\parallel {l}_{n}-x\parallel }^{2}+{\parallel {k}_{n}-y\parallel }^{2}\le {\parallel {u}_{n}-x\parallel }^{2}+{\parallel {v}_{n}-y\parallel }^{2}\hfill \\ \phantom{{C}_{n+1}×{Q}_{n+1}=\left\{}\le {\parallel {x}_{n}-x\parallel }^{2}+{\parallel {y}_{n}-y\parallel }^{2}\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right),\hfill \\ {y}_{n+1}={P}_{{Q}_{n+1}}\left({y}_{1}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$

where $\xi \in \left(0,min\left\{1,\frac{1}{{\parallel B\parallel }^{2}}\right\}\right)$ and $\left\{{r}_{n}\right\}\subset \left(0,+\mathrm{\infty }\right)$ with ${lim inf}_{n\to +\mathrm{\infty }}{r}_{n}>0$, ${P}_{C}$ and ${P}_{Q}$ are two projection operators from ${H}_{1}$ into C and from ${H}_{2}$ into Q, respectively. Suppose that

$\mathrm{\Omega }=\left\{p\in EP\left(f\right)\cap EP\left(g\right):Ap\in EP\left(h\right)\right\}\ne \mathrm{\varnothing }.$

Then there exists $p\in \mathrm{\Omega }$ such that

1. (a)

${x}_{n},{u}_{n}\to p$ as $n\to \mathrm{\infty }$;

2. (b)

${y}_{n},{v}_{n}\to p$ as $n\to \mathrm{\infty }$;

3. (c)

${w}_{n}\to {w}^{\ast }:=Ap$ as $n\to \mathrm{\infty }$.

Putting $A=B=I$ (identical operator), ${H}_{1}={H}_{2}={H}_{3}=H$ and $C=Q=K$, then we have the following result.

Corollary 2.4 Let H be a real Hilbert space. Let C be a closed convex subset of H. Let $f,g,h:C×C\to \mathbb{R}$ be three bi-functions. Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let ${x}_{1},{y}_{1}\in C$, ${C}_{1}=C$, $\left\{{x}_{n}\right\}$, $\left\{{y}_{n}\right\}$, $\left\{{u}_{n}\right\}$, $\left\{{v}_{n}\right\}$ and $\left\{{w}_{n}\right\}$ be sequences generated by

$\left\{\begin{array}{c}{u}_{n}={T}_{{r}_{n}}^{f}{x}_{n},\phantom{\rule{2em}{0ex}}{v}_{n}={T}_{{r}_{n}}^{g}{y}_{n},\phantom{\rule{2em}{0ex}}{w}_{n}={T}_{{r}_{n}}^{h}\left(\frac{1}{2}{u}_{n}+\frac{1}{2}{v}_{n}\right),\hfill \\ {l}_{n}={u}_{n}-\xi \left({u}_{n}-{w}_{n}\right),\phantom{\rule{2em}{0ex}}{k}_{n}={v}_{n}-\xi \left({v}_{n}-{w}_{n}\right),\hfill \\ {C}_{n+1}×{C}_{n+1}=\left\{\left(x,y\right)\in {C}_{n}×{C}_{n}:{\parallel {l}_{n}-x\parallel }^{2}+{\parallel {k}_{n}-y\parallel }^{2}\le {\parallel {u}_{n}-x\parallel }^{2}+{\parallel {v}_{n}-y\parallel }^{2}\hfill \\ \phantom{{C}_{n+1}×{C}_{n+1}=\left\{}\le {\parallel {x}_{n}-x\parallel }^{2}+{\parallel {y}_{n}-y\parallel }^{2}\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right),\hfill \\ {y}_{n+1}={P}_{{C}_{n+1}}\left({y}_{1}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$

where $\xi \in \left(0,1\right)$ and $\left\{{r}_{n}\right\}\subset \left(0,+\mathrm{\infty }\right)$ with ${lim inf}_{n\to +\mathrm{\infty }}{r}_{n}>0$, ${P}_{C}$ is a projection operator from H into C. Suppose that

$\mathrm{\Omega }=\left\{\left(p,q\right)\in EP\left(f\right)×EP\left(g\right):p=q\in EP\left(h\right)\right\}\ne \mathrm{\varnothing }.$

Then there exists $\left(p,q\right)\in \mathrm{\Omega }$ such that

1. (a)

$\left({x}_{n},{y}_{n}\right)\to \left(p,q\right)$ as $n\to \mathrm{\infty }$;

2. (b)

$\left({u}_{n},{v}_{n}\right)\to \left(p,q\right)$ as $n\to \mathrm{\infty }$;

3. (c)

${w}_{n}\to {w}^{\ast }:=p=q\in EP\left(h\right)$ as $n\to \mathrm{\infty }$.

Remark 2.1 In Corollary 2.2, it is obvious that

$\mathrm{\Omega }=\left\{\left(p,q\right)\in EP\left(f\right)×EP\left(g\right):p=q\in EP\left(h\right)\right\}\ne \mathrm{\varnothing }$

implies

$\mathrm{\Omega }=\left\{p\in EP\left(f\right)\cap EP\left(g\right)\cap EP\left(h\right)\right\}\ne \mathrm{\varnothing }.$

Hence, the problem studied in Corollary 2.4 is still the study of a common solution of three equilibrium problems in essence.

If C, Q, K are linear subspaces of a real Hilbert space, then we have the following corollaries from Theorem 2.1 and Corollary 2.1.

Corollary 2.5 Let ${H}_{1}$, ${H}_{2}$ and ${H}_{3}$ be three real Hilbert spaces. Let $C\subset {H}_{1}$, $Q\subset {H}_{2}$ and $K\subset {H}_{3}$ be three linear subspaces. Let $f:C×C\to \mathbb{R}$, $g:Q×Q\to \mathbb{R}$ and $h:K×K\to \mathbb{R}$ be three bi-functions. $A:{H}_{1}\to {H}_{3}$ and $B:{H}_{2}\to {H}_{3}$ are two bounded linear operators with their adjoint operators ${A}^{\ast }$ and ${B}^{\ast }$, respectively. Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let ${x}_{1}\in C$, ${y}_{1}\in Q$, ${C}_{1}=C$, ${Q}_{1}=Q$, $\left\{{x}_{n}\right\}$, $\left\{{y}_{n}\right\}$, $\left\{{u}_{n}\right\}$, $\left\{{v}_{n}\right\}$ and $\left\{{w}_{n}\right\}$ be sequences generated by

$\left\{\begin{array}{c}{u}_{n}={T}_{{r}_{n}}^{f}{x}_{n},\phantom{\rule{2em}{0ex}}{v}_{n}={T}_{{r}_{n}}^{g}{y}_{n},\phantom{\rule{2em}{0ex}}{w}_{n}={T}_{{r}_{n}}^{h}\left(\frac{1}{2}A{u}_{n}+\frac{1}{2}B{v}_{n}\right),\hfill \\ {l}_{n}={u}_{n}-\xi {A}^{\ast }\left(A{u}_{n}-{w}_{n}\right),\phantom{\rule{2em}{0ex}}{k}_{n}={v}_{n}-\xi {B}^{\ast }\left(B{v}_{n}-{w}_{n}\right),\hfill \\ {C}_{n+1}×{Q}_{n+1}=\left\{\left(x,y\right)\in {C}_{n}×{Q}_{n}:{\parallel {l}_{n}-x\parallel }^{2}+{\parallel {k}_{n}-y\parallel }^{2}\le {\parallel {u}_{n}-x\parallel }^{2}+{\parallel {v}_{n}-y\parallel }^{2}\hfill \\ \phantom{{C}_{n+1}×{Q}_{n+1}=\left\{}\le {\parallel {x}_{n}-x\parallel }^{2}+{\parallel {y}_{n}-y\parallel }^{2}\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right),\hfill \\ {y}_{n+1}={P}_{{Q}_{n+1}}\left({y}_{1}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$

where $\xi \in \left(0,min\left(\frac{1}{{\parallel A\parallel }^{2}},\frac{1}{{\parallel B\parallel }^{2}}\right)\right)$ and $\left\{{r}_{n}\right\}\subset \left(0,+\mathrm{\infty }\right)$ with ${lim inf}_{n\to +\mathrm{\infty }}{r}_{n}>0$, ${P}_{C}$ and ${P}_{Q}$ are two projection operators from ${H}_{1}$ into C and from ${H}_{2}$ into Q, respectively. Suppose that

$\mathrm{\Omega }=\left\{\left(p,q\right)\in EP\left(f\right)×EP\left(g\right):Ap=Bq\in EP\left(h\right)\right\}\ne \mathrm{\varnothing }.$

Then there exists $\left(p,q\right)\in \mathrm{\Omega }$ such that

1. (a)

$\left({x}_{n},{y}_{n}\right)\to \left(p,q\right)$ as $n\to \mathrm{\infty }$;

2. (b)

$\left({u}_{n},{v}_{n}\right)\to \left(p,q\right)$ as $n\to \mathrm{\infty }$;

3. (c)

${w}_{n}\to {w}^{\ast }:=Ap=Bq\in EP\left(h\right)$ as $n\to \mathrm{\infty }$.

Corollary 2.6 Let ${H}_{1}$ and ${H}_{2}$ be two real Hilbert spaces. Let $C\subset {H}_{1}$, $Q\subset {H}_{1}$ and $K\subset {H}_{2}$ be three linear subspaces. Let $f:C×C\to \mathbb{R}$, $g:Q×Q\to \mathbb{R}$ and $h:K×K\to \mathbb{R}$ be three bi-functions. $B:{H}_{1}\to {H}_{2}$ is a bounded linear operator with its adjoint operator ${B}^{\ast }$. Suppose that all the bi-functions f, g and h satisfy conditions (A1)-(A4). Let ${x}_{1}\in C$, ${y}_{1}\in Q$, ${C}_{1}=C$, ${Q}_{1}=Q$, $\left\{{x}_{n}\right\}$, $\left\{{y}_{n}\right\}$, $\left\{{u}_{n}\right\}$, $\left\{{v}_{n}\right\}$ and $\left\{{w}_{n}\right\}$ be sequences generated by

$\left\{\begin{array}{c}{u}_{n}={T}_{{r}_{n}}^{f}{x}_{n},\phantom{\rule{2em}{0ex}}{v}_{n}={T}_{{r}_{n}}^{g}{y}_{n},\phantom{\rule{2em}{0ex}}{w}_{n}={T}_{{r}_{n}}^{h}\left(\frac{1}{2}B{u}_{n}+\frac{1}{2}B{v}_{n}\right),\hfill \\ {l}_{n}={u}_{n}-\xi {B}^{\ast }\left(B{u}_{n}-{w}_{n}\right),\phantom{\rule{2em}{0ex}}{k}_{n}={v}_{n}-\xi {B}^{\ast }\left(B{v}_{n}-{w}_{n}\right),\hfill \\ {C}_{n+1}×{Q}_{n+1}=\left\{\left(x,y\right)\in {C}_{n}×{Q}_{n}:{\parallel {l}_{n}-x\parallel }^{2}+{\parallel {k}_{n}-y\parallel }^{2}\le {\parallel {u}_{n}-x\parallel }^{2}+{\parallel {v}_{n}-y\parallel }^{2}\hfill \\ \phantom{{C}_{n+1}×{Q}_{n+1}=\left\{}\le {\parallel {x}_{n}-x\parallel }^{2}+{\parallel {y}_{n}-y\parallel }^{2}\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right),\hfill \\ {y}_{n+1}={P}_{{Q}_{n+1}}\left({y}_{1}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$

where $\xi \in \left(0,\frac{1}{{\parallel B\parallel }^{2}}\right)$ and $\left\{{r}_{n}\right\}\subset \left(0,+\mathrm{\infty }\right)$ with ${lim inf}_{n\to +\mathrm{\infty }}{r}_{n}>0$, ${P}_{C}$ and ${P}_{Q}$ are two projection operators from ${H}_{1}$ into C and from ${H}_{1}$ into Q, respectively. Suppose that

$\mathrm{\Omega }=\left\{\left(p,q\right)\in EP\left(f\right)×EP\left(g\right):Bp=Bq\in EP\left(h\right)\right\}\ne \mathrm{\varnothing }.$

Then there exists $\left(p,q\right)\in \mathrm{\Omega }$ such that

1. (a)

$\left({x}_{n},{y}_{n}\right)\to \left(p,q\right)$ as $n\to \mathrm{\infty }$;

2. (b)

$\left({u}_{n},{v}_{n}\right)\to \left(p,q\right)$ as $n\to \mathrm{\infty }$;

3. (c)

${w}_{n}\to {w}^{\ast }:=Bp=Bq\in EP\left(h\right)$ as $n\to \mathrm{\infty }$.

Remark 2.2 In fact, the problem studied by Corollaries 2.1-2.3 and Corollary 2.6 is (SEP).

In order to prove Theorem 2.1, we need the following crucial known results.

Lemma 2.2 (see [10])

Let K be a nonempty closed convex subset of H, and let F be a bi-function of $K×K$ into satisfying (A1)-(A4). For $r>0$, define a mapping ${T}_{r}^{F}:H\to K$ as follows:

${T}_{r}^{F}\left(x\right)=\left\{z\in K:F\left(z,y\right)+\frac{1}{r}〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in K\right\}$
(2.2)

for all $x\in H$. Then the following hold:

1. (i)

${T}_{r}^{F}$ is single-valued and $\mathcal{F}\left({T}_{r}^{F}\right)=EP\left(F\right)$ for $\mathrm{\forall }r>0$ and $EP\left(F\right)$ is closed and convex;

2. (ii)

${T}_{r}^{F}$ is firmly nonexpansive, that is, for any $x,y\in H$, ${\parallel {T}_{r}^{F}x-{T}_{r}^{F}y\parallel }^{2}\le 〈{T}_{r}^{F}x-{T}_{r}^{F}y,x-y〉$.

Lemma 2.3 ([20])

Let ${F}_{r}^{F}$ be the same as in Lemma  2.2. If $\mathcal{F}\left({T}_{r}^{F}\right)=EP\left(F\right)\ne \mathrm{\varnothing }$, then, for any $x\in H$ and ${x}^{\ast }\in \mathcal{F}\left({T}_{r}^{F}\right)$, ${\parallel {T}_{r}^{F}x-x\parallel }^{2}\le {\parallel x-{x}^{\ast }\parallel }^{2}-{\parallel {T}_{r}^{F}x-{x}^{\ast }\parallel }^{2}$.

Lemma 2.4 ([1, 19])

Let the mapping ${T}_{r}^{F}$ be defined as in Lemma  2.2. Then, for $r,s>0$ and $x,y\in H$,

$\parallel {T}_{r}^{F}\left(x\right)-{T}_{s}^{F}\left(y\right)\parallel \le \parallel x-y\parallel +\frac{|s-r|}{s}\parallel {T}_{s}^{F}\left(y\right)-y\parallel .$

In particular, $\parallel {T}_{r}^{F}\left(x\right)-{T}_{r}^{F}\left(y\right)\parallel \le \parallel x-y\parallel$ for any $r>0$ and $x,y\in H$, that is, ${T}_{r}^{F}$ is nonexpansive for any $r>0$.

Lemma 2.5 (see, e.g., [25])

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

1. (a)

${\parallel x-y\parallel }^{2}={\parallel x\parallel }^{2}+{\parallel y\parallel }^{2}-2〈x,y〉$ for all $x,y\in H$;

2. (b)

${\parallel \alpha x+\left(1-\alpha \right)y\parallel }^{2}=\alpha {\parallel x\parallel }^{2}+\left(1-\alpha \right){\parallel y\parallel }^{2}-\alpha \left(1-\alpha \right){\parallel x-y\parallel }^{2}$ for all $x,y\in H$ and $\alpha \in \left[0,1\right]$.

## 3 Proof of Theorem 2.1

Applying Lemmas 2.1 and 2.2, we know that $\left\{{u}_{n}\right\}$, $\left\{{v}_{n}\right\}$ and $\left\{{w}_{n}\right\}$ are all well defined. It is also easy to verify that ${C}_{n}$, ${Q}_{n}$, ${C}_{n}×{Q}_{n}$ are closed convex sets for $n\in \mathbb{N}$.

Now, we claim ${C}_{n}×{Q}_{n}\ne \mathrm{\varnothing }$ for all $n\in \mathbb{N}$. Indeed, it suffices to prove that $\mathrm{\Omega }\subset {C}_{n}×{Q}_{n}$ for all $n\in \mathbb{N}$. Let $\left({x}^{\ast },{y}^{\ast }\right)\in \mathrm{\Omega }$. Then ${x}^{\ast }\in EP\left(f\right)$, ${y}^{\ast }\in EP\left(g\right)$ and

${w}^{\ast }:=A{x}^{\ast }=B{y}^{\ast }\in EP\left(h\right).$

Let $n\in \mathbb{N}$ be given. By Lemma 2.3, we have

(3.1)

From (2.1), (3.1) and Lemma 2.5, we obtain

$\begin{array}{rl}{\parallel {l}_{n}-{x}^{\ast }\parallel }^{2}=& {\parallel {P}_{C}\left({u}_{n}-\xi {A}^{\ast }\left(A{u}_{n}-{w}_{n}\right)\right)-{x}^{\ast }\parallel }^{2}\le {\parallel {u}_{n}-{x}^{\ast }-\xi {A}^{\ast }\left(A{u}_{n}-{w}_{n}\right)\parallel }^{2}\\ =& {\parallel {u}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel \xi {A}^{\ast }\left(A{u}_{n}-{w}_{n}\right)\parallel }^{2}-2\xi 〈{u}_{n}-{x}^{\ast },{A}^{\ast }\left(A{u}_{n}-{w}_{n}\right)〉\\ =& {\parallel {u}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel \xi {A}^{\ast }\left(A{u}_{n}-{w}_{n}\right)\parallel }^{2}-2\xi 〈A{u}_{n}-A{x}^{\ast },A{u}_{n}-{w}_{n}〉\\ =& {\parallel {u}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel \xi {A}^{\ast }\left(A{u}_{n}-{w}_{n}\right)\parallel }^{2}-2\xi 〈A{u}_{n}-{w}^{\ast },A{u}_{n}-{w}_{n}〉\\ =& {\parallel {u}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel \xi {A}^{\ast }\left(A{u}_{n}-{w}_{n}\right)\parallel }^{2}-\xi {\parallel A{u}_{n}-{w}^{\ast }\parallel }^{2}\\ -\xi {\parallel A{u}_{n}-{w}_{n}\parallel }^{2}+\xi {\parallel {w}_{n}-{w}^{\ast }\parallel }^{2}\\ \le & {\parallel {u}_{n}-{x}^{\ast }\parallel }^{2}-\xi \left(1-\xi {\parallel {A}^{\ast }\parallel }^{2}\right){\parallel A{u}_{n}-{w}_{n}\parallel }^{2}\\ -\xi {\parallel A{u}_{n}-{w}^{\ast }\parallel }^{2}+\xi {\parallel {w}_{n}-{w}^{\ast }\parallel }^{2}\\ \le & {\parallel {u}_{n}-{x}^{\ast }\parallel }^{2}-\xi \left(1-\xi {\parallel {A}^{\ast }\parallel }^{2}\right){\parallel A{u}_{n}-{w}_{n}\parallel }^{2}-\xi {\parallel A{u}_{n}-{w}^{\ast }\parallel }^{2}\\ +\frac{\xi }{2}{\parallel A{u}_{n}-{w}^{\ast }\parallel }^{2}+\frac{\xi }{2}{\parallel B{v}_{n}-{w}^{\ast }\parallel }^{2}\\ =& {\parallel {u}_{n}-{x}^{\ast }\parallel }^{2}-\xi \left(1-\xi {\parallel {A}^{\ast }\parallel }^{2}\right){\parallel A{u}_{n}-{w}_{n}\parallel }^{2}\\ -\frac{\xi }{2}{\parallel A{u}_{n}-{w}^{\ast }\parallel }^{2}+\frac{\xi }{2}{\parallel B{v}_{n}-{w}^{\ast }\parallel }^{2}\end{array}$
(3.2)

and

$\begin{array}{rl}{\parallel {k}_{n}-{y}^{\ast }\parallel }^{2}=& {\parallel {P}_{Q}\left({v}_{n}-\xi {B}^{\ast }\left(B{v}_{n}-{w}_{n}\right)\right)-{y}^{\ast }\parallel }^{2}\le {\parallel {v}_{n}-{y}^{\ast }-\xi {B}^{\ast }\left(B{v}_{n}-{w}_{n}\right)\parallel }^{2}\\ =& {\parallel {v}_{n}-{y}^{\ast }\parallel }^{2}+{\parallel \xi {B}^{\ast }\left(B{v}_{n}-{w}_{n}\right)\parallel }^{2}-2\xi 〈{v}_{n}-{y}^{\ast },{B}^{\ast }\left(B{v}_{n}-{w}_{n}\right)〉\\ =& {\parallel {v}_{n}-{y}^{\ast }\parallel }^{2}+{\parallel \xi {B}^{\ast }\left(B{v}_{n}-{w}_{n}\right)\parallel }^{2}-2\xi 〈B{v}_{n}-B{y}^{\ast },B{v}_{n}-{w}_{n}〉\\ =& {\parallel {v}_{n}-{y}^{\ast }\parallel }^{2}+{\parallel \xi {B}^{\ast }\left(B{v}_{n}-{w}_{n}\right)\parallel }^{2}-2\xi 〈B{v}_{n}-{w}^{\ast },B{v}_{n}-{w}_{n}〉\\ =& {\parallel {v}_{n}-{y}^{\ast }\parallel }^{2}+{\parallel \xi {B}^{\ast }\left(B{v}_{n}-{w}_{n}\right)\parallel }^{2}-\xi {\parallel B{v}_{n}-{w}^{\ast }\parallel }^{2}\\ -\xi {\parallel B{v}_{n}-{w}_{n}\parallel }^{2}+\xi {\parallel {w}_{n}-{w}^{\ast }\parallel }^{2}\\ \le & {\parallel {v}_{n}-{y}^{\ast }\parallel }^{2}-\xi \left(1-\xi {\parallel {B}^{\ast }\parallel }^{2}\right){\parallel B{v}_{n}-{w}_{n}\parallel }^{2}-\xi {\parallel B{v}_{n}-{w}^{\ast }\parallel }^{2}+\xi {\parallel {w}_{n}-{w}^{\ast }\parallel }^{2}\\ \le & {\parallel {v}_{n}-{y}^{\ast }\parallel }^{2}-\xi \left(1-\xi {\parallel {B}^{\ast }\parallel }^{2}\right){\parallel B{v}_{n}-{w}_{n}\parallel }^{2}-\xi {\parallel B{v}_{n}-{w}^{\ast }\parallel }^{2}\\ +\frac{\xi }{2}{\parallel A{u}_{n}-{w}^{\ast }\parallel }^{2}+\frac{\xi }{2}{\parallel B{v}_{n}-{w}^{\ast }\parallel }^{2}\\ =& {\parallel {v}_{n}-{y}^{\ast }\parallel }^{2}-\xi \left(1-\xi {\parallel {B}^{\ast }\parallel }^{2}\right){\parallel B{v}_{n}-{w}_{n}\parallel }^{2}\\ -\frac{\xi }{2}{\parallel B{v}_{n}-{w}^{\ast }\parallel }^{2}+\frac{\xi }{2}{\parallel A{u}_{n}-{w}^{\ast }\parallel }^{2}.\end{array}$
(3.3)

By taking into account inequalities (3.1), (3.2) and (3.3), we obtain

$\begin{array}{rl}{\parallel {l}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel {k}_{n}-{y}^{\ast }\parallel }^{2}\le & {\parallel {u}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel {v}_{n}-{y}^{\ast }\parallel }^{2}-\xi \left(1-\xi {\parallel {A}^{\ast }\parallel }^{2}\right){\parallel A{u}_{n}-{w}_{n}\parallel }^{2}\\ -\xi \left(1-\xi {\parallel {B}^{\ast }\parallel }^{2}\right){\parallel B{v}_{n}-{w}_{n}\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel {y}_{n}-{y}^{\ast }\parallel }^{2}-\xi \left(1-\xi {\parallel {A}^{\ast }\parallel }^{2}\right){\parallel A{u}_{n}-{w}_{n}\parallel }^{2}\\ -\xi \left(1-\xi {\parallel {B}^{\ast }\parallel }^{2}\right){\parallel B{v}_{n}-{w}_{n}\parallel }^{2},\end{array}$
(3.4)

which implies

${\parallel {l}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel {k}_{n}-{y}^{\ast }\parallel }^{2}\le {\parallel {u}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel {v}_{n}-{y}^{\ast }\parallel }^{2}\le {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel {y}_{n}-{y}^{\ast }\parallel }^{2}.$
(3.5)

Inequality (3.5) shows that $\left({x}^{\ast },{y}^{\ast }\right)\in {C}_{n}×{Q}_{n}$. Hence $\mathrm{\Omega }\subset {C}_{n}×{Q}_{n}$ and ${C}_{n}×{Q}_{n}\ne \mathrm{\varnothing }$ for all $n\in \mathbb{N}$. For each $n\in \mathbb{N}$, since $\mathrm{\Omega }\subset {C}_{n}×{Q}_{n}$, ${C}_{n+1}\subset {C}_{n}$, we have

${x}_{n+1}={P}_{{C}_{n+1}}\left({x}_{1}\right)\subset {C}_{n}.$

Similarly, since ${Q}_{n+1}\subset {Q}_{n}$, we have

${y}_{n+1}={P}_{{Q}_{n+1}}\left({y}_{1}\right)\subset {Q}_{n}.$

So, for any $\left({x}^{\ast },{y}^{\ast }\right)\in \mathrm{\Omega }$, we get

$\parallel {x}_{n+1}-{x}_{1}\parallel \le \parallel {x}^{\ast }-{x}_{1}\parallel$

and

$\parallel {y}_{n+1}-{y}_{1}\parallel \le \parallel {y}^{\ast }-{y}_{1}\parallel .$

The last inequalities deduce that $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ are bounded and hence show that $\left\{{k}_{n}\right\}$, $\left\{{l}_{n}\right\}$, $\left\{{u}_{n}\right\}$ and $\left\{{v}_{n}\right\}$ are all bounded. For some $n\in \mathbb{N}$ with $n>1$, from ${x}_{n}={P}_{{C}_{n}}\left({x}_{1}\right)\subset {C}_{n}$, ${y}_{n}={P}_{{Q}_{n}}\left({y}_{1}\right)\subset {Q}_{n}$ and (1.1), we have

$\begin{array}{r}{\parallel {x}_{n+1}-{x}_{n}\parallel }^{2}+{\parallel {x}_{1}-{x}_{n}\parallel }^{2}={\parallel {x}_{n+1}-{P}_{{C}_{n}}\left({x}_{1}\right)\parallel }^{2}+{\parallel {x}_{1}-{P}_{{C}_{n}}\left({x}_{1}\right)\parallel }^{2}\le {\parallel {x}_{n+1}-{x}_{1}\parallel }^{2},\\ {\parallel {y}_{n+1}-{y}_{n}\parallel }^{2}+{\parallel {y}_{1}-{y}_{n}\parallel }^{2}={\parallel {y}_{n+1}-{P}_{{C}_{n}}\left({y}_{1}\right)\parallel }^{2}+{\parallel {y}_{1}-{P}_{{C}_{n}}\left({y}_{1}\right)\parallel }^{2}\le {\parallel {y}_{n+1}-{y}_{1}\parallel }^{2},\end{array}$

which yields that

$\begin{array}{l}\parallel {x}_{1}-{x}_{n}\parallel \le \parallel {x}_{n+1}-{x}_{1}\parallel ,\phantom{\rule{2em}{0ex}}\parallel {y}_{1}-{y}_{n}\parallel \le \parallel {y}_{n+1}-{y}_{1}\parallel .\end{array}$

Together with the boundedness of $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$, we know ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-{x}_{1}\parallel$ and ${lim}_{n\to \mathrm{\infty }}\parallel {y}_{n}-{y}_{1}\parallel$ exist. For some $k,n\in \mathbb{N}$ with $k>n>1$, due to ${x}_{k}={P}_{{C}_{k}}\left({x}_{1}\right)\subset {C}_{n}$, ${y}_{k}={P}_{{Q}_{k}}\left({y}_{1}\right)\subset {Q}_{n}$ and (1.1), we have

$\begin{array}{r}{\parallel {x}_{k}-{x}_{n}\parallel }^{2}+{\parallel {x}_{1}-{x}_{n}\parallel }^{2}={\parallel {x}_{k}-{P}_{{C}_{n}}\left({x}_{1}\right)\parallel }^{2}+{\parallel {x}_{1}-{P}_{{C}_{n}}\left({x}_{1}\right)\parallel }^{2}\le {\parallel {x}_{k}-{x}_{1}\parallel }^{2},\\ {\parallel {y}_{k}-{y}_{n}\parallel }^{2}+{\parallel {y}_{1}-{y}_{n}\parallel }^{2}={\parallel {y}_{k}-{P}_{{Q}_{n}}\left({y}_{1}\right)\parallel }^{2}+{\parallel {y}_{1}-{P}_{{Q}_{n}}\left({y}_{1}\right)\parallel }^{2}\le {\parallel {y}_{k}-{y}_{1}\parallel }^{2}.\end{array}$
(3.6)

By (3.6), we have ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-{x}_{k}\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel {y}_{n}-{y}_{k}\parallel =0$. Hence $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ are all Cauchy sequences. Let ${x}_{n}\to p$ and ${y}_{n}\to q$ for some $\left(p,q\right)\in C×Q$. We want to prove that $\left(p,q\right)\in \mathrm{\Omega }$. For any $n\in \mathbb{N}$, since

$\left({x}_{n+1},{y}_{n+1}\right)\in {C}_{n+1}×{Q}_{n+1}\subset {C}_{n}×{Q}_{n},$

from (2.1), we have

$\begin{array}{rl}{\parallel {l}_{n}-{x}_{n+1}\parallel }^{2}+{\parallel {k}_{n}-{y}_{n+1}\parallel }^{2}& \le {\parallel {u}_{n}-{x}_{n+1}\parallel }^{2}+{\parallel {v}_{n}-{y}_{n+1}\parallel }^{2}\\ \le {\parallel {x}_{n}-{x}_{n+1}\parallel }^{2}+{\parallel {y}_{n}-{y}_{n+1}\parallel }^{2}.\end{array}$
(3.7)

By taking the limit from both sides of (3.7), we obtain

$\begin{array}{r}\underset{n\to \mathrm{\infty }}{lim}\parallel {l}_{n}-{x}_{n+1}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {k}_{n}-{y}_{n+1}\parallel =0,\\ \underset{n\to \mathrm{\infty }}{lim}\parallel {u}_{n}-{x}_{n+1}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {v}_{n}-{y}_{n+1}\parallel =0.\end{array}$
(3.8)

Moreover, by (3.8), we get

$\begin{array}{r}\underset{n\to \mathrm{\infty }}{lim}\parallel {l}_{n}-{u}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {u}_{n}-{x}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {l}_{n}-{x}_{n}\parallel =0,\\ \underset{n\to \mathrm{\infty }}{lim}\parallel {k}_{n}-{v}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {v}_{n}-{y}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel {k}_{n}-{y}_{n}\parallel =0.\end{array}$
(3.9)

Since ${lim}_{n\to \mathrm{\infty }}\parallel {u}_{n}-{x}_{n}\parallel ={lim}_{n\to \mathrm{\infty }}\parallel {v}_{n}-{y}_{n}\parallel =0$, we have ${u}_{n}\to p$ and ${v}_{n}\to q$ as $n\to \mathrm{\infty }$. Moreover, we obtain $A{u}_{n}\to Ap$ and $B{v}_{n}\to Bq$ as $n\to \mathrm{\infty }$.

Now, we claim $p\in EP\left(f\right)$ and $q\in EP\left(g\right)$. In fact, for $r>0$, by Lemma 2.4, we have

$\begin{array}{rcl}\parallel {T}_{r}^{f}p-p\parallel & =& \parallel {T}_{r}^{f}p-{T}_{{r}_{n}}^{f}{x}_{n}+{T}_{{r}_{n}}^{f}{x}_{n}-{x}_{n}+{x}_{n}-p\parallel \\ \le & \parallel {x}_{n}-p\parallel +\frac{|{r}_{n}-r|}{{r}_{n}}\parallel {T}_{{r}_{n}}^{f}{x}_{n}-{x}_{n}\parallel +\parallel {T}_{{r}_{n}}^{f}{x}_{n}-{x}_{n}\parallel +\parallel {x}_{n}-p\parallel \\ =& \parallel {x}_{n}-p\parallel +\frac{|{r}_{n}-r|}{{r}_{n}}\parallel {u}_{n}-{x}_{n}\parallel +\parallel {u}_{n}-{x}_{n}\parallel +\parallel {x}_{n}-p\parallel \to 0\end{array}$

and

$\begin{array}{rcl}\parallel {T}_{r}^{g}q-q\parallel & \le & \parallel {T}_{r}^{g}q-{T}_{{r}_{n}}^{g}{y}_{n}+{T}_{{r}_{n}}^{g}{y}_{n}-{y}_{n}+{y}_{n}-q\parallel \\ \le & \parallel {y}_{n}-q\parallel +\frac{|{r}_{n}-r|}{{r}_{n}}\parallel {T}_{{r}_{n}}^{g}{y}_{n}-{y}_{n}\parallel +\parallel {T}_{{r}_{n}}^{g}{y}_{n}-{y}_{n}\parallel +\parallel {y}_{n}-q\parallel \\ =& \parallel {y}_{n}-q\parallel +\frac{|{r}_{n}-r|}{{r}_{n}}\parallel {v}_{n}-{y}_{n}\parallel +\parallel {v}_{n}-{y}_{n}\parallel +\parallel {y}_{n}-q\parallel \to 0.\end{array}$

So, $p\in EP\left(f\right)$ and $q\in EP\left(g\right)$.

Finally, we prove $Ap=Bq\in EP\left(h\right)$. Setting

$\theta =min\left\{\xi \left(1-\xi {\parallel {A}^{\ast }\parallel }^{2}\right),\xi \left(1-\xi {\parallel {B}^{\ast }\parallel }^{2}\right)\right\}.$

Then, for any $n\in \mathbb{N}$, by (3.4) and (3.9), we have

$\begin{array}{r}\theta {\parallel A{u}_{n}-{w}_{n}\parallel }^{2}+\theta {\parallel B{v}_{n}-{w}_{n}\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel {y}_{n}-{y}^{\ast }\parallel }^{2}-{\parallel {l}_{n}-{x}^{\ast }\parallel }^{2}-{\parallel {k}_{n}-{y}^{\ast }\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}=\left\{\parallel {x}_{n}-{x}^{\ast }\parallel -\parallel {l}_{n}-{x}^{\ast }\parallel \right\}\left\{\parallel {x}_{n}-{x}^{\ast }\parallel +\parallel {l}_{n}-{x}^{\ast }\parallel \right\}\\ \phantom{\rule{2em}{0ex}}+\left\{\parallel {y}_{n}-{y}^{\ast }\parallel -\parallel {k}_{n}-{y}^{\ast }\parallel \right\}\left\{\parallel {y}_{n}-{y}^{\ast }\parallel +\parallel {k}_{n}-{y}^{\ast }\parallel \right\}\\ \phantom{\rule{1em}{0ex}}\le \parallel {l}_{n}-{x}_{n}\parallel \left\{\parallel {x}_{n}-{x}^{\ast }\parallel +\parallel {l}_{n}-{x}^{\ast }\parallel \right\}+\parallel {k}_{n}-{y}_{n}\parallel \left\{\parallel {y}_{n}-{y}^{\ast }\parallel +\parallel {k}_{n}-{y}^{\ast }\parallel \right\}\\ \phantom{\rule{1em}{0ex}}\to 0.\end{array}$
(3.10)

Hence (3.10) implies

$\underset{n\to \mathrm{\infty }}{lim}\parallel A{u}_{n}-{w}_{n}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel B{v}_{n}-{w}_{n}\parallel =0,\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{n\to \mathrm{\infty }}{lim}\parallel A{u}_{n}-B{v}_{n}\parallel =0.$
(3.11)

Since $A{u}_{n}\to Ap$, $B{v}_{n}\to Bq$ and (3.11), we obtain $Ap=Bq$ and ${w}_{n}\to {w}^{\ast }$, where ${w}^{\ast }:=Ap=Bq$. On the other hand, for $r>0$, by Lemma 2.4 again, we have

$\begin{array}{r}\parallel {T}_{r}^{h}{w}^{\ast }-{w}^{\ast }\parallel \\ \phantom{\rule{1em}{0ex}}=\parallel {T}_{r}^{h}{w}^{\ast }-{T}_{{r}_{n}}^{h}\frac{A{u}_{n}+B{v}_{n}}{2}+{T}_{{r}_{n}}^{h}\frac{A{u}_{n}+B{v}_{n}}{2}-\frac{A{u}_{n}+B{v}_{n}}{2}+\frac{A{u}_{n}+B{v}_{n}}{2}-{w}^{\ast }\parallel \\ \phantom{\rule{1em}{0ex}}\le \parallel \frac{A{u}_{n}+B{v}_{n}}{2}-{w}^{\ast }\parallel +\frac{|{r}_{n}-r|}{{r}_{n}}\parallel {T}_{{r}_{n}}^{h}\frac{A{u}_{n}+B{v}_{n}}{2}-\frac{A{u}_{n}+B{v}_{n}}{2}\parallel \\ \phantom{\rule{2em}{0ex}}+\parallel {T}_{{r}_{n}}^{h}\frac{A{u}_{n}+B{v}_{n}}{2}-\frac{A{u}_{n}+B{v}_{n}}{2}\parallel +\parallel \frac{A{u}_{n}+B{v}_{n}}{2}-{w}^{\ast }\parallel \\ \phantom{\rule{1em}{0ex}}=2\parallel \frac{A{u}_{n}+B{v}_{n}}{2}-{w}^{\ast }\parallel +\frac{|{r}_{n}-r|}{{r}_{n}}\parallel {w}_{n}-\frac{A{u}_{n}+B{v}_{n}}{2}\parallel +\parallel {w}_{n}-\frac{A{u}_{n}+B{v}_{n}}{2}\parallel \to 0.\end{array}$

Hence ${w}^{\ast }\in EP\left(h\right)$, namely $Ap=Bq\in EP\left(h\right)$. Therefore, conclusions (a), (b) and (c) are all proved. The proof is completed.

## 4 An example of Theorem 2.1

In this section, we give an example illustrating Theorem 2.1.

Example 4.1 Let ${H}_{1}={R}^{2}$, ${H}_{2}={R}^{3}$ and ${H}_{3}={R}^{4}$ be three real Hilbert spaces with the standard norm and inner product. For each $\alpha =\left({\alpha }_{1},{\alpha }_{2}\right)\in {R}^{2}$ and $\nu =\left({z}_{1},{z}_{2},{z}_{3},{z}_{4}\right)\in {R}^{4}$, define

$A\alpha =\left({\alpha }_{1},{\alpha }_{2},{\alpha }_{1}+{\alpha }_{2},{\alpha }_{1}-{\alpha }_{2}\right)$

and

${A}^{\ast }\nu =\left({z}_{1}+{z}_{3}+{z}_{4},{z}_{2}+{z}_{3}-{z}_{4}\right).$

Then A is a bounded linear operator from ${R}^{2}$ into ${R}^{4}$ with $\parallel A\parallel =\sqrt{3}$, and ${A}^{\ast }$ is an adjoint operator of A with $\parallel {A}^{\ast }\parallel =\sqrt{3}$. For each $\beta =\left({\beta }_{1},{\beta }_{2},{\beta }_{3}\right)\in {R}^{3}$ and $\nu =\left({z}_{1},{z}_{2},{z}_{3},{z}_{4}\right)\in {R}^{4}$, let

$B\beta =\left({\beta }_{1},{\beta }_{2},{\beta }_{3},{\beta }_{1}-{\beta }_{2}\right)$

and

${B}^{\ast }\nu =\left({z}_{1}+{z}_{4},{z}_{2}-{z}_{4},{z}_{3}\right).$

Then B is a bounded linear operator from ${R}^{3}$ into ${R}^{4}$ with $\parallel B\parallel =\sqrt{3}$, and ${B}^{\ast }$ is an adjoint operator of B with $\parallel {B}^{\ast }\parallel =\sqrt{3}$. Put

$\begin{array}{r}C:=\left\{\alpha =\left({\alpha }_{1},{\alpha }_{2}\right)\in {R}^{2}:-1\le {\alpha }_{1}\le 2,3\le {\alpha }_{2}\le 4\right\},\\ Q:=\left\{\beta =\left({\beta }_{1},{\beta }_{2},{\beta }_{3}\right)\in {R}^{3}:-1\le {\beta }_{1}\le 1,3\le {\beta }_{2}\le 4,3\le {\beta }_{3}\le 5\right\}\end{array}$

and

$K:=\left\{z=\left({z}_{1},{z}_{2},{z}_{3},{z}_{4}\right)\in {R}^{4}:0\le {z}_{1}\le 1,3\le {z}_{2}\le 6,3\le {z}_{3}\le 5,-5\le {z}_{4}\le -3\right\}.$

For each $\alpha =\left({\alpha }_{1},{\alpha }_{2}\right)\in C$, $\beta =\left({\beta }_{1},{\beta }_{2},{\beta }_{3}\right)\in Q$ and $z=\left({z}_{1},{z}_{2},{z}_{3},{z}_{4}\right)\in K$, define

$\begin{array}{r}{f}^{\ast }\left(\alpha \right)={\alpha }_{1}^{2}+{\alpha }_{2}^{2},\\ {g}^{\ast }\left(\beta \right)={\beta }_{1}^{2}+{\beta }_{2}^{2}+{\beta }_{3}^{2}\end{array}$

and

${h}^{\ast }\left(z\right)={z}_{1}^{2}+{z}_{2}^{2}+{z}_{3}^{2}+{z}_{4}^{2}.$

For each $\alpha ,x\in C$, let

$f\left(\alpha ,x\right)={f}^{\ast }\left(x\right)-{f}^{\ast }\left(\alpha \right).$

For each $\beta ,y\in Q$, let

$g\left(\beta ,y\right)={g}^{\ast }\left(y\right)-{g}^{\ast }\left(\beta \right).$

For each $\eta ,z\in K$, let

$h\left(\eta ,z\right)={h}^{\ast }\left(z\right)-{h}^{\ast }\left(\eta \right).$

It is not hard to verify that f, g and h satisfy conditions (A1)-(A4) with $EP\left(f\right)=\left\{p=\left(0,3\right)\right\}$, $EP\left(g\right)=\left\{q=\left(0,3,3\right)\right\}$, $EP\left(h\right)=\left\{\left(0,3,3,-3\right)\right\}$ and

$\mathrm{\Omega }=\left\{\left(p,q\right)\in EP\left(f\right)×EP\left(g\right):Ap=Bq\in EP\left(h\right)\right\}\ne \mathrm{\varnothing }.$

Let ${C}_{1}=C$, ${Q}_{1}=Q$, $\xi =\frac{1}{6}$ and ${r}_{n}\equiv 1$ for $n\in \mathbb{N}$. Thus, for each $\overline{x}=\left(a,b\right)\in C$ and $\overline{y}=\left(c,d,e\right)\in Q$ with $c>0$, we have the following:

• $u=\left(\frac{a}{3},3\right)={T}_{{r}_{n}}^{f}\overline{x}$,

• $v=\left(\frac{c}{3},3,3\right)={T}_{{r}_{n}}^{g}\overline{y}$,

• $w=\left(\frac{a+c}{6},3,3,-3\right)={T}_{{r}_{n}}^{h}\left(\frac{1}{2}A\overline{x}+\frac{1}{2}B\overline{y}\right)$,

• $l={P}_{C}\left(u-\frac{1}{6}{A}^{\ast }\left(Au-w\right)\right)=\left(\frac{7a+c}{36},3\right)$,

• $k={P}_{Q}\left(v-\frac{1}{6}{B}^{\ast }\left(Bv-w\right)\right)=\left(\frac{9c+a}{36},3+\frac{c}{18},3\right)$.

For ${x}_{1}=\left({a}_{1},{b}_{1}\right)\in C$ and ${y}_{1}=\left({c}_{1},{d}_{1},{e}_{1}\right)\in Q$ with $15{c}_{1}\ge {a}_{1}>0$, $17{a}_{1}\ge {c}_{1}$ and ${d}_{1}>3+\frac{{c}_{1}}{18}$, we obtain the following:

• ${u}_{1}=\left(\frac{{a}_{1}}{3},3\right)$,

• ${v}_{1}=\left(\frac{{c}_{1}}{3},3,3\right)$,

• ${w}_{1}=\left(\frac{{a}_{1}+{c}_{1}}{6},3,3,-3\right)$,

• ${l}_{1}=\left(\frac{7{a}_{1}+{c}_{1}}{36},3\right)$,

• ${k}_{1}=\left(\frac{9{c}_{1}+{a}_{1}}{36},3+\frac{{c}_{1}}{18},3\right)$,

• ${C}_{2}=\left\{\alpha =\left({\alpha }_{1},{\alpha }_{2}\right)\in {C}_{1}:-1\le {\alpha }_{1}\le \frac{19{a}_{1}+{c}_{1}}{72},3\le {\alpha }_{2}\le \frac{{b}_{1}+3}{2}\right\}$,

• ${x}_{2}={P}_{{C}_{2}}\left({x}_{1}\right)=\left(\frac{19{a}_{1}+{c}_{1}}{72},\frac{{b}_{1}+3}{2}\right):=\left({a}_{2},{b}_{2}\right)$,

• ${Q}_{2}=\left\{\beta =\left({\beta }_{1},{\beta }_{2},{\beta }_{3}\right)\in {Q}_{1}:-1\le {\beta }_{1}\le \frac{21{c}_{1}+{a}_{1}}{72},3+\frac{{c}_{1}}{36}\le {\beta }_{2}\le \frac{{d}_{1}+3}{2},3\le {\beta }_{3}\le \frac{{e}_{1}+3}{2}\right\}$,

• ${y}_{2}={P}_{{Q}_{2}}\left({y}_{1}\right)=\left(\frac{21{c}_{1}+{a}_{1}}{72},\frac{{d}_{1}+3}{2},\frac{{e}_{1}+3}{2}\right):=\left({c}_{2},{d}_{2},{e}_{2}\right)$.

From ${x}_{2}$, ${y}_{2}$, we have ${u}_{2}=\left(\frac{1}{3}{a}_{2},3\right)$, ${v}_{2}=\left(\frac{1}{3}{c}_{2},3,3\right)$ and ${w}_{2}=\left(\frac{1}{6}\left({a}_{2}+{b}_{2}\right),3,3,-3\right)$. Since

$\begin{array}{r}{d}_{2}>3+\frac{{c}_{2}}{18},\\ 15{c}_{2}\ge {a}_{2}>0,\end{array}$

and

$17{a}_{2}\ge {c}_{2},$

we get the following:

• ${C}_{3}=\left\{\alpha =\left({\alpha }_{1},{\alpha }_{2}\right)\in {C}_{2}:-1\le {\alpha }_{1}\le \frac{19{a}_{2}+{c}_{2}}{72},3\le {\alpha }_{2}\le \frac{1}{2}\left({b}_{2}+3\right)\right\}$,

• ${x}_{3}=\left(\frac{19{a}_{2}+{c}_{2}}{72},\frac{1}{2}\left({b}_{2}+3\right)\right):=\left({a}_{3},{b}_{3}\right)$,

• ${Q}_{3}=\left\{\beta =\left({\beta }_{1},{\beta }_{2},{\beta }_{3}\right)\in {Q}_{1}:-1\le {\beta }_{1}\le \frac{21{c}_{2}+{a}_{2}}{72},3+\frac{{c}_{2}}{36}\le {\beta }_{2}\le \frac{{d}_{2}+3}{2},3\le {\beta }_{3}\le \frac{{e}_{2}+3}{2}\right\}$,

• ${y}_{3}=\left(\frac{21{c}_{2}+{a}_{2}}{72},\frac{{d}_{2}+3}{2},\frac{{e}_{2}+3}{2}\right):=\left({c}_{3},{d}_{3},{e}_{3}\right)$.

Similarly, for $n\in \mathbb{N}$ with $n>1$, we obtain

• ${C}_{n+1}=\left\{\alpha =\left({\alpha }_{1},{\alpha }_{2}\right)\in {C}_{n}:-1\le {\alpha }_{1}\le \frac{19{a}_{n-1}+{c}_{n-1}}{72},3\le {\alpha }_{2}\le \frac{1}{2}\left({b}_{n-1}+3\right)\right\}$,

• ${x}_{n+1}=\left(\frac{19{a}_{n-1}+{c}_{n-1}}{72},\frac{1}{2}\left({b}_{n-1}+3\right)\right)$,

• ${Q}_{n+1}=\left\{\beta =\left({\beta }_{1},{\beta }_{2},{\beta }_{3}\right)\in {Q}_{n}:-1\le {\beta }_{1}\le \frac{21{c}_{n-1}+{a}_{n-1}}{72},3+\frac{{c}_{n-1}}{36}\le {\beta }_{2}\le \frac{{d}_{n-1}+3}{2},3\le {\beta }_{3}\le \frac{{e}_{n-1}+3}{2}\right\}$,

• ${y}_{n+1}=\left(\frac{21{c}_{n-1}+{a}_{n-1}}{72},\frac{{d}_{n-1}+3}{2},\frac{{e}_{n-1}+3}{2}\right)$,

• ${u}_{n+1}=\left(\frac{1}{3}{a}_{n-1},3\right)$,

• ${v}_{n+1}=\left(\frac{1}{3}{c}_{n-1},3,3\right)$,

• ${w}_{n+1}=\left(\frac{{c}_{n-1}+{a}_{n-1}}{6},3,3,-3\right)$.

By mathematical induction, we know that $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$, $\left\{{c}_{n}\right\}$, $\left\{{d}_{n}\right\}$ and $\left\{{e}_{n}\right\}$ all are decreasing sequences. Moreover, ${a}_{n}\to 0$, ${b}_{n}\to 3$, ${c}_{n}\to 0$, ${d}_{n}\to 3$ and ${e}_{n}\to 3$ as $n\to \mathrm{\infty }$. So, we have ${u}_{n}\to \left(0,3\right)$, ${v}_{n}\to \left(0,3,3\right)$, ${w}_{n}\to \left(0,3,3,-3\right)$, ${x}_{n}\to \left(0,3\right)$ and ${y}_{n}\to \left(0,3,3\right)$ as $n\to \mathrm{\infty }$.

## 5 Conclusion

In this paper, we first introduce and investigate BSEP which can be regarded as a new development in the field of equilibrium problems. We provide some new iterative algorithms for BSEP and establish strong convergence theorems for these iterative algorithms in different spaces. An example illustrating Theorem 2.1 is also given.

## References

1. He Z: The split equilibrium problems and its convergence algorithms. J. Inequal. Appl. 2012., 2012: Article ID 162

2. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.

3. Flam SD, Antipin AS: Equilibrium programming using proximal-link algorithms. Math. Program. 1997, 78: 29–41.

4. Moudafi A, Théra M: Proximal and dynamical approaches to equilibrium problems. Lecture Notes in Economics and Mathematical Systems 477. In Ill-Posed Variational Problems and Regularization Techniques. Springer, Berlin; 1999:187–201.

5. Moudafi A: Second-order differential proximal methods for equilibrium problems. J. Inequal. Pure Appl. Math. 2003, 18(4):1–17.

6. Lin L-J, Du W-S: Systems of equilibrium problems with applications to new variants of Ekeland’s variational principle, fixed point theorems and parametric optimization problems. J. Glob. Optim. 2008, 40: 663–677. 10.1007/s10898-007-9146-0

7. Du W-S: Hybrid inclusion and disclusion systems with applications to equilibria and parametric optimization. J. Glob. Optim. 2010, 47: 119–132. 10.1007/s10898-009-9461-8

8. Du W-S: Applications of an HIDS theorem to the existence of fixed point, abstract equilibria and optimization problems. Abstr. Appl. Anal. 2011., 2011: Article ID 247236 10.1155/2011/247236

9. Ceng LC, Yao JC: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Appl. Math. Comput. 2008, 198: 729–741. 10.1016/j.amc.2007.09.011

10. Combettes PL, Hirstoaga A: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.

11. He Z: A new iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contractive mappings and its application. Math. Commun. 2012, 17: 411–422.

12. He Z, Du W-S: Strong convergence theorems for equilibrium problems and fixed point problems: a new iterative method, some comments and applications. Fixed Point Theory Appl. 2011., 2011: Article ID 33

13. Jung JS: Strong convergence of composite iterative methods for equilibrium problems and fixed point problems. Appl. Math. Comput. 2009, 213: 498–505. 10.1016/j.amc.2009.03.048

14. Tada A, Takahashi W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 2007, 133: 359–370. 10.1007/s10957-007-9187-z

15. Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 2007, 331: 506–515. 10.1016/j.jmaa.2006.08.036

16. Yao Y, Noor MA, Liou YC: On iterative methods for equilibrium problems. Nonlinear Anal. 2009, 70: 497–509. 10.1016/j.na.2007.12.021

17. Colao V, Marino G: Strong convergence for a minimization problem on points of equilibrium and common fixed points of an infinite family of nonexpansive mappings. Nonlinear Anal. 2010, 73: 3513–3524. 10.1016/j.na.2010.07.011

18. Censor Y, Gibali A, Reich S: Algorithms for the split variational inequality problem. Numer. Algorithms 2012, 59: 301–323. 10.1007/s11075-011-9490-5

19. He Z, Du W-S: Nonlinear algorithms approach to split common solution problems. Fixed Point Theory Appl. 2012., 2012: Article ID 130

20. He Z, Du W-S: On hybrid split problem and its nonlinear algorithms. Fixed Point Theory Appl. 2013., 2013: Article ID 47

21. Moudafi A: The split common fixed-point problem for demicontractive mappings. Inverse Probl. 2010., 26: Article ID 055007

22. Moudafi A: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 2011, 74: 4083–4087. 10.1016/j.na.2011.03.041

23. Moudafi A: A relaxed alternating CQ-algorithm for convex feasibility problems. Nonlinear Anal. 2013, 79: 117–121.

24. Xu H-K: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 2010., 26: Article ID 105018

25. Chang SS, Lee HWJ, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 2009, 70: 3307–3319. 10.1016/j.na.2008.04.035

## Acknowledgements

The first author was supported by the Natural Science Foundation of Yunnan Province (201401CA00262) and the Candidate Foundation of Youth Academic Experts at Honghe University (2014HB0206); the second author was supported by Grant No. MOST 103-2115-M-017-001 of the Ministry of Science and Technology of the Republic of China.

## Author information

Authors

### Corresponding author

Correspondence to Wei-Shih Du.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

He, Z., Du, WS. New feasible iterative algorithms and strong convergence theorems for bilevel split equilibrium problems. Fixed Point Theory Appl 2014, 187 (2014). https://doi.org/10.1186/1687-1812-2014-187

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1687-1812-2014-187

### Keywords

• metric projection