# Split hierarchical variational inequality problems and related problems

- Qamrul Hasan Ansari
^{1, 2}, - Nimit Nimana
^{3}and - Narin Petrot
^{3, 4}Email author

**2014**:208

https://doi.org/10.1186/1687-1812-2014-208

© Ansari et al.; licensee Springer. 2014

**Received: **14 May 2014

**Accepted: **16 September 2014

**Published: **13 October 2014

## Abstract

The main objective of this paper is to introduce a split hierarchical variational inequality problem. Several related problems are also considered. We propose an iterative method for finding a solution of our problem. The weak convergence of the sequence generated by the proposed method is studied.

**MSC:**47H09, 47H10, 47J25, 49J40, 65K10.

## Keywords

## 1 Introduction

*C*and

*Q*are nonempty closed convex subsets of ${\mathbb{R}}^{n}$ and ${\mathbb{R}}^{m}$, respectively, and

*A*is an $m\times n$ matrix. They proposed an algorithm to find the solution of SFP. Their algorithm did not become popular, since it concerns the complicated matrix inverse computations and, subsequently, is considered in the case when $n=m$. Based on these observations, Byrne [2] applied the forward-backward method, a type of projected gradient method, presenting the so-called CQ-iterative procedure, which is defined by

*C*and

*Q*, respectively. The convergence result of the sequence ${\{{x}_{k}\}}_{k=1}^{\mathrm{\infty}}$ to a solution of the considered split feasibility problem was presented. Further, Byrne also proposed an application to dynamic emission tomographic image reconstruction. A few years later, Censor

*et al.*[3, 4] proposed an incredible application of the split feasibility problem to the inverse problem of intensity-modulated radiation therapy treatment planning. Recently, Xu [5] considered SFP in the setting of infinite-dimensional Hilbert spaces and established the following CQ-algorithm: Let ${H}_{1}$ and ${H}_{2}$ be real Hilbert spaces and $A:{H}_{1}\to {H}_{2}$ be a bounded linear operator. For a given ${x}_{1}\in {H}_{1}$, generated a sequence ${\{{x}_{k}\}}_{k=1}^{\mathrm{\infty}}$ by the following iterative scheme:

where $\gamma \in (0,2/{\parallel A\parallel}^{2})$, and ${A}^{\ast}$ is the adjoint operator of *A*. He also proved the weak convergence of the sequence produced by the above procedure to a solution of the SFP. The details of the CQ-algorithm for SFP problem are given in [6]. A comprehensive literature, survey, and references on SFP can be found in [7].

*T*and

*S*are some nonlinear operators on ${\mathbb{R}}^{n}$ and ${\mathbb{R}}^{m}$, respectively, and

*A*is a real $m\times n$ matrix. Based on the properties of operators

*T*and

*S*, called cutter or directed operators, they presented the following algorithm for solving the split common fixed point problem:

where an initial ${x}_{1}\in {\mathbb{R}}^{n}$, $\gamma \in (0,2/{\parallel A\parallel}^{2})$. They also presented a convergence result for this algorithm. Moudafi [9] studied the split common fixed point problem in the context of the demicontractive operators *T* and *S* in the setting of infinite-dimensional Hilbert spaces. He established the weak convergence of the sequence generated by his scheme to a solution of the split common fixed point problem.

On the other hand, the theory of variational inequalities is well known and well developed because of its applications in different areas of science, social science, engineering, and management. There are several monographs on variational inequalities, but we mention here a few [10–12]. Let *C* be a nonempty closed convex subset of a real Hilbert space *H* and $f:H\to H$ be an operator. The variational inequality problem defined by *C* and *f* is to find ${x}^{\ast}\in C$ such that

($VIP(C;f)$) $\u3008f\left({x}^{\ast}\right),x-{x}^{\ast}\u3009\ge 0$, for all $x\in C$.

Another problem closely related to $VIP(C;f)$ is known as the Minty variational inequality problem: find ${x}^{\ast}\in C$ such that

($MVIP(C;f)$) $\u3008f(x),x-{x}^{\ast}\u3009\ge 0$, for all $x\in C$.

The trivial unlikeness of two proposed problems is the linearity of variational inequalities. In fact, the Minty variational inequality $MVIP(C;f)$ is linear but the variational inequality $VIP(C;f)$ is not. However, under the (hemi)continuity and monotonicity of *f*, the solution sets of these problems are the same (see [[13], Lemma 1]).

If the constrained set *C* in variational inequality formulations is a set of fixed points of an operator, then the variational inequality problem is known as hierarchical variational inequality problem.

In the recent past, several methods for solving hierarchical variational inequalities have been investigated in the literature; see, for example, [14–19] and the references therein.

The goal of this paper is to introduce a split-type problem, by combining a split fixed point problem and a hierarchical variational inequality problem. The considered problem can be applied to solve many existing problems. We present an iterative procedure for finding a solution of the proposed problem and show that under some suitable assumptions, the sequence generated by our algorithm converges weakly to a solution of the considered problem.

The rest of this paper is divided into four sections. In Section 2, we recall and state preliminaries on numerous nonlinear operators and their useful properties. Section 3 we divide in two subsections, that is, first, we present a split problem, called the split hierarchical Minty variational inequality problem. We subsequently propose an algorithm for solving our problem and establish a convergence result under some assumptions. Second, we present the split hierarchical variational inequality problem. In the last section, we investigate some related problems, where we can apply the considered problem.

## 2 Preliminaries

Let *H* be a real Hilbert space whose inner product and norm are denoted by $\u3008\cdot \phantom{\rule{0.2em}{0ex}},\cdot \u3009$ and $\parallel \cdot \parallel $, respectively. The strong convergence and weak convergence of a sequence ${\{{x}_{k}\}}_{k=1}^{\mathrm{\infty}}$ to $x\in H$ are denoted by ${x}_{k}\to x$ and ${x}_{k}\rightharpoonup x$, respectively. Let $T:H\to H$ be an operator. We denote by $R(T)$ the range of *T*, and by $Fix(T)$ the set of all fixed points of *T*, that is, $Fix(T)=\{x\in H:x=Tx\}$. The operator *T* is said to be *nonexpansive* if for all $x,y\in H$, $\parallel Tx-Ty\parallel \le \parallel x-y\parallel $; *strongly nonexpansive* [20, 21] if *T* is nonexpansive and for all bounded sequences ${\{{x}_{k}\}}_{k=1}^{\mathrm{\infty}}$, ${\{{y}_{k}\}}_{k=1}^{\mathrm{\infty}}$ in *H*, the condition ${lim}_{k\to \mathrm{\infty}}(\parallel {x}_{k}-{y}_{k}\parallel -\parallel T{x}_{k}-T{y}_{k}\parallel )=0$ implies ${lim}_{k\to \mathrm{\infty}}\parallel ({x}_{k}-{y}_{k})-(T{x}_{k}-T{y}_{k})\parallel =0$; *averaged nonexpansive* if for all $x,y\in H$, $T=(1-\alpha )I+\alpha S$ holds for a nonexpansive operator $S:H\to H$ and $\alpha \in (0,1)$; *firmly nonexpansive* if $2T-I$ is nonexpansive, or equivalently for all $x,y\in H$, ${\parallel Tx-Ty\parallel}^{2}\le \u3008x-y,Tx-Ty\u3009$; *cutter* [21] if $\u3008x-Tx,z-Tx\u3009\le 0$ for all $x\in H$ and all $z\in Fix(T)$; *monotone* if $\u3008Tx-Ty,x-y\u3009\ge 0$, for all $x,y\in H$; *α*-*inverse strongly monotone* (or *α*-cocoercive) if there exists a positive real number *α* such that $\u3008Tx-Ty,x-y\u3009\ge \alpha {\parallel Tx-Ty\parallel}^{2}$, for all $x,y\in H$.

Let $B:H\to {2}^{H}$ be a set-valued operator, we define a graph of *B* by $\{(x,y)\in H\times H:y\in B(x)\}$ and an inverse operator of *B*, denoted ${B}^{-1}$, by $\{(y,x)\in H\times H:x\in B(y)\}$. A set-valued operator $B:H\to {2}^{H}$ is called *monotone* if for all $x,y\in H$, $u\in B(x)$ and $v\in B(y)$ such that $\u3008x-y,u-v\u3009\ge 0$. A monotone operator *B* is said to be *maximal monotone* if there exists no other monotone operator such that its graph properly contains the graph of *B*. For a maximal monotone operator *B*, we know that for each $x\in H$ and a positive real number *σ*, there is a unique $z\in H$ such that $x\in (I+\sigma B)z$. We define the *resolvent* of *B* with parameter *σ* by ${J}_{\sigma}^{B}:={(I+\sigma B)}^{-1}$. It is well known that the resolvent is a single-valued and firmly nonexpansive operator.

**Remark 2.1** It can be seen that the class of averaged nonexpansive operators is a proper subclass of the class of strongly nonexpansive operators. Since any firmly nonexpansive operator is an averaged nonexpansive operator, it is clear that a class of firmly nonexpansive operators is contained in the class of strongly nonexpansive operators. Further, a firmly nonexpansive operator with fixed point is cutter. However, a nonexpansive cutter operator need not to be firmly nonexpansive; for further details, see [21].

The following well-known lemma is due to Opial [22].

**Lemma 2.2** (Demiclosedness principle) [[22], Lemma 2]

*Let* *C* *be a nonempty closed convex subset of a real Hilbert space* *H* *and* $T:C\to H$ *be a nonexpansive operator*. *If the sequence* ${\{{x}_{k}\}}_{k=1}^{\mathrm{\infty}}\subseteq C$ *converges weakly to an element* $x\in C$ *and the sequence* ${\{{x}_{k}-T{x}_{k}\}}_{k=1}^{\mathrm{\infty}}$ *converges strongly to* 0, *then* *x* *is a fixed point of the operator* *T*.

To prove the main theorem of this paper, we need the following lemma.

**Lemma 2.3** [[23], Section 2.2.1, Lemma 2]

*Assume that* ${\{{a}_{k}\}}_{k=1}^{\mathrm{\infty}}$ *and* ${\{{b}_{k}\}}_{k=1}^{\mathrm{\infty}}$ *are nonnegative real sequences such that* ${a}_{k+1}\le {a}_{k}+{b}_{k}$. *If* ${\sum}_{k=1}^{\mathrm{\infty}}{b}_{k}<\mathrm{\infty}$, *then* ${lim}_{k\to \mathrm{\infty}}{a}_{k}$ *exists*.

The following lemma can be immediately obtained by the properties of an inner product.

**Lemma 2.4**

*Let*

*H*

*be a real Hilbert space*

*H*.

*Then*,

*for all*$x,y\in H$,

## 3 Split hierarchical variational inequality problems and convergent results

In this section we introduce split hierarchical variational inequality problems and discuss some related problems. Further, we propose an iterative method for finding a solution of the hierarchical variational inequality problem and prove the convergence result for the sequence generated by the proposed iterative method.

### 3.1 Split hierarchical Minty variational inequality problem

*split hierarchical Minty variational inequality problem*(in short, SHMVIP) is to find ${x}^{\ast}\in Fix(T)$ such that

We note that the problems (3.1) and (3.2) are nothing but hierarchical Minty variational inequality problems. If *f* and *h* are zero operators, that is, $f\equiv h\equiv 0$, then SHMVIP (3.1) and (3.2) reduces to the split fixed point problem (1.2). Moreover, SHMVIP (3.1) and (3.2) can be applied to several existing split-type problems, which we will discuss in Section 4.

Inspired by the iterative scheme (1.3) for solving split common fixed point problem (1.2) and the existing algorithms, a generalization of the projected gradient method, for solving the hierarchical variational inequality problem of Yamada [19] and Iiduka [16], we now present an iterative algorithm for solving HMVIP (3.1)-(3.2).

**Algorithm 3.1** Initialization: Choose ${\{{\alpha}_{k}\}}_{k=1}^{\mathrm{\infty}},{\{{\beta}_{k}\}}_{k=1}^{\mathrm{\infty}}\subset (0,+\mathrm{\infty})$. Take arbitrary ${x}_{1}\in {H}_{1}$.

Update $k:=k+1$.

Rest of the section, unless otherwise specified, we assume that *T* is a strongly nonexpansive operator on ${H}_{1}$ with $Fix(T)\ne \mathrm{\varnothing}$ and *S* is a strongly nonexpansive cutter operator on ${H}_{2}$ with $Fix(S)\ne \mathrm{\varnothing}$, *f* (respectively, *h*) is a monotone and continuous operator on ${H}_{1}$ (respectively, ${H}_{2}$) and $A:{H}_{1}\to {H}_{2}$ is a bounded linear operator with $R(A)\cap Fix(S)\ne \mathrm{\varnothing}$.

**Remark 3.2** (i) The Algorithm 3.1 can be applied to the iterative scheme (1.3) by setting the operators $f\equiv h\equiv 0$.

(ii) The iterative Algorithm 3.1 extends and develops the algorithms in [[24], Algorithm 6.1] and [[25], Algorithm (8)] in many aspects. Indeed, the metric projections ${P}_{C}$ and ${P}_{Q}$ in [[24], Algorithm 6.1] and the resolvent operators ${J}_{\lambda}^{{B}_{1}}$ and ${J}_{\lambda}^{{B}_{2}}$ in [[25], Algorithm (8)] are firmly nonexpansive operators which are special cases of the assumption on the operators *T* and *S*. Second, its involves control sequences ${\{{\alpha}_{k}\}}_{k=1}^{\mathrm{\infty}}$ and ${\{{\beta}_{k}\}}_{k=1}^{\mathrm{\infty}}$, while in [[24], Algorithm 6.1] and [[25], Algorithm (8)] they are constant. Furthermore, we assume $\gamma \in (0,\frac{2}{{\parallel A\parallel}^{2}})$, while in [[24], Algorithm 6.1] and [[25], Algorithm (8)], it was assumed to be in $(0,\frac{1}{{\parallel A\parallel}^{2}})$, which clearly is a more restrictive assumption.

(iii) By setting an operator *A* to be zero operator, the iterative Algorithm 3.1 relates to existing iterative schemes for solving the hierarchical variational inequality problem in, for example, [[15], Algorithm 3.1] and [[17], Algorithm 5].

The following theorem provides the weak convergence of the sequence generated by the Algorithm 3.1 to an element of Γ.

**Theorem 3.3**

*Let a sequence*${\{{x}_{k}\}}_{k=1}^{\mathrm{\infty}}$

*be generated by Algorithm*3.1

*with*${x}_{1}\in {H}_{1}$,

*and let*${\{{\alpha}_{k}\}}_{k=1}^{\mathrm{\infty}}$

*and*${\{{\beta}_{k}\}}_{k=1}^{\mathrm{\infty}}\subset (0,1)$

*be sequences such that*${\sum}_{k=1}^{\mathrm{\infty}}{\alpha}_{k}<\mathrm{\infty}$

*and*${lim}_{k\to \mathrm{\infty}}{\beta}_{k}=0$.

*If*$\mathrm{\Gamma}\ne \mathrm{\varnothing}$,

*then the following statements hold*.

- (i)
*If there exists a natural number*${k}_{0}$*such that*$\mathrm{\Gamma}\subset \bigcap _{k={k}_{0}}^{\mathrm{\infty}}\{z\in {H}_{1}:\u3008h(A{x}_{k}),S(I-{\beta}_{k}h)A{x}_{k}-Az\u3009\ge 0\},$*then*,*for all*$k\ge {k}_{0}$*and*$z\in \mathrm{\Gamma}$,*we have*${\parallel {y}_{k}-z\parallel}^{2}\le {\parallel {x}_{k}-z\parallel}^{2}-\gamma (2-\gamma {\parallel A\parallel}^{2}){\parallel (I-S(I-{\beta}_{k}h))A{x}_{k}\parallel}^{2}.$ - (ii)
*If the sequence*${\{f({y}_{k})\}}_{k=1}^{\mathrm{\infty}}$*is bounded*,*then*${lim}_{k\to \mathrm{\infty}}\parallel {x}_{k}-z\parallel $*exists for all*$z\in \mathrm{\Gamma}$. - (iii)
*If the sequence*${\{h(A{x}_{k})\}}_{k=1}^{\mathrm{\infty}}$*is bounded*,*then*${lim}_{k\to \mathrm{\infty}}\parallel {x}_{k+1}-{x}_{k}\parallel =0$, ${lim}_{k\to \mathrm{\infty}}\parallel {x}_{k}-T{x}_{k}\parallel =0$,*and*${lim}_{k\to \mathrm{\infty}}\parallel A{x}_{k}-SA{x}_{k}\parallel =0$. - (iv)
*If*$\parallel {x}_{k+1}-{x}_{k}\parallel =o({\alpha}_{k})$,*and*${\alpha}_{k}=o({\beta}_{k}^{2})$,*then the sequence*${\{{x}_{k}\}}_{k=1}^{\mathrm{\infty}}$*converges weakly to an element of*Γ.

*Proof*(i) Let $z\in \mathrm{\Gamma}$ be given. Then

*S*is a cutter operator, we have with

(ii) Let ${M}_{1}:=sup\{\parallel f({y}_{k})\parallel :k\ge 1\}$. For $k\ge {k}_{0}$, we have $\parallel {x}_{k+1}-z\parallel \le \parallel {x}_{k}-z\parallel +{\alpha}_{k}{M}_{1}$. Since ${\sum}_{k=1}^{\mathrm{\infty}}{\alpha}_{k}<\mathrm{\infty}$, by using Lemma 2.3, we obtain the result that ${lim}_{k\to \mathrm{\infty}}\parallel {x}_{k}-z\parallel $ exists.

(iii) We first show that ${lim}_{k\to \mathrm{\infty}}\parallel {x}_{k}-{x}_{k+1}\parallel =0$.

*T*and (3.5), we have

*T*and the boundedness of ${\{{w}_{k}\}}_{k=1}^{\mathrm{\infty}}$, we have

as desired.

We now show that ${lim}_{k\to \mathrm{\infty}}\parallel {x}_{k}-T{x}_{k}\parallel =0$.

*T*, we obtain

Next, we show that ${lim}_{k\to \mathrm{\infty}}\parallel A{x}_{k}-SA{x}_{k}\parallel =0$.

*S*is a strongly nonexpansive mapping, we have

- (iv)
Since ${\{{x}_{k}\}}_{k=1}^{\mathrm{\infty}}$ is a bounded sequence, there exist a subsequence ${\{{x}_{{k}_{j}}\}}_{j=1}^{\mathrm{\infty}}$ of ${\{{x}_{k}\}}_{k=1}^{\mathrm{\infty}}$ and $q\in {H}_{1}$ such that ${x}_{{k}_{j}}\rightharpoonup q\in {H}_{1}$. By (iii) and the demiclosed principle of the nonexpansive operator

*T*, we obtain $q\in Fix(T)$. We claim that $q\in Fix(T)$ solves (3.1).

where $M:=sup\{\parallel {x}_{k}-z\parallel +\parallel {x}_{k+1}-z\parallel :k\ge 1\}<\mathrm{\infty}$. Since ${x}_{{k}_{j}}\rightharpoonup q$, ${\alpha}_{k}\to 0$, and $\parallel {x}_{k+1}-{x}_{k}\parallel =o({\alpha}_{k})$, by (3.8), we have $\u3008f(z),q-z\u3009\le 0$ for all $z\in Fix(T)$ which means that $q\in Fix(T)$ solves (3.1).

Next, by $A{x}_{{k}_{j}}\rightharpoonup Aq\in {H}_{2}$ together with (iii) and the demiclosedness of the nonexpansive operator *S*, we know that $Aq\in Fix(S)$. We show that such $Aq\in Fix(S)$ solves (3.2).

where ${M}_{4}:=sup\{\parallel A{x}_{k}-Az\parallel +\parallel S{z}_{k}-SAz\parallel :k\ge 1\}<\mathrm{\infty}$. Using this together with $A{x}_{{k}_{j}}\rightharpoonup Aq\in Fix(S)$ and (3.14), we obtain $\u3008h(Az),Aq-Az\u3009\le 0$ for all $Az\in Fix(S)$. Thus, $Aq\in Fix(S)$ solves (3.2), and therefore, $q\in \mathrm{\Gamma}$.

Finally, it remains to show that ${x}_{k}\rightharpoonup q\in \mathrm{\Gamma}$. By the boundedness of ${\{{x}_{k}\}}_{k=1}^{\mathrm{\infty}}$, it suffices to show that there is no subsequence ${\{{x}_{{k}_{i}}\}}_{i=1}^{\mathrm{\infty}}$ of ${\{{x}_{k}\}}_{k=1}^{\mathrm{\infty}}$ such that ${x}_{{k}_{i}}\rightharpoonup p\in {H}_{1}$ and $p\ne q$.

which leads to a contradiction. Therefore, the sequence ${\{{x}_{k}\}}_{k=1}^{\mathrm{\infty}}$ converges weakly to a point $q\in \mathrm{\Gamma}$. □

### 3.2 Split hierarchical variational inequality problems

*split hierarchical variational inequality problem:*find a point ${x}^{\ast}\in Fix(T)$ such that

The solution set of the SHVIP is denoted by Ω. Since $Fix(T)$ and $Fix(S)$ are nonempty closed convex and *f* and *h* are monotone and continuous, by the Minty lemma [[13], Lemma 1], SHVIP (3.16)-(3.17) and SHMVIP (3.1)-(3.2) are equivalent. Hence, Algorithm 3.1 and Theorem 3.3 are also applicable for SHVIP (3.16)-(3.17).

When $Fix(T)=C$ a closed convex subset of a Hilbert space ${H}_{1}$ and $Fix(S)=Q$ a closed convex subset of a Hilbert space ${H}_{2}$, then SHVIP (3.16)-(3.17) is considered and studied by Censor *et al.* [24].

**Remark 3.4** It is worth to note that, in the context of SHVIP (3.16)-(3.17), if we assume either the finite dimensional settings of ${H}_{1}$ and ${H}_{2}$ or the compactness of spaces ${H}_{1}$ and ${H}_{2}$, then the monotonicity assumptions of operators *f* and *h* can be omitted.

*f*is continuous, by approaching

*j*to infinity, we obtain $\u3008f(q),q-z\u3009\le 0$, for all $z\in Fix(T)$, which means that $q\in Fix(T)$ solves (3.16). Similarly, from the inequality (3.15), we note that, for $k\ge {k}_{0}$,

where ${M}_{4}:=sup\{\parallel A{x}_{k}-Az\parallel +\parallel S{z}_{k}-SAz\parallel :k\ge 1\}<\mathrm{\infty}$. Since $A{x}_{{k}_{j}}\to Aq\in Fix(S)$ and *h* is continuous, we also get $\u3008h(Aq),Aq-Az\u3009\le 0$, for all $Az\in Fix(S)$, which means that $Aq\in Fix(S)$ solves (3.17), and subsequently, $q\in \mathrm{\Omega}$.

**Remark 3.5** The strong nonexpansiveness of operators *T* and *S* in Theorem 3.3 can be applied to the cases when the operators *T* and *S* are not only firmly nonexpansive, but with relaxation of being firmly nonexpansive and averaged nonexpansive but also strictly nonexpansive, that is, $\parallel Tx-Ty\parallel <\parallel x-y\parallel $ or $x-y=Tx-Ty$ for all *x*, *y*; regarding the compactness of the spaces ${H}_{1}$ and ${H}_{2}$, for more details, see [[21], Remark 2.3.3].

## 4 Some related problems

In this section, we present some split-type problems which are special cases of SHVIP (3.1) and (3.2) and can be solved by using Algorithm 3.1.

### 4.1 A split convex minimization problem

*split convex minimization problem*(in short, SCMP): find

where ∇*ϕ* and ∇*φ* denote the gradient of *ϕ* and *φ*, respectively. Since ∇*ϕ* and ∇*φ* are monotone [[12], Proposition 5.3] and continuous, we can apply Algorithm 3.1 to obtain the solution of SCMP (4.1)-(4.2), and Theorem 3.3 will provide the convergence of the sequence ${\{{x}_{k}\}}_{k=1}^{\mathrm{\infty}}$ to a solution of SCMP (4.1)-(4.2). Furthermore, in the context of finite-dimensional cases, we can remove the convexity of *ϕ* and *φ*.

### 4.2 A split variational inequality problem over the solution set of monotone variational inclusion problem

*monotone variational inclusion problem*(in short, MVIP): find

We denote the solution set of MVIP by $SOL(\varphi ,{B}_{1})$.

where ${J}_{\sigma}^{{B}_{1}}$ represents the resolvent of ${B}_{1}$ with parameter *σ*.

By using these facts and adding the assumption that ${J}_{\sigma}^{{B}_{2}}(I-\lambda \phi )$ is cutter, we can apply Algorithm 3.1 and Theorem 3.3 to obtain the solution of SVIP (4.3)-(4.4).

### 4.3 A split variational inequality problem over the solution set of equilibrium problem

*equilibrium problem*defined by

*C*and

*ϕ*is the problem of finding ${x}^{\ast}\in C$ such that

and we denote the solution set of equilibrium problem of *C* and *ϕ* by $EP(C,\varphi )$.

Recall that Blum and Oettli [26] showed that the bifunction *ϕ* satisfies the following conditions:

(A1) $\varphi (x,x)=0$ for all $x\in C$;

(A2) *ϕ* is monotone, that is, $\varphi (x,y)+\varphi (y,x)\le 0$ for all $x,y\in C$;

(A4) for all $x\in C$, $\varphi (x,\cdot )$ is a convex and lower semicontinuous function,

- (i)
${T}_{r}$ is single-valued;

- (ii)
${T}_{r}$ is firmly nonexpansive;

- (iii)
$Fix({T}_{r})=EP(C;\varphi )$.

*Q*and

*φ*hold, we see that, for all $u\in {H}_{2}$ and $s>0$, the set

is a nonempty set. Moreover, we also see that ${T}_{s}$ is single-valued and firmly nonexpansive; and $Fix({T}_{s})=EP(Q;\phi )$.

By using these facts, we obtain the result that the problem (4.5)-(4.6) forms a special case of the SHVIP (3.1)-(3.2).

## Declarations

### Acknowledgements

The authors would like to thank the anonymous referees for their careful reading and suggestions, which allowed us to improve the first version of this paper. This research is partially support by the Center of Excellence in Mathematics the Commission on Higher Education, Thailand. QH Ansari was partially supported by a KFUPM Funded Research Project No. IN121037. N Nimana was supported by the Thailand Research Fund through the Royal Golden Jubilee PhD Program (Grant No. PHD/0079/2554).

## Authors’ Affiliations

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