# An algorithm for finding common solutions of various problems in nonlinear operator theory

- Eric U Ofoedu
^{1}, - Jonathan N Odumegwu
^{1}, - Habtu Zegeye
^{2}and - Naseer Shahzad
^{3}Email author

**2014**:9

https://doi.org/10.1186/1687-1812-2014-9

© Ofoedu et al.; licensee Springer. 2014

**Received: **27 August 2013

**Accepted: **21 November 2013

**Published: **9 January 2014

## Abstract

In this paper, it is our aim to prove strong convergence of a new iterative algorithm to a common element of the set of solutions of a finite family of classical equilibrium problems; a common set of zeros of a finite family of inverse strongly monotone operators; the set of common fixed points of a finite family of quasi-nonexpansive mappings; and the set of common fixed points of a finite family of continuous pseudocontractive mappings in Hilbert spaces on assumption that the intersection of the aforementioned sets is not empty. Moreover, the common element is shown to be the metric projection of the initial guess on the intersection of these sets.

**MSC:**47H06, 47H09, 47J05, 47J25.

## Keywords

*η*-inverse strongly monotone mappingmaximal monotone operatornonexpansive mappingsreal Hilbert spacepseudocontractive mappingsvariational inequality problem

## 1 Introduction

*H*be a real Hilbert space. A mapping

*T*with domain $D(T)$ and range $R(T)$ in

*H*is called an

*L*-

*Lipschitzian*mapping (or simply a

*Lipschitz*mapping) if and only if there exists $L\ge 0$ such that for all $x,y\in D(T)$,

If $L\in [0,1)$, then *T* is called *strict contraction* or simply *a contraction*; and if $L=1$, then *T* is called nonexpansive. A point $x\in D(T)$ is called a *fixed point* of an operator *T* if and only if $Tx=x$. The set of fixed points of an operator *T* is denoted by $Fix(T)$, that is, $Fix(T):=\{x\in D(T):Tx=x\}$.

*T*with domain $D(T)$ and range $R(T)$ in

*H*is called a

*quasi*-

*nonexpansive*mapping if and only if $Fix(T)\ne \mathrm{\varnothing}$ and for any $x\in D(T)$, for any $u\in Fix(T)$,

Every nonexpansive mapping with a nonempty fixed point set is quasi-nonexpansive. The following examples show that the converse is not true.

**Example 1.1** (see [1])

*T*is quasi-nonexpansive. The mapping

*T*is, however, not a nonexpansive mapping since for $x=\frac{\pi}{2}$ and $y=\pi $,

*T*is quasi-nonexpansive. Finally, we show that

*T*is not nonexpansive. To see this, let $x=\frac{2}{3\pi}$ and $y=\frac{1}{\pi}$, then

The concept of quasi-nonexpansive mappings was essentially introduced by Diaz and Metcalf [3]. Although Examples 1.1 and 1.2 guarantee the existence of a quasi-nonexpansive mapping which is not nonexpansive, we must note that a *linear quasi*-*nonexpansive* mapping defined on a subspace of a given vector space is nonexpansive on that subspace.

Another important generalization of the class of nonexpansive mappings is the class of pseudocontractive mappings. These mappings are intimately connected with the important class of nonlinear accretive operators. This connection will be made precise in what follows.

*T*with domain $D(T)$ and range $R(T)$ in

*H*is called

*pseudocontractive*if and only if for all $x,y\in D(T)$, the following inequality holds:

*T*is called

*pseudocontractive*if and only if for all $x,y\in D(T)$, we have that

It now follows trivially from (1.3) that every nonexpansive mapping is pseudocontractive. We note immediately that the class of pseudocontractive mappings is larger than that of nonexpansive mappings. For examples of pseudocontractive mappings which are not nonexpansive, the reader may see [5].

*A*with domain $D(A)$ and range $R(A)$ in

*E*is called

*monotone*if and only if for all $x,y\in D(A)$, the following inequality is satisfied:

*A*is called

*η*-

*inverse strongly monotone*if and only if there exists $\eta \in (0,1)$ such that for all $x,y\in D(A)$, we have that

It is easy to see from inequalities (1.3) and (1.4) that an operator *A* is monotone if and only if the mapping $T:=(I-A)$ is pseudocontractive. Consequently, the fixed point theory for pseudocontractive mappings is intimately connected with the zero of monotone mappings. For the importance of monotone mappings and their connections with evolution equations, the reader may consult any of the references [5, 6].

Due to the above connection, fixed point theory of pseudocontractive mappings became a flourishing area of intensive research for several authors.

*C*be a closed convex nonempty subset of a real Hilbert space

*H*with inner product $\u3008\cdot ,\cdot \u3009$ and norm $\parallel \cdot \parallel $. Let $f:C\times C\to \mathbb{R}$ be a bifunction. The classical

*equilibrium problem*(

*EP*) for a bifunction

*f*is to find ${u}^{\ast}\in C$ such that

*EP*(1.6) is denoted by

The classical equilibrium problem (*EP*) includes as special cases the monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, Nash equilibria in noncooperative games. Furthermore, there are several other problems, for example, the complementarity problems and fixed point problems, which can also be written in the form of the classical equilibrium problem. In other words, the classical equilibrium problem is a unifying model for several problems arising from engineering, physics, statistics, computer science, optimization theory, operations research, economics and countless other fields. For the past 20 years or so, many existence results have been published for various equilibrium problems (see, *e.g.*, [7–10]). Approximation methods for such problems thus become a necessity.

Iterative approximation of fixed points and zeros of nonlinear mappings has been studied extensively by many authors to solve nonlinear mapping equations as well as variational inequality problems and their generalizations (see, *e.g.*, [11–19]). Most published results on nonexpansive mappings (for example) focus on the iterative approximation of their fixed points or approximation of common fixed points of a given family of this class of mappings.

*et al.*[20])). Nakajo and Takahashi [16] formulated the following modification of the Mann iteration method for a nonexpansive mapping

*T*defined on a nonempty bounded closed and convex subset

*C*of a Hilbert space

*H*:

where ${P}_{C}$ denotes the metric projection from *H* onto a closed convex subset *C* of *H*. They proved that if the sequence ${\{{\alpha}_{n}\}}_{n\ge 0}$ is bounded away from 1, then ${\{{x}_{n}\}}_{n\ge 0}$ defined by (1.7) converges strongly to ${P}_{F(T)}({x}_{0})$.

Formulations similar to (1.7) for different classes of nonlinear problems had been presented by Kim and Xu [21], Nilsrakoo and Saejung [22], Ofoedu *et al.* [23], Yang and Su [24], Zegeye and Shahzad [25–27].

In this paper, motivated by the results of the authors mentioned above, it is our aim to prove strong convergence of a new iterative algorithm to a common element of the set of solutions of a finite family of classical equilibrium problems; a common set of zeros of a finite family of inverse strongly monotone mappings; a set of common fixed points of a finite family of quasi-nonexpansive mappings; and a set of common fixed points of a finite family of continuous pseudocontractive mappings in Hilbert spaces on assumption that the intersection of the aforementioned sets is not empty. Moreover, the common element is shown to be the metric projection of the initial guess on the intersection of these sets. Our theorems complement the results of the authors mentioned above and those of several other authors.

## 2 Preliminary

In what follows, we shall make use of the following lemmas.

**Lemma 2.1** (see, *e.g.*, Kopecka and Reich [28])

*Let*

*C*

*be a nonempty closed and convex subset of a real Hilbert space*.

*Let*$x\in H$

*and*${P}_{C}:H\to C$

*be the metric projection of*

*H*

*onto*

*C*,

*then for any*$y\in C$,

**Lemma 2.2** *Let* *C* *be a closed convex nonempty subset of a real Hilbert space* *H*; *and let* ${P}_{C}:H\to C$ *be the metric projection of* *H* *onto* *C*. *Let* $x\in H$, *then* ${x}_{0}={P}_{C}x$ *if and only if* $\u3008z-{x}_{0},x-{x}_{0}\u3009\le 0$ *for all* $z\in C$.

**Lemma 2.3**

*Let*

*H*

*be a real Hilbert space*,

*then for any*$x,y\in H$, $\alpha \in [0,1]$,

**Lemma 2.4** (see Zegeye [29])

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*.

*Let*$T:C\to H$

*be a continuous pseudocontractive mapping*,

*then for all*$r>0$

*and*$x\in H$,

*there exists*$z\in C$

*such that*

**Lemma 2.5** (see Zegeye [29])

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*.

*Let*$T:C\to C$

*be a continuous pseudocontractive mapping*,

*then for all*$r>0$

*and*$x\in H$,

*define a mapping*${F}_{r}:H\to C$

*by*

*then the following hold*:

- (1)
${F}_{r}$

*is single*-*valued*; - (2)${F}_{r}$
*is firmly nonexpansive type mapping*,*i*.*e*.,*for all*$x,y\in H$,${\parallel {F}_{r}x-{F}_{r}y\parallel}^{2}\le \u3008{F}_{r}x-{F}_{r}y,x-y\u3009;$ - (3)
$Fix({F}_{r})$

*is closed and convex*;*and*$Fix({F}_{r})=Fix(T)$*for all*$r>0$.

In the sequel, we shall require that the bifunction $f:C\times C\to \mathbb{R}$ satisfies the following conditions:

(A1) $f(x,x)=0$, $\mathrm{\forall}x\in C$;

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

(A3) ${lim\hspace{0.17em}sup}_{t\to {0}^{+}}f(tz+(1-t)x,y)\le f(x,y)$ for all $x,y,z\in C$;

(A4) the function $y\mapsto f(x,y)$ is convex and lower semicontinuous for all $x\in C$.

**Lemma 2.6** (see, *e.g.*, [7, 30])

*Let*

*C*

*be a closed convex nonempty subset of a real Hilbert space*

*H*.

*Let*$f:C\times C\to \mathbb{R}$

*be a bifunction satisfying conditions*(A1)-(A4),

*then for all*$r>0$

*and*$x\in H$,

*there exists*$u\in C$

*such that*

*Moreover*,

*if for all*$x\in H$

*we define a mapping*${G}_{r}:H\to {2}^{C}$

*by*

*then the following hold*:

- (1)
${G}_{r}$

*is single*-*valued for all*$r>0$; - (2)${G}_{r}$
*is firmly nonexpansive*,*that is*,*for all*$x,z\in H$,${\parallel {G}_{r}x-{G}_{r}z\parallel}^{2}\le \u3008{G}_{r}x-{G}_{r}z,x-z\u3009;$ - (3)
$Fix({G}_{r})=\mathit{EP}(f)$

*for all*$r>0$; - (4)
$\mathit{EP}(f)$

*is closed and convex*.

**Lemma 2.7** (see Ofoedu [31])

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*.

*Let*$T:C\to C$

*be a continuous pseudocontractive mapping*.

*For*$r>0$,

*let*${F}_{r}:H\to C$

*be the mapping in Lemma*2.5,

*then for any*$x\in H$

*and for any*$p,q>0$,

**Lemma 2.8** (Compare with Lemma 13 of Ofoedu [31])

*Let*

*C*

*be a closed convex nonempty subset of a real Hilbert space*

*H*.

*Let*$f:C\times C\to \mathbb{R}$

*be a bifunction satisfying conditions*(A1)-(A4).

*Let*$r>0$

*and let*${G}_{r}$

*be the mapping in Lemma*2.6,

*then for all*$p,q>0$

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

*we have that*

## 3 Main results

*C*be a nonempty closed convex subset of a real Hilbert space

*H*. Let ${T}_{1},{T}_{2},\dots ,{T}_{m}:C\to C$ be

*m*continuous pseudocontractive mappings; let ${S}_{1},{S}_{2},\dots ,{S}_{l}:C\to C$ be

*l*continuous quasi-nonexpansive mappings; let ${A}_{1},{A}_{2},\dots ,{A}_{d}:C\to H$ be

*d*${\gamma}_{j}$-inverse strongly monotone mappings with constants ${\gamma}_{j}\in (0,1)$, $j=1,2,\dots ,d$; let ${f}_{1},{f}_{2},\dots ,{f}_{t}:C\times C\to \mathbb{R}$ be

*t*bifunctions satisfying conditions (A1)-(A4). For all $x\in E$, $i=1,2,\dots ,m$, let

where ${A}_{n}={A}_{n(modd)}$, ${S}_{n}={S}_{n(modl)}$; $\{{r}_{n}\}\subset (0,\mathrm{\infty})$ such that ${lim}_{n\to \mathrm{\infty}}{r}_{n}={r}_{0}>0$; ${\{{\alpha}_{n}\}}_{n\ge 1}$ a sequence in $(0,1)$ such that ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$; ${\{{\beta}_{i}\}}_{i=1}^{m},{\{{\xi}_{h}\}}_{h=1}^{t}\subset (0,1)$ such that ${\sum}_{i=1}^{m}{\beta}_{i}=1={\sum}_{h=1}^{t}{\xi}_{h}$; $\eta \in (0,1)$ and $\{{\lambda}_{n}\}$ is a sequence in $[a,b]$ for some $a,b\in \mathbb{R}$ such that $0<a<b<2\gamma $, $\gamma ={min}_{1\le j\le d}\{{\gamma}_{j}\}$.

**Lemma 3.1** *Let* *C* *be a nonempty closed convex subset of a real Hilbert space* *H*. *Let* ${T}_{1},{T}_{2},\dots ,{T}_{m}:C\to C$ *be* *m* *continuous pseudocontractive mappings*; *let* ${S}_{1},{S}_{2},\dots ,{S}_{l}:C\to C$ *be* *l* *continuous quasi*-*nonexpansive mappings*; *let* ${A}_{1},{A}_{2},\dots ,{A}_{d}:C\to H$ *be* *d* ${\gamma}_{j}$-*inverse strongly monotone mappings with constants* ${\gamma}_{j}\in (0,1)$, $j=1,2,\dots ,d$; *let* ${f}_{1},{f}_{2},\dots ,{f}_{t}:C\times C\to \mathbb{R}$ *be* *t* *bifunctions satisfying conditions* (A1)-(A4). *Let* $F:={\bigcap}_{i=1}^{m}Fix({T}_{i})\cap {\bigcap}_{j=1}^{d}{A}_{j}^{-1}(0)\cap {\bigcap}_{k=1}^{l}Fix({S}_{k})\cap {\bigcap}_{h=1}^{t}\mathit{EP}({f}_{h})\ne \mathrm{\varnothing}$. *Let* $\{{x}_{n}\}$ *be a sequence defined by* (3.1), *then the sequence* $\{{x}_{n}\}$ *is well defined for each* $n\ge 0$.

*Proof*We first show that ${C}_{n}$ is closed and convex for each $n\in \mathbb{N}\cup \{0\}$. From the definitions of ${C}_{n}$ it is obvious that ${C}_{n}$ is closed. Moreover, since $\parallel {w}_{n}-z\parallel \le \parallel {x}_{n}-z\parallel $ is equivalent to $2\u3008z,{x}_{n}-{w}_{n}\u3009-{\parallel {x}_{n}\parallel}^{2}+{\parallel {w}_{n}\parallel}^{2}\le 0$, it follows that ${C}_{n}$ is convex for each $n\in \mathbb{N}\cup \{0\}$. Next, we prove that $F\subset {C}_{n}$ for each $n\in \mathbb{N}\cup \{0\}$. From the assumption, we see that $F\subset {C}_{0}=C$. Suppose that $F\subset {C}_{k}$ for some $k\ge 1$, then for $p\in F$, we obtain that

So, $p\in {C}_{k+1}$. This implies, by induction, that $F\subset {C}_{n}$ so that the sequence generated by (3.1) is well defined for all $n\ge 0$. □

**Theorem 3.2** *Let* *C* *be a nonempty closed convex subset of a real Hilbert space* *H*. *Let* ${T}_{1},{T}_{2},\dots ,{T}_{m}:C\to C$ *be* *m* *continuous pseudocontractive mappings*; *let* ${S}_{1},{S}_{2},\dots ,{S}_{l}:C\to C$ *be* *l* *continuous quasi*-*nonexpansive mappings*; *let* ${A}_{1},{A}_{2},\dots ,{A}_{d}:C\to H$ *be* *d* ${\gamma}_{j}$-*inverse strongly monotone mappings with constants* ${\gamma}_{j}\in (0,1)$, $j=1,2,\dots ,d$; *let* ${f}_{1},{f}_{2},\dots ,{f}_{t}:C\times C\to \mathbb{R}$ *be* *t* *bifunctions satisfying conditions* (A1)-(A4). *Let* $F:={\bigcap}_{i=1}^{m}Fix({T}_{i})\cap {\bigcap}_{j=1}^{d}{A}_{j}^{-1}(0)\cap {\bigcap}_{k=1}^{l}Fix({S}_{k})\cap {\bigcap}_{h=1}^{t}\mathit{EP}({f}_{h})\ne \mathrm{\varnothing}$. *Let* $\{{x}_{n}\}$ *be a sequence defined by* (3.1). *Then the sequence* ${\{{x}_{n}\}}_{n\ge 0}$ *converges strongly to the element of* *F* *nearest to* ${x}_{0}$.

*Proof*From Lemma 3.1, we obtain that $F\subset {C}_{n}$, $\mathrm{\forall}n\ge 0$ and ${x}_{n}$ is well defined for each $n\ge 0$. From ${x}_{n}={P}_{{C}_{n}}({x}_{0})$ and ${x}_{n+1}={P}_{{C}_{n+1}}({x}_{0})\in {C}_{n+1}\subset {C}_{n}$, we obtain that

*μ*,

*C*. Therefore, there exists ${x}^{\ast}\in C$ such that ${lim}_{n\to \mathrm{\infty}}{x}_{n}={x}^{\ast}$. Since ${x}_{n+1}\in {C}_{n+1}$, we have that

and hence $\parallel {x}_{n}-{w}_{n}\parallel \le \parallel {x}_{n}-{x}_{n+1}\parallel +\parallel {x}_{n+1}-{w}_{n}\parallel \to 0$ as $n\to \mathrm{\infty}$, which implies that ${w}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$.

and hence ${z}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$.

Continuing, we obtain that ${S}_{k}{x}^{\ast}={x}^{\ast}$, $k=3,\dots ,l$. Hence, ${x}^{\ast}\in {\bigcap}_{k=1}^{l}F({S}_{k})$.

*γ*-inverse strongly monotone for $j=1,2,\dots ,d$, we have that ${A}_{j}$ is $\frac{1}{\gamma}$-Lipschitz continuous. Thus,

Similarly, we have that ${A}_{j}{x}^{\ast}=0$ for $j=2,\dots ,d$. Thus, ${x}^{\ast}\in {\bigcap}_{j=1}^{d}{A}_{i}^{-1}(0)$.

Now, by Lemma 2.2 we have that ${x}^{\ast}={P}_{F}({x}_{0})$. This completes the proof. □

**Remark 3.3** We note that ${x}^{\ast}={P}_{F}({x}_{0})$ makes sense since it could be easily shown that *F* is closed and convex. In fact, it is enough to show that the set of zeros of *γ*-inverse monotone mappings and a fixed point set of continuous quasi-nonexpansive mappings are convex sets. Closure of the two sets simply follows from the continuity of the mappings involved.

**Remark 3.4**Several authors (see,

*e.g.*, [8, 31] and references therein) have studied the following problem: Let

*C*be a closed convex nonempty subset of a real Hilbert space

*H*with inner product $\u3008\cdot ,\cdot \u3009$ and norm $\parallel \cdot \parallel $. Let $f:C\times C\to \mathbb{R}$ be a bifunction and $\mathrm{\Phi}:C\to \mathbb{R}\cup \{+\mathrm{\infty}\}$ be a proper extended real-valued function, where ℝ denotes the set of real numbers. Let $\mathrm{\Theta}:C\to H$ be a nonlinear monotone mapping. The

*generalized mixed equilibrium problem*(abbreviated

*GMEP*) for

*f*, Φ and Θ is to find ${u}^{\ast}\in C$ such that

*GMEP*(3.24) is denoted by

*equilibrium problem*(abbreviated

*EP*), that is, the problem of finding ${u}^{\ast}\in C$ such that

*GMEP*(1.6) reduces to the classical

*variational inequality problem*and $\mathit{GMEP}(0,0,\mathrm{\Theta})$ is denoted by $\mathit{VI}(\mathrm{\Theta},C)$, where

*GMEP*(3.24) reduces to the following

*minimization problem*:

*mixed equilibrium problem*(abbreviated

*MEP*) and $\mathit{GMEP}(f,\mathrm{\Phi},0)$ is denoted by $\mathit{MEP}(f,\mathrm{\Phi})$, where

*generalized equilibrium problem*(abbreviated

*GEP*) and $\mathit{GMEP}(f,0,\mathrm{\Theta})$ is denoted by $\mathit{GEP}(f,\mathrm{\Theta})$, where

*GMEP*(3.24) reduces to the generalized variational inequality problem (abbreviated

*GVIP*) and $\mathit{GMEP}(0,\mathrm{\Phi},\mathrm{\Theta})$ is denoted by $\mathit{GVIP}(\mathrm{\Phi},\mathrm{\Theta},C)$, where

then it could be easily checked that Γ is a bifunction and satisfies properties (A1)-(A4). Thus, the so-called generalized mixed equilibrium problem reduces to the classical equilibrium problem for the bifunction Γ. Thus, consideration of the so-called generalized mixed equilibrium problem in place of the classical equilibrium problem studied in this paper leads to no further generalization.

## 4 Application (convex differentiable optimization)

In Section 1, we defined a Lipschitz continuous mapping and an inverse strongly monotone mapping. Inverse strongly monotone mappings arise in various areas of optimization and nonlinear analysis (see, for example, [32–38]). It follows from the Cauchy-Schwarz inequality that if a mapping $A:D(A)\subseteq H\to R(A)\subseteq H$ is $\frac{1}{L}$-inverse strongly monotone, then *A* is *L*-Lipschitz continuous. The converse of this statement, however, fails to be true. To see this, take for instance $A=-I$, where *I* is the identity mapping on *H*, then *A* is *L*-Lipschitz continuous (with $L=1$) but not $\frac{1}{L}$-inverse strongly monotone (that is, not firmly nonexpansive in this case).

Baillon and Haddad [39] showed in 1977 that if $D(A)=H$ and *A* is the gradient of a convex functional on *H*, then *A* is $\frac{1}{L}$-inverse strongly monotone if and only if *A* is *L*-Lipschitz continuous. This remarkable result, which has important applications in optimization theory (see, for example, [40–42]), has become known as the Baillon-Haddad theorem. In fact, we have the following theorem.

**Theorem 4.1** (Baillon-Haddad) (see Corollary 10 of [39])

*Let* $\varphi :H\to \mathbb{R}$ *be a convex Fréchet*-*differentiable functional on* *H* *such that* ∇*ϕ* *is* *L*-*Lipschitz continuous for some* $L\in (0,+\mathrm{\infty})$, *then* ∇*ϕ* *is a* $\frac{1}{L}$-*inverse strongly monotone mapping* (*where* ∇*ϕ* *denotes the gradient of the functional* *ϕ*).

Now, let us turn to the problem of minimizing a continuously Fréchet-differentiable convex functional with minimum norm in Hilbert spaces.

*K*be a closed convex subset of a real Hilbert space

*H*, consider the minimization problem given by

*ϕ*is a Fréchet-differentiable convex functional. Let $\mathrm{\Omega}\subseteq K$, the solution set of (4.1), be nonempty. It is known that a point $z\in \mathrm{\Omega}$ if and only if the following optimality condition holds:

It is easy to see that if $K=H$, then optimality condition (4.2) is equivalent to $z\in \mathrm{\Omega}$ if and only if $z\in {(\mathrm{\nabla}\varphi )}^{-1}(0)$.

Thus, we obtain the following as a corollary of Theorem 3.2.

**Theorem 4.2**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*.

*Let*${T}_{1},{T}_{2},\dots ,{T}_{m}:C\to C$

*be*

*m*

*continuous pseudocontractive mappings*;

*let*${S}_{1},{S}_{2},\dots ,{S}_{l}:C\to C$

*be*

*l*

*continuous quasi*-

*nonexpansive mappings*;

*let*${\varphi}_{1},{\varphi}_{2},\dots ,{\varphi}_{d}:H\to H$

*be*

*d*

*convex and Fréchet*-

*differentiable functionals on*

*H*

*such that*${(\mathrm{\nabla}\varphi )}_{j}$

*is*${L}_{j}$-

*Lipschitz continuous for some*${L}_{j}\in (0,+\mathrm{\infty})$, $j=1,2,\dots ,d$;

*let*${f}_{1},{f}_{2},\dots ,{f}_{t}:C\times C\to \mathbb{R}$

*be*

*t*

*bifunctions satisfying conditions*(A1)-(A4).

*Let*$F:={\bigcap}_{i=1}^{m}Fix({T}_{i})\cap {\bigcap}_{j=1}^{d}{(\mathrm{\nabla}{\varphi}_{j})}^{-1}(0)\cap {\bigcap}_{k=1}^{l}Fix({S}_{k})\cap {\bigcap}_{h=1}^{t}\mathit{EP}({f}_{h})\ne \mathrm{\varnothing}$.

*Let*${\{{x}_{n}\}}_{n\ge 0}$

*be a sequence defined by*

*where* ${(\mathrm{\nabla}\varphi )}_{n}={(\mathrm{\nabla}\varphi )}_{n(modd)}$, ${S}_{n}={S}_{n(modl)}$; $\{{r}_{n}\}\subset (0,\mathrm{\infty})$ *such that* ${lim}_{n\to \mathrm{\infty}}{r}_{n}={r}_{0}>0$; ${\{{\alpha}_{n}\}}_{n\ge 1}$ *a sequence in* $(0,1)$ *such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$; ${\{{\beta}_{i}\}}_{i=1}^{m},{\{{\xi}_{h}\}}_{h=1}^{t}\subset (0,1)$ *such that* ${\sum}_{i=1}^{m}{\beta}_{i}=1={\sum}_{h=1}^{t}{\xi}_{h}$; $\eta \in (0,1)$ *and* $\{{\lambda}_{n}\}$ *is a sequence in* $[a,b]$ *for some* $a,b\in \mathbb{R}$ *such that* $0<a<b<\frac{2}{L}$, $L={max}_{1\le j\le d}\{{L}_{j}\}$. *Then the sequence* ${\{{x}_{n}\}}_{n\ge 0}$ *converges strongly to the element of* *F* *nearest to* ${x}_{0}$.

*Proof* Since, by our hypothesis, ${(\mathrm{\nabla}\varphi )}_{j}$ is ${L}_{j}$-Lipschitz continuous for some ${L}_{j}\in (0,+\mathrm{\infty})$, $j=1,2,\dots ,d$, we obtain from Theorem 4.1 that ${(\mathrm{\nabla}\varphi )}_{j}$ is $\frac{1}{{L}_{j}}$-inverse strongly monotone, $j=1,2,\dots ,d$; and since $L={max}_{1\le j\le d}\{{L}_{j}\}$, it is then easy to see that ${(\mathrm{\nabla}\varphi )}_{j}$ is $\frac{1}{L}$-inverse strongly monotone, $j=1,2,\dots ,d$. The rest, therefore, follows as in the proof of Theorem 3.2 with $\gamma =\frac{1}{L}$. This completes the proof. □

## Declarations

### Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. N Shahzad acknowledges with thanks DSR for financial support.

## Authors’ Affiliations

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