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# Strong convergence theorem for the modified generalized equilibrium problem and fixed point problem of strictly pseudo-contractive mappings

Fixed Point Theory and Applications20142014:86

https://doi.org/10.1186/1687-1812-2014-86

• Accepted: 17 March 2014
• Published:

## Abstract

The purpose of this paper is to modify the generalized equilibrium problem introduced by Ceng et al. (J. Glob. Optim. 43:487-502, 2012) and to introduce the K-mapping generated by a finite family of strictly pseudo-contractive mappings and finite real numbers modifying the results of Kangtunyakarn and Suantai (Nonlinear Anal. 71:4448-4460, 2009). Then we prove the strong convergence theorem for finding a common element of the set of fixed points of a finite family of strictly pseudo-contractive mappings and a finite family of the set of solutions of the modified generalized equilibrium problem. Moreover, using our main result, we obtain the additional results related to the generalized equilibrium problem.

## Keywords

• strictly pseudo-contractive mapping
• K-mapping
• the modified generalized equilibrium problem
• equilibrium-like function

## 1 Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product $〈\cdot ,\cdot 〉$ and the norm $\parallel \cdot \parallel$. A mapping $f:C\to C$ is contractive if there exists a constant $\alpha \in \left(0,1\right)$ such that
$\parallel f\left(x\right)-f\left(y\right)\parallel \le \alpha \parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

We now recall some well-known concepts and results as follows.

Definition 1.1 Let $B:C\to C$ be a mapping. Then B is called
1. (i)
monotone if
$〈Bx-By,x-y〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C,$

2. (ii)
υ-strongly monotone if there exists a positive real number υ such that
$〈Bx-By,x-y〉\ge \upsilon {\parallel x-y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C,$

3. (iii)
ξ-inverse strongly monotone if there exists a positive real number ξ such that
$〈x-y,Bx-By〉\ge \xi {\parallel Bx-By\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C,$

4. (iv)
μ-Lipschitz continuous if there exists a nonnegative real number $\mu \ge 0$ such that
$\parallel Bx-By\parallel \le \mu \parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

Definition 1.2 Let $T:C\to C$ be a mapping. Then:
1. (i)

An element $x\in C$ is said to be a fixed point of T if $Tx=x$ and $F\left(T\right)=\left\{x\in C:Tx=x\right\}$ denotes the set of fixed points of T.

2. (ii)
Mapping T is called nonexpansive if
$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

3. (iii)
T is said to be κ-strictly pseudo-contractive if there exists a constant $\kappa \in \left[0,1\right)$ such that
${\parallel Tx-Ty\parallel }^{2}\le {\parallel x-y\parallel }^{2}+\kappa {\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$
(1.1)

Note that the class of κ-strictly pseudo-contractions strictly includes the class of nonexpansive mappings, that is, nonexpansive mapping is a 0-strictly pseudo-contraction mapping. In a real Hilbert space H (1.1) is equivalent to
$〈Tx-Ty,x-y〉\le {\parallel x-y\parallel }^{2}-\frac{1-\kappa }{2}{\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

Remark 1.1 $T:C\to C$ is a κ-strictly pseudo-contraction if and only if $I-T$ is $\frac{1-\kappa }{2}$-inverse strongly monotone.

In the last decades, many researcher have studied fixed point theorems associated with various types of nonlinear mapping; see, for instance, [14]. Fixed point problems arise in many fields such as the vibration of masses attached to strings or nets [5] and a network bandwidth allocation problem [6] which is one of the central issues in modern communication networks. For applications to neural networks, fixed point theorems can be used to design dynamic neural network in order to solve steady state solutions [7]. For general information on neural networks, see for instance, [8, 9].

Let $F:C×C\to \mathbb{R}$ be bifunction. The equilibrium problem for F is to determine its equilibrium point, i.e., the set
$\mathit{EP}\left(F\right)=\left\{x\in C:F\left(x,y\right)\ge 0,\mathrm{\forall }y\in C\right\}.$
(1.2)

Equilibrium problems were introduced by [10] in 1994 where such problems have had a significant impact and influence in the development of several branches of pure and applied sciences. Various problems in physics, optimization, and economics are related to seeking some elements of $\mathit{EP}\left(F\right)$; see [10, 11]. Many authors have been investigating iterative algorithms for the equilibrium problems; see, for example, [1115].

Let $\mathit{CB}\left(H\right)$ be the family of all nonempty closed bounded subsets of H and $\mathcal{H}\left(\cdot ,\cdot \right)$ be the Hausdorff metric on $\mathit{CB}\left(H\right)$ defined as
$\mathcal{H}\left(U,V\right)=max\left\{\underset{u\in U}{sup}d\left(u,V\right),\underset{v\in V}{sup}d\left(U,v\right)\right\},\phantom{\rule{1em}{0ex}}\mathrm{\forall }U,V\in \mathit{CB}\left(H\right),$

where $d\left(u,V\right)={inf}_{v\in V}d\left(u,v\right)$, $d\left(U,v\right)={inf}_{u\in U}d\left(u,v\right)$ and $d\left(u,v\right)=\parallel u-v\parallel$.

Let C be a nonempty closed convex subset of H. Let $\phi :C\to \mathbb{R}$ be a real-valued function, $T:C\to \mathit{CB}\left(H\right)$ a multivalued mapping and $\mathrm{\Phi }:H×C×C\to \mathbb{R}$ an equilibrium-like function, that is, $\mathrm{\Phi }\left(w,u,v\right)+\mathrm{\Phi }\left(w,v,u\right)=0$ for all $\left(w,u,v\right)\in H×C×C$ which satisfies the following conditions with respect to the multivalued mapping $T:C\to \mathit{CB}\left(H\right)$.

(H1) For each fixed $v\in C$, $\left(w,u\right)↦\mathrm{\Phi }\left(w,u,v\right)$ is an upper semicontinuous function from $H×C\to \mathbb{R}$, that is, for $\left(w,u\right)\in H×C$, whenever ${w}_{n}\to w$ and ${u}_{n}\to u$ as $n\to \mathrm{\infty }$,
$\underset{n\to \mathrm{\infty }}{lim sup}\mathrm{\Phi }\left({w}_{n},{u}_{n},v\right)\le \mathrm{\Phi }\left(w,u,v\right).$

(H2) For each fixed $\left(w,v\right)\in H×C$, $u↦\mathrm{\Phi }\left(w,u,v\right)$ is a concave function.

(H3) For each fixed $\left(w,u\right)\in H×C$, $v↦\mathrm{\Phi }\left(w,u,v\right)$ is a convex function.

In 2009, Ceng et al. [16] introduced the generalized equilibrium problem $\left(\mathit{GEP}\right)$ as follows:
(1.3)

The set of such solutions $u\in C$ of $\left(\mathit{GEP}\right)$ is denoted by ${\left(\mathit{GEP}\right)}_{s}\left(\mathrm{\Phi },\phi \right)$. In the case of $\phi =0$ and $\mathrm{\Phi }\left(w,u,v\right)\equiv G\left(u,v\right)$, then ${\left(\mathit{GEP}\right)}_{s}\left(\mathrm{\Phi },\phi \right)$ is denoted by $\mathit{EP}\left(G\right)$.

By using Nadler’s theorem [17], they introduced the following algorithm:

Let ${x}_{1}\in C$ and ${w}_{1}\in T\left({x}_{1}\right)$, there exist sequences $\left\{{w}_{n}\right\}\subseteq H$ and $\left\{{x}_{n}\right\},\left\{{u}_{n}\right\}\subseteq C$ such that
$\left\{\begin{array}{c}{w}_{n}\in T\left({x}_{n}\right),\phantom{\rule{1em}{0ex}}\parallel {w}_{n}-{w}_{n+1}\parallel \le \left(1+\frac{1}{n}\right)\mathcal{H}\left(T\left({x}_{n}\right),T\left({x}_{n+1}\right)\right),\hfill \\ \mathrm{\Phi }\left({w}_{n},{u}_{n},v\right)+\phi \left(v\right)-\phi \left({u}_{n}\right)+\frac{1}{{r}_{n}}〈{u}_{n}-{x}_{n},v-{u}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }v\in C,\hfill \\ {x}_{n+1}={\alpha }_{n}f\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)S{u}_{n},\phantom{\rule{1em}{0ex}}n=1,2,\dots .\hfill \end{array}$
(1.4)

They proved the strong convergence theorem of the sequence $\left\{{x}_{n}\right\}$ generated by (1.4) as follows.

Theorem 1.2 ([16])

Let C be a nonempty, bounded, closed and convex subset of a real Hilbert space H and let $\phi :C\to \mathbb{R}$ be a lower semicontinuous and convex functional. Let $T:C\to \mathit{CB}\left(H\right)$ be -Lipschitz continuous with constant μ, $\mathrm{\Phi }:H×C×C\to \mathbb{R}$ be an equilibrium-like function satisfying (H1)-(H3) and S be a nonexpansive mapping of C into itself such that $F\left(S\right)\cap {\left(\mathit{GEP}\right)}_{s}\left(\mathrm{\Phi },\phi \right)\ne \mathrm{\varnothing }$. Let f be a contraction of C into itself and let $\left\{{x}_{n}\right\}$, $\left\{{w}_{n}\right\}$, and $\left\{{u}_{n}\right\}$ be sequences generated by (1.4), where $\left\{{\alpha }_{n}\right\}\subseteq \left[0,1\right]$ and $\left\{{r}_{n}\right\}\subset \left(0,\mathrm{\infty }\right)$ satisfy
$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim}{\alpha }_{n}=0,\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty },\phantom{\rule{2em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_{n}|<\mathrm{\infty },\hfill \\ \underset{n\to \mathrm{\infty }}{lim inf}{r}_{n}>0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\sum _{n=1}^{\mathrm{\infty }}|{r}_{n+1}-{r}_{n}|<\mathrm{\infty }.\hfill \end{array}$
If there exists a constant $\lambda >0$ such that
$\mathrm{\Phi }\left({w}_{1},{T}_{{r}_{1}}\left({x}_{1}\right),{T}_{{r}_{2}}\left({x}_{2}\right)\right)+\mathrm{\Phi }\left({w}_{2},{T}_{{r}_{2}}\left({x}_{2}\right),{T}_{{r}_{1}}\left({x}_{1}\right)\right)\le -\lambda {\parallel {T}_{{r}_{1}}\left({x}_{1}\right)-{T}_{{r}_{2}}\left({x}_{2}\right)\parallel }^{2},$
for all $\left({r}_{1},{r}_{2}\right)\in \mathrm{\Xi }×\mathrm{\Xi }$, $\left({x}_{1},{x}_{2}\right)\in C×C$ and ${w}_{i}\in T\left({x}_{i}\right)$, $i=1,2$, where $\mathrm{\Xi }=\left\{{r}_{n}:n\ge 1\right\}$, then for $\stackrel{ˆ}{x}={P}_{F\left(S\right)\cap {\left(\mathit{GEP}\right)}_{s}\left(\mathrm{\Phi },\phi \right)}f\left(\stackrel{ˆ}{x}\right)$, there exists $\stackrel{ˆ}{w}\in T\left(\stackrel{ˆ}{x}\right)$ such that $\left(\stackrel{ˆ}{x},\stackrel{ˆ}{w}\right)$ is a solution of $\left(\mathit{GEP}\right)$ and

In 2012, Kangtunyakarn [12] introduced the iterative algorithm as follows.

Algorithm 1.3 ([12])

Let ${T}_{i}:i=1,2,\dots ,N$, be ${\kappa }_{i}$-pseudo-contraction mappings of C into itself and $\kappa =max\left\{{\kappa }_{i}:i=1,2,\dots ,N\right\}$ and let ${S}_{n}$ be the S-mappings generated by ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\alpha }_{1}^{\left(n\right)},{\alpha }_{2}^{\left(n\right)},\dots ,{\alpha }_{N}^{\left(n\right)}$, where ${\alpha }_{j}^{\left(n\right)}=\left({\alpha }_{1}^{n,j},{\alpha }_{2}^{n,j},{\alpha }_{3}^{n,j}\right)\in I×I×I$, $I=\left[0,1\right]$, ${\alpha }_{1}^{n,j}+{\alpha }_{2}^{n,j}+{\alpha }_{3}^{n,j}=1$ and $\kappa for all $j=1,2,\dots ,N-1$, $\kappa \le {\alpha }_{1}^{n,N}\le 1$, $\kappa \le {\alpha }_{3}^{n,N}\le d<1$, $\kappa \le {\alpha }_{2}^{n,N}\le e<1$ for all $j=1,2,\dots ,N$. Let ${x}_{1}\in C={C}_{1}$ and ${w}_{1}^{1}\in T\left({x}_{1}\right)$, ${w}_{1}^{2}\in D\left({x}_{1}\right)$, there exist sequences $\left\{{w}_{n}^{1}\right\},\left\{{w}_{n}^{2}\right\}\in H$, and $\left\{{x}_{n}\right\},\left\{{u}_{n}\right\},\left\{{v}_{n}\right\}\subseteq C$ such that
$\left\{\begin{array}{c}{w}_{n}^{1}\in T\left({x}_{n}\right),\parallel {w}_{n}^{1}-{w}_{n+1}^{1}\parallel \le \left(1+\frac{1}{n}\right)\mathcal{H}\left(T\left({x}_{n}\right),T\left({x}_{n+1}\right)\right),\hfill \\ {w}_{n}^{2}\in D\left({x}_{n}\right),\parallel {w}_{n}^{2}-{w}_{n+1}^{2}\parallel \le \left(1+\frac{1}{n}\right)\mathcal{H}\left(D\left({x}_{n}\right),D\left({x}_{n+1}\right)\right),\hfill \\ \mathrm{\Phi }\left({w}_{n}^{1},{u}_{n},u\right)+{\phi }_{1}\left(u\right)-{\phi }_{1}\left({u}_{n}\right)+\frac{1}{{r}_{n}}〈{u}_{n}-{x}_{n},u-{u}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }u\in C,\hfill \\ \mathrm{\Phi }\left({w}_{n}^{2},{v}_{n},v\right)+{\phi }_{2}\left(v\right)-{\phi }_{2}\left({v}_{n}\right)+\frac{1}{{s}_{n}}〈{v}_{n}-{x}_{n},v-{v}_{n}〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }v\in C,\hfill \\ {z}_{n}={\delta }_{n}{P}_{C}\left(I-\lambda A\right){u}_{n}+\left(1-{\delta }_{n}\right){P}_{C}\left(I-\eta B\right){v}_{n},\hfill \\ {y}_{n}={\alpha }_{n}{z}_{n}+\left(1-{\alpha }_{n}\right){S}_{n}{z}_{n},\hfill \\ {C}_{n+1}=\left\{z\in {C}_{n}:\parallel {y}_{n}-z\parallel \le \parallel {x}_{n}-z\parallel \right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1,\hfill \end{array}$
(1.5)

where $D,T:C\to \mathit{CB}\left(H\right)$ are -Lipschitz continuous with constants ${\mu }_{1}$, ${\mu }_{2}$, respectively, ${\mathrm{\Phi }}_{1},{\mathrm{\Phi }}_{2}:H×C×C\to \mathbb{R}$ are equilibrium-like functions satisfying (H1)-(H3), $A:C\to H$ is an α-inverse strongly monotone mapping and $B:C\to H$ is a β-inverse strongly monotone mapping.

He proved under some control conditions on $\left\{{\delta }_{n}\right\}$, $\left\{{\alpha }_{n}\right\}$, $\left\{{s}_{n}\right\}$, and $\left\{{r}_{n}\right\}$ that the sequence $\left\{{x}_{n}\right\}$ generated by (1.5) converges strongly to ${P}_{\mathcal{F}}{x}_{1}$, where $\mathcal{F}={\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap {\left(\mathit{GEP}\right)}_{s}\left({\mathrm{\Phi }}_{1},{\phi }_{1}\right)\cap {\left(\mathit{GEP}\right)}_{s}\left({\mathrm{\Phi }}_{2},{\phi }_{2}\right)\cap F\left({G}_{1}\right)\cap F\left({G}_{2}\right)$, ${G}_{1},{G}_{2}:C\to C$ are defined by ${G}_{1}\left(x\right)={P}_{C}\left(x-\lambda Ax\right)$, ${G}_{2}\left(x\right)={P}_{C}\left(x-\eta Bx\right)$, $\mathrm{\forall }x\in C$ and ${P}_{\mathcal{F}}{x}_{1}$ is a solution of the following system of variational inequalities:
$\left\{\begin{array}{c}〈A{x}^{\ast },x-{x}^{\ast }〉\ge 0,\hfill \\ 〈B{x}^{\ast },x-{x}^{\ast }〉\ge 0.\hfill \end{array}$
By modifying the generalized equilibrium problem (1.3), we introduced the modified generalized equilibrium problem $\left(\mathit{MGEP}\right)$ as follows:
(1.6)

where $A:C\to C$ is a mapping. The set of such solutions of $\left(\mathit{MGEP}\right)$ is denoted by ${\left(\mathit{MGEP}\right)}_{s}\left(\mathrm{\Phi },\phi ,A\right)$. If $A=0$, (1.6) reduces to (1.3).

In this paper, motivated by Theorem 1.2, Algorithm 1.3 and (1.6), we modify the generalized equilibrium problem introduced by Ceng et al. [16] and introduce the K-mapping generated by a finite family of strictly pseudo-contractive mappings and finite real numbers modifying the results of Kangtunyakarn and Suantai [13]. Then we prove the strong convergence theorem for finding a common element of the set of fixed points of a finite family of strictly pseudo-contractive mappings and a finite family of the set of solutions of the modified generalized equilibrium problem. Moreover, using our main result, we obtain the additional results related to the generalized equilibrium problem.

## 2 Preliminaries

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. We denote weak convergence and strong convergence by the notations ‘’ and ‘→’, respectively.

Recall that the (nearest point) projection ${P}_{C}$ from H onto C assigns to each $x\in H$ the unique point ${P}_{C}x\in C$ satisfying the property
$\parallel x-{P}_{C}x\parallel =\underset{y\in C}{min}\parallel x-y\parallel .$

The following lemmas are needed to prove the main theorem.

Lemma 2.1 ([18])

Let H be a real Hilbert space. Then the following identities hold:
1. (i)

${\parallel x±y\parallel }^{2}={\parallel x\parallel }^{2}±2〈x,y〉+{\parallel y\parallel }^{2}$, $\mathrm{\forall }x,y\in H$;

2. (ii)

${\parallel x+y\parallel }^{2}\le {\parallel x\parallel }^{2}+2〈y,x+y〉$, $\mathrm{\forall }x,y\in H$.

Lemma 2.2 ([19])

Let H be a real Hilbert space. Then for all ${x}_{i}\in H$ and ${\alpha }_{i}\in \left[0,1\right]$ for $i=0,1,2,\dots ,n$ such that ${\sum }_{i=0}^{n}{\alpha }_{i}=1$ the following equality holds:
${\parallel \sum _{i=0}^{n}{\alpha }_{i}{x}_{i}\parallel }^{2}=\sum _{i=0}^{n}{\alpha }_{i}{\parallel {x}_{i}\parallel }^{2}-\sum _{0\le i,j\le n}{\alpha }_{i}{\alpha }_{j}{\parallel {x}_{i}-{x}_{j}\parallel }^{2}.$

Lemma 2.3 ([18])

For a given $z\in H$ and $u\in C$,
$u={P}_{C}z\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}〈u-z,v-u〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }v\in C.$
Furthermore, ${P}_{C}$ is a firmly nonexpansive mapping of H onto C and satisfies
${\parallel {P}_{C}x-{P}_{C}y\parallel }^{2}\le 〈{P}_{C}x-{P}_{C}y,x-y〉,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in H.$

Lemma 2.4 (Demiclosedness principle [20])

Assume that T is a nonexpansive self-mapping of closed convex subset C of a Hilbert space H. If T has a fixed point, then $I-T$ is demiclosed. That is, whenever $\left\{{x}_{n}\right\}$ is a sequence in C weakly converging to some $x\in C$ and the sequence $\left\{\left(I-T\right){x}_{n}\right\}$ strongly converges to some y it follows that $\left(I-T\right)x=y$. Here, I is the identity mapping of H.

Lemma 2.5 ([21])

Let C be a nonempty closed convex subset of a real Hilbert space H and $S:C\to C$ be a self-mapping of C. If S is a κ-strict pseudo-contractive mapping, then S satisfies the Lipschitz condition
$\parallel Sx-Sy\parallel \le \frac{1+\kappa }{1-\kappa }\parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

Lemma 2.6 ([22])

Let $\left\{{s}_{n}\right\}$ be a sequence of nonnegative real numbers satisfying
${s}_{n+1}\le \left(1-{\alpha }_{n}\right){s}_{n}+{\delta }_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 0,$
where ${\alpha }_{n}$ is a sequence in $\left(0,1\right)$ and $\left\{{\delta }_{n}\right\}$ is a sequence such that
1. (1)

${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$;

2. (2)

${lim sup}_{n\to \mathrm{\infty }}\frac{{\delta }_{n}}{{\alpha }_{n}}\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n}|<\mathrm{\infty }$.

Then ${lim}_{n\to \mathrm{\infty }}{s}_{n}=0$.

Definition 2.1 A multivalued mapping $T:C\to \mathit{CB}\left(H\right)$ is said to be -Lipschitz continuous if there exists a constant $\mu >0$ such that
$\mathcal{H}\left(T\left(u\right),T\left(v\right)\right)\le \mu \parallel u-v\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }u,v\in C,$

where $\mathcal{H}\left(\cdot ,\cdot \right)$ is the Hausdorff metric on $\mathit{CB}\left(H\right)$.

Let $\left(X,\parallel \cdot \parallel \right)$ be a normed vector space and $\mathcal{H}\left(\cdot ,\cdot \right)$ is the Hausdorff metric on $\mathit{CB}\left(H\right)$. If $U,V\in \mathit{CB}\left(H\right)$, then for every $ϵ>0$ and $u\in U$, there exists $v\in V$ such that
$\parallel u-v\parallel \le \left(1+ϵ\right)\mathcal{H}\left(U,V\right).$

Theorem 2.8 ([16])

Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H, and let $\phi :C\to \mathbb{R}$ be a lower semicontinuous and convex functional. Let $T:C\to \mathit{CB}\left(H\right)$ be -Lipschitz continuous with constant μ, and $\mathrm{\Phi }:H×C×C\to \mathbb{R}$ be an equilibrium-like function satisfying (H1)-(H3). Let $r>0$ be a constant. For each $x\in C$, take ${w}_{x}\in T\left(x\right)$ arbitrarily and define a mapping ${T}_{r}:C\to C$ as follows:
${T}_{r}\left(x\right)=\left\{u\in C:\mathrm{\Phi }\left({w}_{x},u,v\right)+\phi \left(v\right)-\phi \left(u\right)+\frac{1}{r}〈u-x,v-u〉\ge 0,\mathrm{\forall }v\in C\right\}.$
Then we have the following:
1. (a)

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

2. (b)
${T}_{r}$ is firmly nonexpansive (that is, for any $u,v\in C$, ${\parallel {T}_{r}u-{T}_{r}v\parallel }^{2}\le 〈{T}_{r}u-{T}_{r}v,u-v〉$) if
$\mathrm{\Phi }\left({w}_{1},{T}_{r}\left({x}_{1}\right),{T}_{r}\left({x}_{2}\right)\right)+\mathrm{\Phi }\left({w}_{2},{T}_{r}\left({x}_{2}\right),{T}_{r}\left({x}_{1}\right)\right)\le 0,$

for all $\left({x}_{1},{x}_{2}\right)\in C×C$ and all ${w}_{i}\in T\left({x}_{i}\right)$, $i=1,2$;
1. (c)

$F\left({T}_{r}\right)={\left(\mathit{GEP}\right)}_{s}\left(\mathrm{\Phi },\phi \right)$;

2. (d)

${\left(\mathit{GEP}\right)}_{s}\left(\mathrm{\Phi },\phi \right)$ is closed and convex.

Definition 2.2 ([13])

Let C be a nonempty closed convex subset of a real Banach space. Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of ${\kappa }_{i}$-strictly pseudo-contractive mapping of C into itself and let ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{N}$ be real numbers with $0\le {\lambda }_{i}\le 1$ for every $i=1,2,\dots ,N$. Define a mapping $K:C\to C$ as follows:
$\begin{array}{c}{U}_{1}={\lambda }_{1}{T}_{1}+\left(1-{\lambda }_{1}\right)I,\hfill \\ {U}_{2}={\lambda }_{2}{T}_{2}{U}_{1}+\left(1-{\lambda }_{2}\right){U}_{1},\hfill \\ {U}_{3}={\lambda }_{3}{T}_{3}{U}_{2}+\left(1-{\lambda }_{3}\right){U}_{2},\hfill \\ ⋮\hfill \\ {U}_{N-1}={\lambda }_{N-1}{T}_{N-1}{U}_{N-2}+\left(1-{\lambda }_{N-1}\right){U}_{N-2},\hfill \\ K={U}_{N}={\lambda }_{N}{T}_{N}{U}_{N-1}+\left(1-{\lambda }_{N}\right){U}_{N-1}.\hfill \end{array}$
(2.1)

Such a mapping K is called the K-mapping generated by ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{N}$.

The following lemmas are needed to prove our main result.

Lemma 2.9 Let C be a nonempty closed convex subset of a real Hilbert space H. Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of ${\kappa }_{i}$-strictly pseudo-contractive mapping of C into itself with ${\kappa }_{i}\le {\gamma }_{1}$, for all $i=1,2,\dots ,N$, and ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\ne \mathrm{\varnothing }$. Let ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{N}$ be real numbers with $0<{\lambda }_{i}<{\gamma }_{2}$, for all $i=1,2,\dots ,N$ and ${\gamma }_{1}+{\gamma }_{2}<1$. Let K be the K-mapping generated by ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{N}$. Then the following properties hold:
1. (i)

$F\left(K\right)={\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$;

2. (ii)

K is a nonexpansive mapping.

Proof To prove (i), it is easy to see that ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\subseteq F\left(K\right)$.

Next, we claim that $F\left(K\right)\subseteq {\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$. To show this, let $x\in F\left(K\right)$ and $y\in {\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$.

By the definition of K-mapping, we get
$\begin{array}{c}\parallel x-y\parallel \hfill \\ \phantom{\rule{1em}{0ex}}={\parallel Kx-y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {\lambda }_{N}{T}_{N}{U}_{N-1}x+\left(1-{\lambda }_{N}\right){U}_{N-1}x-y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {\lambda }_{N}\left({T}_{N}{U}_{N-1}x-y\right)+\left(1-{\lambda }_{N}\right)\left({U}_{N-1}x-y\right)\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\lambda }_{N}^{2}{\parallel {T}_{N}{U}_{N-1}x-y\parallel }^{2}+{\left(1-{\lambda }_{N}\right)}^{2}{\parallel {U}_{N-1}x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+2{\lambda }_{N}\left(1-{\lambda }_{N}\right)〈{T}_{N}{U}_{N-1}x-y,{U}_{N-1}x-y〉\hfill \\ \phantom{\rule{1em}{0ex}}={\lambda }_{N}^{2}\left({\parallel {U}_{N-1}x-y\parallel }^{2}+{\kappa }_{N}{\parallel {T}_{N}{U}_{N-1}x-{U}_{N-1}x\parallel }^{2}\right)+{\left(1-{\lambda }_{N}\right)}^{2}{\parallel {U}_{N-1}x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+2{\lambda }_{N}\left(1-{\lambda }_{N}\right)\left({\parallel {U}_{N-1}x-y\parallel }^{2}-\frac{1-{\kappa }_{N}}{2}{\parallel {T}_{N}{U}_{N-1}x-{U}_{N-1}x\parallel }^{2}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=\left({\lambda }_{N}^{2}+{\left(1-{\lambda }_{N}\right)}^{2}+2{\lambda }_{N}\left(1-{\lambda }_{N}\right)\right){\parallel {U}_{N-1}x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+\left({\lambda }_{N}^{2}{\kappa }_{N}-{\lambda }_{N}\left(1-{\lambda }_{N}\right)\left(1-{\kappa }_{N}\right)\right){\parallel {T}_{N}{U}_{N-1}x-{U}_{N-1}x\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\left({\lambda }_{N}+1-{\lambda }_{N}\right)}^{2}{\parallel {U}_{N-1}x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+{\lambda }_{N}\left({\lambda }_{N}{\kappa }_{N}-\left(1-{\lambda }_{N}\right)\left(1-{\kappa }_{N}\right)\right){\parallel {T}_{N}{U}_{N-1}x-{U}_{N-1}x\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {U}_{N-1}x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+{\lambda }_{N}\left({\lambda }_{N}{\kappa }_{N}-\left(1-{\kappa }_{N}\right)+{\lambda }_{N}\left(1-{\kappa }_{N}\right)\right){\parallel {T}_{N}{U}_{N-1}x-{U}_{N-1}x\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {U}_{N-1}x-y\parallel }^{2}+{\lambda }_{N}\left({\kappa }_{N}+{\lambda }_{N}-1\right){\parallel {T}_{N}{U}_{N-1}x-{U}_{N-1}x\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {U}_{N-1}x-y\parallel }^{2}+{\lambda }_{N}\left({\gamma }_{1}+{\gamma }_{2}-1\right){\parallel {T}_{N}{U}_{N-1}x-{U}_{N-1}x\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {U}_{N-1}x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}⋮\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {U}_{2}x-y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {\lambda }_{2}\left({T}_{2}{U}_{1}x-y\right)+\left(1-{\lambda }_{2}\right)\left({U}_{1}x-y\right)\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\lambda }_{2}^{2}{\parallel {T}_{2}{U}_{1}x-y\parallel }^{2}+{\left(1-{\lambda }_{2}\right)}^{2}{\parallel {U}_{1}x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+2{\lambda }_{2}\left(1-{\lambda }_{2}\right)〈{T}_{2}{U}_{1}x-y,{U}_{1}x-y〉\hfill \\ \phantom{\rule{1em}{0ex}}={\lambda }_{2}^{2}\left({\parallel {U}_{1}x-y\parallel }^{2}+{\kappa }_{2}{\parallel {T}_{2}{U}_{1}x-{U}_{1}x\parallel }^{2}\right)+{\left(1-{\lambda }_{2}\right)}^{2}{\parallel {U}_{1}x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+2{\lambda }_{2}\left(1-{\lambda }_{2}\right)\left({\parallel {U}_{1}x-y\parallel }^{2}-\frac{1-{\kappa }_{2}}{2}{\parallel {T}_{2}{U}_{1}x-{U}_{1}x\parallel }^{2}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=\left({\lambda }_{2}^{2}+{\left(1-{\lambda }_{2}\right)}^{2}+2{\lambda }_{2}\left(1-{\lambda }_{2}\right)\right){\parallel {U}_{1}x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+\left({\lambda }_{2}^{2}{\kappa }_{2}-{\lambda }_{2}\left(1-{\lambda }_{2}\right)\left(1-{\kappa }_{2}\right)\right){\parallel {T}_{2}{U}_{1}x-{U}_{1}x\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\left({\lambda }_{2}+1-{\lambda }_{2}\right)}^{2}{\parallel {U}_{1}x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+{\lambda }_{2}\left({\lambda }_{2}{\kappa }_{2}-\left(1-{\lambda }_{2}\right)\left(1-{\kappa }_{2}\right)\right){\parallel {T}_{2}{U}_{1}x-{U}_{1}x\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {U}_{1}x-y\parallel }^{2}+{\lambda }_{2}\left({\kappa }_{2}+{\lambda }_{2}-1\right){\parallel {T}_{2}{U}_{1}x-{U}_{1}x\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {U}_{1}x-y\parallel }^{2}+{\lambda }_{2}\left(\left({\gamma }_{1}+{\gamma }_{2}\right)-1\right){\parallel {T}_{2}{U}_{1}x-{U}_{1}x\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {U}_{1}x-y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {\lambda }_{1}\left({T}_{1}x-y\right)+\left(1-{\lambda }_{1}\right)\left(x-y\right)\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\lambda }_{1}^{2}{\parallel {T}_{1}x-y\parallel }^{2}+{\left(1-{\lambda }_{1}\right)}^{2}{\parallel x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+2{\lambda }_{1}\left(1-{\lambda }_{1}\right)〈{T}_{1}x-y,x-y〉\hfill \\ \phantom{\rule{1em}{0ex}}={\lambda }_{1}^{2}\left({\parallel x-y\parallel }^{2}+{\kappa }_{1}{\parallel {T}_{1}x-x\parallel }^{2}\right)+{\left(1-{\lambda }_{1}\right)}^{2}{\parallel x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+2{\lambda }_{1}\left(1-{\lambda }_{1}\right)\left({\parallel x-y\parallel }^{2}-\frac{1-{\kappa }_{1}}{2}{\parallel {T}_{1}x-x\parallel }^{2}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=\left({\lambda }_{1}^{2}+{\left(1-{\lambda }_{1}\right)}^{2}+2{\lambda }_{1}\left(1-{\lambda }_{1}\right)\right){\parallel x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+\left({\lambda }_{1}^{2}{\kappa }_{1}-{\lambda }_{1}\left(1-{\lambda }_{1}\right)\left(1-{\kappa }_{1}\right)\right){\parallel {T}_{1}x-x\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\left({\lambda }_{1}+1-{\lambda }_{1}\right)}^{2}{\parallel x-y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+{\lambda }_{1}\left({\lambda }_{1}{\kappa }_{1}-\left(1-{\lambda }_{1}\right)\left(1-{\kappa }_{1}\right)\right){\parallel {T}_{1}x-x\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel x-y\parallel }^{2}+{\lambda }_{1}\left({\kappa }_{1}+{\lambda }_{1}-1\right){\parallel {T}_{1}x-x\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel x-y\parallel }^{2}+{\lambda }_{1}\left(\left({\gamma }_{1}+{\gamma }_{2}\right)-1\right){\parallel {T}_{1}x-x\parallel }^{2}.\hfill \end{array}$
(2.2)
From (2.2), it yields
${\lambda }_{1}\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right){\parallel {T}_{1}x-x\parallel }^{2}\le 0.$
This implies that
$\parallel {T}_{1}x-x\parallel =0.$
Therefore $x={T}_{1}x$, that is,
$x\in F\left({T}_{1}\right).$
(2.3)
By the definition of ${U}_{1}$ and (2.3), we have
${U}_{1}x={\lambda }_{1}{T}_{1}x+\left(1-{\lambda }_{1}\right)x=x,$
that is,
$x\in F\left({U}_{1}\right).$
(2.4)
Again by (2.2) and (2.4), we obtain
$\begin{array}{rl}{\parallel x-y\parallel }^{2}& \le {\parallel {U}_{1}x-y\parallel }^{2}+{\lambda }_{2}\left(\left({\gamma }_{1}+{\gamma }_{2}\right)-1\right){\parallel {T}_{2}{U}_{1}x-{U}_{1}x\parallel }^{2}\\ ={\parallel x-y\parallel }^{2}+{\lambda }_{2}\left(\left({\gamma }_{1}+{\gamma }_{2}\right)-1\right){\parallel {T}_{2}x-x\parallel }^{2},\end{array}$
which implies that $x={T}_{2}x$, that is,
$x\in F\left({T}_{2}\right).$
(2.5)
By the definition of ${U}_{2}$, (2.4), and (2.5), we get
${U}_{2}x={\lambda }_{2}{T}_{2}{U}_{1}x+\left(1-{\lambda }_{2}\right){U}_{1}x=x,$
from which it follows that
$x\in F\left({U}_{2}\right).$
Using the same argument, we can conclude that
$x\in F\left({T}_{i}\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}x\in F\left({U}_{i}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall }i=1,2,\dots ,N-1.$
Next, we show that $x\in F\left({T}_{N}\right)$. Since
$\begin{array}{rl}0& =Kx-x\\ ={\lambda }_{N}{T}_{N}{U}_{N-1}x+\left(1-{\lambda }_{N}\right){U}_{N-1}x-x\\ ={\lambda }_{N}\left({T}_{N}x-x\right)\end{array}$
and ${\lambda }_{N}\in \left(0,1\right]$, we obtain
$x\in F\left({T}_{N}\right),$
from which it follows that
$x\in \bigcap _{i=1}^{N}F\left({T}_{i}\right).$
(2.6)
Therefore
$F\left(K\right)\subseteq \bigcap _{i=1}^{N}F\left({T}_{i}\right).$
(2.7)
Hence
$F\left(K\right)=\bigcap _{i=1}^{N}F\left({T}_{i}\right).$
(2.8)

To prove (ii), we claim that K is a nonexpansive mapping.

Let $x,y\in C$. Then we obtain
$\begin{array}{c}{\parallel Kx-Ky\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel \left({\lambda }_{N}{T}_{N}{U}_{N-1}x+\left(1-{\lambda }_{N}\right){U}_{N-1}x\right)-\left({\lambda }_{N}{T}_{N}{U}_{N-1}y+\left(1-{\lambda }_{N}\right){U}_{N-1}y\right)\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel \left({U}_{N-1}x-{\lambda }_{N}\left({U}_{N-1}x-{T}_{N}{U}_{N-1}x\right)\right)-\left({U}_{N-1}y-{\lambda }_{N}\left({U}_{N-1}y-{T}_{N}{U}_{N-1}y\right)\right)\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel \left({U}_{N-1}x-{U}_{N-1}y\right)-{\lambda }_{N}\left(\left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\right)\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {U}_{N-1}x-{U}_{N-1}y\parallel }^{2}+{\lambda }_{N}^{2}{\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-2{\lambda }_{N}〈{U}_{N-1}x-{U}_{N-1}y,\left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y〉\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {U}_{N-1}x-{U}_{N-1}y\parallel }^{2}+{\lambda }_{N}^{2}{\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-2{\lambda }_{N}\left(\frac{1-{\kappa }_{N}}{2}\right){\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {U}_{N-1}x-{U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+{\lambda }_{N}\left({\lambda }_{N}-\left(1-{\kappa }_{N}\right)\right){\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {U}_{N-1}x-{U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+{\lambda }_{N}\left({\gamma }_{1}+{\gamma }_{2}-1\right){\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {U}_{N-1}x-{U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-{\lambda }_{N}\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right){\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel \left({\lambda }_{N-1}{T}_{N-1}{U}_{N-2}x+\left(1-{\lambda }_{N-1}\right){U}_{N-2}x\right)-\left({\lambda }_{N-1}{T}_{N-1}{U}_{N-2}y+\left(1-{\lambda }_{N-1}\right){U}_{N-2}y\right)\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-{\lambda }_{N}\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right){\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel \left({U}_{N-2}x-{\lambda }_{N-1}\left(I-{T}_{N-1}\right){U}_{N-2}x\right)-\left({U}_{N-2}y-{\lambda }_{N-1}\left(I-{T}_{N-1}\right){U}_{N-2}y\right)\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-{\lambda }_{N}\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right){\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel \left({U}_{N-2}x-{U}_{N-2}y\right)-{\lambda }_{N-1}\left(\left(I-{T}_{N-1}\right){U}_{N-2}x-\left(I-{T}_{N-1}\right){U}_{N-2}y\right)\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-{\lambda }_{N}\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right){\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {U}_{N-2}x-{U}_{N-2}y\parallel }^{2}+{\lambda }_{N-1}^{2}{\parallel \left(I-{T}_{N-1}\right){U}_{N-2}x-\left(I-{T}_{N-1}\right){U}_{N-2}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-2{\lambda }_{N-1}〈{U}_{N-2}x-{U}_{N-2}y,\left(I-{T}_{N-1}\right){U}_{N-2}x-\left(I-{T}_{N-1}\right){U}_{N-2}y〉\hfill \\ \phantom{\rule{2em}{0ex}}-{\lambda }_{N}\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right){\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {U}_{N-2}x-{U}_{N-2}y\parallel }^{2}+{\lambda }_{N-1}^{2}{\parallel \left(I-{T}_{N-1}\right){U}_{N-2}x-\left(I-{T}_{N-1}\right){U}_{N-2}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-2{\lambda }_{N-1}\left(\frac{1-{\kappa }_{N-1}}{2}\right){\parallel \left(I-{T}_{N-1}\right){U}_{N-2}x-\left(I-{T}_{N-1}\right){U}_{N-2}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-{\lambda }_{N}\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right){\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {U}_{N-2}x-{U}_{N-2}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+{\lambda }_{N-1}\left({\lambda }_{N-1}-\left(1-{\kappa }_{N-1}\right)\right){\parallel \left(I-{T}_{N-1}\right){U}_{N-2}x-\left(I-{T}_{N-1}\right){U}_{N-2}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-{\lambda }_{N}\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right){\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel {U}_{N-2}x-{U}_{N-2}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+{\lambda }_{N-1}\left({\gamma }_{1}+{\gamma }_{2}-1\right){\parallel \left(I-{T}_{N-1}\right){U}_{N-2}x-\left(I-{T}_{N-1}\right){U}_{N-2}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-{\lambda }_{N}\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right){\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {U}_{N-2}x-{U}_{N-2}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-{\lambda }_{N-1}\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right){\parallel \left(I-{T}_{N-1}\right){U}_{N-2}x-\left(I-{T}_{N-1}\right){U}_{N-2}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-{\lambda }_{N}\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right){\parallel \left(I-{T}_{N}\right){U}_{N-1}x-\left(I-{T}_{N}\right){U}_{N-1}y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel {U}_{N-2}x-{U}_{N-2}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}-\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right)\sum _{i=N-1}^{N}{\lambda }_{i}{\parallel \left(I-{T}_{i}\right){U}_{i-1}x-\left(I-{T}_{i}\right){U}_{i-1}y\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}⋮\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel x-y\parallel }^{2}-\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right)\sum _{i=1}^{N}{\lambda }_{i}{\parallel \left(I-{T}_{i}\right){U}_{i-1}x-\left(I-{T}_{i}\right){U}_{i-1}y\parallel }^{2},\hfill \end{array}$
which implies that
${\parallel Kx-Ky\parallel }^{2}\le {\parallel x-y\parallel }^{2}-\left(1-\left({\gamma }_{1}+{\gamma }_{2}\right)\right)\sum _{i=1}^{N}{\lambda }_{i}{\parallel \left(I-{T}_{i}\right){U}_{i-1}x-\left(I-{T}_{i}\right){U}_{i-1}y\parallel }^{2}.$
(2.9)
From (2.9) and ${\gamma }_{1}+{\gamma }_{2}<1$, we obtain
$\parallel Kx-Ky\parallel \le \parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C,$

that is, K is a nonexpansive mapping. □

Lemma 2.10 Let C be a nonempty closed convex subset of a real Hilbert space H. Let ${\left\{{T}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ be a finite family of ${\kappa }_{i}$-strictly pseudo-contractive mappings of C into itself with ${\kappa }_{i}\le {\gamma }_{1}$ and ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\ne \mathrm{\varnothing }$. For every $i=1,2,\dots ,N$ and $n\in \mathbb{N}$, let ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{N}$ and ${\lambda }_{1}^{n},{\lambda }_{2}^{n},\dots ,{\lambda }_{N}^{n}$ be real numbers with $0<{\lambda }_{i},{\lambda }_{i}^{n}<{\gamma }_{2}$ and ${\gamma }_{1}+{\gamma }_{2}<1$ such that ${\lambda }_{i}^{n}\to {\lambda }_{i}$ as $n\to \mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_{i}^{n+1}-{\lambda }_{i}^{n}|<\mathrm{\infty }$. For every $n\in \mathbb{N}$, let K and ${K}_{n}$ be the K-mappings generated by ${T}_{1},{T}_{1},\dots ,{T}_{N}$ and ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{N}$ and ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\lambda }_{1}^{n},{\lambda }_{2}^{n},\dots ,{\lambda }_{N}^{n}$, respectively. Then, for every bounded sequence $\left\{{x}_{n}\right\}$ in C, the following properties hold:
1. (i)

${lim}_{n\to \mathrm{\infty }}\parallel {K}_{n}{x}_{n}-K{x}_{n}\parallel =0$;

2. (ii)

${\sum }_{n=1}^{\mathrm{\infty }}\parallel {K}_{n}{x}_{n-1}-{K}_{n-1}{x}_{n-1}\parallel <\mathrm{\infty }$.

Proof Let $\left\{{x}_{n}\right\}$ be a bounded sequence in C and let ${U}_{k}$ and ${U}_{n,k}$ be generated by ${T}_{1},{T}_{1},\dots ,{T}_{N}$ and ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{N}$ and ${T}_{1},{T}_{1},\dots ,{T}_{N}$ and ${\lambda }_{1}^{n},{\lambda }_{2}^{n},\dots ,{\lambda }_{N}^{n}$, respectively.

First, we shall prove that (i) holds. For each $n\in \mathbb{N}$, we obtain
$\begin{array}{rl}\parallel {U}_{n,1}{x}_{n}-{U}_{1}{x}_{n}\parallel & =\parallel {\lambda }_{1}^{n}{T}_{1}{x}_{n}+\left(1-{\lambda }_{1}^{n}\right){x}_{n}-\left({\lambda }_{1}{T}_{1}{x}_{n}+\left(1-{\lambda }_{1}\right){x}_{n}\right)\parallel \\ =\parallel {\lambda }_{1}^{n}{T}_{1}{x}_{n}-{\lambda }_{1}^{n}{x}_{n}-{\lambda }_{1}{T}_{1}{x}_{n}+{\lambda }_{1}{x}_{n}\parallel \\ =\parallel \left({\lambda }_{1}^{n}-{\lambda }_{1}\right){T}_{1}{x}_{n}-\left({\lambda }_{1}^{n}-{\lambda }_{1}\right){x}_{n}\parallel \\ =|{\lambda }_{1}^{n}-{\lambda }_{1}|\parallel {T}_{1}{x}_{n}-{x}_{n}\parallel .\end{array}$
(2.10)
For $k\in \left\{2,3,\dots ,N\right\}$, we have
$\begin{array}{c}\parallel {U}_{n,k}{x}_{n}-{U}_{k}{x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {\lambda }_{k}^{n}{T}_{k}{U}_{n,k-1}{x}_{n}+\left(1-{\lambda }_{k}^{n}\right){U}_{n,k-1}{x}_{n}-\left({\lambda }_{k}{T}_{k}{U}_{k-1}{x}_{n}+\left(1-{\lambda }_{k}\right){U}_{k-1}{x}_{n}\right)\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {\lambda }_{k}^{n}{T}_{k}{U}_{n,k-1}{x}_{n}-{\lambda }_{k}{T}_{k}{U}_{k-1}{x}_{n}+\left(1-{\lambda }_{k}^{n}\right){U}_{n,k-1}{x}_{n}-\left(1-{\lambda }_{k}\right){U}_{k-1}{x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {\lambda }_{k}^{n}{T}_{k}{U}_{n,k-1}{x}_{n}-{\lambda }_{k}^{n}{T}_{k}{U}_{k-1}{x}_{n}+{\lambda }_{k}^{n}{T}_{k}{U}_{k-1}{x}_{n}-{\lambda }_{k}{T}_{k}{U}_{k-1}{x}_{n}\hfill \\ \phantom{\rule{2em}{0ex}}+\left(1-{\lambda }_{k}^{n}\right){U}_{n,k-1}{x}_{n}-\left(1-{\lambda }_{k}^{n}\right){U}_{k-1}{x}_{n}+\left(1-{\lambda }_{k}^{n}\right){U}_{k-1}{x}_{n}\hfill \\ \phantom{\rule{2em}{0ex}}-\left(1-{\lambda }_{k}\right){U}_{k-1}{x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {\lambda }_{k}^{n}\left({T}_{k}{U}_{n,k-1}{x}_{n}-{T}_{k}{U}_{k-1}{x}_{n}\right)+\left({\lambda }_{k}^{n}-{\lambda }_{k}\right){T}_{k}{U}_{k-1}{x}_{n}\hfill \\ \phantom{\rule{2em}{0ex}}+\left(1-{\lambda }_{k}^{n}\right)\left({U}_{n,k-1}{x}_{n}-{U}_{k-1}{x}_{n}\right)+\left(1-{\lambda }_{k}^{n}-\left(1-{\lambda }_{k}\right)\right){U}_{k-1}{x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {\lambda }_{k}^{n}\parallel {T}_{k}{U}_{n,k-1}{x}_{n}-{T}_{k}{U}_{k-1}{x}_{n}\parallel +|{\lambda }_{k}^{n}-{\lambda }_{k}|\parallel {T}_{k}{U}_{k-1}{x}_{n}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\left(1-{\lambda }_{k}^{n}\right)\parallel {U}_{n,k-1}{x}_{n}-{U}_{k-1}{x}_{n}\parallel +|{\lambda }_{k}-{\lambda }_{k}^{n}|\parallel {U}_{k-1}{x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {\lambda }_{k}^{n}\frac{1+{\kappa }_{k}}{1-{\kappa }_{k}}\parallel {U}_{n,k-1}{x}_{n}-{U}_{k-1}{x}_{n}\parallel +|{\lambda }_{k}^{n}-{\lambda }_{k}|\parallel {T}_{k}{U}_{k-1}{x}_{n}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\left(1-{\lambda }_{k}^{n}\right)\parallel {U}_{n,k-1}{x}_{n}-{U}_{k-1}{x}_{n}\parallel +|{\lambda }_{k}-{\lambda }_{k}^{n}|\parallel {U}_{k-1}{x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1+{\kappa }_{k}}{1-{\kappa }_{k}}\parallel {U}_{n,k-1}{x}_{n}-{U}_{k-1}{x}_{n}\parallel +\frac{1-{\kappa }_{k}}{1-{\kappa }_{k}}\parallel {U}_{n,k-1}{x}_{n}-{U}_{k-1}{x}_{n}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+|{\lambda }_{k}^{n}-{\lambda }_{k}|\left(\parallel {T}_{k}{U}_{k-1}{x}_{n}\parallel +\parallel {U}_{k-1}{x}_{n}\parallel \right)\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{2}{1-{\kappa }_{k}}\parallel {U}_{n,k-1}{x}_{n}-{U}_{k-1}{x}_{n}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+|{\lambda }_{k}^{n}-{\lambda }_{k}|\left(\parallel {T}_{k}{U}_{k-1}{x}_{n}\parallel +\parallel {U}_{k-1}{x}_{n}\parallel \right).\hfill \end{array}$
(2.11)
By (2.10) and (2.11), we get
$\begin{array}{c}\parallel {K}_{n}{x}_{n}-K{x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {U}_{n,N}{x}_{n}-{U}_{N}{x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{2}{1-{\kappa }_{N}}\parallel {U}_{n,N-1}{x}_{n}-{U}_{N-1}{x}_{n}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+|{\lambda }_{N}^{n}-{\lambda }_{N}|\left(\parallel {T}_{N}{U}_{N-1}{x}_{n}\parallel +\parallel {U}_{N-1}{x}_{n}\parallel \right)\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{2}{1-{\kappa }_{N}}\left(\frac{2}{1-{\kappa }_{N-1}}\parallel {U}_{n,N-2}{x}_{n}-{U}_{N-2}{x}_{n}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+|{\lambda }_{N-1}^{n}-{\lambda }_{N-1}|\left(\parallel {T}_{N-1}{U}_{N-2}{x}_{n}\parallel +\parallel {U}_{N-2}{x}_{n}\parallel \right)\right)\hfill \\ \phantom{\rule{2em}{0ex}}+|{\lambda }_{N}^{n}-{\lambda }_{N}|\left(\parallel {T}_{N}{U}_{N-1}{x}_{n}\parallel +\parallel {U}_{N-1}{x}_{n}\parallel \right)\hfill \\ \phantom{\rule{1em}{0ex}}=\left(\frac{2}{1-{\kappa }_{N}}\right)\left(\frac{2}{1-{\kappa }_{N-1}}\right)\parallel {U}_{n,N-2}{x}_{n}-{U}_{N-2}{x}_{n}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\frac{2}{1-{\kappa }_{N}}|{\lambda }_{N-1}^{n}-{\lambda }_{N-1}|\left(\parallel {T}_{N-1}{U}_{N-2}{x}_{n}\parallel +\parallel {U}_{N-2}{x}_{n}\parallel \right)\hfill \\ \phantom{\rule{2em}{0ex}}+|{\lambda }_{N}^{n}-{\lambda }_{N}|\left(\parallel {T}_{N}{U}_{N-1}{x}_{n}\parallel +\parallel {U}_{N-1}{x}_{n}\parallel \right)\hfill \\ \phantom{\rule{1em}{0ex}}=\prod _{j=N-1}^{N}\left(\frac{2}{1-{\kappa }_{j}}\right)\parallel {U}_{n,N-2}{x}_{n}-{U}_{N-2}{x}_{n}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\sum _{j=N-1}^{N}{\left(\frac{2}{1-{\kappa }_{j+1}}\right)}^{N-j}|{\lambda }_{j}^{n}-{\lambda }_{j}|\left(\parallel {T}_{j}{U}_{j-1}{x}_{n}\parallel +\parallel {U}_{j-1}{x}_{n}\parallel \right)\hfill \\ \phantom{\rule{2em}{0ex}}⋮\hfill \\ \phantom{\rule{1em}{0ex}}\le \prod _{j=2}^{N}\left(\frac{2}{1-{\kappa }_{j}}\right)\parallel {U}_{n,1}{x}_{n}-{U}_{1}{x}_{n}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\sum _{j=2}^{N}{\left(\frac{2}{1-{\kappa }_{j+1}}\right)}^{N-j}|{\lambda }_{j}^{n}-{\lambda }_{j}|\left(\parallel {T}_{j}{U}_{j-1}{x}_{n}\parallel +\parallel {U}_{j-1}{x}_{n}\parallel \right)\hfill \\ \phantom{\rule{1em}{0ex}}=\prod _{j=2}^{N}\left(\frac{2}{1-{\kappa }_{j}}\right)|{\lambda }_{1}^{n}-{\lambda }_{1}|\parallel {T}_{1}{x}_{n}-{x}_{n}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\sum _{j=2}^{N}{\left(\frac{2}{1-{\kappa }_{j+1}}\right)}^{N-j}|{\lambda }_{j}^{n}-{\lambda }_{j}|\left(\parallel {T}_{j}{U}_{j-1}{x}_{n}\parallel +\parallel {U}_{j-1}{x}_{n}\parallel \right).\hfill \end{array}$
(2.12)

By (2.12) and the fact that ${\lambda }_{i}^{n}\to {\lambda }_{i}$ as $n\to \mathrm{\infty }$ for all $i=1,2,\dots ,N$, we deduce that ${lim}_{n\to \mathrm{\infty }}\parallel {K}_{n}{x}_{n}-K{x}_{n}\parallel =0$.

Next, we will claim that (ii) holds. For each $n\in \mathbb{N}$, we obtain
$\begin{array}{c}\parallel {U}_{n,1}{x}_{n-1}-{U}_{n-1,1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {\lambda }_{1}^{n}{T}_{1}{x}_{n-1}+\left(1-{\lambda }_{1}^{n}\right){x}_{n-1}-\left({\lambda }_{1}^{n-1}{T}_{1}{x}_{n-1}+\left(1-{\lambda }_{1}^{n-1}\right){x}_{n-1}\right)\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {\lambda }_{1}^{n}{T}_{1}{x}_{n-1}-{\lambda }_{1}^{n}{x}_{n-1}-{\lambda }_{1}^{n-1}{T}_{1}{x}_{n-1}+{\lambda }_{1}^{n-1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel \left({\lambda }_{1}^{n}-{\lambda }_{1}^{n-1}\right){T}_{1}{x}_{n-1}-\left({\lambda }_{1}^{n}-{\lambda }_{1}^{n-1}\right){x}_{n-1}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=|{\lambda }_{1}^{n}-{\lambda }_{1}^{n-1}|\parallel {T}_{1}{x}_{n-1}-{x}_{n-1}\parallel .\hfill \end{array}$
(2.13)
For $k\in \left\{2,3,\dots ,N\right\}$, we have
$\begin{array}{c}\parallel {U}_{n,k}{x}_{n-1}-{U}_{n-1,k}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {\lambda }_{k}^{n}{T}_{k}{U}_{n,k-1}{x}_{n-1}+\left(1-{\lambda }_{k}^{n}\right){U}_{n,k-1}{x}_{n-1}-\left({\lambda }_{k}^{n-1}{T}_{k}{U}_{n-1,k-1}{x}_{n-1}\hfill \\ \phantom{\rule{2em}{0ex}}+\left(1-{\lambda }_{k}^{n-1}\right){U}_{n-1,k-1}{x}_{n-1}\right)\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {\lambda }_{k}^{n}{T}_{k}{U}_{n,k-1}{x}_{n-1}-{\lambda }_{k}^{n-1}{T}_{k}{U}_{n-1,k-1}{x}_{n-1}+\left(1-{\lambda }_{k}^{n}\right){U}_{n,k-1}{x}_{n-1}\hfill \\ \phantom{\rule{2em}{0ex}}-\left(1-{\lambda }_{k}^{n-1}\right){U}_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {\lambda }_{k}^{n}{T}_{k}{U}_{n,k-1}{x}_{n-1}-{\lambda }_{k}^{n}{T}_{k}{U}_{n-1,k-1}{x}_{n-1}+{\lambda }_{k}^{n}{T}_{k}{U}_{n-1,k-1}{x}_{n-1}\hfill \\ \phantom{\rule{2em}{0ex}}-{\lambda }_{k}^{n-1}{T}_{k}{U}_{n-1,k-1}{x}_{n-1}+\left(1-{\lambda }_{k}^{n}\right){U}_{n,k-1}{x}_{n-1}-\left(1-{\lambda }_{k}^{n}\right){U}_{n-1,k-1}{x}_{n-1}\hfill \\ \phantom{\rule{2em}{0ex}}+\left(1-{\lambda }_{k}^{n}\right){U}_{n-1,k-1}{x}_{n-1}-\left(1-{\lambda }_{k}^{n-1}\right){U}_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {\lambda }_{k}^{n}\left({T}_{k}{U}_{n,k-1}{x}_{n-1}-{T}_{k}{U}_{n-1,k-1}{x}_{n-1}\right)+\left({\lambda }_{k}^{n}-{\lambda }_{k}^{n-1}\right){T}_{k}{U}_{n-1,k-1}{x}_{n-1}\hfill \\ \phantom{\rule{2em}{0ex}}+\left(1-{\lambda }_{k}^{n}\right)\left({U}_{n,k-1}{x}_{n-1}-{U}_{n-1,k-1}{x}_{n-1}\right)\hfill \\ \phantom{\rule{2em}{0ex}}+\left(1-{\lambda }_{k}^{n}-\left(1-{\lambda }_{k}^{n-1}\right)\right){U}_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {\lambda }_{k}^{n}\parallel {T}_{k}{U}_{n,k-1}{x}_{n-1}-{T}_{k}{U}_{n-1,k-1}{x}_{n-1}\parallel +|{\lambda }_{k}^{n}-{\lambda }_{k}^{n-1}|\parallel {T}_{k}{U}_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\left(1-{\lambda }_{k}^{n}\right)\parallel {U}_{n,k-1}{x}_{n-1}-{U}_{n-1,k-1}{x}_{n-1}\parallel +|{\lambda }_{k}^{n}-{\lambda }_{k}^{n-1}|\parallel {U}_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {\lambda }_{k}^{n}\frac{1+{\kappa }_{k}}{1-{\kappa }_{k}}\parallel {U}_{n,k-1}{x}_{n-1}-{U}_{n-1,k-1}{x}_{n-1}\parallel +|{\lambda }_{k}^{n}-{\lambda }_{k}^{n-1}|\parallel {T}_{k}{U}_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\left(1-{\lambda }_{k}^{n}\right)\parallel {U}_{n,k-1}{x}_{n-1}-{U}_{n-1,k-1}{x}_{n-1}\parallel +|{\lambda }_{k}^{n}-{\lambda }_{k}^{n-1}|\parallel {U}_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1+{\kappa }_{k}}{1-{\kappa }_{k}}\parallel {U}_{n,k-1}{x}_{n-1}-{U}_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\frac{1-{\kappa }_{k}}{1-{\kappa }_{k}}\parallel {U}_{n,k-1}{x}_{n-1}-{U}_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+|{\lambda }_{k}^{n}-{\lambda }_{k}^{n-1}|\left(\parallel {T}_{k}{U}_{n-1,k-1}{x}_{n-1}\parallel +\parallel {U}_{n-1,k-1}{x}_{n-1}\parallel \right)\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{2}{1-{\kappa }_{k}}\parallel {U}_{n,k-1}{x}_{n-1}-{U}_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+|{\lambda }_{k}^{n}-{\lambda }_{k}^{n-1}|\left(\parallel {T}_{k}{U}_{n-1,k-1}{x}_{n-1}\parallel +\parallel {U}_{n-1,k-1}{x}_{n-1}\parallel \right).\hfill \end{array}$
(2.14)
From (2.13) and (2.14), we obtain
$\begin{array}{c}\parallel {K}_{n}{x}_{n-1}-{K}_{n-1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {U}_{n,N}{x}_{n-1}-{U}_{n-1,N}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{2}{1-{\kappa }_{N}}\parallel {U}_{n,N-1}{x}_{n-1}-{U}_{n-1,N-1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+|{\lambda }_{N}^{n}-{\lambda }_{N}^{n-1}|\left(\parallel {T}_{N}{U}_{n-1,N-1}{x}_{n-1}\parallel +\parallel {U}_{n-1,N-1}{x}_{n-1}\parallel \right)\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{2}{1-{\kappa }_{N}}\left(\frac{2}{1-{\kappa }_{N-1}}\parallel {U}_{n,N-2}{x}_{n-1}-{U}_{n-1,N-2}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+|{\lambda }_{N-1}^{n}-{\lambda }_{N-1}^{n-1}|\left(\parallel {T}_{N-1}{U}_{n-1,N-2}{x}_{n-1}\parallel +\parallel {U}_{n-1,N-2}{x}_{n-1}\parallel \right)\right)\hfill \\ \phantom{\rule{2em}{0ex}}+|{\lambda }_{N}^{n}-{\lambda }_{N}^{n-1}|\left(\parallel {T}_{N}{U}_{n-1,N-1}{x}_{n-1}\parallel +\parallel {U}_{n-1,N-1}{x}_{n-1}\parallel \right)\hfill \\ \phantom{\rule{1em}{0ex}}=\left(\frac{2}{1-{\kappa }_{N}}\right)\left(\frac{2}{1-{\kappa }_{N-1}}\right)\parallel {U}_{n,N-2}{x}_{n-1}-{U}_{n-1,N-2}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\frac{2}{1-{\kappa }_{N}}|{\lambda }_{N-1}^{n}-{\lambda }_{N-1}^{n-1}|\left(\parallel {T}_{N-1}{U}_{n-1,N-2}{x}_{n-1}\parallel +\parallel {U}_{n-1,N-2}{x}_{n-1}\parallel \right)\hfill \\ \phantom{\rule{2em}{0ex}}+|{\lambda }_{N}^{n}-{\lambda }_{N}^{n-1}|\left(\parallel {T}_{N}{U}_{n-1,N-1}{x}_{n-1}\parallel +\parallel {U}_{n-1,N-1}{x}_{n-1}\parallel \right)\hfill \\ \phantom{\rule{1em}{0ex}}=\prod _{j=N-1}^{N}\left(\frac{2}{1-{\kappa }_{j}}\right)\parallel {U}_{n,N-2}{x}_{n-1}-{U}_{n-1,N-2}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\sum _{j=N-1}^{N}{\left(\frac{2}{1-{\kappa }_{j+1}}\right)}^{N-j}|{\lambda }_{j}^{n}-{\lambda }_{j}^{n-1}|\left(\parallel {T}_{j}{U}_{n-1,j-1}{x}_{n-1}\parallel +\parallel {U}_{n-1,j-1}{x}_{n-1}\parallel \right)\hfill \\ \phantom{\rule{2em}{0ex}}⋮\hfill \\ \phantom{\rule{1em}{0ex}}\le \prod _{j=2}^{N}\left(\frac{2}{1-{\kappa }_{j}}\right)\parallel {U}_{n,1}{x}_{n-1}-{U}_{n-1,1}{x}_{n-1}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\sum _{j=2}^{N}{\left(\frac{2}{1-{\kappa }_{j+1}}\right)}^{N-j}|{\lambda }_{j}^{n}-{\lambda }_{j}^{n-1}|\left(\parallel {T}_{j}{U}_{n-1,j-1}{x}_{n-1}\parallel +\parallel {U}_{n-1,j-1}{x}_{n-1}\parallel \right)\hfill \\ \phantom{\rule{1em}{0ex}}=\prod _{j=2}^{N}\left(\frac{2}{1-{\kappa }_{j}}\right)|{\lambda }_{1}^{n}-{\lambda }_{1}^{n-1}|\parallel {T}_{1}{x}_{n-1}-{x}_{n-1}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+\sum _{j=2}^{N}{\left(\frac{2}{1-{\kappa }_{j+1}}\right)}^{N-j}|{\lambda }_{j}^{n}-{\lambda }_{j}^{n-1}|\left(\parallel {T}_{j}{U}_{n-1,j-1}{x}_{n-1}\parallel +\parallel {U}_{n-1,j-1}{x}_{n-1}\parallel \right)\hfill \\ \phantom{\rule{1em}{0ex}}\le \prod _{j=2}^{N}\left(\frac{2}{1-{\kappa }_{j}}\right)|{\lambda }_{1}^{n}-{\lambda }_{1}^{n-1}|M+2\sum _{j=2}^{N}{\left(\frac{2}{1-{\kappa }_{j+1}}\right)}^{N-j}|{\lambda }_{j}^{n}-{\lambda }_{j}^{n-1}|M,\hfill \end{array}$
(2.15)

where $M={max}_{n\in \mathbb{N}}\left\{\parallel {T}_{1}{x}_{n-1}-{x}_{n-1}\parallel ,\parallel {T}_{j}{U}_{n-1,j-1}{x}_{n-1}\parallel ,\parallel {U}_{n-1,j-1}{x}_{n-1}\parallel \right\}$, for all $j=2,3,\dots ,N$. Hence, by (2.15) and ${\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_{i}^{n+1}-{\lambda }_{i}^{n}|<\mathrm{\infty }$ for all $i=1,2,\dots ,N$, we have ${\sum }_{n=1}^{\mathrm{\infty }}\parallel {K}_{n}{x}_{n-1}-{K}_{n-1}{x}_{n-1}\parallel <\mathrm{\infty }$. □

In 2010, Kangtunyakarn and Suantai [23] introduced the S-mapping generated by the finite family of ${\kappa }_{i}$-strictly pseudo-contractions in Hilbert space as in the following definition.

Definition 2.3 ([23])

Let C be a nonempty closed convex subset of real Hilbert space. Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of ${\kappa }_{i}$-strictly pseudo-contractions of C into itself. For each $j=1,2,\dots ,N$, let ${\alpha }_{j}=\left({\alpha }_{1}^{j},{\alpha }_{2}^{j},{\alpha }_{3}^{j}\right)\in I×I×I$ where $I\in \left[0,1\right]$ and ${\alpha }_{1}^{j}+{\alpha }_{2}^{j}+{\alpha }_{3}^{j}=1$. Define the mappings $S:C\to C$ as follows:
$\begin{array}{c}{U}_{0}=I,\hfill \\ {U}_{1}={\alpha }_{1}^{1}{T}_{1}{U}_{0}+{\alpha }_{2}^{1}{U}_{0}+{\alpha }_{3}^{1}I,\hfill \\ {U}_{2}={\alpha }_{1}^{2}{T}_{2}{U}_{1}+{\alpha }_{2}^{2}{U}_{1}+{\alpha }_{3}^{2}I,\hfill \\ {U}_{3}={\alpha }_{1}^{3}{T}_{3}{U}_{2}+{\alpha }_{2}^{3}{U}_{2}+{\alpha }_{3}^{3}I,\hfill \\ ⋮\hfill \\ {U}_{N-1}={\alpha }_{1}^{N-1}{T}_{N-1}{U}_{N-2}+{\alpha }_{2}^{N-1}{U}_{N-2}+{\alpha }_{3}^{N-1}I,\hfill \\ S={U}_{N}={\alpha }_{1}^{N}{T}_{N}{U}_{N-1}+{\alpha }_{2}^{N}{U}_{N-1}+{\alpha }_{3}^{N}I.\hfill \end{array}$

This mapping is called S-mapping generated by ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{N}$.

Furthermore, they obtained the following important lemma.

Lemma 2.11 ([23])

Let C be a nonempty closed convex subset of real Hilbert space. Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of ${\kappa }_{i}$-strictly pseudo-contractions of C into itself with ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\ne \mathrm{\varnothing }$ and $\kappa =max\left\{{\kappa }_{i}:i=1,2,\dots ,N\right\}$ and let ${\alpha }_{j}=\left({\alpha }_{1}^{j},{\alpha }_{2}^{j},{\alpha }_{3}^{j}\right)\in I×I×I$, $j=1,2,\dots ,N$, where $I=\left[0,1\right]$, ${\alpha }_{1}^{j}+{\alpha }_{2}^{j}+{\alpha }_{3}^{j}=1$, ${\alpha }_{1}^{j},{\alpha }_{2}^{j}\in \left(\kappa ,1\right)$ for all $j=1,2,\dots ,N-1$ and ${\alpha }_{1}^{N}\in \left(\kappa ,1\right]$, ${\alpha }_{3}^{N}\in \left[\kappa ,1\right)$, ${\alpha }_{2}^{j}\in \left[\kappa ,1\right)$ for all $j=1,2,\dots ,N$. Let S be the mapping generated by ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{N}$. Then $F\left(S\right)={\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$ and S is a nonexpansive mapping.

By putting ${\alpha }_{1}^{j}={\lambda }_{j}$ and ${\alpha }_{2}^{j}=0$, for all $j=1,2,\dots ,N$, we see that the S-mapping reduces to the K-mapping as defined in Definition 2.2. Moreover, from Lemma 2.11, we have the following result.

Lemma 2.12 Let C be a nonempty closed convex subset of real Hilbert space. Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of ${\kappa }_{i}$-strictly pseudo-contractions of C into itself with ${\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\ne \mathrm{\varnothing }$ and $\kappa =max\left\{{\kappa }_{i}:i=1,2,\dots ,N\right\}$ and let ${\lambda }_{j}\in \left(\kappa ,1\right)\subset \left[0,1\right]$, for all $j=1,2,\dots ,N-1$ and ${\lambda }_{N}\in \left(\kappa ,1\right]$. Let K be the mapping generated by ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{N}$. Then $F\left(K\right)={\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$ and K is a nonexpansive mapping.

Remark 2.13 For the result of Lemma 2.9 in our work, we obtain some improvement as follows:
1. (i)

We relax the conditions of ${\kappa }_{i}$ and ${\lambda }_{i}$ in Lemma 2.12 in sense that ${\kappa }_{i}$ is not depended on ${\lambda }_{i}$, for all $i=1,2,\dots ,N$.

2. (ii)

We do not assume the condition $\kappa =max\left\{{\kappa }_{i}:i=1,2,\dots ,N\right\}$.

Example 2.14 Let be the set of real numbers and let ${T}_{i}:\mathbb{R}\to \mathbb{R}$ be defined by

and ${\lambda }_{i}=\frac{i+5}{i+6}$, for all $i=1,2,\dots ,5$. Let K be the K-mapping generated by ${T}_{1},{T}_{2},\dots ,{T}_{5}$ and ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{5}$. Then $F\left(K\right)={\bigcap }_{i=1}^{5}F\left({T}_{i}\right)=\left\{0\right\}$.

Solution. It is easy to see that ${T}_{i}$ is ${\kappa }_{i}$-strictly pseudo-contractive mapping with ${\kappa }_{i}=\frac{i}{i+2}$. We obtain $\kappa =max\left\{{\kappa }_{i}:i=1,2,\dots ,5\right\}=\frac{5}{7}$ and ${\lambda }_{i}\in \left(\frac{5}{7},1\right]$, for all $i=1,2,\dots ,5$. By the definition of a K-mapping, we have
$\begin{array}{c}{U}_{1}x=\left(\frac{6}{7}\right)\left(-2x\right)+\left(1-\frac{6}{7}\right)x,\hfill \\ {U}_{2}x=\left(\frac{7}{8}\right)\left(-3{U}_{1}x\right)+\left(1-\frac{7}{8}\right){U}_{1}x,\hfill \\ {U}_{3}x=\left(\frac{8}{9}\right)\left(-4{U}_{2}x\right)+\left(1-\frac{8}{9}\right){U}_{2}x,\hfill \\ {U}_{4}x=\left(\frac{9}{10}\right)\left(-5{U}_{3}x\right)+\left(1-\frac{9}{10}\right){U}_{3}x,\hfill \\ Kx={U}_{5}x=\left(\frac{10}{11}\right)\left(-6{U}_{4}x\right)+\left(1-\frac{10}{11}\right){U}_{4}x.\hfill \end{array}$
(2.16)
Observe that ${\bigcap }_{i=1}^{5}F\left({T}_{i}\right)=\left\{0\right\}$. Then, by Lemma 2.12, we obtain
$F\left(K\right)=\bigcap _{i=1}^{5}F\left({T}_{i}\right)=\left\{0\right\}.$

Next, we give an example for Lemma 2.9.

Example 2.15 Let be the set of real numbers and let ${T}_{i}:\mathbb{R}\to \mathbb{R}$ be defined by

and ${\lambda }_{i}=\frac{i}{5i+1}$, for all $i=1,2,\dots ,5$. Let K be the K-mapping generated by ${T}_{1},{T}_{2},\dots ,{T}_{5}$ and ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{5}$. Choose ${\gamma }_{1}=\frac{11}{14}$ and ${\gamma }_{2}=\frac{11}{52}$, from which it follows that ${\gamma }_{1}+{\gamma }_{2}=\frac{11}{14}+\frac{11}{52}=\frac{726}{728}=\frac{363}{364}<1$. Then, by Lemma 2.9, we obtain $F\left(K\right)={\bigcap }_{i=1}^{5}F\left({T}_{i}\right)=\left\{0\right\}$.

## 3 Strong convergence theorem

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. For every $i=1,2,\dots ,N$, ${S}_{i}:C\to \mathit{CB}\left(H\right)$ be -Lipschitz continuous with coefficients ${\mu }_{i}$, ${\mathrm{\Phi }}_{i}:H×C×C\to \mathbb{R}$ be equilibrium-like function satisfying (H1)-(H3). Let $\phi :C\to \mathbb{R}$ be a lower semicontinuous and convex function and $A:C\to C$ be an α-inverse strongly monotone mapping. Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of ${\kappa }_{i}$-strictly pseudo-contractive mappings and ${\kappa }_{i}\le {\gamma }_{1}$ with $\mathcal{F}:={\bigcap }_{i=1}^{N}F\left({T}_{i}\right)\cap {\bigcap }_{i=1}^{N}{\left(\mathit{MGEP}\right)}_{s}\left({\mathrm{\Phi }}_{i},\phi ,A\right)\ne \mathrm{\varnothing }$. For every $n\in \mathbb{N}$, let ${K}_{n}$ be the K-mapping generated by ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and ${\lambda }_{1}^{n},{\lambda }_{2}^{n},\dots ,{\lambda }_{N}^{n}$ where $0<\varphi \le {\lambda }_{i}^{n}\le \psi <{\gamma }_{2}$, for all $i=1,2,\dots ,N$ and ${\gamma }_{1}+{\gamma }_{2}<1$. For every $i=1,2,\dots ,N$, let $\left\{{x}_{n}\right\}$ be the sequence generated by ${x}_{1}\in C$ and ${w}_{1}^{i}\in {S}_{i}\left(I-{r}_{1}^{i}A\right){x}_{1}$, there exist sequences $\left\{{w}_{n}^{i}\right\}\in H$ and $\left\{{x}_{n}\right\},\left\{{u}_{n}^{i}\right\}\subseteq C$ such that
$\left\{\begin{array}{c}\parallel {w}_{n}^{i}-{w}_{n+1}^{i}\parallel \le \left(1+\frac{1}{n}\right)\mathcal{H}\left({S}_{i}\left(I-{r}_{n}^{i}A\right){x}_{n},{S}_{i}\left(I-{r}_{n+1}^{i}A\right){x}_{n+1}\right),\hfill \\ \phantom{\rule{1em}{0ex}}{w}_{n}^{i}\in {S}_{i}\left(I-{r}_{n}^{i}A\right){x}_{n}\hfill \\ {\mathrm{\Phi }}_{i}\left({w}_{n}^{i},{u}_{n}^{i},y\right)+\phi \left(y\right)-\phi \left({u}_{n}^{i}\right)+\frac{1}{{r}_{n}^{i}}〈{u}_{n}^{i}-{x}_{n},y-{u}_{n}^{i}〉+〈A{x}_{n},y-{u}_{n}^{i}〉\hfill \\ \phantom{\rule{1em}{0ex}}\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C,\hfill \\ {x}_{n+1}={\alpha }_{n}f\left({x}_{n}\right)+{\beta }_{n}\left({\sum }_{i=1}^{N}{a}_{n}^{i}{u}_{n}^{i}\right)+{\delta }_{n}{K}_{n}{x}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1,\hfill \end{array}$
(3.1)
where $f:C\to C$ be a contraction mapping with a constant ξ and $\left\{{\alpha }_{n}\right\},\left\{{\beta }_{n}\right\},\left\{{\delta }_{n}\right\}\subseteq \left(0,1\right)$ with ${\alpha }_{n}+{\beta }_{n}+{\delta }_{n}=1$, $\mathrm{\forall }n\ge 1$. Suppose the following conditions hold:
1. (i)

${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$;

2. (ii)

$0<\tau \le {\beta }_{n},{\delta }_{n}\le \upsilon <1$;

3. (iii)

$0\le \eta \le {a}_{n}^{i}\le \sigma <1$, for all $i=1,2,\dots ,N-1$ and $0<\eta \le {a}_{n}^{N}\le \sigma \le 1$ with ${\sum }_{n=1}^{N}{a}_{n}^{i}=1$;

4. (iv)

$0<ϵ\le {r}_{n}^{i}\le \omega <2\alpha$, for all $n\in \mathbb{N}$ and $i=1,2,\dots ,N$;

5. (v)

${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_{n}|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_{n}|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_{n}|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{r}_{n+1}^{i}-{r}_{n}^{i}|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{a}_{n+1}^{i}-{a}_{n}^{i}|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_{i}^{n+1}-{\lambda }_{i}^{n}|<\mathrm{\infty }$, for all $i=1,2,\dots ,N$;

6. (vi)
for each $i=1,2,\dots ,N$, there exists ${\rho }_{i}>0$ such that
$\begin{array}{c}{\mathrm{\Phi }}_{i}\left({w}_{1}^{i},{T}_{{r}_{1}^{i}}\left({x}_{1}\right),{T}_{{r}_{2}^{i}}\left({x}_{2}\right)\right)+{\mathrm{\Phi }}_{i}\left({w}_{2}^{i},{T}_{{r}_{2}^{i}}\left({x}_{2}\right),{T}_{{r}_{1}^{i}}\left({x}_{1}\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le -{\rho }_{i}{\parallel {T}_{{r}_{1}^{i}}\left({x}_{1}\right)-{T}_{{r}_{2}^{i}}\left({x}_{2}\right)\parallel }^{2},\hfill \end{array}$
(3.2)

for all $\left({r}_{1}^{i},{r}_{2}^{i}\right)\in {\mathrm{\Theta }}_{i}×{\mathrm{\Theta }}_{i}$, $\left({x}_{1},{x}_{2}\right)\in C×C$ and ${w}_{j}^{i}\in {S}_{i}\left({x}_{j}\right)$, for $j=1,2$, where ${\mathrm{\Theta }}_{i}=\left\{{r}_{n}^{i}:n\ge 1\right\}$.

Then $\left\{{x}_{n}\right\}$ and $\left\{{u}_{n}^{i}\right\}$ converges strongly to $q={P}_{\mathcal{F}}f\left(q\right)$, for every $i=1,2,\dots ,N$.

Proof The proof shall be divided into seven steps.

Step 1. We will prove that $I-{r}_{n}^{i}A$ is nonexpansive, for all $i=1,2,\dots ,N$.

From (3.1), we have
${\mathrm{\Phi }}_{i}\left({w}_{n}^{i},{u}_{n}^{i},y\right)+\phi \left(y\right)-\phi \left({u}_{n}^{i}\right)+\frac{1}{{r}_{n}^{i}}〈{u}_{n}^{i}-\left(I-{r}_{n}^{i}A\right){x}_{n},y-{u}_{n}^{i}〉\ge 0,$
(3.3)
for every $y\in C$. From (3.3) and Theorem 2.8, we obtain
${u}_{n}^{i}={T}_{{r}_{n}^{i}}\left(I-{r}_{n}^{i}A\right){x}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }i=1,2,\dots ,N.$
Put ${r}^{i}\in {\mathrm{\Theta }}_{i}$ for all $i=1,2,\dots ,N$. From (3.2), we have
$\begin{array}{c}{\mathrm{\Phi }}_{i}\left({w}_{1}^{i},{T}_{{r}^{i}}\left({x}_{1}\right),{T}_{{r}^{i}}\left({x}_{2}\right)\right)+{\mathrm{\Phi }}_{i}\left({w}_{2}^{i},{T}_{{r}^{i}}\left({x}_{2}\right),{T}_{{r}^{i}}\left({x}_{1}\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le -{\rho }_{i}{\parallel {T}_{{r}^{i}}\left({x}_{1}\right)-{T}_{{r}^{i}}\left({x}_{2}\right)\parallel }^{2}\le 0,\hfill \end{array}$
(3.4)

for all $\left({x}_{1},{x}_{2}\right)\in C×C$ and ${w}_{j}^{i}\in {S}_{i}\left({x}_{j}\right)$, $j=1,2$.

From (3.4), we find the implication that Theorem 2.8 holds.

It obvious to see that $I-{r}_{n}^{i}A$ is a nonexpansive mapping, for every $i=1,2,\dots ,N$.

Indeed, A is α-inverse strongly monotone with ${r}_{n}^{i}\in \left(0,2\alpha \right)$, we get
$\begin{array}{c}{\parallel \left(I-{r}_{n}^{i}A\right)x-\left(I-{r}_{n}^{i}A\right)y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel x-y-{r}_{n}^{i}\left(Ax-Ay\right)\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel x-y\parallel }^{2}-2{r}_{n}^{i}〈x-y,Ax-Ay〉+{\left({r}_{n}^{i}\right)}^{2}{\parallel Ax-Ay\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel x-y\parallel }^{2}-2\alpha {r}_{n}^{i}{\parallel Ax-Ay\parallel }^{2}+{\left({r}_{n}^{i}\right)}^{2}{\parallel Ax-Ay\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\parallel x-y\parallel }^{2}+{r}_{n}^{i}\left({r}_{n}^{i}-2\alpha \right){\parallel Ax-Ay\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\parallel x-y\parallel }^{2}.\hfill \end{array}$

Thus $I-{r}_{n}^{i}A$ is a nonexpansive mapping, for all $i=1,2,\dots ,N$.

Step 2. We will show that $\left\{{x}_{n}\right\}$ is bounded.

Let $z\in \mathcal{F}$. By nonexpansiveness of ${K}_{n}$, we have
$\begin{array}{c}\parallel {x}_{n+1}-z\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {\alpha }_{n}\parallel f\left({x}_{n}\right)-z\parallel +{\beta }_{n}\parallel \sum _{i=1}^{N}{a}_{n}^{i}\left({u}_{n}^{i}-z\right)\parallel +{\delta }_{n}\parallel {K}_{n}{x}_{n}-z\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {\alpha }_{n}\parallel f\left({x}_{n}\right)-f\left(z\right)+f\left(z\right)-z\parallel +{\beta }_{n}\sum _{i=1}^{N}{a}_{n}^{i}\parallel {u}_{n}^{i}-z\parallel +{\delta }_{n}\parallel {x}_{n}-z\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {\alpha }_{n}\left(\parallel f\left({x}_{n}\right)-f\left(z\right)\parallel +\parallel f\left(z\right)-z\parallel \right)+{\beta }_{n}\sum _{i=1}^{N}{a}_{n}^{i}\parallel {T}_{{r}_{n}^{i}}\left(I-{r}_{n}^{i}A\right){x}_{n}-z\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+{\delta }_{n}\parallel {x}_{n}-z\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {\alpha }_{n}\left(\xi \parallel {x}_{n}-z\parallel +\parallel f\left(z\right)-z\parallel \right)+{\beta }_{n}\sum _{i=1}^{N}{a}_{n}^{i}\parallel {x}_{n}-z\parallel +{\delta }_{n}\parallel {x}_{n}-z\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\left(1-{\alpha }_{n}\left(1-\xi \right)\right)\parallel {x}_{n}-z\parallel +{\alpha }_{n}\parallel f\left(z\right)-z\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le max\left\{\parallel {x}_{1}-z\parallel ,\frac{\parallel f\left(z\right)-z\parallel }{1-\xi }\right\}.\hfill \end{array}$

By induction, we have $\parallel {x}_{n}-z\parallel \le max\left\{\parallel {x}_{1}-z\parallel ,\frac{\parallel f\left(z\right)-z\parallel }{1-\xi }\right\}$, $\mathrm{\forall }n\in \mathbb{N}$. It follows that $\left\{{x}_{n}\right\}$ is bounded and so is $\left\{{u}_{n}^{i}\right\}$, $\mathrm{\forall }i=1,2,\dots ,N$.

Step 3. We will show that ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n+1}-{x}_{n}\parallel =0$.

By the definition of ${x}_{n}$, we obtain
$\begin{array}{c}\parallel {x}_{n+1}-{x}_{n}\parallel \hfill \\ \phantom{\rule{1em}{0ex}}=\parallel {\alpha }_{n}f\left({x}_{n}\right)+{\beta }_{n}\left(\sum _{i=1}^{N}{a}_{n}^{i}{u}_{n}^{i}\right)+{\delta }_{n}{K}_{n}{x}_{n}\hfill \\ \phantom{\rule{2em}{0ex}}-\left({\alpha }_{n-1}f\left({x}_{n-1}\right)+{\beta }_{n-1}\left(\sum _{i=1}^{N}{a}_{n-1}^{i}{u}_{n-1}^{i}\right)+{\delta }_{n-1}{K}_{n-1}{x}_{n-1}\right)\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le {\alpha }_{n}\parallel f\left({x}_{n}\right)-f\left({x}_{n-1}\right)\parallel +|{\alpha }_{n}-{\alpha }_{n-1}|\parallel f\left({x}_{n-1}\right)\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+{\beta }_{n}\parallel \sum _{i=1}^{N}{a}_{n}^{i}{u}_{n}^{i}-\sum _{i=1}^{N}{a}_{n-1}^{i}{u}_{n-1}^{i}\parallel +|{\beta }_{n}-{\beta }_{n-1}|\parallel \sum _{i=1}^{N}{a}_{n-1}^{i}{u}_{n-1}^{i}\parallel \hfill \\ \phantom{\rule{2em}{0ex}}+{\delta }_{n}\parallel {K}_{}\hfill \end{array}$