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

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.

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 $\alpha \in (0,1)$ such that

$$\parallel f(x)-f(y)\parallel \le \alpha \parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

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

• (i) monotone if

  $$〈Bx-By,x-y〉\ge 0,\mathrm{\forall }x,y\in C,$$

• (ii) υ-strongly monotone if there exists a positive real number υ such that

  $$〈Bx-By,x-y〉\ge \upsilon {\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in C,$$

• (iii) ξ-inverse strongly monotone if there exists a positive real number ξ such that

  $$〈x-y,Bx-By〉\ge \xi {\parallel Bx-By\parallel }^2,\mathrm{\forall }x,y\in C,$$

• (iv) μ-Lipschitz continuous if there exists a nonnegative real number $\mu \ge 0$ such that

  $$\parallel Bx-By\parallel \le \mu \parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

Definition 1.2 Let $T:C\to C$ be a mapping. Then:

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

• (ii) Mapping T is called nonexpansive if

  $$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

• (iii) T is said to be κ-strictly pseudo-contractive if there exists a constant $\kappa \in [0,1)$ such that

  $${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+\kappa {\parallel (I-T)x-(I-T)y\parallel }^2,\mathrm{\forall }x,y\in C.$$

  (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

$$〈Tx-Ty,x-y〉\le {\parallel x-y\parallel }^2-\frac{1-\kappa }{2}{\parallel (I-T)x-(I-T)y\parallel }^2,\mathrm{\forall }x,y\in C.$$

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, [1–4]. 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 $F:C\times C\to \mathbb{R}$ be bifunction. The equilibrium problem for F is to determine its equilibrium point, i.e., the set

$$\mathit{EP}(F)=\{x\in C:F(x,y)\ge 0,\mathrm{\forall }y\in C\}.$$

(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 $\mathit{EP}(F)$; see [10, 11]. Many authors have been investigating iterative algorithms for the equilibrium problems; see, for example, [11–15].

Let $\mathit{CB}(H)$ be the family of all nonempty closed bounded subsets of H and $\mathcal{H}(\cdot ,\cdot )$ be the Hausdorff metric on $\mathit{CB}(H)$ defined as

$$\mathcal{H}(U,V)=max\{\underset{u\in U}{sup}d(u,V),\underset{v\in V}{sup}d(U,v)\},\mathrm{\forall }U,V\in \mathit{CB}(H),$$

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

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

(H1) For each fixed $v\in C$, $(w,u)\mapsto \mathrm{\Phi }(w,u,v)$ is an upper semicontinuous function from $H\times C\to \mathbb{R}$, that is, for $(w,u)\in H\times C$, whenever $w_n\to w$ and $u_n\to u$ as $n\to \mathrm{\infty }$,

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\mathrm{\Phi }(w_n,u_n,v)\le \mathrm{\Phi }(w,u,v).$$

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

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

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

$$(\mathit{GEP})\{\begin{array}{c}\text{Find }u\in C\text{ and }w\in T(u)\text{ such that}\hfill \\ \mathrm{\Phi }(w,u,v)+\phi (v)-\phi (u)\ge 0,\mathrm{\forall }v\in C.\hfill \end{array}$$

(1.3)

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

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

Let $x_1\in C$ and $w_1\in T(x_1)$, there exist sequences $\{w_n\}\subseteq H$ and $\{x_n\},\{u_n\}\subseteq C$ such that

$$\{\begin{array}{c}{w}_n\in T(x_n),\parallel w_n-w_{n+1}\parallel \le (1+\frac{1}{n})\mathcal{H}(T(x_n),T(x_{n+1})),\hfill \\ \mathrm{\Phi }(w_n,u_n,v)+\phi (v)-\phi (u_n)+\frac{1}{r_n}〈u_n-x_n,v-u_n〉\ge 0,\mathrm{\forall }v\in C,\hfill \\ x_{n+1}={\alpha }_{n}f(x_n)+(1-{\alpha }_n)S{u}_n,n=1,2,\dots .\hfill \end{array}$$

(1.4)

They proved the strong convergence theorem of the sequence $\{x_n\}$ 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 $T:C\to \mathit{CB}(H)$ be ℋ-Lipschitz continuous with constant μ, $\mathrm{\Phi }:H\times C\times C\to \mathbb{R}$ be an equilibrium-like function satisfying (H1)-(H3) and S be a nonexpansive mapping of C into itself such that $F(S)\cap {(\mathit{GEP})}_s(\mathrm{\Phi },\phi )\ne \mathrm{\varnothing }$. Let f be a contraction of C into itself and let $\{x_n\}$, $\{w_n\}$, and $\{u_n\}$ be sequences generated by (1.4), where $\{{\alpha }_n\}\subseteq [0,1]$ and $\{r_n\}\subset (0,\mathrm{\infty })$ satisfy

$$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n=0,\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },\sum _{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty },\hfill \\ \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{r}_n>0\mathit{\text{and}}\sum _{n=1}^{\mathrm{\infty }}|r_{n+1}-r_n|<\mathrm{\infty }.\hfill \end{array}$$

If there exists a constant $\lambda >0$ such that

$$\mathrm{\Phi }(w_1,T_{r_1}(x_1),T_{r_2}(x_2))+\mathrm{\Phi }(w_2,T_{r_2}(x_2),T_{r_1}(x_1))\le -\lambda {\parallel T_{r_1}(x_1)-T_{r_2}(x_2)\parallel }^2,$$

for all $(r_1,r_2)\in \mathrm{\Xi }\times \mathrm{\Xi }$, $(x_1,x_2)\in C\times C$ and $w_i\in T(x_i)$, $i=1,2$, where $\mathrm{\Xi }=\{r_n:n\ge 1\}$, then for $\hat{x}=P_{F(S)\cap {(\mathit{GEP})}_s(\mathrm{\Phi },\phi )}f(\hat{x})$, there exists $\hat{w}\in T(\hat{x})$ such that $(\hat{x},\hat{w})$ is a solution of $(\mathit{GEP})$ and

$$x_n\to \hat{x},w_n\to \hat{w}\mathit{\text{and}}{u}_n\to \hat{x}\mathit{\text{as }}n\to \mathrm{\infty }.$$

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 $\kappa =max\{{\kappa }_i:i=1,2,\dots ,N\}$ and let $S_n$ be the S-mappings generated by $T_1,T_2,\dots ,T_N$ and ${\alpha }_1^{(n)},{\alpha }_2^{(n)},\dots ,{\alpha }_N^{(n)}$, where ${\alpha }_j^{(n)}=({\alpha }_1^{n,j},{\alpha }_2^{n,j},{\alpha }_3^{n,j})\in I\times I\times I$, $I=[0,1]$, ${\alpha }_1^{n,j}+{\alpha }_2^{n,j}+{\alpha }_3^{n,j}=1$ and $\kappa <a\le {\alpha }_1^{n,j},{\alpha }_3^{n,j}\le b<1$ 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 $w_1^1\in T(x_1)$, $w_1^2\in D(x_1)$, there exist sequences $\{w_n^1\},\{w_n^2\}\in H$, and $\{x_n\},\{u_n\},\{v_n\}\subseteq C$ such that

$$\{\begin{array}{c}{w}_n^1\in T(x_n),\parallel w_n^1-w_{n+1}^1\parallel \le (1+\frac{1}{n})\mathcal{H}(T(x_n),T(x_{n+1})),\hfill \\ w_n^2\in D(x_n),\parallel w_n^2-w_{n+1}^2\parallel \le (1+\frac{1}{n})\mathcal{H}(D(x_n),D(x_{n+1})),\hfill \\ \mathrm{\Phi }(w_n^1,u_n,u)+{\phi }_1(u)-{\phi }_1(u_n)+\frac{1}{r_n}〈u_n-x_n,u-u_n〉\ge 0,\mathrm{\forall }u\in C,\hfill \\ \mathrm{\Phi }(w_n^2,v_n,v)+{\phi }_2(v)-{\phi }_2(v_n)+\frac{1}{s_n}〈v_n-x_n,v-v_n〉\ge 0,\mathrm{\forall }v\in C,\hfill \\ z_n={\delta }_n{P}_C(I-\lambda A)u_n+(1-{\delta }_n)P_C(I-\eta B)v_n,\hfill \\ y_n={\alpha }_n{z}_n+(1-{\alpha }_n)S_n{z}_n,\hfill \\ C_{n+1}=\{z\in C_n:\parallel y_n-z\parallel \le \parallel x_n-z\parallel \},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_1,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(1.5)

where $D,T:C\to \mathit{CB}(H)$ are ℋ-Lipschitz continuous with constants ${\mu }_1$, ${\mu }_2$, respectively, ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2:H\times C\times C\to \mathbb{R}$ 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 $\{{\delta }_n\}$, $\{{\alpha }_n\}$, $\{s_n\}$, and $\{r_n\}$ that the sequence $\{x_n\}$ generated by (1.5) converges strongly to $P_{\mathcal{F}}{x}_1$, where $\mathcal{F}={\bigcap }_{i=1}^{N}F(T_i)\cap {(\mathit{GEP})}_s({\mathrm{\Phi }}_1,{\phi }_1)\cap {(\mathit{GEP})}_s({\mathrm{\Phi }}_2,{\phi }_2)\cap F(G_1)\cap F(G_2)$, $G_1,G_2:C\to C$ are defined by $G_1(x)=P_C(x-\lambda Ax)$, $G_2(x)=P_C(x-\eta Bx)$, $\mathrm{\forall }x\in C$ and $P_{\mathcal{F}}{x}_1$ is a solution of the following system of variational inequalities:

$$\{\begin{array}{c}〈A{x}^{\ast },x-x^{\ast }〉\ge 0,\hfill \\ 〈B{x}^{\ast },x-x^{\ast }〉\ge 0.\hfill \end{array}$$

By modifying the generalized equilibrium problem (1.3), we introduced the modified generalized equilibrium problem $(\mathit{MGEP})$ as follows:

$$(\mathit{MGEP})\{\begin{array}{c}\text{Find }u\in C\text{ and }w\in T(I-\lambda A)u,\mathrm{\forall }\lambda >0,\hfill \\ \mathrm{\Phi }(w,u,v)+\phi (v)-\phi (u)+〈v-u,Au〉\ge 0,\mathrm{\forall }v\in C,\hfill \end{array}$$

(1.6)

where $A:C\to C$ is a mapping. The set of such solutions of $(\mathit{MGEP})$ is denoted by ${(\mathit{MGEP})}_s(\mathrm{\Phi },\phi ,A)$. 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:

• (i) ${\parallel x\pm y\parallel }^2={\parallel x\parallel }^2\pm 2〈x,y〉+{\parallel y\parallel }^2$, $\mathrm{\forall }x,y\in H$;

• (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 ${\alpha }_i\in [0,1]$ 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 $u\in C$,

$$u=P_{C}z\iff 〈u-z,v-u〉\ge 0,\mathrm{\forall }v\in C.$$

Furthermore, $P_C$ is a firmly nonexpansive mapping of H onto C and satisfies

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

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 $\{x_n\}$ is a sequence in C weakly converging to some $x\in C$ and the sequence $\{(I-T)x_n\}$ strongly converges to some y it follows that $(I-T)x=y$. 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

$$\parallel Sx-Sy\parallel \le \frac{1+\kappa }{1-\kappa }\parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

Lemma 2.6 ([22])

Let $\{s_n\}$ be a sequence of nonnegative real numbers satisfying

$$s_{n+1}\le (1-{\alpha }_n)s_n+{\delta }_n,\mathrm{\forall }n\ge 0,$$

where ${\alpha }_n$ is a sequence in $(0,1)$ and $\{{\delta }_n\}$ is a sequence such that

• (1) ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (2) ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\frac{{\delta }_n}{{\alpha }_n}\le 0$ 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 $T:C\to \mathit{CB}(H)$ is said to be ℋ-Lipschitz continuous if there exists a constant $\mu >0$ such that

$$\mathcal{H}(T(u),T(v))\le \mu \parallel u-v\parallel ,\mathrm{\forall }u,v\in C,$$

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

Lemma 2.7 (Nadler’s theorem [17])

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

$$\parallel u-v\parallel \le (1+ϵ)\mathcal{H}(U,V).$$

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 $T:C\to \mathit{CB}(H)$ be ℋ-Lipschitz continuous with constant μ, and $\mathrm{\Phi }:H\times C\times C\to \mathbb{R}$ be an equilibrium-like function satisfying (H1)-(H3). Let $r>0$ be a constant. For each $x\in C$, take $w_x\in T(x)$ arbitrarily and define a mapping $T_r:C\to C$ as follows:

$$T_r(x)=\{u\in C:\mathrm{\Phi }(w_x,u,v)+\phi (v)-\phi (u)+\frac{1}{r}〈u-x,v-u〉\ge 0,\mathrm{\forall }v\in C\}.$$

Then we have the following:

• (a) $T_r$ is single-valued;

• (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

  $$\mathrm{\Phi }(w_1,T_r(x_1),T_r(x_2))+\mathrm{\Phi }(w_2,T_r(x_2),T_r(x_1))\le 0,$$

for all $(x_1,x_2)\in C\times C$ and all $w_i\in T(x_i)$, $i=1,2$;

• (c) $F(T_r)={(\mathit{GEP})}_s(\mathrm{\Phi },\phi )$;

• (d) ${(\mathit{GEP})}_s(\mathrm{\Phi },\phi )$ is closed and convex.

Definition 2.2 ([13])

Let C be a nonempty closed convex subset of a real Banach space. Let ${\{T_i\}}_{i=1}^N$ 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:

$$\begin{array}{c}{U}_1={\lambda }_1{T}_1+(1-{\lambda }_1)I,\hfill \\ U_2={\lambda }_2{T}_2{U}_1+(1-{\lambda }_2)U_1,\hfill \\ U_3={\lambda }_3{T}_3{U}_2+(1-{\lambda }_3)U_2,\hfill \\ ⋮\hfill \\ U_{N-1}={\lambda }_{N-1}{T}_{N-1}{U}_{N-2}+(1-{\lambda }_{N-1})U_{N-2},\hfill \\ K=U_N={\lambda }_N{T}_N{U}_{N-1}+(1-{\lambda }_N)U_{N-1}.\hfill \end{array}$$

(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 ${\{T_i\}}_{i=1}^N$ 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 ${\bigcap }_{i=1}^{N}F(T_i)\ne \mathrm{\varnothing }$. 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:

• (i) $F(K)={\bigcap }_{i=1}^{N}F(T_i)$;

• (ii) K is a nonexpansive mapping.

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

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

By the definition of K-mapping, we get

$$\begin{array}{c}\parallel x-y\parallel \hfill \\ ={\parallel Kx-y\parallel }^2\hfill \\ ={\parallel {\lambda }_N{T}_N{U}_{N-1}x+(1-{\lambda }_N)U_{N-1}x-y\parallel }^2\hfill \\ ={\parallel {\lambda }_N(T_N{U}_{N-1}x-y)+(1-{\lambda }_N)(U_{N-1}x-y)\parallel }^2\hfill \\ ={\lambda }_N^2{\parallel T_N{U}_{N-1}x-y\parallel }^2+{(1-{\lambda }_N)}^2{\parallel U_{N-1}x-y\parallel }^2\hfill \\ +2{\lambda }_N(1-{\lambda }_N)〈T_N{U}_{N-1}x-y,U_{N-1}x-y〉\hfill \\ ={\lambda }_N^2({\parallel U_{N-1}x-y\parallel }^2+{\kappa }_N{\parallel T_N{U}_{N-1}x-U_{N-1}x\parallel }^2)+{(1-{\lambda }_N)}^2{\parallel U_{N-1}x-y\parallel }^2\hfill \\ +2{\lambda }_N(1-{\lambda }_N)({\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)\hfill \\ =({\lambda }_N^2+{(1-{\lambda }_N)}^2+2{\lambda }_N(1-{\lambda }_N)){\parallel U_{N-1}x-y\parallel }^2\hfill \\ +({\lambda }_N^2{\kappa }_N-{\lambda }_N(1-{\lambda }_N)(1-{\kappa }_N)){\parallel T_N{U}_{N-1}x-U_{N-1}x\parallel }^2\hfill \\ ={({\lambda }_N+1-{\lambda }_N)}^2{\parallel U_{N-1}x-y\parallel }^2\hfill \\ +{\lambda }_N({\lambda }_N{\kappa }_N-(1-{\lambda }_N)(1-{\kappa }_N)){\parallel T_N{U}_{N-1}x-U_{N-1}x\parallel }^2\hfill \\ ={\parallel U_{N-1}x-y\parallel }^2\hfill \\ +{\lambda }_N({\lambda }_N{\kappa }_N-(1-{\kappa }_N)+{\lambda }_N(1-{\kappa }_N)){\parallel T_N{U}_{N-1}x-U_{N-1}x\parallel }^2\hfill \\ ={\parallel U_{N-1}x-y\parallel }^2+{\lambda }_N({\kappa }_N+{\lambda }_N-1){\parallel T_N{U}_{N-1}x-U_{N-1}x\parallel }^2\hfill \\ \le {\parallel U_{N-1}x-y\parallel }^2+{\lambda }_N({\gamma }_1+{\gamma }_2-1){\parallel T_N{U}_{N-1}x-U_{N-1}x\parallel }^2\hfill \\ \le {\parallel U_{N-1}x-y\parallel }^2\hfill \\ ⋮\hfill \\ ={\parallel U_{2}x-y\parallel }^2\hfill \\ ={\parallel {\lambda }_2(T_2{U}_{1}x-y)+(1-{\lambda }_2)(U_{1}x-y)\parallel }^2\hfill \\ ={\lambda }_2^2{\parallel T_2{U}_{1}x-y\parallel }^2+{(1-{\lambda }_2)}^2{\parallel U_{1}x-y\parallel }^2\hfill \\ +2{\lambda }_2(1-{\lambda }_2)〈T_2{U}_{1}x-y,U_{1}x-y〉\hfill \\ ={\lambda }_2^2({\parallel U_{1}x-y\parallel }^2+{\kappa }_2{\parallel T_2{U}_{1}x-U_{1}x\parallel }^2)+{(1-{\lambda }_2)}^2{\parallel U_{1}x-y\parallel }^2\hfill \\ +2{\lambda }_2(1-{\lambda }_2)({\parallel U_{1}x-y\parallel }^2-\frac{1-{\kappa }_2}{2}{\parallel T_2{U}_{1}x-U_{1}x\parallel }^2)\hfill \\ =({\lambda }_2^2+{(1-{\lambda }_2)}^2+2{\lambda }_2(1-{\lambda }_2)){\parallel U_{1}x-y\parallel }^2\hfill \\ +({\lambda }_2^2{\kappa }_2-{\lambda }_2(1-{\lambda }_2)(1-{\kappa }_2)){\parallel T_2{U}_{1}x-U_{1}x\parallel }^2\hfill \\ ={({\lambda }_2+1-{\lambda }_2)}^2{\parallel U_{1}x-y\parallel }^2\hfill \\ +{\lambda }_2({\lambda }_2{\kappa }_2-(1-{\lambda }_2)(1-{\kappa }_2)){\parallel T_2{U}_{1}x-U_{1}x\parallel }^2\hfill \\ ={\parallel U_{1}x-y\parallel }^2+{\lambda }_2({\kappa }_2+{\lambda }_2-1){\parallel T_2{U}_{1}x-U_{1}x\parallel }^2\hfill \\ \le {\parallel U_{1}x-y\parallel }^2+{\lambda }_2(({\gamma }_1+{\gamma }_2)-1){\parallel T_2{U}_{1}x-U_{1}x\parallel }^2\hfill \\ \le {\parallel U_{1}x-y\parallel }^2\hfill \\ ={\parallel {\lambda }_1(T_{1}x-y)+(1-{\lambda }_1)(x-y)\parallel }^2\hfill \\ ={\lambda }_1^2{\parallel T_{1}x-y\parallel }^2+{(1-{\lambda }_1)}^2{\parallel x-y\parallel }^2\hfill \\ +2{\lambda }_1(1-{\lambda }_1)〈T_{1}x-y,x-y〉\hfill \\ ={\lambda }_1^2({\parallel x-y\parallel }^2+{\kappa }_1{\parallel T_{1}x-x\parallel }^2)+{(1-{\lambda }_1)}^2{\parallel x-y\parallel }^2\hfill \\ +2{\lambda }_1(1-{\lambda }_1)({\parallel x-y\parallel }^2-\frac{1-{\kappa }_1}{2}{\parallel T_{1}x-x\parallel }^2)\hfill \\ =({\lambda }_1^2+{(1-{\lambda }_1)}^2+2{\lambda }_1(1-{\lambda }_1)){\parallel x-y\parallel }^2\hfill \\ +({\lambda }_1^2{\kappa }_1-{\lambda }_1(1-{\lambda }_1)(1-{\kappa }_1)){\parallel T_{1}x-x\parallel }^2\hfill \\ ={({\lambda }_1+1-{\lambda }_1)}^2{\parallel x-y\parallel }^2\hfill \\ +{\lambda }_1({\lambda }_1{\kappa }_1-(1-{\lambda }_1)(1-{\kappa }_1)){\parallel T_{1}x-x\parallel }^2\hfill \\ ={\parallel x-y\parallel }^2+{\lambda }_1({\kappa }_1+{\lambda }_1-1){\parallel T_{1}x-x\parallel }^2\hfill \\ \le {\parallel x-y\parallel }^2+{\lambda }_1(({\gamma }_1+{\gamma }_2)-1){\parallel T_{1}x-x\parallel }^2.\hfill \end{array}$$

(2.2)

From (2.2), it yields

$${\lambda }_1(1-({\gamma }_1+{\gamma }_2)){\parallel T_{1}x-x\parallel }^2\le 0.$$

This implies that

$$\parallel T_{1}x-x\parallel =0.$$

Therefore $x=T_{1}x$, that is,

$$x\in F(T_1).$$

(2.3)

By the definition of $U_1$ and (2.3), we have

$$U_{1}x={\lambda }_1{T}_{1}x+(1-{\lambda }_1)x=x,$$

that is,

$$x\in F(U_1).$$

(2.4)

Again by (2.2) and (2.4), we obtain

$$\begin{array}{rl}{\parallel x-y\parallel }^2& \le {\parallel U_{1}x-y\parallel }^2+{\lambda }_2(({\gamma }_1+{\gamma }_2)-1){\parallel T_2{U}_{1}x-U_{1}x\parallel }^2\\ ={\parallel x-y\parallel }^2+{\lambda }_2(({\gamma }_1+{\gamma }_2)-1){\parallel T_{2}x-x\parallel }^2,\end{array}$$

which implies that $x=T_{2}x$, that is,

$$x\in F(T_2).$$

(2.5)

By the definition of $U_2$, (2.4), and (2.5), we get

$$U_{2}x={\lambda }_2{T}_2{U}_{1}x+(1-{\lambda }_2)U_{1}x=x,$$

from which it follows that

$$x\in F(U_2).$$

Using the same argument, we can conclude that

$$x\in F(T_i)\text{and}x\in F(U_i),\mathrm{\forall }i=1,2,\dots ,N-1.$$

Next, we show that $x\in F(T_N)$. Since

$$\begin{array}{rl}0& =Kx-x\\ ={\lambda }_N{T}_N{U}_{N-1}x+(1-{\lambda }_N)U_{N-1}x-x\\ ={\lambda }_N(T_{N}x-x)\end{array}$$

and ${\lambda }_N\in (0,1]$, we obtain

$$x\in F(T_N),$$

from which it follows that

$$x\in \bigcap _{i=1}^{N}F(T_i).$$

(2.6)

Therefore

$$F(K)\subseteq \bigcap _{i=1}^{N}F(T_i).$$

(2.7)

Hence

$$F(K)=\bigcap _{i=1}^{N}F(T_i).$$

(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 \\ ={\parallel ({\lambda }_N{T}_N{U}_{N-1}x+(1-{\lambda }_N)U_{N-1}x)-({\lambda }_N{T}_N{U}_{N-1}y+(1-{\lambda }_N)U_{N-1}y)\parallel }^2\hfill \\ ={\parallel (U_{N-1}x-{\lambda }_N(U_{N-1}x-T_N{U}_{N-1}x))-(U_{N-1}y-{\lambda }_N(U_{N-1}y-T_N{U}_{N-1}y))\parallel }^2\hfill \\ ={\parallel (U_{N-1}x-U_{N-1}y)-{\lambda }_N((I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y)\parallel }^2\hfill \\ ={\parallel U_{N-1}x-U_{N-1}y\parallel }^2+{\lambda }_N^2{\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ -2{\lambda }_N〈U_{N-1}x-U_{N-1}y,(I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y〉\hfill \\ \le {\parallel U_{N-1}x-U_{N-1}y\parallel }^2+{\lambda }_N^2{\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ -2{\lambda }_N\left(\frac{1-{\kappa }_N}{2}\right){\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ ={\parallel U_{N-1}x-U_{N-1}y\parallel }^2\hfill \\ +{\lambda }_N({\lambda }_N-(1-{\kappa }_N)){\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ \le {\parallel U_{N-1}x-U_{N-1}y\parallel }^2\hfill \\ +{\lambda }_N({\gamma }_1+{\gamma }_2-1){\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ ={\parallel U_{N-1}x-U_{N-1}y\parallel }^2\hfill \\ -{\lambda }_N(1-({\gamma }_1+{\gamma }_2)){\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ ={\parallel ({\lambda }_{N-1}{T}_{N-1}{U}_{N-2}x+(1-{\lambda }_{N-1})U_{N-2}x)-({\lambda }_{N-1}{T}_{N-1}{U}_{N-2}y+(1-{\lambda }_{N-1})U_{N-2}y)\parallel }^2\hfill \\ -{\lambda }_N(1-({\gamma }_1+{\gamma }_2)){\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ ={\parallel (U_{N-2}x-{\lambda }_{N-1}(I-T_{N-1})U_{N-2}x)-(U_{N-2}y-{\lambda }_{N-1}(I-T_{N-1})U_{N-2}y)\parallel }^2\hfill \\ -{\lambda }_N(1-({\gamma }_1+{\gamma }_2)){\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ ={\parallel (U_{N-2}x-U_{N-2}y)-{\lambda }_{N-1}((I-T_{N-1})U_{N-2}x-(I-T_{N-1})U_{N-2}y)\parallel }^2\hfill \\ -{\lambda }_N(1-({\gamma }_1+{\gamma }_2)){\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ ={\parallel U_{N-2}x-U_{N-2}y\parallel }^2+{\lambda }_{N-1}^2{\parallel (I-T_{N-1})U_{N-2}x-(I-T_{N-1})U_{N-2}y\parallel }^2\hfill \\ -2{\lambda }_{N-1}〈U_{N-2}x-U_{N-2}y,(I-T_{N-1})U_{N-2}x-(I-T_{N-1})U_{N-2}y〉\hfill \\ -{\lambda }_N(1-({\gamma }_1+{\gamma }_2)){\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ \le {\parallel U_{N-2}x-U_{N-2}y\parallel }^2+{\lambda }_{N-1}^2{\parallel (I-T_{N-1})U_{N-2}x-(I-T_{N-1})U_{N-2}y\parallel }^2\hfill \\ -2{\lambda }_{N-1}\left(\frac{1-{\kappa }_{N-1}}{2}\right){\parallel (I-T_{N-1})U_{N-2}x-(I-T_{N-1})U_{N-2}y\parallel }^2\hfill \\ -{\lambda }_N(1-({\gamma }_1+{\gamma }_2)){\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ ={\parallel U_{N-2}x-U_{N-2}y\parallel }^2\hfill \\ +{\lambda }_{N-1}({\lambda }_{N-1}-(1-{\kappa }_{N-1})){\parallel (I-T_{N-1})U_{N-2}x-(I-T_{N-1})U_{N-2}y\parallel }^2\hfill \\ -{\lambda }_N(1-({\gamma }_1+{\gamma }_2)){\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ \le {\parallel U_{N-2}x-U_{N-2}y\parallel }^2\hfill \\ +{\lambda }_{N-1}({\gamma }_1+{\gamma }_2-1){\parallel (I-T_{N-1})U_{N-2}x-(I-T_{N-1})U_{N-2}y\parallel }^2\hfill \\ -{\lambda }_N(1-({\gamma }_1+{\gamma }_2)){\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ ={\parallel U_{N-2}x-U_{N-2}y\parallel }^2\hfill \\ -{\lambda }_{N-1}(1-({\gamma }_1+{\gamma }_2)){\parallel (I-T_{N-1})U_{N-2}x-(I-T_{N-1})U_{N-2}y\parallel }^2\hfill \\ -{\lambda }_N(1-({\gamma }_1+{\gamma }_2)){\parallel (I-T_N)U_{N-1}x-(I-T_N)U_{N-1}y\parallel }^2\hfill \\ ={\parallel U_{N-2}x-U_{N-2}y\parallel }^2\hfill \\ -(1-({\gamma }_1+{\gamma }_2))\sum _{i=N-1}^N{\lambda }_i{\parallel (I-T_i)U_{i-1}x-(I-T_i)U_{i-1}y\parallel }^2\hfill \\ ⋮\hfill \\ \le {\parallel x-y\parallel }^2-(1-({\gamma }_1+{\gamma }_2))\sum _{i=1}^N{\lambda }_i{\parallel (I-T_i)U_{i-1}x-(I-T_i)U_{i-1}y\parallel }^2,\hfill \end{array}$$

which implies that

$${\parallel Kx-Ky\parallel }^2\le {\parallel x-y\parallel }^2-(1-({\gamma }_1+{\gamma }_2))\sum _{i=1}^N{\lambda }_i{\parallel (I-T_i)U_{i-1}x-(I-T_i)U_{i-1}y\parallel }^2.$$

(2.9)

From (2.9) and ${\gamma }_1+{\gamma }_2<1$, we obtain

$$\parallel Kx-Ky\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in C,$$

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 ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ be a finite family of ${\kappa }_i$-strictly pseudo-contractive mappings of C into itself with ${\kappa }_i\le {\gamma }_1$ and ${\bigcap }_{i=1}^{N}F(T_i)\ne \mathrm{\varnothing }$. 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 $\{x_n\}$ in C, the following properties hold:

• (i) ${lim}_{n\to \mathrm{\infty }}\parallel K_n{x}_n-K{x}_n\parallel =0$;

• (ii) ${\sum }_{n=1}^{\mathrm{\infty }}\parallel K_n{x}_{n-1}-K_{n-1}{x}_{n-1}\parallel <\mathrm{\infty }$.

Proof Let $\{x_n\}$ 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

$$\begin{array}{rl}\parallel U_{n,1}{x}_n-U_1{x}_n\parallel & =\parallel {\lambda }_1^n{T}_1{x}_n+(1-{\lambda }_1^n)x_n-({\lambda }_1{T}_1{x}_n+(1-{\lambda }_1)x_n)\parallel \\ =\parallel {\lambda }_1^n{T}_1{x}_n-{\lambda }_1^n{x}_n-{\lambda }_1{T}_1{x}_n+{\lambda }_1{x}_n\parallel \\ =\parallel ({\lambda }_1^n-{\lambda }_1)T_1{x}_n-({\lambda }_1^n-{\lambda }_1)x_n\parallel \\ =|{\lambda }_1^n-{\lambda }_1|\parallel T_1{x}_n-x_n\parallel .\end{array}$$

(2.10)

For $k\in \{2,3,\dots ,N\}$, we have

$$\begin{array}{c}\parallel U_{n,k}{x}_n-U_k{x}_n\parallel \hfill \\ =\parallel {\lambda }_k^n{T}_k{U}_{n,k-1}{x}_n+(1-{\lambda }_k^n)U_{n,k-1}{x}_n-({\lambda }_k{T}_k{U}_{k-1}{x}_n+(1-{\lambda }_k)U_{k-1}{x}_n)\parallel \hfill \\ =\parallel {\lambda }_k^n{T}_k{U}_{n,k-1}{x}_n-{\lambda }_k{T}_k{U}_{k-1}{x}_n+(1-{\lambda }_k^n)U_{n,k-1}{x}_n-(1-{\lambda }_k)U_{k-1}{x}_n\parallel \hfill \\ =\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 \\ +(1-{\lambda }_k^n)U_{n,k-1}{x}_n-(1-{\lambda }_k^n)U_{k-1}{x}_n+(1-{\lambda }_k^n)U_{k-1}{x}_n\hfill \\ -(1-{\lambda }_k)U_{k-1}{x}_n\parallel \hfill \\ =\parallel {\lambda }_k^n(T_k{U}_{n,k-1}{x}_n-T_k{U}_{k-1}{x}_n)+({\lambda }_k^n-{\lambda }_k)T_k{U}_{k-1}{x}_n\hfill \\ +(1-{\lambda }_k^n)(U_{n,k-1}{x}_n-U_{k-1}{x}_n)+(1-{\lambda }_k^n-(1-{\lambda }_k))U_{k-1}{x}_n\parallel \hfill \\ \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 \\ +(1-{\lambda }_k^n)\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 \\ \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 \\ +(1-{\lambda }_k^n)\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 \\ \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 \\ +|{\lambda }_k^n-{\lambda }_k|(\parallel T_k{U}_{k-1}{x}_n\parallel +\parallel U_{k-1}{x}_n\parallel )\hfill \\ =\frac{2}{1-{\kappa }_k}\parallel U_{n,k-1}{x}_n-U_{k-1}{x}_n\parallel \hfill \\ +|{\lambda }_k^n-{\lambda }_k|(\parallel T_k{U}_{k-1}{x}_n\parallel +\parallel U_{k-1}{x}_n\parallel ).\hfill \end{array}$$

(2.11)

By (2.10) and (2.11), we get

$$\begin{array}{c}\parallel K_n{x}_n-K{x}_n\parallel \hfill \\ =\parallel U_{n,N}{x}_n-U_N{x}_n\parallel \hfill \\ \le \frac{2}{1-{\kappa }_N}\parallel U_{n,N-1}{x}_n-U_{N-1}{x}_n\parallel \hfill \\ +|{\lambda }_N^n-{\lambda }_N|(\parallel T_N{U}_{N-1}{x}_n\parallel +\parallel U_{N-1}{x}_n\parallel )\hfill \\ \le \frac{2}{1-{\kappa }_N}(\frac{2}{1-{\kappa }_{N-1}}\parallel U_{n,N-2}{x}_n-U_{N-2}{x}_n\parallel \hfill \\ +|{\lambda }_{N-1}^n-{\lambda }_{N-1}|(\parallel T_{N-1}{U}_{N-2}{x}_n\parallel +\parallel U_{N-2}{x}_n\parallel ))\hfill \\ +|{\lambda }_N^n-{\lambda }_N|(\parallel T_N{U}_{N-1}{x}_n\parallel +\parallel U_{N-1}{x}_n\parallel )\hfill \\ =\left(\frac{2}{1-{\kappa }_N}\right)\left(\frac{2}{1-{\kappa }_{N-1}}\right)\parallel U_{n,N-2}{x}_n-U_{N-2}{x}_n\parallel \hfill \\ +\frac{2}{1-{\kappa }_N}|{\lambda }_{N-1}^n-{\lambda }_{N-1}|(\parallel T_{N-1}{U}_{N-2}{x}_n\parallel +\parallel U_{N-2}{x}_n\parallel )\hfill \\ +|{\lambda }_N^n-{\lambda }_N|(\parallel T_N{U}_{N-1}{x}_n\parallel +\parallel U_{N-1}{x}_n\parallel )\hfill \\ =\prod _{j=N-1}^N\left(\frac{2}{1-{\kappa }_j}\right)\parallel U_{n,N-2}{x}_n-U_{N-2}{x}_n\parallel \hfill \\ +\sum _{j=N-1}^N{\left(\frac{2}{1-{\kappa }_{j+1}}\right)}^{N-j}|{\lambda }_j^n-{\lambda }_j|(\parallel T_j{U}_{j-1}{x}_n\parallel +\parallel U_{j-1}{x}_n\parallel )\hfill \\ ⋮\hfill \\ \le \prod _{j=2}^N\left(\frac{2}{1-{\kappa }_j}\right)\parallel U_{n,1}{x}_n-U_1{x}_n\parallel \hfill \\ +\sum _{j=2}^N{\left(\frac{2}{1-{\kappa }_{j+1}}\right)}^{N-j}|{\lambda }_j^n-{\lambda }_j|(\parallel T_j{U}_{j-1}{x}_n\parallel +\parallel U_{j-1}{x}_n\parallel )\hfill \\ =\prod _{j=2}^N\left(\frac{2}{1-{\kappa }_j}\right)|{\lambda }_1^n-{\lambda }_1|\parallel T_1{x}_n-x_n\parallel \hfill \\ +\sum _{j=2}^N{\left(\frac{2}{1-{\kappa }_{j+1}}\right)}^{N-j}|{\lambda }_j^n-{\lambda }_j|(\parallel T_j{U}_{j-1}{x}_n\parallel +\parallel U_{j-1}{x}_n\parallel ).\hfill \end{array}$$

(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

$$\begin{array}{c}\parallel U_{n,1}{x}_{n-1}-U_{n-1,1}{x}_{n-1}\parallel \hfill \\ =\parallel {\lambda }_1^n{T}_1{x}_{n-1}+(1-{\lambda }_1^n)x_{n-1}-({\lambda }_1^{n-1}{T}_1{x}_{n-1}+(1-{\lambda }_1^{n-1})x_{n-1})\parallel \hfill \\ =\parallel {\lambda }_1^n{T}_1{x}_{n-1}-{\lambda }_1^n{x}_{n-1}-{\lambda }_1^{n-1}{T}_1{x}_{n-1}+{\lambda }_1^{n-1}{x}_{n-1}\parallel \hfill \\ =\parallel ({\lambda }_1^n-{\lambda }_1^{n-1})T_1{x}_{n-1}-({\lambda }_1^n-{\lambda }_1^{n-1})x_{n-1}\parallel \hfill \\ =|{\lambda }_1^n-{\lambda }_1^{n-1}|\parallel T_1{x}_{n-1}-x_{n-1}\parallel .\hfill \end{array}$$

(2.13)

For $k\in \{2,3,\dots ,N\}$, we have

$$\begin{array}{c}\parallel U_{n,k}{x}_{n-1}-U_{n-1,k}{x}_{n-1}\parallel \hfill \\ =\parallel {\lambda }_k^n{T}_k{U}_{n,k-1}{x}_{n-1}+(1-{\lambda }_k^n)U_{n,k-1}{x}_{n-1}-({\lambda }_k^{n-1}{T}_k{U}_{n-1,k-1}{x}_{n-1}\hfill \\ +(1-{\lambda }_k^{n-1})U_{n-1,k-1}{x}_{n-1})\parallel \hfill \\ =\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}+(1-{\lambda }_k^n)U_{n,k-1}{x}_{n-1}\hfill \\ -(1-{\lambda }_k^{n-1})U_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ =\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 \\ -{\lambda }_k^{n-1}{T}_k{U}_{n-1,k-1}{x}_{n-1}+(1-{\lambda }_k^n)U_{n,k-1}{x}_{n-1}-(1-{\lambda }_k^n)U_{n-1,k-1}{x}_{n-1}\hfill \\ +(1-{\lambda }_k^n)U_{n-1,k-1}{x}_{n-1}-(1-{\lambda }_k^{n-1})U_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ =\parallel {\lambda }_k^n(T_k{U}_{n,k-1}{x}_{n-1}-T_k{U}_{n-1,k-1}{x}_{n-1})+({\lambda }_k^n-{\lambda }_k^{n-1})T_k{U}_{n-1,k-1}{x}_{n-1}\hfill \\ +(1-{\lambda }_k^n)(U_{n,k-1}{x}_{n-1}-U_{n-1,k-1}{x}_{n-1})\hfill \\ +(1-{\lambda }_k^n-(1-{\lambda }_k^{n-1}))U_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ \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 \\ +(1-{\lambda }_k^n)\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 \\ \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 \\ +(1-{\lambda }_k^n)\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 \\ \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 \\ +\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 \\ +|{\lambda }_k^n-{\lambda }_k^{n-1}|(\parallel T_k{U}_{n-1,k-1}{x}_{n-1}\parallel +\parallel U_{n-1,k-1}{x}_{n-1}\parallel )\hfill \\ =\frac{2}{1-{\kappa }_k}\parallel U_{n,k-1}{x}_{n-1}-U_{n-1,k-1}{x}_{n-1}\parallel \hfill \\ +|{\lambda }_k^n-{\lambda }_k^{n-1}|(\parallel T_k{U}_{n-1,k-1}{x}_{n-1}\parallel +\parallel U_{n-1,k-1}{x}_{n-1}\parallel ).\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 \\ =\parallel U_{n,N}{x}_{n-1}-U_{n-1,N}{x}_{n-1}\parallel \hfill \\ \le \frac{2}{1-{\kappa }_N}\parallel U_{n,N-1}{x}_{n-1}-U_{n-1,N-1}{x}_{n-1}\parallel \hfill \\ +|{\lambda }_N^n-{\lambda }_N^{n-1}|(\parallel T_N{U}_{n-1,N-1}{x}_{n-1}\parallel +\parallel U_{n-1,N-1}{x}_{n-1}\parallel )\hfill \\ \le \frac{2}{1-{\kappa }_N}(\frac{2}{1-{\kappa }_{N-1}}\parallel U_{n,N-2}{x}_{n-1}-U_{n-1,N-2}{x}_{n-1}\parallel \hfill \\ +|{\lambda }_{N-1}^n-{\lambda }_{N-1}^{n-1}|(\parallel T_{N-1}{U}_{n-1,N-2}{x}_{n-1}\parallel +\parallel U_{n-1,N-2}{x}_{n-1}\parallel ))\hfill \\ +|{\lambda }_N^n-{\lambda }_N^{n-1}|(\parallel T_N{U}_{n-1,N-1}{x}_{n-1}\parallel +\parallel U_{n-1,N-1}{x}_{n-1}\parallel )\hfill \\ =\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 \\ +\frac{2}{1-{\kappa }_N}|{\lambda }_{N-1}^n-{\lambda }_{N-1}^{n-1}|(\parallel T_{N-1}{U}_{n-1,N-2}{x}_{n-1}\parallel +\parallel U_{n-1,N-2}{x}_{n-1}\parallel )\hfill \\ +|{\lambda }_N^n-{\lambda }_N^{n-1}|(\parallel T_N{U}_{n-1,N-1}{x}_{n-1}\parallel +\parallel U_{n-1,N-1}{x}_{n-1}\parallel )\hfill \\ =\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 \\ +\sum _{j=N-1}^N{\left(\frac{2}{1-{\kappa }_{j+1}}\right)}^{N-j}|{\lambda }_j^n-{\lambda }_j^{n-1}|(\parallel T_j{U}_{n-1,j-1}{x}_{n-1}\parallel +\parallel U_{n-1,j-1}{x}_{n-1}\parallel )\hfill \\ ⋮\hfill \\ \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 \\ +\sum _{j=2}^N{\left(\frac{2}{1-{\kappa }_{j+1}}\right)}^{N-j}|{\lambda }_j^n-{\lambda }_j^{n-1}|(\parallel T_j{U}_{n-1,j-1}{x}_{n-1}\parallel +\parallel U_{n-1,j-1}{x}_{n-1}\parallel )\hfill \\ =\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 \\ +\sum _{j=2}^N{\left(\frac{2}{1-{\kappa }_{j+1}}\right)}^{N-j}|{\lambda }_j^n-{\lambda }_j^{n-1}|(\parallel T_j{U}_{n-1,j-1}{x}_{n-1}\parallel +\parallel U_{n-1,j-1}{x}_{n-1}\parallel )\hfill \\ \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 $M={max}_{n\in \mathbb{N}}\{\parallel T_1{x}_{n-1}-x_{n-1}\parallel ,\parallel T_j{U}_{n-1,j-1}{x}_{n-1}\parallel ,\parallel U_{n-1,j-1}{x}_{n-1}\parallel \}$, 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 ${\{T_i\}}_{i=1}^N$ be a finite family of ${\kappa }_i$-strictly pseudo-contractions of C into itself. For each $j=1,2,\dots ,N$, let ${\alpha }_j=({\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j)\in I\times I\times I$ where $I\in [0,1]$ 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 ${\{T_i\}}_{i=1}^N$ be a finite family of ${\kappa }_i$-strictly pseudo-contractions of C into itself with ${\bigcap }_{i=1}^{N}F(T_i)\ne \mathrm{\varnothing }$ and $\kappa =max\{{\kappa }_i:i=1,2,\dots ,N\}$ and let ${\alpha }_j=({\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j)\in I\times I\times I$, $j=1,2,\dots ,N$, where $I=[0,1]$, ${\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1$, ${\alpha }_1^j,{\alpha }_2^j\in (\kappa ,1)$ for all $j=1,2,\dots ,N-1$ and ${\alpha }_1^N\in (\kappa ,1]$, ${\alpha }_3^N\in [\kappa ,1)$, ${\alpha }_2^j\in [\kappa ,1)$ 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 $F(S)={\bigcap }_{i=1}^{N}F(T_i)$ 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 ${\{T_i\}}_{i=1}^N$ be a finite family of ${\kappa }_i$-strictly pseudo-contractions of C into itself with ${\bigcap }_{i=1}^{N}F(T_i)\ne \mathrm{\varnothing }$ and $\kappa =max\{{\kappa }_i:i=1,2,\dots ,N\}$ and let ${\lambda }_j\in (\kappa ,1)\subset [0,1]$, for all $j=1,2,\dots ,N-1$ and ${\lambda }_N\in (\kappa ,1]$. Let K be the mapping generated by $T_1,T_2,\dots ,T_N$ and ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_N$. Then $F(K)={\bigcap }_{i=1}^{N}F(T_i)$ 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:

• (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 $i=1,2,\dots ,N$.

• (ii) We do not assume the condition $\kappa =max\{{\kappa }_i:i=1,2,\dots ,N\}$.

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

$$T_{i}x=-(i+1)x,\text{for all }x\in \mathbb{R},$$

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 $F(K)={\bigcap }_{i=1}^{5}F(T_i)=\{0\}$.

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 $\kappa =max\{{\kappa }_i:i=1,2,\dots ,5\}=\frac{5}{7}$ and ${\lambda }_i\in (\frac{5}{7},1]$, for all $i=1,2,\dots ,5$. By the definition of a K-mapping, we have

$$\begin{array}{c}{U}_{1}x=\left(\frac{6}{7}\right)(-2x)+(1-\frac{6}{7})x,\hfill \\ U_{2}x=\left(\frac{7}{8}\right)(-3U_{1}x)+(1-\frac{7}{8})U_{1}x,\hfill \\ U_{3}x=\left(\frac{8}{9}\right)(-4U_{2}x)+(1-\frac{8}{9})U_{2}x,\hfill \\ U_{4}x=\left(\frac{9}{10}\right)(-5U_{3}x)+(1-\frac{9}{10})U_{3}x,\hfill \\ Kx=U_{5}x=\left(\frac{10}{11}\right)(-6U_{4}x)+(1-\frac{10}{11})U_{4}x.\hfill \end{array}$$

(2.16)

Observe that ${\bigcap }_{i=1}^{5}F(T_i)=\{0\}$. Then, by Lemma 2.12, we obtain

$$F(K)=\bigcap _{i=1}^{5}F(T_i)=\{0\}.$$

Next, we give an example for Lemma 2.9.

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

$$T_{i}x=-(i+1)x,\text{for all }x\in \mathbb{R},$$

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 $F(K)={\bigcap }_{i=1}^{5}F(T_i)=\{0\}$.

3 Strong convergence theorem

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. For every $i=1,2,\dots ,N$, $S_i:C\to \mathit{CB}(H)$ be ℋ-Lipschitz continuous with coefficients ${\mu }_i$, ${\mathrm{\Phi }}_i:H\times C\times C\to \mathbb{R}$ 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 ${\{T_i\}}_{i=1}^N$ be a finite family of ${\kappa }_i$-strictly pseudo-contractive mappings and ${\kappa }_i\le {\gamma }_1$ with $\mathcal{F}:={\bigcap }_{i=1}^{N}F(T_i)\cap {\bigcap }_{i=1}^N{(\mathit{MGEP})}_s({\mathrm{\Phi }}_i,\phi ,A)\ne \mathrm{\varnothing }$. 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 $\{x_n\}$ be the sequence generated by $x_1\in C$ and $w_1^i\in S_i(I-r_1^{i}A)x_1$, there exist sequences $\{w_n^i\}\in H$ and $\{x_n\},\{u_n^i\}\subseteq C$ such that

$$\{\begin{array}{c}\parallel w_n^i-w_{n+1}^i\parallel \le (1+\frac{1}{n})\mathcal{H}(S_i(I-r_n^{i}A)x_n,S_i(I-r_{n+1}^{i}A)x_{n+1}),\hfill \\ w_n^i\in S_i(I-r_n^{i}A)x_n\hfill \\ {\mathrm{\Phi }}_i(w_n^i,u_n^i,y)+\phi (y)-\phi (u_n^i)+\frac{1}{r_n^i}〈u_n^i-x_n,y-u_n^i〉+〈A{x}_n,y-u_n^i〉\hfill \\ \ge 0,\mathrm{\forall }y\in C,\hfill \\ x_{n+1}={\alpha }_{n}f(x_n)+{\beta }_n({\sum }_{i=1}^N{a}_n^i{u}_n^i)+{\delta }_n{K}_n{x}_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.1)

where $f:C\to C$ be a contraction mapping with a constant ξ and $\{{\alpha }_n\},\{{\beta }_n\},\{{\delta }_n\}\subseteq (0,1)$ with ${\alpha }_n+{\beta }_n+{\delta }_n=1$, $\mathrm{\forall }n\ge 1$. Suppose the following conditions hold:

• (i) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (ii) $0<\tau \le {\beta }_n,{\delta }_n\le \upsilon <1$;

• (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 ${\sum }_{n=1}^N{a}_n^i=1$;

• (iv) $0<ϵ\le r_n^i\le \omega <2\alpha$, for all $n\in \mathbb{N}$ and $i=1,2,\dots ,N$;

• (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 $i=1,2,\dots ,N$;

• (vi) for each $i=1,2,\dots ,N$, there exists ${\rho }_i>0$ such that

  $$\begin{array}{c}{\mathrm{\Phi }}_i(w_1^i,T_{r_1^i}(x_1),T_{r_2^i}(x_2))+{\mathrm{\Phi }}_i(w_2^i,T_{r_2^i}(x_2),T_{r_1^i}(x_1))\hfill \\ \le -{\rho }_i{\parallel T_{r_1^i}(x_1)-T_{r_2^i}(x_2)\parallel }^2,\hfill \end{array}$$

  (3.2)

for all $(r_1^i,r_2^i)\in {\mathrm{\Theta }}_i\times {\mathrm{\Theta }}_i$, $(x_1,x_2)\in C\times C$ and $w_j^i\in S_i(x_j)$, for $j=1,2$, where ${\mathrm{\Theta }}_i=\{r_n^i:n\ge 1\}$.

Then $\{x_n\}$ and $\{u_n^i\}$ converges strongly to $q=P_{\mathcal{F}}f(q)$, 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

$${\mathrm{\Phi }}_i(w_n^i,u_n^i,y)+\phi (y)-\phi \left(u_n^i\right)+\frac{1}{r_n^i}〈u_n^i-(I-r_n^{i}A)x_n,y-u_n^i〉\ge 0,$$

(3.3)

for every $y\in C$. From (3.3) and Theorem 2.8, we obtain

$$u_n^i=T_{r_n^i}(I-r_n^{i}A)x_n,\mathrm{\forall }i=1,2,\dots ,N.$$

Put $r^i\in {\mathrm{\Theta }}_i$ for all $i=1,2,\dots ,N$. From (3.2), we have

$$\begin{array}{c}{\mathrm{\Phi }}_i(w_1^i,T_{r^i}(x_1),T_{r^i}(x_2))+{\mathrm{\Phi }}_i(w_2^i,T_{r^i}(x_2),T_{r^i}(x_1))\hfill \\ \le -{\rho }_i{\parallel T_{r^i}(x_1)-T_{r^i}(x_2)\parallel }^2\le 0,\hfill \end{array}$$

(3.4)

for all $(x_1,x_2)\in C\times C$ and $w_j^i\in S_i(x_j)$, $j=1,2$.

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 $r_n^i\in (0,2\alpha )$, we get

$$\begin{array}{c}{\parallel (I-r_n^{i}A)x-(I-r_n^{i}A)y\parallel }^2\hfill \\ ={\parallel x-y-r_n^i(Ax-Ay)\parallel }^2\hfill \\ ={\parallel x-y\parallel }^2-2r_n^i〈x-y,Ax-Ay〉+{\left(r_n^i\right)}^2{\parallel Ax-Ay\parallel }^2\hfill \\ \le {\parallel x-y\parallel }^2-2\alpha r_n^i{\parallel Ax-Ay\parallel }^2+{\left(r_n^i\right)}^2{\parallel Ax-Ay\parallel }^2\hfill \\ ={\parallel x-y\parallel }^2+r_n^i(r_n^i-2\alpha ){\parallel Ax-Ay\parallel }^2\hfill \\ \le {\parallel x-y\parallel }^2.\hfill \end{array}$$

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

Step 2. We will show that $\{x_n\}$ is bounded.

Let $z\in \mathcal{F}$. By nonexpansiveness of $K_n$, we have

$$\begin{array}{c}\parallel x_{n+1}-z\parallel \hfill \\ \le {\alpha }_n\parallel f(x_n)-z\parallel +{\beta }_n\parallel \sum _{i=1}^N{a}_n^i(u_n^i-z)\parallel +{\delta }_n\parallel K_n{x}_n-z\parallel \hfill \\ \le {\alpha }_n\parallel f(x_n)-f(z)+f(z)-z\parallel +{\beta }_n\sum _{i=1}^N{a}_n^i\parallel u_n^i-z\parallel +{\delta }_n\parallel x_n-z\parallel \hfill \\ \le {\alpha }_n(\parallel f(x_n)-f(z)\parallel +\parallel f(z)-z\parallel )+{\beta }_n\sum _{i=1}^N{a}_n^i\parallel T_{r_n^i}(I-r_n^{i}A)x_n-z\parallel \hfill \\ +{\delta }_n\parallel x_n-z\parallel \hfill \\ \le {\alpha }_n(\xi \parallel x_n-z\parallel +\parallel f(z)-z\parallel )+{\beta }_n\sum _{i=1}^N{a}_n^i\parallel x_n-z\parallel +{\delta }_n\parallel x_n-z\parallel \hfill \\ =(1-{\alpha }_n(1-\xi ))\parallel x_n-z\parallel +{\alpha }_n\parallel f(z)-z\parallel \hfill \\ \le max\{\parallel x_1-z\parallel ,\frac{\parallel f(z)-z\parallel }{1-\xi }\}.\hfill \end{array}$$

By induction, we have $\parallel x_n-z\parallel \le max\{\parallel x_1-z\parallel ,\frac{\parallel f(z)-z\parallel }{1-\xi }\}$, $\mathrm{\forall }n\in \mathbb{N}$. It follows that $\{x_n\}$ is bounded and so is $\{u_n^i\}$, $\mathrm{\forall }i=1,2,\dots ,N$.

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 \\ =\parallel {\alpha }_{n}f(x_n)+{\beta }_n\left(\sum _{i=1}^N{a}_n^i{u}_n^i\right)+{\delta }_n{K}_n{x}_n\hfill \\ -({\alpha }_{n-1}f(x_{n-1})+{\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})\parallel \hfill \\ \le {\alpha }_n\parallel f(x_n)-f(x_{n-1})\parallel +|{\alpha }_n-{\alpha }_{n-1}|\parallel f(x_{n-1})\parallel \hfill \\ +{\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 \\ +{\delta }_n\parallel K_n{x}_n-K_n{x}_{n-1}\parallel +{\delta }_n\parallel K_n{x}_{n-1}-K_{n-1}{x}_{n-1}\parallel \hfill \\ +|{\delta }_n-{\delta }_{n-1}|\parallel K_{n-1}{x}_{n-1}\parallel \hfill \\ \le {\alpha }_n\xi \parallel x_n-x_{n-1}\parallel +|{\alpha }_n-{\alpha }_{n-1}|\parallel f(x_{n-1})\parallel \hfill \\ +{\beta }_n\parallel \sum _{i=1}^N{a}_n^i{u}_n^i-\sum _{i=1}^N{a}_n^i{u}_{n-1}^i+\sum _{i=1}^N{a}_n^i{u}_{n-1}^i-\sum _{i=1}^N{a}_{n-1}^i{u}_{n-1}^i\parallel \hfill \\ +|{\beta }_n-{\beta }_{n-1}|\sum _{i=1}^N{a}_{n-1}^i\parallel u_{n-1}^i\parallel +{\delta }_n\parallel x_n-x_{n-1}\parallel \hfill \\ +{\delta }_n\parallel K_n{x}_{n-1}-K_{n-1}{x}_{n-1}\parallel +|{\delta }_n-{\delta }_{n-1}|\parallel K_{n-1}{x}_{n-1}\parallel \hfill \\ \le {\alpha }_n\xi \parallel x_n-x_{n-1}\parallel +|{\alpha }_n-{\alpha }_{n-1}|\parallel f(x_{n-1})\parallel +{\beta }_n\parallel \sum _{i=1}^N{a}_n^i(u_n^i-u_{n-1}^i)\parallel \hfill \\ +{\beta }_n\parallel \sum _{i=1}^N(a_n^i-a_{n-1}^i)u_{n-1}^i\parallel +|{\beta }_n-{\beta }_{n-1}|\sum _{i=1}^N{a}_{n-1}^i\parallel u_{n-1}^i\parallel \hfill \\ +{\delta }_n\parallel x_n-x_{n-1}\parallel +{\delta }_n\parallel K_n{x}_{n-1}-K_{n-1}{x}_{n-1}\parallel +|{\delta }_n-{\delta }_{n-1}|\parallel K_{n-1}{x}_{n-1}\parallel \hfill \\ \le {\alpha }_n\xi \parallel x_n-x_{n-1}\parallel +|{\alpha }_n-{\alpha }_{n-1}|\parallel f(x_{n-1})\parallel +{\beta }_n\sum _{i=1}^N{a}_n^i\parallel u_n^i-u_{n-1}^i\parallel \hfill \\ +{\beta }_n\sum _{i=1}^N|a_n^i-a_{n-1}^i|\parallel u_{n-1}^i\parallel +|{\beta }_n-{\beta }_{n-1}|\sum _{i=1}^N{a}_{n-1}^i\parallel u_{n-1}^i\parallel \hfill \\ +{\delta }_n\parallel x_n-x_{n-1}\parallel +{\delta }_n\parallel K_n{x}_{n-1}-K_{n-1}{x}_{n-1}\parallel +|{\delta }_n-{\delta }_{n-1}|\parallel K_{n-1}{x}_{n-1}\parallel .\hfill \end{array}$$

(3.5)

From $u_n^i=T_{r_n^i}(I-r_n^{i}A)x_n$, for all $i=1,2,\dots ,N$, we have

$${\mathrm{\Phi }}_i(w_n^i,u_n^i,y)+\phi (y)-\phi \left(u_n^i\right)+\frac{1}{r_n^i}〈u_n^i-(I-r_n^{i}A)x_n,y-u_n^i〉\ge 0,\mathrm{\forall }y\in C$$

and

$$\begin{array}{c}{\mathrm{\Phi }}_i(w_{n+1}^i,u_{n+1}^i,y)+\phi (y)-\phi \left(u_{n+1}^i\right)\hfill \\ +\frac{1}{r_{n+1}^i}〈u_{n+1}^i-(I-r_{n+1}^{i}A)x_{n+1},y-u_{n+1}^i〉\ge 0,\mathrm{\forall }y\in C.\hfill \end{array}$$

In particular, we obtain

$${\mathrm{\Phi }}_i(w_n^i,u_n^i,u_{n+1}^i)+\phi \left(u_{n+1}^i\right)-\phi \left(u_n^i\right)+\frac{1}{r_n^i}〈u_n^i-(I-r_n^{i}A)x_n,u_{n+1}^i-u_n^i〉\ge 0$$

(3.6)

and

$$\begin{array}{c}{\mathrm{\Phi }}_i(w_{n+1}^i,u_{n+1}^i,u_n^i)+\phi \left(u_n^i\right)-\phi \left(u_{n+1}^i\right)\hfill \\ +\frac{1}{r_{n+1}^i}〈u_{n+1}^i-(I-r_{n+1}^{i}A)x_{n+1},u_n^i-u_{n+1}^i〉\ge 0.\hfill \end{array}$$

(3.7)

Summing up (3.6) with (3.7) and applying (3.4), we get

$$\begin{array}{c}\frac{1}{r_n^i}〈u_n^i-(I-r_n^{i}A)x_n,u_{n+1}^i-u_n^i〉\hfill \\ +\frac{1}{r_{n+1}^i}〈u_{n+1}^i-(I-r_{n+1}^{i}A)x_{n+1},u_n^i-u_{n+1}^i〉\ge 0,\hfill \end{array}$$

which implies that

$$〈u_{n+1}^i-u_n^i,\frac{u_n^i-(I-r_n^{i}A)x_n}{r_n^i}-\frac{u_{n+1}^i-(I-r_{n+1}^{i}A)x_{n+1}}{r_{n+1}^i}〉\ge 0.$$

It follows that

$$〈u_{n+1}^i-u_n^i,u_n^i-u_{n+1}^i+u_{n+1}^i-(I-r_n^{i}A)x_n-\frac{r_n^i}{r_{n+1}^i}(u_{n+1}^i-(I-r_{n+1}^{i}A)x_{n+1})〉\ge 0.$$

(3.8)

From (3.8), we obtain

$$\begin{array}{c}{\parallel u_{n+1}^i-u_n^i\parallel }^2\hfill \\ \le 〈u_{n+1}^i-u_n^i,u_{n+1}^i-(I-r_n^{i}A)x_n-\frac{r_n^i}{r_{n+1}^i}(u_{n+1}^i-(I-r_{n+1}^{i}A)x_{n+1})〉\hfill \\ =〈u_{n+1}^i-u_n^i,(I-r_{n+1}^{i}A)x_{n+1}-(I-r_n^{i}A)x_n\hfill \\ +(1-\frac{r_n^i}{r_{n+1}^i})(u_{n+1}^i-(I-r_{n+1}^{i}A)x_{n+1})〉\hfill \\ \le \parallel u_{n+1}^i-u_n^i\parallel \parallel (I-r_{n+1}^{i}A)x_{n+1}-(I-r_n^{i}A)x_n\hfill \\ +(1-\frac{r_n^i}{r_{n+1}^i})(u_{n+1}^i-(I-r_{n+1}^{i}A)x_{n+1})\parallel \hfill \\ \le \parallel u_{n+1}^i-u_n^i\parallel [\parallel (I-r_{n+1}^{i}A)x_{n+1}-(I-r_{n+1}^{i}A)x_n+(I-r_{n+1}^{i}A)x_n\hfill \\ -(I-r_n^{i}A)x_n\parallel +|1-\frac{r_n^i}{r_{n+1}^i}|\parallel u_{n+1}^i-(I-r_{n+1}^{i}A)x_{n+1}\parallel ]\hfill \\ \le \parallel u_{n+1}^i-u_n^i\parallel [\parallel (I-r_{n+1}^{i}A)x_{n+1}-(I-r_{n+1}^{i}A)x_n\parallel \hfill \\ +\parallel (I-r_{n+1}^{i}A)x_n-(I-r_n^{i}A)x_n\parallel \hfill \\ +\frac{1}{r_{n+1}^i}|r_{n+1}^i-r_n^i|\parallel u_{n+1}^i-(I-r_{n+1}^{i}A)x_{n+1}\parallel ]\hfill \\ \le \parallel u_{n+1}^i-u_n^i\parallel [\parallel x_{n+1}-x_n\parallel +|r_{n+1}^i-r_n^i|\parallel A{x}_n\parallel \hfill \\ +\frac{1}{ϵ}|r_{n+1}^i-r_n^i|\parallel u_{n+1}^i-(I-r_{n+1}^{i}A)x_{n+1}\parallel ],\hfill \end{array}$$

from which it follows that

$$\begin{array}{rcl}\parallel u_{n+1}^i-u_n^i\parallel & \le & \parallel x_{n+1}-x_n\parallel +|r_{n+1}^i-r_n^i|\parallel A{x}_n\parallel \\ +\frac{1}{ϵ}|r_{n+1}^i-r_n^i|\parallel u_{n+1}^i-(I-r_{n+1}^{i}A)x_{n+1}\parallel .\end{array}$$

(3.9)

From (3.9), we have

$$\begin{array}{rcl}\parallel u_n^i-u_{n-1}^i\parallel & \le & \parallel x_n-x_{n-1}\parallel +|r_n^i-r_{n-1}^i|\parallel A{x}_{n-1}\parallel \\ +\frac{1}{ϵ}|r_n^i-r_{n-1}^i|\parallel u_n^i-(I-r_n^{i}A)x_n\parallel .\end{array}$$

(3.10)

From (3.5) and (3.10), we obtain

$$\begin{array}{c}\parallel x_{n+1}-x_n\parallel \hfill \\ \le {\alpha }_n\xi \parallel x_n-x_{n-1}\parallel +|{\alpha }_n-{\alpha }_{n-1}|\parallel f(x_{n-1})\parallel +{\beta }_n\sum _{i=1}^N{a}_n^i[\parallel x_n-x_{n-1}\parallel \hfill \\ +|r_n^i-r_{n-1}^i|\parallel A{x}_{n-1}\parallel +\frac{1}{ϵ}|r_n^i-r_{n-1}^i|\parallel u_n^i-(I-r_n^{i}A)x_n\parallel ]\hfill \\ +{\beta }_n\sum _{i=1}^N|a_n^i-a_{n-1}^i|\parallel u_{n-1}^i\parallel +|{\beta }_n-{\beta }_{n-1}|\sum _{i=1}^N{a}_{n-1}^i\parallel u_{n-1}^i\parallel +{\delta }_n\parallel x_n-x_{n-1}\parallel \hfill \\ +{\delta }_n\parallel K_n{x}_{n-1}-K_{n-1}{x}_{n-1}\parallel +|{\delta }_n-{\delta }_{n-1}|\parallel K_{n-1}{x}_{n-1}\parallel \hfill \\ ={\alpha }_n\xi \parallel x_n-x_{n-1}\parallel +|{\alpha }_n-{\alpha }_{n-1}|\parallel f(x_{n-1})\parallel +{\beta }_n\sum _{i=1}^N{a}_n^i\parallel x_n-x_{n-1}\parallel \hfill \\ +{\beta }_n\sum _{i=1}^N{a}_n^i|r_n^i-r_{n-1}^i|\parallel A{x}_{n-1}\parallel +\frac{{\beta }_n}{ϵ}\sum _{i=1}^N{a}_n^i|r_n^i-r_{n-1}^i|\parallel u_n^i-(I-r_n^{i}A)x_n\parallel \hfill \\ +{\beta }_n\sum _{i=1}^N|a_n^i-a_{n-1}^i|\parallel u_{n-1}^i\parallel +|{\beta }_n-{\beta }_{n-1}|\sum _{i=1}^N{a}_{n-1}^i\parallel u_{n-1}^i\parallel +{\delta }_n\parallel x_n-x_{n-1}\parallel \hfill \\ +{\delta }_n\parallel K_n{x}_{n-1}-K_{n-1}{x}_{n-1}\parallel +|{\delta }_n-{\delta }_{n-1}|\parallel K_{n-1}{x}_{n-1}\parallel \hfill \\ \le (1-{\alpha }_n(1-\xi ))\parallel x_n-x_{n-1}\parallel +|{\alpha }_n-{\alpha }_{n-1}|\parallel f(x_{n-1})\parallel \hfill \\ +\sum _{i=1}^N|r_n^i-r_{n-1}^i|\parallel A{x}_{n-1}\parallel +\frac{1}{ϵ}\sum _{i=1}^N|r_n^i-r_{n-1}^i|\parallel u_n^i-(I-r_n^{i}A)x_n\parallel \hfill \\ +\sum _{i=1}^N|a_n^i-a_{n-1}^i|\parallel u_{n-1}^i\parallel +|{\beta }_n-{\beta }_{n-1}|\sum _{i=1}^N{a}_{n-1}^i\parallel u_{n-1}^i\parallel \hfill \\ +\parallel K_n{x}_{n-1}-K_{n-1}{x}_{n-1}\parallel +|{\delta }_n-{\delta }_{n-1}|\parallel K_{n-1}{x}_{n-1}\parallel .\hfill \end{array}$$

Applying the conditions (i), (v), Lemma 2.6, and Lemma 2.10(ii), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-x_n\parallel =0.$$

(3.11)

Step 4. We will show that ${lim}_{n\to \mathrm{\infty }}\parallel u_n^i-x_n\parallel ={lim}_{n\to \mathrm{\infty }}\parallel K_n{x}_n-x_n\parallel =0$, $\mathrm{\forall }i=1,2,\dots ,N$.

Since $T_{r_n^i}$ is a firmly nonexpansive mapping, for every $i=1,2,\dots ,N$, we obtain

$$\begin{array}{c}{\parallel T_{r_n^i}(I-r_n^{i}A)x_n-z\parallel }^2\hfill \\ ={\parallel T_{r_n^i}(I-r_n^{i}A)x_n-T_{r_n^i}(I-r_n^{i}A)z\parallel }^2\hfill \\ \le 〈(I-r_n^{i}A)x_n-(I-r_n^{i}A)z,u_n^i-z〉\hfill \\ =\frac{1}{2}({\parallel (I-r_n^{i}A)x_n-(I-r_n^{i}A)z\parallel }^2+{\parallel u_n^i-z\parallel }^2\hfill \\ -{\parallel (I-r_n^{i}A)x_n-(I-r_n^{i}A)z-(u_n^i-z)\parallel }^2)\hfill \\ \le \frac{1}{2}({\parallel x_n-z\parallel }^2+{\parallel u_n^i-z\parallel }^2-{\parallel (x_n-u_n^i)-r_n^i(A{x}_n-Az)\parallel }^2)\hfill \\ =\frac{1}{2}({\parallel x_n-z\parallel }^2+{\parallel u_n^i-z\parallel }^2-{\parallel x_n-u_n^i\parallel }^2-{\left(r_n^i\right)}^2{\parallel A{x}_n-Az\parallel }^2\hfill \\ +2r_n^i〈x_n-u_n^i,A{x}_n-Az〉)\hfill \\ \le \frac{1}{2}({\parallel x_n-z\parallel }^2+{\parallel u_n^i-z\parallel }^2-{\parallel x_n-u_n^i\parallel }^2+2r_n^i\parallel x_n-u_n^i\parallel \parallel A{x}_n-Az\parallel ),\hfill \end{array}$$

which implies that

$${\parallel u_n^i-z\parallel }^2\le {\parallel x_n-z\parallel }^2-{\parallel x_n-u_n^i\parallel }^2+2r_n^i\parallel x_n-u_n^i\parallel \parallel A{x}_n-Az\parallel .$$

(3.12)

From the nonexpansiveness of $T_{r_n^i}$ and $u_n^i=T_{r_n^i}(I-r_n^{i}A)x_n$, for every $i=1,2,\dots ,N$, we have

$$\begin{array}{rl}{\parallel u_n^i-z\parallel }^2& ={\parallel T_{r_n^i}(I-r_n^{i}A)x_n-T_{r_n^i}(I-r_n^{i}A)z\parallel }^2\\ \le {\parallel (x_n-z)-r_n^i(A{x}_n-Az)\parallel }^2\\ ={\parallel x_n-z\parallel }^2-2r_n^i〈x_n-z,A{x}_n-Az〉+{\left(r_n^i\right)}^2{\parallel A{x}_n-Az\parallel }^2\\ \le {\parallel x_n-z\parallel }^2-2\alpha r_n^i{\parallel A{x}_n-Az\parallel }^2+{\left(r_n^i\right)}^2{\parallel A{x}_n-Az\parallel }^2\\ ={\parallel x_n-z\parallel }^2-r_n^i(2\alpha -r_n^i){\parallel A{x}_n-Az\parallel }^2.\end{array}$$

(3.13)

From the definition of $x_n$ and (3.13), we get

$$\begin{array}{c}{\parallel x_{n+1}-z\parallel }^2\hfill \\ \le {\alpha }_n{\parallel f(x_n)-z\parallel }^2+{\beta }_n\sum _{i=1}^N{a}_n^i{\parallel u_n^i-z\parallel }^2+{\delta }_n{\parallel x_n-z\parallel }^2\hfill \\ \le {\alpha }_n{\parallel f(x_n)-z\parallel }^2+{\beta }_n\sum _{i=1}^N{a}_n^i({\parallel x_n-z\parallel }^2-r_n^i(2\alpha -r_n^i){\parallel A{x}_n-Az\parallel }^2)+{\delta }_n{\parallel x_n-z\parallel }^2\hfill \\ ={\alpha }_n{\parallel f(x_n)-z\parallel }^2+{\beta }_n\sum _{i=1}^N{a}_n^i{\parallel x_n-z\parallel }^2-{\beta }_n\sum _{i=1}^N{a}_n^i{r}_n^i(2\alpha -r_n^i){\parallel A{x}_n-Az\parallel }^2\hfill \\ +{\delta }_n{\parallel x_n-z\parallel }^2\hfill \\ \le {\parallel x_n-z\parallel }^2+{\alpha }_n{\parallel f(x_n)-z\parallel }^2-{\beta }_n\sum _{i=1}^N{a}_n^i{r}_n^i(2\alpha -r_n^i){\parallel A{x}_n-Az\parallel }^2,\hfill \end{array}$$

from which it follows that

$$\begin{array}{c}{\beta }_n\sum _{i=1}^N{a}_n^i{r}_n^i(2\alpha -r_n^i){\parallel A{x}_n-Az\parallel }^2\hfill \\ \le {\parallel x_n-z\parallel }^2-{\parallel x_{n+1}-z\parallel }^2+{\alpha }_n{\parallel f(x_n)-z\parallel }^2\hfill \\ \le (\parallel x_n-z\parallel +\parallel x_{n+1}-z\parallel )\parallel x_{n+1}-x_n\parallel +{\alpha }_n{\parallel f(x_n)-z\parallel }^2.\hfill \end{array}$$

(3.14)

From (3.11), (3.14), and the conditions (i), (ii), (iii), and (iv), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel A{x}_n-Az\parallel =0.$$

(3.15)

From the definition of $x_n$ and (3.12), we have

$$\begin{array}{c}{\parallel x_{n+1}-z\parallel }^2\hfill \\ \le {\alpha }_n{\parallel f(x_n)-z\parallel }^2+{\beta }_n\sum _{i=1}^N{a}_n^i{\parallel u_n^i-z\parallel }^2+{\delta }_n{\parallel x_n-z\parallel }^2\hfill \\ \le {\alpha }_n{\parallel f(x_n)-z\parallel }^2+{\beta }_n\sum _{i=1}^N{a}_n^i({\parallel x_n-z\parallel }^2-{\parallel x_n-u_n^i\parallel }^2\hfill \\ +2r_n^i\parallel x_n-u_n^i\parallel \parallel A{x}_n-Az\parallel )+{\delta }_n{\parallel x_n-z\parallel }^2\hfill \\ \le {\alpha }_n{\parallel f(x_n)-z\parallel }^2+{\beta }_n\sum _{i=1}^N{a}_n^i{\parallel x_n-z\parallel }^2-{\beta }_n\sum _{i=1}^N{a}_n^i{\parallel x_n-u_n^i\parallel }^2\hfill \\ +2{\beta }_n\sum _{i=1}^N{a}_n^i{r}_n^i\parallel x_n-u_n^i\parallel \parallel A{x}_n-Az\parallel +{\delta }_n{\parallel x_n-z\parallel }^2\hfill \\ \le {\parallel x_n-z\parallel }^2+{\alpha }_n{\parallel f(x_n)-z\parallel }^2-{\beta }_n\sum _{i=1}^N{a}_n^i{\parallel x_n-u_n^i\parallel }^2\hfill \\ +2{\beta }_n\sum _{i=1}^N{a}_n^i{r}_n^i\parallel x_n-u_n^i\parallel \parallel A{x}_n-Az\parallel ,\hfill \end{array}$$

which implies that

$$\begin{array}{c}{\beta }_n\sum _{i=1}^N{a}_n^i{\parallel x_n-u_n^i\parallel }^2\hfill \\ \le {\parallel x_n-z\parallel }^2-{\parallel x_{n+1}-z\parallel }^2+{\alpha }_n{\parallel f(x_n)-z\parallel }^2\hfill \\ +2{\beta }_n\sum _{i=1}^N{a}_n^i{r}_n^i\parallel x_n-u_n^i\parallel \parallel A{x}_n-Az\parallel \hfill \\ \le (\parallel x_n-z\parallel +\parallel x_{n+1}-z\parallel )\parallel x_{n+1}-x_n\parallel +{\alpha }_n{\parallel f(x_n)-z\parallel }^2\hfill \\ +2{\beta }_n\sum _{i=1}^N{a}_n^i{r}_n^i\parallel x_n-u_n^i\parallel \parallel A{x}_n-Az\parallel .\hfill \end{array}$$

(3.16)

From (3.11), (3.15), (3.16), and the conditions (i), (ii), (iii), we get

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-u_n^i\parallel =0,\text{for all }i=1,2,\dots ,N.$$

(3.17)

By the definition of $x_n$, we obtain

$$\begin{array}{rl}{x}_{n+1}-x_n& ={\alpha }_{n}f(x_n)+{\beta }_n\left(\sum _{i=1}^N{a}_n^i{u}_n^i\right)+{\delta }_n{K}_n{x}_n-x_n\\ ={\alpha }_n(f(x_n)-x_n)+{\beta }_n\sum _{i=1}^N{a}_n^i(u_n^i-x_n)+{\delta }_n(K_n{x}_n-x_n).\end{array}$$

From (3.11), (3.17), and the conditions (i) and (ii), we get

$$\underset{n\to \mathrm{\infty }}{lim}\parallel K_n{x}_n-x_n\parallel =0.$$

(3.18)

Step 5. We show that $\{x_n\}$, $\{w_n^i\}$ and $\{r_n^i\}$ are Cauchy sequences, for every $i=1,2,\dots ,N$.

Let $a\in (0,1)$, by (3.11), there exists $N\in \mathbb{N}$ such that

$$\parallel x_{n+1}-x_n\parallel <a^n,\mathrm{\forall }n\ge N.$$

(3.19)

Thus, for any $n\ge N\in \mathbb{N}$ and $p\in \mathbb{N}$, we have

$$\parallel x_{n+p}-x_n\parallel \le \sum _{k=n}^{n+p-1}\parallel x_{k+1}-x_k\parallel \le \sum _{k=n}^{n+p-1}{a}^k<\sum _{k=n}^{\mathrm{\infty }}{a}^k=\frac{a^n}{1-a}.$$

(3.20)

Since $a\in (0,1)$, we get ${lim}_{n\to \mathrm{\infty }}{a}^n=0$. From (3.20), taking $n\to \mathrm{\infty }$, we obtain $\{x_n\}$ is a Cauchy sequence in a Hilbert space H. Let ${lim}_{n\to \mathrm{\infty }}{x}_n=x^{\ast }$. Since $S_i:C\to \mathit{CB}(H)$ be ℋ-Lipschitz continuous on H with coefficients ${\mu }_i$, for every $i=1,2,\dots ,N$, and (3.1), we have

$$\begin{array}{c}\parallel w_n^i-w_{n+1}^i\parallel \hfill \\ \le (1+\frac{1}{n})\mathcal{H}(S_i(I-r_n^{i}A)x_n,S_i(I-r_{n+1}^{i}A)x_{n+1})\hfill \\ \le (1+\frac{1}{n}){\mu }_i\parallel (I-r_n^{i}A)x_n-(I-r_{n+1}^{i}A)x_{n+1}\parallel \hfill \\ \le (1+\frac{1}{n}){\mu }_i(\parallel (I-r_n^{i}A)x_n-(I-r_n^{i}A)x_{n+1}\parallel \hfill \\ +\parallel (I-r_n^{i}A)x_{n+1}-(I-r_{n+1}^{i}A)x_{n+1}\parallel )\hfill \\ \le (1+\frac{1}{n}){\mu }_i(\parallel x_n-x_{n+1}\parallel +|r_{n+1}^i-r_n^i|\parallel A{x}_{n+1}\parallel )\hfill \\ \le (1+\frac{1}{n}){\mu }_i(\parallel x_n-x_{n+1}\parallel +|r_{n+1}^i-r_n^i\left|M\right),\hfill \end{array}$$

(3.21)

where $M={max}_{n\in \mathbb{N}}\{\parallel A{x}_n\parallel \}$. From (3.11), (3.21), and the condition (v), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel w_n^i-w_{n+1}^i\parallel =0,\text{for every }i=1,2,\dots ,N.$$

By continuing the same argument as (3.19) and (3.20), we have $\{w_n^i\}$ is a Cauchy sequence in a Hilbert space H, for all $i=1,2,\dots ,N$. Let ${lim}_{n\to \mathrm{\infty }}{w}_n^i=w_i^{\ast }$, for every $i=1,2,\dots ,N$. Using the same method as above and the condition (v), we see that $\{r_n^i\}$ is a Cauchy sequence, for all $i=1,2,\dots ,N$. Put ${lim}_{n\to \mathrm{\infty }}{r}_n^i=r_i^{\ast }$, for every $i=1,2,\dots ,N$.

Next, we will prove that $w_i^{\ast }\in S_i(I-r_i^{\ast }A)x^{\ast }$, for all $i=1,2,\dots ,N$.

Since $w_n^i\in S_i(I-r_n^{i}A)x_n$, we obtain

$$\begin{array}{c}d(w_n^i,S_i(I-r_i^{\ast }A)x^{\ast })\hfill \\ \le max\{d(w_n^i,S_i(I-r_i^{\ast }A)x^{\ast }),\underset{{\tilde{w}}_i\in S_i(I-r_i^{\ast }A)x^{\ast }}{sup}d(S_i(I-r_n^{i}A)x_n,{\tilde{w}}_i)\}\hfill \\ \le max\{\underset{{\hat{w}}_i\in S_i(I-r_n^{i}A)x_n}{sup}d({\hat{w}}_i,S_i(I-r_i^{\ast }A)x^{\ast }),\underset{{\tilde{w}}_i\in S_i(I-r_i^{\ast }A)x^{\ast }}{sup}d(S_i(I-r_n^{i}A)x_n,{\tilde{w}}_i)\}\hfill \\ =\mathcal{H}(S_i(I-r_n^{i}A)x_n,S_i(I-r_i^{\ast }A)x^{\ast }),\text{for every }i=1,2,\dots ,N.\hfill \end{array}$$

(3.22)

Since

$$\begin{array}{rl}d(w_i^{\ast },S_i(I-r_i^{\ast }A)x^{\ast })& \le \parallel w_i^{\ast }-w_n^i\parallel +d(w_n^i,S_i(I-r_i^{\ast }A)x^{\ast })\\ \le \parallel w_i^{\ast }-w_n^i\parallel +\mathcal{H}(S_i(I-r_n^{i}A)x_n,S_i(I-r_i^{\ast }A)x^{\ast })\\ \le \parallel w_i^{\ast }-w_n^i\parallel +{\mu }_i\parallel (I-r_n^{i}A)x_n-(I-r_i^{\ast }A)x^{\ast }\parallel \\ =\parallel w_i^{\ast }-w_n^i\parallel +{\mu }_i\parallel (x_n-x^{\ast })-(r_n^{i}A{x}_n-r_i^{\ast }A{x}^{\ast })\parallel ,\end{array}$$

taking $n\to \mathrm{\infty }$, we have

$$d(w_i^{\ast },S_i(I-r_i^{\ast }A)x^{\ast })=0,$$

which implies that

$$w_i^{\ast }\in S_i(I-r_i^{\ast }A)x^{\ast },\text{for all }i=1,2,\dots ,N.$$

(3.23)

Step 6. We will show that ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}〈f(q)-q,x_n-q〉\le 0$, where $q=P_{\mathcal{F}}f(q)$.

To show this, choose a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈f(q)-q,x_n-q〉=\underset{k\to \mathrm{\infty }}{lim}〈f(q)-q,x_{n_k}-q〉.$$

Without loss of generality, we can assume that $x_{n_k}\rightharpoonup \tilde{x}$ as $k\to \mathrm{\infty }$.

For every $i=1,2,\dots ,N$, $0<\varphi \le {\lambda }_i^n\le \psi <{\gamma }_2<1$, for all $i=1,2,\dots ,N$, without loss of generality, we may assume that

$${\lambda }_i^{n_k}\to {\lambda }_i\in (0,1)\text{as }k\to \mathrm{\infty },\text{for every }i=1,2,\dots ,N.$$

Let K be the K-mapping generated by $T_1,T_2,\dots ,T_N$ and ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_N$. By Lemma 2.9, we see that K is nonexpansive and $F(K)={\bigcap }_{i=1}^{N}F(T_i)$.

From Lemma 2.10(i), we obtain

$$\underset{k\to \mathrm{\infty }}{lim}\parallel K_{n_k}{x}_{n_k}-K{x}_{n_k}\parallel =0.$$

(3.24)

Since

$$\parallel x_{n_k}-K{x}_{n_k}\parallel \le \parallel x_{n_k}-K_{n_k}{x}_{n_k}\parallel +\parallel K_{n_k}{x}_{n_k}-K{x}_{n_k}\parallel ,$$

by (3.18) and (3.24), we have

$$\underset{k\to \mathrm{\infty }}{lim}\parallel x_{n_k}-K{x}_{n_k}\parallel =0.$$

(3.25)

Since $x_{n_k}\rightharpoonup \tilde{x}$ as $n\to \mathrm{\infty }$, by (3.25) and Lemma 2.4, we have

$$\tilde{x}\in F(K)=\bigcap _{i=1}^{N}F(T_i).$$

(3.26)

Next, we show that $x^{\ast }\in {\bigcap }_{i=1}^N{(\mathit{GEP})}_s({\mathrm{\Phi }}_i,\phi ,A)$.

Since $x_{n_k}\to x^{\ast }$ as $k\to \mathrm{\infty }$ and (3.17), we have

$$u_{n_k}^i\to x^{\ast }\text{as }k\to \mathrm{\infty },\text{for all }i=1,2,\dots ,N.$$

(3.27)

From (3.1), we obtain

$${\mathrm{\Phi }}_i(w_{n_k}^i,u_{n_k}^i,y)+\phi (y)-\phi \left(u_{n_k}^i\right)+\frac{1}{r_{n_k}^i}〈u_{n_k}^i-x_{n_k},y-u_{n_k}^i〉+〈A{x}_{n_k},y-u_{n_k}^i〉\ge 0,$$

for every $y\in C$ and $i=1,2,\dots ,N$. From (3.17), (3.27), the condition (H1), and the lower semicontinuity of φ, we get

$${\mathrm{\Phi }}_i(w_i^{\ast },x^{\ast },y)+\phi (y)-\phi \left(x^{\ast }\right)+〈A{x}^{\ast },y-x^{\ast }〉\ge 0,$$

for every $y\in C$ and $i=1,2,\dots ,N$, from which it follows by (3.23) that

$$x^{\ast }\in {(\mathit{GEP})}_s({\mathrm{\Phi }}_i,\phi ,A),\text{for every }i=1,2,\dots ,N.$$

It implies that

$$x^{\ast }\in \bigcap _{i=1}^N{(\mathit{GEP})}_s({\mathrm{\Phi }}_i,\phi ,A).$$

(3.28)

Since $x_{n_k}\rightharpoonup \tilde{x}$ and $x_{n_k}\to x^{\ast }$ as $n\to \mathrm{\infty }$, then $\tilde{x}=x^{\ast }$. From (3.26) and (3.28), we have

$$x^{\ast }\in \mathcal{F}.$$

(3.29)

Indeed, since $x_{n_k}\to x^{\ast }$ as $k\to \mathrm{\infty }$, by (3.29) and Lemma 2.3, we obtain

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈f(q)-q,x_n-q〉=\underset{k\to \mathrm{\infty }}{lim}〈f(q)-q,x_{n_k}-q〉=〈f(q)-q,x^{\ast }-q〉\le 0.$$

(3.30)

Step 7. Finally, we will prove that $\{x_n\}$ and $\{u_n^i\}$ converges strongly to $q=P_{\mathcal{F}}f(q)$, for every $i=1,2,\dots ,N$.

By Lemma 2.1(ii), we have

$$\begin{array}{c}{\parallel x_{n+1}-q\parallel }^2\hfill \\ ={\parallel {\alpha }_n(f(x_n)-q)+{\beta }_n\sum _{i=1}^N{a}_n^i(u_n^i-q)+{\delta }_n(K_n{x}_n-q)\parallel }^2\hfill \\ \le {\parallel {\beta }_n\sum _{i=1}^N{a}_n^i(u_n^i-q)+{\delta }_n(K_n{x}_n-q)\parallel }^2+2{\alpha }_n〈f(x_n)-q,x_{n+1}-q〉\hfill \\ \le {({\beta }_n\parallel \sum _{i=1}^N{a}_n^i(u_n^i-q)\parallel +{\delta }_n\parallel K_n{x}_n-q\parallel )}^2\hfill \\ +2{\alpha }_n〈f(x_n)-f(q),x_{n+1}-q〉+2{\alpha }_n〈f(q)-q,x_{n+1}-q〉\hfill \\ \le {({\beta }_n\sum _{i=1}^N{a}_n^i\parallel x_n-q\parallel +{\delta }_n\parallel x_n-q\parallel )}^2+2{\alpha }_n\parallel f(x_n)-f(q)\parallel \parallel x_{n+1}-q\parallel \hfill \\ +2{\alpha }_n〈f(q)-q,x_{n+1}-q〉\hfill \\ \le {((1-{\alpha }_n)\parallel x_n-q\parallel )}^2+2{\alpha }_n\xi \parallel x_n-q\parallel \parallel x_{n+1}-q\parallel \hfill \\ +2{\alpha }_n〈f(q)-q,x_{n+1}-q〉\hfill \\ \le {(1-{\alpha }_n)}^2{\parallel x_n-q\parallel }^2+{\alpha }_n\xi ({\parallel x_n-q\parallel }^2+{\parallel x_{n+1}-q\parallel }^2)\hfill \\ +2{\alpha }_n〈f(q)-q,x_{n+1}-q〉,\hfill \end{array}$$

which implies that

$$\begin{array}{c}{\parallel x_{n+1}-q\parallel }^2\hfill \\ \le \frac{{(1-{\alpha }_n)}^2+{\alpha }_n\xi }{1-{\alpha }_n\xi }{\parallel x_n-q\parallel }^2+\frac{2{\alpha }_n}{1-{\alpha }_n\xi }〈f(q)-q,x_{n+1}-q〉\hfill \\ =\frac{1-{\alpha }_n\xi -2{\alpha }_n(1-\xi )}{1-{\alpha }_n\xi }{\parallel x_n-q\parallel }^2+\frac{{\alpha }_n^2}{1-{\alpha }_n\xi }{\parallel x_n-q\parallel }^2\hfill \\ +\frac{2{\alpha }_n}{1-{\alpha }_n\xi }〈f(q)-q,x_{n+1}-q〉\hfill \\ =(1-\frac{2{\alpha }_n(1-\xi )}{1-{\alpha }_n\xi }){\parallel x_n-q\parallel }^2+\frac{{\alpha }_n^2}{1-{\alpha }_n\xi }{\parallel x_n-q\parallel }^2\hfill \\ +\frac{2{\alpha }_n}{1-{\alpha }_n\xi }〈f(q)-q,x_{n+1}-q〉\hfill \\ =(1-\frac{2{\alpha }_n(1-\xi )}{1-{\alpha }_n\xi }){\parallel x_n-q\parallel }^2+\frac{2{\alpha }_n(1-\xi )}{1-{\alpha }_n\xi }(\frac{{\alpha }_n}{2(1-\xi )}{\parallel x_n-q\parallel }^2\hfill \\ +\frac{1}{1-\xi }〈f(q)-q,x_{n+1}-q〉).\hfill \end{array}$$

Applying the condition (i), (3.30), and Lemma 2.6, we have the sequence $\{x_n\}$ converges strongly to $q=P_{\mathcal{F}}f(q)$. From (3.17), we also obtain $\{u_n^i\}$ converges strongly to $q=P_{\mathcal{F}}f(q)$, for every $i=1,2,\dots ,N$. This completes the proof. □

The following corollaries are consequences which are applied by Theorem 3.1. Therefore, we omit the proof.

Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. For every $i=1,2,\dots ,N$, $S_i:C\to \mathit{CB}(H)$ be ℋ-Lipschitz continuous with coefficients ${\mu }_i$, ${\mathrm{\Phi }}_i:H\times C\times C\to \mathbb{R}$ 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 $T:C\to C$ be κ-strictly pseudo-contractive mapping with $\kappa \le {\gamma }_1$ and $\mathcal{F}:=F(T)\cap {\bigcap }_{i=1}^N{(\mathit{MGEP})}_s({\mathrm{\Phi }}_i,\phi ,A)\ne \mathrm{\varnothing }$. For every $n\in \mathbb{N}$, let $\{{\lambda }_n\}$ be a sequence of real numbers where $0<{\lambda }_n<{\gamma }_2$ and ${\gamma }_1+{\gamma }_2<1$. For every $i=1,2,\dots ,N$, let $\{x_n\}$ be the sequence generated by $x_1\in C$ and $w_1^i\in S_i(I-r_1^{i}A)x_1$, there exist sequences $\{w_n^i\}\in H$ and $\{x_n\},\{u_n^i\}\subseteq C$ such that

$$\{\begin{array}{c}\parallel w_n^i-w_{n+1}^i\parallel \le (1+\frac{1}{n})\mathcal{H}(S_i(I-r_n^{i}A)x_n,S_i(I-r_{n+1}^{i}A)x_{n+1}),\hfill \\ w_n^i\in S_i(I-r_n^{i}A)x_n\hfill \\ {\mathrm{\Phi }}_i(w_n^i,u_n^i,y)+\phi (y)-\phi (u_n^i)+\frac{1}{r_n^i}〈u_n^i-x_n,y-u_n^i〉+〈A{x}_n,y-u_n^i〉\hfill \\ \ge 0,\mathrm{\forall }y\in C,\hfill \\ x_{n+1}={\alpha }_{n}f(x_n)+{\beta }_n({\sum }_{i=1}^N{a}_n^i{u}_n^i)+{\delta }_n({\lambda }_{n}T+(1-{\lambda }_n)I)x_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.31)

where $f:C\to C$ be a contraction mapping with a constant ξ and $\{{\alpha }_n\},\{{\beta }_n\},\{{\delta }_n\}\subseteq (0,1)$ with ${\alpha }_n+{\beta }_n+{\delta }_n=1$, $\mathrm{\forall }n\ge 1$. Suppose the following conditions hold:

• (i) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (ii) $0<\tau \le {\beta }_n,{\delta }_n\le \upsilon <1$;

• (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 ${\sum }_{n=1}^N{a}_n^i=1$;

• (iv) $0<ϵ\le r_n^i\le \omega <2\alpha$, for all $n\in \mathbb{N}$ and $i=1,2,\dots ,N$;

• (v) ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|r_{n+1}^i-r_n^i|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|a_{n+1}^i-a_n^i|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_{n+1}-{\lambda }_n|<\mathrm{\infty }$, for all $i=1,2,\dots ,N$;

• (vi) for each $i=1,2,\dots ,N$, there exists ${\rho }_i>0$ such that

  $$\begin{array}{c}{\mathrm{\Phi }}_i(w_1^i,T_{r_1^i}(x_1),T_{r_2^i}(x_2))+{\mathrm{\Phi }}_i(w_2^i,T_{r_2^i}(x_2),T_{r_1^i}(x_1))\hfill \\ \le -{\rho }_i{\parallel T_{r_1^i}(x_1)-T_{r_2^i}(x_2)\parallel }^2,\hfill \end{array}$$

  (3.32)

for all $(r_1^i,r_2^i)\in {\mathrm{\Theta }}_i\times {\mathrm{\Theta }}_i,(x_1,x_2)\in C\times C$ and $w_j^i\in S_i(x_j)$, for $j=1,2$, where ${\mathrm{\Theta }}_i=\{r_n^i:n\ge 1\}$.

Then $\{x_n\}$ and $\{u_n^i\}$ converges strongly to $q=P_{\mathcal{F}}f(q)$, for every $i=1,2,\dots ,N$.

Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H. For every $i=1,2,\dots ,N$, $S_i:C\to \mathit{CB}(H)$ be ℋ-Lipschitz continuous with coefficients ${\mu }_i$, ${\mathrm{\Phi }}_i:H\times C\times C\to \mathbb{R}$ be equilibrium-like function satisfying (H1)-(H3). Let $\phi :C\to \mathbb{R}$ be a lower semicontinuous and convex function. Let ${\{T_i\}}_{i=1}^N$ be a finite family of ${\kappa }_i$-strictly pseudo-contractive mappings and ${\kappa }_i\le {\gamma }_1$ with $\mathcal{F}:={\bigcap }_{i=1}^{N}F(T_i)\cap {\bigcap }_{i=1}^N{(\mathit{GEP})}_s({\mathrm{\Phi }}_i,\phi )\ne \mathrm{\varnothing }$. 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 $0<\varphi \le {\lambda }_i^n\le \psi <{\gamma }_2<1$, for all $i=1,2,\dots ,N$ and ${\gamma }_1+{\gamma }_2<1$. For every $i=1,2,\dots ,N$, let $\{x_n\}$ be the sequence generated by $x_1\in C$ and $w_1^i\in S_i(x_1)$, there exist sequences $\{w_n^i\}\in H$ and $\{x_n\},\{u_n^i\}\subseteq C$ such that

$$\{\begin{array}{c}{w}_n^i\in S_i(x_n),\parallel w_n^i-w_{n+1}^i\parallel \le (1+\frac{1}{n})\mathcal{H}(S_i(x_n),S_i(x_{n+1})),\hfill \\ {\mathrm{\Phi }}_i(w_n^i,u_n^i,y)+\phi (y)-\phi (u_n^i)+\frac{1}{r_n^i}〈u_n^i-x_n,y-u_n^i〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ x_{n+1}={\alpha }_{n}f(x_n)+{\beta }_n({\sum }_{i=1}^N{a}_n^i{u}_n^i)+{\delta }_n{K}_n{x}_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.33)

where $f:C\to C$ is a contraction mapping with a constant ξ and $\{{\alpha }_n\},\{{\beta }_n\},\{{\delta }_n\}\subseteq (0,1)$ with ${\alpha }_n+{\beta }_n+{\delta }_n=1$, $\mathrm{\forall }n\ge 1$. Suppose the following conditions hold:

• (i) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (ii) $0<\tau \le {\beta }_n,{\delta }_n\le \upsilon <1$;

• (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 ${\sum }_{n=1}^N{a}_n^i=1$;

• (iv) $0<ϵ\le r_n^i\le \omega <1$, for all $n\in \mathbb{N}$ and $i=1,2,\dots ,N$;

• (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 $i=1,2,\dots ,N$;

• (vi) for each $i=1,2,\dots ,N$, there exists ${\rho }_i>0$ such that

  $$\begin{array}{c}{\mathrm{\Phi }}_i(w_1^i,T_{r_1^i}(x_1),T_{r_2^i}(x_2))+{\mathrm{\Phi }}_i(w_2^i,T_{r_2^i}(x_2),T_{r_1^i}(x_1))\hfill \\ \le -{\rho }_i{\parallel T_{r_1^i}(x_1)-T_{r_2^i}(x_2)\parallel }^2,\hfill \end{array}$$

  (3.34)

for all $(r_1^i,r_2^i)\in {\mathrm{\Theta }}_i\times {\mathrm{\Theta }}_i,(x_1,x_2)\in C\times C$ and $w_j^i\in S_i(x_j)$, for $j=1,2$, where ${\mathrm{\Theta }}_i=\{r_n^i:n\ge 1\}$.

Then $\{x_n\}$ and $\{u_n^i\}$ converges strongly to $q=P_{\mathcal{F}}f(q)$, for every $i=1,2,\dots ,N$.

Remark 3.4 From Corollary 3.3, put $N=1$, then the iterative scheme (3.33) reduces to

$$\{\begin{array}{c}{w}_n^1\in S_1(x_n),\parallel w_n^1-w_{n+1}^1\parallel \le (1+\frac{1}{n})\mathcal{H}(S_1(x_n),S_1(x_{n+1})),\hfill \\ {\mathrm{\Phi }}_1(w_n^1,u_n^1,y)+\phi (y)-\phi (u_n^1)+\frac{1}{r_n^1}〈u_n^1-x_n,y-u_n^1〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ x_{n+1}={\alpha }_{n}f(x_n)+{\beta }_n{u}_n^1+{\delta }_n({\lambda }_1^n{T}_1+(1-{\lambda }_1^n)I)x_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

which is a modification of iterative scheme (1.4) in the results of Ceng et al. [16]. By assuming the initial condition $x_1\in C$, $w_1^1\in S_1(x_1)$ and the following conditions hold:

• (i) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (ii) $0<\tau \le {\beta }_n,{\delta }_n\le \upsilon <1$;

• (iii) $0<ϵ\le r_n^1\le \omega <1$, for all $n\in \mathbb{N}$;

• (iv) ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|r_{n+1}^1-r_n^1|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_1^{n+1}-{\lambda }_1^n|<\mathrm{\infty }$;

• (v) there exists ${\rho }_1>0$ such that

  $$\begin{array}{c}{\mathrm{\Phi }}_1(w_1^1,T_{r_1^1}(x_1),T_{r_2^1}(x_2))+{\mathrm{\Phi }}_1(w_2^1,T_{r_2^1}(x_2),T_{r_1^1}(x_1))\hfill \\ \le -{\rho }_1{\parallel T_{r_1^1}(x_1)-T_{r_2^1}(x_2)\parallel }^2,\hfill \end{array}$$

for all $(r_1^1,r_2^1)\in {\mathrm{\Theta }}_1\times {\mathrm{\Theta }}_1,(x_1,x_2)\in C\times C$ and $w_j^1\in S_1(x_j)$, for $j=1,2$, where ${\mathrm{\Theta }}_1=\{r_n^1:n\ge 1\}$.

Then $\{x_n\}$ and $\{u_n^1\}$ converge strongly to $q=P_{\mathcal{F}}f(q)$.