Algorithm of a new variational inclusion problem and strictly pseudononspreading mapping with application

Abstract

The purpose of this research is to modify the variational inclusion problems and prove a strong convergence theorem for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problems in Hilbert space. By using our main result, we prove a strong convergence theorem involving a κ-quasi-strictly pseudo-contractive mapping in Hilbert space. We give a numerical example to support some of our results.

1 Introduction

Throughout this article, let H be a real Hilbert space with inner product $〈\cdot ,\cdot 〉$ and norm $\parallel \cdot \parallel$. Let C be a nonempty closed convex subset of H. Let $\mathcal{S}:C\to C$ be a nonlinear mapping. A point $u\in C$ is called a fixed point of $\mathcal{S}u=u$ if $\mathcal{S}u=u$. The set of fixed points of $\mathcal{S}u=u$ is denoted by $Fix(\mathcal{S}):=\{u\in C:\mathcal{S}u=u\}$. A mapping $\mathcal{S}u=u$ is called nonexpansive if

$$\parallel \mathcal{S}u-\mathcal{S}v\parallel \le \parallel u-v\parallel ,\mathrm{\forall }u,v\in C.$$

In 2008, Kohsaka and Takahashi [1] introduced the nonspreading mapping in Hilbert space H as follows:

$$2{\parallel \mathcal{S}u-\mathcal{S}v\parallel }^2\le {\parallel \mathcal{S}u-v\parallel }^2+{\parallel u-\mathcal{S}v\parallel }^2,\mathrm{\forall }u,v\in C.$$

(1.1)

It is shown in [2] that (1.1) is equivalent to

$${\parallel \mathcal{S}u-\mathcal{S}v\parallel }^2\le {\parallel u-v\parallel }^2+2〈u-\mathcal{S}u,v-\mathcal{S}v〉,\mathrm{\forall }u,v\in C.$$

(1.2)

The mapping $\mathcal{S}:C\to C$ is called a κ-strictly pseudononspreading mapping if there exists $\kappa \in [0,1)$ such that

$${\parallel \mathcal{S}u-\mathcal{S}v\parallel }^2\le {\parallel u-v\parallel }^2+\kappa {\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2+2〈u-\mathcal{S}u,v-\mathcal{S}v〉,\mathrm{\forall }u,v\in C.$$

This mapping was introduced by Osilike and Isiogugu [3] in 2011. Clearly every nonspreading mapping is a κ-strictly pseudononspreading mapping.

Remark 1.1 Let C be a nonempty closed convex subset of H. Then a mapping $\mathcal{S}:C\to C$ is a κ-strictly pseudononspreading if and only if

$$\frac{1-\kappa }{2}{\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2\le 〈(I-\mathcal{S})u-(I-\mathcal{S})v,u-v〉+〈(I-\mathcal{S})u,(I-\mathcal{S})v〉,$$

for all $u,v\in C$.

Proof Let $u,v\in C$ and $\mathcal{S}u=u$ be a κ-strictly pseudononspreading mapping, then there exists $\kappa \in [0,1)$ such that

$${\parallel \mathcal{S}u-\mathcal{S}v\parallel }^2\le {\parallel u-v\parallel }^2+\kappa {\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2+2〈u-\mathcal{S}u,v-\mathcal{S}v〉.$$

(1.3)

Since

$${\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2={\parallel \mathcal{S}u-\mathcal{S}v\parallel }^2-2〈\mathcal{S}u-\mathcal{S}v,u-v〉+{\parallel u-v\parallel }^2,$$

(1.4)

then we have

$${\parallel \mathcal{S}u-\mathcal{S}v\parallel }^2={\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2+2〈\mathcal{S}u-\mathcal{S}v,u-v〉-{\parallel u-v\parallel }^2.$$

(1.5)

From (1.3) and (1.5), we have

$$\begin{array}{c}{\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2+2〈\mathcal{S}u-\mathcal{S}v,u-v〉-{\parallel u-v\parallel }^2\hfill \\ \le {\parallel u-v\parallel }^2+\kappa {\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2+2〈u-\mathcal{S}u,v-\mathcal{S}v〉.\hfill \end{array}$$

It follows that

$$\begin{array}{rl}(1-\kappa ){\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2& \le 2{\parallel u-v\parallel }^2-2〈\mathcal{S}u-\mathcal{S}v,u-v〉+2〈u-\mathcal{S}u,v-\mathcal{S}v〉\\ =2〈(I-\mathcal{S})u-(I-\mathcal{S})v,u-v〉+2〈u-\mathcal{S}u,v-\mathcal{S}v〉.\end{array}$$

Then

$$\frac{1-\kappa }{2}{\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2\le 〈(I-\mathcal{S})u-(I-\mathcal{S})v,u-v〉+〈(I-\mathcal{S})u,(I-\mathcal{S})v〉.$$

On the other hand, let $u,v\in C$ and

$$\begin{array}{rl}\frac{1-\kappa }{2}{\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2& \le 〈(I-\mathcal{S})u-(I-\mathcal{S})v,u-v〉+〈u-\mathcal{S}u,v-\mathcal{S}v〉\\ ={\parallel u-v\parallel }^2-〈\mathcal{S}u-\mathcal{S}v,u-v〉+〈u-\mathcal{S}u,v-\mathcal{S}v〉.\end{array}$$

Then

$$(1-\kappa ){\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2\le 2{\parallel u-v\parallel }^2-2〈\mathcal{S}u-\mathcal{S}v,u-v〉+2〈u-\mathcal{S}u,v-\mathcal{S}v〉.$$

It follows that

$$\begin{array}{rcl}2〈\mathcal{S}u-\mathcal{S}u,u-v〉& \le & 2{\parallel u-v\parallel }^2-(1-\kappa ){\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2\\ +2〈u-\mathcal{S}u,v-\mathcal{S}v〉.\end{array}$$

(1.6)

From (1.4), we have

$$2〈\mathcal{S}u-\mathcal{S}v,u-v〉={\parallel \mathcal{S}u-\mathcal{S}v\parallel }^2+{\parallel u-v\parallel }^2-{\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2.$$

(1.7)

From (1.6) and (1.7), we have

$$\begin{array}{c}{\parallel \mathcal{S}u-\mathcal{S}v\parallel }^2+{\parallel u-v\parallel }^2-{\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2\hfill \\ \le 2{\parallel u-v\parallel }^2-(1-\kappa ){\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2+2〈u-\mathcal{S}u,v-\mathcal{S}v〉.\hfill \end{array}$$

Then

$${\parallel \mathcal{S}u-\mathcal{S}v\parallel }^2\le {\parallel u-v\parallel }^2+\kappa {\parallel (I-\mathcal{S})u-(I-\mathcal{S})v\parallel }^2+2〈(I-\mathcal{S})u,(I-\mathcal{S})v〉.$$

 □

Example 1.2 Let $\mathcal{S}:[1,\mathrm{\infty })\to [1,\mathrm{\infty })$ be defined by

$$\mathcal{S}u=sinu,\mathrm{\forall }u\in [1,\mathrm{\infty }).$$

Then $\mathcal{S}u=u$ is a κ-strictly pseudononspreading mapping where $\kappa \in [0,1)$.

Example 1.3 Let $\mathcal{S}:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be defined by

$$\mathcal{S}u=\frac{4u^2}{5+4u},\mathrm{\forall }u\in C.$$

Then $\mathcal{S}u=u$ is a $\frac{23}{25}$-strictly pseudononspreading mapping.

The mapping $A:C\to H$ is called α-inverse strongly monotone if there exists a positive real number α such that

$$〈Au-Av,u-v〉\ge \alpha {\parallel Au-Av\parallel }^2,$$

for all $u,v\in C$.

Let $B:H\to H$ be a mapping and $M:H\to 2^H$ be a multi-valued mapping. The variational inclusion problem is to find $x\in H$ such that

$$\theta \in Bx+Mx,$$

(1.8)

where θ is a zero vector in H. The set of the solution of (1.8) is denoted by $VI(H,B,M)$. It is well known that the variational inclusion problems are widely studied in mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, and game theory, etc. Many authors have increasingly investigated such a problem (1.8); see for instance [4–7] and references therein.

Let $M:H\to 2^H$ be a multi-valued maximal monotone mapping, then the single-valued mapping $J_{M,\lambda }:H\to H$ defined by

$$J_{M,\lambda }(x)={(I+\lambda M)}^{-1}(x),\mathrm{\forall }x\in H,$$

is called the resolvent operator associated with M, where λ is any positive number and I is an identity mapping; see [7].

Let $\mathrm{\Psi }:C\times C\to \mathbb{R}$ be a bifunction. The equilibrium problem for Ψ is to determine its equilibrium point. The set of solution of equilibrium problem is denoted by

$$EP(\mathrm{\Psi })=\{u\in C:\mathrm{\Psi }(u,v)\ge 0,\mathrm{\forall }v\in C\}.$$

(1.9)

Finding a solution of an equilibrium problem can be applied to many problems in physics, optimization, and economics. Many researchers have proposed some methods to solve the equilibrium problem; see, for example, [8, 9] and the references therein.

In 2008, Zhang et al. [7] introduced an iterative scheme for finding a common element of the set of solutions of the variational inclusion problem with multi-valued maximal monotone mapping and inverse strongly monotone mappings and the set of fixed points of nonexpansive mappings in Hilbert space as follows:

$$\{\begin{array}{c}{v}_n=J_{M,\lambda }(w_n-\lambda A{w}_n),\hfill \\ w_{n+1}={\alpha }_{n}w+(1-{\alpha }_n)S{v}_n,\mathrm{\forall }n\ge 0,\hfill \end{array}$$

and they proved a strong convergence theorem of the sequence $\{w_n\}$ under suitable conditions of the parameters $\{{\alpha }_n\}$ and λ.

In 2013, Kangtunyakarn [10] introduced an iterative algorithm for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inequality problems as follows:

$$w_{n+1}={\alpha }_{n}u+{\beta }_n{P}_C(I-{\lambda }_n(I-\mathcal{S}))w_n+{\gamma }_{n}S{w}_n,\mathrm{\forall }n\in \mathbb{N},$$

and proved a strong convergence theorem of the sequence $\{w_n\}$ under suitable conditions of the parameters $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, and $\{{\lambda }_n\}$.

Very recently, Suwannaut and Kangtunyakarn [11] have modified (1.9) as follows:

$$EP\left(\sum _{i=1}^N{a}_i{\mathrm{\Psi }}_i\right)=\{u\in C:\left(\sum _{i=1}^N{a}_i{\mathrm{\Psi }}_i\right)(u,v)\ge 0,\mathrm{\forall }v\in C\},$$

(1.10)

where ${\mathrm{\Psi }}_i:C\times C\to \mathbb{R}$ is for bifunctions and $a_i>0$ with ${\sum }_{i=1}^N{a}_i=1$ for every $i=1,2,\dots ,N$. It is obvious that (1.10) reduces to (1.9), if ${\mathrm{\Psi }}_i=\mathrm{\Psi }$, for all $i=1,2,\dots ,N$. They also introduced an iterative method for finding a common element of the set of fixed points of an infinite family of ${\kappa }_i$-strictly pseudo-contractive mappings and the set of solutions of a finite family of an equilibrium problem and a variational inequalities problem as follows:

$$\{\begin{array}{c}{\sum }_{i=1}^N{a}_i{\mathrm{\Psi }}_i(z_n,y)+\frac{1}{r_n}〈y-z_n,z_n-w_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ w_{n+1}={\beta }_n({\alpha }_n\mu +(1-{\alpha }_n)S_n{w}_n)+(1-{\beta }_n)P_C(I-{\rho }_n{\sum }_{i=1}^N{b}_i{A}_i)z_n,\mathrm{\forall }n\ge 1.\hfill \end{array}$$

Under some appropriate conditions, they proved a strong convergence theorem of the sequence $\{w_n\}$ converging to an element of a set ${\bigcap }_{i=1}^{N}EP({\mathrm{\Psi }}_i)\cap {\bigcap }_{i=1}^{N}VP(A_i,C)\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix({\mathcal{S}}_i)$ where $A_i$ is a strongly positive linear bounded operator for every $i=1,2,\dots ,N$.

For $i=1,2,\dots ,N$, let $A_i:H\to H$ be a single-valued mapping and let $M:H\to 2^H$ be a multi-valued mapping. From the concept of (1.8), we introduce the problem of finding $u\in H$ such that

$$\theta \in \sum _{i=1}^N{a}_i{A}_{i}u+Mu,$$

(1.11)

for all $a_i\in (0,1)$ with ${\sum }_{i=1}^N{a}_i=1$ and θ is a zero vector. This problem is called the modified variational inclusion. The set of solutions of (1.11) is denoted by $VI(H,{\sum }_{i=1}^N{a}_i{A}_i,M)$. If $A_i\equiv A$ for all $i=1,2,\dots ,N$, then (1.11) reduces to (1.8).

In this paper, motivated by the research described above, we prove fixed point theory involving the modified variational inclusion and introduce iterative scheme for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problem. By using the same method as our main theorem, we prove a strong convergence theorem for finding a common element of the set of fixed points of a κ-quasi-strictly pseudo-contractive mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problem in Hilbert space. Applying such a problem, we have a convergence theorem associated with a nonspreading mapping. In the last section, we also give numerical examples to support some of our results.

2 Preliminaries

In this paper, we denote weak and strong convergence by the notations ‘⇀’ and ‘→’, respectively. In a real Hilbert space H, recall that the (nearest point) projection $P_C$ from H onto C assigns to each $u\in H$ the unique point $P_{C}u\in C$ satisfying the property

$$\parallel u-P_{C}v\parallel =\underset{v\in C}{min}\parallel u-v\parallel .$$

For a proof of the main theorem, we will use the following lemmas.

Lemma 2.1 ([12])

Given $u\in H$ and $v\in C$, then $P_{C}u=v$ if and only if we have the inequality

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

Lemma 2.2 ([13])

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

$$p_{n+1}\le (1-a_n)p_n+b_n,\mathrm{\forall }n\ge 0,$$

where $\{a_n\}$ is a sequence in $(0,1)$ and $\{b_n\}$ is a sequence such that

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

• (2) ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\frac{b_n}{a_n}\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}|b_n|<\mathrm{\infty }$.

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

Lemma 2.3 Let H be a real Hilbert space. Then

$${\parallel u+v\parallel }^2\le {\parallel u\parallel }^2+2〈v,u+v〉,$$

for all $u,v\in H$.

Lemma 2.4 ([14])

Let H be a Hilbert space. Then for all $u,v\in H$ and ${\alpha }_i\in [0,1]$ for $i=1,2,\dots ,n$ such that ${\alpha }_0+{\alpha }_1+\cdots +{\alpha }_n=1$ the following equality holds:

$${\parallel {\alpha }_0{w}_0+{\alpha }_1{w}_1+\cdots +{\alpha }_n{w}_n\parallel }^2=\sum _{i=0}^n{\alpha }_i{\parallel w_i\parallel }^2-\sum _{0\le i,j\le n}{\alpha }_i{\alpha }_j{\parallel w_i-w_j\parallel }^2.$$

For finding solutions of the equilibrium problem, assume a bifunction $\mathrm{\Psi }:C\times C\to \mathbb{R}$ to satisfy the following conditions:

• (A1) $\mathrm{\Psi }(u,v)=0$ for all $u\in C$;

• (A2) Ψ is monotone, i.e., $\mathrm{\Psi }(u,v)+\mathrm{\Psi }(v,u)\le 0$ for all $u,v\in C$;

• (A3) for each $u,v,z\in C$,

  $$\underset{t\downarrow 0}{lim}\mathrm{\Psi }(tz+(1-t)u,v)\le \mathrm{\Psi }(u,v);$$

• (A4) for each $u\in C$, $v\mapsto \mathrm{\Psi }(u,v)$ is convex and lower semicontinuous.

Lemma 2.5 ([15])

Let C be a nonempty closed convex subset of H and let Ψ be a bifunction of $C\times C$ into ℝ satisfying (A1)-(A4). Let $r>0$ and $u\in H$. Then there exists $z\in C$ such that

$$\mathrm{\Psi }(z,v)+\frac{1}{r}〈v-z,z-u〉\ge 0,\mathrm{\forall }v\in C.$$

Lemma 2.6 ([8])

Assume that $\mathrm{\Psi }:C\times C\to \mathbb{R}$ satisfies (A1)-(A4). For $r>0$, define a mapping ${\mathrm{\Theta }}_r:H\to C$ as follows:

$${\mathrm{\Theta }}_r(u)=\{z\in C:\mathrm{\Psi }(z,v)+\frac{1}{r}〈v-z,z-u〉\ge 0,\mathrm{\forall }v\in C\},$$

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

• (i) ${\mathrm{\Theta }}_r$ is single-valued;

• (ii) ${\mathrm{\Theta }}_r$ is firmly nonexpansive, i.e., for any $u,v\in H$,

  $${\parallel {\mathrm{\Theta }}_r(u)-{\mathrm{\Theta }}_r(v)\parallel }^2\le 〈{\mathrm{\Theta }}_r(u)-{\mathrm{\Theta }}_r(v),u-v〉;$$

• (iii) $Fix({\mathrm{\Theta }}_r)=EP(\mathrm{\Psi })$;

• (iv) $EP(\mathrm{\Psi })$ is closed and convex.

Lemma 2.7 ([11])

Let C be a nonempty closed convex subset of a real Hilbert space H. For $i=1,2,\dots ,N$, let ${\mathrm{\Psi }}_i:C\times C\to \mathbb{R}$ be bifunctions satisfying (A1)-(A4) with ${\bigcap }_{i=1}^{N}EP({\mathrm{\Psi }}_i)\ne \mathrm{\varnothing }$. Then

$$EP\left(\sum _{i=1}^N{a}_i{\mathrm{\Psi }}_i\right)=\bigcap _{i=1}^{N}EP({\mathrm{\Psi }}_i),$$

where $a_i\in (0,1)$ for every $i=1,2,\dots ,N$ and ${\sum }_{i=1}^N{a}_i=1$.

Remark 2.8 ([11])

From Lemma 2.7,

$$Fix({\mathrm{\Theta }}_r)=EP\left(\sum _{i=1}^N{a}_i{\mathrm{\Psi }}_i\right)=\bigcap _{i=1}^{N}EP({\mathrm{\Psi }}_i),$$

where $a_i\in (0,1)$, for each $i=1,2,\dots ,N$, and ${\sum }_{i=1}^N{a}_i=1$.

Lemma 2.9 Let C be a nonempty closed convex subset of a real Hilbert space H. Let ${\sum }_{i=1}^N{a}_i{\mathrm{\Psi }}_i:C\times C\to \mathbb{R}$ be bifunctions satisfying (A1)-(A4) where $a_i\in (0,1)$, for each $i=1,2,\dots ,N$ and ${\sum }_{i=1}^N{a}_i=1$. For every $n\in \mathbb{N}$, let $0<c\le r_n\le d$ with $r_n\to r$ as $n\to \mathrm{\infty }$. Then $\parallel {\mathrm{\Theta }}_{r_n}u-{\mathrm{\Theta }}_{r}u\parallel \to 0$ as $n\to \mathrm{\infty }$ for all $u\in H$.

Proof For every $n\in \mathbb{N}$, let $0<c\le r_n\le d$ with $r_n\to r$ as $n\to \mathrm{\infty }$, from which it follows that $0<c\le r\le d$. For every $u\in H$, by Lemma 2.6, we have

$$\sum _{i=1}^N{a}_i{\mathrm{\Psi }}_i({\mathrm{\Theta }}_{r_n}u,v)+\frac{1}{r_n}〈v-{\mathrm{\Theta }}_{r_n}u,{\mathrm{\Theta }}_{r_n}u-u〉\ge 0,\mathrm{\forall }v\in C$$

and

$$\sum _{i=1}^N{a}_i{\mathrm{\Psi }}_i({\mathrm{\Theta }}_{r}u,v)+\frac{1}{r}〈v-{\mathrm{\Theta }}_{r}u,{\mathrm{\Theta }}_{r_n}u-u〉\ge 0,\mathrm{\forall }v\in C.$$

In particular, we have

$$\sum _{i=1}^N{a}_i{\mathrm{\Psi }}_i({\mathrm{\Theta }}_{r_n}u,{\mathrm{\Theta }}_{r}u)+\frac{1}{r_n}〈{\mathrm{\Theta }}_{r}u-{\mathrm{\Theta }}_{r_n}u,{\mathrm{\Theta }}_{r_n}u-u〉\ge 0$$

(2.1)

and

$$\sum _{i=1}^N{a}_i{\mathrm{\Psi }}_i({\mathrm{\Theta }}_{r}u,{\mathrm{\Theta }}_{r_n}u)+\frac{1}{r}〈{\mathrm{\Theta }}_{r_n}u-{\mathrm{\Theta }}_{r}u,{\mathrm{\Theta }}_{r}u-u〉\ge 0.$$

(2.2)

Summing up (2.1) and (2.2) and using (A2), we have

$$\frac{1}{r}〈{\mathrm{\Theta }}_{r_n}u-{\mathrm{\Theta }}_{r}u,{\mathrm{\Theta }}_{r}u-u〉+\frac{1}{r_n}〈{\mathrm{\Theta }}_{r}u-{\mathrm{\Theta }}_{r_n}u,{\mathrm{\Theta }}_{r_n}u-u〉\ge 0.$$

It follows that

$$〈{\mathrm{\Theta }}_{r_n}u-{\mathrm{\Theta }}_{r}u,\frac{{\mathrm{\Theta }}_{r}u-u}{r}-\frac{{\mathrm{\Theta }}_{r_n}u-u}{r_n}〉\ge 0.$$

This implies that

$$\begin{array}{rl}0& \le 〈{\mathrm{\Theta }}_{r}u-{\mathrm{\Theta }}_{r_n}u,{\mathrm{\Theta }}_{r_n}u-u-\frac{r_n}{r}({\mathrm{\Theta }}_{r}u-u)〉\\ =〈{\mathrm{\Theta }}_{r}u-{\mathrm{\Theta }}_{r_n}u,{\mathrm{\Theta }}_{r_n}u-{\mathrm{\Theta }}_{r}u+(1-\frac{r_n}{r})({\mathrm{\Theta }}_{r}u-u)〉.\end{array}$$

It follows that

$${\parallel {\mathrm{\Theta }}_{r}u-{\mathrm{\Theta }}_{r_n}u\parallel }^2\le |1-\frac{r_n}{r}|\parallel {\mathrm{\Theta }}_{r}u-{\mathrm{\Theta }}_{r_n}u\parallel (\parallel {\mathrm{\Theta }}_{r}u\parallel +\parallel u\parallel ).$$

Then we have

$$\parallel {\mathrm{\Theta }}_{r}u-{\mathrm{\Theta }}_{r_n}u\parallel \le \frac{1}{r}|r-r_n|L,$$

where $L=sup\{\parallel {\mathrm{\Theta }}_{r}u\parallel +\parallel u\parallel \}$. Since $r_n\to r$ as $n\to \mathrm{\infty }$, we have

$$\parallel {\mathrm{\Theta }}_{r_n}u-{\mathrm{\Theta }}_{r}u\parallel \to 0\text{as }n\to \mathrm{\infty }.$$

 □

Remark 2.10 Let $\mathcal{S}:H\to H$ be a κ-strictly pseudononspreading mapping with $Fix(\mathcal{S})\ne \mathrm{\varnothing }$. Define $T:H\to H$ by $Tu:=((1-\lambda )I+\lambda \mathcal{S})u$, where $\lambda \in (0,1-\kappa )$. Then the following hold:

• (i) $Fix(\mathcal{S})=Fix(T)=Fix(I-\lambda (I-\mathcal{S}))$;

• (ii) for every $u\in H$ and $v\in Fix(\mathcal{S})$,

  $$\parallel Tu-v\parallel \le \parallel u-v\parallel .$$

Proof (i) It is easy to see that $Fix(\mathcal{S})=Fix(T)=Fix(I-\lambda (I-\mathcal{S}))$.

(ii) Next, we show that $\parallel Tu-v\parallel \le \parallel u-v\parallel$. For every $u\in H$ and $v\in Fix(\mathcal{S})$, we have

$$\begin{array}{rcl}{\parallel Tu-v\parallel }^2& =& {\parallel (I-\lambda (I-\mathcal{S}))u-v\parallel }^2\\ =& {\parallel (1-\lambda )(u-v)+\lambda (\mathcal{S}u-v)\parallel }^2\\ =& (1-\lambda ){\parallel u-v\parallel }^2+\lambda {\parallel \mathcal{S}u-v\parallel }^2-\lambda (1-\lambda ){\parallel \mathcal{S}u-u\parallel }^2\\ \le & (1-\lambda ){\parallel u-v\parallel }^2+\lambda ({\parallel u-v\parallel }^2+\kappa {\parallel (I-\mathcal{S})u\parallel }^2)\\ -\lambda (1-\lambda ){\parallel \mathcal{S}u-u\parallel }^2\\ =& {\parallel u-v\parallel }^2+\kappa \lambda {\parallel \mathcal{S}u-u\parallel }^2-\lambda (1-\lambda ){\parallel \mathcal{S}u-u\parallel }^2\\ =& {\parallel u-v\parallel }^2+\lambda (\lambda -(1-\kappa )){\parallel \mathcal{S}u-u\parallel }^2\\ \le & {\parallel u-v\parallel }^2.\end{array}$$

 □

Lemma 2.11 ([7])

$u\in H$ is a solution of variational inclusion (1.8) if and only if $u=J_{M,\lambda }(u-\lambda Bu)$, $\mathrm{\forall }\lambda >0$, i.e.,

$$VI(H,B,M)=Fix(J_{M,\lambda }(I-\lambda B)),\mathrm{\forall }\lambda >0.$$

Further, if $\lambda \in (0,2\alpha ]$, then $VI(H,B,M)$ is a closed convex subset in H.

Lemma 2.12 ([7])

The resolvent operator $J_{M,\lambda }$ associated with M is single-valued, nonexpansive for all $\lambda >0$ and 1-inverse strongly monotone.

Lemma 2.13 Let H be a real Hilbert space and let $M:H\to 2^H$ be a multi-valued maximal monotone mapping. For every $i=1,2,\dots ,N$, let $A_i:H\to H$ be ${\alpha }_i$-inverse strongly monotone mapping with $\eta ={min}_{i=1,2,\dots ,N}\{{\alpha }_i\}$ and ${\bigcap }_{i=1}^{N}VI(H,A_i,M)\ne \mathrm{\varnothing }$. Then

$$VI(H,\sum _{i=1}^N{a}_i{A}_i,M)=\bigcap _{i=1}^{N}VI(H,A_i,M),$$

where ${\sum }_{i=1}^N{a}_i=1$, and $0<a_i<1$ for every $i=1,2,\dots ,N$. Moreover, $J_{M,\lambda }(I-\lambda {\sum }_{i=1}^N{a}_i{A}_i)$ is a nonexpansive mapping, for all $0<\lambda <2\eta$.

Proof Clearly ${\bigcap }_{i=1}^{N}VI(H,A_i,M)\subseteq VI(H,{\sum }_{i=1}^N{a}_i{A}_i,M)$.

Let $u_0\in VI(H,{\sum }_{i=1}^N{a}_i{A}_i,M)$ and let $u^{\ast }\in {\bigcap }_{i=1}^{N}VI(H,A_i,M)$. From Lemma 2.11, we have

$$u_0\in Fix\left(J_{M,\lambda }(I-\lambda \sum _{i=1}^N{a}_i{A}_i)\right).$$

Since ${\bigcap }_{i=1}^{N}VI(H,A_i,M)\subseteq VI(H,{\sum }_{i=1}^N{a}_i{A}_i,M)$, we have $u^{\ast }\in VI(H,{\sum }_{i=1}^N{a}_i{A}_i,M)$. From Lemma 2.11, we have

$$u^{\ast }\in Fix\left(J_{M,\lambda }(I-\lambda \sum _{i=1}^N{a}_i{A}_i)\right).$$

From the nonexpansiveness of $J_{M,\lambda }$, we have

$$\begin{array}{rl}{\parallel u^{\ast }-u_0\parallel }^2& ={\parallel J_{M,\lambda }(I-\lambda \sum _{i=1}^N{a}_i{A}_i)u^{\ast }-J_{M,\lambda }(I-\lambda \sum _{i=1}^N{a}_i{A}_i)u_0\parallel }^2\\ \le {\parallel (I-\lambda \sum _{i=1}^N{a}_i{A}_i)u^{\ast }-(I-\lambda \sum _{i=1}^N{a}_i{A}_i)u_0\parallel }^2\\ ={\parallel (u^{\ast }-u_0)-\lambda (\sum _{i=1}^N{a}_i{A}_i{u}^{\ast }-\sum _{i=1}^N{a}_i{A}_i{u}_0)\parallel }^2\\ \le {\parallel u^{\ast }-u_0\parallel }^2-2\lambda \sum _{i=1}^N{a}_i〈u^{\ast }-u_0,A_i{u}^{\ast }-A_i{u}_0〉+{\lambda }^2\sum _{i=1}^N{a}_i{\parallel A_i{u}^{\ast }-A_i{u}_0\parallel }^2\\ \le {\parallel u^{\ast }-u_0\parallel }^2-2\lambda \sum _{i=1}^N{a}_i{\alpha }_i{\parallel A_i{u}^{\ast }-A_i{u}_0\parallel }^2+{\lambda }^2\sum _{i=1}^N{a}_i{\parallel A_i{u}^{\ast }-A_i{u}_0\parallel }^2\\ \le {\parallel u^{\ast }-u_0\parallel }^2-2\lambda \eta \sum _{i=1}^N{a}_i{\parallel A_i{u}^{\ast }-A_i{u}_0\parallel }^2+{\lambda }^2\sum _{i=1}^N{a}_i{\parallel A_i{u}^{\ast }-A_i{u}_0\parallel }^2\\ ={\parallel u^{\ast }-u_0\parallel }^2+\lambda \sum _{i=1}^N{a}_i(\lambda -2\eta ){\parallel A_i{u}^{\ast }-A_i{u}_0\parallel }^2.\end{array}$$

(2.3)

This implies that

$$\lambda \sum _{i=1}^N{a}_i(2\eta -\lambda ){\parallel A_i{u}^{\ast }-A_i{u}_0\parallel }^2\le 0.$$

Then

$$A_i{u}^{\ast }=A_i{u}_0,\mathrm{\forall }i=1,2,\dots ,N.$$

(2.4)

Since $u_0\in VI(H,{\sum }_{i=1}^N{a}_i{A}_i,M)$, we have

$$\theta \in M{u}_0+\sum _{i=1}^N{a}_i{A}_i{u}_0.$$

(2.5)

From $u^{\ast }\in VI(H,{\sum }_{i=1}^N{a}_i{A}_i,M)$, we have

$$\theta \in M{u}^{\ast }+\sum _{i=1}^N{a}_i{A}_i{u}^{\ast }.$$

(2.6)

From (2.5) and (2.6), we have

$$\theta \in M{u}_0+\sum _{i=1}^N{a}_i{A}_i{u}_0-M{u}^{\ast }-\sum _{i=1}^N{a}_i{A}_i{u}^{\ast }.$$

(2.7)

From (2.4) and (2.7), we have

$$\theta \in M{u}_0-M{u}^{\ast }.$$

(2.8)

Since $u^{\ast }\in {\bigcap }_{i=1}^{N}VI(H,A_i,M)$ and we have (2.4) and (2.8),

$$\theta \in M{u}_0-M{u}^{\ast }+M{u}^{\ast }+A_i{u}^{\ast }=M{u}_0+A_i{u}_0,$$

for all $i=1,2,\dots ,N$. It implies that $u_0\in {\bigcap }_{i=1}^{N}VI(H,A_i,M)$.

Hence

$$VI(H,\sum _{i=1}^N{a}_i{A}_i,M)\subseteq \bigcap _{i=1}^{N}VI(H,A_i,M).$$

Applying (2.3), we can conclude that $J_{M,\lambda }(I-\lambda {\sum }_{i=1}^N{a}_i{A}_i)$ is a nonexpansive mapping for all $i=1,2,\dots ,N$. □

3 Main result

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let $M:H\to 2^H$ be a multi-valued maximal monotone mapping. For every $i=1,2,\dots ,N$, let ${\mathrm{\Psi }}_i:C\times C\to \mathbb{R}$ be a bifunction satisfying (A1)-(A4) and $A_i:H\to H$ be ${\alpha }_i$-inverse strongly monotone mapping with $\eta ={min}_{i=1,2,\dots ,N}\{{\alpha }_i\}$. Let $\mathcal{S}:H\to H$ be a κ-strictly pseudononspreading mapping. Assume $\mathrm{\Phi }:=Fix(\mathcal{S})\cap {\bigcap }_{i=1}^{N}EP({\mathrm{\Psi }}_i)\cap {\bigcap }_{i=1}^{N}VI(H,A_i,M)\ne \mathrm{\varnothing }$. Let the sequences $\{w_n\}$ and $\{z_n\}$ be generated by $w_1,\mu \in H$ and

$$\{\begin{array}{c}{\sum }_{i=1}^N{a}_i{\mathrm{\Psi }}_i(z_n,y)+\frac{1}{r_n}〈y-z_n,z_n-w_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ w_{n+1}={\alpha }_n\mu +{\beta }_n{w}_n+{\gamma }_n{J}_{M,\lambda }(I-\lambda {\sum }_{i=1}^N{b}_i{A}_i)w_n\hfill \\ \phantom{w_{n+1}=}+{\eta }_n(I-{\rho }_n(I-\mathcal{S}))w_n+{\delta }_n{z}_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.1)

where $\{{\alpha }_n\},\{{\beta }_n\},\{{\gamma }_n\},\{{\eta }_n\},\{{\delta }_n\}\subseteq (0,1)$ and $\lambda >0$ with ${\alpha }_n+{\beta }_n+{\gamma }_n+{\eta }_n+{\delta }_n=1$, $0<\alpha <1$, and $0\le a_i,b_i\le 1$, for every $i=1,2,\dots ,N$, $r_n\in [c,d]\subset (0,1)$, $0<p\le {\beta }_n,{\gamma }_n,{\eta }_n,{\delta }_n\le q<1$, ${\rho }_n\in (0,1-\kappa )$ for all $n\ge 1$. 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) ${\sum }_{n=1}^{\mathrm{\infty }}{\rho }_n<\mathrm{\infty }$,

• (iii) $0<\lambda <2\eta$, where $\eta ={min}_{i=1,2,\dots ,N}\{{\alpha }_i\}$,

• (iv) ${\sum }_{i=1}^N{a}_i={\sum }_{i=1}^N{b}_i=1$,

• (v) ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\gamma }_{n+1}-{\gamma }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\rho }_{n+1}-{\rho }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|r_{n+1}-r_n|<\mathrm{\infty }$.

Then the sequences $\{w_n\}$ and $\{z_n\}$ converge strongly to $\omega =P_{\mathrm{\Phi }}\mu$.

Proof The proof of Theorem 3.1 will be divided into five steps:

Step 1. We show that the sequence $\{w_n\}$ is bounded.

Since ${\sum }_{i=1}^N{a}_i{\mathrm{\Psi }}_i$ satisfies (A1)-(A4), and

$$\sum _{i=1}^N{a}_i{\mathrm{\Psi }}_i(z_n,y)+\frac{1}{r_n}〈y-z_n,z_n-w_n〉\ge 0,\mathrm{\forall }y\in C,$$

by Lemma 2.6 and Remark 2.8, we have $z_n={\mathrm{\Theta }}_{r_n}{w}_n$ and $Fix({\mathrm{\Theta }}_{r_n})={\bigcap }_{i=1}^{N}EP({\mathrm{\Psi }}_i)$.

Let $\omega \in \mathrm{\Phi }$. From Lemma 2.11 and Lemma 2.13, we have

$$\omega =J_{M,\lambda }(I-\lambda \sum _{i=1}^N{b}_i{A}_i)\omega .$$

From the nonexpansiveness of $J_{M,\lambda }(I-\lambda {\sum }_{i=1}^N{b}_i{A}_i)$, we have

$$\parallel J_{M,\lambda }(I-\lambda \sum _{i=1}^N{b}_i{A}_i)w_n-\omega \parallel \le \parallel w_n-\omega \parallel .$$

(3.2)

From Remark 2.10, we have

$$\begin{array}{rl}{\parallel (I-{\rho }_n(I-\mathcal{S}))w_n-\omega \parallel }^2& ={\parallel (1-{\rho }_n)w_n+{\rho }_n\mathcal{S}{w}_n-\omega \parallel }^2\\ \le {\parallel w_n-\omega \parallel }^2.\end{array}$$

(3.3)

From the definition of $w_n$, (3.2), and (3.3), we have

$$\begin{array}{rl}\parallel w_{n+1}-\omega \parallel =& \parallel {\alpha }_n\mu +{\beta }_n{w}_n+{\gamma }_n{J}_{M,\lambda }(I-\lambda \sum _{i=1}^N{b}_i{A}_i)w_n\\ +{\eta }_n(I-{\rho }_n(I-\mathcal{S}))w_n+{\delta }_n{z}_n-\omega \parallel \\ \le & {\alpha }_n\parallel \mu -\omega \parallel +{\beta }_n\parallel w_n-\omega \parallel +{\gamma }_n\parallel J_{M,\lambda }(I-\lambda \sum _{i=1}^N{b}_i{A}_i)w_n-\omega \parallel \\ +{\eta }_n\parallel (I-{\rho }_n(I-\mathcal{S}))w_n-\omega \parallel +{\delta }_n\parallel z_n-\omega \parallel \\ \le & {\alpha }_n\parallel \mu -\omega \parallel +(1-{\alpha }_n)\parallel w_n-\omega \parallel \\ \le & max\{\parallel \mu -\omega \parallel ,\parallel w_1-\omega \parallel \}=K.\end{array}$$

By mathematical induction, we have $\parallel w_n-z\parallel \le K$, $\mathrm{\forall }n\in \mathbb{N}$. It implies that $\{w_n\}$ is bounded and so is $\{z_n\}$.

By continuing in the same direction as in Step 1 of Theorem 3.1 in [10], we have

$$\parallel \mathcal{S}{w}_n-\omega \parallel \le \frac{1+\kappa }{1-\kappa }\parallel w_n-\omega \parallel .$$

(3.4)

From (3.4), we can conclude that $\{\mathcal{S}{w}_n\}$ is bounded.

Step 2. Put $G={\sum }_{i=1}^N{b}_i{A}_i$ and $P=I-\mathcal{S}$. We will show that ${lim}_{n\to \mathrm{\infty }}\parallel w_{n+1}-w_n\parallel =0$. From the definition of $w_n$, we have

$$\begin{array}{rcl}\parallel w_{n+1}-w_n\parallel & =& \parallel {\alpha }_n\mu +{\beta }_n{w}_n+{\gamma }_n{J}_{M,\lambda }(I-\lambda G)w_n+{\eta }_n(I-{\rho }_{n}P)w_n\\ +{\delta }_n{z}_n-{\alpha }_{n-1}\mu -{\beta }_{n-1}{w}_{n-1}-{\gamma }_{n-1}{J}_{M,\lambda }(I-\lambda G)w_{n-1}\\ -{\eta }_{n-1}(I-{\rho }_{n-1}P)w_{n-1}-{\delta }_{n-1}{z}_{n-1}\parallel \\ \le & |{\alpha }_n-{\alpha }_{n-1}|\parallel \mu \parallel +{\beta }_n\parallel w_n-w_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel w_{n-1}\parallel \\ +{\gamma }_n\parallel J_{M,\lambda }(I-\lambda G)w_n-J_{M,\lambda }(I-\lambda G)w_{n-1}\parallel \\ +|{\gamma }_n-{\gamma }_{n-1}|\parallel J_{M,\lambda }(I-\lambda G)w_{n-1}\parallel \\ +{\eta }_n\parallel (I-{\rho }_{n}P)w_n-(I-{\rho }_{n-1}P)w_{n-1}\parallel +|{\eta }_n-{\eta }_{n-1}|\parallel (1-{\rho }_{n-1}P)w_{n-1}\parallel \\ +{\delta }_n\parallel z_n-z_{n-1}\parallel +|{\delta }_n-{\delta }_{n-1}|\parallel z_{n-1}\parallel \\ \le & |{\alpha }_n-{\alpha }_{n-1}|\parallel \mu \parallel +{\beta }_n\parallel w_n-w_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel w_{n-1}\parallel +{\gamma }_n\parallel w_n-w_{n-1}\parallel \\ +|{\gamma }_n-{\gamma }_{n-1}|\parallel J_{M,\lambda }(I-\lambda G)w_{n-1}\parallel +|{\eta }_n-{\eta }_{n-1}|\parallel (I-{\rho }_{n-1}P)w_{n-1}\parallel \\ +{\eta }_n(\parallel w_n-w_{n-1}\parallel +{\rho }_n\parallel P{w}_n-P{w}_{n-1}\parallel +|{\rho }_n-{\rho }_{n-1}|\parallel P{w}_{n-1}\parallel )\\ +{\delta }_n\parallel z_n-z_{n-1}\parallel +|{\delta }_n-{\delta }_{n-1}|\parallel z_{n-1}\parallel \\ \le & |{\alpha }_n-{\alpha }_{n-1}|\parallel \mu \parallel +{\beta }_n\parallel w_n-w_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel w_{n-1}\parallel +{\gamma }_n\parallel w_n-w_{n-1}\parallel \\ +|{\gamma }_n-{\gamma }_{n-1}|\parallel J_{M,\lambda }(I-\lambda G)w_{n-1}\parallel +|{\eta }_n-{\eta }_{n-1}|\parallel (I-{\rho }_{n-1}P)w_{n-1}\parallel \\ +{\eta }_n\parallel w_n-w_{n-1}\parallel +{\rho }_n\parallel P{w}_n-P{w}_{n-1}\parallel +|{\rho }_n-{\rho }_{n-1}|\parallel P{w}_{n-1}\parallel \\ +{\delta }_n\parallel z_n-z_{n-1}\parallel +|{\delta }_n-{\delta }_{n-1}|\parallel z_{n-1}\parallel .\end{array}$$

(3.5)

By continuing in the same direction as in Step 2 of Theorem 3.1 in [11], we have

$$\parallel z_n-z_{n-1}\parallel \le \parallel w_n-w_{n-1}\parallel +\frac{1}{d}|r_n-r_{n-1}|\parallel z_n-w_n\parallel .$$

(3.6)

By substituting (3.6) into (3.5), we obtain

$$\begin{array}{rcl}\parallel w_{n+1}-w_n\parallel & \le & |{\alpha }_n-{\alpha }_{n-1}|\parallel \mu \parallel +{\beta }_n\parallel w_n-w_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel w_{n-1}\parallel +{\gamma }_n\parallel w_n-w_{n-1}\parallel \\ +|{\gamma }_n-{\gamma }_{n-1}|\parallel J_{M,\lambda }(I-\lambda G)w_{n-1}\parallel +|{\eta }_n-{\eta }_{n-1}|\parallel (I-{\rho }_{n-1}P)w_{n-1}\parallel \\ +{\eta }_n\parallel w_n-w_{n-1}\parallel +{\rho }_n\parallel P{w}_n-P{w}_{n-1}\parallel +|{\rho }_n-{\rho }_{n-1}|\parallel P{w}_{n-1}\parallel \\ +{\delta }_n\parallel z_n-z_{n-1}\parallel +|{\delta }_n-{\delta }_{n-1}|\parallel z_{n-1}\parallel \\ \le & |{\alpha }_n-{\alpha }_{n-1}|\parallel \mu \parallel +{\beta }_n\parallel w_n-w_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel w_{n-1}\parallel +{\gamma }_n\parallel w_n-w_{n-1}\parallel \\ +|{\gamma }_n-{\gamma }_{n-1}|\parallel J_{M,\lambda }(I-\lambda G)w_{n-1}\parallel +|{\eta }_n-{\eta }_{n-1}|\parallel (I-{\rho }_{n-1}P)w_{n-1}\parallel \\ +{\eta }_n\parallel w_n-w_{n-1}\parallel +{\rho }_n\parallel P{w}_n-P{w}_{n-1}\parallel +|{\rho }_n-{\rho }_{n-1}|\parallel P{w}_{n-1}\parallel \\ +{\delta }_n(\parallel w_n-w_{n-1}\parallel +\frac{1}{d}|r_n-r_{n-1}|\parallel z_n-w_n\parallel )+|{\delta }_n-{\delta }_{n-1}|\parallel z_{n-1}\parallel \\ \le & |{\alpha }_n-{\alpha }_{n-1}|\parallel \mu \parallel +(1-{\alpha }_n)\parallel w_n-w_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel w_{n-1}\parallel \\ +|{\gamma }_n-{\gamma }_{n-1}|\parallel J_{M,\lambda }(I-\lambda G)w_{n-1}\parallel +|{\eta }_n-{\eta }_{n-1}|\parallel (I-{\rho }_{n-1}P)w_{n-1}\parallel \\ +{\rho }_n\parallel P{w}_n-P{w}_{n-1}\parallel +|{\rho }_n-{\rho }_{n-1}|\parallel P{w}_{n-1}\parallel \\ +\frac{1}{d}|r_n-r_{n-1}|\parallel z_n-w_n\parallel +|{\delta }_n-{\delta }_{n-1}|\parallel z_{n-1}\parallel .\end{array}$$

(3.7)

Applying Lemma 2.2, (3.7), and the conditions (i), (ii), (v), we have

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

(3.8)

Step 3. We show that ${lim}_{n\to \mathrm{\infty }}\parallel z_n-w_n\parallel ={lim}_{n\to \mathrm{\infty }}\parallel (I-{\rho }_{n}P)w_n-w_n\parallel ={lim}_{n\to \mathrm{\infty }}\parallel J_{M,\lambda }(I-\lambda G)w_n-w_n\parallel =0$. By the definition of $w_n$, (3.2), and (3.3), we have

$$\begin{array}{rcl}{\parallel w_{n+1}-\omega \parallel }^2& =& \parallel {\alpha }_n\mu +{\beta }_n{w}_n+{\gamma }_n{J}_{M,\lambda }(I-\lambda G)w_n\\ {+{\eta }_n(I-{\rho }_{n}P)w_n+{\delta }_n{z}_n-\omega \parallel }^2\\ \le & {\alpha }_n{\parallel \mu -\omega \parallel }^2+{\beta }_n{\parallel w_n-\omega \parallel }^2+{\gamma }_n{\parallel J_{M,\lambda }(I-\lambda G)w_n-\omega \parallel }^2\\ +{\eta }_n{\parallel (I-{\rho }_{n}P)w_n-\omega \parallel }^2+{\delta }_n{\parallel z_n-\omega \parallel }^2-{\beta }_n{\delta }_n{\parallel w_n-z_n\parallel }^2\\ -{\beta }_n{\gamma }_n{\parallel J_{M,\lambda }(I-\lambda G)w_n-w_n\parallel }^2\\ =& {\alpha }_n{\parallel \mu -\omega \parallel }^2+(1-{\alpha }_n){\parallel w_n-\omega \parallel }^2-{\beta }_n{\delta }_n{\parallel w_n-z_n\parallel }^2\\ -{\beta }_n{\gamma }_n{\parallel J_{M,\lambda }(I-\lambda G)w_n-w_n\parallel }^2\\ \le & {\alpha }_n{\parallel \mu -\omega \parallel }^2+{\parallel w_n-\omega \parallel }^2-{\beta }_n{\delta }_n{\parallel w_n-{\omega }_n\parallel }^2\\ -{\beta }_n{\gamma }_n{\parallel J_{M,\lambda }(I-\lambda G)w_n-w_n\parallel }^2.\end{array}$$

It implies that

$$\begin{array}{rcl}{\beta }_n{\delta }_n{\parallel z_n-w_n\parallel }^2& \le & {\alpha }_n{\parallel \mu -\omega \parallel }^2+{\parallel w_n-\omega \parallel }^2-{\parallel w_{n+1}-\omega \parallel }^2\\ -{\beta }_n{\gamma }_n{\parallel J_{M,\lambda }(I-\lambda G)w_n-w_n\parallel }^2\\ \le & {\alpha }_n{\parallel \mu -\omega \parallel }^2+(\parallel w_n-\omega \parallel +\parallel w_{n+1}-\omega \parallel )\parallel w_{n+1}-w_n\parallel .\end{array}$$

From the condition (i) and (3.8), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel z_n-w_n\parallel =0.$$

(3.9)

By continuing in the same direction as (3.9), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J_{M,\lambda }(I-\lambda G)w_n-w_n\parallel =0.$$

(3.10)

From the definition of $w_n$, we have

$$\begin{array}{rcl}{w}_{n+1}-w_n& =& {\alpha }_n(\mu -w_n)+{\gamma }_n(J_{M,\lambda }(I-\lambda G)w_n-w_n)\\ +{\eta }_n((I-{\rho }_{n}P)w_n-w_n)+{\delta }_n(z_n-w_n).\end{array}$$

From the condition (i), (3.8), (3.9), and (3.10), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel (I-{\rho }_n(I-\mathcal{S}))w_n-w_n\parallel =0.$$

(3.11)

Step 4. We will show that ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}〈\mu -\omega ,w_n-\omega 〉\le 0$, where $\omega =P_{\mathrm{\Phi }}\mu$.

To show this, choose a subsequence $\{w_{n_k}\}$ of $\{w_n\}$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\mu -\omega ,w_n-\omega 〉=\underset{k\to \mathrm{\infty }}{lim}〈\mu -\omega ,w_{n_k}-\omega 〉.$$

(3.12)

Without loss of generality, we can assume that $w_{n_k}\rightharpoonup \xi$ as $k\to \mathrm{\infty }$. From (3.9), we obtain $z_{n_k}\rightharpoonup \xi$ as $k\to \mathrm{\infty }$.

First, we will show that $\xi \in {\bigcap }_{i=1}^{N}VI(H,A_i,M)$. Assume that $\xi \notin {\bigcap }_{i=1}^{N}VI(H,A_i,M)$. By Lemmas 2.11 and 2.13, ${\bigcap }_{i=1}^{N}VI(H,A_i,M)=Fix(J_{M,\lambda }((I-\lambda G)))$. Then $\xi \ne J_{M,\lambda }(I-\lambda G)\xi$, where $G={\sum }_{i=1}^N{b}_i{A}_i$. By the nonexpansiveness of $J_{M,\lambda }((I-\lambda G))$, (3.10), and Opial’s condition, we obtain

$$\begin{array}{rcl}\underset{k\to \mathrm{\infty }}{lim}inf\parallel w_{n_k}-\xi \parallel & <& \underset{k\to \mathrm{\infty }}{lim}inf\parallel w_{n_k}-J_{M,\lambda }((I-\lambda G))\xi \parallel \\ \le & \underset{k\to \mathrm{\infty }}{lim}inf(\parallel w_{n_k}-J_{M,\lambda }((I-\lambda G))w_{n_k}\parallel \\ +\parallel J_{M,\lambda }((I-\lambda G))w_{n_k}-J_{M,\lambda }((I-\lambda G))\xi \parallel )\\ \le & \underset{k\to \mathrm{\infty }}{lim}inf\parallel w_{n_k}-\xi \parallel .\end{array}$$

This is a contradiction. Then we have

$$\xi \in \bigcap _{i=1}^{N}VI(H,A_i,M).$$

(3.13)

Next, we will show that $\xi \in Fix(\mathcal{S})$. Assume that $\xi \notin Fix(\mathcal{S})$. From Remark 2.10(i), we get $Fix(\mathcal{S})=Fix(I-{\rho }_{n_k}(I-\mathcal{S}))$. Then $\xi \ne (I-{\rho }_{n_k}(I-\mathcal{S}))\xi$. From the condition (ii), (3.11), and Opial’s condition, we obtain

$$\begin{array}{rcl}\underset{k\to \mathrm{\infty }}{lim}inf\parallel w_{n_k}-\xi \parallel & <& \underset{k\to \mathrm{\infty }}{lim}inf\parallel w_{n_k}-(I-{\rho }_{n_k}(I-\mathcal{S}))\xi \parallel \\ \le & \underset{k\to \mathrm{\infty }}{lim}inf(\parallel w_{n_k}-(I-{\rho }_{n_k}(I-\mathcal{S}))w_{n_k}\parallel \\ +\parallel (I-{\rho }_{n_k}(I-\mathcal{S}))w_{n_k}-(I-{\rho }_{n_k}(I-\mathcal{S}))\xi \parallel )\\ \le & \underset{k\to \mathrm{\infty }}{lim}inf(\parallel w_{n_k}-(I-{\rho }_{n_k}(I-\mathcal{S}))w_{n_k}\parallel \\ +\parallel w_{n_k}-\xi \parallel +{\rho }_{n_k}\parallel (I-\mathcal{S})w_{n_k}-(I-\mathcal{S})\xi \parallel )\\ =& \underset{k\to \mathrm{\infty }}{lim}inf\parallel w_{n_k}-\xi \parallel .\end{array}$$

This is a contradiction. Then we have

$$\xi \in Fix(\mathcal{S}).$$

(3.14)

Since $0<c\le r_n\le d$, $\mathrm{\forall }n\in \mathbb{N}$, then we have $r_{n_k}\to r$ as $k\to \mathrm{\infty }$ with $0<c\le r\le d$. Applying Lemma 2.9, we have $\parallel {\mathrm{\Theta }}_{r_{n_k}}{w}_{n_k}-{\mathrm{\Theta }}_r{w}_{n_k}\parallel \to 0$ as $k\to \mathrm{\infty }$. Next, we will show that $\xi \in {\bigcap }_{i=1}^{N}EP({\mathrm{\Psi }}_i)$. Assume that $\xi \notin {\bigcap }_{i=1}^{N}EP({\mathrm{\Psi }}_i)$. From Remark 2.8, we have $\xi \notin Fix({\mathrm{\Theta }}_r)$. By Opial’s condition and (3.9), we have

$$\begin{array}{rcl}\underset{k\to \mathrm{\infty }}{lim}inf\parallel w_{n_k}-\xi \parallel & <& \underset{k\to \mathrm{\infty }}{lim}inf\parallel w_{n_k}-{\mathrm{\Theta }}_r\xi \parallel \\ \le & \underset{k\to \mathrm{\infty }}{lim}inf(\parallel w_{n_k}-{\mathrm{\Theta }}_{r_{n_k}}{w}_{n_k}\parallel \\ +\parallel {\mathrm{\Theta }}_{r_{n_k}}{w}_{n_k}-{\mathrm{\Theta }}_r{w}_{n_k}\parallel +\parallel {\mathrm{\Theta }}_r{w}_{n_k}-{\mathrm{\Theta }}_r\xi \parallel )\\ \le & \underset{k\to \mathrm{\infty }}{lim}inf\parallel w_{n_k}-\xi \parallel .\end{array}$$

This is a contradiction. Then we have

$$\xi \in \bigcap _{i=1}^{N}EP({\mathrm{\Psi }}_i).$$

(3.15)

From (3.13), (3.14), and (3.15), we can conclude that $\xi \in \mathrm{\Phi }$.

Since $w_{n_k}\rightharpoonup \xi$ as $k\to \mathrm{\infty }$ and $\xi \in \mathrm{\Phi }$. By (3.12) and Lemma 2.1, we have

$$\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\mu -\omega ,w_n-\omega 〉& =\underset{k\to \mathrm{\infty }}{lim}〈\mu -\omega ,w_{n_k}-\omega 〉\\ =〈\mu -\omega ,\xi -\omega 〉\\ \le 0.\end{array}$$

(3.16)

Step 5. Finally, we will show that ${lim}_{n\to \mathrm{\infty }}{w}_n=\omega$, where $\omega =P_{\mathrm{\Phi }}\mu$. From the definition of $w_n$, we have

$$\begin{array}{rcl}{\parallel w_{n+1}-\omega \parallel }^2& =& {\parallel {\alpha }_n\mu +{\beta }_n{w}_n+{\gamma }_n{J}_{M,\lambda }(I-\lambda G)w_n+{\eta }_n(I-{\rho }_{n}P)w_n+{\delta }_n{z}_n-\omega \parallel }^2\\ \le & \parallel {\alpha }_n(\mu -\omega )+{\beta }_n(w_n-\omega )+{\gamma }_n(J_{M,\lambda }(I-\lambda G)w_n-\omega )\\ {+{\eta }_n((I-{\rho }_{n}P)w_n-\omega )+{\delta }_n(z_n-\omega )\parallel }^2\\ \le & ({\beta }_n\parallel w_n-\omega \parallel +{\gamma }_n\parallel J_{M,\lambda }(I-\lambda G)w_n-\omega \parallel +{\eta }_n\parallel (I-{\rho }_{n}P)w_n-\omega \parallel \\ {+{\delta }_n\parallel z_n-\omega \parallel )}^2+2{\alpha }_n〈\mu -\omega ,w_{n+1}-\omega 〉\\ \le & (1-{\alpha }_n){\parallel w_n-\omega \parallel }^2+2{\alpha }_n〈\mu -\omega ,w_{n+1}-\omega 〉.\end{array}$$

From the condition (i), (3.16), and Lemma 2.2, we can conclude that the sequence $\{w_n\}$ converges strongly to $\omega =P_{\mathrm{\Phi }}\mu$. By (3.9), we find that $\{z_n\}$ converges strongly to $\omega =P_{\mathrm{\Phi }}\mu$. This completes the proof. □

As a direct proof of Theorem 3.1, we obtain the following results.

Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H and let $M:H\to 2^H$ be a multi-valued maximal monotone mapping. For every $i=1,2,\dots ,N$, let ${\mathrm{\Psi }}_i:C\times C\to \mathbb{R}$ be a bifunction satisfying (A1)-(A4) and let $A:H\to H$ be an α-inverse strongly monotone mapping. Let $\mathcal{S}:H\to H$ be a κ-strictly pseudononspreading mapping. Assume $\mathrm{\Phi }:=Fix(\mathcal{S})\cap {\bigcap }_{i=1}^{N}EP({\mathrm{\Psi }}_i)\cap VI(H,A,M)\ne \mathrm{\varnothing }$. Let the sequences $\{w_n\}$ and $\{z_n\}$ be generated by $w_1,\mu \in H$ and

$$\{\begin{array}{c}{\sum }_{i=1}^N{a}_i{\mathrm{\Psi }}_i(z_n,y)+\frac{1}{r_n}〈y-z_n,z_n-w_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ w_{n+1}={\alpha }_n\mu +{\beta }_n{w}_n+{\gamma }_n{J}_{M,\lambda }(I-\lambda A)w_n\hfill \\ \phantom{w_{n+1}=}+{\eta }_n(I-{\rho }_n(I-\mathcal{S}))w_n+{\delta }_n{z}_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.17)

where $\{{\alpha }_n\},\{{\beta }_n\},\{{\gamma }_n\},\{{\eta }_n\},\{{\delta }_n\}\subseteq (0,1)$ and $\lambda >0$ with ${\alpha }_n+{\beta }_n+{\gamma }_n+{\eta }_n+{\delta }_n=1$, $0<\alpha <1$, and $0\le a_i\le 1$, for every $i=1,2,\dots ,N$, $r_n\in [c,d]\subset (0,1)$, $0<p\le {\beta }_n,{\gamma }_n,{\eta }_n,{\delta }_n\le q<1$, ${\rho }_n\in (0,1-\kappa )$ for all $n\ge 1$. 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) ${\sum }_{n=1}^{\mathrm{\infty }}{\rho }_n<\mathrm{\infty }$,

• (iii) $0<\lambda <2\alpha$,

• (iv) ${\sum }_{i=1}^N{a}_i=1$,

• (v) ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\gamma }_{n+1}-{\gamma }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\rho }_{n+1}-{\rho }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|r_{n+1}-r_n|<\mathrm{\infty }$.

Then the sequences $\{w_n\}$ and $\{z_n\}$ converge strongly to $\omega =P_{\mathrm{\Phi }}\mu$.

Proof Put $A_i\equiv A$ for all $i=1,2,\dots ,N$ in Theorem 3.1. So, from Theorem 3.1, we obtain the desired result. □

Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H and let $M:H\to 2^H$ be a multi-valued maximal monotone mapping. Let $\mathrm{\Psi }:C\times C\to \mathbb{R}$ be a bifunction satisfying (A1)-(A4). For every $i=1,2,\dots ,N$, $A_i:H\to H$ be ${\alpha }_i$-inverse strongly monotone mapping with $\eta ={min}_{i=1,2,\dots ,N}\{{\alpha }_i\}$. Let $\mathcal{S}:H\to H$ be a κ-strictly pseudononspreading mapping. Assume $\mathrm{\Phi }:=Fix(\mathcal{S})\cap EP(F)\cap {\bigcap }_{i=1}^{N}VI(H,A_i,M)\ne \mathrm{\varnothing }$. Let the sequences $\{w_n\}$ and $\{z_n\}$ be generated by $w_1,\mu \in H$ and

$$\{\begin{array}{c}\mathrm{\Psi }(z_n,y)+\frac{1}{r_n}〈y-z_n,z_n-w_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ w_{n+1}={\alpha }_n\mu +{\beta }_n{w}_n+{\gamma }_n{J}_{M,\lambda }(I-\lambda {\sum }_{i=1}^N{b}_i{A}_i)w_n\hfill \\ \phantom{w_{n+1}=}+{\eta }_n(I-{\rho }_n(I-\mathcal{S}))w_n+{\delta }_n{z}_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.18)

where $\{{\alpha }_n\},\{{\beta }_n\},\{{\gamma }_n\},\{{\eta }_n\},\{{\delta }_n\}\subseteq (0,1)$ and $\lambda >0$ with ${\alpha }_n+{\beta }_n+{\gamma }_n+{\eta }_n+{\delta }_n=1$, $0<\alpha <1$, and $0\le b_i\le 1$, for every $i=1,2,\dots ,N$, $r_n\in [c,d]\subset (0,1)$, $0<p\le {\beta }_n,{\gamma }_n,{\eta }_n,{\delta }_n\le q<1$, ${\rho }_n\in (0,1-\kappa )$ for all $n\ge 1$. 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) ${\sum }_{n=1}^{\mathrm{\infty }}{\rho }_n<\mathrm{\infty }$,

• (iii) $0<\lambda <2\eta$, where $\eta ={min}_{i=1,2,3,\dots ,N}\{{\alpha }_i\}$,

• (iv) ${\sum }_{i=1}^N{b}_i=1$,

• (v) ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\gamma }_{n+1}-{\gamma }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\rho }_{n+1}-{\rho }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|r_{n+1}-r_n|<\mathrm{\infty }$.

Then the sequences $\{w_n\}$ and $\{z_n\}$ converge strongly to $\omega =P_{\mathrm{\Phi }}\mu$.

Proof Take $\mathrm{\Psi }={\mathrm{\Psi }}_i$, $\mathrm{\forall }i=1,2,\dots ,N$. By Theorem 3.1, we obtain the desired conclusion. □

4 Applications

In this section, we utilize our main theorem to prove a strong convergence theorem for finding a common element of the set of fixed points of a κ-quasi-strictly pseudo-contractive mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problem in Hilbert space. To obtain this result, we recall some definitions, lemmas, and remarks as follows.

Definition 4.1 Let C be a subset of a real Hilbert space H and let $\mathcal{S}:C\to C$ be a mapping. Then $\mathcal{S}u=u$ is said to be κ-quasi-strictly pseudo-contractive if there exists a constant $\kappa \in [0,1)$ such that

$${\parallel \mathcal{S}u-p\parallel }^2\le {\parallel u-p\parallel }^2+\kappa {\parallel u-\mathcal{S}u\parallel }^2,\mathrm{\forall }u\in C\text{ and }\mathrm{\forall }p\in Fix(\mathcal{S}).$$

$\mathcal{S}u=u$ is said to be quasi-nonexpansive if

$$\parallel \mathcal{S}u-p\parallel \le \parallel u-p\parallel ,\mathrm{\forall }u\in C\text{ and }\mathrm{\forall }p\in Fix(\mathcal{S}).$$

The class of κ-quasi-strictly pseudo-contractions includes the class of quasi-nonexpansive mappings.

Remark 4.1 If $\mathcal{S}:C\to C$ be a κ-strictly pseudononspreading mapping with $Fix(\mathcal{S})\ne \mathrm{\varnothing }$, then $\mathcal{S}u=u$ is a κ-quasi-strictly pseudo-contractive mapping.

Example 4.2 Let $\mathcal{S}:[0,1]\to [0,1]$ be defined by

$$\mathcal{S}u=\frac{2u+1}{3},\text{for all }u\in [0,1].$$

Then $\mathcal{S}u=u$ is a κ-strictly pseudononspreading mapping where $\kappa \in [0,1)$. Since $1\in Fix(\mathcal{S})$, $\mathcal{S}u=u$ is also κ-quasi-strictly pseudo-contractive mapping.

Next, we give the example to show that the converse of Remark 4.1 is not true.

Example 4.3 Let $\mathcal{S}:[-2,2]\to [-2,2]$ be defined by

$$\mathcal{S}u=-\frac{5}{3}u,\mathrm{\forall }u\in [-2,2].$$

First, show that $\mathcal{S}u=u$ is a κ-quasi-strictly pseudo-contractive mapping for all $u\in [-2,2]$.

Observe that $Fix(\mathcal{S})=\{0\}$. Let $u\in [-2,2]$, we have

$${|\mathcal{S}u-\mathcal{S}0|}^2=|-\frac{5}{3}u-0{|}^2=\frac{25}{9}{|u|}^2$$

and

$$\begin{array}{rl}{|u-0|}^2+\frac{1}{4}|(I-\mathcal{S})u{|}^2& ={|u|}^2+\frac{1}{4}|u+\frac{5}{3}u{|}^2\\ ={|u|}^2+\frac{1}{4}|\frac{8}{3}u{|}^2\\ ={|u|}^2+\frac{64}{9}\left(\frac{1}{4}\right){|u|}^2\\ =\left(\frac{25}{9}\right){|u|}^2.\end{array}$$

Then $\mathcal{S}u=u$ is a $\frac{1}{4}$-quasi-strictly pseudo-contractive mapping. Next, we show that $\mathcal{S}u=u$ is not a $\frac{1}{4}$-strictly pseudononspreading mapping.

Choose $u=\frac{3}{2}$ and $v=\frac{-3}{2}$, we have

$$\begin{array}{c}|\mathcal{S}\left(\frac{3}{2}\right)-\mathcal{S}(-\frac{3}{2}){|}^2=|-\frac{5}{3}\left(\frac{3}{2}\right)+\frac{5}{3}(-\frac{3}{2}){|}^2\hfill \\ \phantom{|\mathcal{S}\left(\frac{3}{2}\right)-\mathcal{S}(-\frac{3}{2}){|}^2}=|-\frac{10}{2}{|}^2\hfill \\ \phantom{|\mathcal{S}\left(\frac{3}{2}\right)-\mathcal{S}(-\frac{3}{2}){|}^2}=25,\hfill \\ {|u-v|}^2=|\frac{3}{2}+\frac{3}{2}{|}^2=9,\hfill \\ \frac{1}{4}|(I-\mathcal{S})\left(\frac{3}{2}\right)-(I-\mathcal{S})\left(\frac{-3}{2}\right){|}^2=\frac{1}{4}|\left(\frac{3}{2}\right)+\frac{5}{3}\left(\frac{3}{2}\right)-((-\frac{3}{2})+\frac{5}{3}(-\frac{3}{2})){|}^2\hfill \\ \phantom{\frac{1}{4}|(I-\mathcal{S})\left(\frac{3}{2}\right)-(I-\mathcal{S})\left(\frac{-3}{2}\right){|}^2}=\frac{1}{4}{|8|}^2\hfill \\ \phantom{\frac{1}{4}|(I-\mathcal{S})\left(\frac{3}{2}\right)-(I-\mathcal{S})\left(\frac{-3}{2}\right){|}^2}=16\hfill \end{array}$$

and

$$\begin{array}{rl}2〈(I-\mathcal{S})\left(\frac{3}{2}\right),(I-\mathcal{S})\left(\frac{-3}{2}\right)〉& =2〈\left(\frac{3}{2}\right)+\frac{5}{3}\left(\frac{3}{2}\right),((-\frac{3}{2})+\frac{5}{3}(-\frac{3}{2}))〉\\ =2(4)(-4)=-32.\end{array}$$

Then we have

$${|\mathcal{S}u-\mathcal{S}v|}^2>{|u-v|}^2+\frac{1}{4}|(I-\mathcal{S})u-(I-\mathcal{S})v{|}^2+2〈u-\mathcal{S}u,v-\mathcal{S}v〉.$$

By changing $\mathcal{S}u=u$ from being a κ-strictly pseudononspreading mapping with $Fix(\mathcal{S})\ne \mathrm{\varnothing }$ into a κ-quasi-strictly pseudo-contractive mapping, we obtain the same result as in Remark 2.10.

Remark 4.4 Let $\mathcal{S}:H\to H$ be a κ-quasi-strictly pseudo-contractive mapping with $Fix(\mathcal{S})\ne \mathrm{\varnothing }$. Define $T:H\to H$ by $Tu:=((1-\lambda )I+\lambda \mathcal{S})u$, where $\lambda \in (0,1-\kappa )$. Then the following hold:

• (i) $Fix(\mathcal{S})=Fix(T)=Fix(I-\lambda (I-\mathcal{S}))$;

• (ii) for every $u\in H$ and $v\in Fix(\mathcal{S})$,

  $$\parallel Tu-v\parallel \le \parallel u-v\parallel .$$

In 2009, Kangtunyakarn and Suantai [16] introduced the S-mapping generated by ${\mathcal{S}}_1,{\mathcal{S}}_2,\dots ,{\mathcal{S}}_N$ and ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_N$ as follows.

Definition 4.2 ([16])

Let C be a nonempty convex subset of a real Banach space. Let ${\{{\mathcal{S}}_i\}}_{i=1}^N$ be a finite family of (nonexpansive) mappings of C into itself. For each $j=1,2,\dots$ , let ${\alpha }_j=({\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j)\in I\times I\times I$ where $I=[0,1]$ and ${\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1$. Define the mapping $S:C\to C$ as follows:

$$\begin{array}{c}{U}_0=I,\hfill \\ U_1={\alpha }_1^1{\mathcal{S}}_1{U}_0+{\alpha }_2^1{U}_0+{\alpha }_3^{1}I,\hfill \\ U_2={\alpha }_1^2{\mathcal{S}}_2{U}_1+{\alpha }_2^2{U}_1+{\alpha }_3^{2}I,\hfill \\ U_3={\alpha }_1^3{\mathcal{S}}_3{U}_2+{\alpha }_2^3{U}_2+{\alpha }_3^{3}I,\hfill \\ ⋮\hfill \\ U_{n-1}={\alpha }_1^{N-1}{\mathcal{S}}_{N-1}{U}_{N-2}+{\alpha }_2^{N-1}{U}_{N-2}+{\alpha }_3^{N-1}I,\hfill \\ S=U_n={\alpha }_1^N{\mathcal{S}}_N{U}_{n-1}+{\alpha }_2^N{U}_{n-1}+{\alpha }_3^{N}I.\hfill \end{array}$$

This mapping is called the S-mapping generated by ${\mathcal{S}}_1,{\mathcal{S}}_2,\dots ,{\mathcal{S}}_N$ and ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_N$.

Lemma 4.5 ([17])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let ${\{{\mathcal{S}}_i\}}_{i=1}^N$ be a finite family of nonspreading mappings of C into itself with ${\bigcap }_{i=1}^{N}Fix({\mathcal{S}}_i)\ne \mathrm{\varnothing }$ and let ${\alpha }_j=({\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j)\in I\times I\times I$ where $I=[0,1]$, ${\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1$, ${\alpha }_1^j,{\alpha }_3^j\in (0,1)$ for all $j=1,2,\dots ,N-1$ and ${\alpha }_1^N\in (0,1]$, ${\alpha }_3^N\in [0,1)$, ${\alpha }_2^j\in (0,1)$ for all $j=1,2,\dots ,N$. Let S be the S-mapping generated by ${\mathcal{S}}_1,{\mathcal{S}}_2,\dots ,{\mathcal{S}}_N$ and ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_N$. Then $Fix(S)={\bigcap }_{i=1}^{N}Fix({\mathcal{S}}_i)$ and S is a quasi-nonexpansive mapping.

Remark 4.6 From Lemma 4.5 it still holds if $C\equiv H$.

Theorem 4.7 Let C be a nonempty closed convex subset of a real Hilbert space H. Let $M:H\to 2^H$ be a multi-valued maximal monotone mapping. For every $i=1,2,\dots ,N$, let ${\mathrm{\Psi }}_i:C\times C\to \mathbb{R}$ be a bifunction satisfying (A1)-(A4) and $A_i:H\to H$ be ${\alpha }_i$-inverse strongly monotone mapping with $\eta ={min}_{i=1,2,\dots ,N}\{{\alpha }_i\}$. Let $\mathcal{S}:H\to H$ be a κ-quasi-strictly pseudo-contractive mapping. Assume $\mathrm{\Phi }:=Fix(\mathcal{S})\cap {\bigcap }_{i=1}^{N}EP({\mathrm{\Psi }}_i)\cap {\bigcap }_{i=1}^{N}VI(H,A_i,M)\ne \mathrm{\varnothing }$. Let the sequences $\{w_n\}$ and $\{z_n\}$ be generated by $w_1,\mu \in H$ and

$$\{\begin{array}{c}{\sum }_{i=1}^N{a}_i{\mathrm{\Psi }}_i(z_n,y)+\frac{1}{r_n}〈y-z_n,z_n-w_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ w_{n+1}={\alpha }_n\mu +{\beta }_n{w}_n+{\gamma }_n{J}_{M,\lambda }(I-\lambda {\sum }_{i=1}^N{b}_i{A}_i)w_n\hfill \\ \phantom{w_{n+1}=}+{\eta }_n(I-{\rho }_n(I-\mathcal{S}))w_n+{\delta }_n{z}_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(4.1)

where $\{{\alpha }_n\},\{{\beta }_n\},\{{\gamma }_n\},\{{\eta }_n\},\{{\delta }_n\}\subseteq (0,1)$, and $\lambda >0$ with ${\alpha }_n+{\beta }_n+{\gamma }_n+{\eta }_n+{\delta }_n=1$, $0<\alpha <1$ and $0\le a_i,b_i\le 1$, for every $i=1,2,\dots ,N$, $r_n\in [c,d]\subset (0,1)$, $0<p\le {\beta }_n,{\gamma }_n,{\eta }_n,{\delta }_n\le q<1$, ${\rho }_n\in (0,1-\kappa )$ for all $n\ge 1$. 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) ${\sum }_{n=1}^{\mathrm{\infty }}{\rho }_n<\mathrm{\infty }$,

• (iii) $0<\lambda <2\eta$, where $\eta ={min}_{i=1,2,\dots ,N}\{{\alpha }_i\}$,

• (iv) ${\sum }_{i=1}^N{a}_i={\sum }_{i=1}^N{b}_i=1$,

• (v) ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\gamma }_{n+1}-{\gamma }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\rho }_{n+1}-{\rho }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|r_{n+1}-r_n|<\mathrm{\infty }$.

Then the sequences $\{w_n\}$ and $\{z_n\}$ converge strongly to $\omega =P_{\mathrm{\Phi }}\mu$.

Proof Using Remark 4.4 and the same method of proof in Theorem 3.1, we have the desired conclusion. □

Theorem 4.8 Let C be a nonempty closed convex subset of a real Hilbert space H. Let $M:H\to 2^H$ be a multi-valued maximal monotone mapping. For every $i=1,2,\dots ,N$, let ${\mathrm{\Psi }}_i:C\times C\to \mathbb{R}$ be a bifunction satisfying (A1)-(A4), and let $A_i:H\to H$ be ${\alpha }_i$-inverse strongly monotone mapping with $\eta ={min}_{i=1,2,\dots ,N}\{{\alpha }_i\}$. Let ${\mathcal{S}}_i:H\to H$, for $i=1,2,\dots ,N$ be a finite family of nonspreading mappings with $\mathrm{\Psi }:={\bigcap }_{i=1}^{N}Fix({\mathcal{S}}_i)\cap {\bigcap }_{i=1}^{N}EP({\mathrm{\Psi }}_i)\cap {\bigcap }_{i=1}^{N}VI(H,A_i,M)\ne \mathrm{\varnothing }$. Let ${\theta }_j=({\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j)\in I\times I\times I$, $j=1,2,\dots ,N$, where $I=[0,1]$, ${\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1$, ${\alpha }_1^j,{\alpha }_3^j\in (0,1)$ for all $j=1,2,\dots ,N-1$, and ${\alpha }_1^N\in (0,1]$, ${\alpha }_3^N\in [0,1)$, ${\alpha }_2^j\in (0,1)$ for all $j=1,2,\dots ,N$, and let S be the S-mapping generated by ${\mathcal{S}}_1,{\mathcal{S}}_2,\dots ,{\mathcal{S}}_N$ and ${\theta }_1,{\theta }_2,\dots ,{\theta }_N$. Let the sequences $\{w_n\}$ and $\{z_n\}$ be generated by $w_1,\mu \in H$ and

$$\{\begin{array}{c}{\sum }_{i=1}^N{a}_i{\mathrm{\Psi }}_i(z_n,y)+\frac{1}{r_n}〈y-z_n,z_n-w_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ w_{n+1}={\alpha }_n\mu +{\beta }_n{w}_n+{\gamma }_n{J}_{M,\lambda }(I-\lambda {\sum }_{i=1}^N{b}_i{A}_i)w_n\hfill \\ \phantom{w_{n+1}=}+{\eta }_n(I-{\rho }_n(I-S))w_n+{\delta }_n{z}_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(4.2)

where $\{{\alpha }_n\},\{{\beta }_n\},\{{\gamma }_n\},\{{\eta }_n\},\{{\delta }_n\}\subseteq (0,1)$ and $\lambda >0$ with ${\alpha }_n+{\beta }_n+{\gamma }_n+{\eta }_n+{\delta }_n=1$, $0<\alpha <1$ and $0\le a_i,b_i\le 1$, for every $i=1,2,\dots ,N$, $r_n\in [c,d]\subset (0,1)$, $0<p\le {\beta }_n,{\gamma }_n,{\eta }_n,{\delta }_n\le q<1$, ${\rho }_n\in (0,1)$ for all $n\ge 1$. 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) ${\sum }_{n=1}^{\mathrm{\infty }}{\rho }_n<\mathrm{\infty }$,

• (iii) $0<\lambda <2\eta$, where $\eta ={min}_{i=1,2,\dots ,N}\{{\alpha }_i\}$,

• (iv) ${\sum }_{i=1}^N{a}_i={\sum }_{i=1}^N{b}_i=1$,

• (v) ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\beta }_{n+1}-{\beta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\gamma }_{n+1}-{\gamma }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\rho }_{n+1}-{\rho }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n+1}-{\delta }_n|<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|r_{n+1}-r_n|<\mathrm{\infty }$.

Then the sequences $\{w_n\}$ and $\{z_n\}$ converge strongly to $\omega =P_{\mathrm{\Phi }}\mu$.

Proof From Theorem 4.7 and Remark 4.6, we obtain the desired conclusion. □

5 Numerical results

The purpose of this section we give a numerical example to support our some result. The following example is given for supporting Theorem 3.1.

Example 5.1 Let ℝ be the set of real numbers. For every $i=1,2,\dots ,N$, let ${\mathrm{\Psi }}_i:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$, $A_i:\mathbb{R}\to \mathbb{R}$ be defined by

$$\begin{array}{c}{\mathrm{\Psi }}_i(u,v)=i(v-u)(3u+v),\hfill \\ A_{i}u=\frac{iu}{10},\hfill \end{array}$$

for all $u,v\in \mathbb{R}$ and let $\mathcal{S}:\mathbb{R}\to \mathbb{R}$ be defined by

$$\mathcal{S}u=\{\begin{array}{cc}\frac{-3u}{5}\hfill & \text{if }u\in [0,\mathrm{\infty }),\hfill \\ u\hfill & \text{if }u\in (-\mathrm{\infty },0).\hfill \end{array}$$

For every $i=1,2,\dots ,N$, suppose that $J_{M,\lambda }=I$, $\lambda =\frac{1}{N}$, $a_i=\frac{3}{4^i}+\frac{1}{N{4}^N}$, $b_i=\frac{8}{9^i}+\frac{1}{N{9}^N}$. Let $\{w_n\}$ and $\{z_n\}$ be generated by (3.1), where ${\alpha }_n=\frac{1}{20n}$, ${\beta }_n=\frac{3(20n-1)}{220n}$, ${\gamma }_n=\frac{2(20n-1)}{220n}$, ${\eta }_n=\frac{5(20n-1)}{220n}$, ${\delta }_n=\frac{20n-1}{220n}$, $r_n=\frac{3n}{5n+6}$, and ${\rho }_n=\frac{1}{2n^2}$ for every $n\in \mathbb{N}$. Then the sequences $\{w_n\}$ and $\{z_n\}$ converge strongly to 0.

Solution. It is easy to see that $\mathcal{S}u=u$ is a κ-strictly pseudononspreading mapping. Since $a_i=\frac{3}{4^i}+\frac{1}{N{4}^N}$, we obtain

$$\sum _{i=1}^N{a}_i{\mathrm{\Psi }}_i(u,v)=\sum _{i=1}^N(\frac{3}{4^i}+\frac{1}{N{4}^N})i(v-u)(v+3u).$$

It is easy to check that ${\mathrm{\Psi }}_i$ satisfies all the conditions of Theorem 3.1 and $EP({\sum }_{i=1}^N{a}_i{\mathrm{\Psi }}_i)={\bigcap }_{i=1}^{N}EP({\mathrm{\Psi }}_i)=\{0\}$. Then we have

$$Fix(\mathcal{S})\cap \bigcap _{i=1}^{N}EP({\mathrm{\Psi }}_i)=\{0\}.$$

(5.1)

Put $S_1={\sum }_{i=1}^N(\frac{3}{4^i}+\frac{1}{N{4}^N})i$, then we have

$$\begin{array}{c}0\le \sum _{i=1}^N{a}_i{\mathrm{\Psi }}_i(z_n,y)+\frac{1}{r_n}〈y-z_n,z_n-w_n〉\hfill \\ \phantom{0}=S_1(y-z_n)(y+3z_n)+\frac{1}{r_n}(y-z_n)(z_n-w_n)\hfill \\ \iff 0\le S_1{r}_n(y-z_n)(y+3z_n)+(y-z_n)(z_n-w_n)\hfill \\ \phantom{\iff 0}=S_1{r}_n{y}^2+(z_n+2r_n{S}_1{z}_n-w_n)y+z_n{w}_n-3r_n{S}_1{z}_n^2-z_n^2.\hfill \end{array}$$

Let $G(y)=S_1{r}_n{y}^2+(z_n+2r_n{S}_1{z}_n-w_n)y+z_n{w}_n-3r_n{S}_1{z}_n^2-z_n^2$. $G(y)$ is a quadratic function of y with coefficient $a=S_1{r}_n$, $b=z_n+2r_n{S}_1{z}_n-w_n$, and $c=z_n{w}_n-3r_n{S}_1{z}_n^2-z_n^2$. Determine the discriminant Δ of G as follows:

$$\begin{array}{rl}\mathrm{\Delta }& =b^2-4ac\\ ={(z_n+2r_n{S}_1{z}_n-w_n)}^2-4(S_1{r}_n)(z_n{w}_n-3r_n{S}_1{z}_n^2-z_n^2)\\ =z_n^2+8r_n{S}_1{z}_n^2+16r_n^2{S}_1^2{z}_n^2-2z_n{w}_n-8r_n{S}_1{z}_n{w}_n+w_n^2\\ ={(z_n+4S_1{r}_n{z}_n-w_n)}^2.\end{array}$$

We know that $G(y)\ge 0$, $\mathrm{\forall }y\in \mathbb{R}$. If it has at most one solution in ℝ, then $\mathrm{\Delta }\le 0$, so we obtain

$$z_n=\frac{w_n}{1+4S_1{r}_n},$$

(5.2)

where $S_1={\sum }_{i=1}^N(\frac{3}{4^i}+\frac{1}{N{4}^N})i$.

Since $A_{i}u=\frac{iu}{10}$ and $b_i=\frac{8}{9^i}+\frac{1}{N{9}^N}$,

$$\sum _{i=1}^N{b}_i{A}_{i}u=\sum _{i=1}^N(\frac{8}{9^i}+\frac{1}{N{9}^N})\frac{iu}{10}.$$

From (5.1) and the definition of $A_i$, we have

$$Fix(\mathcal{S})\cap \bigcap _{i=1}^{N}EP({\mathrm{\Psi }}_i)\cap \bigcap _{i=1}^{N}VI(H,A_i,M)=\{0\}.$$

(5.3)

For every $n\in \mathbb{N}$, ${\alpha }_n=\frac{1}{20n}$, ${\beta }_n=\frac{3(20n-1)}{220n}$, ${\gamma }_n=\frac{2(20n-1)}{220n}$, ${\eta }_n=\frac{20n-1}{220n}$, ${\delta }_n=\frac{5(20n-1)}{220}$