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# The modification of system of variational inequalities for fixed point theory in Banach spaces

- Atid Kangtunyakarn
^{1}Email author

**2014**:123

https://doi.org/10.1186/1687-1812-2014-123

© Kangtunyakarn; licensee Springer. 2014

**Received:**2 February 2014**Accepted:**5 May 2014**Published:**19 May 2014

## Abstract

In this paper, we use methods different from extragradient methods to prove a strong convergence theorem for the sets of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings and the set of solutions of modification of a system of variational inequalities problems in a uniformly convex and 2-uniformly smooth Banach space. Applying the main result we obtain a strong convergence theorem involving two sets of solutions of variational inequalities problems introduced by Aoyama *et al.* (Fixed Point Theory Appl. 2006:35390, 2006, doi:10.1155/FPTA/2006/35390) in a uniformly convex and 2-uniformly smooth Banach space. We also give a numerical example to support our result.

## Keywords

- nonexpansive mapping
- strictly pseudo-contractive mapping
- the modification of system of variational inequalities problems

## 1 Introduction

Let *E* be a real Banach space with its dual space ${E}^{\ast}$ and let *C* be a nonempty closed convex subset of *E*. Throughout this paper, we denote the norm of *E* and ${E}^{\ast}$ by the same symbol $\parallel \cdot \parallel $. We use the symbols ‘→’ and ‘⇀’ to denote strong and weak convergence, respectively. Recall the following definitions.

**Definition 1.1** A Banach space *E* is said to be *uniformly convex* iff for any *ϵ*, $0<\u03f5\le 2$, the inequalities $\parallel x\parallel \le 1$, $\parallel y\parallel \le 1$ and $\parallel x-y\parallel \ge \u03f5$ imply there exists a $\delta >0$ such that $\parallel \frac{x+y}{2}\parallel \le 1-\delta $.

**Definition 1.2**Let

*E*be a Banach space. Then a function ${\rho}_{E}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is said to be

*the modulus of smoothness of*

*E*if

*E*is said to be

*uniformly smooth*if

*E*is said to be

*q*-uniformly smooth if there exists a fixed constant $c>0$ such that ${\rho}_{E}(t)\le c{t}^{q}$. It is easy to see that, if

*E*is

*q*-uniformly smooth, then $q\le 2$ and

*E*is uniformly smooth. Hilbert space, ${L}_{P}$ (or ${l}_{p}$) spaces, $0<p<\mathrm{\infty}$ and the Sobolev spaces, ${W}_{m}^{p}$, $0<p<\mathrm{\infty}$ are

*q*-uniformly smooth. Hilbert spaces are 2-uniformly smooth, while

**Definition 1.3**A mapping

*J*from

*E*onto ${E}^{\ast}$ satisfying the condition

is called the normalized duality mapping of *E*. The duality pair $\u3008x,f\u3009$ represents $f(x)$ for $f\in {E}^{\ast}$ and $x\in E$.

It is well known that if *E* is smooth, then *J* is a single value, which we denote by *j*.

**Definition 1.4**Let

*C*be a nonempty subset of a Banach space

*E*and $T:C\to C$ be a self-mapping.

*T*is called a nonexpansive mapping if

for all $x,y\in C$.

*T*is called an

*η*-strictly pseudo-contractive mapping if there exists a constant $\eta \in (0,1)$ such that

for every $x,y\in C$ and for some $j(x-y)\in J(x-y)$.

**Example 1.1**Let ℝ be a real line endowed with Euclidean norm and let the mapping $T:(0,\frac{1}{2})\to (0,\frac{1}{2})$ defined by

for all $x\in (0,\frac{1}{2})$. Then *T* is $\frac{3}{4}$-strictly pseudo-contractive mapping.

**Example 1.2** Let *E* be 2-uniformly smooth Banach space and let $T:E\to E$ be *λ*-strictly pseudo-contractive mapping. Let *K* be the 2-uniformly smooth constant of *E* and $0\le d\le \frac{\lambda}{{K}^{2}}$, then $(I-d(I-T))$ is a nonexpansive mapping.

**Definition 1.5** Let $C\subseteq E$ be closed convex and ${Q}_{C}$ be a mapping of *E* onto *C*. The mapping ${Q}_{C}$ is said to be *sunny* if ${Q}_{C}({Q}_{C}x+t(x-{Q}_{C}x))={Q}_{C}x$ for all $x\in E$ and $t\ge 0$. A mapping ${Q}_{C}$ is called *retraction* if ${Q}_{C}^{2}={Q}_{C}$. A subset *C* of *E* is called a sunny nonexpansive retract of *E* if there exists a sunny nonexpansive retraction of *E* onto *C*.

*A*of

*C*into

*E*is said to be

*accretive*if there exists $j(x-y)\in J(x-y)$ such that

*α*-

*inverse strongly accretive*if there exist $j(x-y)\in J(x-y)$ and $\alpha >0$ such that

**Remark 1.1** From (1.2) and (1.3), if *T* is an *η*-strictly pseudo-contractive mapping, then $I-T$ is an *η*-inverse strongly accretive.

In 2000, Ansari and Yao [1] introduced the system of generalized implicit variational inequalities and proved the existence of its solution. They derived the existence results for a solution of system of generalized variational inequalities and used their results as tools to establish the existence of a solution of system of optimization problems.

Ansari *et al.* [2] introduced the system of vector equilibrium problems and prove the existence of its solution. Moreover, they also applied their result to the system of vector variational inequalities. The results of [1] and [2] were used as tools to solve Nash problem for vector-value functions and (non)convex vector valued function.

*et al.*[3] introduced the system of general variational inequalities problem for finding $({x}^{\ast},{y}^{\ast})\in C\times C$ such that

They proved fixed points theorem by using modification of extragradient methods as follows.

**Theorem 1.2**

*Let*

*C*

*be a nonempty closed convex subset of a uniformly convex and*2-

*uniformly smooth Banach space*

*E*

*which admits a weakly sequentially continuous duality mapping*.

*Let*${Q}_{C}$

*be the sunny nonexpansive retraction from*

*X*

*into*

*C*.

*Let the mappings*$A,B:C\to E$

*be*

*α*-

*inverse strongly accretive with*$\alpha \ge {K}^{2}$

*and*

*β*-

*inverse strongly accretive with*$\beta \ge {K}^{2}$,

*respectively*.

*Define the mapping by*$Gx={Q}_{C}({Q}_{C}(x-Bx)-\lambda A{Q}_{C}(x-Bx))$

*for all*$x\in C$

*and the set of fixed point of*

*G*

*denoted by*$\mathrm{\Omega}\ne \mathrm{\varnothing}$.

*For given*${x}_{0}\in C$,

*let the sequence*$\{{x}_{n}\}$

*be generated by*

*where*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$,

*and*$\{{\gamma}_{n}\}$

*are three sequences in*$(0,1)$.

*Suppose the sequences*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$,

*and*$\{{\gamma}_{n}\}$

*satisfy the following conditions*:

*Then* $\{{x}_{n}\}$ *converges strongly to* ${Q}^{\prime}u$, *where* ${Q}^{\prime}$ *is the sunny nonexpansive retraction of* *C* *onto* Ω.

*et al.*[5]. If $A=B$, then (1.6) reduces to a problem for finding $({x}^{\ast},{y}^{\ast})\in C\times C$ such that

Variational inequality theory is one of very important mathematical tools for solving many problems in economic, engineering, physical, pure and applied science *etc.*

Many authors have studied the iterative scheme for finding the solutions of a variational inequality problem; see for example [7–10].

By using the extragradient methods, Cai and Bu [4] proved a strong convergence theorem for finding the solutions of (1.5) as follows.

**Theorem 1.3**

*Let*

*C*

*be a nonempty closed convex subset of a*2-

*uniformly smooth and uniformly convex Banach space*

*E*

*such that*$C\pm C\subset C$.

*Let*${P}_{C}$

*be the sunny nonexpansive retraction from*

*E*

*to*

*C*.

*Let the mapping*$A,B:C\to E$

*be*

*α*-

*inverse strongly accretive and*

*β*-

*inverse strongly accretive*,

*respectively*.

*Let*${\{{T}_{i}:C\to C\}}_{i=0}^{\mathrm{\infty}}$

*be an infinite family of nonexpansive mapping with*$F={\bigcap}_{i=0}^{\mathrm{\infty}}\cap {\mathrm{\Omega}}^{\prime}\ne \mathrm{\varnothing}$.

*Let*$S:C\to C$

*be a nonexpansive mapping and*$D:C\to C$

*be a strongly positive linear bounded operator with the coefficient*$\overline{\gamma}$

*such that*$0<\gamma <\overline{\gamma}$.

*For arbitrarily given*${x}_{0}\in C$,

*let the sequence*$\{{x}_{n}\}$

*be generated iteratively by*

*where*$0<\lambda <\frac{\alpha}{{K}^{2}}$

*and*$0<\mu <\frac{\beta}{{K}^{2}}$.

*Assume that*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$,

*and*$\{{\gamma}_{n}\}$

*are three sequences in*$[0,1]$

*satisfying the following conditions*:

*Suppose that for any bounded subset*${D}^{\prime}$

*of*

*C*

*there exists an increasing*,

*continuous*,

*and convex function*${h}_{{D}^{\prime}}$

*from*${\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$

*such that*${h}_{{D}^{\prime}}(0)=0$

*and*${lim}_{k,l\to \mathrm{\infty}}sup\{{h}_{{D}^{\prime}}(\parallel {T}_{k}z-{T}_{l}z\parallel ):z\in {D}^{\prime}\}=0$.

*Let*

*T*

*be a mapping from*

*C*

*into*

*C*

*defined by*$Tx={lim}_{n\to \mathrm{\infty}}{T}_{n}x$

*for all*$x\in C$

*and suppose that*$F(T)={\bigcap}_{i=0}^{\mathrm{\infty}}F({T}_{i})$.

*Then*$\{{x}_{n}\}$

*converges strongly to*$z\in F$,

*which also solves the following variational inequality*:

For the research related to the extragradient methods, some additional references are [11–13].

for all $x\in C$, ${\lambda}_{A},{\lambda}_{B}>0$ and $a\in [0,1]$. This problem is called *the modification of a system of variational inequalities problems* in Banach space. If $a=0$, then (1.8) reduces to (1.5).

Motivated by Theorems 1.2 and 1.3, we use the methods different from extragradient methods to prove a strong convergence theorem for finding the solutions of (1.8) and an element of the set of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings in a uniformly convex and 2-uniformly smooth Banach space. Applying the main result, we obtain a strong convergence theorem involving two sets of solutions of variational inequalities problems introduced by Aoyama *et al*. [14] in a uniformly convex and 2-uniformly smooth Banach space. Moreover, we also give a numerical example to support our main results in the last section.

## 2 Preliminaries

The following lemmas and definitions are important tools to prove the results in the next sections.

**Definition 2.1** ([15])

*C*be a nonempty convex subset of a Banach space. Let ${\{{S}_{i}\}}_{i=1}^{N}$ and ${\{{T}_{i}\}}_{i=1}^{N}$ be two finite families of mappings 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 mapping ${S}^{A}:C\to C$ as follows:

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

**Lemma 2.1** ([15])

*Let* *C* *be a nonempty closed convex subset of a uniformly convex and* 2-*uniformly smooth Banach space*. *Let* ${\{{S}_{i}\}}_{i=1}^{N}$ *be a finite family of* ${\kappa}_{i}$-*strict pseudo*-*contractions of* *C* *into itself and let* ${\{{T}_{i}\}}_{i=1}^{N}$ *be a finite family of nonexpansive mappings of* *C* *into itself with* ${\bigcap}_{i=1}^{N}F({S}_{i})\cap {\bigcap}_{i=1}^{N}F({T}_{i})\ne \mathrm{\varnothing}$ *and* $\kappa =min\{{\kappa}_{i}:i=1,2,\dots ,N\}$ *with* ${K}^{2}\le \kappa $, *where* *K* *is the* 2-*uniformly smooth constant of* *E*. *Let* ${\alpha}_{j}=({\alpha}_{1}^{j},{\alpha}_{2}^{j},{\alpha}_{3}^{j})\in I\times I\times I$, *where* $I=[0,1]$, ${\alpha}_{1}^{j}+{\alpha}_{2}^{j}+{\alpha}_{3}^{j}=1$, ${\alpha}_{1}^{j}\in (0,1]$, ${\alpha}_{2}^{j}\in [0,1]$ *and* ${\alpha}_{3}^{j}\in (0,1)$ *for all* $j=1,2,\dots ,N$. *Let* ${S}^{A}$ *be the* ${S}^{A}$-*mapping generated by* ${S}_{1},{S}_{2},\dots ,{S}_{N}$, ${T}_{1},{T}_{2},\dots ,{T}_{N}$, *and* ${\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{N}$. *Then* $F({S}^{A})={\bigcap}_{i=1}^{N}F({S}_{i})\cap {\bigcap}_{i=1}^{N}F({T}_{i})$ *and* ${S}^{A}$ *is a nonexpansive mapping*.

**Lemma 2.2** ([16])

*Let*$\{{s}_{n}\}$

*be a sequence of nonnegative real numbers satisfying*

*where*$\{{\alpha}_{n}\}$

*is a sequence in*$(0,1)$

*and*$\{{\delta}_{n}\}$

*is a sequence such that*

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

**Lemma 2.3** ([17])

*Let*

*E*

*be a real*2-

*uniformly smooth Banach space with the best smooth constant*

*K*.

*Then the following inequality holds*:

*for any* $x,y\in E$.

**Lemma 2.4** ([18])

*Let* *C* *be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space* *E* *and let* *T* *be a nonexpansive mapping of* *C* *into itself with* $F(T)\ne \mathrm{\varnothing}$. *Then* $F(T)$ *is a sunny nonexpansive retract of* *C*.

**Lemma 2.5** ([19])

*Let*

*C*

*be a nonempty closed convex subset of a smooth Banach space and*${Q}_{C}$

*be a retraction from*

*E*

*onto*

*C*.

*Then the following are equivalent*:

- (i)
${Q}_{C}$

*is both sunny and nonexpansive*; - (ii)
$\u3008x-{Q}_{C}x,J(y-{Q}_{C}x)\u3009\le 0$

*for all*$x\in E$*and*$y\in C$.

It is obvious that if *E* is a Hilbert space, we find that a sunny nonexpansive retraction ${Q}_{C}$ is coincident with the metric projection from *E* onto *C*. From Lemma 2.5, let $x\in E$ and ${x}_{0}\in C$. Then we have ${x}_{0}={Q}_{C}x$ if and only if $\u3008x-{x}_{0},J(y-{x}_{0})\u3009\le 0$, for all $y\in C$, where ${Q}_{C}$ is a sunny nonexpansive retraction from *E* onto *C*.

**Lemma 2.6** ([20])

*Let*

*E*

*be a uniformly convex Banach space and*${B}_{r}=\{x\in E:\parallel x\parallel \le r\}$, $r>0$.

*Then there exists a continuous*,

*strictly increasing*,

*and convex function*$g:[0,\mathrm{\infty}]\to [0,\mathrm{\infty}]$, $g(0)=0$

*such that*

*for all* $x,y,z\in {B}_{r}$ *and all* $\alpha ,\beta ,\gamma \in [0,1]$ *with* $\alpha +\beta +\gamma =1$.

**Lemma 2.7** ([21])

*Let*

*C*

*be a closed and convex subset of a real uniformly smooth Banach space*

*E*

*and let*$T:C\to C$

*be a nonexpansive mapping with a nonempty fixed point*$F(T)$.

*If*$\{{x}_{n}\}\subset C$

*is a bounded sequence such that*${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-T{x}_{n}\parallel =0$.

*Then there exists a unique sunny nonexpansive retraction*${Q}_{F(T)}:C\to F(T)$

*such that*

*for any given* $u\in C$.

**Lemma 2.8** ([17])

*Let* $r>0$. *If* *E* *is uniformly convex*, *then there exists a continuous*, *strictly increasing*, *and convex function* $g:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$, $g(0)=0$ *such that for all* $x,y\in {B}_{r}(0)=\{x\in E:\parallel x\parallel \le r\}$ *and for any* $\alpha \in [0,1]$, *we have* ${\parallel \alpha x+(1-\alpha )y\parallel}^{2}\le \alpha {\parallel x\parallel}^{2}+(1-\alpha ){\parallel y\parallel}^{2}-\alpha (1-\alpha )g(\parallel x-y\parallel )$.

**Lemma 2.9** ([22])

*Let*

*C*

*be a closed convex subset of a strictly convex Banach space*

*E*.

*Let*${T}_{1}$

*and*${T}_{2}$

*be two nonexpansive mappings from*

*C*

*into itself with*$F({T}_{1})\cap F({T}_{2})\ne \mathrm{\varnothing}$.

*Define a mapping*

*S*

*by*

*where* *λ* *is a constant in* $(0,1)$. *Then* *S* *is nonexpansive and* $F(S)=F({T}_{1})\cap F({T}_{2})$.

**Lemma 2.10**

*Let*

*C*

*be a nonempty closed convex subset of a smooth Banach space*

*E*

*and let*$A,B:C\to E$

*be mappings*.

*Let*${Q}_{C}$

*be a sunny nonexpansive retraction of*

*E*

*onto*

*C*.

*For every*${\lambda}_{A},{\lambda}_{B}>0$

*and*$a\in [0,1]$.

*The following are equivalent*:

- (a)
$({x}^{\ast},{z}^{\ast})$

*is a solution of*(1.8); - (b)${x}^{\ast}$
*is a fixed point of mapping*$G:C\to C$,*i*.*e*., ${x}^{\ast}\in F(G)$,*defined by*$Gx={Q}_{C}(I-{\lambda}_{A}A)(aI+(1-a){Q}_{C}(I-{\lambda}_{B}B))x,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in C,$

*where* ${z}^{\ast}={Q}_{C}(I-{\lambda}_{B}B){x}^{\ast}$.

*Proof*First we show that (a) ⇒ (b). Let $({x}^{\ast},{z}^{\ast})$ is a solution of (1.8), and we have

and ${z}^{\ast}={Q}_{C}(I-{\lambda}_{B}B){x}^{\ast}$.

Then ${x}^{\ast}\in F(G)$, where ${z}^{\ast}={Q}_{C}(I-{\lambda}_{B}B){x}^{\ast}$.

for all $x\in C$. Then we find that $({x}^{\ast},{z}^{\ast})$ is a solution of (1.8). □

**Example 2.1**Let ℝ be a real line with the Euclidean norm and let $A,B:\mathbb{R}\to \mathbb{R}$ defined by $Ax=\frac{x-1}{4}$ and $Bx=\frac{x-1}{2}$ for all $x\in \mathbb{R}$. The mapping $G:\mathbb{R}\to \mathbb{R}$ defined by

for all $x\in \mathbb{R}$. Then $1\in F(G)$ and $(1,1)$ is a solution of (1.8).

## 3 Main results

**Theorem 3.1**

*Let*

*C*

*be a nonempty closed convex subset of a uniformly convex and*2-

*uniformly smooth Banach space*

*E*

*and let*${Q}_{C}$

*be a sunny nonexpansive retraction of*

*E*

*onto C*.

*Let*$A,B:C\to E$

*be*

*α*-

*and*

*β*-

*inverse strongly accretive operators*,

*respectively*.

*Define the mapping*$G:C\to C$

*by*$Gx={Q}_{C}(I-{\lambda}_{A}A)(aI+(1-a){Q}_{C}(I-{\lambda}_{B}B))x$

*for all*$x\in C$, ${\lambda}_{A}\in (0,\frac{\alpha}{{K}^{2}})$, ${\lambda}_{B}\in (0,\frac{\beta}{{K}^{2}})$

*and*$a\in [0,1]$,

*where*

*K*

*is the*2-

*uniformly smooth constant of*

*E*.

*Let*${\{{S}_{i}\}}_{i=1}^{N}$

*be a finite family of*${\kappa}_{i}$-

*strict pseudo*-

*contractions of*

*C*

*into itself and let*${\{{T}_{i}\}}_{i=1}^{N}$

*be a finite family of nonexpansive mappings of*

*C*

*into itself and*$\kappa =min\{{\kappa}_{i}:i=1,2,\dots ,N\}$

*with*${K}^{2}\le \kappa $.

*Let*${\alpha}_{j}=({\alpha}_{1}^{j},{\alpha}_{2}^{j},{\alpha}_{3}^{j})\in I\times I\times I$,

*where*$I=[0,1]$, ${\alpha}_{1}^{j}+{\alpha}_{2}^{j}+{\alpha}_{3}^{j}=1$, ${\alpha}_{1}^{j}\in (0,1]$, ${\alpha}_{2}^{j}\in [0,1]$,

*and*${\alpha}_{3}^{j}\in (0,1)$

*for all*$j=1,2,\dots ,N$.

*Let*${S}^{A}$

*be the*${S}^{A}$-

*mapping generated by*${S}_{1},{S}_{2},\dots ,{S}_{N}$, ${T}_{1},{T}_{2},\dots ,{T}_{N}$,

*and*${\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{N}$.

*Assume that*$\mathcal{F}=F(G)\cap {\bigcap}_{i=1}^{N}F({S}_{i})\cap {\bigcap}_{i=1}^{N}F({T}_{i})\ne \mathrm{\varnothing}$.

*Let the sequence*$\{{x}_{n}\}$

*be generated by*$u,{x}_{1}\in C$

*and*

*where*$\{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subseteq [0,1]$

*with*${\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1$.

*Suppose that the following conditions are satisfied*:

*Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${x}_{0}={Q}_{\mathcal{F}}u$ *and* $({x}_{0},{z}_{0})$ *is a solution of* (1.8), *where* ${z}_{0}={Q}_{C}(I-{\lambda}_{B}B){x}_{0}$.

*Proof*First, we show that ${Q}_{C}(I-{\lambda}_{A}A)$ and ${Q}_{C}(I-{\lambda}_{B}B)$ are nonexpansive mappings. Let $x,y\in C$; we have

*G*, we see that

*G*is a nonexpansive mapping. Let ${x}^{\ast}\in \mathcal{F}$. Put ${y}_{n}={\alpha}_{n}u+{\beta}_{n}{x}_{n}+{\gamma}_{n}{S}^{A}{x}_{n}$ for all $n\ge 1$. From the definition of ${x}_{n}$ and Lemma 2.10, we have

Applying mathematical induction, we can conclude that the sequence $\{{x}_{n}\}$ is bounded and so is $\{{y}_{n}\}$.

*g*, we have

*B*, (3.7) and (3.8), we have

*G*and ${S}^{A}$ are nonexpansive mappings, we have

*B*is a nonexpansive mapping. From Lemma 2.7, we have

where ${x}_{0}={Q}_{\mathcal{F}}u$.

Applying Lemma 2.2, the condition (i) and (3.10), we can conclude that the sequence $\{{x}_{n}\}$ converges strongly to ${x}_{0}={Q}_{\mathcal{F}}u$ and $({x}_{0},{z}_{0})$ is a solution of (1.8), where ${z}_{0}={Q}_{C}(I-{\lambda}_{B}B){x}_{0}$. This completes the proof. □

The following corollary is a strong convergence theorem involving problem (1.5). This result is a direct proof from Theorem 3.1. We, therefore, omit the proof.

**Corollary 3.2**

*Let*

*C*

*be a nonempty closed convex subset of a uniformly convex and*2-

*uniformly smooth Banach space*

*E*

*and let*${Q}_{C}$

*be a sunny nonexpansive retraction of*

*E*

*onto*

*C*.

*Let*$A,B:C\to E$

*be*

*α*-

*and*

*β*-

*inverse strongly accretive operators*,

*respectively*.

*Define the mapping*$G:C\to C$

*by*$Gx={Q}_{C}(I-{\lambda}_{A}A)({Q}_{C}(I-{\lambda}_{B}B))x$

*for all*$x\in C$, ${\lambda}_{A}\in (0,\frac{\alpha}{{K}^{2}})$, ${\lambda}_{B}\in (0,\frac{\beta}{{K}^{2}})$,

*where*

*K*

*is the*2-

*uniformly smooth constant of*

*E*.

*Let*${\{{S}_{i}\}}_{i=1}^{N}$

*be a finite family of*${\kappa}_{i}$-

*strict pseudo*-

*contractions of*

*C*

*into itself and let*${\{{T}_{i}\}}_{i=1}^{N}$

*be a finite family of nonexpansive mappings of*

*C*

*into itself and*$\kappa =min\{{\kappa}_{i}:i=1,2,\dots ,N\}$

*with*${K}^{2}\le \kappa $.

*Let*${\alpha}_{j}=({\alpha}_{1}^{j},{\alpha}_{2}^{j},{\alpha}_{3}^{j})\in I\times I\times I$,

*where*$I=[0,1]$, ${\alpha}_{1}^{j}+{\alpha}_{2}^{j}+{\alpha}_{3}^{j}=1$, ${\alpha}_{1}^{j}\in (0,1]$, ${\alpha}_{2}^{j}\in [0,1]$,

*and*${\alpha}_{3}^{j}\in (0,1)$

*for all*$j=1,2,\dots ,N$.

*Let*${S}^{A}$

*be the*${S}^{A}$-

*mapping generated by*${S}_{1},{S}_{2},\dots ,{S}_{N}$, ${T}_{1},{T}_{2},\dots ,{T}_{N}$,

*and*${\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{N}$.

*Assume that*$\mathcal{F}=F(G){\bigcap}_{i=1}^{N}F({S}_{i})\cap {\bigcap}_{i=1}^{N}F({T}_{i})\ne \mathrm{\varnothing}$.

*Let the sequence*$\{{x}_{n}\}$

*be generated by*$u,{x}_{1}\in C$

*and*

*where*$\{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subseteq [0,1]$

*with*${\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1$.

*Suppose that the following conditions are satisfied*:

*Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${x}_{0}={Q}_{\mathcal{F}}u$ *and* $({x}_{0},{z}_{0})$ *is a solution of* (1.5), *where* ${z}_{0}={Q}_{C}(I-{\lambda}_{B}B){x}_{0}$.

## 4 Applications

In this section, we prove a strong convergence theorem involving two sets of solutions of variational inequalities in Banach space. We give some useful lemmas and definitions to prove Theorem 4.4.

*et al*. [14]. The set of solutions of the variational inequality in a Banach space is denoted by $S(C,A)$, that is,

The variational inequalities problems have been studied by many authors; see, for example, [11, 23].

**Lemma 4.1** ([14])

*Let*

*C*

*be a nonempty closed convex subset of a smooth Banach space*

*E*.

*Let*${Q}_{C}$

*be a sunny nonexpansive retraction from*

*E*

*onto*

*C*

*and let*

*A*

*be an accretive operator of*

*C*

*into*

*E*.

*Then*,

*for all*$\lambda >0$,

**Lemma 4.2** *Let* *C* *be a nonempty closed convex subset of a uniformly convex Banach space E*. *Let* $T,S:C\to C$ *be nonexpansive mappings with* $F(T)\cap F(S)\ne \mathrm{\varnothing}$. *Define the mapping* ${T}_{a}:C\to C$ *by* ${T}_{a}x=S(ax+(1-a)Tx)$ *for all* $x\in C$ *and* $a\in (0,1)$. *Then* $F({T}_{a})=F(T)\cap F(S)$ *and* ${T}_{a}$ *is a nonexpansive mapping*.

*Proof*It is easy to see that $F(T)\cap F(S)\subseteq F({T}_{a})$. Let ${x}_{0}\in F({T}_{a})$ and ${x}^{\ast}\in F(S)\cap F(T)$. From the definition of ${T}_{a}$, we have

*g*, we have ${x}_{0}=T{x}_{0}$, that is, ${x}_{0}\in F(T)$. Since ${x}_{0}\in F({T}_{a})$ and ${x}_{0}\in F(T)$, we have

It follows that ${x}_{0}\in F(S)$. Hence $F({T}_{a})\subseteq F(T)\cap F(S)$. Applying (4.3), we have ${T}_{a}$ is a nonexpansive mapping. □

**Lemma 4.3** *Let* *C* *be a nonempty closed convex subset of a uniformly convex and* 2-*uniformly smooth Banach space* *E* *and let* ${Q}_{C}$ *be a sunny nonexpansive retraction from* *E* *onto* *C*. *Let* $A,B:C\to E$ *be* *α*- *and* *β*-*inverse strongly accretive operators*, *respectively*. *Define a mapping* *G* *as in Lemma * 2.10 *and for every* ${\lambda}_{A}\in (0,\frac{\alpha}{{K}^{2}})$, ${\lambda}_{B}\in (0,\frac{\beta}{{K}^{2}})$ *and* $a\in (0,1)$ *where* *K* *is* 2-*uniformly smooth constant*. *If* $S(C,A)\cap S(C,B)\ne \mathrm{\varnothing}$, *then* $F(G)=S(C,A)\cap S(C,B)$.

*Proof*From Lemma 4.1, we have

Using the same method as Theorem 3.1, we find that ${Q}_{C}(I-{\lambda}_{A}A)$ and ${Q}_{C}(I-{\lambda}_{B}B)$ are nonexpansive mappings.

*G*and Lemma 4.2, we have

□

From Theorem 3.1 and Lemma 4.3, we have the following theorem.

**Theorem 4.4**

*Let*

*C*

*be a nonempty closed convex subset of a uniformly convex and*2-

*uniformly smooth Banach space*

*E*

*and let*${Q}_{C}$

*be a sunny nonexpansive retraction of*

*E*

*onto*

*C*.

*Let*$A,B:C\to E$

*be*

*α*-

*and*

*β*-

*inverse strongly accretive operators*,

*respectively*.

*Let*${\{{S}_{i}\}}_{i=1}^{N}$

*be a finite family of*${\kappa}_{i}$-

*strict pseudo*-

*contractions of*

*C*

*into itself and let*${\{{T}_{i}\}}_{i=1}^{N}$

*be a finite family of nonexpansive mappings of*

*C*

*into itself and*$\kappa =min\{{\kappa}_{i}:i=1,2,\dots ,N\}$

*with*${K}^{2}\le \kappa $,

*where*

*K*

*is the*2-

*uniformly smooth constant of*

*E*.

*Let*${\alpha}_{j}=({\alpha}_{1}^{j},{\alpha}_{2}^{j},{\alpha}_{3}^{j})\in I\times I\times I$,

*where*$I=[0,1]$, ${\alpha}_{1}^{j}+{\alpha}_{2}^{j}+{\alpha}_{3}^{j}=1$, ${\alpha}_{1}^{j}\in (0,1]$, ${\alpha}_{2}^{j}\in [0,1]$,

*and*${\alpha}_{3}^{j}\in (0,1)$

*for all*$j=1,2,\dots ,N$.

*Let*${S}^{A}$

*be the*${S}^{A}$-

*mapping generated by*${S}_{1},{S}_{2},\dots ,{S}_{N}$, ${T}_{1},{T}_{2},\dots ,{T}_{N}$,

*and*${\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{N}$.

*Assume that*$\mathcal{F}=S(C,A)\cap S(C,B){\bigcap}_{i=1}^{N}F({S}_{i})\cap {\bigcap}_{i=1}^{N}F({T}_{i})\ne \mathrm{\varnothing}$.

*Let the sequence*$\{{x}_{n}\}$

*be generated by*$u,{x}_{1}\in C$,

*and*

*where*$\{{\alpha}_{n}\},\{{\beta}_{n}\},\{{\gamma}_{n}\}\subseteq [0,1]$

*and*$a\in (0,1)$

*with*${\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1$, ${\lambda}_{A}\in (0,\frac{\alpha}{{K}^{2}})$, ${\lambda}_{B}\in (0,\frac{\beta}{{K}^{2}})$.

*Suppose that the following conditions are satisfied*:

*Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${x}_{0}={Q}_{\mathcal{F}}u$ *and* $({x}_{0},{z}_{0})$ *is a solution of* (1.8), *where* ${z}_{0}={Q}_{C}(I-{\lambda}_{B}B){x}_{0}$.

From Theorem 4.4, we have the following result.

**Example 4.1** Let ${l}^{2}=\{x={({x}_{i})}_{i=1}^{\mathrm{\infty}}:{\sum}_{i=1}^{\mathrm{\infty}}{|{x}_{i}|}^{2}<\mathrm{\infty}\}$ with norm define by $\parallel x\parallel ={({\sum}_{i=1}^{\mathrm{\infty}}|{x}_{i}|)}^{\frac{1}{2}}$. Define the mappings $A,B:{l}^{2}\to {l}^{2}$ by $Ax=2x$ and $Bx=3x$ for all $x={({x}_{i})}_{i=1}^{\mathrm{\infty}}\in {l}^{2}$.

where ${Q}_{{l}^{2}}$ is a sunny nonexpansive retraction of ${l}^{2}$ onto ${l}^{2}$. Then the sequence $\{{x}_{n}\}$ converges strongly to 0 and $(0,0)$ is a solution of (1.8).

**Remark 4.5** If $E={l}_{p}$ ($p\ge 2$), then Theorem 4.4 also holds.

## 5 Example and numerical results

In this section, we give a numerical example to support the main result.

**Example 5.1** Let ℝ be the real line with Euclidean norm and let $C=[0,\frac{\pi}{2}]$ and $A,B:C\to \mathbb{R}$ be mappings defined by $Ax=\frac{x}{2}$ and $Bx=\frac{x}{4}$ for all $x\in C$. For every $i=1,2,\dots ,N$, define the mapping ${S}_{i},{T}_{i}:C\to C$ by ${T}_{i}x=\frac{sinx}{i}$ and ${S}_{i}x=\frac{{x}^{2}}{x+i}$ for all $x\in C$ and $\frac{1}{{(N+1)}^{2}}\le \frac{1}{{N}^{2}}$.

Suppose that ${S}^{A}$ is the ${S}^{A}$-mapping generated by ${S}_{1},{S}_{2},\dots ,{S}_{N}$, ${T}_{1},{T}_{2},\dots ,{T}_{N}$, and ${\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{N}$ where ${\alpha}_{j}=({\alpha}_{1}^{j},{\alpha}_{2}^{j},{\alpha}_{3}^{j})$ and ${\alpha}_{1}^{j}={\alpha}_{2}^{j}={\alpha}_{3}^{j}=\frac{1}{3}$ for all $j=1,2,\dots ,N$. Define the mapping $G:C\to C$ by $Gx={Q}_{C}(I-\frac{1}{5}A)(\frac{1}{2}I+\frac{1}{2}{Q}_{C}(I-\frac{1}{17}B))x$ for all $x\in C$. Let the sequence $\{{x}_{n}\}$ be generated by (3.1), where ${\alpha}_{n}=\frac{1}{7n}$, ${\beta}_{n}=\frac{6n-1}{14n}$, and ${\gamma}_{n}=\frac{8n-1}{14n}$ for all $n\ge 1$. Then $\{{x}_{n}\}$ converges strongly to 0 and $(0,0)$ is a solution of (1.8).

*Solution*. For every $i=1,2,\dots ,N$, it is easy to see that ${T}_{i}$ is a nonexpansive mapping and ${S}_{i}$ is $\frac{1}{{i}^{2}}$-strictly pseudo-contractive mappings with ${\bigcap}_{i=1}^{N}F({S}_{i})\cap {\bigcap}_{i=1}^{N}F({T}_{i})=\{0\}$. Then *A* is $\frac{1}{4}$-inverse strongly accretive and *B* is $\frac{1}{16}$-inverse strongly accretive. From the definition of *G*, we have $F(G)=\{0\}$ and $(0,0)$ is a solution of (1.8). Then $\mathcal{F}={\bigcap}_{i=1}^{N}F({S}_{i})\cap {\bigcap}_{i=1}^{N}F({T}_{i})\cap F(G)=\{0\}$.

For every $n\ge 1$ and $i=1,2,\dots ,N$, the mappings ${T}_{i}$, ${S}_{i}$, *G*, *A*, *B* and sequences $\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$ satisfy all conditions in Theorem 3.1. Since the sequence $\{{x}_{n}\}$ is generated by (3.1), from Theorem 3.1, we find that the sequence $\{{x}_{n}\}$ converges strongly to 0 and $(0,0)$ is a solution in (1.8).

- (i)
${x}_{1}=\frac{\pi}{2}$, $u=\frac{\pi}{4}$ and $n=N=20$,

- (ii)
${x}_{1}=\frac{\pi}{4}$, $u=\frac{\pi}{6}$ and $n=N=20$.

**The values of**
$\mathbf{\{}{\mathit{x}}_{\mathit{n}}\mathbf{\}}$
**with**
${\mathit{x}}_{\mathbf{1}}\mathbf{=}\frac{\mathit{\pi}}{\mathbf{2}}$
**,**
$\mathit{u}\mathbf{=}\frac{\mathit{\pi}}{\mathbf{4}}$
**, and**
${\mathit{x}}_{\mathbf{1}}\mathbf{=}\frac{\mathit{\pi}}{\mathbf{4}}$
**,**
$\mathit{u}\mathbf{=}\frac{\mathit{\pi}}{\mathbf{6}}$

n | ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\frac{\mathit{\pi}}{\mathbf{2}}$, $\mathit{u}\mathbf{=}\frac{\mathit{\pi}}{\mathbf{4}}$ | ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\frac{\mathit{\pi}}{\mathbf{4}}$, $\mathit{u}\mathbf{=}\frac{\mathit{\pi}}{\mathbf{6}}$ |
---|---|---|

${\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}}$ | |

1 | 1.5707963268 | 0.7853981634 |

2 | 0.6127630899 | 0.3232983687 |

3 | 0.2701199079 | 0.1495005671 |

4 | 0.1333284242 | 0.0775756830 |

5 | 0.0750990544 | 0.0458214332 |

⋮ | ⋮ | ⋮ |

10 | 0.0200438855 | 0.0133281318 |

⋮ | ⋮ | ⋮ |

16 | 0.0114504755 | 0.0076335361 |

17 | 0.0107016897 | 0.0071344156 |

18 | 0.0100455821 | 0.0066970377 |

19 | 0.0094657530 | 0.0063104954 |

20 | 0.0089495227 | 0.0059663460 |

### Conclusion

- (i)
Table 1 and Figure 1 show that the sequences $\{{x}_{n}\}$ converge to 0, where $\{0\}={\bigcap}_{i=1}^{N}F({S}_{i})\cap {\bigcap}_{i=1}^{N}F({T}_{i})\cap F(G)$.

- (ii)
Theorem 3.1 guarantees the convergence of $\{{x}_{n}\}$ in Example 5.1.

- (iii)
If the sequence $\{{x}_{n}\}$ is generated by (4.4), from Theorem 4.4 and Example 5.1, we also see that the sequence $\{{x}_{n}\}$ converges to 0, where $\{0\}=S(C,A)\cap (C,B){\bigcap}_{i=1}^{N}F({S}_{i})\cap {\bigcap}_{i=1}^{N}F({T}_{i})$.

## Declarations

### Acknowledgements

This research was supported by the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang.

## Authors’ Affiliations

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