- Review
- Open Access

# Strong convergence theorems for solutions of fixed point and variational inequality problems

- Lanxiang Yu
^{1}Email author and - Jianmin Song
^{2}

**2014**:215

https://doi.org/10.1186/1029-242X-2014-215

© Yu and Song; licensee Springer. 2014

**Received: **16 March 2014

**Accepted: **21 May 2014

**Published: **29 May 2014

## Abstract

The purpose of this paper is to investigate viscosity approximation methods for finding a common element in the set of fixed points of a strict pseudocontraction and in the set of solutions of a generalized variational inequality in the framework of Banach spaces.

## Keywords

- sunny nonexpansive retraction
- inverse-strongly accretive mapping
- nonexpansive mapping
- variational inequality

## 1 Introduction

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

*H*, and let ${P}_{C}$ be the metric projection of

*H*onto

*C*. Recall that a mapping $A:C\to H$ is said to be monotone iff

For such a case, *A* is said to be *α*-inverse-strongly monotone.

It is clear that variational inequality problem (1.1) is equivalent to a fixed point problem. *u* is a solution of the above inequality iff it is a fixed point of the mapping ${P}_{C}(I-rA)$, where *I* is the identity and *r* is some positive real number.

Variational inequality problems have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, and network. Recently, many authors studied the solutions of inequality (1.1) based on iterative methods; see [1–17] and the references therein.

Let $S:C\to C$ be a mapping. In this paper, we denote by $F(S)$ the set of fixed points of the mapping *S*.

*S*is said to be nonexpansive iff

*S*is said to be a strict pseudocontraction iff there exits a positive constant

*λ*such that

It is clear that the class of strict pseudocontractions includes the class of nonexpansive mappings as a special case.

Recently, many authors have investigated the problems of finding a common element in the set of solution of variational inequalities for an inverse-strongly monotone mapping and in the set of fixed points of nonexpansive mappings or strict pseudocontractions; see [18–25] and the references therein. However, most of the results are in the framework of Hilbert spaces. In this paper, we investigate a common element problem in the framework of Banach spaces. A strong convergence theorem for common solutions to fixed point problems of strict pseudocontractions and solution problems of variational inequality (1.1) is established in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve and extend the corresponding results announced by Iiduka and Takahashi [5] and Hao [26].

## 2 Preliminaries

*C*be a nonempty closed and convex subset of a Banach space

*E*. Let ${E}^{\ast}$ be the dual space of

*E*, and let $\u3008\cdot ,\cdot \u3009$ denote the pairing between

*E*and ${E}^{\ast}$. For $q>1$, the generalized duality mapping ${J}_{q}:E\to {2}^{{E}^{\ast}}$ is defined by

*E*is a Hilbert space, then $J=I$, the identity mapping. The normalized duality mapping

*J*has the following properties:

- (1)
if

*E*is smooth, then*J*is single-valued; - (2)
if

*E*is strictly convex, then it is one-to-one and $\u3008x-y,{x}^{\ast}-{y}^{\ast}\u3009>0$ holds for all $(x,{x}^{\ast}),(y,{y}^{\ast})\in J$ with $x\ne y$; - (3)
if

*E*is reflexive, then*J*is surjective; - (4)
if

*E*is uniformly smooth, then*J*is uniformly norm-to-norm continuous on each bounded subset of*E*.

*E*is said to be uniformly convex if, for any $\u03f5\in (0,2]$, there exists $\delta >0$ such that, for any $x,y\in U$,

*E*is said to be smooth if the limit

*E*is said to be Fréchet differentiable if, for any $x\in U$, the limit (2.1) is attained uniformly for all $y\in U$. The modulus of smoothness of

*E*is defined by

where $\rho :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is a function. It is known that *E* is uniformly smooth if and only if ${lim}_{\tau \to 0}\frac{\rho (\tau )}{\tau}=0$. Let *q* be a fixed real number with $1<q\le 2$. A Banach space *E* is said to be *q*-uniformly smooth if there exists a constant $c>0$ such that $\rho (\tau )\le c{\tau}^{q}$ for all $\tau >0$.

We remark that all Hilbert spaces, ${L}_{p}$ (or ${l}_{p}$) spaces ($p\ge 2$) and the Sobolev spaces ${W}_{m}^{p}$ ($p\ge 2$) are 2-uniformly smooth, while ${L}_{p}$ (or ${l}_{p}$) and ${W}_{m}^{p}$ spaces ($1<p\le 2$) are *p*-uniformly smooth. Typical examples of both uniformly convex and uniformly smooth Banach spaces are ${L}^{p}$, where $p>1$. More precisely, ${L}^{p}$ is $min\{p,2\}$-uniformly smooth for every $p>1$.

*S*is said to be

*λ*-strictly pseudocontractive iff there exist a constant $\lambda \in (0,1)$ and $j(x-y)\in J(x-y)$ such that

*E*is a smooth Banach space. Let

*C*be a nonempty closed convex subset of

*E*. Recall that an operator

*A*of

*C*into

*E*is said to be accretive iff

An accretive operator *A* is said to be *m*-accretive if the range of $I+rA$ is *E* for all $r>0$. In a real Hilbert space, an operator *A* is *m*-accretive if and only if *A* is maximal monotone.

*A*of

*C*into

*E*is said to be

*α*-inverse strongly accretive iff there exits a real constant $\alpha >0$ such that

Evidently, the definition of an inverse-strongly accretive operator is based on that of an inverse-strongly monotone operator.

*D*be a subset of

*C*and

*Q*be a mapping of

*C*into

*D*. Then

*Q*is said to be sunny if

whenever $Qx+t(x-Qx)\in C$ for $x\in C$ and $t\ge 0$. A mapping *Q* of *C* into itself is called a retraction if ${Q}^{2}=Q$. If a mapping *Q* of *C* into itself is a retraction, then $Qz=z$ for all $z\in R(Q)$, where $R(Q)$ is the range of *Q*. A subset *D* of *C* is called a sunny nonexpansive retract of *C* if there exists a sunny nonexpansive retraction from *C* onto *D*.

The following result describes a characterization of sunny nonexpansive retractions on a smooth Banach space.

**Proposition 2.1** [27]

*Let*

*E*

*be a smooth Banach space*,

*and let*

*C*

*be a nonempty subset of E*.

*Let*$Q:E\to C$

*be a retraction*,

*and let*

*J*

*be the normalized duality mapping on*

*E*.

*Then the following are equivalent*:

- (1)
${Q}_{C}$

*is sunny and nonexpansive*; - (2)
${\parallel {Q}_{C}x-{Q}_{C}y\parallel}^{2}\le \u3008x-y,J({Q}_{C}x-{Q}_{C}y)\u3009$ $\mathrm{\forall}x,y\in E$;

- (3)
$\u3008x-{Q}_{C}x,J(y-{Q}_{C}x)\u3009\le 0$ $\mathrm{\forall}x\in E,y\in C$.

**Proposition 2.2** [28]

*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 the set* $F(T)$ *is a sunny nonexpansive retract of* *C*.

Recently, Aoyama *et al.* [29] considered the following generalized variational inequality problem.

*C*be a nonempty closed convex subset of

*E*, and let

*A*be an accretive operator of

*C*into

*E*. Find a point $u\in C$ such that

Next, we use $BVI(C,A)$ to denote the set of solutions of variational inequality problem (2.4).

Aoyama *et al.* [29] proved that variational inequality (2.4) is equivalent to a fixed point problem. The element $u\in C$ is a solution of variational inequality (2.4) iff $u\in C$ is a fixed point of the mapping ${Q}_{C}(I-rA)$, where $r>0$ is a constant and ${Q}_{C}$ is a sunny nonexpansive retraction from *E* onto *C*.

The following lemmas also play an important role in this paper.

**Lemma 2.3** [30]

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

*is a sequence of nonnegative real numbers such that*

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

*is a sequence in*$(0,1)$, $\{{e}_{n}\}$

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

*are sequences such that*

- (1)
${\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}=\mathrm{\infty}$;

- (2)
${\sum}_{n=1}^{\mathrm{\infty}}{e}_{n}<\mathrm{\infty}$;

- (3)
${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\delta}_{n}/{\gamma}_{n}\le 0$

*or*${\sum}_{n=1}^{\mathrm{\infty}}|{\delta}_{n}|<\mathrm{\infty}$.

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

**Lemma 2.4** [31]

*Let*

*E*

*be a real*2-

*uniformly smooth Banach space with the best smooth constant*

*K*.

*Then the following inequality holds*:

**Lemma 2.5** [29]

*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 2.6** [32]

*Let* *C* *be a closed convex subset of a real strictly convex Banach space* *E* *and* ${S}_{i}:C\to C$ ($i=1,2$) *be two nonexpansive mappings such that* $F=F({S}_{1})\cap F({S}_{2})\ne \mathrm{\varnothing}$. *Define* $Sx=\delta {S}_{1}x+(1-\delta ){S}_{2}x$, *where* $\delta \in (0,1)$. *Then* $S:C\to C$ *is a nonexpansive mapping with* $F(S)=F\ne \mathrm{\varnothing}$.

**Lemma 2.7** [33]

*Let* *C* *be a nonempty subset of a real* 2-*uniformly smooth Banach space* *E*, *and let* $T:C\to C$ *be a* *κ*-*strict pseudocontraction*. *For* $\alpha \in (0,1)$, *we define* ${T}_{\alpha}x=(1-\alpha )x+\alpha Tx$ *for every* $x\in C$. *Then*, *as* $\alpha \in (0,\frac{\kappa}{{K}^{2}}]$, ${T}_{\alpha}$ *is nonexpansive such that* $F({T}_{\alpha})=F(T)$.

**Lemma 2.8** [34]

*Let* *E* *be a real uniformly smooth Banach space*, *and let* *C* *be a nonempty closed convex subset of* *E*. *Let* $T:C\to C$ *be a nonexpansive mapping with a fixed point*, *and let* $f:C\to C$ *be a contraction*. *For each* $t\in (0,1)$, *let* ${z}_{t}$ *be the unique solution of the equation* $x=tf(x)+(1-t)Tx$. *Then* $\{{z}_{t}\}$ *converges to a fixed point of* *T* *as* $t\to 0$ *and* $Q(f)=s\text{-}{lim}_{t\to 0}{z}_{t}$ *defines the unique sunny nonexpansive retraction from* *C* *onto* $F(T)$.

## 3 Main results

**Theorem 3.1**

*Let*

*E*

*be a uniformly convex and*2-

*uniformly smooth Banach space with the best smooth constant*

*K*,

*and let*

*C*

*be a nonempty*,

*closed and convex subset of*

*E*.

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

*be a sunny nonexpansive retraction from*

*E*

*onto*

*C*,

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

*be an*

*α*-

*inverse strongly accretive mapping*.

*Let*$S:C\to C$

*be a*

*λ*-

*strict pseudocontraction with a fixed point*.

*Assume that*$F:=F(S)\cap BVI(C,A)\ne \mathrm{\varnothing}$.

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

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

*be real number sequences in*$(0,1)$.

*Suppose that*${x}_{1}=x\in C$

*and that*$\{{x}_{n}\}$

*is given by*

*where*${S}_{t}=(1-t)x+tSx$, $t\in (0,\frac{\lambda}{{K}^{2}}]$, $f:C\to C$

*is a*

*κ*-

*contractive mapping*, $\{{e}_{n}\}$

*is a bounded computational error in*

*E*, $\lambda \in (0,\alpha /{K}^{2}]$

*and*$\mu \in (0,1)$.

*Assume that the following restrictions are satisfied*:

- (a)
${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$, ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$

*and*${\sum}_{n=1}^{\mathrm{\infty}}|{\alpha}_{n+1}-{\alpha}_{n}|<\mathrm{\infty}$; - (b)
${\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}<\mathrm{\infty}$.

*Then* $\{{x}_{n}\}$ *converges strongly to* $x={Q}_{F}f(x)$, *where* ${Q}_{F}$ *is a sunny nonexpansive retraction from* *C* *onto* *F*.

*Proof*Fixing ${x}^{\ast}\in F$, we find that ${x}^{\ast}={Q}_{C}({x}^{\ast}-\lambda A{x}^{\ast})$ and $S{x}^{\ast}={x}^{\ast}$. It follows from Lemma 2.7 that ${S}_{t}{x}^{\ast}={x}^{\ast}$. Put ${y}_{n}={Q}_{C}({x}_{n}-\lambda A{x}_{n})$. In view of Lemma 2.4, we find that

*V*by

*V*is a nonexpansive mapping with

*C*onto

*F*, the strong limit of the sequence ${z}_{t}$ defined by

*E*is 2-uniformly smooth, $j:E\to {E}^{\ast}$ is uniformly continuous on any bounded sets of

*E*, which ensures that the ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}$ and ${lim\hspace{0.17em}sup}_{t\to 0}$ are interchangeable, hence

This shows that (3.6) holds.

This completes the proof. □

**Remark 3.2** The framework of the space in Theorem 3.1 can be applicable to ${L}^{p}$, $p\ge 2$.

## 4 Applications

In this section, we always assume that *E* is a uniformly convex and 2-uniformly smooth Banach space. Let *C* be a nonempty, closed and convex subset of *E*.

First, we consider common fixed points of two strict pseudocontractions.

**Theorem 4.1**

*Let*

*E*

*be a uniformly convex and*2-

*uniformly smooth Banach space with the best smooth constant*

*K*,

*and let*

*C*

*be a nonempty closed convex subset of*

*E*.

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

*be a sunny nonexpansive retraction from*

*E*

*onto*

*C*,

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

*be an*

*α*-

*strict pseudocontraction*.

*Let*$S:C\to C$

*be a*

*λ*-

*strict pseudocontraction*.

*Assume that*$F:=F(S)\cap F(T)\ne \mathrm{\varnothing}$.

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

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

*be real number sequences in*$(0,1)$.

*Suppose that*${x}_{1}=x\in C$

*and that*$\{{x}_{n}\}$

*is given by*

*where*${S}_{t}=(1-t)x+tSx$, $t\in (0,\frac{\lambda}{{K}^{2}}]$, $f:C\to C$

*is a*

*κ*-

*contractive mapping*, $\{{e}_{n}\}$

*is a bounded computational error in*

*E*, $\lambda \in (0,\alpha /{K}^{2}]$

*and*$\mu \in (0,1)$.

*Assume that the following restrictions are satisfied*:

- (a)
${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$, ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$

*and*${\sum}_{n=1}^{\mathrm{\infty}}|{\alpha}_{n+1}-{\alpha}_{n}|<\mathrm{\infty}$; - (b)
${\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}<\mathrm{\infty}$.

*Then* $\{{x}_{n}\}$ *converges strongly to* $x={Q}_{F}f(x)$, *where* ${Q}_{F}$ *is a sunny nonexpansive retraction from* *C* *onto* *F*.

*Proof* Since $(I-T)$ is an *α*-inverse strongly accretive mapping, we find from Theorem 3.1 the desired conclusion. □

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

*E*is accretive if for each ${x}_{i}\in D(B)$ and ${y}_{i}\in B{x}_{i}$ ($i=1,2$),

*B*is

*m*-accretive if $R(I+rB)=E$ for each $r>0$. Next, we assume that

*B*is

*m*-accretive and has a zero (

*i.e.*, the inclusion $0\in B(z)$ is solvable). The set of zeros of

*B*is denoted by Ω. Hence,

For each $r>0$, we denote by ${J}_{r}$ the resolvent of *B*, *i.e.*, ${J}_{r}={(I+rB)}^{-1}$. Note that if *B* is *m*-accretive, then ${J}_{r}:E\to E$ is nonexpansive and $F({J}_{r})=\mathrm{\Omega}$ for all $r>0$.

From the above, we have the following theorem.

**Theorem 4.2**

*Let*

*E*

*be a uniformly convex and*2-

*uniformly smooth Banach space with the best smooth constant*

*K*,

*and let*

*C*

*be a nonempty*,

*closed and convex subset of*

*E*.

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

*be a sunny nonexpansive retraction from*

*E*

*onto*

*C*,

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

*be an*

*α*-

*inverse strongly accretive mapping*.

*Let*$B:C\to C$

*be an*

*m*-

*accretive operator*.

*Assume that*$F:={A}^{-1}(0)\cap {B}^{-1}(0)\ne \mathrm{\varnothing}$.

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

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

*be real number sequences in*$(0,1)$.

*Suppose that*${x}_{1}=x\in C$

*and that*$\{{x}_{n}\}$

*is given by*

*where*${J}_{r}={(I+rB)}^{-1}$, $f:C\to C$

*is a*

*κ*-

*contractive mapping*, $\{{e}_{n}\}$

*is a bounded computational error in*

*E*, $\lambda \in (0,\alpha /{K}^{2}]$

*and*$\mu \in (0,1)$.

*Assume that the following restrictions are satisfied*:

- (a)
${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$, ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$

*and*${\sum}_{n=1}^{\mathrm{\infty}}|{\alpha}_{n+1}-{\alpha}_{n}|<\mathrm{\infty}$; - (b)
${\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}<\mathrm{\infty}$.

*Then* $\{{x}_{n}\}$ *converges strongly to* $x={Q}_{F}f(x)$, *where* ${Q}_{F}$ *is a sunny nonexpansive retraction from* *C* *onto* *F*.

## Declarations

### Acknowledgements

The authors are grateful to the reviewers for useful suggestions which improved the contents of this paper.

## Authors’ Affiliations

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