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

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.

1 Introduction

Let 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

$$〈Ax-Ay,x-y〉\ge 0\mathrm{\forall }x,y\in C.$$

Recall that a mapping $A:C\to H$ is said to be inverse-strongly monotone iff there exists a positive real number $\alpha >0$ such that

$$〈Ax-Ay,x-y〉\ge \alpha {\parallel Ax-Ay\parallel }^2\mathrm{\forall }x,y\in C.$$

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

Recall that the classical variational inequality problem, denoted by $VI(C,A)$, is to find $u\in C$ such that

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

(1.1)

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.

Recall that S is said to be nonexpansive iff

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

Recall that S is said to be a strict pseudocontraction iff there exits a positive constant λ such that

$${\parallel Sx-Sy\parallel }^2\le {\parallel x-y\parallel }^2+\lambda {\parallel (I-S)x-(I-S)y\parallel }^2\mathrm{\forall }x,y\in C.$$

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

Let C be a nonempty closed and convex subset of a Banach space E. Let $E^{\ast }$ be the dual space of E, and let $〈\cdot ,\cdot 〉$ 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

$$J_q(x)=\{f\in E^{\ast }:〈x,f〉={\parallel x\parallel }^q,\parallel f\parallel ={\parallel x\parallel }^{q-1}\}$$

for all $x\in E$. In particular, $J=J_2$ is called the normalized duality mapping. It is known that $J_q(x)={\parallel x\parallel }^{q-2}J(x)$ for all $x\in E$. If 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 $〈x-y,x^{\ast }-y^{\ast }〉>0$ 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.

Let $U=\{x\in X:\parallel x\parallel =1\}$. A Banach space E is said to be uniformly convex if, for any $ϵ\in (0,2]$, there exists $\delta >0$ such that, for any $x,y\in U$,

$$\parallel x-y\parallel \ge ϵ\text{implies}\parallel \frac{x+y}{2}\parallel \le 1-\delta .$$

It is known that a uniformly convex Banach space is reflexive and strictly convex. Hilbert spaces are 2-uniformly convex, while $L^p$ is $max\{p,2\}$-uniformly convex for every $p>1$. A Banach space E is said to be smooth if the limit

$$\underset{t\to 0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$$

(2.1)

exists for all $x,y\in U$. It is also said to be uniformly smooth if the limit (2.1) is attained uniformly for $x,y\in U$. The norm of 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

$$\rho (\tau )=sup\{\frac{1}{2}(\parallel x+y\parallel +\parallel x-y\parallel )-1:x,y\in X,\parallel x\parallel =1,\parallel y\parallel =\tau \},$$

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 $p>1$.

Recall that a mapping 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

$$〈Sx-Sy,j(x-y)〉\le {\parallel x-y\parallel }^2-\lambda {\parallel (I-S)x-(I-S)y\parallel }^2\mathrm{\forall }x,y\in C.$$

(2.2)

It is clear that (2.2) is equivalent to the following:

$$〈(I-S)x-(I-S)y,j(x-y)〉\ge \lambda {\parallel (I-S)x-(I-S)y\parallel }^2\mathrm{\forall }x,y\in C.$$

(2.3)

Next, we assume 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

$$〈Ax-Ay,J(x-y)〉\ge 0\mathrm{\forall }x,y\in C.$$

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.

Recall that an operator A of C into E is said to be α-inverse strongly accretive iff there exits a real constant $\alpha >0$ such that

$$〈Ax-Ay,J(x-y)〉\ge \alpha {\parallel Ax-Ay\parallel }^2\mathrm{\forall }x,y\in C.$$

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

Let D be a subset of C and Q be a mapping of C into D. Then Q is said to be sunny if

$$Q(Qx+t(x-Qx))=Qx,$$

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) ${\parallel Q_{C}x-Q_{C}y\parallel }^2\le 〈x-y,J(Q_{C}x-Q_{C}y)〉$ $\mathrm{\forall }x,y\in E$;

• (3) $〈x-Q_{C}x,J(y-Q_{C}x)〉\le 0$ $\mathrm{\forall }x\in E,y\in C$.

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.

Let 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

$$〈Au,J(v-u)〉\ge 0\mathrm{\forall }v\in C.$$

(2.4)

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

$${\alpha }_{n+1}\le (1-{\gamma }_n){\alpha }_n+{\delta }_n+e_n,$$

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:

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

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 $\lambda >0$,

$$BVI(C,A)=F(Q_C(I-\lambda A)).$$

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

$$x_{n+1}={\alpha }_{n}f(x_n)+{\beta }_n[\mu S_t{x}_n+(1-\mu )Q_C(x_n-\lambda A{x}_n)]+{\gamma }_n{Q}_C{e}_n,n\ge 1,$$

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

$$\begin{array}{rcl}{\parallel y_n-x^{\ast }\parallel }^2& \le & {\parallel (x_n-x^{\ast })-\lambda (A{x}_n-A{x}^{\ast })\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2-2\lambda 〈A{x}_n-A{x}^{\ast },J(x_n-x^{\ast })〉\\ +2K^2{\lambda }^2{\parallel A{x}_n-A{x}^{\ast }\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2-2\lambda \alpha {\parallel A{x}_n-A{x}^{\ast }\parallel }^2+2K^2{\lambda }^2{\parallel A{x}_n-A{x}^{\ast }\parallel }^2\\ =& {\parallel x_n-x^{\ast }\parallel }^2+2\lambda (\lambda K^2-\alpha ){\parallel A{x}_n-A{x}^{\ast }\parallel }^2.\end{array}$$

Since $\lambda \in (0,\alpha /K^2]$, we have that

$$\parallel y_n-x^{\ast }\parallel \le \parallel x_n-x^{\ast }\parallel .$$

This implies that $Q_C(I-\lambda A)$ is a nonexpansive mapping. Hence, we have

$$\begin{array}{c}\parallel x_{n+1}-x^{\ast }\parallel \hfill \\ =\parallel {\alpha }_{n}f(x_n)+{\beta }_n[\mu S{x}_n+(1-\mu )Q_C(x_n-\lambda A{x}_n)]+{\gamma }_n{Q}_C{e}_n-x^{\ast }\parallel \hfill \\ \le {\alpha }_n\parallel f(x_n)-x^{\ast }\parallel +{\beta }_n\parallel \mu S{x}_n+(1-\mu )Q_C(x_n-\lambda A{x}_n)-x^{\ast }\parallel \hfill \\ +{\gamma }_n\parallel Q_C{e}_n-x^{\ast }\parallel \hfill \\ \le {\alpha }_n\parallel f(x_n)-x^{\ast }\parallel +{\beta }_n\mu \parallel x_n-x^{\ast }\parallel +{\beta }_n(1-\mu )\parallel x_n-x^{\ast }\parallel \hfill \\ +{\gamma }_n\parallel Q_C{e}_n-x^{\ast }\parallel \hfill \\ \le {\alpha }_n\kappa \parallel x_n-x^{\ast }\parallel +{\alpha }_n\parallel f\left(x^{\ast }\right)-x^{\ast }\parallel +{\beta }_n\parallel x_n-x^{\ast }\parallel \hfill \\ +{\gamma }_n\parallel e_n-x^{\ast }\parallel \hfill \\ \le (1-{\alpha }_n(1-\kappa ))\parallel x_n-x^{\ast }\parallel +{\alpha }_n\parallel f\left(x^{\ast }\right)-x^{\ast }\parallel +{\gamma }_n\parallel e_n-x^{\ast }\parallel \hfill \\ \le max\{\parallel x_n-x^{\ast }\parallel ,\frac{\parallel f(x^{\ast })-x^{\ast }\parallel }{1-\kappa }\}+{\gamma }_n\parallel e_n-x^{\ast }\parallel ,\hfill \end{array}$$

which implies that the sequence $\{x_n\}$ is bounded, so is $\{y_n\}$. Define

$$t_n=\mu S_t{x}_n+(1-\mu )Q_C(x_n-\lambda A{x}_n).$$

It follows that

$$\begin{array}{c}\parallel t_n-t_{n-1}\parallel \hfill \\ =\parallel \mu S_t{x}_n+(1-\mu )Q_C(I-\lambda A)x_n-[\mu S_t{x}_{n-1}+(1-\mu )Q_C(I-\lambda A)x_{n-1}]\parallel \hfill \\ \le \mu \parallel S_t{x}_n-S_t{x}_{n-1}\parallel +(1-\mu )\parallel Q_C(I-\lambda A)x_n-Q_C(I-\lambda A)x_{n-1}\parallel \hfill \\ \le \mu \parallel x_n-x_{n-1}\parallel +(1-\mu )\parallel x_n-x_{n-1}\parallel \hfill \\ =\parallel x_n-x_{n-1}\parallel .\hfill \end{array}$$

(3.1)

On the other hand, we have

$$\begin{array}{c}\parallel x_{n+1}-x_n\parallel \hfill \\ \le {\alpha }_n\parallel f(x_n)-f(x_{n-1})\parallel +|{\alpha }_n-{\alpha }_{n-1}|\parallel f(x_{n-1})\parallel \hfill \\ +{\beta }_n\parallel t_n-t_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel t_{n-1}\parallel +{\gamma }_n\parallel Q_C{e}_n-Q_C{e}_{n-1}\parallel \hfill \\ +|{\gamma }_n-{\gamma }_{n-1}|\parallel Q_C{e}_n\parallel .\hfill \end{array}$$

(3.2)

Substituting (3.1) into (3.2), we see that

$$\begin{array}{c}\parallel x_{n+1}-x_n\parallel \hfill \\ \le (1-{\alpha }_n(1-\kappa ))\parallel x_n-x_{n-1}\parallel +|{\alpha }_n-{\alpha }_{n-1}|\parallel f(x_{n-1})\parallel \hfill \\ +|{\beta }_n-{\beta }_{n-1}|\parallel t_{n-1}\parallel +{\gamma }_n\parallel Q_C{e}_n-Q_C{e}_{n-1}\parallel \hfill \\ +|{\gamma }_n-{\gamma }_{n-1}|\parallel Q_C{e}_n\parallel \hfill \\ \le (1-{\alpha }_n(1-\kappa ))\parallel x_n-x_{n-1}\parallel +|{\alpha }_n-{\alpha }_{n-1}|(\parallel f(x_{n-1})\parallel +\parallel t_{n-1}\parallel )\hfill \\ +|{\gamma }_n-{\gamma }_{n-1}|(\parallel t_{n-1}\parallel +\parallel Q_C{e}_n\parallel )+{\gamma }_n\parallel Q_C{e}_n-Q_C{e}_{n-1}\parallel .\hfill \end{array}$$

In view of Lemma 2.3, we find from the restrictions (a) and (b) that

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

(3.3)

Note that

$$\begin{array}{rcl}\parallel x_n-t_n\parallel & \le & \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-t_n\parallel \\ \le & \parallel x_n-x_{n+1}\parallel +{\alpha }_n\parallel f(x_n)-t_n\parallel +{\gamma }_n\parallel Q_C{e}_n-t_n\parallel .\end{array}$$

Using (3.3), we find from the restrictions (a) and (b) that

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

(3.4)

Define a mapping V by

$$Vx=\mu S_{t}x+(1-\mu )Q_C(I-\lambda A)x\mathrm{\forall }x\in C.$$

Using Lemma 2.6, we see that the mapping V is a nonexpansive mapping with

$$F(V)=F(S_t)\cap F(Q_C(I-\lambda A))=F(S_t)\cap BVI(C,A)=F(S)\cap BVI(C,A)=F.$$

From (3.4), we see that

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

(3.5)

Next, we show that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈f(x)-x,J(x_n-x)〉\le 0,$$

(3.6)

where $x=Q_{F}f(x)$, and $Q_F$ is a sunny nonexpansive retraction from C onto F, the strong limit of the sequence $z_t$ defined by

$$z_t=tf(z_t)+(1-t)V{z}_t.$$

It follows that

$$\parallel z_t-x_n\parallel =\parallel (1-t)(V{z}_t-x_n)+t(f(z_t)-x_n)\parallel .$$

For any $t\in (0,1)$, we see that

$$\begin{array}{rcl}{\parallel z_t-x_n\parallel }^2& \le & {(1-t)}^2{\parallel V{z}_t-x_n\parallel }^2+2t〈f(z_t)-x_n,j(z_t-x_n)〉\\ \le & {(1-t)}^2({\parallel V{z}_t-V{x}_n\parallel }^2+{\parallel V{x}_n-x_n\parallel }^2\\ +2\parallel V{z}_t-V{x}_n\parallel \parallel V{x}_n-x_n\parallel )+2t〈f(z_t)-z_t,j(z_t-x_n)〉\\ +2t〈z_t-x_n,j(z_t-x_n)〉\\ \le & {(1-t)}^2{\parallel z_t-x_n\parallel }^2+{\lambda }_n(t)+2t〈f(z_t)-z_t,j(z_t-x_n)〉\\ +2t{\parallel z_t-x_n\parallel }^2,\end{array}$$

(3.7)

where

$${\lambda }_n(t)={\parallel V{x}_n-x_n\parallel }^2+2\parallel z_t-x_n\parallel \parallel V{x}_n-x_n\parallel .$$

It follows from (3.7) that

$$〈z_t-f(z_t),j(z_t-x_n)〉\le \frac{t}{2}{\parallel z_t-x_n\parallel }^2+\frac{1}{2t}{\lambda }_n(t).$$

This implies that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈z_t-f(z_t),j(z_t-x_n)〉\le \frac{t}{2}{\parallel z_t-x_n\parallel }^2.$$

Since 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

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈f(x)-x,j(x_n-x)〉\le 0.$$

This shows that (3.6) holds.

Finally, we show that $x_n\to x$ as $n\to \mathrm{\infty }$. Note that

$$\begin{array}{c}{\parallel x_{n+1}-x\parallel }^2\hfill \\ ={\alpha }_n〈f(x_n)-x,j(x_{n+1}-x)〉+{\beta }_n〈t_n-x,j(x_{n+1}-x)〉\hfill \\ +{\gamma }_n〈Q_C{e}_n-x,j(x_{n+1}-x)〉\hfill \\ \le {\alpha }_n〈f(x_n)-x,j(x_{n+1}-q)〉+{\beta }_n\parallel t_n-x\parallel \parallel x_{n+1}-x\parallel \hfill \\ +{\gamma }_n\parallel Q_C{e}_n-x\parallel \parallel x_{n+1}-x\parallel \hfill \\ \le {\alpha }_n\kappa \parallel x_n-x\parallel \parallel x_{n+1}-q\parallel +{\alpha }_n〈f(x)-x,j(x_{n+1}-q)〉\hfill \\ +{\beta }_n\parallel x_n-x\parallel \parallel x_{n+1}-x\parallel +{\gamma }_n\parallel Q_C{e}_n-x\parallel \parallel x_{n+1}-x\parallel \hfill \\ \le \frac{{\alpha }_n\kappa +{\beta }_n}{2}({\parallel x_n-x\parallel }^2+{\parallel x_{n+1}-x\parallel }^2)+{\alpha }_n〈f(x)-x,j(x_{n+1}-x)〉\hfill \\ +\frac{{\gamma }_n}{2}({\parallel e_n-x\parallel }^2+{\parallel x_{n+1}-x\parallel }^2)\hfill \\ \le \frac{{\alpha }_n\kappa +{\beta }_n}{2}{\parallel x_n-x\parallel }^2+\frac{1-{\alpha }_n(1-\kappa )}{2}{\parallel x_{n+1}-x\parallel }^2\hfill \\ +{\alpha }_n〈f(x)-x,j(x_{n+1}-x)〉+\frac{{\gamma }_n}{2}{\parallel e_n-x\parallel }^2.\hfill \end{array}$$

It follows that

$$\begin{array}{rcl}{\parallel x_{n+1}-x\parallel }^2& \le & (1-{\alpha }_n(1-\kappa )){\parallel x_n-x\parallel }^2\\ +2{\alpha }_n〈f(x)-x,j(x_{n+1}-x)〉+{\gamma }_n{\parallel e_n-x\parallel }^2.\end{array}$$

Using Lemma 2.3, we find from the restrictions (a) and (b) that

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

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

$$x_{n+1}={\alpha }_{n}f(x_n)+(1-{\alpha }_n)[\mu S_t{x}_n+(1-\mu )((1-\alpha )x_n+\alpha T{x}_n)],n\ge 1,$$

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. □

Closely related to the class of pseudocontractive mappings is the class of accretive mappings. Recall that an operator B with domain $D(B)$ and range $R(B)$ in E is accretive if for each $x_i\in D(B)$ and $y_i\in B{x}_i$ ($i=1,2$),

$$〈y_2-y_1,J(x_2-x_1)〉\ge 0.$$

An accretive operator 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,

$$\mathrm{\Omega }=\{z\in D(B):0\in B(z)\}=B^{-1}(0).$$

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

$$x_{n+1}={\alpha }_{n}f(x_n)+{\beta }_n[\mu J_r{x}_n+(1-\mu )(x_n-\lambda A{x}_n)]+{\gamma }_n{Q}_C{e}_n,n\ge 1,$$

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.