Common fixed points of a family of strictly pseudocontractive mappings

Abstract

In this article, fixed point problems of a family of strictly pseudocontractive mappings are investigated based on a viscosity iterative process. Strong convergence theorems are established in a real q-uniformly Banach space.

MSC:47H09, 47J05, 47J25.

1 Introduction

Fixed point problems of nonlinear mappings as an important branch of nonlinear analysis theory have been applied in many disciplines, including economics, optimization, image recovery, mechanics, quantum physics, transportation and control theory; for more details, see [1–31] and the references therein.

Strictly pseudocontractive mappings, which act as a link between nonexpansive mappings and pseudocontractive mappings, have been extensively studied by many authors; see [20–31] and the references therein. The computation of fixed points is important in the study of many real world problems, including inverse problems; for instance, it is not hard to show that the split feasibility problem and the convex feasibility problem in signal processing and image reconstruction can both be formulated as a problem of finding fixed points of certain operators, respectively.

Recently, many authors studied the following convex feasibility problem (CFP): $x\in {\bigcap }_{i=1}^N{\mathrm{\Omega }}_i$, where $N\ge 1$ is an integer, and each ${\mathrm{\Omega }}_i$ is assumed to be the fixed point set of a nonlinear mapping $T_i$, $i=1,2,\dots ,N$. There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [32], computer tomography [33] and radiation therapy treatment planning [34].

In this paper, we investigate the problem of finding a common fixed point of a finite family of strictly pseudocontractive mappings based on a viscosity approximation iterative process. Strong convergence theorems of common fixed points are established in a real q-uniformly Banach space.

2 Preliminaries

Throughout this paper, we always assume that E is a real Banach space. Let $E^{\ast }$ be the dual space of E. Let $J_q$ ($q>1$) denote the generalized duality mapping from E into $2^{E^{\ast }}$ given by

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

where $〈\cdot ,\cdot 〉$ denotes the generalized duality pairing. In particular, $J_2$ is called the normalized duality mapping, which is usually denoted by J. In this paper, we use j to denote the single-valued normalized duality mapping. It is known that $J_q(x)={\parallel x\parallel }^{q-2}J(x)$ if $x\ne 0$. If E is a Hilbert space, then $J=I$, the identity mapping. Further, we have the following properties of the generalized duality mapping $J_q$:

• (1) $J_q(tx)=t^{q-1}{J}_q(x)$ for all $x\in E$ and $t\in [0,\mathrm{\infty })$;

• (2) $J_q(-x)=-J_q(x)$ for all $x\in E$.

A Banach space E is said to be smooth if the limit

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

exists for all $x,y\in U_E$. It is also said to be uniformly smooth if the limit is attained uniformly for all $x,y\in U_E$. The norm of E is said to be Fréchet differentiable if, for any $x\in U_E$, the above limit is attained uniformly for all $y\in U_E$. The modulus of smoothness of E is the function ${\rho }_E:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ defined by

$${\rho }_E(\tau )=sup\{\frac{1}{2}(\parallel x+y\parallel +\parallel x-y\parallel )-1:\parallel x\parallel \le 1,\parallel y\parallel \le \tau \},\mathrm{\forall }\tau \ge 0.$$

The Banach space E is uniformly smooth if and only if ${lim}_{\tau \to \mathrm{\infty }}\frac{{\rho }_E(\tau )}{\tau }=0$. Let $q>1$. The Banach space E is said to be q-uniformly smooth if there exists a constant $c>0$ such that ${\rho }_E(\tau )\le c{\tau }^q$. Indeed, there is no Banach space which is q-uniformly smooth with $q>2$. Hilbert spaces, $L^p$ (or $l^p$) spaces and Sobolev spaces $W_m^p$, where $p\ge 2$, are 2-uniformly smooth.

Let C be a nonempty closed convex subset of E and $T:C\to C$ be a mapping. In this paper, we use $F(T)$ to denote the fixed point set of T. A mapping T is said to be κ-contractive iff there exists a constant $\kappa \in (01)$ such that

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

A mapping T is said to be nonexpansive iff

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

A mapping T is said to be κ-strictly pseudocontractive iff there exist a constant $\kappa \in (0,1)$ and $j_q(x-y)\in J_q(x-y)$ such that

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

(2.1)

It is clear that (2.1) is equivalent to the following:

$$〈(I-T)x-(I-T)y,j_q(x-y)〉\ge \kappa {\parallel (I-T)x-(I-T)y\parallel }^q,\mathrm{\forall }x,y\in C.$$

(2.2)

The class of κ-strictly pseudocontractive mappings was first introduced by Browder and Petryshyn [35] in Hilbert spaces.

One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping. More precisely, take $t\in (0,1)$ and define a contraction $T_t:C\to C$ by

$$T_{t}x=tu+(1-t)Tx,\mathrm{\forall }x\in C,$$

(2.3)

where $u\in C$ is a fixed point. Banach’s contraction mapping principle guarantees that $T_t$ has a unique fixed point $x_t$ in C. In the case of T having a fixed point, Browder [36] proved that $x_t$ converges strongly to a fixed point of T in the framework of Hilbert spaces. Reich [37] extended Browder’s result to the setting of Banach spaces and proved that if E is a uniformly smooth Banach space, then $x_t$ converges strongly to a fixed point of T, and the limit defines the (unique) sunny nonexpansive retraction from C onto $F(T)$; for more details, see [37] and the reference therein.

Recently, Xu [38] investigated the viscosity approximation process in a smooth Banach space. Let $f:C\to C$ be a contraction. Take $t\in (0,1)$ and define a mapping $T_t:C\to C$ by

$$T_{t}x=tf(x)+(1-t)Tx,\mathrm{\forall }x\in C.$$

(2.4)

It is not hard to see that $T_t$ also enjoys a unique fixed point. Xu proved that $\{z_t\}$ converges to a fixed point of T as $t\to 0$, and $Q(f)=s-{lim}_{t\to 0}{z}_t$ defines the unique sunny nonexpansive retraction from C onto $F(T)$.

Recently, construction of fixed points for nonexpansive mappings via the normal Mann iterative process has been extensively investigated by many authors. The normal Mann iterative process generates a sequence $\{x_n\}$ in the following manner:

$$\{\begin{array}{c}{x}_1\in C\text{chosen arbitrarily},\hfill \\ x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(2.5)

where the sequence $\{{\alpha }_n\}$ is in the interval $(0,1)$.

In an infinite-dimensional Hilbert space, the normal Mann iteration algorithm has only weak convergence. In many disciplines, including economics, image recovery and control theory, problems arise in infinite dimension spaces. In such problems, strong convergence is often much more desirable than weak convergence, for it translates the physically tangible property that the energy $\parallel x_n-x\parallel$ of the error between the iterate $x_n$ and the solution x eventually becomes arbitrarily small. We also remark here that many authors have been instigating the problem of modifying the normal Mann iteration process to have strong convergence for κ-strictly pseudocontractive mappings; see [24–27] and the references therein.

Let D be a nonempty subset of C. Let $Q:C\to D$. Q is said to be a contraction iff $Q^2=Q$; sunny iff for each $x\in C$ and $t\in (0,1)$, we have $Q(tx+(1-t)Qx)=Qx$; sunny nonexpansive retraction iff Q is sunny, nonexpansive and a contraction. K is said to be a nonexpansive retract of C if there exists a nonexpansive retraction from C onto D. The following result, which was established in [39], describes a characterization of sunny nonexpansive retractions on a smooth Banach space.

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 is sunny and nonexpansive;

• (2) ${\parallel Qx-Qy\parallel }^2\le 〈x-y,j(Qx-Qy)〉$, $\mathrm{\forall }x,y\in E$;

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

In this paper, we investigate the problem of modifying the normal Mann iteration process for a family of κ-strictly pseudocontractive mappings. Strong convergence of the purposed iterative process is obtained in a real q-uniformly Banach space. In order to prove our main results, we need the following tools.

Lemma 2.1 [27]

Let C be a nonempty subset of a real q-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 $\alpha \in (0,\mu ]$, where $\mu =min\{1,{\{\frac{q\kappa }{D}\}}^{\frac{1}{q-1}}\}$, $T_{\alpha }$ is nonexpansive such that $F(T_{\alpha })=F(T)$.

Lemma 2.2 [40]

Let E be a real q-uniformly smooth Banach space. Then the following inequality holds:

$${\parallel x+y\parallel }^q\le {\parallel x\parallel }^q+q〈y,J_{q}x〉+D{\parallel y\parallel }^q,\mathrm{\forall }x,y\in E,$$

where D is some fixed positive constant.

Lemma 2.3 [41]

Assume that $\{a_n\}$ is a sequence of nonnegative real numbers such that

$$a_{n+1}\le (1-t_n)a_n+b_n+0(t_n),$$

where $\{b_n\}$ and $\{t_n\}$ satisfy the following restrictions:

• (i) $t_n\to 0$, ${\sum }_{n=1}^{\mathrm{\infty }}{t}_n=\mathrm{\infty }$;

• (ii) ${\sum }_{n=1}^{\mathrm{\infty }}|b_n|<\mathrm{\infty }$.

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

Lemma 2.4 [42]

Let $\{x_n\}$ and $\{y_n\}$ be bounded sequences in a Banach space X, and let $\{{\beta }_n\}$ be a sequence in $[0,1]$ with

$$0<\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\beta }_n\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\beta }_n<1.$$

Suppose $x_{n+1}=(1-{\beta }_n)y_n+{\beta }_n{x}_n$ for all integers $n\ge 0$ and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}(\parallel y_{n+1}-y_n\parallel -\parallel x_{n+1}-x_n\parallel )\le 0.$$

Then ${lim}_{n\to \mathrm{\infty }}\parallel y_n-x_n\parallel =0$.

Lemma 2.5 [25]

Let E be a smooth Banach space, and let C be a nonempty convex subset of E. Given an integer $N\ge 1$, assume that ${\{T_i\}}_{i=1}^N:C\to C$ is a finite family of ${\kappa }_i$-strict pseudocontractions such that ${\bigcap }_{i=1}^{N}F(T_i)\ne \mathrm{\varnothing }$. Assume that ${\{{\lambda }_i\}}_{i=1}^r$ is a positive sequence such that ${\sum }_{i=1}^N{\lambda }_i=1$. Then $F({\sum }_{i=1}^{N}F(T_i))={\bigcap }_{i=1}^{N}F(T_i)$.

Lemma 2.6 [43]

Let $q>1$. Then the following inequality holds:

$$ab\le \frac{a^q}{q}+\frac{(q-1)b^{\frac{q}{q-1}}}{q}$$

for arbitrary positive real numbers a and b.

3 Main results

Theorem 3.1 Let C be a nonempty closed convex subset of a real q-uniformly smooth Banach space E, and let N be some positive integer. Let $T_i:C\to C$ be a ${\kappa }_i$-strictly pseudocontractive mapping for each $1\le i\le N$. Assume that ${\bigcap }_{i=1}^{N}F(T_i)\ne \mathrm{\varnothing }$. Let f be an α-contractive mapping. Let $\{x_n\}$ be a sequence generated in the following process:

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{arbitrarily chosen}},\hfill \\ y_n=(1-{\delta }_n)x_n+{\delta }_n{\sum }_{i=1}^N{\lambda }_i{T}_i{x}_n,\hfill \\ x_{n+1}={\alpha }_{n}f(x_n)+{\beta }_n{x}_n+{\gamma }_n{y}_n,n\ge 0,\hfill \end{array}$$

where $\{{\alpha }_n\}$, $\{{\beta }_n\}$ and $\{{\lambda }_i\}$ are real number sequences in $[0,1]$ satisfying the following restrictions:

• (a) ${\alpha }_n+{\beta }_n+{\gamma }_n=1$;

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

• (c) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n<1$;

• (d) ${lim}_{n\to \mathrm{\infty }}\parallel {\delta }_{n+1}-{\delta }_n\parallel =0$, $\delta \le {\delta }_n\le min\{1,{\{\frac{q\kappa }{D}\}}^{\frac{1}{q-1}}\}$;

• (e) ${\sum }_{i=1}^N{\lambda }_i=1$,

where $\delta >0$ is some real number, and $\kappa :=min\{{\kappa }_i:1\le i\le N\}$. Then $\{x_n\}$ converges strongly as $n\to \mathrm{\infty }$ to some point in ${\bigcap }_{i=1}^{N}F(T_i)$, which is the unique solution in ${\bigcap }_{i=1}^{N}F(T_i)$ to the following variational inequality:

$$〈f(z)-z,j_q(z-p)〉\ge 0,\mathrm{\forall }p\in \bigcap _{i=1}^{N}F(T_i).$$

Proof First, we show that $\{x_n\}$ and $\{y_n\}$ are bounded. Putting $T:={\sum }_{i=1}^N{\lambda }_i{T}_i$, we see that T is a κ-strictly pseudocontractive mapping. Indeed, we have the following:

$$\begin{array}{r}〈Tx-Ty,j(x-y)〉\\ ={\lambda }_1〈T_{1}x-T_{1}y,j(x-y)〉+{\lambda }_2〈T_{2}x-T_{2}y,j(x-y)〉+\cdots +{\lambda }_N〈T_{N}x-T_{N}y,j(x-y)〉\\ \le {\lambda }_1({\parallel x-y\parallel }^2-{\kappa }_1{\parallel (I-T_1)x-(I-T_1)y\parallel }^2)\\ +{\lambda }_2({\parallel x-y\parallel }^2-{\kappa }_2{\parallel (I-T_2)x-(I-T_2)y\parallel }^2)+\cdots \\ +{\lambda }_N({\parallel x-y\parallel }^2-{\kappa }_N{\parallel (I-T_N)x-(I-T_N)y\parallel }^2)\\ \le {\parallel x-y\parallel }^2-\kappa ({\lambda }_1{\parallel (I-T_1)x-(I-T_1)y\parallel }^2\\ +{\lambda }_2{\parallel (I-T_2)x-(I-T_2)y\parallel }^2+\cdots +{\lambda }_N{\parallel (I-T_N)x-(I-T_N)y\parallel }^2)\\ \le {\parallel x-y\parallel }^2-\kappa {\parallel (I-T)x-(I-T)y\parallel }^2.\end{array}$$

This proves that T is a κ-strictly pseudocontractive mapping. Fix $p\in {\bigcap }_{i=1}^{N}F(T_i)$ and put $T_{{\delta }_n}x=(1-{\delta }_n)x+{\delta }_{n}Tx$, $\mathrm{\forall }x\in C$. It follows from Lemma 2.1 that $T_{{\delta }_n}$ is nonexpansive. This in turn implies that

$$\parallel y_n-p\parallel \le \parallel x_n-p\parallel .$$

(3.1)

It follows that

$$\begin{array}{rl}\parallel x_{n+1}-p\parallel & =\parallel {\alpha }_n(f(x_n)-p)+{\beta }_n(x_n-p)+{\gamma }_n(y_n-p)\parallel \\ \le {\alpha }_n\parallel f(x_n)-f(p)\parallel +{\alpha }_n\parallel p-f(p)\parallel +{\beta }_n\parallel x_n-p\parallel +{\gamma }_n\parallel y_n-p\parallel \\ \le {\alpha }_n\alpha \parallel x_n-p\parallel +{\alpha }_n\parallel p-f(p)\parallel +{\beta }_n\parallel x_n-p\parallel +{\gamma }_n\parallel y_n-p\parallel \\ \le (1-{\alpha }_n(1-\alpha ))\parallel x_n-p\parallel +{\alpha }_n\parallel p-f(p)\parallel \\ \le max\{\parallel x_n-p\parallel ,\frac{\parallel f(p)-p\parallel }{1-\alpha }\}.\end{array}$$

This in turn implies that

$$\parallel x_n-p\parallel \le max\{\parallel x_0-p\parallel ,\frac{\parallel p-f(p)\parallel }{1-\alpha }\},$$

which gives that the sequence $\{x_n\}$ is bounded, so is $\{y_n\}$. Notice that

$$\begin{array}{rl}\parallel y_{n+1}-y_n\parallel & =\parallel T_{{\delta }_{n+1}}{x}_{n+1}-T_{{\delta }_n}{x}_n\parallel \\ \le \parallel T_{{\delta }_{n+1}}{x}_{n+1}-T_{{\delta }_{n+1}}{x}_n\parallel +\parallel T_{{\delta }_{n+1}}{x}_n-T_{{\delta }_n}{x}_n\parallel \\ \le \parallel x_{n+1}-x_n\parallel +\parallel {\delta }_{n+1}{x}_n+(1-{\delta }_{n+1})T{x}_n-{\delta }_n{x}_n-(1-{\delta }_n)T{x}_n\parallel \\ \le \parallel x_{n+1}-x_n\parallel +|{\delta }_{n+1}-{\delta }_n|\parallel x_n-T{x}_n\parallel .\end{array}$$

(3.2)

Putting $t_n=\frac{x_{n+1}-{\beta }_n{x}_n}{1-{\beta }_n}$, we see that

$$x_{n+1}=(1-{\beta }_n)t_n+{\beta }_n{x}_n.$$

(3.3)

Now, we compute $t_{n+1}-t_n$. Noticing that

$$\begin{array}{rl}{t}_{n+1}-t_n& =\frac{{\alpha }_{n+1}f(x_{n+1})+{\gamma }_{n+1}{y}_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_{n}f(x_n)+{\gamma }_n{y}_n}{1-{\beta }_n}\\ =\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}f(x_{n+1})+\frac{1-{\beta }_{n+1}-{\alpha }_{n+1}}{1-{\beta }_{n+1}}{y}_{n+1}-\frac{{\alpha }_n}{1-{\beta }_n}f(x_n)-\frac{1-{\beta }_n-{\alpha }_n}{1-{\beta }_n}{y}_n\\ =\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}(f(x_{n+1})-y_{n+1})+\frac{{\alpha }_n}{1-{\beta }_n}(y_n-f(x_n))+y_{n+1}-y_n,\end{array}$$

we have

$$\parallel t_{n+1}-t_n\parallel \le \frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\parallel f(x_{n+1})-y_{n+1}\parallel +\frac{{\alpha }_n}{1-{\beta }_n}\parallel y_n-f(x_n)\parallel +\parallel y_{n+1}-y_n\parallel .$$

(3.4)

Substituting (3.2) into (3.4), we arrive at

$$\begin{array}{rl}\parallel t_{n+1}-t_n\parallel -\parallel x_{n+1}-x_n\parallel \le & \frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\parallel f(x_{n+1})-y_{n+1}\parallel +\frac{{\alpha }_n}{1-{\beta }_n}\parallel y_n-f(x_n)\parallel \\ +|{\delta }_{n+1}-{\delta }_n|\parallel x_n-T{x}_n\parallel .\end{array}$$

It follows from the restrictions (b) and (c) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel t_{n+1}-t_n\parallel -\parallel x_{n+1}-x_{n+1}\parallel <0.$$

In view of Lemma 2.4, we obtain that ${lim}_{n\to \mathrm{\infty }}\parallel t_n-x_n\parallel =0$. This implies from the restriction (c) that

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

(3.5)

Notice that

$$x_{n+1}-x_n={\alpha }_n(f(x_n)-x_n)+{\gamma }_n(y_n-x_n).$$

It follows that ${lim}_{n\to \mathrm{\infty }}\parallel y_n-x_n\parallel =0$. On the other hand, we have $y_n-x_n={\delta }_n(T{x}_n-x_n)$. It follows that ${lim}_{n\to \mathrm{\infty }}\parallel T{x}_n-x_n\parallel =0$. This in turn implies that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel T_{\mu }{x}_n-x_n\parallel =0,$$

(3.6)

where $\mu =min\{1,{\{\frac{q\kappa }{D}\}}^{\frac{1}{q-1}}\}$. Next, we show that

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

(3.7)

where $z=Qf(z)$, where Q is a sunny nonexpansive retraction from C onto ${\bigcap }_{i=1}^{N}F(T_i)$, is the strong limit of the sequence $z_t$ defined by

$$z_t=tf(z_t)+(1-t)T_{\mu }{z}_t,\mathrm{\forall }t\in (0,1).$$

It follows that

$$z_t-x_n=(1-t)(T_{\mu }{z}_t-x_n)+t(f(z_t)-x_n),\mathrm{\forall }t\in (0,1).$$

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

$$\begin{array}{rl}{\parallel z_t-x_n\parallel }^q& =t〈f(z_t)-x_n,j_q(z_t-x_n)〉+(1-t)〈T_{\mu }{z}_t-x_n,j_q(z_t-x_n)〉\\ \le t〈f(z_t)-x_n,j_q(z_t-x_n)〉+(1-t){\parallel z_t-x_n\parallel }^q+M\parallel T_{\mu }{x}_n-x_n\parallel ,\end{array}$$

where $M=sup\{{\parallel x_n-z_t\parallel }^{q-1}:t\in (0,1),n\ge 0\}$. It follows that

$$〈f(z_t)-x_t,j_q(x_n-z_t)〉\le \frac{M}{t}\parallel T_{\mu }{x}_n-x_n\parallel .$$

Fixing t and letting $n\to \mathrm{\infty }$ yields that

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

Since E is q-uniformly smooth, $J_q:E\to E^{\ast }$ is uniformly continuous on any bounded sets of E, which ensures that ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}$ and ${lim\hspace{0.17em}sup}_{t\to 0}$ are interchangeable, hence

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

Finally, we show that $x_n\to z$ as $n\to \mathrm{\infty }$. In view of Lemma 2.6, we see that

$$\begin{array}{rl}{\parallel x_{n+1}-z\parallel }^q=& {\alpha }_n〈f(x_n)-z,j_q(x_{n+1}-z)〉+{\beta }_n〈x_n-z,j_q(x_{n+1}-z)〉\\ +{\gamma }_n〈y_n-z,j_q(x_{n+1}-z)〉\\ \le & {\alpha }_n〈f(x_n)-z,j_q(x_{n+1}-z)〉+{\beta }_n\parallel x_n-z\parallel {\parallel x_{n+1}-z\parallel }^{q-1}\\ +{\gamma }_n\parallel y_n-z\parallel {\parallel x_{n+1}-z\parallel }^{q-1}\\ \le & {\alpha }_n{\lambda }_n+(1-{\alpha }_n)\parallel x_n-z\parallel {\parallel x_{n+1}-z\parallel }^{q-1}\\ \le & {\alpha }_n{\lambda }_n+(1-{\alpha }_n)\frac{{\parallel x_n-z\parallel }^q}{q}+\frac{q-1}{q}{\parallel x_{n+1}-z\parallel }^q,\end{array}$$

where ${\lambda }_n=max\{〈f(x_n)-z,j_q(x_{n+1}-z)〉,0\}$. This implies that

$${\parallel x_{n+1}-z\parallel }^q\le q{\alpha }_n{\lambda }_n+(1-{\alpha }_n){\parallel x_n-z\parallel }^q.$$

In view of Lemma 2.3, we find the desired conclusion immediately. This completes the proof. □

Remark 3.2 Theorem 3.1 mainly improves the corresponding results in Yuan et al. [22] from 2-uniformly smooth Banach spaces to q-uniformly smooth Banach spaces. Theorem 3.1 is applicable to the spaces $l^p$ and $L^p$ for all $q>1$.

From Theorem 3.1, we have the following result immediately.

Corollary 3.3 Let C be a nonempty closed convex subset of a real q-uniformly smooth Banach space E, and let N be some positive integer. Let $T_i:C\to C$ be a ${\kappa }_i$-strictly pseudocontractive mapping for each $1\le i\le N$. Assume that ${\bigcap }_{i=1}^{N}F(T_i)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be a sequence generated in the following process:

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{arbitrarily chosen}},\hfill \\ y_n=(1-{\delta }_n)x_n+{\delta }_n{\sum }_{i=1}^N{\lambda }_i{T}_i{x}_n,\hfill \\ x_{n+1}={\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_n{y}_n,n\ge 0,\hfill \end{array}$$

where u is a fixed element in C, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ and $\{{\lambda }_i\}$ are real number sequences in $[0,1]$ satisfying the following restrictions:

• (a) ${\alpha }_n+{\beta }_n+{\gamma }_n=1$;

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

• (c) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n<1$;

• (d) ${lim}_{n\to \mathrm{\infty }}\parallel {\delta }_{n+1}-{\delta }_n\parallel =0$, $\delta \le {\delta }_n\le min\{1,{\{\frac{q\kappa }{D}\}}^{\frac{1}{q-1}}\}$;

• (e) ${\sum }_{i=1}^N{\lambda }_i=1$,

where $\delta >0$ is some real number, and $\kappa :=min\{{\kappa }_i:1\le i\le N\}$. Then $\{x_n\}$ converges strongly as $n\to \mathrm{\infty }$ to some point in ${\bigcap }_{i=1}^{N}F(T_i)$, which is the unique solution in ${\bigcap }_{i=1}^{N}F(T_i)$ to the following variational inequality:

$$〈u-z,j(z-p)〉\ge 0,\mathrm{\forall }p\in \bigcap _{i=1}^{N}F(T_i).$$

Remark 3.4 Corollary 3.3 improves the corresponding results in Zhou [26] from 2-uniformly smooth Banach spaces to q-uniformly smooth Banach spaces and relaxes the restrictions imposed on the parameter $\{{\lambda }_i\}$ in Zhang and Su [27].