Convergence theorems of a new iteration for asymptotically nonexpansive mappings in Banach spaces

Abstract

In this paper, the problem of modified iterative approximation of common fixed points of asymptotically nonexpansive is investigated in the framework of Banach spaces. Weak convergence theorems are established.

MSC:47H09, 47J05, 47J25, 47H25.

1 Introduction

Fixed-point theory as an important branch of nonlinear analysis has been applied in the study of nonlinear phenomena. The theory itself is a beautiful mixture of analysis, topology, and geometry. Recently, iterative algorithms for finding common fixed points of nonlinear mappings have been considered by many authors. The well-known convex feasibility problem capture application in various disciplines such as image restorations, and radiation therapy treatment planning is to find a point in the intersection of common fixed-point sets of nonlinear mappings (see, [1–6]).

From the method of generating iterative sequence, we can divide iterative algorithms into explicit algorithms. Recently, both explicit Mann iterative algorithms and implicit Mann-iterative algorithms have been extensively studied for approximating common fixed points of nonlinear mappings (see [7–17]).

In this paper, we consider the problem of approximating a common fixed point of asymptotically nonexpansive mappings based on a general implicit iterative algorithm, which includes an explicit process as a special case. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, weak convergence theorems are established in a uniformly convex Banach space.

2 Preliminaries

Let E be a real Banach space. E is said to be uniformly convex if for any two sequences $\{x_n\}$ and $\{y_n\}$ in E such that $\parallel x_n\parallel =\parallel y_n\parallel =1$ and ${lim}_{n\to \mathrm{\infty }}\parallel x_n+y_n\parallel =2$, then ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$ holds. It is known that a uniformly convex Banach space is reflexive.

In this paper, we use the symbols ⇀ and → denote weak convergence and strong convergence, respectively. E is said to have Opial’s condition (see [18]) if, for each sequence $\{x_n\}$ in E, $x_n\rightharpoonup x$ implies that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_n-x\parallel <\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_n-y\parallel ,\mathrm{\forall }y\in E(y\ne x).$$

Let C be a nonempty subset of E, and $T:C\to C$ a mapping. In this paper, the symbol $F(T)$ stands for the fixed point set of T. T is said to be nonexpansive if

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

T is said to be asymptotically nonexpansive if there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ as $n\to \mathrm{\infty }$ such that

$$\parallel T^{n}x-T^{n}y\parallel \le k_n\parallel x-y\parallel ,\mathrm{\forall }x,y\in C,\mathrm{\forall }n\ge 1.$$

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [19] as a generalization of the class of nonexpansive mappings. They proved that if C is a nonempty, closed, convex, and bounded subset of a real uniformly convex Banach space, then every asymptotically nonexpansive self mapping has a fixed point (see [19]).

In order to prove our main results, we still need the following lemmas.

Lemma 2.1 [20]

Let C be a nonempty, closed, and convex subset of a uniformly convex Banach space E. Let $T:C\to C$ be an asymptotically nonexpansive mapping. Then $I-T$ is demiclosed at zero, that is, $x_n\rightharpoonup x$ and $x_n-T{x}_n\to 0$ imply that $x=Tx$.

Lemma 2.2 [21]

Let $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ be three nonnegative sequences satisfying the following condition:

$$a_{n+1}\le (1+b_n)a_n+c_n,\mathrm{\forall }n\ge n_0,$$

where $n_0$ is some nonnegative integer, ${\sum }_{n=1}^{\mathrm{\infty }}{b}_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{c}_n<\mathrm{\infty }$. Then the ${lim}_{n\to \mathrm{\infty }}{a}_n$ exists.

Lemma 2.3 [15]

Let E be a uniformly convex Banach space, $r>0$ a positive number and $B_r(0)$ a closed ball of E with the center at zero. Then there exists a continuous, strictly increasing, and convex function $g:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $g(0)=0$ such that

$${\parallel \sum _{s=1}^m({\alpha }_s{x}_s)\parallel }^2\le \sum _{s=1}^m\left({\alpha }_s{\parallel x_s\parallel }^2\right)-{\alpha }_i{\alpha }_{j}g(\parallel x_i-x_j\parallel ),\mathrm{\forall }i,j\in \{1,2,\dots ,r\},$$

where $x_1,x_2,\dots ,x_m\in B_r(0)$, and ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m\in (0,1)$ with ${\sum }_{i=1}^m{\alpha }_i=1$.

3 Main Results

Before starting the main results in this paper, we give the implicit iterative process first. Let C be a nonempty, closed, and convex subset of a Banach space E. Let $T:C\to C$ be an asymptotically nonexpansive mapping with the sequence $\{k_n\}$. For every $u\in C$ and $t_n\in (0,1)$, Define a mapping $T_n:C\to C$ below

$$T_n=t_{n}u+(1-t_n)T^{n}x,\mathrm{\forall }x\in C,\mathrm{\forall }n\ge 1.$$

If $(1-t_n)k_n<1$, for every $n\ge 1$, then $T_n$ is a contraction. In the light of the Banach contraction principle, we see that there exists a unique fixed point of $T_n$, for every $n\ge 1$.

Let $x_0$ be chosen arbitrarily and $r\ge 1$ a positive integer. Let $\{{\alpha }_n\},\{{\beta }_{n,1}\},\{{\beta }_{n,2}\},\dots ,\{{\beta }_{n,r}\},\{{\gamma }_{n,1}\},\{{\gamma }_{n,2}\},\dots$ , and $\{{\gamma }_{n,r}\},\{a_n\},\{b_{n,1}\},\{b_{n,2}\},\dots ,\{b_{n,r}\},\{c_{n,1}\},\{c_{n,2}\},\dots$ , and $\{c_{n,r}\}$ be real number sequences in $(0,1)$ such that

$${\alpha }_n+\sum _{m=1}^r{\beta }_{n,m}+\sum _{m=1}^r{\gamma }_{n,m}=a_n+\sum _{m=1}^r{b}_{n,m}+\sum _{m=1}^r{c}_{n,m}=1.$$

Let $S_m,T_m:C\to C$ be asymptotically nonexpansive mappings, for every $m\in \{1,2,\dots ,r\}$.

Find $x_1$, $y_1$ by solving the following equations:

Find $x_2$, $y_2$ by solving the following equations:

Find $x_n$, $y_n$ by solving the following equations:

In view of the above, we have the following implicit iterative algorithm:

Now we show that (ϒ) can be employed to approximate fixed points of asymptotically nonexpansive mapplings, which are assumed to be Lipschitz continuous. Let $S_m:C\to C$ be an asymptotically nonexpansive mapping with the sequence $\{s_{n,m}\}$, and $T_m:C\to C$ an asymptotically nonexpansive mapping with the sequence $\{t_{n,m}\}$, for every $m\in \{1,2,\dots ,r\}$, where $r\ge 1$ is some positive integer. Define a mapping $C_n:C\to C$ by

$$\begin{array}{rcl}{C}_n(x)& =& {\alpha }_n{x}_{n-1}+\sum _{m=1}^r{\beta }_{n,m}{S}_m^n{x}_{n-1}\\ +\sum _n^m{\gamma }_{n,m}{T}_m^n(a_{n}x+\sum _{m=1}^r{b}_{n,m}{S}_m^{n}x+\sum _m^n{c}_{n,m}{T}_m^{n}x),\mathrm{\forall }n\ge 1.\end{array}$$

It follows that

where $t_n=max\{t_{n,m}:1\le m\le r\}$ and $s_n=max\{s_{n,m}:1\le m\le r\}$.

If ${\sum }_{m=1}^r{\gamma }_{n,m}{t}_n(a_n+{\sum }_{m=1}^r{b}_{n,m}{s}_n+{\sum }_{m=1}^r{c}_{n,m}{t}_n)<1$ for all $1\le m\le r$, $n\ge 1$, then $C_n$ is a contraction. Hence, by the Banach contraction principle, there exists a unique fixed point $x_n\in C$ such that

$$\begin{array}{rcl}{x}_n& =& C_n(x)={\alpha }_n{x}_{n-1}+\sum _{m=1}^r{\beta }_{n,m}{S}_m^n{x}_{n-1}\\ +\sum _n^m{\gamma }_{n,m}{T}_m^n(a_{n}x+\sum _{m=1}^r{b}_{n,m}{S}_m^{n}x+\sum _m^n{c}_{n,m}{T}_m^{n}x),\mathrm{\forall }n\ge 1.\end{array}$$

That is, the implicit iterative algorithm (ϒ) is well defined.

Now, we are in a position to give our main results.

Theorem 3.1 Let C be a nonempty, closed, and convex subset of a uniformly convex Banach space E. Let $S_m:C\to C$ be an asymptotically nonexpansive mapping with the sequence $\{s_{n,m}\}$, and $T_m:C\to C$ an asymptotically nonexpansive mapping with the sequence $\{t_{n,m}\}$, for every $m\in \{1,2,\dots ,r\}$, where $r\ge 1$ is some positive integer. Assume that

$$\mathcal{F}=\bigcap _{m=1}^{r}F(S_m)\cap \bigcap _{m=1}^{r}F(T_m)\ne \mathrm{\varnothing }.$$

Let $t_n=max\{t_{n,m}:1\le m\le r\}$ and $s_n=max\{s_{n,m}:1\le m\le r\}$. Assume that ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$, where $k_n=max\{s_n,t_n:1\le m\le r\}$. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence generated by (ϒ), where $\{{\alpha }_n\},\{{\beta }_{n,1}\},\{{\beta }_{n,2}\},\dots ,\{{\beta }_{n,r}\},\{{\gamma }_{n,1}\},\{{\gamma }_{n,2}\},\dots ,\{{\gamma }_{n,r}\},\{a_n\},\{b_{n,1}\},\{b_{n,2}\},\dots ,\{b_{n,r}\},\{c_{n,1}\},\{c_{n,2}\},\dots ,\{c_{n,r}\}$ be real number sequences in $(0,1)$ such that ${\alpha }_n+{\sum }_{m=1}^r{\beta }_{n,m}+{\sum }_{m=1}^r{\gamma }_{n,m}=a_n+{\sum }_{m=1}^r{b}_{n,m}+{\sum }_{m=1}^r{c}_{n,m}=1$. Assume that the following restrictions imposed on the control sequence $\{{\alpha }_n\},\{{\beta }_{n,1}\},\{{\beta }_{n,2}\},\dots ,\{{\beta }_{n,r}\},\{{\gamma }_{n,1}\},\{{\gamma }_{n,2}\},\dots ,\{{\gamma }_{n,r}\},\{a_n\},\{b_{n,1}\},\{b_{n,2}\},\dots ,\{b_{n,r}\},\{c_{n,1}\},\{c_{n,2}\},\dots ,\{c_{n,r}\}$ are satisfied

• (a) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\beta }_{n,m}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\gamma }_{n,m}>0$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_n{b}_{n,m}>0$, $\mathrm{\forall }m\in \{1,2,\dots ,r\}$;

• (b) ${\sum }_{m=1}^r{\gamma }_{n,m}{t}_n(a_n+{\sum }_{m=1}^r{b}_{n,m}{s}_n+{\sum }_{m=1}^r{c}_{n,m}{t}_n)<1$.

Then

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-S_m{x}_n\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T_m{x}_n\parallel =0,\mathrm{\forall }m\in \{1,2,\dots ,r\}.$$

Proof Step 1. Taking $p\in \mathcal{F}$, we see that

$$\begin{array}{rcl}\parallel x_n-p\parallel & \le & {\alpha }_n\parallel x_{n-1}-p\parallel +\sum _{m=1}^r{\beta }_{n,m}\parallel S_m^n{x}_{n-1}-p\parallel +\sum _{m=1}^r{\gamma }_{n,m}\parallel T_m^n{y}_n-p\parallel \\ \le & ({\alpha }_n+\sum _{m=1}^r{\beta }_{n,m}{k}_n)\parallel x_{n-1}-p\parallel +\sum _{m=1}^r{\gamma }_{n,m}{k}_n\parallel y_n-p\parallel \end{array}$$

(3.1)

and

$$\begin{array}{rcl}\parallel y_n-p\parallel & \le & a_n\parallel x_n-p\parallel +\sum _{m=1}^r{b}_{n,m}\parallel S_m^n{x}_n-p\parallel +\sum _{m=1}^r{c}_{n,m}\parallel T_m^n{x}_n-p\parallel \\ \le & a_n{k}_n\parallel x_n-p\parallel +\sum _{m=1}^r{b}_{n,m}{k}_n\parallel x_n-p\parallel +\sum _{m=1}^r{c}_{n,m}{k}_n\parallel x_n-p\parallel \\ \le & k_n\parallel x_n-p\parallel .\end{array}$$

(3.2)

Substituting (3.2) into (3.1), we have

$$\parallel x_n-p\parallel \le ({\alpha }_n+\sum _{m=1}^r{\beta }_{n,m}{k}_n)\parallel x_{n-1}-p\parallel +\sum _{m=1}^r{\gamma }_{n,m}{k}_n^2\parallel x_n-p\parallel .$$

(3.3)

In view of ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\beta }_{n,m}>0$, and ${\alpha }_n+{\sum }_{m=1}^r{\beta }_{n,m}+{\sum }_{m=1}^r{\gamma }_{n,m}=1$, we see that there exists some positive integer $n_1$, and a real number h, where $h\in (0,1)$, such that

$$\sum _{m=1}^r{\gamma }_{n,m}\le h,n\ge n_1.$$

Since ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$, we find that there exists some positive integer $n_2$ such that $k_n^2\le 1+\frac{1-h}{2h}$, $\mathrm{\forall }n\ge n_2$. It follows that

$$\sum _{m=1}^r{\gamma }_{n,m}{k}_n^2\le u<1,\mathrm{\forall }n\ge n_3,$$

where $u=h(1+\frac{1-h}{2h})$, and $n_3\ge max\{n_1,n_2\}$. It follows (3.3) that

$$\begin{array}{rcl}\parallel x_n-p\parallel & \le & \left(\frac{{\alpha }_n+{\sum }_{m=1}^r{\beta }_{n,m}{k}_n}{1-{\sum }_{m=1}^r{\gamma }_{n,m}{k}_n^2}\right)\parallel x_{n-1}-p\parallel \\ \le & (1+\frac{{\alpha }_n+{\sum }_{m=1}^r{\beta }_{n,m}{k}_n+{\sum }_{m=1}^r{\gamma }_{n,m}{k}_n^2-1}{1-{\sum }_{m=1}^r{\gamma }_{n,m}{k}_n^2})\parallel x_{n-1}-p\parallel \\ \le & (1+\frac{k_n^2-1}{1-u})\parallel x_{n-1}-p\parallel .\end{array}$$

(3.4)

It follows from Lemma 2.2 that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exists. This implies that the sequence $\{x_n\}$ is bounded.

On the other hand, we find from Lemma 2.3 that

It implies that

Since ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exists, we find from restriction (a) that

$$\underset{n\to \mathrm{\infty }}{lim}P\left(\parallel x_{n-1}-S_m^n{x}_{n-1}\parallel \right)=0,$$

for every $m\in \{1,2,\dots ,r\}$. It follows that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n-1}-S_m^n{x}_{n-1}\parallel =0,\mathrm{\forall }m\in \{1,2,\dots ,r\}.$$

(3.5)

In view of Lemma 2.3, we have

$$\begin{array}{rcl}{\parallel x_n-p\parallel }^2& \le & {\alpha }_n{\parallel x_{n-1}-p\parallel }^2+\sum _{m=1}^r{\beta }_{n,m}{\parallel S_m^n{x}_{n-1}-p\parallel }^2\\ +\sum _{m=1}^r{\gamma }_{n,m}{\parallel T_m^n{y}_n-p\parallel }^2-{\alpha }_n{\gamma }_{n,m}P\left(\parallel x_{n-1}-T_m^n{y}_n\parallel \right)\\ \le & ({\alpha }_n+\sum _{m=1}^r{\beta }_{n,m}{k}_n^2){\parallel x_{n-1}-p\parallel }^2+\sum _{m=1}^r{\gamma }_{n,m}{k}_n^3{\parallel x_n-p\parallel }^2\\ -{\alpha }_n{\gamma }_{n,m}P\left(\parallel x_{n-1}-T_m^n{y}_n\parallel \right),\mathrm{\forall }m\in \{1,2,\dots ,r\}.\end{array}$$

It implies that

Since ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exists, from the condition (a), we have that

$$\underset{n\to \mathrm{\infty }}{lim}P\left(\parallel x_{n-1}-T_m^n{y}_n\parallel \right)=0,$$

for every $m\in \{1,2,\dots ,r\}$. It follows that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n-1}-T_m^n{y}_n\parallel =0,\mathrm{\forall }m\in \{1,2,\dots ,r\}.$$

(3.6)

Notice that

$$\parallel x_n-x_{n-1}\parallel \le \sum _{m=1}^r{\beta }_{n,m}\parallel S_m^n{x}_{n-1}-x_{n-1}\parallel +\sum _{m=1}^r{\gamma }_{n,m}\parallel T_m^n{y}_n-x_{n-1}\parallel$$

In the light of (3.5), and (3.6), we find that

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

(3.7)

Step 2. Notice that

$$\parallel x_n-T_m^n{y}_n\parallel \le \parallel x_n-x_{n-1}\parallel +\parallel x_{n-1}-T_m^n{y}_n\parallel ,\mathrm{\forall }m\in \{1,2,\dots ,r\}.$$

It implies from (3.6), and (3.7) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T_m^n{y}_n\parallel =0,\mathrm{\forall }m\in \{1,2,\dots ,r\}.$$

(3.8)

On the other hand, we have

$$\begin{array}{rcl}\parallel x_n-S_m^n{x}_n\parallel & \le & \parallel x_n-x_{n-1}\parallel +\parallel x_{n-1}-S_m^n{x}_{n-1}\parallel \\ +\parallel S_m^n{x}_{n-1}-S_m^n{x}_n\parallel ,\mathrm{\forall }m\in \{1,2,\dots ,r\}.\end{array}$$

Since $S_m$ is Lipschitz for every $m\in \{1,2,\dots ,r\}$, we see from (3.5) and (3.7) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-S_m^n{x}_n\parallel =0,\mathrm{\forall }m\in \{1,2,\dots ,r\}.$$

(3.9)

Step 3. In view of ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_n{b}_{n,m}>0$, and

$$a_n+\sum _{m=1}^r{b}_{n,m}+\sum _{m=1}^r{c}_{n,m}=1,$$

we see that there exists some positive integer $n_4$, and a real number $h^{\prime }$, where $h^{\prime }\in (0,1)$, such that

$$\sum _{m=1}^r{c}_{n,m}\le h^{\prime },n\ge n_4.$$

Since ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$, we find that there exists a positive integer $n_5$ such that $k_n\le 1+\frac{1-h^{\prime }}{2h^{\prime }}$, $\mathrm{\forall }n\ge n_5$. It follows that

$$\sum _{m=1}^r{c}_{n,m}{k}_n\le u^{\prime }<1,\mathrm{\forall }n\ge n_6,$$

where $u^{\prime }=h^{\prime }(1+\frac{1-h^{\prime }}{2h^{\prime }})$, and $n_6\ge max\{n_4,n_5\}$. It follows that

$$\begin{array}{rcl}\parallel x_n-y_n\parallel & \le & \sum _{m=1}^r{b}_{n,m}\parallel S_m^n{x}_n-x_n\parallel +\sum _{m=1}^r{c}_{n,m}\parallel T_m^n{x}_n-x_n\parallel \\ \le & \sum _{m=1}^r{b}_{n,m}\parallel S_m^n{x}_n-x_n\parallel +\sum _{m=1}^r{c}_{n,m}\parallel T_m^n{x}_n-T_m^n{y}_n\parallel +\sum _{m=1}^r{c}_{n,m}\parallel T_m^n{y}_n-x_n\parallel \\ \le & \sum _{m=1}^r{b}_{n,m}\parallel S_m^n{x}_n-x_n\parallel +\sum _{m=1}^r{c}_{n,m}{k}_n\parallel x_n-y_n\parallel +\sum _{m=1}^r{c}_{n,m}\parallel T_m^n{y}_n-x_n\parallel .\end{array}$$

This implies that

It follows from (3.8) and (3.9) that

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

(3.10)

Step 4. Notice that

$$\begin{array}{rcl}\parallel y_n-y_{n-1}\parallel & \le & a_n\parallel x_n-x_{n-1}\parallel +\sum _{m=1}^r{b}_{n,m}\parallel S_m^n{x}_n-x_{n-1}\parallel \\ +\sum _{m=1}^r{c}_{n,m}\parallel T_m^n{x}_n-x_{n-1}\parallel +\parallel x_{n-1}-y_{n-1}\parallel \\ \le & a_n\parallel x_n-x_{n-1}\parallel +\sum _{m=1}^r{b}_{n,m}\parallel S_m^n{x}_n-x_n\parallel \\ +\sum _{m=1}^r{b}_{n,m}\parallel x_n-x_{n-1}\parallel +\sum _{m=1}^r{c}_{n,m}\parallel T_m^n{x}_n-T_m^n{y}_n\parallel \\ +\sum _{m=1}^r{c}_{n,m}\parallel T_m^n{y}_n-x_{n-1}\parallel +\parallel x_{n-1}-y_{n-1}\parallel .\end{array}$$

Since $T_m$ is Lipschitz for every $m\in \{1,2,\dots ,r\}$, we see from (3.6), (3.7), (3.9), and (3.10) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-y_{n-1}\parallel =0.$$

(3.11)

Step 5. Notice that

$$\begin{array}{rcl}\parallel x_n-S_m{x}_n\parallel & \le & \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-S_m^{n+1}{x}_{n+1}\parallel \\ +\parallel S_m^{n+1}{x}_{n+1}-S_m^{n+1}{x}_n\parallel +\parallel S_m^{n+1}{x}_n-S_m{x}_n\parallel \\ \le & (1+M)\parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-S_m^{n+1}{x}_{n+1}\parallel \\ +M\parallel S_m^n{x}_n-x_n\parallel ,\end{array}$$

where $M={sup}_{n\ge 1}\{k_n\}$. It follows from (3.7) and (3.9) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-S_m{x}_n\parallel =0,\mathrm{\forall }m\in \{1,2,\dots ,r\}.$$

(3.12)

On the other hand, we have

$$\begin{array}{rcl}\parallel x_n-T_m{x}_n\parallel & \le & \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-T_m^{n+1}{y}_{n+1}\parallel \\ +\parallel T_m^{n+1}{y}_{n+1}-T_m^{n+1}{y}_n\parallel +\parallel T_m^{n+1}{y}_n-T_m{x}_n\parallel \\ \le & \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-T_m^{n+1}{y}_{n+1}\parallel +M\parallel y_{n+1}-y_n\parallel \\ +M\parallel T_m^n{y}_n-x_n\parallel .\end{array}$$

It follows from (3.7), (3.8), and (3.11) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T_m{x}_n\parallel =0,\mathrm{\forall }m\in \{1,2,\dots ,r\}.$$

(3.13)

This completes the proof. □

Next, we give the following weak convergence theorems with the help of Opial’s condition.

Theorem 3.2 Let C be a nonempty, closed, and convex subset of a uniformly convex Banach space E with Opial’s condition. Let $S_m:C\to C$ be an asymptotically nonexpansive mapping with the sequence $\{s_{n,m}\}$, and $T_m:C\to C$ an asymptotically nonexpansive mapping with the sequence $\{t_{n,m}\}$, for every $m\in \{1,2,\dots ,r\}$, where $r\ge 1$ is some positive integer. Assume that

$$\mathcal{F}=\bigcap _{m=1}^{r}F(S_m)\cap \bigcap _{m=1}^{r}F(T_m)\ne \mathrm{\varnothing }.$$

Let $t_n=max\{t_{n,m}:1\le m\le r\}$, and $s_n=max\{s_{n,m}:1\le m\le r\}$. Assume that ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$, where $k_n=max\{s_n,t_n:1\le m\le r\}$. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence generated by (ϒ), where $\{{\alpha }_n\},\{{\beta }_{n,1}\},\{{\beta }_{n,2}\},\dots ,\{{\beta }_{n,r}\},\{{\gamma }_{n,1}\},\{{\gamma }_{n,2}\},\dots$ , and $\{{\gamma }_{n,r}\},\{a_n\},\{b_{n,1}\},\{b_{n,2}\},\dots ,\{b_{n,r}\},\{c_{n,1}\},\{c_{n,2}\},\dots$ , and $\{c_{n,r}\}$ be real number sequences in $(0,1)$ such that

$${\alpha }_n+\sum _{m=1}^r{\beta }_{n,m}+\sum _{m=1}^r{\gamma }_{n,m}=a_n+\sum _{m=1}^r{b}_{n,m}+\sum _{m=1}^r{c}_{n,m}=1.$$

Assume that restrictions (a) and (b) as in Theorem  3.1 are satisfied. Then $\{x_n\}$ converges weakly to some point in ℱ.

Proof Since $\{x_n\}$ is bounded, there exists a subsequence $\{x_{n_i}\}\subset \{x_n\}$ such that $\{x_{n_i}\}$ converges weakly to a point $\overline{x}\in C$. It follows from Lemma 2.1 that $\overline{x}\in \mathcal{F}$. Assume that there exists another subsequence $\{x_{n_j}\}\subset \{x_n\}$ such that $\{x_{n_j}\}$ converges weakly to a point $\hat{x}\in C$. It follows from Lemma 2.1 that $\hat{x}\in \mathcal{F}$. If $\overline{x}\ne \hat{x}$, then

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-\overline{x}\parallel & =& \underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_i}-\overline{x}\parallel <\underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_i}-\hat{x}\parallel \\ =& \underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_j}-\hat{x}\parallel <\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_j}-\overline{x}\parallel \\ =& \underset{n\to \mathrm{\infty }}{lim}\parallel x_n-\overline{x}\parallel .\end{array}$$

This is a contradiction. Hence, $\overline{x}=\hat{x}$. This completes the proof. □

If $r=1$, then Theorem 3.2 is reduced to the following.

Corollary 3.1 Let C be a nonempty, closed, and convex subset of a uniformly convex Banach space E with Opial’s condition. Let $S:C\to C$ be an asymptotically nonexpansive mapping with the sequence $\{s_n\}$, and $T:C\to C$ an asymptotically nonexpansive mapping with the sequence $\{t_n\}$. Assume that

$$\mathcal{F}=F(S)\cap F(T)\ne \mathrm{\varnothing }.$$

Assume that ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$, where $k_n=max\{s_n,t_n:1\le m\le r\}$. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence generated by the following:

$$\{\begin{array}{c}{x}_0\in C,\hfill \\ x_n={\alpha }_n{x}_{n-1}+{\beta }_n{S}^n{x}_{n-1}+{\gamma }_n{T}^n{y}_n,\hfill \\ y_n=a_n{x}_n+b_n{S}^n{x}_n+c_n{T}^n{x}_n,\mathrm{\forall }n\ge 1.\hfill \end{array}$$

where $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ are real number sequences in $(0,1)$ such that ${\alpha }_n+{\beta }_n+{\gamma }_n=a_n+b_n+c_n=1$. Assume that the following restrictions imposed on the control sequences $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ are satisfied:

• (a) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\beta }_n>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\gamma }_n>0$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_n{b}_n>0$;

• (b) ${\gamma }_n{t}_n(a_n+b_n{s}_n+c_n{t}_n)<1$.

Then $\{x_n\}$ converges weakly to some point in ℱ.

If $b_{n,m}=c_{n,m}=0$, than Theorem 3.2 reduced the following.

Corollary 3.2 Let C be a nonempty, closed, and convex subset of a uniformly convex Banach space E with Opial’s condition. Let $S_m:C\to C$ be an asymptotically nonexpansive mapping with the sequence $\{s_{n,m}\}$, and $T_m:C\to C$ an asymptotically nonexpansive mapping with the sequence $\{t_{n,m}\}$, for every $m\in \{1,2,\dots ,r\}$. where $r\ge 1$ is some positive integer. Assume that

$$\mathcal{F}=\bigcap _{m=1}^{r}F(S_m)\cap \bigcap _{m=1}^{r}F(T_m)\ne \mathrm{\varnothing }.$$

Let $t_n=max\{t_{n,m}:1\le m\le r\}$, and $s_n=max\{s_{n,m}:1\le m\le r\}$. Assume that ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$, where $k_n=max\{s_n,t_n:1\le m\le r\}$. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence generated by the following:

$$x_0\in C,x_n={\alpha }_n{x}_{n-1}+\sum _{m=1}^r{\beta }_{n,m}{S}_m^n{x}_{n-1}+\sum _{m=1}^r{\gamma }_{n,m}{T}_m^n{x}_n,\mathrm{\forall }n\ge 1$$

where $\{{\alpha }_n\},\{{\beta }_{n,1}\},\{{\beta }_{n,2}\},\dots ,\{{\beta }_{n,r}\},\{{\gamma }_{n,1}\},\{{\gamma }_{n,2}\},\dots$ , and $\{{\gamma }_{n,r}\}$, are real number sequences in $(0,1)$ such that ${\alpha }_n+{\sum }_{m=1}^r{\beta }_{n,m}+{\sum }_{m=1}^r{\gamma }_{n,m}=1$. Assume that the following restrictions imposed on the control sequences $\{{\alpha }_n\},\{{\beta }_{n,1}\},\{{\beta }_{n,2}\},\dots ,\{{\beta }_{n,r}\},\{{\gamma }_{n,1}\},\{{\gamma }_{n,2}\},\dots$ , and $\{{\gamma }_{n,r}\}$ are satisfied

• (a) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\beta }_{n,m}>0$, and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\gamma }_{n,m}>0$, $\mathrm{\forall }m\in \{1,2,\dots r\}$;

• (b) ${\sum }_{m=1}^r{\gamma }_{n,m}{t}_n<1$.

Then $\{x_n\}$ converges weakly to some point in ℱ.

If ${\beta }_{n,m}=b_{n,m}=0$, than Theorem 3.2 reduced the following.

Corollary 3.3 Let C be a nonempty, closed, and convex subset of a uniformly convex Banach space E with Opial’s condition. Let $T_m:C\to C$ be an asymptotically nonexpansive mapping with the sequence $\{t_{n,m}\}$, for every $m\in \{1,2,\dots ,r\}$, where $r\ge 1$ is some positive integer. Assume that

$$\mathcal{F}=\bigcap _{m=1}^{r}F(T_m)\ne \mathrm{\varnothing }.$$

Assume that ${\sum }_{n=1}^{\mathrm{\infty }}(t_n-1)<\mathrm{\infty }$, where $t_n=max\{t_{n,m}:1\le m\le r\}$. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence generated by the following:

$$\{\begin{array}{c}{x}_0\in C,\hfill \\ x_n={\alpha }_n{x}_{n-1}+{\sum }_{m=1}^r{\gamma }_{n,m}{T}_m^n{y}_n,\hfill \\ y_n=a_n{x}_n+{\sum }_{m=1}^r{c}_{n,m}{T}_m^n{x}_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

where $\{{\alpha }_n\},\{{\gamma }_{n,1}\},\{{\gamma }_{n,2}\},\dots ,\{{\gamma }_{n,r}\},\{a_n\},\{c_{n,1}\},\{c_{n,2}\},\dots ,\{c_{n,r}\}$ be real number sequences in $(0,1)$ such that

$${\alpha }_n+\sum _{m=1}^r{\gamma }_{n,m}=a_n+\sum _{m=1}^r{c}_{n,m}=1.$$

Assume that the following restrictions imposed on the control sequence $\{{\alpha }_n\},\{{\gamma }_{n,1}\},\{{\gamma }_{n,2}\},\dots ,\{{\gamma }_{n,r}\},\{a_n\},\{c_{n,1}\},\{c_{n,2}\},\dots ,\{c_{n,r}\}$ are satisfied

• (a) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\gamma }_{n,m}>0$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_n>0$, $\mathrm{\forall }m\in \{1,2,\dots ,r\}$;

• (b) ${\sum }_{m=1}^r{\gamma }_{n,m}{t}_n(a_n+{\sum }_{m=1}^r{c}_{n,m}{t}_n)<1$.

Then $\{x_n\}$ converges weakly to some point in ℱ.

If $S_m=I$, where I stands for the identity mappings, then Theorem 3.2 reduced the following.

Corollary 3.4 Let C be a nonempty, closed, and convex subset of a uniformly convex Banach space E with Opial’s condition. Let $T_m:C\to C$ be an asymptotically nonexpansive mapping with the sequence $\{t_{n,m}\}$, for every $m\in \{1,2,\dots ,r\}$, where $r\ge 1$ is some positive integer. Assume that

$$\mathcal{F}=\bigcap _{m=1}^{r}F(T_m)\ne \mathrm{\varnothing }.$$

Asume that ${\sum }_{n=1}^{\mathrm{\infty }}(t_n-1)<\mathrm{\infty }$, where $t_n=max\{t_{n,m}:1\le m\le r\}$. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence generated by the following:

$$\{\begin{array}{c}{x}_0\in C,\hfill \\ x_n=({\alpha }_n+{\sum }_{m=1}^r{\beta }_{n,m})x_{n-1}+{\sum }_{m=1}^r{\gamma }_{n,m}{T}_m^n{y}_n,\hfill \\ y_n=(a_n+{\sum }_{m=1}^r{b}_{n,m})x_n+{\sum }_{m=1}^r{c}_{n,m}{T}_m^n{x}_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

where $\{{\alpha }_n\},\{{\beta }_{n,1}\},\{{\beta }_{n,2}\},\dots ,\{{\beta }_{n,r}\},\{{\gamma }_{n,1}\},\{{\gamma }_{n,2}\},\dots ,\{{\gamma }_{n,r}\},\{a_n\},\{b_{n,1}\},\{b_{n,2}\},\dots ,\{b_{n,r}\},\{c_{n,1}\},\{c_{n,2}\},\dots ,\{c_{n,r}\}$ be real number sequences in $(0,1)$ such that ${\alpha }_n+{\sum }_{m=1}^r{\beta }_{n,m}+{\sum }_{m=1}^r{\gamma }_{n,m}=a_n+{\sum }_{m=1}^r{b}_{n,m}+{\sum }_{m=1}^r{c}_{n,m}=1$. Assume that the following restrictions imposed on the control sequences $\{{\alpha }_n\},\{{\beta }_{n,1}\},\{{\beta }_{n,2}\},\dots ,\{{\beta }_{n,r}\},\{{\gamma }_{n,1}\},\{{\gamma }_{n,2}\},\dots ,\{{\gamma }_{n,r}\},\{a_n\},\{b_{n,1}\},\{b_{n,2}\},\dots ,\{b_{n,r}\},\{c_{n,1}\},\{c_{n,2}\},\dots ,\{c_{n,r}\}$ are satisfied

• (a) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\beta }_{n,m}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\gamma }_{n,m}>0$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_n{b}_{n,m}>0$, $\mathrm{\forall }m\in \{1,2,\dots ,r\}$;

• (b) ${\sum }_{m=1}^r{\gamma }_{n,m}{t}_n(a_n+{\sum }_{m=1}^r{b}_{n,m}+{\sum }_{m=1}^r{c}_{n,m}{t}_n)<1$.

Then $\{x_n\}$ converges weakly to some point in ℱ.

If $T_m=I$, $b_{n,m}=0$, where I stands for the identity mappings, then Theorem 3.2 reduced the following.

Corollary 3.5 Let C be a nonempty, closed, and convex subset of a uniformly convex Banach spaces E with Opial’s condition. Let $S_m:C\to C$ be an asymptotically nonexpansive mapping with the sequence $\{s_{n,m}\}$, for every $m\in \{1,2,\dots ,r\}$, where $r\ge 1$ is some positive integer. Assume that

$$\mathcal{F}=\bigcap _{m=1}^{r}F(S_m)\ne \mathrm{\varnothing }.$$

Assume that ${\sum }_{n=1}^{\mathrm{\infty }}(s_n-1)<\mathrm{\infty }$, where $s_n=max\{s_{n,m}:1\le m\le r\}$. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence generated by the following:

$$\{\begin{array}{c}{x}_0\in C,\hfill \\ x_n=\frac{{\alpha }_n}{1-{\sum }_{m=1}^r(a_n+{\sum }_{m=1}^r{c}_{n,m})}{x}_{n-1}+\frac{{\sum }_{m=1}^r{\beta }_{n,m}{S}_m^n}{1-{\sum }_{m=1}^r(a_n+{\sum }_{m=1}^r{c}_{n,m})}{x}_{n-1},\mathrm{\forall }n\ge 1,\hfill \end{array}$$

where $\{{\alpha }_n\},\{{\beta }_{n,1}\},\{{\beta }_{n,2}\},\dots ,\{{\beta }_{n,r}\},\{{\gamma }_{n,1}\},\{{\gamma }_{n,2}\},\dots ,\{{\gamma }_{n,r}\},\{a_n\},\{c_{n,1}\},\{c_{n,2}\},\dots ,\{c_{n,r}\}$ be real number sequences in $(0,1)$ such that

$${\alpha }_n+\sum _{m=1}^r{\beta }_{n,m}+\sum _{m=1}^r{\gamma }_{n,m}=a_n+\sum _{m=1}^r{c}_{n,m}=1.$$

Assume that the following restrictions imposed on the control sequences $\{{\alpha }_n\},\{{\beta }_{n,1}\},\{{\beta }_{n,2}\},\dots ,\{{\beta }_{n,r}\},\{{\gamma }_{n,1}\},\{{\gamma }_{n,2}\},\dots ,\{{\gamma }_{n,r}\},\{a_n\},\{c_{n,1}\},\{c_{n,2}\},\dots ,\{c_{n,r}\}$ are satisfied ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\beta }_{n,m}>0$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\gamma }_{n,m}>0$ for all $m\in \{1,2,\dots ,r\}$. Then $\{x_n\}$ converges weakly to some point in ℱ.