# A new iterative process for a hybrid pair of generalized asymptotically nonexpansive single-valued and generalized nonexpansive multi-valued mappings in Banach spaces

## Abstract

In this paper, we construct an iterative process involving a hybrid pair of a finite family of generalized asymptotically nonexpansive single-valued mappings and a finite family of generalized nonexpansive multi-valued mappings and prove weak and strong convergence theorems of the proposed iterative process in Banach spaces. Our main results extend and generalize many results in the literature.

## Introduction

Throughout this paper we denote by $$\mathbb{N}$$ the set of all positive integers. Let X be a Banach space and let D be a nonempty subset of X. Let $$CB(D)$$ and $$KC(D)$$ denote the families of nonempty, closed, and bounded subsets and nonempty, compact, and convex subsets of D, respectively. The Hausdorff metric on $$CB(D)$$ is defined by

$$H(A,B) = \max \Bigl\{ \sup_{x\in A} \operatorname{dist}(x,B), \sup_{y\in B} \operatorname{dist}(y,A) \Bigr\} \quad\mbox{for }A,B\in CB(D),$$

where $$\operatorname{dist}(x,D)=\inf\{\|x-y\|:y\in D\}$$ is the distance from a point x to a subset D. Let t be a single-valued mapping of D into D and T be a multi-valued mapping of D into $$CB(D)$$. The set of fixed points of t and T will be denoted by $$F(t)=\{x\in D:x=tx\}$$ and $$F(T)=\{x\in D:x\in Tx\}$$, respectively. A point x is called a common fixed point of t and T if $$x=tx\in Tx$$.

### Definition 1.1

A single-valued mapping $$t:D\to D$$ is said to be generalized asymptotically nonexpansive if there exist sequences $$\{k_{n}\}\subset[1,\infty)$$ and $$\{s_{n}\}\subset[0,\infty)$$ with $$\lim_{n\to\infty}k_{n} =1$$, $$\lim_{n\to\infty}s_{n} =0$$ such that

$$\bigl\| t^{n}x-t^{n}y\bigr\| \leq k_{n}\|x-y \|+s_{n},$$

for all $$x,y\in D$$ and $$n\in\mathbb{N}$$.

In the case of $$s_{n}=0$$, for all $$n\in\mathbb{N}$$, a single-valued mapping t is called an asymptotically nonexpansive mapping. In particular, if $$k_{n}=1$$ and $$s_{n}=0$$, for all $$n\in\mathbb{N}$$, a single-valued mapping t reduce to a nonexpansive mapping. The fixed point property for generalized asymptotically nonexpansive single-valued mappings can be found in . The following example shows that the fixed point set of a generalized asymptotically nonexpansive mapping is not necessarily closed; see also .

### Example 1.2

()

Define a single-valued mapping $$t: [-\frac{2}{3},\frac{2}{3} ]\to [-\frac{2}{3},\frac{2}{3} ]$$ by

$$\textstyle tx = \left \{ \begin{array}{@{}l@{\quad}l} x, & \mbox{if }x \in [ { - \frac{2}{3},0} ), \\ \frac{16}{81}, & \mbox{if }x =0, \\ x^{4} , & \mbox{if }x \in ( {0,\frac{2}{3}} ]. \end{array} \right .$$

Then t is generalized asymptotically nonexpansive and $$F(t)=[ { - \frac{2}{3},0} )$$ which is not closed.

### Definition 1.3

A multi-valued mapping $$T:D\to CB(D)$$ is said to be

1. (i)

nonexpansive if $$H(Tx,Ty)\leq\|x-y\|$$, for all $$x,y\in D$$;

2. (ii)

quasi-nonexpansive if $$F(T)\ne\emptyset$$ and $$H(Tx,Tp)\leq\|x-p\|$$, for all $$x\in D$$ and $$p\in F(T)$$.

The study of fixed points for nonexpansive multi-valued mappings using the Hausdorff metric was initiated by Markin . Different iterative processes have been used to approximate fixed points of nonexpansive and quasi-nonexpansive multi-valued mappings; in particular, Sastry and Babu  considered Mann and Ishikawa iterates for a multi-valued mapping T with a fixed point p and proved that these iterates converge to a fixed point q of T under certain conditions. Moreover, they illustrated that the fixed point q may be different from p. Later in 2007, Panyanak  generalized results of Sastry and Babu  to uniformly convex Banach spaces and proved a convergence theorem of Mann iterates for a mapping defined on a noncompact domain. In 2009, Shahzad and Zegeye  proved strong convergence theorems for the Ishikawa iteration scheme involving quasi-nonexpansive multi-valued mappings. They constructed an iterative process which removes the restriction of T, namely end-point condition, i.e., $$Tp = \{p\}$$ for any $$p\in F(T)$$; see also [7, 8].

In 2011, Garcia-Falset et al.  introduced a new condition on single-valued mappings, called condition (E), which is weaker than nonexpansiveness. Later, Abkar and Eslamian  used a modified condition for multi-valued mappings as follows.

### Definition 1.4

A multi-valued mapping $$T:D\to CB(D)$$ is said to satisfy condition ($$E_{\mu}$$) where $$\mu\geq0$$ if for each $$x,y\in D$$,

$$\operatorname{dist}(x,Ty)\leq\mu\operatorname{dist}(x,Tx)+\|x-y\|.$$

We say that T satisfies condition (E) whenever T satisfies ($$E_{\mu}$$) for some $$\mu\geq1$$.

### Remark 1.5

From the above definitions, it is clear that if T is nonexpansive, then T satisfies the condition ($$E_{1}$$).

In 2011, Sokhuma and Kaewkhao  introduced the following iterative process for approximating a common fixed point of a pair of a nonexpansive single-valued mapping t and a nonexpansive multi-valued mapping T:

\begin{aligned} \left \{ \begin{array}{@{}l} y_{n} = (1-\alpha_{n})x_{n} + \alpha_{n} z_{n},\\ x_{n+1} = (1-\beta_{n})x_{n} + \beta_{n} ty_{n},\quad n\in\mathbb{N}, \end{array} \right . \end{aligned}
(1.1)

where $$x_{1}\in D$$, $$z_{n}\in Tx_{n}$$, and $$0< a\leq\alpha_{n},\beta_{n}\leq b<1$$. They also proved a strong convergence theorem for the iterative process (1.1) in uniformly convex Banach spaces.

In 2013, Eslamian  extended the results of [11, 13] in uniformly convex Banach spaces. He used the following iterative process for a pair of a finite family of asymptotically nonexpansive single-valued mappings $$\{t_{i}\}_{i=1}^{N}$$ and a finite family of quasi-nonexpansive multi-valued mapping $$\{T_{i}\}_{i=1}^{N}$$:

\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} y_{n} = \beta_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\beta_{n}^{(i)} z_{n}^{(i)}, \\ x_{n+1} = \alpha_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\alpha_{n}^{(i)}t_{i}^{n}y_{n},\quad n\in\mathbb{N}, \end{array} \right . \end{aligned}
(1.2)

where $$x_{1}\in D$$, $$z_{n}^{(i)}\in T_{i}x_{n}$$, and $$\{\alpha_{n}^{(i)}\}$$, $$\{\beta_{n}^{(i)}\}$$ are sequences in $$[0,1]$$ for all $$i=1,2,\ldots,N$$ such that $$\sum_{i=0}^{N}\alpha_{n}^{(i)}=\sum_{i=0}^{N}\beta_{n}^{(i)}=1$$.

In this paper, motivated by the above results, we propose an iterative process for approximating a common fixed point of a pair of a finite family of generalized asymptotically nonexpansive single-valued mappings and a finite family of quasi-nonexpansive multi-valued mappings and prove weak and strong convergence theorems of the proposed iterative process in Banach spaces.

## Preliminaries

A Banach space X is called uniformly convex if for each $$\varepsilon>0$$ there is a $$\delta>0$$ such that for $$x,y \in X$$ with $$\Vert x \Vert \leq1$$, $$\Vert y \Vert \leq1$$, and $$\Vert x - y \Vert \geq\varepsilon$$, $$\Vert x + y \Vert \leq2(1-\delta)$$ holds. The following result was proved by Xu .

### Proposition 2.1

Let X be a uniformly convex Banach space and let $$r > 0$$. Then there exists a strictly increasing, continuous, and convex function $$g: [0,\infty) \to[0,\infty)$$ with $$g(0)=0$$ such that

$$\bigl\| \lambda x + (1-\lambda)y\bigr\| ^{2}\leq\lambda\|x\|^{2} + (1- \lambda)\|y\|^{2}-\lambda(1-\lambda)g\bigl(\|x-y\|\bigr)$$

for all $$x,y\in B_{r}=\{z\in X:\|z\|\leq r\}$$ and $$\lambda\in[0,1]$$.

A Banach space X is said to satisfy the Opial property (see ) if it is given that whenever $$\{x_{n}\}$$ converges weakly to $$x\in X$$,

$$\mathop{\lim\sup}\limits _{n \to\infty} \Vert {x_{n} - x} \Vert < \mathop{\lim\sup}\limits _{n \to\infty} \Vert {x_{n} - y} \Vert$$

for each $$y\in X$$ with $$y \ne x$$. The examples of Banach spaces which satisfy the Opial property are Hilbert spaces and all $$L^{p}[0, 2\pi]$$ with $$1 < p \ne2$$ fail to satisfy the Opial property.

The following results are needed for proving our results.

### Definition 2.2

(see )

Let F be a nonempty subset of a Banach space X and let $$\{x_{n}\}$$ be a sequence in X. We say that $$\{x_{n}\}$$ is of monotone type (I) with respect to F if there exist sequences $$\{\delta_{n}\}$$ and $$\{\varepsilon_{n}\}$$ of nonnegative real numbers such that $$\sum_{n = 1}^{\infty}{\delta_{n} } < \infty$$, $$\sum_{n = 1}^{\infty}{\varepsilon_{n} } < \infty$$, and $$\|x_{n+1}-p\| \leq (1+\delta_{n})\|x_{n}-p\| + \varepsilon_{n}$$ for all $$n\in\mathbb{N}$$ and $$p\in F$$.

### Proposition 2.3

(see )

Let F be a nonempty subset of a Banach space X and let $$\{x_{n}\}$$ be a sequence in X. If $$\{x_{n}\}$$ is of monotone type (I) with respect to F and $$\liminf_{n\to\infty} \operatorname{dist}(x_{n},F) = 0$$, then $$\lim_{n\to\infty} x_{n} = p$$ for some $$p\in X$$ satisfying $$\operatorname{dist}(p,F)=0$$. In particular, if F is closed, then $$p\in F$$.

### Lemma 2.4

(see )

Let $$\{a_{n}\}$$, $$\{b_{n}\}$$, and $$\{c_{n}\}$$ be sequences of nonnegative real numbers satisfy

$$a_{n+1}\leq(1+c_{n})a_{n} + b_{n}, \quad\textit{for all }n\in\mathbb{N},$$

where $$\sum_{n=1}^{\infty}b_{n}<\infty$$ and $$\sum_{n=1}^{\infty}c_{n}<\infty$$. Then:

1. (i)

$$\lim_{n\to\infty}a_{n}$$ exists.

2. (ii)

If $$\liminf_{n\to\infty}a_{n} =0$$, then $$\lim_{n\to\infty}a_{n}=0$$.

### Lemma 2.5

(see )

Let X be a uniformly convex Banach space, let $$\{\lambda_{n}\}$$ be a sequence of real numbers such that $$0< a\leq\lambda_{n}\leq b<1$$, for all $$n\in\mathbb{N}$$, and let $$\{x_{n}\}$$ and $$\{y_{n}\}$$ be sequences of X satisfying, for some $$r\geq0$$,

1. (i)

$$\limsup_{n\to\infty}\|x_{n}\|\leq r$$,

2. (ii)

$$\limsup_{n\to\infty}\|y_{n}\|\leq r$$ and

3. (iii)

$$\lim_{n\to\infty}\|\lambda_{n} x_{n} +(1-\lambda_{n})y_{n}\|= r$$.

Then $$\lim_{n\to\infty}\|x_{n}-y_{n}\|=0$$.

### Lemma 2.6

(see )

Let X be a Banach space which satisfies the Opial property and $$\{x_{n}\}$$ be a sequence in X. Let $$u, v\in X$$ be such that $$\lim_{n\to\infty}\|x_{n} - u\|$$ and $$\lim_{n\to\infty}\|x_{n} - v\|$$ exist. If $$\{x_{n_{i}}\}$$ and $$\{x_{n_{j}}\}$$ are subsequences of $$\{x_{n}\}$$ which converge weakly to u and v, respectively, then $$u = v$$.

## Main results

In this section, we prove weak and strong convergence theorems of the proposed iterative process in Banach spaces. We first note that if $$\{t_{i}\}_{i=1}^{N}$$ is a finite family of generalized asymptotically nonexpansive single-valued mappings of D into itself, where D is a nonempty convex subset of a Banach space X. Then we have $$\|t_{i}^{n}x - t_{i}^{n}y\|\leq k_{n}^{(i)}\|x-y\|+s_{n}^{(i)}$$, for all $$x,y\in D$$ and all $$i=1,2,\ldots,N$$, where $$\{k_{n}^{(i)}\}\subset[1,\infty)$$ and $$\{s_{n}^{(i)}\}\subset [0,\infty)$$ with $$\lim_{n\to\infty}k_{n}^{(i)}=1$$ and $$\lim_{n\to\infty}s_{n}^{(i)}=0$$. Put $$k_{n}=\max_{1\leq i\leq N}\{k_{n}^{(i)}\}$$ and $$s_{n}=\max_{1\leq i\leq N}\{s_{n}^{(i)}\}$$. It is clear that $$\lim_{n\to\infty}k_{n}=1$$ and $$\lim_{n\to\infty}s_{n}=0$$ and

$$\bigl\| t_{i}^{n}x - t_{i}^{n}y\bigr\| \leq k_{n}\|x-y\|+s_{n}$$

for all $$x,y\in D$$, $$i=1,2,\ldots,N$$, and all $$n\in\mathbb{N}$$.

In order to prove our main results, the following lemma is needed.

### Lemma 3.1

Let D be a nonempty, closed, and convex subset of a Banach space X. Let $$\{t_{i}\}_{i=1}^{N}$$ be a finite family of generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences $$\{k_{n}\}\subset[1,\infty)$$ and $$\{s_{n}\}\subset[0,\infty)$$ such that $$\sum_{n=1}^{\infty}(k_{n}-1)<\infty$$ and $$\sum_{n=1}^{\infty}s_{n}<\infty$$. Let $$\{T_{i}\}_{i=1}^{N}$$ be a finite family of quasi-nonexpansive multi-valued mappings of D into $$CB(D)$$. Assume that $$\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap\bigcap_{i=1}^{N} F(T_{i})$$ is nonempty closed and $$T_{i}p=\{p\}$$ for all $$p\in\mathcal{F}$$ and $$i=1,2,\ldots,N$$. Let $$x_{1}\in D$$ and the sequence $$\{x_{n}\}$$ be generated by

\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} y_{n} = \beta_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\beta_{n}^{(i)} z_{n}^{(i)},\quad z_{n}^{(i)}\in T_{i}x_{n}, \\ x_{n+1} = \alpha_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\alpha_{n}^{(i)}t_{i}^{n}y_{n},\quad n\in\mathbb{N}, \end{array} \right . \end{aligned}

where $$\{\alpha_{n}^{(i)}\}$$ and $$\{\beta_{n}^{(i)}\}$$ are sequences in $$[0,1]$$ for all $$i=1,2,\ldots,N$$ such that $$\sum_{i=0}^{N}\alpha_{n}^{(i)}=1$$ and $$\sum_{i=0}^{N}\beta_{n}^{(i)}=1$$. Then $$\lim_{n\to\infty} \|x_{n}-p\|$$ exists for all $$p\in \mathcal{F}$$.

### Proof

Let $$p\in \mathcal{F}$$, for $$i=1,2,\ldots,N$$, we have

\begin{aligned} \|x_{n+1}-p\| &\leq\alpha_{n}^{(0)}\|x_{n} - p\| + \sum_{i=1}^{N}\alpha_{n}^{(i)} \bigl\| t_{i}^{n}y_{n}-p\bigr\| \\ &\leq\alpha_{n}^{(0)}\|x_{n} - p\| + \sum _{i=1}^{N}\alpha_{n}^{(i)} \bigl(k_{n}\|y_{n}-p\| + s_{n}\bigr) \\ &= \alpha_{n}^{(0)}\|x_{n} - p\| + k_{n} \sum_{i=1}^{N}\alpha_{n}^{(i)} \|y_{n}-p\| + s_{n}\sum_{i=1}^{N} \alpha_{n}^{(i)} \\ &\leq\alpha_{n}^{(0)}\|x_{n} - p\| + k_{n} \sum_{i=1}^{N}\alpha_{n}^{(i)} \|y_{n}-p\| +s_{n} \\ &\leq\alpha_{n}^{(0)}\|x_{n} - p\| + k_{n} \sum_{i=1}^{N}\alpha_{n}^{(i)} \Biggl(\beta_{n}^{(0)}\|x_{n}-p\|+\sum _{i=1}^{N}\beta_{n}^{(i)} \bigl\| z_{n}^{(i)}-p\bigr\| \Biggr) + s_{n} \\ &= \Biggl(\alpha_{n}^{(0)} + k_{n} \beta_{n}^{(0)}\sum_{i=1}^{N} \alpha_{n}^{(i)} \Biggr)\|x_{n}-p\| + k_{n} \sum_{i=1}^{N}\alpha_{n}^{(i)} \sum_{i=1}^{N}\beta_{n}^{(i)} \bigl\| z_{n}^{(i)}-p\bigr\| + s_{n} \\ &= \Biggl(\alpha_{n}^{(0)} + k_{n} \beta_{n}^{(0)}\sum_{i=1}^{N} \alpha_{n}^{(i)} \Biggr)\|x_{n}-p\| + k_{n} \sum_{i=1}^{N}\alpha_{n}^{(i)} \sum_{i=1}^{N}\beta_{n}^{(i)} \operatorname {dist}\bigl(z_{n}^{(i)},T_{i}p\bigr)+ s_{n} \\ &\leq \Biggl(\alpha_{n}^{(0)} + k_{n} \beta_{n}^{(0)}\sum_{i=1}^{N} \alpha_{n}^{(i)} \Biggr)\|x_{n}-p\| + k_{n} \sum_{i=1}^{N}\alpha_{n}^{(i)} \sum_{i=1}^{N}\beta_{n}^{(i)}H(T_{i}x_{n},T_{i}p) + s_{n} \\ &\leq \Biggl(\alpha_{n}^{(0)} + k_{n} \beta_{n}^{(0)}\sum_{i=1}^{N} \alpha_{n}^{(i)} \Biggr)\|x_{n}-p\| + k_{n} \sum_{i=1}^{N}\alpha_{n}^{(i)} \sum_{i=1}^{N}\beta_{n}^{(i)} \|x_{n}-p\| + s_{n} \\ &= \Biggl(\alpha_{n}^{(0)} + k_{n}\sum _{i=1}^{N}\alpha_{n}^{(i)} \Biggr) \|x_{n}-p\| + s_{n} \\ &\leq k_{n}\|x_{n}-p\| + s_{n} \\ &= \bigl(1+(k_{n}-1)\bigr)\|x_{n}-p\| + s_{n}. \end{aligned}

By Lemma 2.4, $$\sum_{n=1}^{\infty}(k_{n}-1)<\infty$$ and $$\sum_{n=1}^{\infty}s_{n}<\infty$$, we conclude that $$\lim_{n\to\infty} \|x_{n}-p\|$$ exists for all $$p\in\mathcal{F}$$. □

### Theorem 3.2

Let D be a nonempty, closed, and convex subset of a Banach space X. Let $$\{t_{i}\}_{i=1}^{N}$$ be a finite family of generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences $$\{k_{n}\}\subset[1,\infty)$$ and $$\{s_{n}\}\subset[0,\infty)$$ such that $$\sum_{n=1}^{\infty}(k_{n}-1)<\infty$$ and $$\sum_{n=1}^{\infty}s_{n}<\infty$$. Let $$\{T_{i}\}_{i=1}^{N}$$ be a finite family of quasi-nonexpansive multi-valued mappings of D into $$CB(D)$$. Assume that $$\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap\bigcap_{i=1}^{N} F(T_{i})$$ is nonempty closed and $$T_{i}p=\{p\}$$ for all $$p\in\mathcal{F}$$ and $$i=1,2,\ldots,N$$. Let $$x_{1}\in D$$ and the sequence $$\{x_{n}\}$$ be generated by

\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} y_{n} = \beta_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\beta_{n}^{(i)} z_{n}^{(i)},\quad z_{n}^{(i)}\in T_{i}x_{n}, \\ x_{n+1} = \alpha_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\alpha_{n}^{(i)}t_{i}^{n}y_{n},\quad n\in\mathbb{N}, \end{array} \right . \end{aligned}

where $$\{\alpha_{n}^{(i)}\}$$ and $$\{\beta_{n}^{(i)}\}$$ are sequences in $$[0,1]$$ for all $$i=1,2,\ldots,N$$ such that $$\sum_{i=0}^{N}\alpha_{n}^{(i)}=1$$ and $$\sum_{i=0}^{N}\beta_{n}^{(i)}=1$$. Then the sequence $$\{x_{n}\}$$ converges strongly to a point in $$\mathcal{F}$$ if and only if $$\liminf_{n\to\infty} \operatorname{dist}(x_{n},\mathcal{F})=0$$.

### Proof

The necessity is obvious and thus we prove only the sufficiency. Suppose that $$\liminf_{n\to\infty} \operatorname{dist}(x_{n},\mathcal{F})=0$$. In the proof of Lemma 3.1, we see that the sequence $$\{x_{n}\}$$ is of monotone type (I) with respect to $$\mathcal{F}$$. It follows by Proposition 2.3 that $$\{x_{n}\}$$ converges to a point in $$\mathcal{F}$$. □

The closedness of $$\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap \bigcap_{i=1}^{N} F(T_{i})$$ can be dropped if $$t_{i}$$ is asymptotically nonexpansive for all $$i = 1,2,\ldots,N$$. Then the following corollary is obtained directly from Theorem 3.2.

### Corollary 3.3

Let D be a nonempty, closed, and convex subset of a Banach space X. Let $$\{t_{i}\}_{i=1}^{N}$$ be a finite family of asymptotically nonexpansive single-valued mappings of D into itself with a sequence $$\{k_{n}\}\subset[1,\infty)$$ such that $$\sum_{n=1}^{\infty}(k_{n}-1)<\infty$$. Let $$\{T_{i}\}_{i=1}^{N}$$ be a finite family of quasi-nonexpansive multi-valued mappings of D into $$CB(D)$$. Assume that $$\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap\bigcap_{i=1}^{N} F(T_{i})$$ is nonempty and $$T_{i}p=\{p\}$$ for all $$p\in\mathcal{F}$$ and $$i=1,2,\ldots,N$$. Let $$x_{1}\in D$$ and the sequence $$\{x_{n}\}$$ be generated by

\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} y_{n} = \beta_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\beta_{n}^{(i)} z_{n}^{(i)},\quad z_{n}^{(i)}\in T_{i}x_{n}, \\ x_{n+1} = \alpha_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\alpha_{n}^{(i)}t_{i}^{n}y_{n},\quad n\in\mathbb{N}, \end{array} \right . \end{aligned}

where $$\{\alpha_{n}^{(i)}\}$$ and $$\{\beta_{n}^{(i)}\}$$ are sequences in $$[0,1]$$ for all $$i=1,2,\ldots,N$$ such that $$\sum_{i=0}^{N}\alpha_{n}^{(i)}=1$$ and $$\sum_{i=0}^{N}\beta_{n}^{(i)}=1$$. Then the sequence $$\{x_{n}\}$$ converges strongly to a point in $$\mathcal{F}$$ if and only if $$\liminf_{n\to\infty} \operatorname{dist}(x_{n},\mathcal{F})=0$$.

Recall that a mapping $$t : D \to D$$ is called uniformly L-Lipschitzian if there exists a constant $$L>0$$ such that $$\|t^{n} x-t^{n} y\| \leq L \|x-y\|$$ for all $$x,y \in D$$ and $$n\in\mathbb{N}$$. Next, we prove a strong convergence theorem in a uniformly convex Banach space.

### Lemma 3.4

Let D be a nonempty, closed, and convex subset of a uniformly convex Banach space X. Let $$\{t_{i}\}_{i=1}^{N}$$ be a finite family of uniformly L-Lipschitzian and generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences $$\{k_{n}\}\subset[1,\infty)$$ and $$\{s_{n}\}\subset [0,\infty)$$ such that $$\sum_{n=1}^{\infty}(k_{n}-1)<\infty$$ and $$\sum_{n=1}^{\infty}s_{n}<\infty$$. Let $$\{T_{i}\}_{i=1}^{N}$$ be a finite family of quasi-nonexpansive multi-valued mappings of D into $$CB(D)$$. Assume that $$\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap \bigcap_{i=1}^{N} F(T_{i})$$ is nonempty and $$T_{i}p=\{p\}$$ for all $$p\in \mathcal{F}$$ and $$i=1,2,\ldots,N$$. Let $$x_{1}\in D$$ and the sequence $$\{x_{n}\}$$ be generated by

\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} y_{n} = \beta_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\beta_{n}^{(i)} z_{n}^{(i)}, \quad z_{n}^{(i)}\in T_{i}x_{n}, \\ x_{n+1} = \alpha_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\alpha_{n}^{(i)}t_{i}^{n}y_{n},\quad n\in\mathbb{N}, \end{array} \right . \end{aligned}

where $$\{\alpha_{n}^{(i)}\}$$ and $$\{\beta_{n}^{(i)}\}$$ are sequences in $$[0,1]$$ for all $$i=1,2,\ldots,N$$ such that $$0< a\leq \alpha_{n}^{(i)},\beta_{n}^{(i)}\leq b<1$$, $$\sum_{i=0}^{N}\alpha_{n}^{(i)}=1$$, and $$\sum_{i=0}^{N}\beta_{n}^{(i)}=1$$. Then we have the following:

1. (i)

$$\lim_{n\to\infty} \|x_{n}-z_{n}^{(i)}\|=0$$ for all $$i=1,2,\ldots,N$$;

2. (ii)

$$\lim_{n\to\infty} \|x_{n}-t_{i}x_{n}\|=0$$ for all $$i=1,2,\ldots,N$$.

### Proof

(i) By Lemma 3.1, $$\lim_{n\to\infty} \|x_{n}-p\|$$ exists. Put $$\lim_{n\to\infty} \|x_{n}-p\|=c$$. By the definition of $$\{x_{n}\}$$, we have

\begin{aligned} \bigl\| t_{i}^{n}y_{n} -p\bigr\| &\leq k_{n} \|y_{n}-p\|+s_{n} \\ &\leq k_{n} \Biggl(\beta_{n}^{(0)} \|x_{n}-p\| + \sum_{i=1}^{N} \beta_{n}^{(i)} \bigl\| z_{n}^{(i)}-p\bigr\| \Biggr) +s_{n} \\ &= k_{n}\beta_{n}^{(0)}\|x_{n}-p\|+ k_{n}\sum_{i=1}^{N} \beta_{n}^{(i)}\bigl\| z_{n}^{(i)}-p\bigr\| + s_{n} \\ &= k_{n}\beta_{n}^{(0)}\|x_{n}-p\|+ k_{n}\sum_{i=1}^{N} \beta_{n}^{(i)}\operatorname{dist}\bigl(z_{n}^{(i)},T_{i}p \bigr)+ s_{n} \\ &\leq k_{n}\beta_{n}^{(0)}\|x_{n}-p\|+ k_{n}\sum_{i=1}^{N} \beta_{n}^{(i)}H(T_{i}x_{n},T_{i}p)+ s_{n} \\ &\leq k_{n}\beta_{n}^{(0)}\|x_{n}-p\|+ k_{n}\sum_{i=1}^{N} \beta_{n}^{(i)}\|x_{n}-p\|+ s_{n} \\ &= k_{n} \Biggl(\beta_{n}^{(0)} +\sum _{i=1}^{N}\beta_{n}^{(i)} \Biggr) \|x_{n}-p\| +s_{n} \\ &= k_{n}\|x_{n}-p\|+s_{n}. \end{aligned}

Then we have

\begin{aligned} \limsup_{n\to\infty}\bigl\| t_{i}^{n}y_{n} -p\bigr\| \leq\limsup_{n\to\infty} \bigl(k_{n}\|y_{n}-p \|+s_{n}\bigr) \leq\limsup_{n\to\infty} \bigl(k_{n}\|x_{n}-p \|+s_{n}\bigr). \end{aligned}

By $$\lim_{n\to\infty}k_{n}= 1$$ and $$\lim_{n\to\infty}s_{n}= 0$$, we have

\begin{aligned} \limsup_{n\to\infty}\bigl\| t_{i}^{n}y_{n} -p\bigr\| \leq\limsup_{n\to\infty} \|y_{n}-p\| \leq\limsup _{n\to\infty} \|x_{n}-p\|=c. \end{aligned}
(3.1)

Since $$c=\lim_{n\to\infty} \|x_{n+1}-p\|=\lim_{n\to\infty} \|\alpha_{n}^{(0)} (x_{n}-p)+ \sum_{i=1}^{N}\alpha_{n}^{(i)} (t_{i}^{n}y_{n}-p)\|$$, it follows by Lemma 2.5 that

\begin{aligned} \lim_{n\to\infty}\bigl\| x_{n}-t_{i}^{n}y_{n} \bigr\| =0 \quad\mbox{for all }i=1,2,\ldots,N. \end{aligned}
(3.2)

Consider

\begin{aligned} \|x_{n+1}-p\| &\leq\alpha_{n}^{(0)} \|x_{n}-p\| +\sum_{i=1}^{N} \alpha_{n}^{(i)}\bigl\| t_{i}^{n}y_{n}-p \bigr\| \\ &= \Biggl(1-\sum_{i=1}^{N} \alpha_{n}^{(i)} \Biggr)\|x_{n}-p\|+\sum _{i=1}^{N}\alpha_{n}^{(i)} \bigl\| t_{i}^{n}y_{n}-p\bigr\| \\ &\leq \Biggl(1-\sum_{i=1}^{N} \alpha_{n}^{(i)} \Biggr)\|x_{n}-p\|+\sum _{i=1}^{N}\alpha_{n}^{(i)}\bigl(k_{n} \|y_{n}-p\|+s_{n}\bigr). \end{aligned}

This implies that

\begin{aligned} \|x_{n+1}-p\| - \|x_{n}-p\|\leq\sum _{i=1}^{N}\alpha_{n}^{(i)} \bigl(k_{n}\|y_{n}-p\| - \|x_{n}-p\| + s_{n}\bigr). \end{aligned}

Therefore,

\begin{aligned} \frac{\|x_{n+1}-p\| - \|x_{n}-p\|}{bN} +\|x_{n}-p\|&\leq\frac{\|x_{n+1}-p\| - \|x_{n}-p\|}{\sum_{i=1}^{N}\alpha_{n}^{(i)}} + \|x_{n}-p\| \\ &\leq k_{n}\|y_{n}-p\| + s_{n}. \end{aligned}

By (3.1), we obtain

\begin{aligned} c &=\liminf_{n\to\infty} \biggl(\frac{\|x_{n+1}-p\| - \|x_{n}-p\|}{bN} + \|x_{n}-p\| \biggr) \\ &\leq\liminf_{n\to\infty}\bigl(k_{n}\|y_{n}-p\| + s_{n}\bigr) \\ &=\liminf_{n\to\infty}\|y_{n}-p\| \\ &\leq\limsup_{n\to\infty}\|y_{n}-p\|\leq c. \end{aligned}

Thus,

\begin{aligned} c=\lim_{n\to\infty}\|y_{n}-p\|=\lim_{n\to\infty} \Biggl\Vert \beta_{n}^{(0)}(x_{n}-p)+ \sum _{i=1}^{N}\beta_{n}^{(i)} \bigl(z_{n}^{(i)}-p\bigr)\Biggr\Vert . \end{aligned}

Since

\begin{aligned} \bigl\| z_{n}^{(i)}-p\bigr\| = \operatorname{dist}\bigl(z_{n}^{(i)}, T_{i}p\bigr)\leq H(T_{i}x_{n},T_{i}p) \leq\|x_{n}-p\|, \end{aligned}

it implies that

\begin{aligned} \limsup_{n\to\infty}\bigl\| z_{n}^{(i)}-p\bigr\| \leq \limsup _{n\to\infty}\|x_{n}-p\|=c. \end{aligned}

Hence, by Lemma 2.5, we have

\begin{aligned} \lim_{n\to\infty}\bigl\| x_{n}-z_{n}^{(i)}\bigr\| =0 \quad\mbox{for all }i=1,2,\ldots,N. \end{aligned}

(ii) Since $$t_{i}$$ is generalized asymptotically nonexpansive, for all $$i=1,2,\ldots,N$$, we get

\begin{aligned} \bigl\| t_{i}^{n}x_{n}-x_{n}\bigr\| &\leq \bigl\| t_{i}^{n}x_{n}-t_{i}^{n}y_{n} \bigr\| + \bigl\| t_{i}^{n}y_{n}-x_{n}\bigr\| \leq k_{n}\|x_{n}-y_{n}\| +s_{n}+ \bigl\| t_{i}^{n}y_{n}-x_{n}\bigr\| . \end{aligned}

By the definition of $$\{x_{n}\}$$, we have $$y_{n}-x_{n}=\sum_{i=1}^{N}\beta_{n}^{(i)}(z_{n}^{(i)}-x_{n})$$. This implies that

\begin{aligned} \bigl\| t_{i}^{n}x_{n}-x_{n}\bigr\| &\leq k_{n}\sum_{i=1}^{N} \beta_{n}^{(i)}\bigl\| z_{n}^{(i)}-x_{n} \bigr\| +\bigl\| t_{i}^{n}y_{n}-x_{n} \bigr\| +s_{n} \\ &\leq k_{n}\bigl\| z_{n}^{(i)}-x_{n}\bigr\| + \bigl\| t_{i}^{n}y_{n}-x_{n}\bigr\| +s_{n}. \end{aligned}

Then, by (i) and (3.2), we get

\begin{aligned} \lim_{n\to\infty} \bigl\| x_{n}-t_{i}^{n}x_{n} \bigr\| =0 \quad\mbox{for all }i=1,2,\ldots,N. \end{aligned}
(3.3)

For $$i=1,2,\ldots,N$$, we have

\begin{aligned} \|x_{n}-t_{i}x_{n}\| \leq{}&\|x_{n}-x_{n+1} \| + \bigl\| x_{n+1}-t_{i}^{n+1}x_{n+1}\bigr\| + \bigl\| t_{i}^{n+1}x_{n+1}-t_{i}^{n+1}x_{n} \bigr\| + \bigl\| t_{i}^{n+1}x_{n}-t_{i}x_{n} \bigr\| \\ \leq{}&(1+L)\|x_{n}-x_{n+1}\| + \bigl\| x_{n+1}-t_{i}^{n+1}x_{n+1} \bigr\| + L\bigl\| t_{i}^{n}x_{n}-x_{n}\bigr\| \\ \leq{}&(1+L)\sum_{i=1}^{N} \alpha_{n}^{(i)}\bigl\| x_{n}-t_{i}^{n}y_{n} \bigr\| + \bigl\| x_{n+1}-t_{i}^{n+1}x_{n+1}\bigr\| + L \bigl\| t_{i}^{n}x_{n}-x_{n}\bigr\| . \end{aligned}

By (3.2) and (3.3), we conclude that $$\lim_{n\to\infty} \|x_{n}-t_{i}x_{n}\|=0$$ for all $$i=1,2,\ldots,N$$. □

### Theorem 3.5

Let D be a nonempty, compact, and convex subset of a uniformly convex Banach space X. Let $$\{t_{i}\}_{i=1}^{N}$$ be a finite family of uniformly L-Lipschitzian and generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences $$\{k_{n}\}\subset[1,\infty)$$ and $$\{s_{n}\}\subset [0,\infty)$$ such that $$\sum_{n=1}^{\infty}(k_{n}-1)<\infty$$ and $$\sum_{n=1}^{\infty}s_{n}<\infty$$. Let $$\{T_{i}\}_{i=1}^{N}$$ be a finite family of quasi-nonexpansive multi-valued mappings of D into $$CB(D)$$ satisfying condition (E). Assume that $$\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap\bigcap_{i=1}^{N} F(T_{i})$$ is nonempty and $$T_{i}p=\{p\}$$ for all $$p\in\mathcal{F}$$ and $$i=1,2,\ldots,N$$. Let $$x_{1}\in D$$ and the sequence $$\{x_{n}\}$$ be generated by

\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} y_{n} = \beta_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\beta_{n}^{(i)} z_{n}^{(i)},\quad z_{n}^{(i)}\in T_{i}x_{n}, \\ x_{n+1} = \alpha_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\alpha_{n}^{(i)}t_{i}^{n}y_{n},\quad n\in\mathbb{N}, \end{array} \right . \end{aligned}

where $$\{\alpha_{n}^{(i)}\}$$ and $$\{\beta_{n}^{(i)}\}$$ are sequences in $$[0,1]$$ for all $$i=1,2,\ldots,N$$ such that $$0< a\leq \alpha_{n}^{(i)},\beta_{n}^{(i)}\leq b<1$$, $$\sum_{i=0}^{N}\alpha_{n}^{(i)}=1$$, and $$\sum_{i=0}^{N}\beta_{n}^{(i)}=1$$. Then the sequence $$\{x_{n}\}$$ converges strongly to a point in $$\mathcal{F}$$.

### Proof

By Lemma 3.1, we have $$\{x_{n}\}$$ is bounded. Since D is compact, there exists a subsequence $$\{x_{n_{j}}\}$$ of $$\{x_{n}\}$$ converging strongly to $$p\in D$$. By condition (E), there exists $$\mu\geq1$$ such that for $$i=1,2,\ldots,N$$,

\begin{aligned} \operatorname{dist}(p,T_{i}p) &\leq\|p-x_{n_{j}}\| + \operatorname{dist}(x_{n_{j}},T_{i}p) \\ &\leq\|x_{n_{j}}-p\| + \mu\operatorname{dist}(x_{n_{j}},T_{i}x_{n_{j}}) + \|x_{n_{j}}-p\| \\ &= 2\|x_{n_{j}}-p\| + \mu\operatorname{dist}(x_{n_{j}},T_{i}x_{n_{j}}) \\ &\leq2\|x_{n_{j}}-p\| + \mu\bigl\| x_{n_{j}}-z_{n_{j}}^{(i)} \bigr\| . \end{aligned}

Then, by Lemma 3.4(i), we have $$p\in T_{i}p$$ for all $$i=1,2,\ldots,N$$. So $$p\in\bigcap_{i=1}^{N}F(T_{i})$$.

Since $$t_{i}$$ is uniformly L-Lipschitzian, for all $$i=1,2,\ldots,N$$, we have

\begin{aligned} \|t_{i}p-p\| &\leq\|t_{i}p-t_{i}x_{n_{j}} \| + \|t_{i}x_{n_{j}}-x_{n_{j}}\| + \|x_{n_{j}}-p\|\\ &\leq(L+1)\|x_{n_{j}}-p\| + \|t_{i}x_{n_{j}}-x_{n_{j}}\|. \end{aligned}

By Lemma 3.4(ii), it implies that $$t_{i}p=p$$ for all $$i=1,2,\ldots,N$$. Thus, $$p\in\bigcap_{i=1}^{N}F(t_{i})$$. Therefore, $$p\in\mathcal{F}$$. Since $$\lim_{n\to\infty}\|x_{n}-p\|$$ exists, we get $$\lim_{n\to\infty}\|x_{n}-p\|=\lim_{j\to\infty}\|x_{n_{j}}-p\|=0$$. This shows that $$\{x_{n}\}$$ converges strongly to a point in $$\mathcal{F}$$. □

Next, we give a numerical example to support Theorem 3.5.

### Example 3.6

Let $$\mathbb{R}$$ be the real line with the usual norm $$|\cdot|$$ and let $$D=[0,3]$$. Define two single-valued mappings $$t_{1}$$ and $$t_{2}$$ on D as follows:

$$t_{1}x = \sin x, \qquad t_{2}x = x.$$

Also we define two multi-valued mappings $$T_{1}$$ and $$T_{2}$$ on D as follows:

$$T_{1}x = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} [0,\frac{x}{3} ], & x\ne3; \\ \{1\}, & x=3; \end{array} \right .\qquad T_{2}x = \biggl[\frac{x}{4},\frac{x}{2} \biggr].$$

Let $$\{x_{n}\}$$ and $$\{y_{n}\}$$ be generated by

\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} y_{n} = \beta_{n}^{(0)} x_{n} + \sum_{i=1}^{2}\beta_{n}^{(i)} z_{n}^{(i)},\quad z_{n}^{(i)}\in T_{i}x_{n}, \\ x_{n+1} = \alpha_{n}^{(0)} x_{n} + \sum_{i=1}^{2}\alpha_{n}^{(i)}t_{i}^{n}y_{n},\quad n\in\mathbb{N}, \end{array} \right . \end{aligned}
(3.4)

where $$\alpha_{n}^{(0)}=\frac{3n+4}{10n}$$, $$\alpha_{n}^{(1)}=\frac{2n-1}{5n}$$, $$\alpha_{n}^{(2)}=\frac{3n-2}{10n}$$, $$\beta_{n}^{(0)}=\frac{15n+7}{60n}$$, $$\beta_{n}^{(1)}=\frac{5n-1}{20n}$$, $$\beta_{n}^{(2)}=\frac{15n-2}{30n}$$, for all $$n\in\mathbb{N}$$. Then the sequences $$\{x_{n}\}$$ and $$\{y_{n}\}$$ converge strongly to 0, where $$\{0\}=\bigcap_{i=1}^{2} F(t_{i}) \cap\bigcap_{i=1}^{2} F(T_{i})$$.

### Solution

It is shown in  that both $$t_{1}$$ and $$t_{2}$$ are generalized asymptotically nonexpansive single-valued mappings. Moreover, they are uniformly L-Lipschitzian mappings and $$\bigcap_{i=1}^{2} F(t_{i})=\{0\}$$. It is easy to see that both $$T_{1}$$ and $$T_{2}$$ are quasi-nonexpansive multi-valued mappings satisfying condition (E) and $$\bigcap_{i=1}^{2} F(T_{i})=\{0\}$$. Thus, $$\bigcap_{i=1}^{2} F(t_{i}) \cap\bigcap_{i=1}^{2} F(T_{i})=\{0\}$$. For every $$n\in\mathbb{N}$$, $$\alpha_{n}^{(0)}=\frac{3n+4}{10n}$$, $$\alpha_{n}^{(1)}=\frac{2n-1}{5n}$$, $$\alpha_{n}^{(2)}=\frac{3n-2}{10n}$$, $$\beta_{n}^{(0)}=\frac{15n+7}{60n}$$, $$\beta_{n}^{(1)}=\frac{5n-1}{20n}$$, $$\beta_{n}^{(2)}=\frac{15n-2}{30n}$$. Then the sequences $$\{\alpha_{n}^{(0)}\}$$, $$\{\alpha_{n}^{(1)}\}$$, $$\{\alpha_{n}^{(2)}\}$$, $$\{\beta_{n}^{(0)}\}$$, $$\{\beta_{n}^{(1)}\}$$, and $$\{\beta_{n}^{(2)}\}$$ satisfy all the conditions of Theorem 3.5. Put $$z_{n}^{(1)}=\frac{x_{n}}{2}$$ and $$z_{n}^{(2)}=\frac{x_{n}}{3}$$ for all $$n\in\mathbb{N}$$. Then the algorithm (3.4) becomes

\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} y_{n} = (\frac{13}{24}+\frac{5}{72n} )x_{n}, \\ x_{n+1} = (\frac{37}{80}+\frac{5}{16n}-\frac{1}{72n^{2}} )x_{n} + (\frac{2n-1}{5n} )t_{1}^{n}y_{n},\quad n\in\mathbb{N}. \end{array} \right . \end{aligned}
(3.5)

Using the algorithm (3.5) with the initial point $$x_{1}=2.5$$, we have numerical results in Table  1.

Finally, we prove a weak convergence theorem in uniformly convex Banach spaces.

### Theorem 3.7

Let D be a nonempty, closed, and convex subset of a uniformly convex Banach space X with the Opial property. Let $$\{t_{i}\}_{i=1}^{N}$$ be a finite family of uniformly L-Lipschitzian and generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences $$\{k_{n}\}\subset[1,\infty)$$ and $$\{s_{n}\}\subset[0,\infty)$$ such that $$\sum_{n=1}^{\infty}(k_{n}-1)<\infty$$ and $$\sum_{n=1}^{\infty}s_{n}<\infty$$. Let $$\{T_{i}\}_{i=1}^{N}$$ be a finite family of quasi-nonexpansive multi-valued mappings of D into $$KC(D)$$ satisfying the condition (E). Assume that $$\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap \bigcap_{i=1}^{N} F(T_{i})$$ is nonempty and $$T_{i}p=\{p\}$$ for all $$p\in \mathcal{F}$$ and $$i=1,2,\ldots,N$$. Let $$x_{1}\in D$$ and the sequence $$\{x_{n}\}$$ be generated by

\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} y_{n} = \beta_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\beta_{n}^{(i)} z_{n}^{(i)}, \quad z_{n}^{(i)}\in T_{i}x_{n}, \\ x_{n+1} = \alpha_{n}^{(0)} x_{n} + \sum_{i=1}^{N}\alpha_{n}^{(i)}t_{i}^{n}y_{n},\quad n\in\mathbb{N}, \end{array} \right . \end{aligned}

where $$\{\alpha_{n}^{(i)}\}$$ and $$\{\beta_{n}^{(i)}\}$$ are sequences in $$[0,1]$$ for all $$i=1,2,\ldots,N$$ such that $$0< a\leq \alpha_{n}^{(i)},\beta_{n}^{(i)}\leq b<1$$, $$\sum_{i=0}^{N}\alpha_{n}^{(i)}=1$$, and $$\sum_{i=0}^{N}\beta_{n}^{(i)}=1$$. Then the sequence $$\{x_{n}\}$$ converges weakly to a point in  $$\mathcal{F}$$.

### Proof

By Lemma 3.1, $$\{x_{n}\}$$ is bounded. Since X is uniformly convex, there exists a subsequence $$\{x_{n_{j}}\}$$ of $$\{x_{n}\}$$ converging weakly to $$p\in D$$. By Lemma 3.4, we have $$\lim_{j\to\infty} \|x_{n_{j}}-z_{n_{j}}^{(i)}\|=0$$ and $$\lim_{j\to\infty} \|x_{n_{j}}-t_{i}x_{n_{j}}\|=0$$ for all $$i=1,2,\ldots,N$$. We will show that $$p\in\mathcal{F}$$. Since $$T_{1}p$$ is compact, for all $$j\in\mathbb{N}$$, we can choose $$w_{n_{j}}\in Tp$$ such that $$\|x_{n_{j}}-w_{n_{j}}\|=\operatorname{dist}(x_{n_{j}},T_{1}p)$$ and the sequence $$\{w_{n_{j}}\}$$ has a convergent subsequence $$\{w_{n_{k}}\}$$ with $$\lim_{k\to\infty}w_{n_{k}}=w\in T_{1}p$$. By condition (E), we have

$$\operatorname{dist}(x_{n_{k}},T_{1}p)\leq \mu \operatorname{dist}(x_{n_{k}},T_{1}x_{n_{k}})+ \|x_{n_{k}}-p\|.$$

Then we have

\begin{aligned}[b] \|x_{n_{k}}-w\| &\leq\|x_{n_{k}}-w_{n_{k}}\| + \|w_{n_{k}}-w\| \\ &=\operatorname{dist}(x_{n_{k}},T_{1}p)+ \|w_{n_{k}}-w \| \\ &\leq\mu\operatorname{dist}(x_{n_{k}},T_{1}x_{n_{k}})+ \|x_{n_{k}}-p\|+ \|w_{n_{k}}-w\| \\ &\leq\mu\bigl\| x_{n_{k}}-z_{n_{k}}^{(i)}\bigr\| +\|x_{n_{k}}-p \|+ \|w_{n_{k}}-w\|. \end{aligned}

This implies that

\begin{aligned} \limsup_{k\to\infty} \|x_{n_{k}}-w\|\leq \limsup _{k\to\infty}\|x_{n_{k}}-p\|. \end{aligned}

From the Opial property, we have $$p=w\in T_{1}p$$. Similarly, it can be shown that $$p\in T_{i}p$$ for all $$i = 2,\ldots,N$$. Thus, $$p\in \bigcap_{i=1}^{N} F(T_{i})$$.

Next, by mathematical induction, we can prove that, for $$i=1,2,\ldots,N$$,

\begin{aligned} \lim_{j\to\infty}\bigl\| x_{n_{j}}-t_{i}^{m}x_{n_{j}} \bigr\| =0 \quad\mbox{for each }m\in\mathbb{N}. \end{aligned}
(3.6)

Indeed, it is obvious that the conclusion it true for $$m=1$$. Suppose the conclusion holds for $$m\geq1$$. Since $$t_{i}$$ is uniformly L-Lipschitzian, we have

\begin{aligned} \bigl\| x_{n_{j}}-t_{i}^{m+1}x_{n_{j}}\bigr\| &\leq \bigl\| x_{n_{j}}-t_{i}^{m}x_{n_{j}}\bigr\| +\bigl\| t_{i}^{m}x_{n_{j}}-t_{i}^{m+1}x_{n_{j}} \bigr\| \\ &\leq\bigl\| x_{n_{j}}-t_{i}^{m}x_{n_{j}}\bigr\| + L \|x_{n_{j}}-t_{i}x_{n_{j}}\|. \end{aligned}

This shows that $$\lim_{j\to\infty}\|x_{n_{j}}-t_{i}^{m+1}x_{n_{j}}\|=0$$ for all $$i=1,2,\ldots,N$$. Hence, (3.6) holds.

From (3.6), we have for each $$x\in D$$, $$m\in\mathbb{N}$$ and $$i=1,2,\ldots,N$$,

\begin{aligned} \limsup_{j\to\infty}\|x_{n_{j}}-x\|=\limsup _{j\to\infty}\bigl\| t_{i}^{m}x_{n_{j}}-x\bigr\| . \end{aligned}
(3.7)

Since $$t_{i}$$ is generalized asymptotically nonexpansive, we get

\begin{aligned} \limsup_{j\to\infty}\bigl\| t_{i}^{m}x_{n}-t_{i}^{m}p \bigr\| \leq \limsup_{j\to\infty}\bigl(k_{m}\|x_{n_{j}}-p \|+s_{m}\bigr). \end{aligned}

Then we have

\begin{aligned} \limsup_{m\to\infty} \Bigl(\limsup_{j\to\infty} \bigl\| t_{i}^{m}x_{n_{j}}-t_{i}^{m}p\bigr\| \Bigr)\leq \limsup_{j\to\infty}\|x_{n_{j}}-p\|. \end{aligned}
(3.8)

By Proposition 2.1, we have

\begin{aligned} \biggl\Vert x_{n_{j}}-\frac{p+t_{i}^{m}p}{2}\biggr\Vert ^{2} &= \biggl\Vert \frac{1}{2}(x_{n_{j}}-p)+\frac{1}{2} \bigl(x_{n_{j}}-t_{i}^{m}p\bigr)\biggr\Vert ^{2} \\ &\leq\frac{1}{2}\|x_{n_{j}}-p\|^{2} + \frac{1}{2} \bigl\| x_{n_{j}}-t_{i}^{m}p\bigr\| ^{2}- \frac{1}{4}g\bigl(\bigl\| p-t_{i}^{m}p\bigr\| \bigr). \end{aligned}

It implies that

\begin{aligned} \limsup_{j\to\infty}\biggl\Vert x_{n_{j}}- \frac{p+t_{i}^{m}p}{2}\biggr\Vert ^{2} \leq{}&\frac{1}{2}\limsup _{j\to\infty}\|x_{n_{j}}-p\|^{2} + \frac{1}{2} \limsup_{j\to\infty}\bigl\| x_{n_{j}}-t_{i}^{m}p \bigr\| ^{2} \\ &{} -\frac{1}{4}g\bigl(\bigl\| p-t_{i}^{m}p\bigr\| \bigr). \end{aligned}
(3.9)

By the Opial property and $$\{x_{n_{j}}\}$$ converging weakly to p, we obtain

\begin{aligned} \limsup_{j\to\infty}\|x_{n_{j}}-p\|^{2} &\leq\limsup _{j\to\infty}\biggl\Vert x_{n_{j}}-\frac{p+t_{i}^{m}p}{2}\biggr\Vert ^{2}. \end{aligned}

Then, by (3.9), we have

\begin{aligned} g\bigl(\bigl\| p-t^{m}p\bigr\| \bigr) \leq2\limsup _{j\to\infty}\bigl\| x_{n_{j}}-t_{i}^{m}p \bigr\| ^{2} - 2\limsup_{j\to\infty}\|x_{n_{j}}-p \|^{2}. \end{aligned}
(3.10)

It implies by (3.7), (3.8), and (3.10) that

\begin{aligned} \limsup_{m\to\infty}g\bigl(\bigl\| p-t_{i}^{m}p\bigr\| \bigr)&\leq 2\limsup_{m\to\infty} \Bigl(\limsup_{j\to\infty} \bigl\| x_{n_{j}}-t_{i}^{m}p\bigr\| ^{2} \Bigr)-2\limsup _{j\to\infty}\|x_{n_{j}}-p\|^{2}\\ &\leq0. \end{aligned}

This shows that $$\lim_{m\to\infty}g(\|p-t_{i}^{m}p\|)=0$$ for all $$i=1,2,\ldots,N$$. Then the properties of g yield $$\lim_{m\to\infty}\|p-t_{i}^{m}p\|=0$$ for all $$i=1,2,\ldots,N$$. So we have

\begin{aligned} \|t_{i}p-p\| &\leq\bigl\| t_{i}p-t_{i}^{m+1}p \bigr\| + \bigl\| t_{i}^{m+1}p-p\bigr\| \\ &\leq L\bigl\| p-t_{i}^{m}p\bigr\| + \bigl\| t_{i}^{m+1}p-p \bigr\| \to0 \quad\mbox{as }m\to\infty. \end{aligned}

This implies that $$t_{i}p=p$$ for all $$i=1,2,\ldots,N$$. Thus, $$p\in \bigcap_{i=1}^{N} F(t_{i})$$.

Hence, we obtain $$p\in\mathcal{F}$$.

Finally, we show that $$\{x_{n}\}$$ converges weakly to p. To show this, suppose not. Then there exists a subsequence $$\{x_{n_{l}}\}$$ of $$\{x_{n}\}$$ such that $$\{x_{n_{l}}\}$$ converges weakly to $$q\in D$$ and $$q\ne p$$. By the same method as given above, we can prove that $$q\in \mathcal{F}$$. By Lemma 3.1, $$\lim_{n\to\infty}\|x_{n}-p\|$$ and $$\lim_{n\to\infty}\|x_{n}-q\|$$ exist. It follows by Lemma 2.6 that $$q=p$$. Thus, $$\{x_{n}\}$$ converges weakly to a point in $$\mathcal{F}$$. □

### Remark 3.8

Theorem 3.5 extends and generalizes the results of Sokhuma and Kaewkhao  to a pair of a finite family of generalized asymptotically nonexpansive single-valued mappings and a finite family of quasi-nonexpansive multi-valued mappings satisfying condition (E). Theorems 3.5 and 3.7 extend and generalize the results of Eslamian  and Eslamian and Abkar  to a pair of a finite family of generalized asymptotically nonexpansive single-valued mappings and a finite family of quasi-nonexpansive multi-valued mappings satisfying condition (E).

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## Acknowledgements

This paper was supported by the Thailand Research Fund under the project RTA5780007.

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Correspondence to Withun Phuengrattana.

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The authors declare that they have no competing interests.

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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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