Open Access

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

Fixed Point Theory and Applications20152015:58

https://doi.org/10.1186/s13663-015-0304-7

Received: 21 January 2015

Accepted: 1 April 2015

Published: 22 April 2015

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.

Keywords

fixed pointgeneralized asymptotically nonexpansive mappingnonexpansive mappingBanach space

MSC

47H0947H1054H2554E40

1 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 [1]. The following example shows that the fixed point set of a generalized asymptotically nonexpansive mapping is not necessarily closed; see also [2].

Example 1.2

([1])

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 [3]. Different iterative processes have been used to approximate fixed points of nonexpansive and quasi-nonexpansive multi-valued mappings; in particular, Sastry and Babu [4] 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 [5] generalized results of Sastry and Babu [4] 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 [6] 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. [9] introduced a new condition on single-valued mappings, called condition (E), which is weaker than nonexpansiveness. Later, Abkar and Eslamian [10] 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 [11] 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 [12] 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.

2 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 [14].

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 [15]) 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 [2])

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 [2])

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 [16])

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 [17])

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 [18])

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 \(u = v\).

3 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 [19] 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.
Table 1

The values of the sequences \(\pmb{\{x_{n}\}}\) and \(\pmb{\{y_{n}\}}\) in Example 3.6

n

\(\boldsymbol {x_{n}}\)

\(\boldsymbol {y_{n}}\)

1

2.5000000

1.5277778

2

2.1025927

1.2119111

3

1.5352877

0.8671533

4

1.0799923

0.6037457

5

0.7544605

0.4191447

21

0.0023377

0.0012740

38

0.0000040

0.0000022

39

0.0000027

0.0000015

40

0.0000019

0.0000010

41

0.0000013

0.0000007

42

0.0000009

0.0000005

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 [11] 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 [12] and Eslamian and Abkar [13] 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).

Declarations

Acknowledgements

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

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Chiang Mai University
(2)
Department of Mathematics, Faculty of Science and Technology, Nakhon Pathom Rajabhat University
(3)
Research Center for Pure and Applied Mathematics, Research and Development Institute, Nakhon Pathom Rajabhat University

References

  1. Phuengrattana, W, Suantai, S: Existence theorems for generalized asymptotically nonexpansive mappings in uniformly convex metric spaces. J. Convex Anal. 20(3), 753-761 (2013) MATHMathSciNetGoogle Scholar
  2. Saejung, S, Suantai, S, Yotkaew, P: A note on “Common Fixed Point of Multistep Noor Iteration with Errors for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mapping”. Abstr. Appl. Anal. 2009, Article ID 283461 (2009) View ArticleMathSciNetMATHGoogle Scholar
  3. Markin, JT: Continuous dependence of fixed point sets. Proc. Am. Math. Soc. 38, 545-547 (1973) View ArticleMATHMathSciNetGoogle Scholar
  4. Sastry, KPR, Babu, GVR: Convergence of Ishikawa iterates for a multivalued mapping with a fixed point. Czechoslov. Math. J. 55, 817-826 (2005) View ArticleMATHMathSciNetGoogle Scholar
  5. Panyanak, B: Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Comput. Math. Appl. 54, 872-877 (2007) View ArticleMATHMathSciNetGoogle Scholar
  6. Shahzad, N, Zegeye, H: On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces. Nonlinear Anal. 71, 838-844 (2009) View ArticleMATHMathSciNetGoogle Scholar
  7. Khan, SH, Yildirim, I: Fixed points of multivalued nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2012, Article ID 73 (2012) View ArticleMathSciNetMATHGoogle Scholar
  8. Khan, SH, Abbas, M, Ali, S: Fixed points of multivalued quasi-nonexpansive mappings using a faster iterative process. Acta Math. Sin. Engl. Ser. 30(7), 1231-1241 (2014) View ArticleMATHMathSciNetGoogle Scholar
  9. Garcia-Falset, J, Llorens-Fuster, E, Suzuki, T: Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl. 375, 185-195 (2011) View ArticleMATHMathSciNetGoogle Scholar
  10. Abkar, A, Eslamian, M: Common fixed point results in CAT(0) spaces. Nonlinear Anal. 74, 1835-1840 (2011) View ArticleMATHMathSciNetGoogle Scholar
  11. Sokhuma, K, Kaewkhao, A: Ishikawa iterative process for a pair of single-valued and multivalued nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2011, Article ID 618767 (2011) MathSciNetMATHGoogle Scholar
  12. Eslamian, M: Weak and strong convergence theorems of iterative process for two finite families of mappings. Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys. 75(4), 81-90 (2013) MathSciNetMATHGoogle Scholar
  13. Eslamian, M, Abkar, A: One-step iterative process for a finite family of multivalued mappings. Math. Comput. Model. 54, 105-111 (2011) View ArticleMATHMathSciNetGoogle Scholar
  14. Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127-1138 (1991) View ArticleMATHMathSciNetGoogle Scholar
  15. Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 595-597 (1967) View ArticleMathSciNetMATHGoogle Scholar
  16. Xu, HK: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 65, 109-113 (2002) View ArticleMATHMathSciNetGoogle Scholar
  17. Schu, J: Weak and strong convergence of fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43, 153-159 (1991) View ArticleMATHMathSciNetGoogle Scholar
  18. Suantai, S: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 311, 506-517 (2005) View ArticleMATHMathSciNetGoogle Scholar
  19. Kiziltunc, H, Yolacan, E: Strong convergence theorems for two total asymptotically nonexpansive nonself mappings in Banach spaces. Fixed Point Theory Appl. 2013, Article ID 90 (2013) View ArticleMathSciNetMATHGoogle Scholar

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© Suantai and Phuengrattana; licensee Springer. 2015