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Viscosity approximation method for generalized asymptotically quasi-nonexpansive mappings in a convex metric space

Abstract

A general viscosity iterative method for a finite family of generalized asymptotically quasi-nonexpansive mappings in a convex metric space is introduced. Special cases of the new iterative method are the viscosity iterative method of Chang et al. (Appl. Math. Comput. 212:51-59, 2009), an analogue of the viscosity iterative method of Fukhar-ud-din et al. (J. Nonlinear Convex Anal. 16:47-58, 2015) and an extension of the multistep iterative method of Yildirim and Özdemir (Arab. J. Sci. Eng. 36:393-403, 2011). Our results generalize and extend the corresponding known results in uniformly convex Banach spaces and \(\operatorname{CAT}(0)\) spaces simultaneously.

Introduction and preliminaries

Let C be a nonempty subset of a metric space X and \(T:C\rightarrow C\) be a mapping. We assume that \(F(T)\), the set of fixed points of T, is nonempty and \(I= \{ 1,2,3,\ldots,r \} \). The mapping T is (i) quasi-nonexpansive if \(d ( Tx,Ty ) \leq d ( x,y ) \) for \(x\in C\), \(y\in F ( T ) \); (ii) asymptotically quasi-nonexpansive if there exists a sequence of real numbers \(\{u_{n}\}\) in \([0,\infty)\) with \(\lim_{n\rightarrow\infty}u_{n}=0\) such that \(d ( T^{n}x,p ) \leq ( 1+u_{n} ) d ( x,p ) \) for all \(x\in C\), \(p\in F ( T ) \) and \(n\geq1\); (iii) generalized asymptotically quasi-nonexpansive [1] if there exist two sequences of real numbers \(\{ u_{n}\}\) and \(\{c_{n}\}\) in \([0,\infty)\) with \(\lim_{n\rightarrow\infty }u_{n}=0=\lim_{n\rightarrow\infty}c_{n}\) such that \(d ( T^{n}x,p ) \leq d ( x,p ) +u_{n}d ( x,p ) +c_{n} \) for all \(x\in C\), \(p\in F ( T ) \) and \(n\geq1\); (iv) uniformly L-Lipschitzian if there exists a constant \(L>0\) such that \(d ( T^{n}x,T^{n}y ) \leq Ld ( x,y ) \) for all \(x,y\in C\) and \(n\geq1\); (v) uniformly Hölder continuous if there are constants \(L>0\), \(\gamma>0\) such that \(d ( T^{n}x,T^{n}y ) \leq Ld ( x,y ) ^{\gamma}\) for all \(x,y\in C\) and \(n\geq1\); and (vi) semi-compact if for a sequence \(\{x_{n}\}\) in C with \(\lim_{n\rightarrow \infty}d ( x_{n},Tx_{n} ) =0\), there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{i}}\) converges to a point in C.

Clearly, the class of generalized asymptotically quasi-nonexpansive mappings includes the class of asymptotically quasi-nonexpansive mappings.

The following example improves and extends Example 3.2 in [1] to a finite family of generalized asymptotically quasi-nonexpansive mappings.

Example 1.1

Let \(E= \mathbb{R} \) and \(C = [- \frac{1}{\pi} , \frac{1}{\pi} ]\) and define \(T_{i}x=\frac{x}{i+1}\sin(\frac{1}{x}) \) if \(x\neq0\) and \(T_{i}x=0\) if \(x=0\) for all \(x\in C\) and \(i\in I\). Then \(T_{i}^{n} x\rightarrow0\) uniformly (see [2]). For each fixed n, define \(f_{ in}(x)=\| T _{i}^{n} x\|-\|x\|\) for all x in C and \(i\in I\). Set \(c _{in}= \sup_{x\in C}\{f _{in}(x),0\}\). Then \(\lim_{n\rightarrow\infty}c _{in} =0\) and

$$ \bigl\Vert T _{i }^{n} x\bigr\Vert \leq\|x\|+c _{in}. $$

This shows that \(\{T_{i}: i\in I\}\) is a finite family of generalized asymptotically quasi-nonexpansive mappings with \(\bigcap_{i\in I}F(T_{i})\neq \emptyset\).

Convergence theorems for various mappings through different iterative methods have been obtained by a number of authors (e.g., [1, 3, 4] and the references therein). For more on the study of fixed point iteration process, the interested reader is referred to Berinde [5] and Ciric [6, 7].

Let C be a convex subset of a normed space. Yildirim and Özdemir [8] introduced the following multistep iterative method:

$$ \begin{aligned} &x_{1} \in C, \\ &x_{n+1} =(1-a_{1n})y_{n+r-2}+a_{1n}T_{1}^{n} y_{n+r-2}, \\ &y_{n+r-2} =(1-a_{2 n})y_{n+r-3}+a_{2n}T_{2}^{n}y_{n+r-3}, \\ &\vdots \\ &y_{n+1} =(1-a_{(r-1)n})y_{n}+a_{(r-1)n}T_{(r-1)}^{n} y_{n}, \\ &y_{n} =(1-a_{ r n})x_{n}+a_{rn}T_{r}^{n} x_{n},\quad r\geq2, n\geq1, \end{aligned} $$
(1.1)

where \(\{ T_{i}:i\in I \} \) is a family of self-mappings of C, \(a_{in} \in[\epsilon,1-\epsilon]\), for some \(\epsilon\in(0,\frac{1}{2})\), for all \(n\geq1\).

If \(T_{1}=T_{2}=\cdots=T_{r}\) and \(\alpha_{jn}=0\) for \(j=1,\ldots,r\) and \(r\geq1\), then the iterative method (1.1) reduces to the Mann iterative method [9]. Let us note that the scheme (1.1) and multistep scheme (1.3) in [10] are independent of each other.

Moudafi [11] proposed a viscosity iterative method by selecting a particular fixed point of a given nonexpansive mapping. The so-called viscosity iterative method has been studied by many authors (see, for example, [3, 12]). These methods are very important because of their applicability to convex optimization, linear programming, monotone inclusions and elliptic differential equations [11].

Recently, Chang et al. [13] introduced and studied the following viscosity iterative method:

$$ \begin{aligned} &x_{n+1} =(1-\alpha_{n})f ( x_{n} ) +\alpha_{n}T^{n} y_{n}, \\ &y_{n} =(1-\beta_{n})x_{n}+\beta_{n}T^{n}x_{n}, \quad n\geq1, \end{aligned} $$
(1.2)

where T is an asymptotically nonexpansive mapping [14] and f is a fixed contraction.

The iterative methods in (1.1) and (1.2) involve convex combinations, and so a convex structure is needed to define them on a nonlinear domain.

A mapping \(W: X^{2}\times J\rightarrow X\) is a convex structure [15] on a metric space X if

$$ d \bigl( u,W ( x,y,\alpha ) \bigr) \leq\alpha d(u,x)+(1-\alpha )d(u,y) $$

for all \(x,y,u\in X\) and \(\alpha\in J=[0,1]\). The metric space X together with a convex structure W is known as a convex metric space. A nonempty subset C of a convex metric space X is convex if \(W(x,y,\alpha)\in C\) for all \(x,y\in C\) and \(\alpha\in J\). All normed linear spaces are convex metric spaces, but there are convex metric spaces which are not linear; for example, a \(\operatorname{CAT} ( 0 ) \) space [16, 17].

A convex metric space X is uniformly convex if for any \(\varepsilon>0\), there exists \(\delta=\delta ( \varepsilon ) >0\) such that for all \(r>0\) and \(x,y,z\in X\) with \(d ( z,x ) \leq r\), \(d ( z,y ) \leq r\) and \(d ( x,y ) \geq r\varepsilon\) imply that \(d ( z,W ( x,y,\frac{1}{2} ) ) \leq ( 1-\delta ) r\).

A mapping \(\eta:(0,\infty)\times(0,2]\rightarrow(0,1]\) which provides such \(\delta=\eta(r,\epsilon)\) for given \(r>0\) and \(\varepsilon\in (0,2]\) is called modulus of uniform convexity. We call η monotone if it decreases with r (for a fixed ϵ).

Obviously, uniformly convex Banach spaces are uniformly convex metric spaces.

In general, a convex structure W is not continuous [18]. Throughout this paper, we assume that W is continuous.

We now devise a general iterative method which extends the methods in (1.1) and (1.2) simultaneously in a convex metric space.

We define an \(S_{n}\)-mapping generated by a family \(\{T_{i}:i\in I\}\) of generalized asymptotically quasi-nonexpansive mappings on C as

$$ S_{n}x=U_{rn}x, $$
(1.3)

where \(U_{0n}=I\) (the identity mapping), \(U_{1n}x=W(T_{r}^{n}U_{0n}x,U_{0n}x,a_{rn}), U_{2n}x=W(T_{r-1}^{n}U_{1n}x, U_{1n}x,a_{(r-1)n}),\ldots,U_{rn}x=W(T_{1}^{n}U_{ ( r-1 ) n}x,U_{ ( r-1 ) n}x,a_{1n})\).

For \(\{\alpha_{n}\}\subset J\), a fixed contractive mapping f on C and \(S_{n}\) given in (1.3), we define \(\{ x_{n} \} \) as follows:

$$ x_{1}\in C,\quad x_{n+1}=W\bigl(f ( x_{n} ) ,S_{n}x_{n},\alpha_{n}\bigr) $$
(1.4)

and call it a general viscosity iterative method in a convex metric space.

The purpose of this paper is to:

  1. (i)

    establish a necessary and sufficient condition for convergence of iterative method (1.4) to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive mappings on a convex metric space;

  2. (ii)

    prove strong convergence and -convergence results for the iterative method (1.4) to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive mappings on a uniformly convex metric space.

We now assume that \(F=\bigcap_{i\in I}F(T_{i})\neq\emptyset\).

We need the following known results for our convergence analysis.

Lemma 1.1

(cf. [19])

Let the sequences \(\{a_{n}\}\) and \(\{u_{n}\}\) of real numbers satisfy

$$ a_{n+1}\leq(1+u_{n})a_{n},\quad a_{n} \geq0, u_{n}\geq0,\sum_{n=1}^{\infty }u_{n}< + \infty. $$

Then (i) \(\lim_{n\rightarrow\infty}a_{n}\) exists; (ii) if \(\liminf_{n\rightarrow\infty}a_{n}=0\), then \(\lim_{n\rightarrow\infty}a_{n}=0\).

Lemma 1.2

([20])

Let X be a uniformly convex metric space. Let \(x\in X\) and \(\{a_{n}\}\) be a sequence in \([b,c]\) for some \(b,c\in(0,1)\). If \(\{u_{n}\}\) and \(\{v_{n}\}\) are sequences in X such that \(\limsup_{n\rightarrow\infty}d(u_{n},x)\leq r\), \(\limsup_{n\rightarrow\infty}d(v_{n},x)\leq r\) and \(\lim_{n\rightarrow\infty}d(W(u_{n},v_{n},a_{n}),x)=r\) for some \(r\geq0\), then \(\lim_{n\rightarrow\infty}d(u_{n},v_{n})=0\).

Convergence in convex metric spaces

In this section, we prove some results for the viscosity iterative method (1.4) to converge to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive mappings in a convex metric space.

Lemma 2.1

Let C be a nonempty, closed and convex subset of a convex metric space X and \(\{T_{i}:i\in I\}\) be a family of generalized asymptotically quasi-nonexpansive self-mappings of C, i.e., \(d ( T_{i}^{n}x,p_{i} ) \leq(1+u_{in})d ( x,p_{i} ) +c_{in}\) for all \(x\in C\) and \(p_{i}\in F(T_{i})\), \(i\in I\), where \(\{u_{in}\}\) and \(\{ c_{in} \} \) are sequences in \([0,\infty)\) with \(\sum_{n=1}^{\infty}u_{in}<\infty\), \(\sum_{n=1}^{\infty}c_{in}<\infty\) for each i. Then, for the sequence \(\{x_{n}\}\) in (1.4) with \(\sum_{n=1}^{\infty}\alpha_{n}<\infty\), there are sequences \(\{\nu _{n}\}\) and \(\{\xi_{n}\}\) in \([0,\infty)\) satisfying \(\sum_{n=1}^{\infty}\nu _{n}<\infty\), \(\sum_{n=1}^{\infty}\xi_{n}<\infty\) such that

  1. (a)

    \(d ( x_{n+1},p ) \leq ( 1+\nu_{n} ) ^{r}d ( x_{n},p ) +\xi_{n}\) for all \(p\in F\) and all \(n\geq1\);

  2. (b)

    \(d ( x_{n+m},p ) \leq M_{1} ( d ( x_{n},p ) +\sum_{n=1}^{\infty}\xi_{n} ) \) for all \(p\in F\) and \(n\geq 1\), \(m\geq 1\), \(M_{1}>0\).

Proof

(a) Let \(p\in F\) and \(\nu_{n}=\max_{i\in I}u_{in}\) for all \(n\geq1\). Since \(\sum_{n=1}^{\infty}u_{in}<\infty\) for each i, therefore \(\sum_{n=1}^{\infty}\nu_{n}<\infty\).

Now we have

$$\begin{aligned} d ( U_{1n}x_{n},p ) =&d \bigl( W\bigl(T_{r}^{n}U_{0n}x_{n},U_{0n}x_{n},a_{rn} \bigr),p \bigr) \\ \leq&(1- a _{rn})d ( x_{n},p ) + a _{rn}d \bigl( T_{r}^{n}x_{n},p \bigr) \\ \leq&(1- a _{rn})d ( x_{n},p ) + a _{rn} \bigl[ (1+u_{rn})d ( x_{n},p ) +c_{rn} \bigr] \\ \leq&(1+u_{rn})d ( x_{n},p ) +c_{rn} \\ \leq&(1+\nu_{n})^{1}d ( x_{n},p ) +c_{rn}. \end{aligned}$$

Assume that \(d ( U_{kn}x_{n},p ) \leq(1+\nu_{n})^{k}d ( x_{n},p ) +(1+\nu_{n})^{k-1}\sum_{i=1}^{k}c_{(r-i+1)n}\) holds for some \(1< k\).

Consider

$$\begin{aligned} d ( U_{ ( k+1 ) n}x_{n},p ) =&d \bigl( W\bigl(T_{r-k }^{n}U_{kn}x_{n},U_{kn}x_{n},a_{ ( r-k ) n} \bigr),p \bigr) \\ \leq&(1-a_{ ( r-k ) n})d (U_{kn} x_{n},p ) +a_{ ( r-k ) n}d \bigl( T_{r-k}^{n}U_{kn}x_{n},p \bigr) \\ \leq&(1-a_{ ( r-k ) n})d (U_{kn} x_{n},p ) +a_{ ( r-k ) n}\bigl[(1+u_{ ( r-k ) n})d ( U_{kn}x_{n},p ) +c_{(r-k)n}\bigr] \\ \leq&(1+\nu_{ n})d ( U_{kn}x_{n},p ) +c_{(r-k)n} \\ \leq&(1+\nu_{ n})\Biggl[(1+\nu_{n})^{k}d ( x_{n},p ) +(1+\nu _{n})^{k-1}\sum _{i=1}^{k}c_{(r-i+1)n}\Biggr] +c_{(r-k)n} \\ \leq& (1+\nu_{n})^{k+1}d ( x_{n},p ) +(1+\nu _{n})^{k}\sum_{i=1}^{k+1}c_{(r-i+1)n}. \end{aligned}$$

By mathematical induction, we have

$$ d ( U_{jn}x_{n},p ) \leq(1+\nu_{n})^{j}d ( x_{n},p ) +(1+\nu_{n})^{j-1}\sum _{i=1}^{j}c_{(r-i+1)n},\quad 1\leq j\leq r. $$
(2.1)

Hence

$$ d ( S_{n}x_{n},p ) =d ( U_{rn}x_{n},p ) \leq(1+\nu _{n})^{r}d ( x_{n},p ) +(1+\nu _{n})^{r-1}\sum_{i=1}^{r}c_{(r-i+1)n}. $$
(2.2)

Now, by (1.4) and (2.2), we obtain

$$\begin{aligned} d ( x_{n+1},p ) =&d \bigl( W\bigl(f ( x_{n} ) ,S_{n}x_{n},\alpha_{n}\bigr),p \bigr) \\ \leq&\alpha_{n}d \bigl( f ( x_{n} ) ,p \bigr) + ( 1-\alpha _{n} ) d ( S_{n}x_{n},p ) \\ \leq&\alpha_{n} d ( x_{n},p ) +\alpha_{n}d \bigl( f ( p ) ,p \bigr) \\ &{}+ ( 1-\alpha_{n} ) \Biggl( (1+\nu_{n})^{r}d ( x_{n},p ) +(1+\nu_{n})^{r-1}\sum _{i=1}^{r}c_{(r-i+1)n} \Biggr) \\ \leq&(1+\nu_{n})^{r}d ( x_{n},p ) + ( 1-\alpha _{n} ) (1+\nu_{n})^{r-1}\sum _{i=1}^{r}c_{(r-i+1)n} +\alpha_{n}d \bigl( f ( p ) ,p \bigr) \\ \leq&(1+\nu_{n})^{r}d ( x_{n},p ) + \alpha_{n}d \bigl( f ( p ) ,p \bigr) +(1+\nu_{n})^{r-1} \sum_{i=1}^{r}c_{(r-i+1)n}. \end{aligned}$$

Setting \(\max \{ d ( f ( p ) ,p ) ,\sup(1+\nu _{n})^{r-1} \} =M\), we get that

$$ d ( x_{n+1},p ) \leq(1+\nu_{n})^{r}d ( x_{n},p ) +M \Biggl( \alpha_{n}+\sum _{i=1}^{r}c_{(r-i+1)n} \Biggr) . $$

That is,

$$ d ( x_{n+1},p ) \leq(1+\nu_{n})^{r}d ( x_{n},p ) +\xi _{n}, $$

where \(\xi_{n}=M ( \alpha_{n}+\sum_{i=1}^{r}c_{(r-i+1)n} ) \) and \(\sum_{n=1}^{\infty}\xi_{n}<\infty\).

(b) We know that \(1+t\leq e^{t}\) for \(t\geq0\). Thus, by part (a), we have

$$\begin{aligned} d ( x_{n+m},p ) \leq&(1+\nu_{n+m-1})^{r}d ( x_{n+m-1},p ) +\xi_{n+m-1} \\ \leq&e^{r\nu_{n+m-1}}d ( x_{n+m-1},p ) +\xi_{n+m-1} \\ \leq&e^{r ( \nu_{n+m-1}+\nu_{n+m-2} ) }d ( x_{n+m-2},p ) +\xi_{n+m-1}+ \xi_{n+m-2} \\ &\vdots \\ \leq&e^{r\sum_{i=n}^{n+m-1}v_{i}}d ( x_{n},p ) +\sum _{i=n+1}^{n+m-1}v_{i}\sum _{i=n}^{n+m-1}\xi_{i} \\ \leq&e^{r\sum_{i=1}^{\infty}v_{i}} \Biggl( d ( x_{n},p ) +\sum _{i=1}^{\infty}\xi_{i} \Biggr) \\ =&M_{1} \Biggl( d ( x_{n},p ) +\sum _{i=1}^{\infty}\xi _{i} \Biggr) ,\quad \text{where }M_{1}=e^{r\sum_{i=1}^{\infty}v_{i}}. \end{aligned}$$

 □

The next result deals with a necessary and sufficient condition for the convergence of \(\{x_{n}\}\) in (1.4) to a point of F.

Theorem 2.1

Let C, \(\{T_{i}:i\in I\}\), F, \(\{ u_{in} \} \) and \(\{ c_{in} \} \) be as in Lemma  2.1. Let X be complete. The sequence \(\{x_{n}\}\) in (1.4) with \(\sum_{n=1}^{\infty}\alpha _{n}<\infty\) converges strongly to a point in F if and only if \(\liminf_{n\rightarrow\infty}d(x_{n},F)=0\), where \(d(x,F)=\inf_{p\in F} ( x,p ) \).

Proof

The necessity is obvious; we only prove the sufficiency. By Lemma 2.1(a), we have

$$ d ( x_{n+1},p ) \leq(1+\nu_{n})^{r}d ( x_{n},p ) +\xi _{n}\quad \text{for all }p\in F\text{ and }n \geq1. $$

Therefore,

$$\begin{aligned} d(x_{n+1},F) \leq&(1+\nu_{n})^{r}d(x_{n},F)+ \xi_{n} \\ =& \Biggl( 1+\sum_{k=1}^{r} \frac{r(r-1)\cdots(r-k+1)}{k!}\nu _{n}^{k} \Biggr) d(x_{n},F)+ \xi_{n}. \end{aligned}$$

As \(\sum_{n=1}^{\infty}\nu_{n}<+\infty\), so \(\sum_{n=1}^{\infty }\sum_{k=1}^{r}\frac{r(r-1)\cdots(r-k+1)}{k!}\nu_{n}^{k}<\infty\). Now \(\sum_{n=1}^{\infty}\xi_{n}<\infty\) in Lemma 2.1(a), so by Lemma 1.1 and \(\liminf_{n\rightarrow\infty}d(x_{n},F)=0\), we get that \(\lim_{n\rightarrow\infty}d(x_{n},F)=0\). Next, we prove that \(\{x_{n}\}\) is a Cauchy sequence in X. Let \(\varepsilon>0\). From the proof of Lemma 2.1(b), we have

$$ d ( x_{n+m},x_{n} ) \leq d ( x_{n+m},F ) +d ( x_{n},F ) \leq ( 1+M_{1} ) d ( x_{n},F ) +M_{1}\sum_{i=n}^{\infty} \xi_{i}. $$
(2.3)

As \(\lim_{n\rightarrow\infty}d(x_{n},F)=0\) and \(\sum_{i=1}^{\infty }\xi _{i}<\infty\), so there exists a natural number \(n_{0}\) such that

$$ d(x_{n},F)\leq\frac{\varepsilon}{2 ( 1+M_{1} ) }\quad \text{and}\quad \sum _{i=n}^{\infty}\xi_{i}< \frac{\varepsilon}{2M_{1}}\quad \text{for all } n\geq n_{0}. $$

So, for all integers \(n\geq n_{0}\), \(m\geq1\), we obtain from (2.3) that

$$ d ( x_{n+m},x_{n} ) < ( M_{1}+1 ) \frac{\varepsilon}{ 2 ( 1+M_{1} ) }+M_{1}\frac{\varepsilon}{2M_{1}}=\varepsilon. $$

Thus, \(\{x_{n}\}\) is a Cauchy sequence in X and so it converges to \(q\in X\). Finally, we show that \(q\in F\). For any \(\overline{\varepsilon}>0\), there exists a natural number \(n_{1}\) such that

$$ d(x_{n},F)=\inf_{p\in F}d ( x_{n},p ) < \frac{\overline {\varepsilon}}{3}\quad \text{and}\quad d ( x_{n},q ) < \frac{\overline{\varepsilon }}{2} \quad \text{for all }n\geq n_{1}. $$

There must exist \(p^{\ast}\in F\) such that \(d ( x_{n},p^{\ast } ) <\frac{\overline{\varepsilon}}{2}\) for all \(n\geq n_{1}\); in particular, \(d ( x_{n_{1}},p^{\ast} ) <\frac{\overline{\varepsilon}}{2}\) and \(d ( x_{n_{1}},q ) <\frac{\overline{\varepsilon}}{2}\).

Hence

$$ d \bigl( p^{\ast},q \bigr) \leq d \bigl( x_{n_{1}},p^{\ast} \bigr) +d ( x_{n_{1}},q ) < \overline{\varepsilon}. $$

Since ε̅ is arbitrary, therefore \(d ( p^{\ast },q ) =0\). That is, \(q=p^{\ast}\in F\). □

Remark 2.1

A generalized asymptotically nonexpansive mapping is a generalized asymptotically quasi-nonexpansive mapping. So Theorem 2.1 holds good for the class of generalized asymptotically nonexpansive mappings.

Results in a uniformly convex metric space

The aim of this section is to establish some convergence results for the iterative method (1.4) of generalized asymptotically quasi-nonexpansive mappings on a uniformly convex metric space.

Lemma 3.1

Let C be a nonempty, closed and convex subset of a uniformly convex metric space X and \(\{T_{i}:i\in I\}\) be a family of uniformly Hölder continuous and generalized asymptotically quasi-nonexpansive self-mappings of C, i.e., \(d ( T_{i}^{n}x,p_{i} ) \leq (1+u_{in})d ( x,p_{i} ) +c_{in}\) for all \(x\in C\) and \(p_{i}\in F(T_{i})\), where \(\{u_{in}\}\) and \(\{c_{in}\}\) are sequences in \([0,\infty)\) with \(\sum_{n=1}^{\infty}u_{in}<\infty\) and \(\sum_{n=1}^{\infty }c_{in}<\infty\), respectively, for each \(i\in I\). Then, for the sequence \(\{x_{n}\}\) in (1.4) with \(a_{in}\in{}[\delta,1-\delta]\) for some \(\delta\in ( 0,\frac{1}{2} ) \) and \(\sum_{n=1}^{\infty }\alpha_{n}<\infty\), we have \(\lim_{n\rightarrow\infty}d ( x_{n},T_{j}x_{n} ) =0\) for each \(j\in I\).

Proof

Let \(p\in F\) and \(\nu_{n}=\max_{i\in I}u_{in}\) for all \(n\geq1\). By Lemma 1.1(i) and Lemma 2.1(a), it follows that \(\lim_{n\rightarrow\infty}d ( x_{n},p ) \) exists for all \(p\in F\). Assume that

$$ \lim_{n\rightarrow\infty}d ( x_{n},p ) =c. $$
(3.1)

Inequality (2.1) together with (3.1) gives that

$$ \limsup_{n\rightarrow\infty}d ( U_{jn}x_{n},p ) \leq c, \quad 1\leq j\leq r. $$
(3.2)

By (1.4), we have

$$\begin{aligned} d ( x_{n+1},p ) & =d \bigl( W\bigl(f ( x_{n} ) ,S_{n}x_{n},\alpha_{n}\bigr),p \bigr) \\ & \leq\alpha_{n}d \bigl( f ( x_{n} ) ,p \bigr) + ( 1-\alpha _{n} ) d ( S_{n}x_{n},p ) \\ & \leq\alpha_{n}d \bigl( f ( x_{n} ) ,p \bigr) +\alpha _{n}d \bigl( f ( p ) ,p \bigr) + ( 1-\alpha_{n} ) d ( U_{rn}x_{n},p ) , \end{aligned}$$

and hence

$$ c\leq\liminf_{n\rightarrow\infty}d ( U_{rn}x_{n},p ). $$
(3.3)

Combining (3.2) and (3.3), we get

$$ \lim_{n\rightarrow\infty}d ( U_{rn}x_{n},p ) =c. $$

Note that

$$\begin{aligned} d ( U_{rn}x_{n},p ) =&d\bigl(W\bigl(T_{1}^{n}U_{ ( r-1 ) n}x_{n},U_{ ( r-1 ) n}x_{n},a_{1n} \bigr),p\bigr) \\ \leq&a_{1n}d\bigl( T_{1}^{n}U_{ ( r-1 ) n}x_{n},p \bigr)+(1-a_{1n})d( U_{ ( r-1 ) n}x_{n},p) \\ \leq&a_{1n}\bigl[(1+u_{1n})d ( U_{ ( r-1 ) n}x_{n},p ) +c_{1n} \bigr]+(1-a_{1n})d( U_{ ( r-1 ) n}x_{n},p) \\ \leq&a_{1n}(1+\nu_{ n})d ( U_{ ( r-1 ) n}x_{n},p )+a_{1n} c_{1n} \\ \leq&a_{1n}(1+\nu_{ n})\bigl[a_{2n}(1+ \nu_{ n})d ( U_{ ( r-2 ) n}x_{n},p )+a_{2n} c_{2 n}\bigr]+a_{1n}(1+\nu_{ n})c_{1n} \\ \leq&a_{1n}a_{2n}(1+\nu_{ n})^{2} d ( U_{ ( r-2 ) n}x_{n},p )+a_{1n}a_{2n}(1+ \nu_{ n}) c_{2 n} + a_{1n} c_{1 n} \\ &\vdots \\ \leq&a_{1n}a_{2n}\cdots a_{(j-1)n}(1+ \nu_{ n})^{j-1} d ( U_{ ( r-(j-1) ) n}x_{n},p ) \\ &{}+a_{1n}a_{2n}\cdots a_{(j-1)n}(1+\nu_{ n}) ^{(j-1)-1}c_{(j-1) n} \\ &{}+a_{1n}a_{2n}\cdots a_{((j-1)-1)n}(1+\nu_{ n}) ^{(j-1)-2}c_{((j-1)-1) n}+\cdots \\ &{}+a_{1n}a_{2n}(1+\nu_{ n}) c_{2 n} + a_{1n} c_{1 n}. \end{aligned}$$

Hence

$$ c\leq\liminf_{n\rightarrow\infty}d ( U_{(r-(j-1))n}x_{n},p ) , \quad 1\leq j\leq r . $$
(3.4)

Using (3.2) and (3.4), we have

$$ \lim_{n\rightarrow\infty}d ( U_{(r-(j-1))n}x_{n},p ) =c. $$

That is,

$$ \lim_{n\rightarrow\infty}d \bigl( W\bigl(T_{j}^{n}U_{ ( r-j ) n}x_{n},U_{ ( r-j ) n}x_{n},a_{jn} \bigr),p \bigr) =c\quad \text{for }1\leq j\leq r. $$

This together with (3.1), (3.2) and Lemma 1.2 gives that

$$ \lim_{n\rightarrow\infty}d \bigl( T_{j}^{n}U_{ ( r-j ) n}x_{n},U_{ ( r-j ) n}x_{n}, \bigr) =0\quad \text{for }1\leq j\leq r. $$
(3.5)

If \(j=r\),we have by (3.5)

$$ \lim_{n\rightarrow\infty}d \bigl( T_{r}^{n}x_{n},x_{n} \bigr) =0. $$

In case \(j\in \{ 1,2,3,\ldots,r-1 \} \), we observe that

$$\begin{aligned} d ( x_{n},U_{ ( r-j ) n}x_{n} ) =&d \bigl( x_{n},W \bigl( T_{j+1}^{n}U_{ (r-( j+1) ) n}x_{n},U_{ (r-( j+1) ) n}x_{n},a_{ ( j+1 ) n} \bigr) \bigr) \\ \leq&a_{ ( j+1 ) n}d \bigl( T_{j+1}^{n}U_{ ( r-( j+1) ) n}x_{n},x_{n} \bigr)+(1- a_{ ( j+1 ) n})d ( U_{ ( r-( j+1) ) n}x_{n},x_{n} ) \\ \leq& (1+\nu_{n})d ( U_{ ( r-( j+1) ) n}x_{n},x_{n} )+c_{ ( j+1 ) n} \\ &\vdots \\ \leq&(1+\nu_{n})^{r-j}d ( U_{0 n}x_{n},x_{n} )+(1+\nu_{n})^{r-j-1}c_{r n} \\ &{}+(1+\nu_{n})^{r-j-2}c_{ ( r-1 ) n}+\cdots+(1+ \nu_{n})c_{ ( j+2 ) n}+ c_{ ( j+1 ) n}. \end{aligned}$$

Hence,

$$ \lim_{n\rightarrow\infty}d ( x_{n},U_{ ( r-j ) n}x_{n} )=0. $$
(3.6)

Since \(T_{j}\) is uniformly Hölder continuous, therefore the inequality

$$\begin{aligned} d \bigl( T_{j}^{n}x_{n},x_{n} \bigr) \leq&d \bigl( T_{j}^{n}x_{n},T_{j}^{n}U_{ ( r-j ) n}x_{n} \bigr) +d \bigl( T_{j}^{n}U_{ ( r-j ) n}x_{n},U_{ ( r-j ) n}x_{n} \bigr) \\ &{}+d( U_{ ( r-j ) n}x_{n},x_{n}) \\ \leq&Ld ( x_{n},U_{ ( r-j ) n}x_{n} ) ^{\gamma }+d ( x_{n},U_{ ( r-j ) n}x_{n} )+d \bigl( T_{j}^{n}U_{ ( r-j ) n}x_{n},U_{ ( r-j ) n}x_{n} \bigr) \end{aligned}$$

together with (3.5) and (3.6) gives that

$$ \lim_{n\rightarrow\infty}d \bigl( T_{j}^{n}x_{n},x_{n} \bigr) =0. $$

Hence,

$$ d \bigl( T_{j}^{n}x_{n},x_{n} \bigr) \rightarrow0\quad \text{as }n\rightarrow \infty\text{ for }1\leq j\leq r. $$
(3.7)

As before, we can show that

$$\begin{aligned} d ( x_{n},x_{n+1} ) =&d \bigl( x_{n},W\bigl(f ( x_{n} ) ,S_{n}x_{n},\alpha_{n}\bigr) \bigr) \\ \leq&\alpha_{n} ( 1+\alpha ) d ( x_{n},p ) +\alpha _{n}d \bigl( p,f ( p ) \bigr) \\ &{}+ (1-\alpha_{n})\bigl[ a_{1n}d \bigl( U_{ ( r-1 ) n}x_{n},T_{1}^{n}U_{ ( r-1 ) n}x_{n} \bigr)+ d ( x_{n}, U_{ ( r-1 ) n}x_{n} )\bigr]. \end{aligned}$$

Therefore, by (3.5) and (3.6), we get

$$ \lim_{n\rightarrow\infty}d ( x_{n},x_{n+1} ) =0. $$
(3.8)

Let us observe that

$$\begin{aligned} d ( x_{n},T_{j}x_{n} ) \leq&d ( x_{n},x_{n+1} ) +d \bigl( x_{n+1},T_{j}^{n+1}x_{n+1} \bigr) \\ &{}+d \bigl( T_{j}^{n+1}x_{n+1},T_{j}^{n+1}x_{n} \bigr) +d \bigl( T_{j}^{n+1}x_{n},T_{j}x_{n} \bigr) \\ \leq&d ( x_{n},x_{n+1} ) +d \bigl( x_{n+1},T_{j}^{n+1}x_{n+1} \bigr) \\ &{}+Ld ( x_{n+1},x_{n} ) ^{\gamma}+Ld \bigl( T_{j}^{n}x_{n},x_{n} \bigr) ^{\gamma}. \end{aligned}$$

By the uniform Hölder continuity of \(T_{j}\), (3.7) and (3.8), we get

$$ \lim_{n\rightarrow\infty}d ( x_{n},T_{j}x_{n} ) =0,\quad 1\leq j\leq r. $$
(3.9)

 □

Theorem 3.1

Under the hypotheses of Lemma  3.1, assume, for some \(1\leq j\leq r\), that \(T_{j}^{m}\) is semi-compact for some positive integer m. If X is complete, then \(\{x_{n}\}\) in (1.4) converges strongly to a point in F.

Proof

Fix \(j\in I\) and suppose \(T_{j}^{m}\) to be semi-compact for some \(m\geq1\). By (3.9), we obtain

$$\begin{aligned} d \bigl( T_{j}^{m}x_{n},x_{n} \bigr) \leq&d \bigl( T_{j}^{m}x_{n},T_{j}^{m-1}x_{n} \bigr) +d \bigl( T_{j}^{m-1}x_{n},T_{j}^{m-2}x_{n} \bigr) \\ &{}+\cdots+d \bigl( T_{j}^{2}x_{n},T_{j}x_{n} \bigr) +d ( T_{j}x_{n},x_{n} ) \\ \leq&d ( T_{j}x_{n},x_{n} ) + ( m-1 ) Ld ( T_{j}x_{n},x_{n} ) ^{\gamma}\rightarrow0. \end{aligned}$$

Since \(\{x_{n}\}\) is bounded and \(T_{j}^{m}\) is semi-compact, \(\{x_{n}\}\) has a convergent subsequence \(\{x_{n_{i}}\}\) such that \(x_{n_{i}}\rightarrow q\in C\). Hence, by (3.9), we have

$$ d ( q,T_{i}q ) =\lim_{n\rightarrow\infty}d ( x_{n_{j}},T_{i}x_{n_{j}} ) =0,\quad i\in I. $$

Thus \(q\in F\), and so by Theorem 2.1, \(\{x_{n}\}\) converges strongly to a common fixed point q of the family \(\{T_{i}:i\in I\}\). □

An immediate consequence of Lemma 3.1 and Theorem 3.1 is the following strong convergence result in uniformly convex metric spaces.

Theorem 3.2

Let C, \(\{T_{i}:i\in I\}\), F, \(\{ u_{in} \} \) and \(\{ c_{in} \} \) be as in Lemma  3.1. If there exists a constant M such that \(d ( x_{n},T_{j}x_{n} ) \geq Md(x_{n},F)\) for all \(n\geq1\) and X is complete, then the sequence \(\{x_{n}\}\) in (1.4) converges strongly to a point in F.

The concept of -convergence in a metric space was introduced by Lim [21] and its analogue in \(\operatorname{CAT}(0)\) spaces was investigated by Dhompongsa and Panyanak [22]. Here we study -convergence in uniformly convex metric spaces.

For this, we collect some basic concepts.

Let \(\{x_{n}\}\) be a bounded sequence in a uniformly convex metric space X. For \(x\in X\), define a continuous functional \(r(\cdot,\{x_{n}\} ):X\rightarrow {}[0,\infty)\) by

$$ r\bigl(x,\{x_{n}\}\bigr)=\limsup_{n\rightarrow\infty}d(x,x_{n}). $$

The asymptotic radius \(\rho=r(\{x_{n}\})\) of \(\{x_{n}\}\) is given by

$$ \rho=\inf\bigl\{ r\bigl(x,\{x_{n}\}\bigr):x\in X\bigr\} . $$

The asymptotic center of a bounded sequence \(\{x_{n}\}\) with respect to a subset C of X is defined as follows:

$$ A_{C}\bigl(\{x_{n}\}\bigr)=\bigl\{ x\in X:r\bigl(x, \{x_{n}\}\bigr)\leq r\bigl(y,\{x_{n}\}\bigr)\text{ for any }y \in C\bigr\} . $$

If the asymptotic center is taken with respect to X, then it is simply denoted by \(A(\{x_{n}\})\). A sequence \(\{x_{n}\}\) in X is said to -converge to \(x\in X\) if x is the unique asymptotic center of \(\{u_{n}\}\) for every subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\). In this case, we write \(\triangle\mbox{-} \lim_{n}x_{n}=x\) and call x as -limit of \(\{x_{n}\}\).

Lemma 3.2

([23])

Let \((X,d)\) be a complete uniformly convex metric space with monotone modulus of uniform convexity. Then every bounded sequence \(\{x_{n}\}\) in X has a unique asymptotic center with respect to any nonempty closed convex subset C of X.

Lemma 3.3

([20])

Let C be a nonempty closed convex subset of a uniformly convex metric space and \(\{x_{n}\}\) be a bounded sequence in C such that \(A(\{x_{n}\})=\{y\}\) and \(r(\{x_{n}\})=\rho\). If \(\{ y_{m}\} \) is another sequence in C such that \(\lim_{m\rightarrow\infty }r(y_{m},\{x_{n}\})=\rho\), then \(\lim_{m\rightarrow\infty}y_{m}=y\).

Now, we establish -convergence of the iterative method (1.4).

Theorem 3.3

Let C be a nonempty, closed and convex subset of a complete uniformly convex metric space X with monotone modulus of uniform convexity η, and let \(\{T_{i}:i\in I\}\) be a family of uniformly L-Lipschitzian and generalized asymptotically nonexpansive self-mappings of C such that \(F\neq\phi\), i.e., \(d ( T_{i}^{n}x,T_{i}^{n}y ) \leq (1+u_{in})d ( x,y ) +c_{in}\) for all \(x,y\in C\), where \(\{ u_{in}\}\) and \(\{c_{in}\}\) are sequences in \([0,\infty)\) with \(\sum_{n=1}^{\infty }u_{in}<\infty\) and \(\sum_{n=1}^{\infty}c_{in}<\infty\), respectively, for each \(i\in I\). Then the sequence \(\{x_{n}\}\) in (1.4) with \(a_{in}\in {}[\delta,1-\delta]\) for some \(\delta\in ( 0,\frac {1}{2} ) \) and \(\sum_{n=1}^{\infty}\alpha_{n}<\infty\), -converges to a common fixed point of \(\{T_{j}:j\in I\}\).

Proof

By Lemma 3.1, \(\{x_{n}\}\) is bounded, and so by Lemma 3.2, \(\{ x_{n}\}\) has a unique asymptotic center, that is, \(A(\{x_{n}\})=\{x\}\). Let \(\{z_{n}\} \) be any subsequence of \(\{x_{n}\}\) such that \(A(\{z_{n}\} )=\{z\}\). Also by Lemma 3.1, we have \(\lim_{n\rightarrow\infty}d ( z_{n},T_{j}z_{n} ) =0\) for each \(j\in I\).

We claim that z is a common fixed point of \(\{T_{j}:j\in I\}\). To show this, we define a sequence \(\{w_{k}\}\) in C by \(w_{k}=T^{k}_{j}z\),

$$\begin{aligned} d (w_{k},z_{n} ) =& d \bigl(T^{k}_{j}z,z_{n} \bigr) \\ \leq&d \bigl(T^{k}_{j}z,T^{k}_{j}z_{n} \bigr)+\sum_{i=1}^{k}d \bigl(T^{i}_{j}z_{n},T^{i-1}_{j}z_{n} \bigr) \\ \leq&(1+u_{jn})d ( z, z_{n} )+c_{jn}+kLd (T_{j} z_{n},z_{n} ). \end{aligned}$$

Taking lim sup,

$$ \limsup_{n\rightarrow\infty} d (w_{k},z_{n} )\leq \limsup _{n\rightarrow\infty}d ( z, z_{n} ), $$

i.e., \(r(T^{k}_{j}z,{z_{n}})\leq r( z,{z_{n}})\). It follows from Lemma 3.3 that \(\lim_{k\rightarrow\infty}T^{k}_{j}z=z\). As \(T _{j}\) is uniformly continuous, we have \(T _{j}z=T _{j} (\lim_{k\rightarrow \infty}T^{k}_{j}z )=\lim_{k\rightarrow\infty}T^{k+1}_{j}z =z\). Therefore, z is a common fixed point of \(\{T_{j} :j\in I\} \).

Recall that \(\lim_{n\rightarrow\infty}d(x_{n},z)\) exists by Lemma 3.1.

Suppose \(x\neq z\). By the uniqueness of asymptotic centers, we obtain

$$\begin{aligned} \limsup_{n\rightarrow\infty}d(z_{n},z) < &\limsup _{n\rightarrow\infty }d(z_{n},x) \\ \leq&\limsup_{n\rightarrow\infty}d(x_{n},x) \\ < &\limsup_{n\rightarrow\infty}d(x_{n},z) \\ =&\limsup_{n\rightarrow\infty}d(z_{n},z), \end{aligned}$$

a contradiction. Hence \(x=z\). Since \(\{z_{n}\}\) is an arbitrary subsequence of \(\{x_{n}\}\), therefore \(A(\{z _{n}\})=\{z\}\) for all subsequences \(\{z_{n}\} \) of \(\{x_{n}\}\). This proves that \(\{x_{n}\}\) -converges to a common fixed point of \(\{T_{j} :j\in I\}\). □

Remark 3.1

  1. (i)

    Lemma 3.1, Theorems 3.1 and 3.3 set an analogue of Theorems 2.8-2.10 in [24] and Lemma 3.2, Theorems 3.4 and 3.5 in [25], in uniformly convex metric spaces.

  2. (ii)

    Lemma 3.1 and Theorem 3.1 provide an analogue of Lemma 3.7 and Theorem 3.8 in [1] and Lemma 2.6 and Theorem 2.7 in [4] in uniformly convex metric spaces.

  3. (iii)

    Theorems 2.1 and 3.3 extend Theorems 3.2, 3.6, and 3.7 in [8], to convex metric spaces.

  4. (iv)

    Our results give an analogue of the results in [26].

Open problem

Assume that the initial point is the same in scheme (1.1) and multistep scheme (1.3) in [10]. Under what conditions are these schemes equivalent?

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Acknowledgements

The author AR Khan is grateful to KACST for supporting research project ARP-32-34. The third and the fourth authors are grateful to KFUPM for supporting research project IN121055.

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Khan, A.R., Yasmin, N., Fukhar-ud-din, H. et al. Viscosity approximation method for generalized asymptotically quasi-nonexpansive mappings in a convex metric space. Fixed Point Theory Appl 2015, 196 (2015). https://doi.org/10.1186/s13663-015-0447-6

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MSC

  • 47H09
  • 47H10
  • 47J25

Keywords

  • convex metric space
  • viscosity iterative method
  • generalized asymptotically quasi-nonexpansive mapping
  • uniformly Hölder continuous function
  • common fixed point
  • strong convergence
  • -convergence