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.

1 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:

• (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;

• (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 $\lim_{n\rightarrow\infty}d(u_{n},v_{n})=0$.

2 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

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

• (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.

3 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

• (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.

• (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.

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

• (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?