Modified Picard-Mann hybrid iteration process for total asymptotically nonexpansive mappings

Abstract

In this paper, using the modified hybrid Picard-Mann iteration process, we establish Δ-convergence and strong convergence theorems for total asymptotically nonexpansive mappings on a $CAT(0)$ space. Results established in the paper extend and improve a number of results in the literature. A numerical example is also given to examine the fastness of the proposed iteration process under different control conditions and initial points.

1 Introduction

Let K be a nonempty, closed and convex subset of a normed linear space E. A mapping $T\colon K\to K$ is said to be nonexpansive if $\Vert Tx-Ty\Vert \leq \Vert x-y\Vert$ for every x, y in K. In the last four decades, many papers have appeared in the literature on the iteration methods to approximate fixed points of a nonexpansive mapping, cf. [1–6] and the references therein. In the meantime, some generalizations of nonexpansive mappings have appeared, namely asymptotically nonexpansive mapping [7], asymptotically nonexpansive type mapping [8], asymptotically nonexpansive mappings in the intermediate sense [9].

Recently Alber et al. [10] made an effort to unify some generalization of nonexpansive mappings and introduced the notion of total asymptotically nonexpansive mappings. A mapping $T\colon K\to K$ is said to be total asymptotically nonexpansive if there exist nonnegative real sequences $\{k_{n}^{(1)} \}$ and $\{k_{n}^{(2)} \}$, $n\geq 1$, $k_{n}^{(1)}, k_{n}^{(2)} \to0$ as $n\to\infty$, and a strictly increasing and continuous function $\phi\colon{\mathbb{R}}^{+}\to{\mathbb{R}}^{+}$ with $\phi(0)=0$ such that

$$\bigl\Vert T^{n}x-T^{n}y \bigr\Vert \leq \Vert x-y \Vert + k_{n}^{(1)}\phi \bigl(\Vert x-y\Vert \bigr)+k_{n}^{(2)} $$

for every x, y in K. They further studied the iterative approximation of fixed point of total asymptotically nonexpansive mappings using a modified Mann iteration process. In the modified Mann iteration process [11] the sequence $\{x_{n} \}$ is generated by

$$ \left . \textstyle\begin{array}{l@{}} x_{1}\in K,\\ x_{n+1}=(1-\alpha_{n})x_{n}+\alpha_{n} T^{n}x_{n},\quad n\geq1, \end{array}\displaystyle \right \} $$

(1.1)

where $\{\alpha_{n} \}$ is a real sequence in $(0,1)$ satisfying certain conditions.

Iterative approximation of fixed points of total nonexpansive mappings has also been studied by [12–15].

In 2013, Khan [16] introduced a new iteration process for nonexpansive mappings, which he called ‘Picard-Mann hybrid iteration process’ and claimed that this process is independent of Picard and Mann iterative process and the convergence process is faster than Picard and Mann iteration process. The sequence $\{x_{n} \}$ in this process is given by

$$ \left . \textstyle\begin{array}{l@{}} x_{1}\in K,\\ x_{n+1}=Ty_{n},\\ y_{n}=(1-\alpha_{n})x_{n}+\alpha_{n} T x_{n},\quad n\geq1, \end{array}\displaystyle \right \} $$

(1.2)

where $\{\alpha_{n} \}$ is in $(0,1)$.

On the other hand, in 2003, Kirk [17, 18] initiated the study of fixed point theory in metric spaces with nonpositive curvature. He showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete $CAT(0)$ space always has a fixed point. After his work, fixed point theory in $CAT(0)$ spaces has been rapidly developed and many papers have appeared [19–24]. Nanjaras and Panyanak [25] proved the demiclosed principle for asymptotically nonexpansive mappings in $CAT(0)$ space and obtained a Δ-convergence theorem for the Mann iteration. Abbas et al. [26] proved the demiclosed principle for asymptotically nonexpansive mappings in the intermediate sense and established convergence theorems, Tang et al. [27] proved the demiclosed principle for total asymptotically nonexpansive mappings in a $CAT(0)$ space and obtained Δ-convergence theorem.

Motivated by the above recorded studies, in this paper, we propose a ‘modified hybrid Picard-Mann’ iteration process for iterative approximation of fixed points of total asymptotically nonexpansive mappings in $CAT(0)$ spaces. The sequence $\{x_{n} \}$ in this iteration is given by

$$ \left . \textstyle\begin{array}{l@{}} x_{1} \in C,\\ x_{n+1} =T^{n}y_{n},\\ y_{n} = (1-\alpha_{n}) x_{n} \oplus\alpha_{n} T^{n}x_{n},\quad n \in \mathbb{N}, \end{array}\displaystyle \right \} $$

(1.3)

where C is a nonempty bounded closed and convex subset of a complete $CAT(0)$ space X and $\{\alpha_{n} \}$ is in $(0,1)$.

We establish some strong and Δ-convergence results of the iterative process (1.3) for total asymptotically nonexpansive mappings on a $CAT(0)$ space. Our results extend and improve the corresponding results of Chang et al. [28], Nanjaras and Panyanak [25] and others.

2 Preliminaries

Throughout the paper, we denote by $\mathbb{N}$ the set of positive integers and by $\mathbb{R}$ the set of real numbers.

The following lemma plays an important role in our paper.

Lemma 2.1

([23], Lemma 2.4)

Let X be a $CAT(0)$ space, then

$$ d \bigl((1-t)x \oplus ty,z \bigr) \leq(1-t)d(x,z) + td(y,z) $$

for all $t\in[0, 1]$ and x, y, z are points in X.

Let C be a nonempty subset of a $CAT(0)$ space X.

Definition 2.1

A mapping $T \colon C \rightarrow C$ is said to be a nonexpansive mapping if

$$ d(Tx,Ty) \leq d(x,y), \quad\forall x,y \in C. $$

Definition 2.2

A mapping $T \colon C \rightarrow C$ is said to be asymptotically nonexpansive if there is a sequence $\{k_{n}\} \subset [1,\infty)$ with $k_{n} \rightarrow1$ such that

$$ d \bigl(T^{n}x,T^{n}y \bigr) \leq k_{n}d(x,y), \quad\forall n\geq1, x,y \in C. $$

Definition 2.3

A mapping $T \colon C \rightarrow C$ is said to be uniformly L-Lipschitzian if there exists a constant $L > 0$ such that

$$ d \bigl(T^{n}x,T^{n}y \bigr) \leq Ld(x,y), \quad \forall n\geq1, x,y \in C. $$

Definition 2.4

A mapping $T \colon C \rightarrow C$ is said to be $(\{\nu_{n}\}, \{\mu _{n}\}, \zeta)$-total asymptotically nonexpansive if there exist nonnegative sequences $\{\nu_{n}\}$, $\{\mu_{n}\}$ with $\{\nu_{n}\}\rightarrow0$, $\{\mu_{n}\} \rightarrow0$ and a strictly increasing continuous function $\zeta:[0,\infty) \rightarrow[0,\infty)$ with $\zeta(0) = 0$ such that

$$ d \bigl(T^{n}x,T^{n}y \bigr) \leq d(x,y) + \nu_{n} \zeta \bigl(d(x,y) \bigr)+\mu_{n},\quad\forall n\geq1, x,y \in C. $$

From the definitions, we see that each nonexpansive mapping is an asymptotically nonexpansive mapping with a sequence $\{k_{n} =1\}$, and each asymptotically nonexpansive mapping is a $(\{\nu_{n}\}, \{\mu_{n}\}, \zeta)$- total asymptotically nonexpansive mapping with $\mu_{n}=0$, $\nu_{n}=k_{n}-1$, $n \geq1$ and $\zeta(t)=t$, $t\geq0$, and each asymptotically nonexpansive mapping is a uniformly L-Lipschitzian mapping with $L=\sup\{k_{n}\}$, $n\geq1$.

Let $\{x_{n}\}$ be a bounded sequence in a $CAT(0)$ space X. For $x\in X$, we set

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

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

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

The asymptotic center $A(\{x_{n}\})$ of $\{x_{n}\}$ is the set

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

In a $CAT(0)$ space, $A(\{x_{n}\})$ consists of exactly one point [22], Proposition 7.

A sequence $\{x_{n}\}$ in X is said to Δ-converge to $p \in X$ if p is the unique asymptotic center of $\{u_{n}\}$ for every subsequence $\{u_{n}\}$ of $\{x_{n}\}$. In this case, we write $\Delta\mbox{-}\!\lim x_{n}=p$ and call p the Δ-limit of $\{x_{n}\}$.

Lemma 2.2

In a complete $CAT(0)$ space X, the following hold:

• (i) Every bounded sequence in a complete $CAT(0)$ space always has a Δ-convergent subsequence [24], p.3690.

• (ii) If $\{x_{n}\}$ is a bounded sequence in a closed convex subset C of X, then the asymptotic center of $\{x_{n}\}$ is in C [29], Proposition 2.1.

• (iii) If $\{x_{n}\}$ is a bounded sequence in X with $A(\{x_{n}\}) = \{p\}$, $\{u_{n}\}$ is a subsequence of $\{x_{n}\}$ with $A(\{u_{n}\}) = \{ u\}$ and the sequence $\{d(x_{n},u)\}$ converges, then $p = u$ [23], Lemma 2.8.

The following results are useful to prove our main result.

Lemma 2.3

([25], Lemma 4.5)

Let X be a $CAT(0)$ space, $x \in X$ be a given point and $\{t_{n}\}$ be a sequence in $[b, c]$ with $b, c \in(0, 1)$ and $0< b(1-c) \leq \frac{1}{2}$. Let $\{x_{n}\}$ and $\{y_{n}\}$ be any sequences in X such that

$$ \textstyle\begin{cases} \limsup_{n \rightarrow\infty}d(x_{n},x)\leq r,\\ \limsup_{n \rightarrow\infty}d(y_{n},x)\leq r\textit{ and}\\ \limsup_{n \rightarrow\infty}d((1-t_{n})x_{n} \oplus t_{n}y_{n},x) = r, \end{cases} $$

for some $r\geq0$. Then $\lim_{n \rightarrow\infty}d(x_{n},y_{n}) = 0$.

Lemma 2.4

([30], Lemma 2)

Let $\{a_{n}\}$, $\{\lambda_{n}\}$ and $\{c_{n}\}$ be the sequences of nonnegative numbers such that

$$ a_{n+1} \leq(1+ \lambda_{n})a_{n} + c_{n}, \quad\forall n \geq1. $$

If $\sum_{n=1}^{\infty}\lambda_{n} < {\infty}$ and $\sum_{n=1}^{\infty} c_{n} < {\infty}$, then $\lim_{n \rightarrow \infty} a_{n}$ exists. If there exists a subsequence $\{a_{n_{i}}\}\subset\{a_{n}\}$ such that $a_{n_{i}} \rightarrow0$, then $\lim_{n\rightarrow\infty} a_{n} = 0$.

Lemma 2.5

([28], Theorem 2.8)

Let C be a closed convex subset of a complete $CAT(0)$ space X and let $T \colon C\to C$ be a total asymptotically nonexpansive and uniformly L-Lipschitzian mapping. Let $\{x_{n}\}$ be a bounded sequence in C such that $\lim_{n\to\infty}d(x_{n}, Tx_{n}) = 0$ and $\Delta\mbox{-}\!\lim_{n\to\infty}x_{n} = p$. Then $Tp = p$.

The following existence result is also needed.

Lemma 2.6

([31], Corollary 3.2)

Let C be a nonempty bounded closed convex subset of a complete $CAT(0)$ space X. If $T \colon C\to C$ is a continuous total asymptotically nonexpansive mapping, then T has a fixed point.

3 Main results

We now establish a △-convergence result for the modified Picard-Mann hybrid iterative process.

Theorem 3.1

Let C be a bounded closed and convex subset of a complete $CAT(0)$ space X. Let $T \colon C \rightarrow C$ be a uniformly L-Lipschitzian and $(\{\nu_{n}\}, \{\mu_{n}\}, \zeta)$-total asymptotically nonexpansive mapping. If the following conditions are satisfied:

• (1) $\sum_{n=1}^{\infty}\nu_{n} <\infty$; $\sum_{n=1}^{\infty }\mu_{n} <\infty$;

• (2) there exist constants $a, b \in(0,1)$ with $0 < b(1- a) \leq \frac{1}{2}$ such that $\{\alpha_{n}\} \subset[a, b]$;

• (3) there exists a constant $M^{*} > r$ such that $\zeta(r) \leq M^{*}r$, $r \geq0$;

then the sequence $\{x_{n}\}$ defined by (1.3) Δ-converges to some point $p\in F(T)$, where $F(T)$ is the set of fixed points of T.

Proof

Since T is Lipschitz continuous, $F(T)\neq\emptyset$ by Lemma 2.6. We divide the proof of Theorem 3.1 into three steps.

Step-I. First we prove that $\lim_{n\to\infty}d(x_{n}, p)$ exists for each $p \in F(T)$.

Step-II. Next we prove that

$$ \lim_{n\to\infty}d(x_{n},Tx_{n}) =0. $$

Step-III. Finally to show that the sequence $\{x_{n}\}$ Δ-converges to a fixed point of T, we prove that

$$W_{\Delta}(x_{n})=\bigcup_{\{u_{n}\}\subset\{x_{n}\}}A \bigl(\{u_{n}\} \bigr)\subseteq F(T) $$

and $W_{\Delta}(x_{n})$ consists of exactly one point.

Proof of Step-I: For each $p \in F(T)$, we have

$$\begin{aligned} d(y_{n},p)&=d \bigl((1-\alpha_{n}) x_{n} \oplus\alpha_{n}T^{n} x_{n},p \bigr) \\ &\leq(1-\alpha_{n}) d(x_{n},p) + \alpha_{n} d \bigl(T^{n}x_{n},p \bigr) \\ &\leq(1-\alpha_{n}) d(x_{n},p) + \alpha_{n} \bigl\{ d(x_{n},p) +\nu_{n}\zeta d(x_{n},p) + \mu_{n} \bigr\} \\ &\leq d(x_{n},p) + \beta_{n} \nu_{n}\zeta d(x_{n},p) +\mu_{n} \\ &\leq \bigl(1 +\nu_{n} M^{*} \bigr) d(x_{n},p) + \mu_{n}. \end{aligned}$$

(3.1)

Also,

$$\begin{aligned} d(x_{n+1},p)&=d \bigl(T^{n} y_{n},p \bigr) \\ &\leq d(y_{n},p) +\nu_{n}\zeta d(y_{n},p) + \mu_{n} \\ &\leq \bigl(1 +\nu_{n} M^{*} \bigr) d(y_{n},p) + \mu_{n} \\ &\leq \bigl(1+\nu_{n} M^{*} \bigr) \bigl(1 +\nu_{n} M^{*} \bigr) d(x_{n},p) + \bigl(1+\nu_{n} M^{*} \bigr)\mu_{n} +\mu _{n} \\ &\leq \bigl(1+\nu_{n}M^{*} \bigl(2+\nu_{n}M^{*} \bigr) \bigr)d(x_{n},p)+ \bigl(2+\nu_{n}M^{*} \bigr)\mu_{n} \\ &=(1+\sigma_{n})d(x_{n},p)+\rho_{n}, \end{aligned}$$

where $\sigma_{n}=\nu_{n}M^{*}(2+\nu_{n}M^{*})$ and $\rho_{n}=(2+\nu_{n}M^{*})\mu_{n}$.

It follows from condition (i) and Lemma 2.4 that $\lim_{n\to \infty} d(x_{n}, p)$ exists.

Proof of Step-II. It follows from Step-I that $\lim_{n\to \infty} d(x_{n}, p)$ exists for each given $p \in F$. Without loss of generality, we can assume that

$$ \lim_{n\rightarrow\infty}d(x_{n},p) =r \geq0. $$

From (3.1) we have

$$ \liminf_{n\rightarrow\infty}d(y_{n},p) \leq\limsup _{n\rightarrow \infty }d(y_{n},p) \leq\lim_{n\rightarrow\infty} \bigl\{ \bigl(1 +\nu_{n} M^{*} \bigr) d(x_{n},p) + \mu_{n} \bigr\} =r\geq0. $$

(3.2)

Also,

$$\begin{aligned} d \bigl(T^{n}y_{n},p \bigr) &= d \bigl(T^{n}y_{n}, T^{n}p \bigr) \\ &\leq d(y_{n},p) +\nu_{n}\zeta d(y_{n},p) + \mu_{n} \\ &\leq \bigl(1+\nu_{n}M^{*} \bigr)d(y_{n},p) + \mu_{n}, \quad\forall n\geq1, \end{aligned}$$

so we have

$$ \limsup_{n\to\infty}d \bigl(T^{n}y_{n}, p \bigr) \leq r. $$

Similarly,

$$ \limsup_{n\to\infty}d \bigl(T^{n}x_{n}, p \bigr) \leq r. $$

Now,

$$d(x_{n+1},p)\leq \bigl(1 +\nu_{n} M^{*} \bigr) d(y_{n},p) +\mu_{n} , $$

taking limit infimum, we have

$$\liminf_{n\to\infty}d(x_{n+1},p)\leq\liminf _{n\to\infty}d(y_{n},p), $$

i.e.,

$$ r\leq\liminf_{n\to\infty}d(y_{n},p). $$

(3.3)

From (3.2) and (3.3), we have

$$\lim_{n\to\infty}d(y_{n},p)=r , $$

i.e.,

$$ r=\lim_{n\to\infty}d(y_{n},p)=\lim _{n\to\infty}d \bigl((1-\alpha_{n})x_{n} \oplus \alpha_{n}T^{n}x_{n},p \bigr) . $$

Therefore, by Lemma 2.3, we obtain

$$ \lim_{n\to\infty}d \bigl(x_{n},T^{n}x_{n} \bigr)=0. $$

(3.4)

Also,

$$\begin{aligned} d \bigl(T^{n}y_{n},T^{n}x_{n} \bigr)& \leq d(y_{n},x_{n}) +\nu_{n}\zeta d(y_{n},x_{n}) +\mu_{n} \\ &\leq \bigl(1+\nu_{n}M^{*} \bigr)d(y_{n},x_{n}) + \mu_{n} \\ &\leq \bigl(1+\nu_{n}M^{*} \bigr)d \bigl((1-\alpha_{n})T^{n}x_{n} \oplus\alpha_{n} x_{n},x_{n} \bigr) + \mu_{n} \\ &\leq \bigl(1+\nu_{n}M^{*} \bigr) \bigl\{ (1-\alpha_{n})d \bigl(T^{n}x_{n},x_{n} \bigr) + \alpha_{n}d(x_{n},x_{n}) \bigr\} +\mu_{n}, \end{aligned}$$

from (3.4), we have

$$ \lim_{n\to\infty}d \bigl(T^{n}y_{n},T^{n}x_{n} \bigr)=0, $$

(3.5)

and

$$\begin{aligned} d(x_{n+1},x_{n})&\leq d \bigl(x_{n+1},T^{n}y_{n} \bigr)+d \bigl(T^{n}y_{n},T^{n}x_{n} \bigr)+d \bigl(T^{n}x_{n},x_{n} \bigr) \\ &=d \bigl(T^{n}y_{n},T^{n}y_{n} \bigr)+d \bigl(T^{n}y_{n},T^{n}x_{n} \bigr)+d \bigl(T^{n}x_{n},x_{n} \bigr), \end{aligned}$$

using (3.4) and (3.5) we have

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

Finally,

$$\begin{aligned} d(x_{n},Tx_{n})&\leq d(x_{n},x_{n+1})+d \bigl(x_{n+1},T^{n+1}x_{n+1} \bigr)+d \bigl(T^{n+1}x_{n+1}, T^{n+1}x_{n} \bigr)+d \bigl(T^{n+1}x_{n},Tx_{n} \bigr) \\ &\leq d(x_{n},x_{n+1})+d \bigl(x_{n+1}, T^{n+1}x_{n+1} \bigr)+Ld(x_{n},x_{n+1})+Ld \bigl(T^{n}x_{n},x_{n} \bigr) \\ & \to0 \quad\mbox{as }n\to\infty. \end{aligned}$$

Hence, Step-II is proved.

Proof of Step-III. Let $u \in W_{\Delta}(x_{n})$. Then there exists a subsequence $\{u_{n}\}$ of $\{x_{n}\}$ such that $A(\{u_{n}\}) = \{u\}$. By Lemma 2.2(i), there exists a subsequence $\{v_{n}\}$ of $\{u_{n}\}$ such that $\Delta\mbox{-}\!\lim_{n\to\infty} v_{n} = v \in C$. By Lemma 2.5, $v \in F(T)$. Since $\{d (u_{n}, v)\}$ converges, by Lemma 2.2(iii), $u = v$. This shows that $W_{\Delta}(x_{n}) \subseteq F(T)$.

Now we prove that $W_{\Delta}(x_{n})$ consists of exactly one point. Let $\{u_{n}\}$ be a subsequence of $\{x_{n}\}$ with $A(\{u_{n}\}) = \{u\}$ and let $A(\{x_{n}\}) = \{x\}$. We have already seen that $u = v$ and $v\in F(T)$. Finally, since $\{d(x_{n}, v)\}$ converges, by Lemma 2.2(iii), we have $x = v \in F(T)$. This shows that $W_{\Delta}(x_{n}) = \{x\}$. □

We now establish some strong convergence results.

Theorem 3.2

Let X, C, T, $\{\alpha_{n}\}$, $\{\beta_{n}\}$, $\{x_{n}\}$ satisfy the hypothesis of Theorem  3.1. Then the sequence $\{x_{n}\}$ generated by (1.3) converges strongly to a fixed point of T if and only if

$$\liminf_{n\to\infty}d \bigl(x_{n}, F(T) \bigr) = 0, $$

where $d(x, F(T)) = \inf \{d(x, p) : p \in F(T)\}$.

Proof

Necessity is obvious. Conversely, suppose that $\liminf_{n\to \infty}d(x_{n},F(T))= 0$. As proved in Step-I of Theorem 3.1, $\lim_{n\to\infty} d(x_{n}, F(T))$ exists for all $p \in F(T)$. Thus, by hypothesis, $\lim_{n\to\infty} d(x_{n}, F(T)) = 0$.

Next, we show that $\{x_{n}\}$ is a Cauchy sequence in C. Let $\varepsilon> 0$ be arbitrarily chosen. Since $\lim_{n\to\infty }d(x_{n}, F(T)) = 0$, there exists a positive integer $n_{0}$ such that for all $n \geq n_{0}$,

$$d \bigl(x_{n}, F(T) \bigr) < \frac{\varepsilon}{4} . $$

In particular, $\inf\{d(x_{n_{0}} , p) : p \in F(T)\} < \frac {\varepsilon }{4}$. Thus, there exists $p^{*} \in F(T)$ such that

$$d \bigl(x_{n_{0}}, p^{*} \bigr)< \frac{\varepsilon}{2} . $$

Now, for all $m, n \geq n_{0}$, we have

$$d(x_{n+m}, x_{n})\leq d \bigl(x_{n+m}, p^{*} \bigr)+d \bigl(x_{n}, p^{*} \bigr)\leq2d \bigl(x_{n_{0}}, p^{*} \bigr) < 2 \biggl(\frac{\varepsilon}{2} \biggr) = \varepsilon, $$

i.e., $\{x_{n}\}$ is a Cauchy sequence in the closed subset C of a complete $CAT(0)$ space and hence it converges to a point q in C. Now, $\lim_{n\to\infty} d(x_{n}, F(T)) = 0$ gives that $d(q, F(T)) = 0$ and closedness of $F(T)$ forces q to be in $F(T)$. This completes the proof. □

Senter and Dotson [32], p.375, introduced the concept of Condition (I) as follows.

Definition 3.1

A mapping $T \colon C \to C$ is said to satisfy Condition (I) if there exists a non-decreasing function $f \colon[0,\infty) \to[0,\infty)$ with $f(0) = 0$ and $f(r) > 0$ for all $r > 0$ such that

$$d(x, Tx) \geq f \bigl(d \bigl(x, F(T) \bigr) \bigr) $$

for all $x\in C$.

It is weaker than demicompactness for a nonexpansive mapping T defined on a bounded set. Since every completely continuous mapping $T\colon K\to K$ is continuous and demicompact, so it satisfies Condition (I). Recently, Kim [33] gave an interesting example of total asymptotically nonexpansive self-mapping satisfying Condition (I).

Example 1

([33], Example 3.7)

Let $X:=\mathbb{R}$ and $C:=[0,2]$. Define $T\colon C\to C$ by the formula

$$ Tx=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 1, &x\in[0,1];\\ \frac{1}{\sqrt{3}} \sqrt{4-x^{2}}, & x\in[1,2] . \end{array}\displaystyle \right . $$

Here T is a uniformly continuous and total asymptotically nonexpansive mapping with $F(T)= \{1 \}$. Also T satisfies Condition (I), but T is not Lipschitzian and hence it is not an asymptotically nonexpansive mapping.

Using Condition (I), we now establish the following strong convergence result for total asymptotically nonexpansive mapping.

Theorem 3.3

Let X, C, T, $\{\alpha_{n}\}$, $\{\beta_{n}\}$, $\{x_{n}\}$ satisfy the hypothesis of Theorem  3.1 and let T be a mapping satisfying Condition (I). Then the sequence $\{x_{n}\}$ generated by (1.3) converges strongly to a fixed point of T.

Proof

As proved in Theorem 3.2, $\lim_{n\to\infty} d(x_{n}, F(T))$ exists. Also, by Step-II of Theorem 3.1, we have $\lim_{n\to \infty} d(x_{n}, Tx_{n}) = 0$. It follows from Condition (I) that

$$\lim_{n\to\infty}f\bigl(d \bigl(x_{n}, F(T) \bigr)\bigr) \leq\lim _{n\to\infty}d(x_{n}, Tx_{n}) = 0. $$

That is, $\lim_{n\to\infty}f(d(x_{n}, F(T))) = 0$. Since $f:[0,\infty) \to[0,\infty)$ is a non-decreasing function satisfying $f(0) = 0$ and $f(r)>0$ for all $r > 0$, we obtain

$$\lim_{n\to\infty}d \bigl(x_{n}, F(T) \bigr) = 0. $$

Now all the conditions of Theorem 3.3 are satisfied, therefore by its conclusion $\{x_{n}\}$ converges strongly to a point of $F(T)$. □

4 Numerical example

In this section, using Example 1 of a total asymptotically nonexpansive mapping, we compare the convergence of modified Picard-Mann hybrid iteration process (1.3) with the modified Mann iteration process (1.1).

Set $\alpha_{n}=\frac{n}{n+1}$ for all $n\geq1$. Using MATLAB, we computed the iterates of (1.1) and (1.3) for three different initial points $x_{1}=1.1$, $x_{1}=1.5$ and $x_{1}=1.9$. Both iterations converge to the fixed point 1, the detailed observation is given in Table 1.

Table 1 Iterates of modified Mann and modified Picard-Mann hybrid iterations

Next we examine the fastness of both iterations for different control conditions. Summary of the findings is given in Table 2, where the least number of iterates required to obtain the fixed point for different initial conditions is given under various control conditions.

Table 2 Comparison of fastness for different control conditions

From Table 1 and Table 2, it is clear that for a total asymptotically nonexpansive mapping, modified Mann iteration (1.1) is very much sensitive about the choice of initial point and control condition $\alpha_{n}$, whereas the behavior of proposed iteration (1.3) is consistent.