# A new iteration scheme for a hybrid pair of generalized nonexpansive mappings

## Abstract

In this paper, we construct an iteration scheme involving a hybrid pair of nonexpansive mappings and utilize the same to prove some convergence theorems. In the process, we remove a restricted condition (called end-point condition) in Sokhuma and Kaewkhao’s results (Fixed Point Theory Appl. 2010:618767, 2010). Thus, our results generalized and improved several results contained in Sokhuma and Kaewkhao (Fixed Point Theory Appl. 2010:618767, 2010), Akkasriworn et al. (Int. J. Math. Anal. 6(19):923-932, 2012), Uddin et al. (Bull. Malays. Math. Soc., accepted) and Sokhuma (J. Math. Anal. 4(2):23-31, 2013).

MSC:47H10, 54H25.

## 1 Introduction

Let X be a Banach space and K be a nonempty subset of X. Let $CB\left(K\right)$ be the family of nonempty closed bounded subsets of K, while $C\left(K\right)$ be the family of nonempty compact convex subsets of K. A subset K of X is called proximinal if for each $x\in X$, there exists an element $k\in K$ such that

$d\left(x,k\right)=d\left(x,K\right)=inf\left\{\parallel x-y\parallel :y\in K\right\}.$

It is well known that every closed convex subset of a uniformly convex Banach space is proximinal. We shall denote by $PB\left(K\right)$, the family of nonempty bounded proximinal subsets of K.

The Hausdorff metric H on $CB\left(K\right)$ is defined as

A mapping $f:K\to K$ is said to be nonexpansive if

while a multivalued mapping $T:K\to CB\left(K\right)$ is said to be nonexpansive if

We use the notation $F\left(T\right)$ for the set of fixed points of the mapping T, while $F\left(f,T\right)$ denotes the set of common fixed points of f and T, i.e., a point x is said to be a common fixed point of f and T if $fx=x\in Tx$.

On the other hand, in 2008, Suzuki [1] introduced a new class of mappings which is larger than the class of nonexpansive mappings and named the defining condition condition (C) and utilized them to prove some existence and convergence fixed point theorems.

It is well known that the sequence of the Picard iteration (cf. [2]) defined as (for any ${x}_{1}\in K$)

${x}_{n+1}={f}^{n}x,\phantom{\rule{1em}{0ex}}n\in \mathbb{N},$
(1.1)

does not need to be convergent with respect to a nonexpansive mapping, e.g., the sequence of iterates ${x}_{n+1}=f{x}_{n}$ for the mapping $f:\left[-1,1\right]\to \left[-1,1\right]$ defined by $fx=-x$ does not converge to 0 for any choice of non-zero initial point which is indeed the fixed point of f. In an attempt to construct a convergent sequence of iterates with respect to a nonexpansive mapping, Mann [3] defined an iteration method by (for any ${x}_{1}\in K$)

${x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}f{x}_{n},\phantom{\rule{1em}{0ex}}n\in \mathbb{N},$
(1.2)

where ${\alpha }_{n}\in \left(0,1\right)$.

In 1974, with a view to approximate fixed point of pseudo-contractive compact mappings in Hilbert spaces Ishikawa [4] introduced a new iteration procedure as follows: (for ${x}_{1}\in K$)

$\left\{\begin{array}{c}{y}_{n}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}f{x}_{n},\hfill \\ {x}_{n+1}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}f{y}_{n},\phantom{\rule{1em}{0ex}}n\in \mathbb{N},\hfill \end{array}$
(1.3)

where ${\alpha }_{n},{\beta }_{n}\in \left(0,1\right)$.

The study of fixed points for multivalued contractions as well as multivalued nonexpansive mappings was initiated by Nadler [5] and Markin [6] and by now there exists an extensive literature on multivalued fixed point theory which has applications in diverse areas, such as control theory, convex optimization, differential inclusion, and economics (see [7] and references cited therein). Moreover, the existence of fixed points for multivalued nonexpansive mappings in uniformly convex Banach spaces was proved by Lim [8]. In recent years, different iterative processes have been used to approximate the fixed points of multivalued nonexpansive mappings. Among these iterative procedures, iteration schemes due to Sastry and Babu [9], Panyanak [10], Song and Wang [11], and Shahzad and Zegeye [12] are notable generalizations of Mann and Ishikawa iteration process especially in the case of multivalued mappings. By now, there exists an extensive literature on the iterative fixed points for various classes of mappings. For an almost up to date account of the literature on iterative fixed points, we refer the readers to Berinde [13].

Recently, Sokhuma and Kaewkhao [14] introduced the following modified Ishikawa iteration scheme for a pair of single valued and multivalued mapping.

Let K be a nonempty closed and bounded convex subset of Banach space X and let $f:K\to K$ be a single valued nonexpansive mapping and let $T:K\to CB\left(K\right)$ be a multivalued nonexpansive mapping. The sequence $\left\{{x}_{n}\right\}$ of the modified Ishikawa iteration is defined by

$\left\{\begin{array}{c}{y}_{n}={\beta }_{n}{z}_{n}+\left(1-{\beta }_{n}\right){x}_{n},\hfill \\ {x}_{n+1}={\alpha }_{n}f{y}_{n}+\left(1-{\alpha }_{n}\right){x}_{n},\hfill \end{array}$
(1.4)

where ${x}_{0}\in K$, ${z}_{n}\in T{x}_{n}$ and $0.

This scheme has been studied by several authors [1418] with respect to various classes of mappings in different classes of spaces. All the authors proved their results with the end-point condition $Tw=w$ for all $w\in F\left(T\right)$, where T is multivalued mapping. With a motivation to remove this strong condition, in this paper we introduce a new iteration scheme for a pair of hybrid mapping and prove some convergence theorems for generalized nonexpansive mappings. In this way, we are not only able to remove a restricted condition but also to generalize the class of functions. In the process several relevant results, especially those contained in Sokhuma and Kaewkhao [14], Akkasriworn et al. [15], Uddin et al. [17], Sokhuma [16], and Sokhuma [18] are generalized and improved.

## 2 Preliminaries

With a view to make our presentation self contained, we collect some relevant basic definitions, results, and iterative methods, which will be used frequently in the text later.

In 2005, Sastry and Babu [9] defined Ishikawa iteration scheme for multivalued mappings. Let $T:K\to PB\left(K\right)$ be a multivalued mapping and $p\in F\left(T\right)$. Then the sequence of Ishikawa iteration is defined as follows:

Choose ${x}_{0}\in K$,

${y}_{n}={\beta }_{n}{z}_{n}+\left(1-{\beta }_{n}\right){x}_{n},\phantom{\rule{1em}{0ex}}{\beta }_{n}\in \left[0,1\right],n\ge 0,$

where ${z}_{n}\in T{x}_{n}$ such that $\parallel {z}_{n}-p\parallel =d\left(p,T{x}_{n}\right)$ and

${x}_{n+1}={\alpha }_{n}{\stackrel{´}{z}}_{n}+\left(1-{\alpha }_{n}\right){x}_{n},\phantom{\rule{1em}{0ex}}{\alpha }_{n}\in \left[0,1\right],n\ge 0,$

where ${\stackrel{´}{z}}_{n}\in T{y}_{n}$ such that $\parallel {\stackrel{´}{z}}_{n}-p\parallel =d\left(p,T{y}_{n}\right)$.

Sastry and Babu [9] proved that the Ishikawa iteration scheme for a multivalued nonexpansive mapping T with a fixed point p converges to a fixed point q of T under certain conditions. In 2007, Panyanak [10] extended the results of Sastry and Babu to uniformly convex Banach space for multivalued nonexpansive mappings. Panyanak also modified the iteration scheme of Sastry and Babu and posed the question of convergence of this scheme. He introduced the following modified Ishikawa iteration method:

For ${x}_{0}\in K$, write

${y}_{n}={\beta }_{n}{z}_{n}+\left(1-{\beta }_{n}\right){x}_{n},\phantom{\rule{1em}{0ex}}{\beta }_{n}\in \left[a,b\right],0

where ${z}_{n}\in T{x}_{n}$ is such that $\parallel {z}_{n}-{u}_{n}\parallel =dist\left({u}_{n},T{x}_{n}\right)$, and ${u}_{n}\in F\left(T\right)$ such that $\parallel {x}_{n}-{u}_{n}\parallel =dist\left({x}_{n},F\left(T\right)\right)$, and

${x}_{n+1}={\alpha }_{n}{\stackrel{´}{z}}_{n}+\left(1-{\alpha }_{n}\right){x}_{n},\phantom{\rule{1em}{0ex}}{\alpha }_{n}\in \left[a,b\right],$

where ${\stackrel{´}{z}}_{n}\in T{y}_{n}$ such that $\parallel {\stackrel{´}{z}}_{n}-{v}_{n}\parallel =dist\left({v}_{n},T{y}_{n}\right)$, and ${v}_{n}\in F\left(T\right)$ such that $\parallel {y}_{n}-{v}_{n}\parallel =dist\left({y}_{n},F\left(T\right)\right)$.

In 2009, Song and Wang [11] pointed out a gap in the result of Panyanak [10]. In an attempt to remove this gap, they gave a partial answer to the question raised by Panyanak by using the following iteration scheme.

Let ${\alpha }_{n},{\beta }_{n}\in \left[0,1\right]$ and ${\gamma }_{n}\in \left(0,\mathrm{\infty }\right)$ such that ${lim}_{n\to \mathrm{\infty }}{\gamma }_{n}=0$ with ${x}_{0}\in K$, write

$\begin{array}{r}{y}_{n}={\beta }_{n}{z}_{n}+\left(1-{\beta }_{n}\right){x}_{n},\\ {x}_{n+1}={\alpha }_{n}{\stackrel{´}{z}}_{n}+\left(1-{\alpha }_{n}\right){x}_{n},\end{array}$

where $\parallel {z}_{n}-{\stackrel{´}{z}}_{n}\parallel \le H\left(T{x}_{n},T{y}_{n}\right)+{\gamma }_{n}$ and $\parallel {z}_{n+1}-{\stackrel{´}{z}}_{n}\parallel \le H\left(T{x}_{n+1},T{y}_{n}\right)+{\gamma }_{n}$ for ${z}_{n}\in T{x}_{n}$ and ${\stackrel{´}{z}}_{n}\in T{y}_{n}$.

Simultaneously, Shahzad and Zegeye [12] extended the corresponding results of Sastry and Babu [9], Panyanak [10], and Song and Wang [11] to quasi-nonexpansive multivalued mappings and also relaxed the end-point condition and compactness of the domain by using the following modified iteration scheme and gave an affirmative answer to the Panyanak question in a more general setting wherein

$\begin{array}{c}{y}_{n}={\beta }_{n}{z}_{n}+\left(1-{\beta }_{n}\right){x}_{n},\phantom{\rule{1em}{0ex}}{\beta }_{n}\in \left[0,1\right],n\ge 0,\hfill \\ {x}_{n+1}={\alpha }_{n}{\stackrel{´}{z}}_{n}+\left(1-{\alpha }_{n}\right){x}_{n},\phantom{\rule{1em}{0ex}}{\alpha }_{n}\in \left[0,1\right],n\ge 0,\hfill \end{array}$

where ${z}_{n}\in T{x}_{n}$ and ${\stackrel{´}{z}}_{n}\in T{y}_{n}$.

Now, we collect some relevant definitions and results.

Definition 2.1 ([1])

A mapping f defined on a subset K of a Banach space X is said to satisfy condition (C) if (for all $x,y\in K$)

$\frac{1}{2}\parallel x-fx\parallel \le \parallel x-y\parallel \phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\parallel fx-fy\parallel \le \parallel x-y\parallel .$

Every nonexpansive mapping satisfies condition (C). If f satisfies condition (C) and has a fixed point, then f is a quasi-nonexpansive mapping. But the converse of the above statements does not need to be true in general. The following examples justify the converse fact.

Example 2.2 ([1])

Define a mapping f on $\left[0,3\right]$ by

Then f satisfies condition (C) but f is not a nonexpansive mapping.

Example 2.3 ([1])

Define a mapping f on $\left[0,3\right]$ by

Then $F\left(f\right)\ne \mathrm{\varnothing }$ and f is a quasi-nonexpansive mapping but does not satisfy condition (C).

The following is a multivalued version of condition (C).

Definition 2.4 ([19])

Let T be a mapping defined on a subset K of a Banach space X. Then T is said to satisfy condition (C) if

$\frac{1}{2}d\left(x,Tx\right)\le \parallel x-y\parallel \phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}H\left(Tx,Ty\right)\le \parallel x-y\parallel$

for all $x,y\in X$.

Similar to the case of a single valued mapping, every multivalued nonexpansive mapping satisfies condition (C). If T satisfies condition (C) and has a fixed point, then T is a quasi-nonexpansive mapping. But the converse of the above statements does not need to be true in general. The following examples justify the converse fact.

Example 2.5 ([19])

Define a mapping $T:\left[0,3\right]\to CB\left(\left[0,3\right]\right)$ by

Then T satisfies condition (C) but T is not a nonexpansive mapping.

Example 2.6 ([19])

Define a mapping $T:\left[0,3\right]\to CB\left(\left[0,3\right]\right)$ by

Then $F\left(T\right)\ne \mathrm{\varnothing }$ and T is a quasi-nonexpansive mapping but it does not satisfy condition (C).

The following result is very important and will be used repeatedly.

Lemma 2.7 ([19])

Let K be a subset of a Banach space X and $f:K\to K$ be a mapping which satisfies condition (C), then for all $x,y\in K$ the following holds:

$\parallel x-fy\parallel \le 3\parallel x-fx\parallel +\parallel x-y\parallel .$

García-Falset et al. [20] used the concept of strongly demiclosedness to prove a weaker version of the famous principle of demiclosedness for generalized nonexpansive mappings in which weak convergence has been replaced by strong convergence. The definition runs as follows.

Definition 2.8 ([20])

If $f:K\to K$ is a mapping, then $\left(I-f\right)$ is said to be strongly demiclosed at 0 if for every sequence $\left\{{x}_{n}\right\}$ in K strongly convergent to $z\in K$ and such that ${x}_{n}-f{x}_{n}\to 0$ we have $z=fz$.

Proposition 2.9 Let K be a nonempty subset of a Banach space X. If $f:K\to K$ satisfies condition (C), then $\left(I-f\right)$ is strongly demiclosed at 0.

Proof Let $\left\{{x}_{n}\right\}$ be a sequence such that ${x}_{n}\to z$ and ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-f{x}_{n}\parallel =0$. In view of Lemma 2.7,

$\parallel {x}_{n}-fz\parallel \le 3\parallel {x}_{n}-f{x}_{n}\parallel +\parallel {x}_{n}-z\parallel .$

On letting $n\to \mathrm{\infty }$ we get ${x}_{n}\to fz$ and hence $fz=z$. □

Now, we list the following important property of a uniformly convex Banach space essentially due to Schu [21] and an important lemma due to Khaewkhao and Sokhuma [14].

Lemma 2.10 ([21])

Let X be a uniformly convex Banach space, let $\left\{{u}_{n}\right\}$ be a sequence of real numbers such that $0 for all $n\ge 1$, and let $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ be sequences in X such that ${lim sup}_{n\to \mathrm{\infty }}\parallel {x}_{n}\parallel \le a$, ${lim sup}_{n\to \mathrm{\infty }}\parallel {y}_{n}\parallel \le a$, and ${lim}_{n\to \mathrm{\infty }}\parallel {u}_{n}{x}_{n}+\left(1-{u}_{n}\right){y}_{n}\parallel =a$ for some $a>0$. Then ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-{y}_{n}\parallel =0$.

Lemma 2.11 ([14])

Let X be a Banach space, and let K be a nonempty, closed, convex subset of X. Then

$d\left(y,Ty\right)\le \parallel y-x\parallel +d\left(x,Tx\right)+H\left(Tx,Ty\right),$

where $x,y\in K$ and T is a multivalued nonexpansive mapping from K into $CB\left(K\right)$.

The following very useful theorem is due to Song and Cho [22].

Lemma 2.12 Let $T:K\to P\left(K\right)$ be a multivalued mapping and ${P}_{T}\left(x\right)=\left\{y\in Tx:\parallel x-y\parallel =d\left(x,Tx\right)\right\}$. Then the following are equivalent.

1. (i)

$x\in F\left(T\right)$,

2. (ii)

${P}_{T}\left(x\right)=\left\{x\right\}$,

3. (iii)

$x\in F\left({P}_{T}\right)$.

Moreover, $F\left(T\right)=F\left({P}_{T}\right)$.

## 3 Main results

In this paper, we introduce the following iteration scheme: Let K be a nonempty closed, bounded, and convex subset of Banach space X and let $f:K\to K$ be a single valued nonexpansive mapping and let $T:K\to CB\left(K\right)$ be a multivalued nonexpansive mapping. The sequence $\left\{{x}_{n}\right\}$ of the modified Ishikawa iteration is defined by

$\left\{\begin{array}{c}{y}_{n}={\alpha }_{n}{z}_{n}+\left(1-{\alpha }_{n}\right){x}_{n},\hfill \\ {x}_{n+1}={\beta }_{n}f{y}_{n}+\left(1-{\beta }_{n}\right){x}_{n},\hfill \end{array}$
(3.1)

where ${x}_{0}\in K$, ${z}_{n}\in {P}_{T}{x}_{n}$, and $0.

Lemma 3.1 Let f be a self-mapping of a nonempty closed convex subset K of a uniformly convex Banach space X which satisfies condition (C), and let $T:K\to P\left(K\right)$ be a multivalued mapping with $F\left(f,T\right)\ne \mathrm{\varnothing }$ such that ${P}_{T}$ enjoys condition (C). If $\left\{{x}_{n}\right\}$ is the sequence of the modified Ishikawa iteration defined by (3.1), then ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-w\parallel$ exists for all $w\in F\left(f,T\right)$.

Proof Let $w\in F\left(f,T\right)$, in view of Lemma 2.12 we have

$w\in {P}_{T}w=\left\{w\right\}.$

Also,

$\frac{1}{2}\parallel w-fw\parallel =0\le \parallel f{y}_{n}-w\parallel ,$

owing to condition (C), we get

$\parallel f{y}_{n}-fw\parallel \le \parallel {y}_{n}-w\parallel .$

Similarly, in view of $\frac{1}{2}d\left(w,Tw\right)=0\le \parallel {x}_{n}-w\parallel$, we have $H\left({P}_{T}{x}_{n},{P}_{T}w\right)\le \parallel {x}_{n}-w\parallel$.

Now, consider

$\begin{array}{rl}\parallel {x}_{n+1}-w\parallel & =\parallel \left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}f{y}_{n}-w\parallel \\ \le \left(1-{\beta }_{n}\right)\parallel {x}_{n}-w\parallel +{\beta }_{n}\parallel f{y}_{n}-fw\parallel \\ \le \left(1-{\beta }_{n}\right)\parallel {x}_{n}-w\parallel +{\beta }_{n}\parallel {y}_{n}-w\parallel .\end{array}$
(3.2)

But

$\begin{array}{rl}\parallel {y}_{n}-w\parallel & =\parallel \left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}{z}_{n}-w\parallel \\ \le \left(1-{\alpha }_{n}\right)\parallel {x}_{n}-w\parallel +{\alpha }_{n}\parallel {z}_{n}-w\parallel \\ =\left(1-{\alpha }_{n}\right)\parallel {x}_{n}-w\parallel +{\alpha }_{n}d\left({z}_{n},{P}_{T}w\right)\\ \le \left(1-{\alpha }_{n}\right)\parallel {x}_{n}-w\parallel +{\alpha }_{n}H\left({P}_{T}{x}_{n},{P}_{T}w\right)\\ \le \left(1-{\alpha }_{n}\right)\parallel {x}_{n}-w\parallel +{\alpha }_{n}\parallel {x}_{n}-w\parallel \\ =\parallel {x}_{n}-w\parallel .\end{array}$
(3.3)

In view of (3.2) and (3.3), we have

$\parallel {x}_{n+1}-w\parallel \le \parallel {x}_{n}-w\parallel ,$
(3.4)

which shows that $\left\{\parallel {x}_{n}-p\parallel \right\}$ is a decreasing sequence of non-negative reals. Thus in all, the sequence $\left\{\parallel {x}_{n}-p\parallel \right\}$ is bounded below and decreasing, therefore it remains convergent. □

Lemma 3.2 Let f be a self-mapping of a nonempty closed convex subset K of a uniformly convex Banach space X which satisfies condition (C), and let $T:K\to P\left(K\right)$ be a multivalued mapping with $F\left(f,T\right)\ne \mathrm{\varnothing }$ such that ${P}_{T}$ enjoys condition (C). If $\left\{{x}_{n}\right\}$ is the sequence of the modified Ishikawa iteration defined by (3.1), then ${lim}_{n\to \mathrm{\infty }}\parallel f{y}_{n}-{x}_{n}\parallel =0$.

Proof In view of Lemma 3.1, ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-w\parallel$ exists for all $w\in F\left(f,T\right)$.

Write ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-w\parallel =c$.

Now, consider

$\begin{array}{rl}\parallel f{y}_{n}-w\parallel & \le \parallel {y}_{n}-w\parallel \\ \le \parallel \left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}{z}_{n}-w\parallel \\ \le \left(1-{\alpha }_{n}\right)\parallel {x}_{n}-w\parallel +{\alpha }_{n}\parallel {z}_{n}-w\parallel \\ =\left(1-{\alpha }_{n}\right)\parallel {x}_{n}-w\parallel +{\alpha }_{n}d\left({z}_{n},{P}_{T}w\right)\\ \le \left(1-{\alpha }_{n}\right)\parallel {x}_{n}-w\parallel +{\alpha }_{n}H\left({P}_{T}{x}_{n},{P}_{T}w\right)\\ \le \left(1-{\alpha }_{n}\right)\parallel {x}_{n}-w\parallel +{\alpha }_{n}\parallel {x}_{n}-w\parallel \\ =\parallel {x}_{n}-w\parallel .\end{array}$
(3.5)

On taking lim sup of both sides, we obtain

$\underset{n\to \mathrm{\infty }}{lim sup}\parallel f{y}_{n}-w\parallel \le c.$
(3.6)

Also,

$\begin{array}{rl}c& =\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n+1}-w\parallel \\ =\underset{n\to \mathrm{\infty }}{lim}\parallel \left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}f{y}_{n}-w\parallel \\ =\underset{n\to \mathrm{\infty }}{lim}\parallel \left(1-{\beta }_{n}\right)\left({x}_{n}-w\right)+{\beta }_{n}\left(f{y}_{n}-w\right)\parallel .\end{array}$
(3.7)

In view of (3.5), (3.6), (3.7), and Lemma 2.12, we get

$\underset{n\to \mathrm{\infty }}{lim}\parallel \left(f{y}_{n}-w\right)-\left({x}_{n}-w\right)\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel f{y}_{n}-{x}_{n}\parallel =0.$

□

Lemma 3.3 Let f be a self-mapping of a nonempty closed convex subset K of a uniformly convex Banach space X which satisfies condition (C) and let $T:K\to P\left(K\right)$ be a multivalued mapping with $F\left(f,T\right)\ne \mathrm{\varnothing }$ such that ${P}_{T}$ enjoys condition (C). If $\left\{{x}_{n}\right\}$ is the sequence of the modified Ishikawa iteration defined by (3.1), then ${lim}_{n\to \mathrm{\infty }}\parallel {z}_{n}-{x}_{n}\parallel =0$.

Proof Let $w\in F\left(f,T\right)$ and $\left\{{x}_{n}\right\}$ be the sequence described by (3.1). Then, in view of Lemma 2.12, we have

$w\in {P}_{T}\left(w\right)=\left\{w\right\}.$

Now, consider

$\begin{array}{rl}\parallel {x}_{n+1}-w\parallel & =\parallel \left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}f{y}_{n}-w\parallel \\ \le \left(1-{\beta }_{n}\right)\parallel {x}_{n}-w\parallel +{\beta }_{n}\parallel f{y}_{n}-fw\parallel \\ \le \left(1-{\beta }_{n}\right)\parallel {x}_{n}-w\parallel +{\beta }_{n}\parallel {y}_{n}-w\parallel ,\end{array}$
(3.8)

so that

$\begin{array}{r}\parallel {x}_{n+1}-w\parallel -\parallel {x}_{n}-w\parallel \le {\beta }_{n}\left(\parallel {y}_{n}-w\parallel -\parallel {x}_{n}-w\parallel \right),\phantom{\rule{1em}{0ex}}\text{or}\\ \frac{\parallel {x}_{n+1}-w\parallel -\parallel {x}_{n}-w\parallel }{{\beta }_{n}}\le \parallel {y}_{n}-w\parallel -\parallel {x}_{n}-w\parallel .\end{array}$

Since $0, we have

$\underset{n\to \mathrm{\infty }}{lim inf}\left\{\left(\frac{\parallel {x}_{n+1}-w\parallel -\parallel {x}_{n}-w\parallel }{{\beta }_{n}}\right)+\parallel {x}_{n}-w\parallel \right\}\le \underset{n\to \mathrm{\infty }}{lim inf}\parallel {y}_{n}-w\parallel .$

It follows that

$c\le \underset{n\to \mathrm{\infty }}{lim inf}\parallel {y}_{n}-w\parallel .$

Owing to (3.3) ${lim sup}_{n\to \mathrm{\infty }}\parallel {y}_{n}-w\parallel \le c$ so that

$\begin{array}{rl}c& =\underset{n\to \mathrm{\infty }}{lim}\parallel {y}_{n}-w\parallel \\ =\underset{n\to \mathrm{\infty }}{lim}\parallel \left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}{z}_{n}-w\parallel \\ =\underset{n\to \mathrm{\infty }}{lim}\parallel \left(1-{\alpha }_{n}\right)\left({x}_{n}-w\right)+{\alpha }_{n}\left({z}_{n}-w\right)\parallel .\end{array}$
(3.9)

As $\parallel {z}_{n}-w\parallel =d\left({z}_{n},{P}_{T}w\right)\le H\left({P}_{T}{x}_{n},{P}_{T}w\right)\le \parallel {x}_{n}-w\parallel$, we have

$\underset{n\to \mathrm{\infty }}{lim sup}\parallel {z}_{n}-w\parallel \le \underset{n\to \mathrm{\infty }}{lim sup}\parallel {x}_{n}-w\parallel =c.$
(3.10)

Owing to Lemma 2.12, (3.9), and (3.10) we obtain ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-{z}_{n}\parallel =0$. □

Lemma 3.4 Let f be a self-mapping of a nonempty closed convex subset K of a uniformly convex Banach space X which satisfies condition (C) and let $T:K\to P\left(K\right)$ be a multivalued mapping with $F\left(f,T\right)\ne \mathrm{\varnothing }$ such that ${P}_{T}$ enjoys condition (C). If $\left\{{x}_{n}\right\}$ is the sequence of the modified Ishikawa iteration defined by (3.1), then ${lim}_{n\to \mathrm{\infty }}\parallel f{x}_{n}-{x}_{n}\parallel =0$.

Proof Owing to Lemma 2.7, we get

$\begin{array}{rl}\parallel f{x}_{n}-{x}_{n}\parallel & =\parallel f{x}_{n}-{y}_{n}+{y}_{n}-{x}_{n}\parallel \\ \le \parallel f{x}_{n}-{y}_{n}\parallel +\parallel {y}_{n}-{x}_{n}\parallel \\ \le 3\parallel f{y}_{n}-{y}_{n}\parallel +\parallel {y}_{n}-{x}_{n}\parallel +\parallel {y}_{n}-{x}_{n}\parallel \\ \le 3\parallel f{y}_{n}-{x}_{n}\parallel +5\parallel {y}_{n}-{x}_{n}\parallel \\ =3\parallel f{y}_{n}-{x}_{n}\parallel +5\parallel {\alpha }_{n}{z}_{n}+\left(1-{\alpha }_{n}\right){x}_{n}-{x}_{n}\parallel \\ \le 3\parallel f{y}_{n}-{x}_{n}\parallel +5{\alpha }_{n}\parallel {z}_{n}-{x}_{n}\parallel ;\end{array}$

therefore,

$\underset{n\to \mathrm{\infty }}{lim}\parallel f{x}_{n}-{x}_{n}\parallel \le \underset{n\to \mathrm{\infty }}{lim}5{\alpha }_{n}\parallel {x}_{n}-{z}_{n}\parallel +\underset{n\to \mathrm{\infty }}{lim}3\parallel f{y}_{n}-{x}_{n}\parallel .$

On using Lemma 3.2 and Lemma 3.3, we get

$\underset{n\to \mathrm{\infty }}{lim}\parallel f{x}_{n}-{x}_{n}\parallel =0.$

□

Theorem 3.5 Let f be a self-mapping of a nonempty closed convex subset K of a uniformly convex Banach space X which satisfies condition (C) and let $T:K\to P\left(K\right)$ be a multivalued mapping with $F\left(f,T\right)\ne \mathrm{\varnothing }$ such that ${P}_{T}$ enjoys condition (C). If $\left\{{x}_{n}\right\}$ is the sequence of the modified Ishikawa iteration defined by (3.1), then $\left\{{x}_{{n}_{i}}\right\}\to y$ for some subsequence $\left\{{x}_{{n}_{i}}\right\}$ of $\left\{{x}_{n}\right\}$ implies $y\in F\left(f,T\right)$.

Proof Assume that ${lim}_{i\to \mathrm{\infty }}\parallel {x}_{{n}_{i}}-y\parallel =0$. By Lemma 3.4, we obtain $0={lim}_{i\to \mathrm{\infty }}\parallel f{x}_{{n}_{i}}-{x}_{{n}_{i}}\parallel ={lim}_{i\to \mathrm{\infty }}\parallel \left(I-f\right)\left({x}_{{n}_{i}}\right)\parallel$. Since $\left(I-f\right)$ is strongly demiclosed at 0 so that we have $\left(I-f\right)\left(y\right)=0$. Thus $y=fy$, i.e., $y\in F\left(f\right)$. By Lemma 2.11, we have

$\begin{array}{rl}d\left(y,{P}_{T}y\right)& \le \parallel y-{x}_{{n}_{i}}\parallel +d\left({x}_{{n}_{i}},{P}_{T}{x}_{{n}_{i}}\right)+H\left({P}_{T}{x}_{{n}_{i}},{P}_{T}y\right)\\ \le \parallel y-{x}_{{n}_{i}}\parallel +\parallel {x}_{{n}_{i}}-{z}_{{n}_{i}}\parallel +\parallel {x}_{{n}_{i}}-y\parallel \to 0\end{array}$

as $i\to \mathrm{\infty }$. It follows that $y\in F\left({P}_{T}\right)=F\left(T\right)$. Thus $y\in F\left(f,T\right)$. □

Theorem 3.6 Let f be a self-mapping of a nonempty compact convex subset K of a uniformly convex Banach space X which satisfies condition (C) and let $T:K\to P\left(K\right)$ be a multivalued mapping with $F\left(f,T\right)\ne \mathrm{\varnothing }$ such that ${P}_{T}$ enjoys condition (C). If $\left\{{x}_{n}\right\}$ is the sequence of the modified Ishikawa iteration defined by (3.1), then $\left\{{x}_{n}\right\}$ converges strongly to a common fixed point of f and T.

Proof Since $\left\{{x}_{n}\right\}$ is contained in a compact subset K, there exists a subsequence $\left\{{x}_{{n}_{i}}\right\}$ of $\left\{{x}_{n}\right\}$ such that $\left\{{x}_{{n}_{i}}\right\}$ converges strongly to some point $y\in K$, that is, ${lim}_{i\to \mathrm{\infty }}\parallel {x}_{{n}_{i}}-y\parallel =0$. Now, in view of Theorem 3.5, $y\in F\left(f,T\right)$, while owing to Lemma 3.1 ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-y\parallel$ exists. Thus, in all, ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-y\parallel ={lim}_{i\to \mathrm{\infty }}\parallel {x}_{{n}_{i}}-y\parallel =0$, so that $\left\{{x}_{n}\right\}$ converges strongly to $y\in F\left(f,T\right)$. □

Khan and Fukhar-ud-din [23] introduced the so-called condition (${A}^{\prime }$) for two mappings and gave an improved version in [24] of condition (I) of Senter and Dotson [25]. A hybrid version of condition (${A}^{\prime }$), involving a pair of single valued and multivalued mappings, which is weaker than compactness of the domain, is given as follows:

A pair of a single valued mapping $f:K\to K$ and a multivalued mapping $T:K\to CB\left(K\right)$ is said to satisfy condition (${A}^{\prime }$) if there exists a nondecreasing function $g:\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ with $g\left(0\right)=0$, $g\left(r\right)>0$ for all $r\in \left(0,\mathrm{\infty }\right)$ such that either $d\left(x,fx\right)\ge g\left(d\left(x,F\left(f,T\right)\right)\right)$ or $d\left(x,Tx\right)\ge g\left(d\left(x,F\left(f,T\right)\right)\right)$ for all $x\in K$.

Theorem 3.7 Let f be a self-mapping of a nonempty closed convex subset K of a uniformly convex Banach space X which satisfies condition (C), and let $T:K\to P\left(K\right)$ be a multivalued mapping with $F\left(f,T\right)\ne \mathrm{\varnothing }$ such that ${P}_{T}$ enjoys condition (C). If $\left\{{x}_{n}\right\}$ is the sequence of the modified Ishikawa iteration defined by (3.1) and the pair $\left(f,T\right)$ satisfies condition (${A}^{\prime }$), then $\left\{{x}_{n}\right\}$ converges strongly to a common fixed point of f and T.

Proof Firstly, we show that $F\left(f,T\right)$ is closed. Let $\left\{{x}_{n}\right\}$ be a sequence in $F\left(f,T\right)$ converging to some point $z\in K$. We have

$\begin{array}{rl}\parallel {x}_{n}-fz\parallel & =\parallel f{x}_{n}-fz\parallel \\ \le \parallel {x}_{n}-z\parallel ,\end{array}$

so that

$\underset{n}{lim sup}\parallel {x}_{n}-fz\parallel \le lim sup\parallel {x}_{n}-z\parallel =0.$

Owing to the uniqueness of the limit, we have $fz=z$. Also,

This implies that $\left\{{x}_{n}\right\}$ converges to some point of ${P}_{T}z$ and hence $z\in F\left({P}_{T}\right)=F\left(T\right)$.

By Lemma 3.1, ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-p\parallel$ exists for all $p\in F\left(f,T\right)$ and let us take it to be c. If $c=0$, then there is nothing to prove. If $c>0$, then in view of (3.4) for all $p\in F\left(f,T\right)$, we have

$\parallel {x}_{n+1}-p\parallel \le \parallel {x}_{n}-p\parallel ,$

so that

$\underset{p\in F\left(f,T\right)}{inf}\parallel {x}_{n+1}-p\parallel \le \underset{p\in F\left(f,T\right)}{inf}\parallel {x}_{n}-p\parallel ,$

which amounts to saying that

$d\left({x}_{n+1},F\left(f,T\right)\right)\le d\left({x}_{n},F\left(f,T\right)\right),$

and hence ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},F\left(f,T\right)\right)$ exists. Owing to condition (${A}^{\prime }$) there exists a nondecreasing function g such that

$\underset{n\to \mathrm{\infty }}{lim}g\left(d\left({x}_{n},F\left(f,T\right)\right)\right)\le \underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-f{x}_{n}\parallel =0$

or

$\underset{n\to \mathrm{\infty }}{lim}g\left(d\left({x}_{n},F\left(f,T\right)\right)\right)\le \underset{n\to \mathrm{\infty }}{lim}d\left({x}_{n},{P}_{T}{x}_{n}\right)\le \underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-{z}_{n}\parallel =0,$

so that in both cases ${lim}_{n\to \mathrm{\infty }}g\left(d\left({x}_{n},F\left(f,T\right)\right)\right)=0$. Since g is a nondecreasing function and $g\left(0\right)=0$, we have ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},F\left(f,T\right)\right)=0$.

This implies that there exists a subsequence $\left\{{x}_{{n}_{k}}\right\}$ of $\left\{{x}_{n}\right\}$ such that

wherein $\left\{{p}_{k}\right\}$ is in $F\left(f,T\right)$. By Lemma 3.1, we have

$\parallel {x}_{{n}_{k+1}}-{p}_{k}\parallel \le \parallel {x}_{{n}_{k}}-{p}_{k}\parallel \le \frac{1}{{2}^{k}},$

so that

$\begin{array}{rl}\parallel {p}_{k+1}-{p}_{k}\parallel & \le \parallel {p}_{k+1}-{x}_{{n}_{k+1}}\parallel +\parallel {x}_{{n}_{k+1}}-{p}_{k}\parallel \\ \le \frac{1}{{2}^{k+1}}+\frac{1}{{2}^{k}}<\frac{1}{{2}^{k-1}},\end{array}$

which implies that $\left\{{p}_{k}\right\}$ is a Cauchy sequence. Since $F\left(f,T\right)$ is closed, therefore $\left\{{p}_{k}\right\}$ is a convergent sequence. Write ${lim}_{k\to \mathrm{\infty }}{p}_{k}=p$. Now, in order to show that $\left\{{x}_{n}\right\}$ converges to p let us proceed as follows:

so that ${lim}_{k\to \mathrm{\infty }}\parallel {x}_{{n}_{k}}-p\parallel =0$. Since ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-p\parallel$ exists, we have ${x}_{n}\to p$. □

## References

1. Suzuki T: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 2008, 340: 1088–1095. 10.1016/j.jmaa.2007.09.023

2. Picard E: Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives. J. Math. Pures Appl. 1890, 6: 145–210.

3. Mann WR: Mean value methods in iterations. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3

4. Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5

5. Nadler SB Jr.: Multivalued contraction mappings. Pac. J. Math. 1969, 30: 475–488. 10.2140/pjm.1969.30.475

6. Markin JT: Continuous dependence of fixed point sets. Proc. Am. Math. Soc. 1973, 38: 545–547. 10.1090/S0002-9939-1973-0313897-4

7. Gorniewicz L: Topological Fixed Point Theory of Multivalued Mappings. Kluwer Academic, Dordrecht; 1999.

8. Lim TC: A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach spaces. Bull. Am. Math. Soc. 1974, 80: 1123–1126. 10.1090/S0002-9904-1974-13640-2

9. Sastry KPR, Babu GVR: Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point. Czechoslov. Math. J. 2005, 55: 817–826. 10.1007/s10587-005-0068-z

10. Panyanak B: Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Comput. Math. Appl. 2007, 54: 872–877. 10.1016/j.camwa.2007.03.012

11. Song Y, Wang H: Convergence of iterative algorithms for multivalued mappings in Banach spaces. Nonlinear Anal. 2009, 70: 1547–1556. 10.1016/j.na.2008.02.034

12. Shahzad N, Zegeye H: On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces. Nonlinear Anal. 2009, 71: 838–844. 10.1016/j.na.2008.10.112

13. Berinde V Lecture Notes in Mathematics 1912. In Iterative Approximation of Fixed Points. Springer, Berlin; 2007.

14. Sokhuma K, Kaewkhao A: Ishikawa iterative process for a pair of single-valued and multivalued nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 618767

15. Akkasriworn N, Sokhuma K, Chuikamwong K: Ishikawa iterative process for a pair of Suzuki generalized nonexpansive single valued and multivalued mappings in Banach spaces. Int. J. Math. Anal. 2012, 6(19):923–932.

16. Sokhuma K: Δ-Convergence theorems for a pair of single-valued and multivalued nonexpansive mappings in CAT(0) spaces. J. Math. Anal. 2013, 4(2):23–31.

17. Uddin, I, Imdad, M, Ali, J: Convergence theorems for a hybrid pair of generalized nonexpansive mappings in Banach spaces. Bull. Malays. Math. Soc. (accepted)

18. Sokhuma K: Convergence theorems for a pair of asymptotically and multi-valued nonexpansive mapping in Banach spaces. Int. J. Math. Anal. 2013, 7(19):927–936.

19. Kaewcharoen A, Panyanak B: Fixed point theorems for some generalized multivalued nonexpansive mappings. Nonlinear Anal. 2011, 74: 5578–5584. 10.1016/j.na.2011.05.042

20. García-Falset J, Llorens-Fuster E, Suzuki T: Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl. 2011, 375: 185–195. 10.1016/j.jmaa.2010.08.069

21. Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884

22. Song Y, Cho YJ: Some notes on Ishikawa iteration for multivalued mappings. Bull. Korean Math. Soc. 2011, 48(3):575–584. 10.4134/BKMS.2011.48.3.575

23. Khan SH, Fukhar-ud-din H: Weak and strong convergence of a scheme with errors for two nonexpansive mappings. Nonlinear Anal. 2005, 8: 1295–1301.

24. Fukhar-ud-din H, Khan SH: Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications. J. Math. Anal. Appl. 2007, 328: 821–829. 10.1016/j.jmaa.2006.05.068

25. Senter HF, Dotson WG: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 1974, 44(2):375–380. 10.1090/S0002-9939-1974-0346608-8

## Acknowledgements

The first author is grateful to University Grants Commission, India for providing financial assistance in the form of the Maulana Azad National fellowship. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.

## Author information

Authors

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors have contributed in this work on an equal basis. All authors read and approved the final manuscript.

## Rights and permissions

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

Uddin, I., Abdou, A.A. & Imdad, M. A new iteration scheme for a hybrid pair of generalized nonexpansive mappings. Fixed Point Theory Appl 2014, 205 (2014). https://doi.org/10.1186/1687-1812-2014-205