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Convergence theorems for some multivalued generalized nonexpansive mappings
Fixed Point Theory and Applications volume 2014, Article number: 33 (2014)
Abstract
In this paper, we propose an algorithms for finding a common fixed point of an infinite family of multivalued generalized nonexpansive mappings in uniformly convex Banach spaces. Under suitable conditions some strong and weak convergence theorems for such mappings are proved. The results presented in the paper improve and extend the corresponding results of Suzuki (J. Math. Anal. Appl. 340:10881095, 2008), Eslamian and Abkar (Math. Comput. Model. 54:105111, 2011), Abbas et al. (Appl. Math. Lett. 24:97102, 2011) and others.
MSC:47J05, 47H09, 49J25.
1 Introduction
The study of fixed points for multivalued contractions and nonexpansive mappings using the Hausdorff metric was initiated by Markin [1] and Nadler [2]. Since then the metric fixed point theory of multivalued mappings has been rapidly developed. The theory of multivalued mappings has been applied to control theory, convex optimization, differential equations, and economics. Different iterative processes have been used to approximate fixed points of multivalued nonexpansive mappings [3–7]. Recently Abbas et al. [8] introduced an onestep iterative process to approximate a common fixed point of two multivalued nonexpansive mappings in uniformly convex Banach spaces.
On the other hand, in 2008 Suzuki [9] introduced a class of mappings satisfying the condition (C) which is weaker than nonexpansive mappings (sometimes, such a mapping is called a generalized nonexpansive mapping). He then proved some fixed point and convergence theorems for such mappings. Very recently, Eslamian and Abkar [10, 11] generalized it to multivalued case, and they proved some fixed point results in uniformly convex Banach spaces.
The aim of this paper is to introduce an iterative process for approximating a common fixed point of an infinite family of multivalued mappings satisfying the condition (C). Under suitable conditions some weak and strong convergence theorems for such iterative process are proved in uniformly convex Banach spaces.
2 Preliminaries
Throughout this paper, we assume that X is a real Banach space, K is a nonempty subset of X. We denote by $\mathcal{N}$ the set of all positive integers. We denote by ‘${x}_{n}\to x$’ and ‘${x}_{n}\rightharpoonup x$’ the strong and weak convergence of $\{{x}_{n}\}$, respectively.
Recall that a subset K of X is called proximinal if, for each $x\in X$, there exists an element ${k}^{\ast}\in K$ such that
Remark 2.1 It is wellknown that weakly compact convex subsets of a Banach space and closed convex subsets of a uniformly convex Banach space are proximinal.
We shall denote the family of nonempty bounded proximinal subsets of X by $P(X)$, the family of nonempty compact subsets of X by $C(X)$ and the family of nonempty bounded and closed subsets of X by $CB(X)$. Let H be the Hausdorff metric induced by the metric d of X defined by
A point $x\in K$ is called a fixed point of a multivalued mapping T, if $x\in Tx$. We denote the set of fixed point of T by $F(T)$. A multivalued mapping $T:K\to CB(X)$ is said to be

(i)
contraction, if there exists a constant $\alpha \in [0,1)$ such that for any $x,y\in K$
$$H(Tx,Ty)\le \alpha \parallel xy\parallel ;$$ 
(ii)
nonexpansive, if for all $x,y\in K$
$$H(Tx,Ty)\le \parallel xy\parallel ;$$ 
(iii)
quasinonexpansive, if $F(T)\ne \mathrm{\varnothing}$ and
$$H(Tx,Tp)\le \parallel xp\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}p\in F(T),x\in K.$$
Definition 2.2 A multivalued mapping $T:X\to CB(X)$ is said to satisfy the condition (C) provided that
Lemma 2.3 Let $T:X\to CB(X)$ be a multivalued mapping.

(1)
If T is nonexpansive, then T satisfies the condition (C).

(2)
If T satisfies the condition (C) and $F(T)\ne \mathrm{\varnothing}$, then T is a quasinonexpansive mapping.
Proof The conclusion (1) is obvious. Now we prove the conclusion (2).
In fact, for any $p\in F(T)$, we have $\frac{1}{2}d(p,Tp)=0\le \parallel xp\parallel $, $\mathrm{\forall}x\in X$. Since T satisfies the condition (C), we have
□
We mention that there exist singlevalued and multivalued mappings satisfying the condition (C) which are not nonexpansive, for example:
Example 1 [9]
Define a mapping T on $[0,3]$ by
Then T is a singlevalued mapping satisfying condition (C), but T is not nonexpansive.
Example 2 [10]
Define a mapping $T:[0,5]\to [0,5]$ by
then it is easy to prove that T is a multivalued mapping satisfying condition (C), but T is not nonexpansive.
Definition 2.4 A Banach space X is said to satisfy Opial condition, if ${x}_{n}\rightharpoonup z$ (as $n\to \mathrm{\infty}$) and $z\ne y$ imply that
Lemma 2.5 Let K be a nonempty subset of a uniformly convex Banach space X and $T:K\to CB(K)$ be a multivalued mapping with convexvalued and satisfying the condition (C), then
Proof Let $x\in K$, since Tx is a nonempty closed and convex subset of K. By Remark 2.1, it is proximal, hence there exists $z\in Tx$ such that $\parallel zx\parallel =d(x,Tx)$. Since $\frac{1}{2}d(x,Tx)\le \parallel zx\parallel $ and T satisfies the condition (C), we have
Also since Tz is proximal, there exists $u\in Tz$ such that $\parallel zu\parallel =d(z,Tz)$. This together with (2.1) shows that
Now we show that either $\frac{1}{2}d(x,Tx)\le \parallel xy\parallel $ or $\frac{1}{2}d(z,Tz)\le \parallel zy\parallel $ holds. Suppose to the contrary, we may assume that
From (2.2) we have
which is a contradiction. If $\frac{1}{2}d(x,Tx)\le \parallel xy\parallel $, then by the fact that T satisfies the condition (C), we have $H(Tx,Ty)\le \parallel xy\parallel $. In the other case, if $\frac{1}{2}d(z,Tz)\le \parallel zy\parallel $, again by the assumption that T satisfies the condition (C) we obtain $H(Tz,Ty)\le \parallel zy\parallel $. Hence, we get
This completes the proof of Lemma 2.5. □
Lemma 2.6 [12]
Let X be a uniformly convex Banach space, ${B}_{r}(0):=\{x\in X:\parallel x\parallel \le r\}$ be a closed ball with center 0 and radius $r>0$. For any given sequence $\{{x}_{1},{x}_{2},\dots ,{x}_{n},\dots \}\subset {B}_{r}(0)$ and any given number sequence $\{{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{n},\dots \}$ with ${\lambda}_{i}\ge 0$, ${\sum}_{i=1}^{\mathrm{\infty}}{\lambda}_{i}=1$, then there exists a continuous strictly increasing and convex function $g:[0,2r)\to [0,\mathrm{\infty})$ with $g(0)=0$ such that for any $i,j\in \mathcal{N}$, $i<j$ the following holds:
Lemma 2.7 Let X be a strictly convex Banach space, K be a nonempty closed and convex subset of X and $T:K\to CB(K)$ be a multivalued mapping satisfying the condition (C). If $F(T)$ is nonempty, then it is a closed and convex subset of K.
Proof Let $\{{x}_{n}\}$ be a sequence in $F(T)$ converging to some point $p\in K$, i.e., ${x}_{n}\in T({x}_{n})$, $\mathrm{\forall}n\ge 1$ and ${x}_{n}\to p\in K$. Since T satisfies the condition (C) and
we have
This implies that $d(p,Tp)=0$. Since Tp is closed, we have $p\in Tp$, i.e., $p\in F(T)$ and so $F(T)$ is closed.
Next we prove that $F(T)$ is a convex subset in K. In fact, for any given $\lambda \in (0,1)$, $x,y\in F(T)$ with $x\ne y$ and put $w=\lambda x+(1\lambda )y$. Since T satisfies the condition (C), we have
This implies that
Since X is strictly convex, this implies that there exist $\mu \in (0,1)$ and a point $u\in Tw$ such that $u=\mu x+(1\mu )y$. Since
and
from (2.4), we have $1\mu \le (1\lambda )$ and from (2.5), we have $\mu \le \lambda $. These implies that $\mu =\lambda $. Therefore $u=w$ and $w\in Tw$, i.e., $w\in F(T)$. This completes the proof of Lemma 2.7. □
Lemma 2.8 (Demiclosed principle)
Let X be a uniformly convex Banach space satisfying the Opial condition, K be a nonempty closed and convex subset of X. Let $T:K\to CB(K)$ be a multivalued mapping with convexvalues and satisfying the condition (C). Let $\{{x}_{n}\}$ be a sequence in K such that ${x}_{n}\rightharpoonup p\in K$, and let ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$, then $p\in Tp$, i.e., $IT$ is demiclosed at zero.
Proof By the assumption that $T:K\to CB(K)$ is a multivalued mapping with convexvalues, hence Tp is a nonempty closed and convex subset of K. By Remark 2.1, it is proximal. Therefore for each ${x}_{n}$, $n\ge 1$, there exists a point ${u}_{n}\in Tp$ such that
On the other hand, it follows from Lemma 2.5 that
Taking the superior limit on the both sides of the above inequality, we have
By virtue of the Opial condition, we have ${u}_{n}=p$, $\mathrm{\forall}n\ge 1$. And so $p\in Tp$.
This completes the proof of Lemma 2.8. □
3 Weak convergence theorems
We are now in a position to give the following theorem.
Theorem 3.1 Let X be a real uniformly convex Banach space with Opial condition and K be a nonempty closed and convex subset of X. Let ${T}_{i}:K\to CB(K)$, $i=1,2,\dots $ be an infinite family of multivalued mappings with nonempty convexvalues and satisfying the condition (C). For given ${x}_{1}\in K$, let $\{{x}_{n}\}$ be the sequence in K defined by
where $\{{\alpha}_{i,n}\}\subset (0,1)$. If the following conditions are satisfied:

(i)
${\sum}_{i=0}^{\mathrm{\infty}}{\alpha}_{i,n}=1$, for each $n\ge 1$;

(ii)
for each $i\ge 1$, $lim{inf}_{n\to \mathrm{\infty}}{\alpha}_{0,n}{\alpha}_{i,n}>0$;

(iii)
$\mathcal{F}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\varnothing}$ and ${T}_{i}p=\{p\}$, $\mathrm{\forall}i\ge 1$ and $p\in \mathcal{F}$,
then the sequence $\{{x}_{n}\}$ converges weakly to some point ${p}^{\ast}\in \mathcal{F}$.
Proof (I) First we claim that
In fact, since ${\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\varnothing}$, it follows from Lemma 2.3(2) that for each $i\ge 1$, ${T}_{i}$ is a multivalued quasinonexpansive mapping. Hence for each $p\in \mathcal{F}$, by condition (iii) we have
and
This shows that $\{\parallel {x}_{n}p\parallel \}$ is decreasing and bounded below. Hence the limit ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel $ exists for each $p\in \mathcal{F}$. And so $\{\parallel {x}_{n}p\parallel \}$ and $\{\parallel {w}_{i,n}p\parallel \}$ both are bounded.
(II) Next we prove that
Since $\{\parallel {x}_{n}p\parallel \}$ and $\{\parallel {w}_{i,n}p\parallel \}$ both are bounded, from Lemma 2.6 and (3.3), for each $l\ge 1$ we have
And so
By condition (ii) we have
Since g is continuous and strictly increasing with $g(0)=0$, this implies that
Thus, we have

(III)
Finally we prove that ${x}_{n}\rightharpoonup {p}^{\ast}$ (some point in ℱ).
In fact, since $\{{x}_{n}\}$ is bounded, there exists a subsequence ${x}_{{n}_{i}}\subset \{{x}_{n}\}$ such that ${x}_{{n}_{i}}\rightharpoonup {p}^{\ast}\in K$. By Lemma 2.8, $I{T}_{l}$ is demiclosed at zero. Hence from (3.8), ${p}^{\ast}\in F({T}_{l})$. By the arbitrariness of $l\ge 1$, we have ${p}^{\ast}\in \mathcal{F}$.
If there exists another subsequence $\{{x}_{{n}_{j}}\}\subset \{{x}_{n}\}$ such that ${x}_{{n}_{j}}\rightharpoonup {q}^{\ast}\in K$ and ${p}^{\ast}\ne {q}^{\ast}$. By the same method as given above we can also prove that ${q}^{\ast}\in \mathcal{F}$. Since X has the Opial property, we have
This is a contradiction. Therefore ${p}^{\ast}={q}^{\ast}$ and ${x}_{n}\rightharpoonup {p}^{\ast}\in \mathcal{F}$.
This completes the proof of Theorem 3.1. □
The following theorem can be obtained from Theorem 3.1 immediately.
Theorem 3.2 Let X be a real uniformly convex Banach space with Opial condition and K be a nonempty closed and convex subset of X. Let ${T}_{i}:K\to K$, $i=1,2,\dots $ be an infinite family of singlevalued mappings satisfying the condition (C). For given ${x}_{1}\in K$, let $\{{x}_{n}\}$ be the sequence in K defined by
where $\{{\alpha}_{i,n}\}\subset (0,1)$ is the sequence as given in Theorem 3.1. If $\mathcal{F}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\varnothing}$, then the sequence $\{{x}_{n}\}$ converges weakly to some point ${p}^{\ast}\in \mathcal{F}$.
Remark 3.3 Theorem 3.1 improves and extends the corresponding results in Eslamian and Abkar [[11], Theorem 3.3], Suzuki [[9], Theorem 3] and Abbas et al. [[8], Theorem 2].
4 Some strong convergence theorems
Theorem 4.1 Let X be a real uniformly convex Banach space and K be a nonempty closed and convex subset of X. Let ${T}_{i}:K\to CB(K)$, $i=1,2,\dots $ be an infinite family of multivalued mappings satisfying the condition (C). For given ${x}_{1}\in K$, let $\{{x}_{n}\}$ be the sequence defined by (3.1). If the conditions (i), (ii), and (iii) in Theorem 3.1 are satisfied, then $\{{x}_{n}\}$ converges strongly to some point ${p}^{\ast}\in \mathcal{F}$, if and only if the following condition is satisfied:
Proof The necessity of condition (4.1) is obvious.
Next we prove the sufficiency of condition (4.1).
In fact, as in the proof of Theorem 3.1, for each $i\ge 1$, we have $limd({x}_{n},{T}_{i}{x}_{n})=0$ (see (3.8)), and for each $p\in \mathcal{F}$ the limit ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel $ exists. Hence by condition (4.1) we have
Therefore we can choose a subsequence $\{{x}_{{n}_{k}}\}\subset \{{x}_{n}\}$ and a subsequence $\{{p}_{k}\}\subset \mathcal{F}$ such that for all positive integer $k\ge 1$
Since the sequence $\{\parallel {x}_{n}p\parallel \}$, $p\in \mathcal{F}$ is decreasing, we obtain
Hence
This implies that $\{{p}_{k}\}$ is a Cauchy sequence in K. Without loss of generality, we can assume that ${p}_{k}\to {p}^{\ast}\in K$. Since for each $i\ge 1$
This implies that ${p}^{\ast}\in {T}_{i}{p}^{\ast}$, for all $i\ge 1$. Therefore ${p}^{\ast}\in \mathcal{F}$ and ${x}_{n}\to {p}^{\ast}$.
This completes the proof of Theorem 4.1. □
The following theorem can be obtained from Theorem 4.1 immediately.
Theorem 4.2 Let X be a real uniformly convex Banach space and K be a nonempty closed and convex subset of X. Let ${T}_{i}:K\to CB(K)$, $i=1,2,\dots $ be an infinite family of multivalued mappings satisfying the condition (C). For given ${x}_{1}\in K$, let $\{{x}_{n}\}$ be the sequence defined by (3.1). If the conditions (i), (ii), and (iii) in Theorem 3.1 and the following condition (iv) are satisfied:

(iv)
there exists an increasing function $f:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with $f(r)>0$, $\mathrm{\forall}r>0$ such that for some $m\ge 1$
$$d({x}_{n},{T}_{m}({x}_{n}))\ge f(d({x}_{n},\mathcal{F})),$$(4.3)
then the sequence $\{{x}_{n}\}$ converges strongly to some point ${p}^{\ast}\in \mathcal{F}$.
Proof As in the proof of Theorem 3.1, for each $i\ge 1$, ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{T}_{i}{x}_{n})=0$. Especially we have ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{T}_{m}{x}_{n})=0$. Hence from (4.3) we obtain ${lim}_{n\to \mathrm{\infty}}d({x}_{n},\mathcal{F})=0$. The conclusion of Theorem 4.2 can be obtained from Theorem 4.1 immediately.
We now intend to remove the condition that ${T}_{i}(p)=\{p\}$ for each $p\in \mathcal{F}$ and each $i\ge 1$.
Let X be a real uniformly convex Banach space and K be a nonempty closed and convex subset of X. Let ${T}_{i}:K\to CB(K)$, $i=1,2,\dots $ be an infinite family of multivalued mappings with convexvalues. Then for each $i\ge 1$ and for each $x\in K$, ${T}_{i}x$ is a nonempty closed and convex subset in K. Hence by Remark 2.1, it is proximinal. Now we define a multivalued mapping ${P}_{{T}_{i}}:K\to CB(K)$ by
For any given ${x}_{1}\in K$ define a sequence $\{{x}_{n}\}$ by
where $\{{\alpha}_{i,n}\}\subset (0,1)$. □
We have the following.
Theorem 4.3 Let X, K, $\{{T}_{i}\}$, $\{{P}_{{T}_{i}}\}$ and $\{{x}_{n}\}$ be the same as above. If the following conditions are satisfied:

(i)
${\sum}_{i=0}^{\mathrm{\infty}}{\alpha}_{i,n}=1$, for each $n\ge 1$;

(ii)
for each $i\ge 1$, $lim{inf}_{n\to \mathrm{\infty}}{\alpha}_{0,n}{\alpha}_{i,n}>0$;

(iii)
$\mathcal{F}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\varnothing}$, and, for each $i\ge 1$, the mapping ${P}_{{T}_{i}}:K\to CB(K)$ satisfies the condition (C);

(iv)
there exists an increasing function $f:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with $f(r)>0$ for all $r>0$ such that for some $m\ge 1$
$$d({x}_{n},{T}_{m}({x}_{n}))\ge f(d({x}_{n},\mathcal{F})),$$
then the sequence $\{{x}_{n}\}$ converges strongly to some point ${p}^{\ast}\in \mathcal{F}$.
Proof Let $p\in \mathcal{F}$, then we have
Moreover, by the same method as give in the proof of Theorem 3.1 we can also prove that the limit ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel $ exists for each $p\in \mathcal{F}$ and
Therefore, we can choose a subsequence $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$ and a sequence $\{{p}_{k}\}$ in ℱ such that for any positive integer k
As in the proof of Theorem 3.1, $\{{p}_{k}\}$ is a Cauchy sequence in K and hence it converges to some point $q\in K$. By virtue of the definition of mapping ${P}_{{T}_{i}}$, we have ${P}_{{T}_{i}}(q)\subset {T}_{i}(q)$, $i\ge 1$. Hence from (4.6) we have
Since ${p}_{k}\to q$ (as $k\to \mathrm{\infty}$), it follows that $d(q,{T}_{i}(q))=0$ for $i\ge 1$. Hence $q\in \mathcal{F}$ and $\{{x}_{{n}_{k}}\}$ converges strongly to q. Since ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}q\parallel $ exists, we conclude that $\{{x}_{n}\}$ converges strongly to q.
This completes the proof of Theorem 4.3. □
Remark 4.4 Theorems 4.1, 4.2 and 4.3 improve and extend the corresponding results in Eslamian et al. [[11], Theorems 3.1, 3.2, 3.4], Suzuki [[9], Theorem 2] and Abbas et al. [[8], Theorems 1, 3, 4].
5 Applications
The convex feasibility problem (CFP) was first introduced by Censor and Elfving [13] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [14]. Recently, it has been found that the CFP can also be used in various disciplines such as image restoration, computer tomograph and radiation therapy treatment planning.
Let X be a real Banach space, K be a nonempty closed and convex subset of X and $\{{K}_{i}\}$ be a countable family of subset of K. The ‘so called’ convex feasibility problem for the family of subsets $\{{K}_{i}\}$ is to find a point ${x}^{\ast}\in {\bigcap}_{i=1}^{\mathrm{\infty}}{K}_{i}$.
In this section, we shall utilize Theorem 3.2 to study the convex feasibility problem for an infinite family of singlevalued mappings satisfying the condition (C). We have the following result.
Theorem 5.1 Let X be a real uniformly convex Banach space with Opial condition and K be a nonempty closed and convex subset of X. Let ${T}_{i}:K\to K$, $i=1,2,\dots $ be an infinite family of mappings satisfying the condition (C). Let $\{{K}_{i}=F({T}_{i}),i=1,2,\dots \}$. For given ${x}_{1}\in K$, let $\{{x}_{n}\}$ be the sequence in K defined by
where $\{{\alpha}_{i,n}\}\subset (0,1)$ is the sequence as given in Theorem 3.2. If $\mathcal{F}:={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\varnothing}$, then there exists a point ${x}^{\ast}\in \mathcal{F}$ which is a solution of the convex feasibility problem for the family of subsets $\{{K}_{i}\}$, and the sequence $\{{x}_{n}\}$ defined by (5.1) converges weakly to ${x}^{\ast}$.
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The authors would like to express their thanks to the Reviewers and the Editors for their helpful suggestions and advices. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).
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Chang, S., Tang, Y., Wang, L. et al. Convergence theorems for some multivalued generalized nonexpansive mappings. Fixed Point Theory Appl 2014, 33 (2014). https://doi.org/10.1186/16871812201433
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Keywords
 multivalued mapping satisfying condition (C)
 multivalued generalized nonexpansive mapping
 weak and strong convergence theorem