# Krasnoselskii-type algorithm for family of multi-valued strictly pseudo-contractive mappings

- CE Chidume
^{1}Email author and - JN Ezeora
^{1, 2}

**2014**:111

https://doi.org/10.1186/1687-1812-2014-111

© Chidume and Ezeora; licensee Springer. 2014

**Received: **24 November 2013

**Accepted: **31 March 2014

**Published: **7 May 2014

## Abstract

A Krasnoselskii-type algorithm is constructed and the sequence of the algorithm is proved to be an approximate fixed point sequence for a common fixed point of a suitable finite family of multi-valued strictly pseudo-contractive mappings in a real Hilbert space. Under some mild additional compactness-type condition on the operators, the sequence is proved to converge strongly to a common fixed point of the family.

**MSC:**47H04, 47H06, 47H15, 47H17, 47J25.

### Keywords

*k*-strictly pseudo-contractive mappings multi-valued mappings Hilbert spaces

## 1 Introduction

For several years, the study of fixed point theory for *multi-valued nonlinear mappings* has attracted, and continues to attract, the interest of several well known mathematicians (see, for example, Brouwer [1], Chang [2], Chidume *et al.* [3], Denavari and Frigon [4], Yingtaweesittikul [5], Kakutani [6], Nash [7, 8], Geanakoplos [9], Nadler [10], Downing and Kirk [11]).

Interest in such studies stems, perhaps, mainly from the usefulness of such fixed point theory in real-world applications, such as in *Game Theory* and *Market Economy* and in other areas of mathematics, such as in *Non-Smooth Differential Equations* (see *e.g.*, [12]).

*Game theory* is perhaps the most successful area of application of fixed point theory for multi-valued mappings. However, it has been remarked that the applications of this theory to *equilibrium problems* in game theory are mostly static in the sense that while they enhance the understanding of conditions under which equilibrium may be achieved, they do not indicate how to construct a process starting from a non-equilibrium point that will converge to an equilibrium solution. Iterative methods for fixed points of multi-valued mappings are designed to address this problem. For more details, one may consult [12–14].

*K*be a nonempty subset of a normed space

*E*. The set

*K*is called

*proximinal*(see

*e.g.*, [15–17]) if for each $x\in E$, there exists $u\in K$ such that

where $d(x,y)=\parallel x-y\parallel $ for all $x,y\in E$. Every nonempty, closed and convex subset of a real Hilbert space is proximinal.

*K*, respectively. The

*Hausdorff metric*on $CB(K)$ is defined by

Let $T:D(T)\subseteq E\to CB(E)$ be a *multi-valued mapping* on *E*. A point $x\in D(T)$ is called a *fixed point of* *T* if $x\in Tx$. The fixed point set of *T* is denoted by $F(T):=\{x\in D(T):x\in Tx\}$.

*L*-

*Lipschitzian*if there exists $L>0$ such that

When $L\in (0,1)$ in (1.1), we say that *T* is a *contraction*, and *T* is called *nonexpansive* if $L=1$.

Several papers deal with the problem of approximating fixed points of *multi-valued nonexpansive* mappings (see, for example [15–20] and the references therein) and their generalizations (see *e.g.*, [21, 22]).

*et al.*[18], introduced a one-step iterative process as follows, ${x}_{1}\in K$:

Using (1.2), Abbas *et al.* proved weak and strong convergence theorems for approximation of common fixed point of *two multi-valued nonexpansive mappings* in real Banach spaces.

Very recently, Chidume *et al.* [12], introduced the class of *multi-valued* *k-strictly pseudo-contractive* maps defined on a real Hilbert space *H* as follows.

**Definition 1.1**A multi-valued map $T:D(T)\subset H\to CB(H)$ is called

*k-strictly pseudo-contractive*if there exists $k\in (0,1)$ such that for all $x,y\in D(T)$,

In the case that *T* is single-valued, definition 1.1 reduces to the definition introduced and studied by Browder and Petryshn [23] as an important generalization of the class of nonexpansive mappings. Chidume *et al.* [12], proved strong convergence theorems for approximating fixed points of this class of mappings using a *Krasnoselskii-type algorithm*, [24] which is well known to be superior to the recursion formula of Mann [25] or Ishikawa [26].

In this paper, motivated by the results of Chidume *et al.* [12], Abbas *et al.* [18], Khastan [27], Eslamian [28], Shahzad and Zegeye [29] and Song and Wong [17], we introduce a new *Krasnoselskii-type* algorithm and prove strong convergence theorems for the sequence of the algorithm for approximating a common fixed point of a *finite family of multi-valued strictly pseudo-contractive mappings in a real Hilbert space.* Our results, under the setting of our theorems, generalize those of Abbas *et al.* [18] and Chidume *et al.* [12], which are themselves generalizations of many important results, from *two multi-valued nonexpansive mappings*, and a single *multi-valued strictly pseudo-contractive mapping*, respectively, to *a finite family* of *multi-valued strictly pseudo-contractive mappings*.

## 2 Main result

In the sequel, we shall need the following lemma whose proof can be found in Eslamian [28]. We reproduce the proof here for completeness.

**Lemma 2.1**

*Let*

*H*

*be a real Hilbert space*.

*Let*$\{{x}_{i},i=1,\dots ,m\}\subset H$.

*For*${\alpha}_{i}\in (0,1)$, $i=1,\dots ,m$

*such that*${\sum}_{i=1}^{m}{\alpha}_{i}=1$,

*the following identity holds*:

*Proof*The proof is by induction. For $m=2$, (2.1) reduces to the standard identity in Hilbert spaces. Assume that (2.1) is true for some $k\ge 2$, that is,

Therefore, by induction, we find that (2.1) is true. This completes the proof. □

We now prove the following theorem.

**Theorem 2.2**

*Let*

*K*

*be a nonempty*,

*closed and convex subset of a real Hilbert space*

*H*

*and*${T}_{i}:K\to CB(K)$

*be a finite family of multi*-

*valued*${k}_{i}$-

*strictly pseudo*-

*contractive mappings*, ${k}_{i}\in (0,1)$, $i=1,\dots ,m$

*such that*${\bigcap}_{i=1}^{m}F({T}_{i})\ne \mathrm{\varnothing}$.

*Assume that for*$p\in {\bigcap}_{i=1}^{m}F({T}_{i})$, ${T}_{i}p=\{p\}$.

*Let*$\{{x}_{n}\}$

*be a sequence defined by*${x}_{0}\in K$,

*where* ${y}_{n}^{i}\in {T}_{i}{x}_{n}$, $n\ge 1$ *and* ${\lambda}_{i}\in (k,1)$, $i=0,1,\dots ,m$, *such that* ${\sum}_{i=0}^{m}{\lambda}_{i}=1$ *and* $k:=max\{{k}_{i},i=1,\dots ,m\}$. *Then* ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{T}_{i}{x}_{n})=0$ $\mathrm{\forall}i=0,\dots ,m$.

*Proof*Let $p\in {\bigcap}_{i=1}^{m}F({T}_{i})$. Then

where ${z}_{0}=({x}_{n}-p)$, ${z}_{i}=({y}_{n}^{i}-p)$, $i=1,\dots ,m$, and ${\sum}_{i=1}^{m}{\lambda}_{i}{z}_{i}={\sum}_{i=1}^{m}{\lambda}_{i}({y}_{n}^{i}-p)$.

This completes the proof. □

**Definition 2.3** A mapping $T:K\to CB(K)$ is called *hemicompact* if, for any sequence $\{{x}_{n}\}$ in *K* such that $d({x}_{n},T{x}_{n})\to 0$ as $n\to \mathrm{\infty}$, there exists a subsequence $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$ such that ${x}_{{n}_{k}}\to p\in K$. We note that if *K* is compact, then every multi-valued mapping $T:K\to CB(K)$ is hemicompact.

**Theorem 2.4**

*Let*

*K*

*be a nonempty*,

*closed and convex subset of a real Hilbert space*

*H*

*and*${T}_{i}:K\to CB(K)$

*be a finite family of multi*-

*valued*${k}_{i}$-

*strictly pseudo*-

*contractive mappings*, ${k}_{i}\in (0,1)$, $i=1,\dots ,m$

*such that*${\bigcap}_{i=1}^{m}F({T}_{i})\ne \mathrm{\varnothing}$.

*Assume that for*$p\in {\bigcap}_{i=1}^{m}F({T}_{i})$, ${T}_{i}p=\{p\}$

*and*${T}_{i}$, $i=1,\dots ,m$

*is hemicompact and continuous*.

*Let*$\{{x}_{n}\}$

*be a sequence defined by*${x}_{0}\in K$,

*where* ${y}_{n}^{i}\in {T}_{i}{x}_{n}$, $n\ge 1$ *and* ${\lambda}_{i}\in (k,1)$, $i=0,1,\dots ,m$ *such that* ${\sum}_{0=1}^{m}{\lambda}_{i}=1$ *with* $k:=max\{{k}_{i},i=1,\dots ,m\}$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to an element of* ${\bigcap}_{i=1}^{m}F({T}_{i})$.

*Proof* From Theorem 2.2, we have ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{T}_{i}{x}_{n})=0$, $i=1,\dots ,m$. Since ${T}_{i}$, $i=1,\dots ,m$, is hemicompact, there exists a subsequence $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$ such that ${x}_{{n}_{k}}\to q$ as $k\to \mathrm{\infty}$ for some $q\in K$. Moreover, by continuity of ${T}_{i}$, $i=1,\dots ,m$, we also have $d({x}_{{n}_{k}},{T}_{i}{x}_{{n}_{k}})\to d(q,{T}_{i}q)$, $i=1,\dots ,m$ as $k\to \mathrm{\infty}$. Therefore, $d(q,{T}_{i}q)=0$, $i=1,\dots ,m$ and so $q\in F({T}_{i})$. Setting $p=q$ in the proof of Theorem 2.2, it follows from inequality (2.6) that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-q\parallel $ exists. So, $\{{x}_{n}\}$ converges strongly to $q\in {\bigcap}_{i=1}^{m}F({T}_{i})$. This completes the proof. □

The following is an immediate corollary of Theorem 2.4. The proof basically follows as the proof of its analog for single multi-valued strictly pseudo-contractive map in Chidume *et al.* [12]. The proof is therefore omitted.

**Corollary 2.5**

*Let*

*K*

*be a nonempty*,

*compact and convex subset of a real Hilbert space*

*H*

*and*${T}_{i}:K\to CB(K)$

*be a finite family of multi*-

*valued*${k}_{i}$-

*strictly pseudo*-

*contractive mappings*, ${k}_{i}\in (0,1)$, $i=1,\dots ,m$

*such that*${\bigcap}_{i=1}^{m}F({T}_{i})\ne \mathrm{\varnothing}$.

*Assume that for*$p\in {\bigcap}_{i=1}^{m}F({T}_{i})$, ${T}_{i}p=\{p\}$

*and*${T}_{i}$, $i=1,\dots ,m$

*is continuous*.

*Let*$\{{x}_{n}\}$

*be a sequence defined by*${x}_{0}\in K$,

*where* ${y}_{n}^{i}\in {T}_{i}{x}_{n}$, $n\ge 1$ *and* ${\lambda}_{i}\in (k,1)$, $i=0,1,\dots ,m$ *such that* ${\sum}_{0=1}^{m}{\lambda}_{i}=1$ *with* $k:=max\{{k}_{i},i=1,\dots ,m\}$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to an element of* ${\bigcap}_{i=1}^{m}F({T}_{i})$.

**Remark 2.6** In Theorem 2.4, the continuity assumption on ${T}_{i}$, $i=1,\dots ,m$ can be dispensed with if we assume that for every $x\in K$, ${T}_{i}x$, $i=1,\dots ,m$ is proximinal and weakly closed.

**Remark 2.7** If we set $i=1$ in all the results obtained in this paper, we recover the results of Chidume *et al.* [12].

**Remark 2.8** The recursion formulas studied in this paper are of the Krasnoselkii type (see *e.g.* [24]) which is well known to be superior to the recursion formula of either the Mann algorithm or the *so-called* Ishikawa-type algorithm.

**Remark 2.9** Our theorems and corollary improve the results of Chidume *et al.* [12] from *single* multi-valued strictly pseudo-contractive mapping to *finite family* of multi-valued strictly pseudo-contractive mappings. Furthermore, under the setting of Hilbert space, our theorems and corollary improve the convergence theorems for multi-valued *nonexpansive mappings* to the more general class of multi-valued *strictly pseudo-contractive mappings* studied in Sastry and Babu [15], Panyanak [16], Song and Wang [17], Shahzad and Zegeye [29] and Abbas *et al.* [18]. Also, in all our algorithms, ${y}_{n}\in T{x}_{n}$ is arbitrary and is not required to satisfy the very restrictive condition, ‘${y}_{n}\in T({x}_{n})$ such that $\parallel {y}_{n}-{x}^{\ast}\parallel =d({x}^{\ast},T{x}_{n})$’ imposed in [15–18, 20, 29].

For examples of multi-valued maps such that, for each $x\in K$, the set *Tx* is proximinal and weakly closed, the reader may consult Chidume *et al.* [12].

## Declarations

## Authors’ Affiliations

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