# 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 mappingsmulti-valued mappingsHilbert 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

## References

- Brouwer LEJ: Über Abbildung von Mannigfaltigkeiten.
*Math. Ann.*1912, 71(4):598.MATHGoogle Scholar - Chang KC,
*et al*.: Convergence theorems for some multi-valued generalized nonexpansive mappings.*Fixed Point Theory Appl.*2014., 2014: Article ID 33Google Scholar - Chidume CE,
*et al*.: Krasnoselskii-type algorithm for fixed points of multivalued strictly pseudo-contractive mappings.*Fixed Point Theory Appl.*2013., 2013: Article ID 58Google Scholar - Denavari T, Frigon M: Fixed point results for multivalued contractions on a metric space with a graph.
*J. Math. Anal. Appl.*2013, 405: 507–517. 10.1016/j.jmaa.2013.04.014View ArticleMathSciNetGoogle Scholar - Yingtaweesittikul H: Suzuki type fixed point theorems for generalized multi-valued mappings in
*b*-metric spaces.*Fixed Point Theory Appl.*2013., 2013: Article ID 215Google Scholar - Kakutani S: A generalization of Brouwer’s fixed point theorem.
*Duke Math. J.*1941, 8(3):457–459. 10.1215/S0012-7094-41-00838-4View ArticleMathSciNetGoogle Scholar - Nash JF: Non-cooperative games.
*Ann. Math. (2)*1951, 54: 286–295. 10.2307/1969529View ArticleMathSciNetMATHGoogle Scholar - Nash JF: Equilibrium points in
*n*-person games.*Proc. Natl. Acad. Sci. USA*1950, 36(1):48–49. 10.1073/pnas.36.1.48View ArticleMathSciNetMATHGoogle Scholar - Geanakoplos J: Nash and Walras equilibrium via Brouwer.
*Econ. Theory*2003, 21: 585–603. 10.1007/s001990000076View ArticleMathSciNetMATHGoogle Scholar - Nadler SB Jr: Multivalued contraction mappings.
*Pac. J. Math.*1969, 30: 475–488. 10.2140/pjm.1969.30.475View ArticleMathSciNetGoogle Scholar - Downing D, Kirk WA: Fixed point theorems for set-valued mappings in metric and Banach spaces.
*Math. Jpn.*1977, 22(1):99–112.MathSciNetMATHGoogle Scholar - Chidume CE, Chidume CO, Djitte N, Minjibir MS: Convergence theorems for fixed points of multivalued strictly pseudo-contractive mappings in Hilbert spaces.
*Abstr. Appl. Anal.*2013., 2013: Article ID 629468Google Scholar - Chang KC: The obstacle problem and partial differential equations with discontinuous nonlinearities.
*Commun. Pure Appl. Math.*1980, 33: 117–146. 10.1002/cpa.3160330203View ArticleMATHGoogle Scholar - Erbe L, Krawcewicz W: Existence of solutions to boundary value problems for impulsive second order differential inclusions.
*Rocky Mt. J. Math.*1992, 22: 1–20. 10.1216/rmjm/1181072792View ArticleMathSciNetGoogle Scholar - Babu GVR, Sastry KPR: 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-zView ArticleMathSciNetMATHGoogle Scholar - Panyanak B: Mann and Ishikawa iteration processes for multi-valued mappings in Banach spaces.
*Comput. Math. Appl.*2007, 54: 872–877. 10.1016/j.camwa.2007.03.012View ArticleMathSciNetMATHGoogle Scholar - Song Y, Wang H: Erratum to “Mann and Ishikawa iterative processes for multi-valued mappings in Banach spaces”.
*Comput. Math. Appl.*2007, 54: 872–877. 10.1016/j.camwa.2007.03.012View ArticleMathSciNetGoogle Scholar - Abbas M, Khan SH, Khan AR, Agarwal RP: Common fixed points of two multi-valued nonexpansive mappings by one-step iterative scheme.
*Appl. Math. Lett.*2011, 24: 97–102. 10.1016/j.aml.2010.08.025View ArticleMathSciNetMATHGoogle Scholar - Yildirim I, Khan SH: Fixed points of multivalued nonexpansive mappings in Banach spaces.
*Fixed Point Theory Appl.*2012., 2012: Article ID 73 10.1186/1687-1812-2012-73Google Scholar - Khan SH, Yildirim I, Rhoades BE: A one-step iterative scheme for two multi-valued nonexpansive mappings in Banach spaces.
*Comput. Math. Appl.*2011, 61: 3172–3178. 10.1016/j.camwa.2011.04.011View ArticleMathSciNetMATHGoogle Scholar - Daffer PZ, Kaneko H: Fixed points of generalized contractive multi-valued mappings.
*J. Math. Anal. Appl.*1995, 192: 655–666. 10.1006/jmaa.1995.1194View ArticleMathSciNetMATHGoogle Scholar - Garcia-Falset J, Lorens-Fuster E, Suzuki T: Fixed point theory for a class of generalised nonexpansive mappings.
*J. Math. Anal. Appl.*2011, 375: 185–195. 10.1016/j.jmaa.2010.08.069View ArticleMathSciNetMATHGoogle Scholar - Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces.
*J. Math. Anal. Appl.*1967, 20: 197–228. 10.1016/0022-247X(67)90085-6View ArticleMathSciNetMATHGoogle Scholar - Krasnosel’skiĭ MA: Two observations about the method of successive approximations.
*Usp. Mat. Nauk*1955, 10: 123–127.Google Scholar - Mann WR: Mean value methods in iterations.
*Proc. Am. Math. Soc.*1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3View ArticleMATHGoogle Scholar - Ishikawa S: Fixed points by a new iteration method.
*Proc. Am. Math. Soc.*1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5View ArticleMathSciNetMATHGoogle Scholar - Khastan A,
*et al*.: Schauder fixed point theorem in semilinear spaces and its application to fractional differential equations with uncertainty.*Fixed Point Theory Appl.*2014., 2014: Article ID 21Google Scholar - Eslamian M: Hybrid method for equilibrium problems and fixed point problems of finite families of nonexpansive semigroups.
*Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat.*2003, 107(2):299–307.View ArticleMathSciNetGoogle Scholar - 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.2008.10.112View ArticleMathSciNetMATHGoogle Scholar

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