# The Theory of Reich's Fixed Point Theorem for Multivalued Operators

- Tania Lazăr
^{1}, - Ghiocel Moţ
^{2}Email author, - Gabriela Petruşel
^{3}and - Silviu Szentesi
^{4}

**2010**:178421

**DOI: **10.1155/2010/178421

© Tania Lazăr et al. 2010

**Received: **12 April 2010

**Accepted: **18 July 2010

**Published: **4 August 2010

## Abstract

The purpose of this paper is to present a theory of Reich's fixed point theorem for multivalued operators in terms of fixed points, strict fixed points, multivalued weakly Picard operators, multivalued Picard operators, data dependence of the fixed point set, sequence of multivalued operators and fixed points, Ulam-Hyers stability of a multivalued fixed point equation, well-posedness of the fixed point problem, and the generated fractal operator.

## 1. Introduction

is the (generalized) Pompeiu-Hausdorff functional.

It is well known that if is a complete metric space, then the pair is a complete generalized metric space. (See [1, 2]).

Definition 1.1.

Reich proved that any Reich-type multivalued -contraction on a complete metric space has at least one fixed point (see [3]).

In a recent paper Petruşel and Rus introduced the concept of "theory of a metric fixed point theorem" and used this theory for the case of multivalued contraction (see [4]). For the singlevalued case, see [5].

The purpose of this paper is to extend this approach to the case of Reich-type multivalued -contraction. We will discuss Reich's fixed point theorem in terms of

(i)fixed points and strict fixed points,

(ii)multivalued weakly Picard operators,

(iii)multivalued Picard operators,

(iv)data dependence of the fixed point set,

(v)sequence of multivalued operators and fixed points,

(vi)Ulam-Hyers stability of a multivaled fixed point equation,

(vii)well-posedness of the fixed point problem;

(viii)fractal operators.

Notice also that the theory of fixed points and strict fixed points for multivalued operators is closely related to some important models in mathematical economics, such as optimal preferences, game theory, and equilibrium of an abstract economy. See [6] for a nice survey.

## 2. Notations and Basic Concepts

Throughout this paper, the standard notations and terminologies in nonlinear analysis are used (see the papers by Kirk and Sims [7], Granas and Dugundji [8], Hu and Papageorgiou [2], Rus et al. [9], Petruşel [10], and Rus [11]).

is called the fractal operator generated by . For a well-written introduction on the theory of fractals see the papers of Barnsley [12], Hutchinson [13], Yamaguti et al. [14].

It is known that if is a metric space and , then the following statements hold:

(a)if is upper semicontinuous, then , for every ;

(b)the continuity of implies the continuity of .

A sequence of successive approximations of starting from is a sequence of elements in with , for .

we denote the graph of the multivalued operator .

If , then denote the iterate operators of .

Definition 2.1 (see [15]).

Let be a metric space. Then, is called a multivalued weakly Picard operator (briefly MWP operator) if for each and each there exists a sequence in such that

(iii)the sequence is convergent and its limit is a fixed point of .

For the following concepts see the papers by Rus et al. [15], Petruşel [10], Petruşel and Rus [16], and Rus et al. [9].

Definition 2.2.

Let be a metric space, and let be an MWP operator. The multivalued operator is defined by the formula there exists a sequence of successive approximations of starting from that converges to .

Definition 2.3.

Let be a metric space and an MWP operator. Then is said to be a -multivalued weakly Picard operator (briefly -MWP operator) if and only if there exists a selection of such that for all .

We recall now the notion of multivalued Picard operator.

Definition 2.4.

Let be a metric space and . By definition, is called a multivalued Picard operator (briefly MP operator) if and only if

In [10] other results on MWP operators are presented. For related concepts and results see, for example, [1, 17–23].

## 3. A Theory of Reich's Fixed Point Principle

We recall the fixed point theorem for a single-valued Reich-type operator, which is needed for the proof of our first main result.

Theorem 3.1 (see [3]).

Then is a Picard operator, that is, we have:

(ii)for each the sequence converges in to

Our main result concerning Reich's fixed point theorem is the following.

Theorem 3.2.

Let be a complete metric space, and let be a Reich-type multivalued -contraction. Let . Then one has the following

(ii) is a -multivalued weakly Picard operator;

(iii)let be a Reich-type multivalued -contraction and such that for each , then

(iv)let ( ) be a sequence of Reich-type multivalued -contraction, such that uniformly as . Then, as .

If, moreover for each , then one additionally has:

(vi) , is a set-to-set -contraction and (thus) ;

- (i)

If we choose , then by (3.4) we get that the sequence is Cauchy and hence convergent in to some

- (iv)
follows immediately from (iii).

- (v)

Hence, is a Reich-type single-valued -contraction on the complete metric space . From Theorem 3.1 we obtain that

(viii)-(ix) Let be an arbitrary. Then Hence , for each . Moreover, . From (vii), we immediately get that . Hence . The proof is complete.

A second result for Reich-type multivalued -contractions formulates as follows.

Theorem 3.3.

A third result for the case of -contraction is the following.

Theorem 3.4.

Let be a complete metric space, and let be a Reich-type multivalued -contraction such that . Then one has

(xv)If is a sequence such that as and is -continuous, then as .

- (xiii)
- (xiv)
- (xv)

For compact metric spaces we have the following result.

Theorem 3.5.

- (xvi)

Remark 3.6.

For we obtain the results given in [4]. On the other hand, our results unify and generalize some results given in [12, 13, 17, 26–34]. Notice that, if the operator is singlevalued, then we obtain the well-posedness concept introduced in [35].

Remark 3.7.

An open question is to present a theory of the Ćirić-type multivalued contraction theorem (see [36]). For some problems for other classes of generalized contractions, see for example, [17, 21, 27, 34, 37].

## Declarations

### Acknowledgments

The second and the forth authors wish to thank National Council of Research of Higher Education in Romania (CNCSIS) by "Planul National, PN II (2007–2013)—Programul IDEI-1239" for the provided financial support. The authors are grateful for the reviewer(s) for the careful reading of the paper and for the suggestions which improved the quality of this work.

## Authors’ Affiliations

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