Fixed Point Theory and Applications

, 2010:178421 | Cite as

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

  • Tania Lazăr
  • Ghiocel Moţ
  • Gabriela Petruşel
  • Silviu Szentesi
Open Access
Research Article

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.

Keywords

Fixed Point Theorem Fractal Operator Data Dependence Invariant Subset Fixed Point Problem 

1. Introduction

Let Open image in new window be a metric space and consider the following family of subsets Open image in new window We also consider the following (generalized) functionals:
Open image in new window is called the (generalized) excess functional:

Open image in new window is the (generalized) Pompeiu-Hausdorff functional.

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

Definition 1.1.

If Open image in new window is a metric space, then a multivalued operator Open image in new window is said to be a Reich-type multivalued Open image in new window -contraction if and only if there exist Open image in new window with Open image in new window such that

Reich proved that any Reich-type multivalued Open image in new window -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 Open image in new window -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]).

Let Open image in new window be a nonempty set. Then we denote.
Let Open image in new window be a multivalued operator. Then the operator Open image in new window , which is defined by

is called the fractal operator generated by Open image in new window . 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 Open image in new window is a metric space and Open image in new window , then the following statements hold:

(a)if Open image in new window is upper semicontinuous, then Open image in new window , for every Open image in new window ;

(b)the continuity of Open image in new window implies the continuity of Open image in new window .

The set of all nonempty invariant subsets of Open image in new window is denoted by Open image in new window , that is,

A sequence of successive approximations of Open image in new window starting from Open image in new window is a sequence Open image in new window of elements in Open image in new window with Open image in new window , for Open image in new window .

If Open image in new window then Open image in new window denotes the fixed point set of Open image in new window and Open image in new window denotes the strict fixed point set of Open image in new window . By

we denote the graph of the multivalued operator Open image in new window .

If Open image in new window , then Open image in new window denote the iterate operators of Open image in new window .

Definition 2.1 (see [15]).

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

(i) Open image in new window and Open image in new window ;

(ii) Open image in new window for all Open image in new window ;

(iii)the sequence Open image in new window is convergent and its limit is a fixed point of Open image in new window .

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

Definition 2.3.

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

We recall now the notion of multivalued Picard operator.

Definition 2.4.

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

(i) Open image in new window ;

(ii) Open image in new window as Open image in new window , for each Open image in new window .

In [10] other results on MWP operators are presented. For related concepts and results see, for example, [1, 17, 18, 19, 20, 21, 22, 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]).

Let Open image in new window be a complete metric space, and let Open image in new window be a Reich-type single-valued Open image in new window -contraction, that is, there exist Open image in new window with Open image in new window such that

Then Open image in new window is a Picard operator, that is, we have:

(i) Open image in new window ;

(ii)for each Open image in new window the sequence Open image in new window converges in Open image in new window to Open image in new window

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

Theorem 3.2.

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

(i) Open image in new window ;

(ii) Open image in new window is a Open image in new window -multivalued weakly Picard operator;

(iii)let Open image in new window be a Reich-type multivalued Open image in new window -contraction and Open image in new window such that Open image in new window for each Open image in new window , then Open image in new window

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

If, moreover Open image in new window for each Open image in new window , then one additionally has:

  1. (v)

    (Ulam-Hyers stability of the inclusion Open image in new window ) Let Open image in new window and Open image in new window be such that Open image in new window then there exists Open image in new window such that Open image in new window ;

     

(vi) Open image in new window , Open image in new window is a set-to-set Open image in new window -contraction and (thus) Open image in new window ;

(vii) Open image in new window as Open image in new window , for each Open image in new window ;

(viii) Open image in new window and Open image in new window are compact;

(ix) Open image in new window for each Open image in new window .

Proof.
  1. (i)
    Let Open image in new window and Open image in new window be arbitrarily chosen. Then, for each arbitrary Open image in new window there exists Open image in new window such that Open image in new window . Hence
     
Denote Open image in new window By an inductive procedure, we obtain a sequence of successive approximations for Open image in new window starting from Open image in new window such that, for each Open image in new window , we have Open image in new window Then

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

Hence Open image in new window .
  1. (ii)
    Let Open image in new window in (3.4). Then we get that
     
Hence Open image in new window is a Open image in new window -multivalued weakly Picard operator.
  1. (iii)
    Let Open image in new window be arbitrarily chosen. Then, by (ii), we have that
     
Let Open image in new window be an arbitrary. Then, there exists Open image in new window such that
In a similar way, we can prove that for each Open image in new window there exists Open image in new window such that
Thus, (3.11) and (3.12) together imply that Open image in new window for every Open image in new window . Let Open image in new window and we get the desired conclusion.
  1. (iv)

    follows immediately from (iii).

     
  2. (v)
    Let Open image in new window and Open image in new window be such that Open image in new window . Then, since Open image in new window is compact, there exists Open image in new window such that Open image in new window . From the proof of (i), we have that
     
Since Open image in new window , we get that Open image in new window .
  1. (vi)
    We will prove for any Open image in new window that
     
For this purpose, let Open image in new window and let Open image in new window . Then, there exists Open image in new window such that Open image in new window . Since the sets Open image in new window are compact, there exists Open image in new window such that
In a similar way we obtain that
Thus, (3.16) and (3.17) together imply that

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

(a) Open image in new window and

(b) Open image in new window as Open image in new window , for each Open image in new window .

  1. (vii)
     

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

A second result for Reich-type multivalued Open image in new window -contractions formulates as follows.

Theorem 3.3.

Let Open image in new window be a complete metric space and Open image in new window a Reich-type multivalued Open image in new window -contraction with Open image in new window . Then, the following assertions hold:
  1. (x)
     
  2. (xi)

    (Well-posedness of the fixed point problem with respect to Open image in new window [24]) If Open image in new window is a sequence in Open image in new window such that Open image in new window as Open image in new window , then Open image in new window as Open image in new window ;

     
  3. (xii)

    (Well-posedness of the fixed point problem with respect to Open image in new window [24]) If Open image in new window is a sequence in Open image in new window such that Open image in new window as Open image in new window , then Open image in new window as Open image in new window .

     

A third result for the case of Open image in new window -contraction is the following.

Theorem 3.4.

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

(xiii) Open image in new window as Open image in new window , for each Open image in new window ;

(xiv) Open image in new window for each Open image in new window ;

(xv)If Open image in new window is a sequence such that Open image in new window as Open image in new window and Open image in new window is Open image in new window -continuous, then Open image in new window as Open image in new window .

Proof.
  1. (xiii)

    From the fact that Open image in new window and Theorem 3.2 (vi) we have that Open image in new window . The conclusion follows by Theorem 3.2 (vii).

     
  2. (xiv)
     
  3. (xv)

    Let Open image in new window be a sequence such that Open image in new window as Open image in new window . Then, we have Open image in new window as Open image in new window . The proof is complete.

     

For compact metric spaces we have the following result.

Theorem 3.5.

Let Open image in new window be a compact metric space, and let Open image in new window be a Open image in new window -continuous Reich-type multivalued Open image in new window -contraction. Then
  1. (xvi)

    if Open image in new window is such that Open image in new window as Open image in new window , then there exists a subsequence Open image in new window of Open image in new window such that Open image in new window as Open image in new window (generalized well-posedness of the fixed point problem with respect to Open image in new window [24, 25]).

     

as Open image in new window Hence Open image in new window .

Remark 3.6.

For Open image in new window we obtain the results given in [4]. On the other hand, our results unify and generalize some results given in [12, 13, 17, 26, 27, 28, 29, 30, 31, 32, 33, 34]. Notice that, if the operator Open image in new window 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].

Notes

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.

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Copyright information

© Tania Lazăr et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Tania Lazăr
    • 1
  • Ghiocel Moţ
    • 2
  • Gabriela Petruşel
    • 3
  • Silviu Szentesi
    • 4
  1. 1.Commercial Academy of Satu MareSatu MareRomania
  2. 2.Aurel Vlaicu University of AradAradRomania
  3. 3.Department of BusinessBabeş-Bolyai UniversityCluj-NapocaRomania
  4. 4.Aurel Vlaicu University of AradAradRomania

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