The Theory of Reich's Fixed Point Theorem for Multivalued Operators
© Tania Lazăr et al. 2010
Received: 12 April 2010
Accepted: 18 July 2010
Published: 4 August 2010
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.
Reich proved that any Reich-type multivalued -contraction on a complete metric space has at least one fixed point (see ).
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 ). For the singlevalued case, see .
(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;
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  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 , Granas and Dugundji , Hu and Papageorgiou , Rus et al. , Petruşel , and Rus ).
Definition 2.1 (see ).
We recall now the notion of multivalued Picard operator.
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 ).
Our main result concerning Reich's fixed point theorem is the following.
follows immediately from (iii).
For compact metric spaces we have the following result.
For we obtain the results given in . 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 .
An open question is to present a theory of the Ćirić-type multivalued contraction theorem (see ). For some problems for other classes of generalized contractions, see for example, [17, 21, 27, 34, 37].
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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