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.
Let be a metric space and consider the following family of subsets We also consider the following (generalized) functionals:
is called the gap functional between and . In particular, if then :
is called the (generalized) excess functional:
is the (generalized) Pompeiu-Hausdorff functional.
If is a metric space, then a multivalued operator is said to be a Reich-type multivalued -contraction if and only if there exist with such that
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 .
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;
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 ).
Let be a nonempty set. Then we denote.
Let be a metric space. Then and
Let be a multivalued operator. Then the operator , which is defined by
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 .
The set of all nonempty invariant subsets of is denoted by , that is,
A sequence of successive approximations of starting from is a sequence of elements in with , for .
If then denotes the fixed point set of and denotes the strict fixed point set of . By
we denote the graph of the multivalued operator .
If , then denote the iterate operators of .
Definition 2.1 (see ).
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
(i) and ;
(ii) for all ;
(iii)the sequence is convergent and its limit is a fixed point of .
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 .
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.
Let be a metric space and . By definition, is called a multivalued Picard operator (briefly MP operator) if and only if
(ii) as , for each .
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 ).
Let be a complete metric space, and let be a Reich-type single-valued -contraction, that is, there exist with such that
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.
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:
(v) (Ulam-Hyers stability of the inclusion ) Let and be such that then there exists such that ;
(vi), is a set-to-set -contraction and (thus) ;
(vii) as , for each ;
(viii) and are compact;
(ix) for each .
(i) Let and be arbitrarily chosen. Then, for each arbitrary there exists such that . Hence
Denote By an inductive procedure, we obtain a sequence of successive approximations for starting from such that, for each , we have Then
If we choose , then by (3.4) we get that the sequence is Cauchy and hence convergent in to some
Notice that, by , we obtain that
(ii) Let in (3.4). Then we get that
For we get
Let in (3.8), then
Hence is a -multivalued weakly Picard operator.
(iii) Let be arbitrarily chosen. Then, by (ii), we have that
Let be an arbitrary. Then, there exists such that
In a similar way, we can prove that for each there exists such that
Thus, (3.11) and (3.12) together imply that for every . Let and we get the desired conclusion.
(iv) follows immediately from (iii).
(v) Let and be such that . Then, since is compact, there exists such that . From the proof of (i), we have that
Since , we get that .
(vi) We will prove for any that
For this purpose, let and let . Then, there exists such that . Since the sets are compact, there exists such that
From (3.15) we get that . Hence
In a similar way we obtain that
Thus, (3.16) and (3.17) together imply that
Hence, is a Reich-type single-valued -contraction on the complete metric space . From Theorem 3.1 we obtain that
(b) as , for each .
(vii) From (vi)-(b) we get that as , for each .
(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.
Let be a complete metric space and a Reich-type multivalued -contraction with . Then, the following assertions hold:
(xi) (Well-posedness of the fixed point problem with respect to ) If is a sequence in such that as , then as ;
(xii) (Well-posedness of the fixed point problem with respect to ) If is a sequence in such that as , then as .
(x) Let . Note that . Indeed, if , then . Thus .
Let us show now that . Suppose that . Then, . Thus . Hence . Since , we get that .
(xi) Let be a sequence in such that as . Then, . Then as .
(xii) follows by (xi) since as .
A third result for the case of -contraction is the following.
Let be a complete metric space, and let be a Reich-type multivalued -contraction such that . Then one has
(xiii) as , for each ;
(xiv) for each ;
(xv)If is a sequence such that as and is -continuous, then as .
(xiii) From the fact that and Theorem 3.2 (vi) we have that . The conclusion follows by Theorem 3.2 (vii).
(xiv) Let be an arbitrary. Then , and thus . On the other hand . Thus , for each .
(xv) Let be a sequence such that as . Then, we have as . The proof is complete.
For compact metric spaces we have the following result.
Let be a compact metric space, and let be a -continuous Reich-type multivalued -contraction. Then
(xvi) Let be a sequence in such that as . Let be a subsequence of such that as . Then, there exists , such that as . Then . Hence
as Hence .
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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