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

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.

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.

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 [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.

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 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

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 .

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 [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

(i) and ;

(ii) for all ;

(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

(i);

(ii) as , for each .

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

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 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:

(i);

(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

(i);

(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 .

Proof.

(i) Let and be arbitrarily chosen. Then, for each arbitrary there exists such that . Hence

Thus

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

Hence .

(ii) Let in (3.4). Then we get that

For we get

Then

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

(a) and

(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.

Theorem 3.3.

Let be a complete metric space and a Reich-type multivalued -contraction with . Then, the following assertions hold:

(x) ;

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

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

Proof.

(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.

Theorem 3.4.

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 .

Proof.

(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.

Theorem 3.5.

Let be a compact metric space, and let be a -continuous Reich-type multivalued -contraction. Then

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

Proof.

(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 .

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].

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.

1. Covitz, H, Nadler,, SB Jr..: Multi-valued contraction mappings in generalized metric spaces. Israel Journal of Mathematics. 8, 5–11 (1970). Publisher Full Text

2. Hu, S, Papageorgiou, NS: Handbook of Multivalued Analysis. Vol. I, Mathematics and Its Applications,p. xvi+964. Kluwer Academic Publishers, Dordrecht, The Netherlands (1997)

3. Reich, S: Fixed points of contractive functions. Bollettino della Unione Matematica Italiana. 5, 26–42 (1972)

4. Petruşel, A, Rus, IA: The theory of a metric fixed point theorem for multivalued operators. Proceedings of the 9th International Conference on Fixed Point Theory and Its Applications, Yokohama Publishing to appear

5. Rus, IA: The theory of a metrical fixed point theoremml: theoretical and applicative relevances. Fixed Point Theory. 9(2), 541–559 (2008)

6. Border, KC: Fixed Point Theorems with Applications to Economics and Game Theory,p. viii+129. Cambridge University Press, Cambridge, UK (1985)

7. Kirk WA, Sims B (eds.): Handbook of Metric Fixed Point Theory,p. xiv+703. Kluwer Academic Publishers, Dordrecht, The Netherlands (2001)

8. Granas, A, Dugundji, J: Fixed Point Theory, Springer Monographs in Mathematics,p. xvi+690. Springer, New York, NY, USA (2003)

9. Rus, IA, Petruşel, A, Petruşel, G: Fixed Point Theory, Cluj University Press, Cluj-Napoca, Romania (2008)

10. Petruşel, A: Multivalued weakly Picard operators and applications. Scientiae Mathematicae Japonicae. 59(1), 169–202 (2004)

11. Rus, IA: Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, Romania (2001)

12. Barnsley, MF: Fractals Everywhere,p. xiv+534. Academic Press, Boston, Mass, USA (1988)

13. Hutchinson, JE: Fractals and self-similarity. Indiana University Mathematics Journal. 30(5), 713–747 (1981). Publisher Full Text

14. Yamaguti, M, Hata, M, Kigami, J: Mathematics of Fractals, Translations of Mathematical Monographs,p. xii+78. American Mathematical Society, Providence, RI, USA (1997)

15. Rus, IA, Petruşel, A, Sîntămărian, Alina: Data dependence of the fixed point set of some multivalued weakly Picard operators. Nonlinear Analysis. Theory, Methods & Applications. 52(8), 1947–1959 (2003). PubMed Abstract | Publisher Full Text

16. Petruşel, A, Rus, IA: Dynamics on generated by a finite family of multi-valued operators on (X, d). Mathematica Moravica. 5, 103–110 (2001)

17. Agarwal, RP, Dshalalow, JH, O'Regan, D: Fixed point and homotopy results for generalized contractive maps of Reich type. Applicable Analysis. 82(4), 329–350 (2003). Publisher Full Text

18. Andres, J, Górniewicz, L: On the Banach contraction principle for multivalued mappings. Approximation, Optimization and Mathematical Economics (Pointe-à-Pitre, 1999), pp. 1–23. Physica, Heidelberg, Germany (2001)

19. Andres, J, Fišer, J: Metric and topological multivalued fractals. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 14(4), 1277–1289 (2004). Publisher Full Text

20. Espínola, R, Petruşel, A: Existence and data dependence of fixed points for multivalued operators on gauge spaces. Journal of Mathematical Analysis and Applications. 309(2), 420–432 (2005). Publisher Full Text

21. Frigon, M: Fixed point results for multivalued contractions on gauge spaces. Set Valued Mappings with Applications in Nonlinear Analysis, pp. 175–181. Taylor & Francis, London, UK (2002)

22. Frigon, M: Fixed point and continuation results for contractions in metric and gauge spaces. Fixed Point theory and Its Applications, pp. 89–114. Polish Academy of Sciences, Warsaw, Poland (2007)

23. Jachymski, J: Continuous dependence of attractors of iterated function systems. Journal of Mathematical Analysis and Applications. 198(1), 221–226 (1996). Publisher Full Text

24. Petruşel, A, Rus, IA: Well-posedness of the fixed point problem for multivalued operators. In: Cârja O, Vrabie II (eds.) Applied Analysis and Differential Equations, pp. 295–306. World Scientific Publishing, Hackensack, NJ, USA (2007)

25. Petruşel, A, Rus, IA, Yao, J-C: Well-posedness in the generalized sense of the fixed point problems for multivalued operators. Taiwanese Journal of Mathematics. 11(3), 903–914 (2007)

26. Fraser, RB Jr.., Nadler, SB Jr..: Sequences of contractive maps and fixed points. Pacific Journal of Mathematics. 31, 659–667 (1969)

27. Jachymski, J, Józwik, I: Nonlinear contractive conditions: a comparison and related problems. Fixed Point Theory and Its Applications, Banach Center Publications, pp. 123–146. Polish Academy of Sciences, Warsaw, Poland (2007)

28. Lim, T-C: On fixed point stability for set-valued contractive mappings with applications to generalized differential equations. Journal of Mathematical Analysis and Applications. 110(2), 436–441 (1985). Publisher Full Text

29. Markin, JT: Continuous dependence of fixed point sets. Proceedings of the American Mathematical Society. 38, 545–547 (1973). Publisher Full Text

30. Nadler, SB Jr..: Multi-valued contraction mappings. Pacific Journal of Mathematics. 30, 475–488 (1969)

31. Papageorgiou, NS: Convergence theorems for fixed points of multifunctions and solutions of differential inclusions in Banach spaces. Glasnik Matematički. Serija III. 23(2), 247–257 (1988)

32. Reich, S: Kannan's fixed point theorem. Bollettino della Unione Matematica Italiana. 4, 1–11 (1971)

33. Reich, S: Some remarks concerning contraction mappings. Canadian Mathematical Bulletin. 14, 121–124 (1971). Publisher Full Text

34. Rus, IA: Fixed point theorems for multi-valued mappings in complete metric spaces. Mathematica Japonica. 20, 21–24 (1975)

35. Reich, S, Zaslavski, AJ: Well-posedness of fixed point problems. Far East Journal of Mathematical Sciences. 393–401 (2001)

36. Ćirić, LB: Fixed points for generalized multi-valued contractions. Matematički Vesnik. 9(24), 265–272 (1972)

37. Xu, H-K: Metric fixed point theory for multivalued mappings. Dissertationes Mathematicae. 389, 39 (2000)