# On the topological dimension of the solutions sets for some classes of operator and differential inclusions

Ralf Bader; Boris D. Gel'man; Mikhail Kamenskii; Valeri Obukhovskii

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2002)

- Volume: 22, Issue: 1, page 17-32
- ISSN: 1509-9407

## Access Full Article

top## Abstract

top## How to cite

topRalf Bader, et al. "On the topological dimension of the solutions sets for some classes of operator and differential inclusions." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 22.1 (2002): 17-32. <http://eudml.org/doc/271523>.

@article{RalfBader2002,

abstract = {In the present paper, we give the lower estimation for the topological dimension of the fixed points set of a condensing continuous multimap in a Banach space. The abstract result is applied to the fixed point set of the multioperator of the form $ = S _F$ where $_F$ is the superposition multioperator generated by the Carathéodory type multifunction F and S is the shift of a linear injective operator. We present sufficient conditions under which this set has the infinite topological dimension. In the last section of the paper, we consider the applications of the solutions sets for Cauchy and periodic problems for semilinear differential inclusions in a Banach space.},

author = {Ralf Bader, Boris D. Gel'man, Mikhail Kamenskii, Valeri Obukhovskii},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {solutions set; fixed points set; topological dimension; multivalued map; condensing map; topological degree; differential inclusion; periodic problem; fixed point; condensing mapping; multivalued mapping},

language = {eng},

number = {1},

pages = {17-32},

title = {On the topological dimension of the solutions sets for some classes of operator and differential inclusions},

url = {http://eudml.org/doc/271523},

volume = {22},

year = {2002},

}

TY - JOUR

AU - Ralf Bader

AU - Boris D. Gel'man

AU - Mikhail Kamenskii

AU - Valeri Obukhovskii

TI - On the topological dimension of the solutions sets for some classes of operator and differential inclusions

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2002

VL - 22

IS - 1

SP - 17

EP - 32

AB - In the present paper, we give the lower estimation for the topological dimension of the fixed points set of a condensing continuous multimap in a Banach space. The abstract result is applied to the fixed point set of the multioperator of the form $ = S _F$ where $_F$ is the superposition multioperator generated by the Carathéodory type multifunction F and S is the shift of a linear injective operator. We present sufficient conditions under which this set has the infinite topological dimension. In the last section of the paper, we consider the applications of the solutions sets for Cauchy and periodic problems for semilinear differential inclusions in a Banach space.

LA - eng

KW - solutions set; fixed points set; topological dimension; multivalued map; condensing map; topological degree; differential inclusion; periodic problem; fixed point; condensing mapping; multivalued mapping

UR - http://eudml.org/doc/271523

ER -

## References

top- [1] P.S. Aleksandrov and B.A. Pasynkov, Introduction to Dimension Theory, Nauka, Moscow 1973 (in Russian).
- [2] J.P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo 1984.
- [3] Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis and V.V. Obukhovskii, Topological methods in the fixed-point theory of multivalued maps, Russian Math. Surveys 35 (1980), 65-143.
- [4] Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis and V.V. Obukhovskii, Multivalued Mappings, J. Sov. Math. 24 (1984), 719-791.
- [5] Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis and V.V. Obukhovskii, Introduction to the Theory of Multivalued Maps, Voronezh Gos. Univ., Voronezh 1986 (in Russian).
- [6] J.F. Couchouron, M. Kamenski, A Unified Topological Point of View for Integro-Differential Inclusions, Differential Inclusions and Opt. Control (J. Andres, L. Górniewicz, and P. Nistri eds.), Lecture Notes in Nonlinear Anal. 2 (1998), 123-137. Zbl1105.45005
- [7] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lect. Notes in Math. 580, Springer-Verlag, Berlin-Heidelberg-New York 1977. Zbl0346.46038
- [8] R. Dragoni, J.W. Macki, P. Nistri and P. Zecca, Solution Sets of Differential Equations in Abstract Spaces, Pitman Res. Notes in Math., 342, Longman 1996. Zbl0847.34004
- [9] Z. Dzedzej and B. Gelman, Dimension of a set of solutions for differential inlcusions, Demonstratio Math. 26 (1993), 149-158. Zbl0783.34008
- [10] R. Engelking, Dimension Theory, PWN, Warszawa 1978.
- [11] B.D. Gelman, Topological properties of the set of fixed points of a multivalued map, Mat. Sbornik 188 (1997), 33-56 (in Russian); English transl.: Sbornik: Mathem. 188 (1997), 1761-1782.
- [12] B.D. Gelman, On topological dimension of a set of solutions of functional inclusions, Differential Inclusions and Opt. Control (J. Andres, L. Górniewicz and P. Nistri eds.), Lecture Notes in Nonlinear Anal. 2 (1998), 163-178.
- [13] S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory, Kluwer Acad. Publ. Dordrecht-Boston-London 1997. Zbl0887.47001
- [14] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Ser. in Nonlinear Analysis and Appl. 7, Walter de Gruyter, Berlin-New York 2001. Zbl0988.34001
- [15] M. Kisielewicz, Differential Inclusions and Optimal Control, PWN, Warszawa, Kluwer Academic Publishers, Dordrecht-Boston-London 1991. Zbl0731.49001
- [16] M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustylnik and P.E. Sobolevskii, Integral Operators in Spaces of Summable Functions, Noordhoff International Publishing, Leyden 1976.
- [17] E. Michael, Continuous selections I, Ann. Math. 63 (1956), 361-382. Zbl0071.15902
- [18] B. Ricceri, On the topological dimension of the solution set of a class of nonlinear equations, C.R. Acad. Sci. Paris Ser. I Math. 325 (1997), 65-70. Zbl0884.47043
- [19] J. Saint Raymond, Points fixes des multiapplications a valeurs convex, C.R. Acad. Sci. Paris Ser. I Math. 298 (1984), 71-74. Zbl0561.54042

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.