A nonlinear periodic averanging principle

3 years ago

Abstract

This paper is devoted to a nonlinear averaging principle for periodic solutions of a class of second order inclusions. In addition an existence theorem for periodic solutions of such inclusions is established. This work which complements the abstract nonlinear averaging principle worked out in Couchouron and Kamenski (Nonlin. Anal. 42 (2000) 1101) makes a synthesis of the methods contained in Couchouron and Kamenski and Couchouron and Precup (Electron. J. Differential, Equations 4 (2002) 1) and represents a (nonvariational) topological approach for boundary values problems.

Authors

Jean-Francois Couchourn
Université de Metz, Mathématiques INRIA Lorraine, Ile du Saulcy, 57045 Metz, France

Mihail Kamenski
University of Voronezh, Faculty of Mathematics, 394693 Voronezh, Russia

Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

Keywords

Periodic solution; Averaging principle; Evolution equation; Dissipative operator, Multivalued mapping, Acyclic set, Fredholm inclusion; Fixed point

Paper coordinates

J.-F. Couchouron, M. Kamenski, R. Precup, A nonlinear periodic averanging principle, Nonlinear Anal. 54 (2003), 1439-1467, https://doi.org/10.1016/S0362-546X(03)00196-2

PDF

(requires subscription) https://doi.org/10.1016/S0362-546X(03)00196-2

About this paper

Journal

Nonlinear Analysis Theory Methods&Applications

Publisher Name

Elsevier

DOI

https://doi.org/10.1016/S0362-546X(03)00196-2

Print ISSN

Online ISSN

MR 1997229, Zbl 1034.34074.

google scholar link

[1] R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, A.E. Rodkina, B.N. Sadovskii, Measures of Noncompactness and Condensing Operators, Birkhauser, Basel-Boston-Berlin, 1992.
[2] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Academiei & Noordhoff International Publishing, Bucuresti-Leiden, 1976.
[3] Ph. Benilan, M.G. Crandall, A. Pazy, Evolution Equations Governed by Accretive Operators, to appear.
[4] Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis, V.V. Obukhovskii, Multivaluedanalysis andoperator inclusions, J. Soviet Math. 39 (1987) 2772–2811.
[5] J.-F. Couchouron, M. Kamenski, A unified topological point of view for integro-differential inclusions, in: J. Andres, L. G Xorniewicz, P. Nistri (Eds.), Di8erential Inclusions and Optimal Control, Lecture Notes in Nonlinear Analysis, Vol. 2, 1998, pp. 123–137.
[6] J.-F. Couchouron, M. Kamenski, An abstract topological point of view anda general averaging principle in the theory of differential inclusions, Nonlin. Anal. 42 (2000) 1101–1129.
[7] J.-F. Couchouron, R. Precup, Existence principles for inclusions of Hammerstein type involving noncompact acyclic multivalued maps, Electron. J. Di8erential Equations (4) (2002) 1–21.
[8] K. Deimling, MultivaluedDi8erential Equations, Walter de Gruyter, Berlin-New York, 1992.
[9] J. Diestel, W.M. Ruess, W. Schachermayer, Weak compactness in L1(F; X ), Proc. Amer. Math. Soc. 118 (1993) 447–453.
[10] P.M. Fitzpatrick, W.V. Petryshyn, Fixedpoint theorems for multivaluednoncompact acyclic mappings, Pacific J. Math. 54 (1974) 17–23.
[11] L. Gorniewicz, Evolutions governedby m-accretive plus compact operators, Nonlin. Anal. 7 (1983) 707–715.
[12] L. Gorniewicz, Topological FixedPoint Theory of MultivaluedMappings, Kluwer Academic Publishers, Dordrecht-Boston-London, 1999.
[13] S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory, Kluwer Academic Publishers, Dordrecht-Boston-London, 1997.
[14] M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, Walter de Gruyter, Berlin-New York, 2001.
[15] V.V. Obukhovskij, On the topological degree for a class of non-compact multivalued mappings, linear evolution equations, Funkts. Anal. 23 (1984) 82–93 (in Russian).
[16] D. O’Regan, R. Precup, Fixedpoint theorems for set-valued maps and existence principles for integral inclusions, J. Math. Anal. Appl. 245 (2000) 594–612.
[17] D. O’Regan, R. Precup, Integrable solutions of Hammerstein integral inclusions in Banach spaces, Dynam. Contin. Discrete Impuls. Systems 9 (2002) 165–176.
[18] I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Longman Scientific & Technical Systems, 1987.