Anti-periodic solutions for second order differential inclusions


In this paper, we extend the existence results presented in [9] for \(L^p\) spaces to operator inclusions of Hammerstein type in \(W^{1,p}\) spaces. We also show an application of our results to anti-periodic boundary-value problems of second-order differential equations with nonlinearities depending on \(u^{\prime}\).


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

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


Anti-periodic solution; nonlinear boundary-value problem; dissipative operator; multivalued mapping; fixed point.

Paper coordinates

J.-F. Couchouron, R. Precup, Anti-periodic solutions for second order differential inclusions, Electron. J. Differential Equations 2004 (2004), 1-17.


About this paper


Electronic Journal of Differential Equations

Publisher Name
Print ISSN
Online ISSN


MR 93a:58028, Zbl 745.54018

google scholar link

[1] A. R. Aftabizadeh, S. Aizicovici and N.H. Pavel, Anti-periodic boundary value problems for higher order differential equations in Hilbert spaces, Nonlinear Anal. 18 (1992), 253-267.
[2] A.R. Aftabizadeh, S. Aizicovici and N.H. Pavel, On a class of second-order anti-periodic boundary value problems, J. Math. Anal. Appl. 171 (1992), 301-320.
[3] S. Aizicovici and N.H. Pavel, Anti-periodic solutions to a class of nonlinear differential equations in Hilbert spaces, J. Funct. Anal. 99 (1991), 387-408.
[4] S. Aizicovici, N.H. Pavel and I.I. Vrabie, Anti-periodic solutions to strongly nonlinear evolution equations in Hilbert spaces, An. S¸tiint¸. Univ. Al.I. Cuza Ia¸si Mat. 44 (1998), 227-234.
[5] V. Barbu, “Nonlinear Semigroups and Differential Equations in Banach Spaces”, Ed. Academiei & Noordhoff International Publishing, Bucure¸sti-Leyden, 1976.
[6] Z. Cai and N.H. Pavel, Generalized periodic and anti-periodic solutions for the heat equation in R1, Libertas Math. 10 (1990), 109-121.
[7] J.-M. Coron, Periodic solutions of a nonlinear wave equation without assumption of monotonicity, Math. Ann. 262 (1983), 272-285.
[8] J.-F. Couchouron and M. Kamenski, A unified topological point of view for integro-differential inclusions, in “Differential Inclusions and Optimal Control” (eds. J. Andres, L. G´orniewicz and P. Nistri), Lecture Notes in Nonlinear Anal., Vol. 2, 1998, 123-137.
[9] J.-F. Couchouron, M. Kamenski and R. Precup, A nonlinear periodic averaging principle, Nonlinear Anal. 54 (2003), 1439-1467.
[10] J.-F. Couchouron and R. Precup, Existence principles for inclusions of Hammerstein type involving noncompact acyclic multivalued maps, Electronic J. Differential Equations 2002 (2002), No. 04, 1-21.
[11] K. Deimling, “Multivalued Differential Equations”, Walter de Gruyter, Berlin-New York, 1992.
[12] J. Diestel, W.M. Ruess and W. Schachermayer, Weak compactness in L1(µ, X), Proc. Amer. Math. Soc. 118 (1993), 447-453.
[13] S. Eilenberg and D. Montgomery, Fixed point theorems for multivalued transformations, Amer. J. Math. 68 (1946), 214-222.
[14] P.M. Fitzpatrick and W.V. Petryshyn, Fixed point theorems for multivalued noncompactacyclic mappings, Pacific J. Math. 54 (1974), 17-23.
[15] M. Frigon, Theoremes d’existence de solutions d’inclusions diff´erentielles, in “Topological Methods in Differential Equations and Inclusions” (eds. A. Granas and M. Frigon), NATO ASI Series C, Vol. 472, Kluwer Academic Publishers, Dordrecht-Boston-London, 1995, 51-87.
[16] D. Guo, V. Lakshmikantham and X. Liu, “Nonlinear Integral Equations in Abstract Spaces”, Kluwer Academic Publishers, Dordrecht-Boston-London, 1996.
[17] A. Haraux, Anti-periodic solutions of some nonlinear evolution equations, Manuscripta Math. 63 (1989), 479-505.
[18] S. Hu and N.S. Papageorgiou, “Handbook of Multivalued Analysis, Vol. I: Theory”, Kluwer Academic Publishers, Dordrecht-Boston-London, 1997.
[19] H. Okochi, On the existence of periodic solutions to nonlinear abstract parabolic equations, J. Math. Soc. Japan 40 (1988), 541-553.
[20] H. Okochi, On the existence of anti-periodic solutions to nonlinear evolution equations associated with odd subdifferential operators, J. Funct. Anal. 91 (1990), 246-258.

Related Posts