Existence Results for a Class of Periodic Evolution Variational Inequalities*

Abstract

In this paper, using the Brouwer topological degree, the authors prove an existence result for finite variational inequalities. This approach is also used to obtain the existence of periodic solutions for a class of evolution variational inequalities.

Keywords

Variational inequalities / Differential inclusions / Topological degree / Guiding functions / Periodic solutions / 49J40 / 49J20 / 35K85

References

[1] Adly Journal of Math. Anal. and Appl., 2006, 322: 1055

[2] Adly J. Math. Pures Appl., 2004, 83: 17

[3] Amann Proc. Amer. Math. Soc., 1982, 85: 591

[4] Brézis J. Math. Pures Appl., 1972, 51: 1

[5] Brézis, H., Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North Holland, Amsterdam, 1973

[6] de Blasi Nonlinear Anal. Ser. A: Theory Methods, 1999, 37: 217

[7] Goeleven Nonlinear Analysis, 2005, 61: 961

[8] Goeleven Advances in Nonlinear Variational Inequalities, 2003, 6: 1

[9] Goeleven IEEE Transaction on Automatic Control, 2004, 49: 1

[10] Górniewicz, L., Topological approach to differential inclusions, Topological Methods in Differential Equa- tions and Inclusions, A. Granas and M. Frigon (eds.), NATO ASI Series, Mathematical and Physical Sciences, Vol. 472, Kluwer Academic Publishers, 1994

[11] Hiriart-Urruty, J. B. and Lemaréchal, C., Fundamentals of Convex Analysis, Springer, Berlin, 1993

[12] Kinderlehrer, D. and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980

[13] Krasonel’skii, M. A., The Operator of Translation along the Trajectories of Differential Equations, Nauka, Moscow, 1966 (in Russian); English translation: American Math. Soc., Translations of Math. Monographs, Vol. 19, Providence, 1968

[14] Krasonel’skii, M. A. and Zabreiko, P. P., Geometrical Methods of Nonlinear Analysis, Nauka, Moscow, 1975 (in Russian); English translation: Springer, Berlin, 1984

[15] Lions, J. L., Quelques Méthodes de Résolution des Problèmes aux Limites Non-linéaires, Dunod, Paris, 1969

[16] Lloyd, N. G., Degree Theory, Cambridge University Press, Cambridge, 1978

[17] Mawhin, J., Continuation theorems and periodic solutions of ordinary differential equations, Topological Methods in Differential Equations and Inclusions, A. Granas and M. Frigon (eds.), NATO ASI Series, Mathematical and Physical Sciences, Vol. 472, Kluwer Academic Publishers, 1994

[18] Showalter, R. E., Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, American Mathematical Society, Providence, RI, USA, 1997

[19] Zeidler, E., Nonlinear Functional Analysis and Its Applications, I, Fixed-Point Theorems, Springer-Verlag, New York, 1986