# American Institute of Mathematical Sciences

August  2012, 5(4): 779-788. doi: 10.3934/dcdss.2012.5.779

## Three nonzero periodic solutions for a differential inclusion

 1 Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy, Italy

Received  May 2011 Revised  July 2011 Published  November 2011

We prove the existence of three non-zero periodic solutions for an ordinary differential inclusion. Our approach is variational and based on a multiplicity theorem for the critical points of a nonsmooth functional, which extends a recent result of Ricceri.
Citation: Francesca Faraci, Antonio Iannizzotto. Three nonzero periodic solutions for a differential inclusion. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 779-788. doi: 10.3934/dcdss.2012.5.779
##### References:
 [1] G. Bonanno, A minimax inequality and its applications to ordinary differential equations,, J. Math. Anal. Appl., 270 (2002), 210. doi: 10.1016/S0022-247X(02)00068-9. [2] A. Boucherif and S. M. Bouguima, Periodic solutions of second [order] ordinary differential equations with a discontinuous nonlinearity,, in, 343 (1996), 54. [3] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations,, J. Math. Anal. Appl., 80 (1981), 102. doi: 10.1016/0022-247X(81)90095-0. [4] L. H. Erbe and W. Krawcewicz, Nonlinear boundary value problems for differential inclusions $y''\in F(t,y,y')$,, Ann. Polon. Math., 54 (1991), 195. [5] M. Frigon and A. Granas, Problèmes aux limites pour des inclusions différentielles de type semi-continues inférieurement,, Riv. Mat. Univ. Parma (4), 17 (1991), 87. [6] A. Iannizzotto, Three critical points for perturbed nonsmooth functionals and applications,, Nonlinear Anal., 72 (2010), 1319. doi: 10.1016/j.na.2009.08.001. [7] D. Kandilakis, N. C. Kourogenis and N. S. Papageorgiou, Two nontrivial critical points for nonsmooth functionals via local linking and applications,, J. Global Optim., 34 (2006), 219. doi: 10.1007/s10898-005-3884-7. [8] M. Krastanov, N. Ribarska and T. Tsachev, A note on: "On a critical point theory for multivalued functionals and application to partial differential inclusions,'', Nonlinear Anal., 43 (2001), 153. [9] R. Livrea and S. A. Marano, On a min-max principle for non-smooth functions and applications,, Commun. Appl. Anal., 13 (2009), 411. [10] D. Motreanu and P. D. Panagiotopoulos, "Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities,", Nonconvex Optimization and its Applications, 29 (1999). [11] N. S. Papageorgiou and F. Papalini, Existence of two solutions for quasilinear periodic differential equations with discontinuities,, Arch. Math. (Brno), 38 (2002), 285. [12] B. Ricceri, Multiplicity of global minima for parametrized functions,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 21 (2010), 47. [13] B. Ricceri, A class of nonlinear eigenvalue problems with four solutions,, J. Nonlinear Convex Anal., 11 (2010), 503.

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##### References:
 [1] G. Bonanno, A minimax inequality and its applications to ordinary differential equations,, J. Math. Anal. Appl., 270 (2002), 210. doi: 10.1016/S0022-247X(02)00068-9. [2] A. Boucherif and S. M. Bouguima, Periodic solutions of second [order] ordinary differential equations with a discontinuous nonlinearity,, in, 343 (1996), 54. [3] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations,, J. Math. Anal. Appl., 80 (1981), 102. doi: 10.1016/0022-247X(81)90095-0. [4] L. H. Erbe and W. Krawcewicz, Nonlinear boundary value problems for differential inclusions $y''\in F(t,y,y')$,, Ann. Polon. Math., 54 (1991), 195. [5] M. Frigon and A. Granas, Problèmes aux limites pour des inclusions différentielles de type semi-continues inférieurement,, Riv. Mat. Univ. Parma (4), 17 (1991), 87. [6] A. Iannizzotto, Three critical points for perturbed nonsmooth functionals and applications,, Nonlinear Anal., 72 (2010), 1319. doi: 10.1016/j.na.2009.08.001. [7] D. Kandilakis, N. C. Kourogenis and N. S. Papageorgiou, Two nontrivial critical points for nonsmooth functionals via local linking and applications,, J. Global Optim., 34 (2006), 219. doi: 10.1007/s10898-005-3884-7. [8] M. Krastanov, N. Ribarska and T. Tsachev, A note on: "On a critical point theory for multivalued functionals and application to partial differential inclusions,'', Nonlinear Anal., 43 (2001), 153. [9] R. Livrea and S. A. Marano, On a min-max principle for non-smooth functions and applications,, Commun. Appl. Anal., 13 (2009), 411. [10] D. Motreanu and P. D. Panagiotopoulos, "Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities,", Nonconvex Optimization and its Applications, 29 (1999). [11] N. S. Papageorgiou and F. Papalini, Existence of two solutions for quasilinear periodic differential equations with discontinuities,, Arch. Math. (Brno), 38 (2002), 285. [12] B. Ricceri, Multiplicity of global minima for parametrized functions,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 21 (2010), 47. [13] B. Ricceri, A class of nonlinear eigenvalue problems with four solutions,, J. Nonlinear Convex Anal., 11 (2010), 503.
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