Article Contents
Article Contents

# Three nonzero periodic solutions for a differential inclusion

• 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.
Mathematics Subject Classification: Primary: 34A60; Secondary: 34C25.

 Citation:

•  [1] G. Bonanno, A minimax inequality and its applications to ordinary differential equations, J. Math. Anal. Appl., 270 (2002), 210-229.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 "Nonlinear Partial Differential Equations" (Fès, 1994), Pitman Res. Notes Math. Ser., 343, Longman, Harlow, (1996), 54-60. [3] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.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), 195-226. [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), 87-97. [6] A. Iannizzotto, Three critical points for perturbed nonsmooth functionals and applications, Nonlinear Anal., 72 (2010), 1319-1338.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), 219-244.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), 153-158. [9] R. Livrea and S. A. Marano, On a min-max principle for non-smooth functions and applications, Commun. Appl. Anal., 13 (2009), 411-430. [10] D. Motreanu and P. D. Panagiotopoulos, "Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities," Nonconvex Optimization and its Applications, 29, Kluwer Academic Publishers, Dordrecht, 1999. [11] N. S. Papageorgiou and F. Papalini, Existence of two solutions for quasilinear periodic differential equations with discontinuities, Arch. Math. (Brno), 38 (2002), 285-296. [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), 47-57. [13] B. Ricceri, A class of nonlinear eigenvalue problems with four solutions, J. Nonlinear Convex Anal., 11 (2010), 503-511.