Three nonzero periodic solutions for a differential inclusion

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

Three nonzero periodic solutions for a differential inclusion

Francesca Faraci ^1, and Antonio Iannizzotto ^1,

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

Abstract
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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.

Keywords: Multiplicity, Variational methods., Ordinary differential inclusions, Periodic solutions.

Mathematics Subject Classification: Primary: 34A60; Secondary: 34C2.

Citation: Francesca Faraci, Antonio Iannizzotto. Three nonzero periodic solutions for a differential inclusion. Discrete and 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), 210-229. 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.

show all references

References:

[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.

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