Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
multiple solutions; periodic problem; one-dimensional $p$-Laplacian; discontinuous vector field; nonsmooth Palais-Smale condition; locally Lipschitz function; generalized subdifferential; critical point; Saddle Point Theorem; Ekeland variational principle
multiple solutions; periodic problem; one-dimensional $p$-Laplacian; discontinuous vector field; nonsmooth Palais-Smale condition; locally Lipschitz function; generalized subdifferential; critical point; Saddle Point Theorem; Ekeland variational principle
Summary:
In this paper we consider a periodic problem driven by the one dimensional $p$-Laplacian and with a discontinuous right hand side. We pass to a multivalued problem, by filling in the gaps at the discontinuity points. Then for the multivalued problem, using the nonsmooth critical point theory, we establish the existence of at least three distinct periodic solutions.
References:
[1] Boccardo L., Drábek P., Giachetti D., Kučera M.: Generalization of Fredholm alternative for nonlinear differential operators. Nonlinear Anal. 10 (1986), 1083–1103. MR 0857742
[2] Chang K. C.: Variational methods for non-differentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80 (1981), 102–129. MR 0614246 | Zbl 0487.49027
[3] Clarke F. H.: Optimization and Nonsmooth Analysis. Wiley, New York 1983. MR 0709590 | Zbl 0582.49001
[4] Dang H., Oppenheimer S. F.: Existence and uniqueness results for some nonlinear boundary value problems. J. Math. Anal. Appl. 198 (1996), 35–48. MR 1373525
[5] De Coster C.: On pairs of positive solutions for the one dimensional $p$-Laplacian. Nonlinear Anal. 23 (1994), 669–681. MR 1297285
[6] Del Pino M., Elgueta M., Manasevich R.: A homotopic deformation along p of a Leray-Schauder degree result and existence for $(|u^{\prime }|^{p-2}u^{\prime })^{\prime }+f(t,u)=0,\;u(0)=u(T)=0$. J. Differential Equations 80 (1989), 1–13. MR 1003248 | Zbl 0708.34019
[7] Del Pino M., Manasevich R., Murua A.: Existence and multiplicity of solutions with prescribed period for a second order quasilinear ode. Nonlinear Anal. 18 (1992), 79–92. MR 1138643
[8] Drábek P., Invernizzi S.: On the periodic bvp for the forced Duffing equation with jumping nonlinearity. Nonlinear Anal. 10 (1986), 643–650. MR 0849954 | Zbl 0616.34010
[9] Fabry C., Fayyad D.: Periodic solutions of second order differential equations with a $p$-Laplacian and assymetric nonlinearities. Rend. Istit. Mat. Univ. Trieste 24 (1992), 207–227. MR 1310080
[10] Fabry C., Mawhin J., Nkashama M.: A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations. Bull. London Math. Soc. 18 (1986), 173–180. MR 0818822 | Zbl 0586.34038
[11] Guo Z.: Boundary value problems of a class of quasilinear ordinary differential equations. Differential Integral Equations 6 (1993), 705–719. MR 1202567 | Zbl 0784.34018
[12] Hu S., Papageorgiou N. S.: Handbook of Multivalued Analysis. Vol I: Theory. Kluwer, The Netherlands, 1997. MR 1485775
[13] Hu S., Papageorgiou N. S.: Handbook of Multivalued Analysis. Vol II: Applications. Kluwer, The Netherlands, 2000. MR 1741926
[14] Manasevich R., Mawhin J.: Periodic solutions for nonlinear systems with $p$-Laplacian-like operators. J. Differential Equations 145 (1998), 367–393. MR 1621038
[15] Papageorgiou N. S., Yannakakis N.: Nonlinear boundary value problems. Proc. Indian Acad. Sci. Math. Sci. 109 (1999), 211–230. MR 1687731 | Zbl 0952.34035
[16] Szulkin A.: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincarè Non Linèaire 3 (1986), 77–109. MR 0837231 | Zbl 0612.58011
[17] Tang C.-L.: Existence and multiplicity of periodic solutions for nonautonomous second order systems. Nonlinear Anal. 32 (1998), 299–304. MR 1610641 | Zbl 0949.34032
[18] Zhang M.: Nonuniform nonresonance at the first eigenvalue of the $p$-Laplacian. Nonlinear Anal. 29 (1997), 41–51. MR 1447568 | Zbl 0876.35039
[19] Mawhin J. M., Willem M.: Critical Point Theory and Hamiltonian Systems. Springer, Berlin (1989). MR 0982267 | Zbl 0676.58017
Search
Partner of
