# Multiple solutions for nonlinear periodic problems with discontinuities

Nikolaos S. Papageorgiou; Nikolaos Yannakakis

Archivum Mathematicum (2002)

- Volume: 038, Issue: 3, page 171-182
- ISSN: 0044-8753

## Access Full Article

top## Abstract

top## How to cite

topPapageorgiou, Nikolaos S., and Yannakakis, Nikolaos. "Multiple solutions for nonlinear periodic problems with discontinuities." Archivum Mathematicum 038.3 (2002): 171-182. <http://eudml.org/doc/248941>.

@article{Papageorgiou2002,

abstract = {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.},

author = {Papageorgiou, Nikolaos S., Yannakakis, Nikolaos},

journal = {Archivum Mathematicum},

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 -Laplacian; discontinuous vector field; nonsmooth Palais-Smale condition; locally Lipschitz function; generalized subdifferential; critical point},

language = {eng},

number = {3},

pages = {171-182},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {Multiple solutions for nonlinear periodic problems with discontinuities},

url = {http://eudml.org/doc/248941},

volume = {038},

year = {2002},

}

TY - JOUR

AU - Papageorgiou, Nikolaos S.

AU - Yannakakis, Nikolaos

TI - Multiple solutions for nonlinear periodic problems with discontinuities

JO - Archivum Mathematicum

PY - 2002

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 038

IS - 3

SP - 171

EP - 182

AB - 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.

LA - eng

KW - 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 -Laplacian; discontinuous vector field; nonsmooth Palais-Smale condition; locally Lipschitz function; generalized subdifferential; critical point

UR - http://eudml.org/doc/248941

ER -

## References

top- Boccardo L., Drábek P., Giachetti D., Kučera M., Generalization of Fredholm alternative for nonlinear differential operators, Nonlinear Anal. 10 (1986), 1083–1103. (1986) MR0857742
- Chang K. C., Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102–129. (1981) Zbl0487.49027MR0614246
- Clarke F. H., Optimization and Nonsmooth Analysis, Wiley, New York 1983. (1983) Zbl0582.49001MR0709590
- Dang H., Oppenheimer S. F., Existence and uniqueness results for some nonlinear boundary value problems, J. Math. Anal. Appl. 198 (1996), 35–48. (198) MR1373525
- De Coster C., On pairs of positive solutions for the one dimensional $p$-Laplacian, Nonlinear Anal. 23 (1994), 669–681. (1994) MR1297285
- Del Pino M., Elgueta M., Manasevich R., A homotopic deformation along p of a Leray-Schauder degree result and existence for $\left(\right|{u}^{\text{'}}{{|}^{p-2}{u}^{\text{'}})}^{\text{'}}+f(t,u)=0,\phantom{\rule{0.277778em}{0ex}}u\left(0\right)=u\left(T\right)=0$, J. Differential Equations 80 (1989), 1–13. (1989) Zbl0708.34019MR1003248
- 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. (1992) MR1138643
- Drábek P., Invernizzi S., On the periodic bvp for the forced Duffing equation with jumping nonlinearity, Nonlinear Anal. 10 (1986), 643–650. (1986) Zbl0616.34010MR0849954
- 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. (1992) MR1310080
- 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. (1986) Zbl0586.34038MR0818822
- Guo Z., Boundary value problems of a class of quasilinear ordinary differential equations, Differential Integral Equations 6 (1993), 705–719. (1993) Zbl0784.34018MR1202567
- Hu S., Papageorgiou N. S., Handbook of Multivalued Analysis. Vol I: Theory, Kluwer, The Netherlands, 1997. (1997) MR1485775
- Hu S., Papageorgiou N. S., Handbook of Multivalued Analysis. Vol II: Applications, Kluwer, The Netherlands, 2000. MR1741926
- Manasevich R., Mawhin J., Periodic solutions for nonlinear systems with $p$-Laplacian-like operators, J. Differential Equations 145 (1998), 367–393. (1998) MR1621038
- Papageorgiou N. S., Yannakakis N., Nonlinear boundary value problems, Proc. Indian Acad. Sci. Math. Sci. 109 (1999), 211–230. (1999) Zbl0952.34035MR1687731
- 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. (1986) Zbl0612.58011MR0837231
- Tang C.-L., Existence and multiplicity of periodic solutions for nonautonomous second order systems, Nonlinear Anal. 32 (1998), 299–304. (1998) Zbl0949.34032MR1610641
- Zhang M., Nonuniform nonresonance at the first eigenvalue of the $p$-Laplacian, Nonlinear Anal. 29 (1997), 41–51. (1997) Zbl0876.35039MR1447568
- Mawhin J. M., Willem M., Critical Point Theory and Hamiltonian Systems, Springer, Berlin (1989). (1989) Zbl0676.58017MR0982267

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.