# Article

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Keywords:
one dimensional $p$-Laplacian; maximal monotone operator; pseudomonotone operator; generalized pseudomonotonicity; coercive operator; first nonzero eigenvalue; upper solution; lower solution; truncation map; penalty function; multiplicity result
Summary:
In this paper we examine a quasilinear periodic problem driven by the one- dimensional $p$-Laplacian and with discontinuous forcing term $f$. By filling in the gaps at the discontinuity points of $f$ we pass to a multivalued periodic problem. For this second order nonlinear periodic differential inclusion, using variational arguments, techniques from the theory of nonlinear operators of monotone type and the method of upper and lower solutions, we prove the existence of at least two non trivial solutions, one positive, the other negative.
References:
[1] Ahmad S., Lazer A.: Critical point theory and a theorem of Amaral and Pera. Bollettino U.M.I. 6 (1984), 583–598. MR 0774464 | Zbl 0603.34036
[2] Boccardo L., Drábek P., Giacchetti D., Kučera M.: Generalization of Fredholm alternative for some nonlinear boundary value problem. Nonlinear Anal. T.M.A. 10 (1986), 1083–1103. MR 0857742
[3] 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
[4] Del Pino M., Elgueta M., Manasevich R.: A homotopic deformation along p of a Leray Schauder degree result and existence for $(| u^{\prime } |^{p-2}u^{\prime })^{\prime }+ f(t,u)=0, u(0)=u(T)=0, p>1^*$. J. Differential Equations 80 (1989), 1–13. MR 1003248 | Zbl 0708.34019
[5] Del Pino M., Manasevich R., Murua A.: Existence and multiplicity of solutions with prescribed period for a second order quasilinear o.d.e. Nonlinear Anal. T.M.A. 18 (1992), 79–92. MR 1138643 | Zbl 0761.34032
[6] Drábek P.: Solvability of boundary value problems with homogeneous ordinary differential operator. Rend. Istit. Mat. Univ. Trieste 8 (1986), 105–124. MR 0928322
[7] Fabry C., Fayyad D.: Periodic solutions of second order differential equations with a $p$-Laplacian and asymmetric nonlinearities. Rend. Istit. Mat. Univ. Trieste 24 (1992), 207–227. MR 1310080 | Zbl 0824.34026
[8] Fabry C., Mawhin J., Nkashama M. N.: 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
[9] Fonda A., Lupo D.: Periodic solutions of second order ordinary differential differential equations. Bollettino U.M.I. 7 (1989), 291–299. MR 1026756
[10] Gilbarg D., Trudinger N.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York (1977). MR 0473443 | Zbl 0361.35003
[11] Gossez J.-P., Omari P.: A note on periodic solutions for second order ordinary differential equation. Bollettino U.M.I. 7 (1991), 223–231.
[12] Guo Z.: Boundary value problems of a class of quasilinear differential equations. Diff. Integral Equations 6 (1993), 705–719. MR 1202567
[13] Hu S., Papageorgiou N. S.: Handbook of Multivalued Analysis, Volume I: Theory. Kluwer, Dordrecht, The Netherlands (1997). MR 1485775 | Zbl 0887.47001
[14] Hu S., Papageorgiou N. S.: Handbook of Multivalued Analysis, Volume II: Applications. Kluwer, Dordrecht, The Netherlands (2000). MR 1741926 | Zbl 0943.47037
[15] Kesavan S.: Topics in Functional Analysis and Applications. Wiley, New York (1989). MR 0990018 | Zbl 0666.46001
[16] Manasevich R., Mawhin J.: Periodic solutions for nonlinear systems with $p$-Laplacian like operators. J. Differential Equations 145 (1998), 367–393. MR 1621038
[17] Mawhin J., Willem M.: Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. J. Differential Equations 2 (1984), 264–287. MR 0741271 | Zbl 0557.34036
[18] Zeidler E.: Nonlinear Functional Analysis and its Applications II. Springer-Verlag, New York (1985). MR 0768749

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