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measurable multifunction; usc and lsc multifunction; maximal monotone operator; pseudomonotone operator; generalized pseudomonotone operator; coercive operator; surjective operator; eigenvalue; eigenfunction; Rayleigh quotient; $p$-Laplacian; Yosida approximation; periodic problem.
In this paper we study two boundary value problems for second order strongly nonlinear differential inclusions involving a maximal monotone term. The first is a vector problem with Dirichlet boundary conditions and a nonlinear differential operator of the form $x\mapsto a(x,x^{\prime })^{\prime }$. In this problem the maximal monotone term is required to be defined everywhere in the state space $\mathbb{R}^N$. The second problem is a scalar problem with periodic boundary conditions and a differential operator of the form $x\mapsto (a(x)x^{\prime })^{\prime }$. In this case the maximal monotone term need not be defined everywhere, incorporating into our framework differential variational inequalities. Using techniques from multivalued analysis and from nonlinear analysis, we prove the existence of solutions for both problems under convexity and nonconvexity conditions on the multivalued right-hand side.
[1] R.  Bader: A topological fixed point index theory for evolution inclusions. Zeitsh. Anal. Anwend. 20 (2001), 3–15. DOI 10.4171/ZAA/1001 | MR 1826317 | Zbl 0985.34053
[2] L.  Boccardo, P.  Drábek, D.  Giachetti and M.  Kučera: Generalization of the Fredholm alternative for nonlinear differential operators. Nonlin. Anal. 10 (1986), 1083–1103. DOI 10.1016/0362-546X(86)90091-X | MR 0857742
[3] H.  Brezis: Operateurs Maximaux Monotones. North-Holland, Amsterdam, 1973. Zbl 0252.47055
[4] F.  Browder and P.  Hess: Nonlinear mappings of monotone type in Banach spaces. J.  Funct. Anal. 11 (1972), 251–254. DOI 10.1016/0022-1236(72)90070-5 | MR 0365242
[5] F. H.  Clarke: Optimization and Nonsmooth Analysis. Wiley, New York, 1983. MR 0709590 | Zbl 0582.49001
[6] D.  Cohn: Measure Theory. Birkhauser-Verlag, Boston, 1980. MR 0578344 | Zbl 0436.28001
[7] H.  Dang and S. F.  Oppenheimer: Existence and uniqueness results for some nonlinear boundary value problems. J.  Math. Anal. Appl. 198 (1996), 35–48. DOI 10.1006/jmaa.1996.0066 | MR 1373525
[8] M.  Del Pino, M. Elgueta and R.  Manasevich: 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. DOI 10.1016/0022-0396(89)90093-4 | MR 1003248
[9] P.  Drábek: Solvability of boundary value problems with homogeneous ordinary differential operator. Rend. Ist. Mat. Univ. Trieste 8 (1986), 105–124. MR 0928322
[10] L.  Erbe and W.  Krawcewicz: Nonlinear boundary value problems for differential inclusions $y^{\prime \prime }\in F(t,y,y^{\prime })$. Ann. Pol. Math. 54 (1991), 195–226. MR 1114171
[11] L.  Erbe and W.  Krawcewicz: Boundary value problems differential inclusions. Lect. Notes Pure Appl. Math., No.  127, Marcel-Dekker, New York, 1990, pp. 115–135. MR 1096748
[12] L.  Erbe and W.  Krawcewicz: Existence of solutions to boundary value problems for impulsive second order differential inclusions. Rocky Mountain J.  Math. 22 (1992), 519–539. DOI 10.1216/rmjm/1181072746 | MR 1180717
[13] L.  Erbe, W. Krawcewicz and G. Peschke: Bifurcation of a parametrized family of boundary value problems for second order differential inclusions. Ann. Mat. Pura Appl. 166 (1993), 169–195. DOI 10.1007/BF01765848 | MR 1271418
[14] C.  Fabry and D.  Fayyad: 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
[15] M.  Frigon: Application de la theorie de la transversalite topologique a des problemes non lineaires pour des equations differentielles ordinaires. Dissertationes Math. 269 (1990). MR 1075674 | Zbl 0728.34017
[16] M.  Frigon: Theoremes d’existence des solutions d’inclusions differentielle. In: Topological Methods in Diferential Equations and Inclusions. NATO ASI Series, Section  C, Vol. 472, Kluwer, Dordrecht, 1995, pp. 51–87. MR 1368670
[17] M.  Frigon and A.  Granas: Problemes aux limites pour des inclusions differentielles de type semi-continues inferieurement. Rivista Mat. Univ. Parma 17 (1991), 87–97. MR 1174938
[18] S.  Fučík, J.  Nečas, J.  Souček and V.  Souček: Spectral Analysis of Nonlinear Operators. Lecture Notes in Math., Vol.  346. Springer-Verlag, Berlin, 1973. MR 0467421
[19] Z.  Guo: Boundary value problems of a class of quasilinear differential equations. Diff. Intergral Eqns 6 (1993), 705–719.
[20] N.  Halidias and N. S.  Papageorgiou: Existence and relaxation results for nonlinear second order multivalued boundary value problems in  $\mathbb{R}^N$. J.  Diff. Eqns 147 (1998), 123–154. DOI 10.1006/jdeq.1998.3439 | MR 1632661
[21] N.  Halidias and N. S.  Papageorgiou: Existence of solutions for quasilinear second order differential inclusions with nonlinear boundary conditions. J.  Comput. Appl. Math. 113 (2000), 51–64. DOI 10.1016/S0377-0427(99)00243-5 | MR 1735812
[22] P.  Hartman: Ordinary Differential Equations, 2nd Edition. Birkhauser-Verlag, Boston-Basel-Stuttgart, 1982. MR 0658490
[23] S.  Hu and N. S.  Papageorgiou: Handbook of Multivalued Analysis. Volume  I: Theory. Kluwer, Dordrecht, 1997. MR 1485775
[24] S.  Hu and N. S.  Papageorgiou: Handbook of Multivalued Analysis. Volume  II: Applications. Kluwer, Dordrecht, 2000. MR 1741926
[25] D.  Kandilakis and N. S.  Papageorgiou: Existence theorems for nonlinear boundary value problems for second order differential inclusions. J.  Differential Equations 132 (1996), 107–125. DOI 10.1006/jdeq.1996.0173 | MR 1418502
[26] E.  Klein and A.  Thompson: Theory of Correspondences. Wiley, New York, 1984. MR 0752692
[27] S. Th.  Kyritsi, N.  Matzakos and N. S.  Papageorgiou: Periodic problems for strongly nonlinear second order differential inclusions. J.  Differential Equations 183 (2002), 279–302. DOI 10.1006/jdeq.2001.4110 | MR 1919781
[28] R.  Manasevich and J.  Mawhin: Periodic solutions for nonlinear systems with $p$-Laplacian-like operators. J.  Differential Equations 145 (1998), 367–393. DOI 10.1006/jdeq.1998.3425 | MR 1621038
[29] R.  Manasevich and J.  Mawhin: Boundary value problems for nonlinear perturbations of vector $p$-Laplacian-like operators. J.  Korean Math. Soc. 37 (2000), 665–685. MR 1783579
[30] M.  Marcus and V.  Mizel: Absolute continuity on tracks and mappings of Sobolev spaces. Arch. Rational Mech. Anal. 45 (1972), 294–320. DOI 10.1007/BF00251378 | MR 0338765
[31 J.  Mawhin and M.  Willem] Critical Point Theory and Hamiltonian Systems. Springer-Verlag, New York, 1989. MR 0982267 | Zbl 0676.58017
[32] Z.  Naniewicz and P.  Panagiotopoulos: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York, 1994. MR 1304257
[33] N. S.  Papageorgiou: Convergence theorems for Banach soace valued integrable multifunctions. Intern. J.  Math. Sc. 10 (1987), 433–442. DOI 10.1155/S0161171287000516 | MR 0896595
[34] T.  Pruszko: Some applications of the topological deggre theory to multivalued boundary value problems. Dissertationes Math. 229 (1984). MR 0741752
[35] D.  Wagner: Survey of measurable selection theorems. SIAM J.  Control Optim. 15 (1977), . DOI 10.1137/0315056 | MR 0486391 | Zbl 0407.28006
[36] E.  Zeidler: Nonlinear Functional Analysis and its Applications  II. Springer-Verlag, New York, 1990. MR 0816732 | Zbl 0684.47029
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