×

zbMATH — the first resource for mathematics

Nonlinear boundary value problems for second order differential inclusions. (English) Zbl 1081.34020
Summary: We study two boundary value problems for second-order strongly nonlinear differential inclusions involving a maximal monotone term. The first one is a vector problem with Dirichlet boundary conditions and a nonlinear differential operator of the form \(x\mapsto a(x,x')'\). 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 \(x\mapsto (a(x)x')'\). In this case, the maximal monotone term need not be defined everywhere, incorporating into our framework differential variational inequalities. Using techniques from multi-valued analysis and from nonlinear analysis, we prove the existence of solutions for both problems under convexity and nonconvexity conditions on the multi-valued right-hand side.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] R. Bader: A topological fixed point index theory for evolution inclusions. Zeitsh. Anal. Anwend. 20 (2001), 3–15. · Zbl 0985.34053
[2] L. Boccardo, P. Drabek, D. Giachetti and M. Kucera: Generalization of the Fredholm alternative for nonlinear differential operators. Nonlin. Anal. 10 (1986), 1083–1103. · Zbl 0623.34031
[3] H. Brezis: Operateurs Maximaux Monotones. North-Holland, Amsterdam, 1973.
[4] F. Browder and P. Hess: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11 (1972), 251–254. · Zbl 0249.47044
[5] F. H. Clarke: Optimization and Nonsmooth Analysis. Wiley, New York, 1983. · Zbl 0582.49001
[6] D. Cohn: Measure Theory. Birkhauser-Verlag, Boston, 1980.
[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. · Zbl 0855.34021
[8] M. Del Pino, M. Elgueta and R. Manasevich: A homotopic deformation along p of a Leray-Schauder degree result and existence for (|u’| p u’)’ + f(t, u) = 0, u(0) = u(T) = 0. J. Differential Equations 80 (1989), 1–13. · Zbl 0708.34019
[9] P. Drabek: Solvability of boundary value problems with homogeneous ordinary differential operator. Rend. Ist. Mat. Univ. Trieste 8 (1986), 105–124. · Zbl 0633.34015
[10] L. Erbe and W. Krawcewicz: Nonlinear boundary value problems for differential inclusions y” F(t, y, y’). Ann. Pol. Math. 54 (1991), 195–226. · Zbl 0731.34078
[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. · Zbl 0714.34040
[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. · Zbl 0784.34012
[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. · Zbl 0798.34017
[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. · Zbl 0824.34026
[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).
[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. · Zbl 0834.34021
[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. · Zbl 0756.34021
[18] S. Fucik, J. Necas, J. Soucek and V. Soucek: Spectral Analysis of Nonlinear Operators. Lecture Notes in Math., Vol. 346. Springer-Verlag, Berlin, 1973. · Zbl 0268.47056
[19] Z. Guo: Boundary value problems of a class of quasilinear differential equations. Diff. Intergral Eqns 6 (1993), 705–719. · Zbl 0784.34018
[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. · Zbl 0912.34020
[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. · Zbl 0941.34008
[22] P. Hartman: Ordinary Differential Equations, 2nd Edition. Birkhauser-Verlag, Boston-Basel-Stuttgart, 1982. · Zbl 0476.34002
[23] S. Hu and N. S. Papageorgiou: Handbook of Multivalued Analysis. Volume I: Theory. Kluwer, Dordrecht, 1997. · Zbl 0887.47001
[24] S. Hu and N. S. Papageorgiou: Handbook of Multivalued Analysis. Volume II: Applications. Kluwer, Dordrecht, 2000. · Zbl 0943.47037
[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. · Zbl 0859.34011
[26] E. Klein and A. Thompson: Theory of Correspondences. Wiley, New York, 1984.
[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. · Zbl 1022.34008
[28] R. Manasevich and J. Mawhin: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Differential Equations 145 (1998), 367–393. · Zbl 0910.34051
[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. · Zbl 0976.34013
[30] M. Marcus and V. Mizel: Absolute continuity on tracks and mappings of Sobolev spaces. Arch. Rational Mech. Anal. 45 (1972), 294–320. · Zbl 0236.46033
[31] J. Mawhin and M. Willem Critical Point Theory and Hamiltonian Systems. Springer-Verlag, New York, 1989. · Zbl 0676.58017
[32] Z. Naniewicz and P. Panagiotopoulos: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York, 1994. · Zbl 0968.49008
[33] N. S. Papageorgiou: Convergence theorems for Banach soace valued integrable multifunctions. Intern. J. Math. Sc. 10 (1987), 433–442. · Zbl 0619.28009
[34] T. Pruszko: Some applications of the topological deggre theory to multivalued boundary value problems. Dissertationes Math. 229 (1984). · Zbl 0543.34008
[35] D. Wagner: Survey of measurable selection theorems. SIAM J. Control Optim. 15 (1977). · Zbl 0407.28006
[36] E. Zeidler: Nonlinear Functional Analysis and its Applications II. Springer-Verlag, New York, 1990. · Zbl 0684.47029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.