Czechoslovak Mathematical Journal

, Volume 55, Issue 3, pp 545–579

Nonlinear Boundary Value Problems for Second Order Differential Inclusions

  • Sophia Th. Kyritsi
  • Nikolaos Matzakos
  • Nikolaos Papageorgiou
Article

Abstract

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 xa(x, x′)′. In this problem the maximal monotone term is required to be defined everywhere in the state space ℝN. The second problem is a scalar problem with periodic boundary conditions and a differential operator of the form x ↦ (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 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.

Keywords

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Bader: A topological fixed point index theory for evolution inclusions. Zeitsh. Anal. Anwend. 20 (2001), 3–15.Google Scholar
  2. [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.CrossRefGoogle Scholar
  3. [3]
    H. Brezis: Operateurs Maximaux Monotones. North-Holland, Amsterdam, 1973.Google Scholar
  4. [4]
    F. Browder and P. Hess: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11 (1972), 251–254.CrossRefGoogle Scholar
  5. [5]
    F. H. Clarke: Optimization and Nonsmooth Analysis. Wiley, New York, 1983.Google Scholar
  6. [6]
    D. Cohn: Measure Theory. Birkhauser-Verlag, Boston, 1980.Google Scholar
  7. [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.CrossRefGoogle Scholar
  8. [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−2 u′)′ + f(t, u) = 0, u(0) = u(T) = 0. J. Differential Equations 80 (1989), 1–13.CrossRefGoogle Scholar
  9. [9]
    P. Drabek: Solvability of boundary value problems with homogeneous ordinary differential operator. Rend. Ist. Mat. Univ. Trieste 8 (1986), 105–124.Google Scholar
  10. [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.Google Scholar
  11. [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.Google Scholar
  12. [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.Google Scholar
  13. [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.CrossRefGoogle Scholar
  14. [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.Google Scholar
  15. [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).Google Scholar
  16. [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.Google Scholar
  17. [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.Google Scholar
  18. [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.Google Scholar
  19. [19]
    Z. Guo: Boundary value problems of a class of quasilinear differential equations. Diff. Intergral Eqns 6 (1993), 705–719.Google Scholar
  20. [20]
    N. Halidias and N. S. Papageorgiou: Existence and relaxation results for nonlinear second order multivalued boundary value problems in ℝN. J. Diff. Eqns 147 (1998), 123–154.CrossRefGoogle Scholar
  21. [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.CrossRefGoogle Scholar
  22. [22]
    P. Hartman: Ordinary Differential Equations, 2nd Edition. Birkhauser-Verlag, Boston-Basel-Stuttgart, 1982.Google Scholar
  23. [23]
    S. Hu and N. S. Papageorgiou: Handbook of Multivalued Analysis. Volume I: Theory. Kluwer, Dordrecht, 1997.Google Scholar
  24. [24]
    S. Hu and N. S. Papageorgiou: Handbook of Multivalued Analysis. Volume II: Applications. Kluwer, Dordrecht, 2000.Google Scholar
  25. [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.CrossRefGoogle Scholar
  26. [26]
    E. Klein and A. Thompson: Theory of Correspondences. Wiley, New York, 1984.Google Scholar
  27. [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.CrossRefGoogle Scholar
  28. [28]
    R. Manasevich and J. Mawhin: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Differential Equations 145 (1998), 367–393.CrossRefGoogle Scholar
  29. [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.Google Scholar
  30. [30]
    M. Marcus and V. Mizel: Absolute continuity on tracks and mappings of Sobolev spaces. Arch. Rational Mech. Anal. 45 (1972), 294–320.CrossRefGoogle Scholar
  31. [31]
    J. Mawhin and M. Willem Critical Point Theory and Hamiltonian Systems. Springer-Verlag, New York, 1989.Google Scholar
  32. [32]
    Z. Naniewicz and P. Panagiotopoulos: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York, 1994.Google Scholar
  33. [33]
    N. S. Papageorgiou: Convergence theorems for Banach soace valued integrable multifunctions. Intern. J. Math. Sc. 10 (1987), 433–442.CrossRefGoogle Scholar
  34. [34]
    T. Pruszko: Some applications of the topological deggre theory to multivalued boundary value problems. Dissertationes Math. 229 (1984).Google Scholar
  35. [35]
    D. Wagner: Survey of measurable selection theorems. SIAM J. Control Optim. 15 (1977).Google Scholar
  36. [36]
    E. Zeidler: Nonlinear Functional Analysis and its Applications II. Springer-Verlag, New York, 1990.Google Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2005

Authors and Affiliations

  • Sophia Th. Kyritsi
    • 1
  • Nikolaos Matzakos
    • 1
  • Nikolaos Papageorgiou
    • 1
  1. 1.Dept. of MathematicsNational Technical UniversityAthensGreece

Personalised recommendations