Periodic Solutions of Systems with p-Laplacian-like Operators

  • Jean Mawhin
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 43)

Abstract

A great deal of attention has recently been given to extending spectral, bifurcation or existence results for semilinear equations of the second order, in both ordinary and partial differential cases, to the case of nonlinear perturbations of the so-called p-Laplacian operator, defined by, or of some suitable generalization.

Keywords

Manifold 

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References

  1. 1.
    T. Bartsch and J. Mawhin, The Leray-Schauder degree of Slequivariant operators associated to autonomous neutral equations in spaces of periodic functionsJ. Differential Equations92 (1991), 90–99.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    A. Cabada and R.L. Pouso, Existence result for the problem (0(u’))’ =f (t uu’) with periodic and Neumann boundary conditions,Nonlinear Anal. T.M.A. 30 (1997), 1733–1742.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systemsTrans. Amer. Math. Soc.329 (1992), 41–72.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    M. Cuesta, Etude de la résonance et du spectre de F u’eik des opérateurs laplacien et p-laplacien, Ph.D. Thesis, Université de Bruxelles, 1993.Google Scholar
  5. 5.
    K. DeimlingNonlinear Functional AnalysisSpringer-Verlag, Berlin, 1985.MATHCrossRefGoogle Scholar
  6. 6.
    M. Del Pino, R. Manásevich and A. Murua, Existence and multiplicity of solutions with prescribed period for a second order quasilinear o.d.e.Nonlinear Analysis T.M.A.18 (1992), 79–92.MATHCrossRefGoogle Scholar
  7. 7.
    H. Dang and S.F. Oppenheimer, Existence and uniqueness results for some nonlinear boundary value problemsJ. Math. Anal. Appl. 198(1996), 35–48.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    C. Fabry and D. Fayyad, Periodic solutions of second order differential equations with a p-Laplacian and asymmetric nonlinearitiesRend. Ist. Mat. Univ. Trieste24 (1992), 207–227.MathSciNetMATHGoogle Scholar
  9. 9.
    Z. Guo, Boundary value problems of a class of quasilinear ordinary differential equationsDifferential and Integral Equations 6(1993), 705–719.MathSciNetMATHGoogle Scholar
  10. 10.
    Ph. Hartman, On boundary value problems for systems of ordinary nonlinear second order differential equationsTrans. Amer. Math. Soc.96 (1960), 493–509.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Y.X. Huang and G. Metzen, The existence of solutions to a class of semilinear equationsDifferential and Integral Equations 8(1995), 429–452.MathSciNetMATHGoogle Scholar
  12. 12.
    H.W. Knobloch, On the existence of periodic solutions for second order vector differential equationsJ. Differential Equations 9(1971), 67–85.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operatorsJ. Differential Equations 145(1998), 367–393.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    R. Manásevich and J. Mawhin, The spectrum of p-Laplaciansystemswith various boundary conditions and applications,Advances in Differential Equations 5(2000), 1289–1318.MathSciNetMATHGoogle Scholar
  15. 15.
    J. Mawhin, An extension of a theorem of A.C. Lazer on forced nonlinear oscillationsJ. Math. Anal. Appl.40 (1972), 20–29.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spacesJ. Differential Equations 12(1972), 610–636.Google Scholar
  17. 17.
    J. MawhinTopological Methods in Nonlinear Boundary Value ProblemsCBMS Regional Conf. Ser in Math., vol. 40, AMS, Providence, 1979.Google Scholar
  18. 18.
    J. MawhinPoints fixes points critiques et problèmes aux limitesSémin. Math. Sup., vol. 92, Université de Montréal, 1985.Google Scholar
  19. 19.
    J. MawhinProblèmes de Dirichlet variationnels non linéairesSémin. Math. Sup., vol. 104, Université de Montréal, 1987.MATHGoogle Scholar
  20. 20.
    J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations. In:Topological Methods for Ordinary Differential Equations (M. Furi, P. Zecca eds.)Lecture Notes in Math. 1537, Springer-Verlag, Berlin, 1993, 74–142.CrossRefGoogle Scholar
  21. 21.
    J. Mawhin, Continuation theorems and periodic solutions of ordinary differential equations. In: Topological Methods in Differential Equations and Inclusions (A. Granas, M. Frigon eds.), NATO ASI Series C 472, Kluwer, Dordrecht, 1995, 291–375.CrossRefGoogle Scholar
  22. 22.
    N. Rouche and J. MawhinOrdinary Differential Equations. Stability and Periodic SolutionsPitman, Boston, 1980.MATHGoogle Scholar
  23. 23.
    L. Véron, Première valeur propre non nulle du p-Laplacien et équations quasi linéaires elliptiques sur une variété riemannienne compacteC.R. Acad. Sci. Paris 314(1992), 271–276.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Jean Mawhin
    • 1
  1. 1.Institut de Mathématique Pure et AppliquéeUniversité Catholique de LouvainLouvain-la-NeuveBelgique

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