Results in Mathematics

, Volume 38, Issue 1–2, pp 9–17 | Cite as

An Existence Result for Second Order Functional Differential Inclusions

  • M. Benchohra
  • S. K. Ntouyas


In this paper we investigate the existence of mild solutions on a compact interval for second order functional differential inclusions in Banach spaces. We shall rely on a fixed point theorem for condensing maps due to Martelli.

1991 Mathematics Subject Classification

34 A60 34 G20 34 K10 

Key words and phrases

Initial value problems Convex multivalued map Mild solution Functional differential inclusion Existence Fixed point Abstract space 


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  1. [1]
    E.P. Avgerinos and N.S. Papageorgiou, On quasilinear evolution inclusions, Glas. Mat. Ser.III 28(48) No. 1 (1993), 35–52.MathSciNetMATHGoogle Scholar
  2. [2]
    J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel-Dekker, New York, 1980.MATHGoogle Scholar
  3. [3]
    M. Benchohra, Existence of mild solutions on infinite intervals to first order initial value problems for a class of differential inclusions in Banach spaces, Differential Inclusions and Optimal Control, to appear.Google Scholar
  4. [4]
    M. Benchohra and S. K. Ntouyas, Existence results for functional differential and integrodifferential inclusions in Banach spaces, submitted.Google Scholar
  5. [5]
    K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin-New York, 1992.MATHCrossRefGoogle Scholar
  6. [6]
    J. Dugundji and A. Granas, Fixed Point Theory, Monografie Mat. PWN, Warsaw, 1982.MATHGoogle Scholar
  7. [7]
    L.H. Erbe, Q. Kong and B.G. Zhang, Oscillation Theory for Functional Differential Equations, Pure and Applied Mathematics, 1994.Google Scholar
  8. [8]
    H.O. Fattorini, Ordinary differential equations in linear topological spaces, I, J. Diff. Eq. 5 (1968), 72–105.MathSciNetCrossRefGoogle Scholar
  9. [9]
    H.O. Fattorini, Ordinary differential equations in linear topological spaces, II, J. Diff. Eq. 6 (1969), 50–70.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    J.A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, New York, 1985.MATHGoogle Scholar
  11. [11]
    S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker, New York, 1994.Google Scholar
  12. [12]
    J. Henderson, Boundary Value Problems for Functional Differenial Equations, World Scientific, 1982.Google Scholar
  13. [13]
    S.G. Hristova and D.D. Bainov, Application of the monotone iterative techniques of Lakshmikantham to the solution of the initial value problem for functional differential equations, J. Math. Phys. Sci. 24 (1990), 405–413.MathSciNetMATHGoogle Scholar
  14. [14]
    Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Kluwer, Dordrecht, Boston, London, 1997.MATHGoogle Scholar
  15. [15]
    T. Kusano and S. Oharu, Semilinear evolution equations with singularities in ordered Banach Spaces, Diff. Int. Equs 5(6) (1992), 1383–1405.MathSciNetMATHGoogle Scholar
  16. [16]
    G.S. Ladde, V. Lakshmikantham and A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston M. A. (1985).MATHGoogle Scholar
  17. [17]
    A. Lasota and Z. Opial, An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781–786.MathSciNetMATHGoogle Scholar
  18. [18]
    E. Liz and J. J. Nieto, Periodic BVPs for a class of functional differential equations, L. Mat. Anal. Appl. 200 (1996), 680–686.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    M. Martelli, A Royhe’s type theorem for noncompact acyclic-valued map, Boll. Un. Math. Ital. 4 (1975), 70–76.MathSciNetGoogle Scholar
  20. [20]
    J.J. Nieto, Y. Jiang and Y. Jurang, Monotone iterative method for functional differential equations, Nonl. Anal. 32(6) (1998), 741–749.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    S.K. Ntouyas, Initial and boundary value problems for functional differential equations via the topological transversality method: A survey, Bull. Greek Math. Soc. 40 (1998), 3–41.MathSciNetMATHGoogle Scholar
  22. [22]
    N.S. Papageorgiou, Mild solutions of semilinear evolution inclusions, Indian J. Pure Appl. Math. 26(3) (1995), 189–216.MathSciNetMATHGoogle Scholar
  23. [23]
    N.S. Papageorgiou, Boundary value problems for evolution inclusions, Comment. Math. Univ. Carol. 29 (1988), 355–363.MATHGoogle Scholar
  24. [24]
    H. Schaefer, Uber die methode der a priori schranken, Math. Ann. 129, (1955), 415–416.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    C.C. Travis and G.F. Webb, Second order differential equations in Banach spaces, Proc. Int. Symp. on Nonlinear Equations in Abstract Spaces, Academic Press, New York (1978), 331–361.Google Scholar
  26. [26]
    C.C. Travis and G.F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hung. 32 (1978), 75–96.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    K. Yosida, Functional Analysis, 6th edn. Springer-Verlag, Berlin, 1980.MATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 2000

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de Sidi Bel Abbès, BP 89Algérie
  2. 2.Department of MathematicsUniversity of IoanninaIoanninaGreece

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