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Nonlinear integrodifferential equations anda priori bounds on periodic solutions

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This paper studies the existence of aperiodic solution of a nonlinear integrodifferential system of the form\(x'(t) = Dx(t) + f\left( {x(t)} \right) + \int\limits_{ - \infty }^t {k(t,s)g\left( {x(s)} \right)ds + p(t)} ,\), for each continuous periodic function p and under suitable assumptions on f, k and g. A topological transversality method is employed to obtain the existence of periodic solutions. This method relies ona priori bounds on periodic solutions. Several examples are provided where a variant of Liapunov's direct method is employed to obtaina priori bounds on periodic solutions.


  1. [1]

    O. A. Arino -T. A. Burton -J. R. Haddock,Periodic solutions to functional differential equations, Proc. Roy. Soc. Edinburgh A,101 (1985), pp. 253–271.

  2. [2]

    J. Cronin,Fixed Points and Topological Degree in Nonlinear Analysis, Math. Surveys,11, Amer. Math. Soc., Providence, R.I. (1964).

  3. [3]

    A. Granas,Sur la méthode de continuité de Poincaré, C. R. Acad. Sci., Paris,282 (1976), pp. 983–985.

  4. [4]

    A. Granas -R. B. Guenther -J. W. Lee,Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky Mountain J. Math.,10 (1980), pp. 35–58.

  5. [5]

    A. Granas -R. B. Guenther -J. W. Lee,Nonlinear Boundary Value Problems for Ordinary Differential Equations, Dissertation Mathematicae,244, Polish Scientific Publications, Warsaw (1985).

  6. [6]

    J. Hale,Theory of Functional Differential Equations, Springer, New York (1977).

  7. [7]

    J. Mawhin,Topological Degree Methods in Nonlinear Boundary Value Problems, Regional Conference Series in Math.,40, Amer. Math. Soc., Providence, R.I. (1979).

  8. [8]

    J. Mawhin,An estension of a theorem of A. C. Lazer on forced nonlinear oscillations, J. Math. Anal. Appl.,40 (1970), pp. 20–29.

  9. [9]

    J. Mawhin,Periodic solutions of nonlinear functional differential equations, J. Diff. Equat.,10 (1971), pp. 240–261.

  10. [10]

    I. P. Natanson,Theory of Functions of a Real Variable, Vol. 11, Frederick Ungar Publishing Co., New York (1955).

  11. [11]

    R. Reissig,Extension of some results concerning the generalized Lienard equation, Ann. Mat. Pura Appl.,104 (1975), pp. 269–281.

  12. [12]

    T. Yoshizawa,Stability Theory by Lyapunov's Second Method, Math. Soc. Japan, Tokyo (1966).

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Burton, T.A., Eloe, P.W. & Islam, M.N. Nonlinear integrodifferential equations anda priori bounds on periodic solutions. Annali di Matematica 161, 271–283 (1992).

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  • Periodic Solution
  • Functional Differential Equation
  • Nonlinear Functional Differential Equation
  • Lienard Equation
  • Riodic Solution