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Nonlinear integrodifferential equations anda priori bounds on periodic solutions

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Summary

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.

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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). https://doi.org/10.1007/BF01759641

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Keywords

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