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PERIODICITY OF SOLUTIONS FOR A SYSTEM OF NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS

In this paper, we study the existence of periodic solutions of the nonlinear system of integro-differential equations d dt x (t) = − ∫ t t−τ(t) A(t, s)x(s)ds+Q (t, x(t), x(t− τ (t))) , t ∈ R. In the process we use the fundamental matrix solution to convert the given integro-differential equation into an equivalent integral equation. Then by using Krasnoselskii’s fixed point theorem we obtain the existence of periodic solution and by contraction mapping principle we obtain the uniqueness. An application in two dimension is given.