Existence of periodic and positive solutions of nonlinear integro-differential equations with variable delay

Abstract

In this paper, we study the existence of periodic and positive periodic
solutions of the nonlinear neutral integro-differential equation%
x^{\prime }\left( t\right) =-\int_{t-\tau \left( t\right) }^{t}a(t,s)x(s)ds+
\frac{d}{dt}Q\left( t,x\left( t-\tau \left( t\right) \right) \right)
+G\left( t,x\left( t\right) ,x\left( t-\tau \left( t\right) \right) \right) .
We invert this equation to obtain an integral equation. By using Krasnoselski's fixed point Theorem we obtain the existence of periodic solution and by contraction mapping principle we obtain the uniqueness, an example is included to show the feasibility of our results. The positive solutions are studied.