Periodic nonnegative solutions in neutral nonlinear integro-differential equations with functional delay (Q2517010)

The authors study the existence of nonnegative periodic solutions to the following nonlinear neutral integro-differential equation, \[ x'(t)=-\int_{t-\tau(t)}^t a(t,s)g(x(s))ds+c(t)x'(t-\tau(t))+G(t,x(t),x(t-\tau(t))), \tag{\(*\)} \] which can be used to describe population dynamics. Under suitable assumptions, first (\(*\)) is transformed into an integral equation written as a sum of two mappings. One of them is a large contraction while the other is compact. Thus the existence of nonnegative periodic solution is equivalent to the existence of fixed points. This is achieved by using a variant of the Krasnoselskii fixed point theorem.