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

Abstract

We use a variation of Krasnoselskii’s fixed point theorem due to Burton to investigate the existence of solutions of the nonlinear integro-differential neutral equation $$\begin{aligned} x^{\prime }\left( t\right) =-\int _{t-\tau \left( t\right) }^{t}a\left( t,s\right) g(x\left( s\right) )ds+c\left( t\right) x^{\prime }\left( t-\tau \left( t\right) \right) +G\left( t,x(t),x(t-\tau (t)\right) ). \end{aligned}$$ We transform and then invert this equation to obtain a fixed point mapping. We, carely, express this mapping as sum of a compact map and a large contraction and choose suitable hypotheses to show that this fits very nicely into the framework of the modification of Krasnoselskii’s theorem so that existence of nonnegative T-periodic solutions is concluded.