Periodic solutions of nonlinear integrodifferential equations (Q581802)

The existence of a solution to the nonlinear integor-differential equation \[ (1)\quad \dot x(t)=f(t,x(t))+\int^{t}_{- \infty}E(t,s,x(s),x(t))ds \] is considered. The following assumptions on (1) are imposed: (a) \(f:R\times R^ n\to R^ r\) is a continuous function and E(t,s,x,y) is defined and continuous for \(-\infty <s\leq t<\infty\), \(x\in R^ n\), \(y\in R^ n,\) (b) There is a \(T>0\) such that \(f(t+T,x)=f(t,x)\) for all \(t\in R\), \(x\in R^ n\) and \(E(t+T,s+T,x,y)=E(t,s,x,y)\) for all \(t\in R,s\leq t,x\in R^ n,y\in R^ n.\) (c) For any \(r>0\), there exists an \(L_ 1(r)>0\) such that \(\int^{t}_{-\infty}| E(t,s,x(s),x(t)| ds\leq L_ 1(r)\) for all t, whenever x(s) is continuous and \(| x(s)| \leq r\) for all \(s\leq t.\) (d) For any \(\epsilon >0\) and \(r>0\), there exists an \(S>0\) such that \(\int^{t-s}_{-\infty}| E(t,s,x(s),x(t))| ds\leq \epsilon\) for all \(t\in R\), whenever x(s) is continuous and \(| x(s)| \leq r\) for all \(s\leq t.\) (e) The equation (1) has a bounded solution defined on [0,\(\infty).\) Sufficient conditions for the existence of an asymptotically almost periodic solution or T-periodic solution to (1) are established.