Periodic and asymptotically periodic solutions of neutral integral equations

Abstract

Many results have been obtained for periodic solutions of Volterra integral equations (for instance, [1-3] and references cited therein). Here we consider two systems of neutral integral equations \begin{eqnarray} x(t)=a(t)+\int_0^t D(t,s,x(s))ds+\int_t^\infty E(t,s,x(s))ds, \ t\in R^+ \end{eqnarray} and \begin{eqnarray} x(t)=p(t)+\int_{-\infty}^t P(t,s,x(s))ds+\int_t^\infty Q(t,s,x(s))ds, \ t\in R,\ \end{eqnarray} where $a, \ p, \ D, \ P, \ E$ and $Q$ are at least continuous. Under suitable conditions, if $\phi$ is a given $R^n$-valued bounded and continuous initial function on $[0,t_0)$ or $(-\infty,t_0)$, then both Eq.(1) and Eq.(2) have solutions denoted by $x(t,t_0,\phi)$ with $x(t,t_0,\phi)=\phi(t)$ for $t<t_0$, satisfying Eq.(1) or Eq.(2) on $[t_0,\infty)$. (cf. Burton-Furumochi [4].) A solution $x(t,t_0,\phi)$ may have a discontinuity at $t_0$.