Periodic and asymptotically periodic solutions of neutral integral equations

T. Furumochi, Shimane University, Matsue, Japan
T. A. Burton, Northwest Research Institute, Port Angeles, WA, U.S.A.

E. J. Qualitative Theory of Diff. Equ., Proc. 6'th Coll. Qualitative Theory of Diff. Equ., No. 10. (2000), pp. 1-19.

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 $t_0$.

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