## Periodic solutions of nonlinear integrodifferential equations.(English)Zbl 0689.45017

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.
Reviewer: I.Foltyńska

### MSC:

 45J05 Integro-ordinary differential equations 45G10 Other nonlinear integral equations
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### References:

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