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Positive pseudo almost periodic solutions for some nonlinear infinite delay integral equations. (English) Zbl 1189.45014

The paper deals with the study of nonlinear infinite delay integral equation of the form

$x\left(t\right)={\int }_{-\infty }^{t}a\left(t,t-s\right)f\left(s,x\left(s\right)\right)\phantom{\rule{0.166667em}{0ex}}ds,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $f:ℝ×{ℝ}_{+}\to ℝ$ is continuous and $a:ℝ×{ℝ}_{+}\to {ℝ}_{+}$ and the function $s↦a\left(t,s\right)$ is integrable over ${ℝ}_{+}$ for any fixed $t$. Under some additional assumptions a few sufficient conditions are established which guarantee the existence of almost periodic, asymptotically almost period and pseudo almost periodic solutions of (1). Some applications to other types of integral or differential equations of the obtained results are indicated.

##### MSC:
 45M15 Periodic solutions of integral equations 45G10 Nonsingular nonlinear integral equations 34K14 Almost and pseudo-periodic solutions of functional differential equations
##### References:
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