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Bohr–Neugebauer type theorem for some partial neutral functional differential equations. (English) Zbl 1118.35052

The authors study the existence of almost periodic solutions for the following partial neutral functional-differential equations

$\left\{\begin{array}{c}\frac{d}{dt}𝒟{u}_{t}=A𝒟{u}_{t}+L\left({u}_{t}\right)+f\left(t\right)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}t\ge \sigma ,\hfill \\ {u}_{\sigma }=\varphi \in C:=C\left(\left[-r,0\right];X\right),\hfill \end{array}\right\$

where $A$ is a linear operator on a Banach space $X$, not necessarily densely defined and satisfies the known Hille-Yosida condition; $C$ is endowed with the uniform topology; $𝒟:C\to X$ is a bounded linear operator having the form $𝒟\varphi :=\varphi \left(0\right)-{\int }_{-r}^{0}\left[d\eta \left(\theta \right)\right]\varphi \left(\theta \right)$. By using the variation of constants formula and the spectral decomposition of the phase space developed in M. Adimy et al. [Can. Appl. Math. Q. 9, No. 1, 1–34 (2001; Zbl 1112.34341)], they prove that the existence of an almost periodic solution is equivalent to the existence of a bounded solution on ${ℝ}^{+}$.

##### MSC:
 35R10 Partial functional-differential equations 35B15 Almost and pseudo-almost periodic solutions of PDE