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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 $$\cases\frac{d}{dt}\mathcal{D}u_t=A\mathcal{D}u_t+L(u_t)+f(t)\quad\text{for }t\geq \sigma,\\ u_{\sigma}=\varphi\in C:=C([-r,0];X),\endcases$$ 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; $\mathcal{D}:C\to X$ is a bounded linear operator having the form $\mathcal{D}\varphi:=\varphi(0)-\int_{-r}^0[d\eta(\theta)]\varphi(\theta)$. By using the variation of constants formula and the spectral decomposition of the phase space developed in {\it 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 $\Bbb R^+$.

35R10Partial functional-differential equations
35B15Almost and pseudo-almost periodic solutions of PDE
Full Text: DOI
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