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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 \[ \begin{cases}\frac{d}{dt}\mathcal{D}u_t=A\mathcal{D}u_t+L(u_t)+f(t)\quad\text{for }t\geq \sigma,\\ u_{\sigma}=\varphi\in C:=C([-r,0];X),\end{cases} \] 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 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 \(\mathbb R^+\).

MSC:

35R10 Partial functional-differential equations
35B15 Almost and pseudo-almost periodic solutions to PDEs

Citations:

Zbl 1112.34341
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References:

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