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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

d dt𝒟u t =A𝒟u t +L(u t )+f(t)fortσ,u σ =φC:=C([-r,0];X),

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:CX is a bounded linear operator having the form 𝒟φ:=φ(0)- -r 0 [dη(θ)]φ(θ). 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 + .

35R10Partial functional-differential equations
35B15Almost and pseudo-almost periodic solutions of PDE