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Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces

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Bu S., Fang Y.

Studia Mathematica

2008

tom 184

nr 2

103-119

We study the maximal regularity on different function spaces of the second order integro-differential equations with infinite delay $(P) u''(t) + αu'(t) + d/dt (∫^{t}_{-∞} b(t-s)u(s)ds) = Au(t) - ∫^{t}_{-∞} a(t-s)Au(s)ds + f(t)$ (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), u'(0) = u'(2π), where A is a closed operator in a Banach space X, α ∈ ℂ, and a,b ∈ L¹(ℝ₊). We use Fourier multipliers to characterize maximal regularity for (P). Using known results on Fourier multipliers, we find suitable conditions on the kernels a and b under which necessary and sufficient conditions are given for the problem (P) to have maximal regularity on $L^{p}(𝕋,X)$, periodic Besov spaces $B_{p,q}^{s}(𝕋,X)$ and periodic Triebel-Lizorkin spaces $F_{p,q}^{s}(𝕋,X)$

Ograniczanie wyników

1 Studia Mathematica

1 Bu S.

1 Fang Y.

1 2008

Wyniki wyszukiwania

Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces