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Studia Mathematica

2008 | 184 | 2 | 103-119

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

EN

Abstrakty

EN
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 $F_{p,q}^{s}(𝕋,X)$

103-119

wydano
2008

Twórcy

autor
• Department of Mathematical Science, University of Tsinghua, Beijing 100084, China
autor
• Department of Mathematical Science, University of Tsinghua, Beijing 100084, China