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

Shangquan Bu, and Yi Fang. "Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces." Studia Mathematica 184.2 (2008): 103-119. <http://eudml.org/doc/286397>.

@article{ShangquanBu2008,
abstract = {We study the maximal regularity on different function spaces of the second order integro-differential equations with infinite delay $(P) u^\{\prime \prime \}(t) + αu^\{\prime \}(t) + d/dt (∫^\{t\}_\{-∞\} b(t-s)u(s)ds) = Au(t) - ∫^\{t\}_\{-∞\} a(t-s)Au(s)ds + f(t)$ (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 $L^\{p\}(,X)$, periodic Besov spaces $B_\{p,q\}^\{s\}(,X)$ and periodic Triebel-Lizorkin spaces $F_\{p,q\}^\{s\}(,X)$},
author = {Shangquan Bu, Yi Fang},
journal = {Studia Mathematica},
keywords = {Fourier multiplier; maximal regularity; Besov space; Triebel-Lizorkin space},
language = {eng},
number = {2},
pages = {103-119},
title = {Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces},
url = {http://eudml.org/doc/286397},
volume = {184},
year = {2008},
}

TY - JOUR
AU - Shangquan Bu
AU - Yi Fang
TI - Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces
JO - Studia Mathematica
PY - 2008
VL - 184
IS - 2
SP - 103
EP - 119
AB - We study the maximal regularity on different function spaces of the second order integro-differential equations with infinite delay $(P) u^{\prime \prime }(t) + αu^{\prime }(t) + d/dt (∫^{t}_{-∞} b(t-s)u(s)ds) = Au(t) - ∫^{t}_{-∞} a(t-s)Au(s)ds + f(t)$ (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 $L^{p}(,X)$, periodic Besov spaces $B_{p,q}^{s}(,X)$ and periodic Triebel-Lizorkin spaces $F_{p,q}^{s}(,X)$
LA - eng
KW - Fourier multiplier; maximal regularity; Besov space; Triebel-Lizorkin space
UR - http://eudml.org/doc/286397
ER -