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

Shangquan Bu¹ ; Yi Fang¹

¹Department of Mathematical Science University of Tsinghua Beijing 100084, China

Studia Mathematica, Tome 184 (2008) no. 2, pp. 103-119

Résumé

We study the maximal regularity on different function spaces of the second order integro-differential equations with infinite delay $$(P)\quad\ u''(t)+\alpha {u}'(t)+\frac{d}{dt}\biggl(\int^{t}_{-\infty}b(t-s)u(s)\,ds\biggr) =Au(t)-\int^{t}_{-\infty}a(t-s)Au(s)\,ds+f(t) $$ $(0\leq t\leq2\pi)$ with periodic boundary conditions $ u(0)=u(2\pi), u'(0)=u'(2\pi)$, where $A$ is a closed operator in a Banach space $X$, $\alpha\in \mathbb C$, and $a, b\in L^1(\mathbb R_+)$. 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 $L^p(\mathbb T, X)$, periodic Besov spaces $B_{p,q}^s(\mathbb T, X)$ and periodic Triebel–Lizorkin spaces $F_{p,q}^s(\mathbb T, X)$

Détail

DOI : 10.4064/sm184-2-1

Keywords: study maximal regularity different function spaces second order integro differential equations infinite delay quad alpha frac biggl int infty t s biggr int infty t s leq leq periodic boundary conditions where closed operator banach space nbsp alpha mathbb mathbb fourier multipliers characterize maximal regularity using known results fourier multipliers suitable conditions kernels under which necessary sufficient conditions given problem have maximal regularity mathbb periodic besov spaces mathbb periodic triebel lizorkin spaces mathbb