Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces
Volume 184 / 2008
Abstract
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 $F_{p,q}^s(\mathbb T, X)$
