Second order Banach space valued differential equations: a semigroup approach

Abstract

Using semigroup techniques we obtain solutions of the differential equation $$ (1)\qquad u^{\prime\prime}(s) = -Bu(s) + f(s)\;\; \textrm{for}\;\; f \in C_{0}(\mathbb{R},\; X)\;\; \textrm{and}\;\; s \in \mathbb{R}, $$ where (B, D(B)) is the generator of a C 0-semigroup $(T(t))_{t \geq 0}$ on a Banach space X. The main idea is to study a semigroup on $C_{0}(\mathbb{R},\; X)$ given as the product of the Gaussian semigroup and the multiplication semigroup induced by $(T(t))_{t \geq 0}$. We use spectral properties of this semigroup in order to solve (1).