Degenerate Volterra equations in Banach spaces

Abstract

This paper is concerned with degenerate Volterra equations $Mu(t)+\int_0^tk(t-s)Lu(s)\, ds=f(t)$ in Banach spaces, both in the hyperbolic case and the parabolic one. The key assumption is played by the representation of the underlying space $X$ as a direct sum $X=N(T)\oplus\overline{R(T)}$, where $T$ is the bounded linear operator $T=ML^{-1}$. Hyperbolicity means that the part $\tilde{T}$ of $T$ in $\overline{R(T)}$ is an abstract potential operator, i.e., $-\tilde{T}^{-1}$ generates a $C_0$-semigroup, and parabolicity means that $-\tilde{T}^{-1}$ generates an analytic semigroup. A maximal regularity result is obtained for parabolic equations. We will also investigate the cases where the kernel $k(\cdot)$ is degenerate or singular at $t=0$ using the results of Prüss [8] on analytic resolvents. Finally we consider the case where $\lambda$ is a pole for $(\lambda L+M)^{-1}$.