On some nonlocal evolution equations in Banach spaces

Abstract

This note is concerned with the initial value problem for the abstract nonlocal equation \( (Au)' + (Bu) \ni f \) where A is a maximal monotone operator from a reflexive Banach space E to its dual E *, while B is a nonlocal maximal monotone operator from \( L^p (0, T; E)$ to $L^q (0, T; E^*)$ with $p^{-1} + q^{-1}=1, p \in (1,\infty)\). Under proper boundedness and coercivity assumptions on the operators, a solution is achieved by means of a discretization argument. Uniqueness and continuous dependence are also discussed and we prove some estimates for the discretization error. Finally, we deal with the approximation of linear Volterra integrodifferential operators.