Multivalued Integral Manifolds in Banach Spaces

Abstract

We consider a differential inclusion $$\dot{x} \in A(t)x + f(t,x) + g(t,x,X_1)$$ in an arbitrary Banach space $X$ with a general exponential dichotomy, where $X_1$ is the closed unit ball of $X.$ The right-hand side is strongly measurable in the time variable and Lipschitz continuous in the others. We prove the existence and uniqueness of quasibounded solutions corresponding to suitable selectors. The stable and unstable sets of these quasibounded solutions are characterised as graphs of certain multifunctions. Exponential dichotomy criteria are also presented.