Existence Results for First and Second Order Nonconvex Sweeping Processes with Perturbations and with Delay: Fixed Point Approach

Abstract

Abstract We are interested in existence results for nonconvex functional differential inclusions. First, we prove an existence result, in separable Hilbert spaces, for first order nonconvex sweeping processes with perturbation and with delay. Then, by using this result and a fixed point theorem we prove an existence result for second order nonconvex sweeping processes with perturbation and with delay of the form 𝑢˙ (𝑡) ∈ 𝐶(𝑢(𝑡)), 𝑢¨(𝑡) ∈ –𝑁𝑃(𝐶(𝑢(𝑡)); 𝑢˙(𝑡)) + 𝐹(𝑡, 𝑢˙𝑡) when 𝐶 is a nonconvex bounded Lipschitz set-valued mapping and 𝐹 is a set-valued mapping with convex compact values taking their values in finite dimensional spaces.