Existence Results for Nonconvex Nonautonomous Differential Inclusions in Hilbert Spaces

We establish a solvability criterion for nonautonomous time-evolution inclusions governed by the right-hand side without the convexity assumption. In this study, we examine the problem x˙(t)∈F(t,x(t))a.e.onI0:=[0,T0], for some T0>0, where F has a closed graph, constrained within the subdifferential operator of a convex, time-dependent potential g(t,·). This work extends the existing literature, which has primarily focused on the autonomous case or time-dependent mappings with a time-independent convex potential in finite-dimensional spaces. Under novel assumptions that the potential g is convex in the state variable and Lipschitz in time, we establish a solvability criterion. An example of the applicability of our result for nonconvex nonautonomous differential inclusions is stated. As a significant application of our main theorem, we demonstrate that a certain class of implicit nonconvex sweeping processes with an unbounded perturbative term admits solutions.