Work

Existence Results for Nonconvex Nonautonomous Differential Inclusions in Hilbert Spaces

Year: 2025

Type: article

Abstract: 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.

Source: Mathematics

DOI: 10.3390/math13243929

Author: Messaoud Bounkhel

Institution: King Saud University

Language: English

Topic: Nonlinear Differential Equations Analysis

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funder: King Saud University