On the Application of a Lusin-Type Theorem to Differential Inclusions

Résumé

In this paper, we establish a special version of Lusin’s theorem and study its application to dif- ferential inclusions. Lusin’s theorem expresses the realization of Littlewood’s second principle and establishes a connection between measurability and continuity. We prove this result and examine its use in the analysis of differential inclusions. As an application, we consider a differential inclusion problem with compact-valued right-hand side and an additional initial derivative condition, which naturally arises in the theory of differential inclusions. By combining the obtained Lusin result with selection methods, we derive sufficient conditions under which classical solutions can be constructed without imposing convexity assumptions on the value sets. The results clarify how Lusin arguments are important in the theory of classical solutions for differential inclusions.