The existence theorems for (1) a differential inclusion in a Banach space and (2) a variational problem geverned by it are presented. In order to solve this problem, some implications of the weak convergence in the space of vector-valued absolutely continuous functions are also explored.
Toru Maruyama. "A generalization of the weak convergence theorem in Sobolev spaces with application to differential inclusions in a Banach space." Proc. Japan Acad. Ser. A Math. Sci. 77 (1) 5 - 10, Jan. 2001. https://doi.org/10.3792/pjaa.77.5