regularity properties for convex-like $c(t)$-valued functions on hilbert spaces

In this paper, we introduce several regularity properties for the non-differentiable convex-like c(t)-valued functions which are defined on a hilbert space. the relationships with various regularity properties are investigated. all results are given in terms of the convex subdifferential. non-trivial numerical examples are incorporated to demonstrate the validity of the results established in this paper. to the best of our knowledge, this paper is the first to investigate the regularity properties for the c(t)-valued functions, even in the differentiable case of finite-dimensional spaces.