Differential Inclusions with Mixed Semicontinuity Properties in a Banach Space

A differential inclusion whose right-hand side is the sum of two set-valued mappings in a separable Banach space is considered. The values of the first mapping are bounded and closed but not necessarily convex, and this mapping is Lipschitz continuous in the phase variable. The values of the second one are closed, and this mapping has mixed semicontinuity properties: given any phase point, it either has closed graph and takes a convex value at this point or is lower semicontinuous in its neighborhood. Under additional assumptions related to measurability and growth conditions, the existence of a solution is proved.