Existence of solutions to differential inclusions with primal lower nice functions

We prove the existence of absolutely continuous solutions to the differential inclusion $$ \dot{x}(t)\in F(x(t))+h(t,x(t)), $$ where F is an upper semi-continuous set-valued function with compact values such that $F(x(t))\subset \partial f(x(t))$ on [0,T], where f is a primal lower nice function, and h a single valued Caratheodory perturbation.