Our goal is to establish some sufficient conditions for the solvability of the nonlocal final value problem involving a class of partial differential equations, which describes the anomalous diffusion phenomenon. Our analysis is based on the theory of completely positive functions, resolvent operators and fixed point arguments in suitable function spaces. Especially, utilizing the regularity of resolvent operators, we are able to deal with non-Lipschitz cases. The obtained results, in particular, extend recent ones proved for fractional diffusion equations.