Delay evolution equations with mixed nonlocal plus local initial conditions

We consider the delay differential equation u' (t) is an element of Au(t) + f(t, u(t)), t is an element of R+, where A is the infinitesimal generator of a nonlinear semigroup of contractions in a Banach space X and f is continuous, subjected to a general mixed nonlocal + local initial condition of the form u(t) = g(u)(t) + psi(t), t is an element of[-tau, 0]. We prove that under natural conditions on A, f, g and psi the problem above has at least one C-0-solution. Applications to T-periodic and to T-anti-periodic problems, as well as an illustrative example referring to the nonlinear diffusion equation are also included.