Delay evolution equations with mixed nonlocal plus local initial conditions

Abstract

We consider the delay differential equation u′(t) ∈ Au(t) + f(t, ut), t ∈ 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) + ψ(t), t ∈ [-τ, 0]. We prove that under natural conditions on A, f, g and ψ the problem above has at least one C0-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.