The Basic Theory for Partial Functional Differential Equations and Applications

Let (X, |·|) be an infinite dimensional Banach space and L(X) be the space of bounded linear operators from X into X. Suppose that r > 0 is a given real number. C ([−r, 0] , X) denotes the space of continuous functions from [−r, 0] to X with the uniform convergence topology and we will use simply CX for C ([−r, 0] , X). For u ∈ C ([−r, b] , X), b > 0 and t ∈ [0, b], let ut denote the element of CX defined by ut(θ) = u(t + θ), −r ≤ θ ≤ 0. By an abstract semilinear functional differential equation on the space X, we mean an evolution equation of the type { du dt (t) = A0u(t) + F (t, ut), t ≥ 0, u0 = φ, (1)