Existence Results for Partial Neutral Functional Differential Equations with Nonlocal Conditions

where A is the infinitesimal generator of an analytic semigroup of bounded linear operators, (T (t))t≥0, on a Banach space X; the histories xt : (−∞, 0] → X, xt(θ) = x(t + θ), belongs to some abstract phase space B defined axiomatically; Ω ⊂ B is open; 0 ≤ σ < T ; σ < t0 < t1 < .... < tn ≤ T ; and q : Bn → B; F, G : [σ, T ] × Ω → X are appropriate continuous functions. There exist a extensive literature of differential equations with nonlocal conditions. Motivated by physical applications, Byszewski studied in [1] a nonlocal Cauchy problem modeled in the form ẋ(t) = Ax(t) + f(t, x(t)), t ∈ (σ, T ], x(0) = x0 + q(t1, t2, t3, ..., tn, u(·)) ∈ X, (2)