Viability for a class of semilinear differential equations of retarded type

Let X be a Banach space, A: D(A) ⊂ X → X the generator of a compact C ₀-semigroup S(t): X → X, t ≥ 0, D a locally closed subset in X, and f: (a, b)×X → X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u′(t) = Au(t) + f(t, u(t − q)), t ∈ t ₀, t ₀ + T], with initial condition = ϕ ∈ C(−q, 0];X), is the tangency condition lim inf _h↓0 h ⁻¹ d(S(h)v(0)+hf(t, v(−q));D) = 0 for almost every t ∈ (a, b) and every v ∈ C(−q, 0];X) with v(0), v(−q) ∈ D. Supported by the National Natural Science Foundation of China (10571150) and the Natural Science Foundation of Jiangsu Education Committee (07KJB110131)