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Viability for a class of semilinear differential equations of retarded type

Research paper by Qi-xiang Dong, Gang Li

Indexed on: 23 Mar '09Published on: 23 Mar '09Published in: Applied Mathematics-A Journal of Chinese Universities



Abstract

Let X be a Banach space, A: D(A) ⊂ X → X the generator of a compact C0-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 ∈ [t0, t0 + T], with initial condition \( u_{t_0 } \) = ϕ ∈ C([−q, 0];X), is the tangency condition lim infh↓0h−1d(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.