Let X be a reflexive Banach space, A: D( A) subset of X --> X the infinitesimal generator of a compact C-0-semigroup S(t): X--> X, t greater than or equal to 0, D a locally closed subset in X and (t, x) --> f(t, x) a function on [a, b) x D which is locally integrably bounded, measurable with respect to t and continuous with respect to x. The main result of the paper is: THEOREM. Under the general assumptions above a necessary and sufficient condition in order that for each (tau, xi) epsilon [a, b) x D there exists at least one mild solution u: [tau, T] --> D of du/dt(t) = Au(t) + f(t, u(t)) satisfying u(tau) = xi is the tangency condition below: There is a negligible subset Z of [a, b,) such that for each t epsilon [a, b)Z and each x epsilon D, lim inf 1/h d(S(h) x + hf (t, x), D) = 0. (C) 2000 Academic Press. [References: 25]
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