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Viability for nonautonomous semilinear differential equations. (English) Zbl 0966.34053

Summary: Let \(X\) be a reflexive Banach space, \(A: D(A)\subset X\to X\) the infinitesimal generator of a compact \(C_0\)-semigroup \(S(t): X\to X\), \(t\geq 0\), \(D\) a locally closed subset in \(X\) and \((t,x)\mapsto f(t,x)\) a function on \([a,b)\times D\) which is locally integrably bounded, measurable with respect to \(t\) and continuous with respect to \(x\). The main result of the paper is the theorem:
Under the general assumption above a necessary and sufficient condition in order that for each \((\tau,\xi)\in [a,b)\times D\) there exists at least one mild solution \(u: [\tau, T]\to D\) to \[ {du\over dt}(t)= Au(t)+ f(t, u(t)) \] satisfying \(u(\tau)= \xi\) is the tangency condition below:
There is a negligible subset \(Z\) of \([a,b)\) such that for each \(t\in [a,b)\setminus Z\) and each \(x\in D\), \[ \liminf_{h\downarrow 0} {1\over h} d(S(h)x+ hf(t,x), D)= 0. \]

MSC:

34G20 Nonlinear differential equations in abstract spaces
93C15 Control/observation systems governed by ordinary differential equations
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