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Existence results on semi-infinite intervals for nonlocal evolutions equations and inclusions via semigroup theory. (English) Zbl 1152.34039

The authors establish the existence of mild solutions defined on a semi-infinite interval for first and second order evolution equations and inclusions in a real Banach spaces with nonlocal initial conditions. The authors establish also controllability results for first and second order semilinear non local initial value problems. Mild solutions are obtained. The proofs rely on the theory of semigroups, cosine families, the nonlinear alternative of Leray-Schauder for compact maps, and the nonlinear alternative of Kakutani for compact upper semi-continuous multivalued maps with convex compact values. Solutions are obtained by a diagonalization process as a limit of a subsequence of solutions \(x_n\) which are fixed points of a suitable sequence of compact operators.

MSC:

34G25 Evolution inclusions
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
93B05 Controllability
47N20 Applications of operator theory to differential and integral equations
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