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Existence of mild solutions for semilinear differential equations with nonlocal and impulsive conditions. (English) Zbl 1417.34057

Summary: This paper is concerned with the existence of mild solutions for impulsive semilinear differential equations with nonlocal conditions. Using the technique of measures of noncompactness in Banach and Fréchet spaces of piecewise continuous functions, existence results are obtained both on bounded and unbounded intervals, when the impulsive functions and the nonlocal item are not compact in the space of piecewise continuous functions but they are continuous and Lipschitzian with respect to some measure of noncompactness, and the linear part generates only a strongly continuous evolution system.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B37 Boundary value problems with impulses for ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34G25 Evolution inclusions
47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
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