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Existence of solutions of semilinear time varying differential equations with impulses, delay, and nonlocal conditions

Abstract:

In this work, we prove the existence of solutions for semilinear time varying systems of ordinary differential equations with impulses, delay, and nonlocal conditions. To solve this problem, some ideas are taken from a similar problem which is studied in Hilbert spaces in the autonomous case A(t) = A, where A generates a strongly continuous semi group T(t). In this paper, the state matrix A(t) is a n× n continuous matrix and rather than a semigroup, we have an evolution operator U(t, s) to describe the solution through an integral equation. Here, the existence of solutions is proved using the Karakostas fixed point theorem and also an uniqueness theorem of the solution is presented. Both theorems, existence and uniqueness, are proved on the interval [- r, τ]. Indeed, through a prolongation theorem which is also presented, the results are extended to the interval [- r, + ∞). Moreover, an application of this work is discussed.