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Generalized linear differential equations in a Banach space: continuous dependence on a parameter. (English) Zbl 1268.45009
This paper is concerned with the solution of generalised linear differential equations of the form \[ x(t)= \tilde{x} + \int_a^t d[A]x +f(t)-f(a), \quad t \in [a,b], \] in a Banach space \(X\). The overall aim is to provide results on the continuous dependence of the solutions on a parameter.
Following preliminaries, in which existing results are reviewed, the main part of the paper presents a sequence of results covering existence, uniqueness, and various technical results on representations of solutions. Section 5 of the paper illustrates how the methods can be applied to deal with linear equations on time scales and among the closing remarks are some suggestions about applications to linear functional differential equations with impulses.

MSC:
45N05 Abstract integral equations, integral equations in abstract spaces
45A05 Linear integral equations
45D05 Volterra integral equations
34N05 Dynamic equations on time scales or measure chains
34K45 Functional-differential equations with impulses
34K06 Linear functional-differential equations
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