| Abstrakt: |
Abstract: We present a variation-of-constants formula for functional differential equations of the form y˙ = L(t)yt + f(yt, t), yt0 = ϕ, where L is a bounded linear operator and ϕ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application t 7→ f(yt, t) is Kurzweil integrable with t in an interval of R, for each regulated function y. This means that t 7→ f(yt, t) may admit not only many discontinuities, but it can also be highly oscillating and yet, we are able to obtain a variation-of-constants formula. Our main goal is achieved via theory of generalized ordinary differential equations introduced by J. Kurzweil (1957). As a matter of fact, we establish a variation-of-constants formula for general linear generalized ordinary differential equations in Banach spaces where the functions involved are Kurzweil integrable. We start by establishing a relation between the solutions of the Cauchy problem for a linear generalized ODE of type dx dτ = D[A(t)x], x(t0) = xe and the solutions of the perturbed Cauchy problem dx dτ = D[A(t)x + F(x, t)], x(t0) = x. e Then we prove that there exists a one-to-one correspondence between a certain class of linear generalized ODE and the Cauchy problem for a linear functional differential equations of the form y˙ = L(t)yt, yt0 = ϕ, where L is a bounded linear operator and ϕ is a regulated function. The main result comes as a consequence of such results. Finally, because of the extent of generalized ODEs, we are also able to describe the variation-of-constants formula for both impulsive FDEs and measure neutral FDEs. |
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