Abstract
We present a variation-of-constants formula for functional differential equations of the form
, where \(\mathcal{L}\) 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 \(t \mapsto f\left( {y_t,t} \right)\) is Kurzweil integrable with t in an interval of ℝ, for each regulated function y. This means that t \(t \mapsto f\left( {y_t,t} \right)\) 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
and the solutions of the perturbed Cauchy problem
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
, where \(\mathcal{L}\) 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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The research has been supported by FAPESP grant 2011/01316-6 and CNPq grants 304424/2011-0 and 152258/2010-8, Brazil.
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Collegari, R., Federson, M. & Frasson, M. Linear FDEs in the frame of generalized ODEs: variation-of-constants formula. Czech Math J 68, 889–920 (2018). https://doi.org/10.21136/CMJ.2018.0023-17
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DOI: https://doi.org/10.21136/CMJ.2018.0023-17