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Linear FDEs in the frame of generalized ODEs: variation-of-constants formula

Rodolfo Collegari, Márcia Federson, Miguel Frasson (2018)

Czechoslovak Mathematical Journal

We present a variation-of-constants formula for functional differential equations of the form y ˙ = ( t ) y t + f ( y t , t ) , y t 0 = ϕ , where 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 f ( y t , t ) is Kurzweil integrable with t in an interval of , for each regulated function y . This means that t f ( y t , t ) may admit not only many discontinuities, but it can also be highly oscillating and yet, we are able to obtain...

Linearized comparison criteria for a nonlinear neutral differential equation

Xinping Guan, Sui Sun Cheng (1996)

Annales Polonici Mathematici

A class of nonlinear neutral differential equations with variable coefficients and delays is considered. Conditions for the existence of eventually positive solutions are obtained which extend some of the criteria existing in the literature. In particular, a linearized comparison theorem is obtained which establishes a connection between our nonlinear equations and a class of linear neutral equations with constant coefficients.

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