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Study about oscillation for functional differential equations

Abstract

The differential equations in measure allow us to treat situations more general than the differential equations with retardation and even more general than the impulsive ordinary ones. The study of this type of equation began with W. Shchedeke, P. Das and R. Sharma in the early 1970s. Since then several authors have devoted themselves to studying this class of equations. Also, with the objective of generalizing certain results on continuous dependence of solutions of differential equations with initial data. J. Kurzweil introduced in 1957 the notion of generalized ordinary differential equations for functions in Euclidean and Banach spaces. This generalization included the notion of generalized Perron integral, as it is now called the integral of Kurzweil. We intend, with this project, to study a class of differential equations, especially with respect to oscillation, through the study of integral equations, through the integral of Kurzweil, or with generalized ordinary differential equations. When working with this type of integral, we can consider more general functions, which may have discontinuities, being very oscillating, that is, of unlimited variation. With this study we want to establish oscillation and non-oscillation criteria for a much larger class of functions than those usually considered. We intend to obtain similar results for differential nonlinear differential equations with discrete delay. (AU)

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