Métodos da média periódico e não-periódico para equações diferenciais funcionais em medida com impulsos e equações dinâmicas funcionais com impulsos

Abstract

The area of differential equations occupies a central role in mathematics. From a theoretical point of view, it is a field that intersects several other distinct areas and connects them. From an applied mathematics point of view, it is an area with numerous applications in the natural sciences and in computational modeling. Due to this, it is an area that attracts many researchers and in which there is great academic production. The objective of this work is to study certain types of measure functional differential equations and functional dynamic equations on time scales, as well as averaging methods for these equations. For this, we studied the Kurzweil integral and its properties, which were developed in the 1950s for solving problems within the ODE area, and the time scales, which were introduced in 1988, in Stefan Hilger’s doctoral thesis, in order to unify discrete and continuous analysis. We use these theories to relate functional differential equations to dynamic equations and to prove results on both. Finally, we use these results and relationships to study the averaging method, which allows us to approximate the solutions of these equations by solutions of simpler equations.