On qualitative properties of generalized ODEs

• Autor(a) principal: Acuña, Rogelio Grau
• Data de Publicação: 2016
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: http://www.teses.usp.br/teses/disponiveis/55/55135/tde-26102016-090644/
• topic: Boundedness; Dynamic equations on time scales; Equações diferenciais em medida; Equações diferenciais ordinárias generalizadas; Equações dinâmicas em escalas temporais; Estabilidade de Lyapunov; Funcionais de Lyapunov; Generalized ordinary differential equations; Integral de Kurzweil-Henstock-Stieltjes; Kurzweil-Henstock-Stieltjes integral; Limitação; Lyapunov functionals; Lyapunov stability; Measure differential equations; Prolongamento; Prolongation

In this work, our goal is to prove results on prolongation of solutions, uniform boundedness of solutions, uniform stability as well uniform asymptotic stability (in the classical sense of Lyapunov) for measure differential equations and for dynamic equations on time scales. In order to get our results, we employ the theory of generalized ODEs, since these equations encompass measure differential equations and dynamic equations on time scales. Therefore, to get our results, we start by proving the expected result for abstract generalized ODEs. Then, using the correspondence between the solutions of these equations and the solutions of measure differential equations (see [38]), we extend all the results to these the latter. After that, using the correspondence between the solutions of measure differential equations and the solutions of dynamic equations on time scales (see [21]), we extend all the results to these last equations. Finally, we investigate autonomous generalized ODEs and show that these equations do not enlarge the class of classical autonomous ODEs, even when we consider a more general class of functions as right-hand sides. All the new results presented in this work are contained in papers [16, 17, 18, 19].