Boundedness of solutions

This chapter presents the concepts of uniform boundedness, quasiuniform boundedness, and uniform ultimate boundedness in the scenery of generalized ordinary differential equations (ODEs). It includes criteria of uniform boundedness and uniform ultimate boundedness for the generalized ODE. The chapter presents some results concerning the boundedness of solutions for the generalized ODE using Lyapunov functionals. The correspondence between the solutions of the generalized ODE and the solutions of the measure differential equation (MDE) allows us to conclude that the MDE is also uniformly bounded. By using the results and the correspondence between generalized ODEs and MDEs, the chapter presents the results on boundedness for MDEs. It concludes by showing the extension of one of these results to a certain impulsive differential equation.

Palavras-chave

Lyapunov functional, Measure differential equation, Ordinary differential equations, Quasiuniform boundedness, Uniform boundedness, Uniform ultimate boundedness