Stability for nonlinear generalized ODEs and for retarded Volterra-Stieltjes integral equations and control theory for these equations and for dynamic equations on time scale - Silva, Fernanda Andrade da (2021)
This work has two main purposes. The first one is to prove Lyapunov-type theorems and converse Lyapunov theorems on boundedness of solutions, regular stability and uniform stability for generalized ODEs and retarded Volterra-Stieltjes integral equations. As an application, we establish necessary and sufficient conditions for a system of perturbed generalized ODEs and for a system of perturbed retarded Volterra-Stieltjes integral equations, defined in a Banach space, to be asymptotically controllable. The second purpose is to investigate the existence and uniqueness of a solution for a linear Volterra-Stieltjes integral equation of the second kind, as well as for a homogeneous and a nonhomogeneous linear dynamic equations on time scales, whose integral forms contain Perron -integrals defined in Banach spaces. We also provide a variation-of-constant formula for a nonhomogeneous linear dynamic equations on time scales and we establish results on controllability for these equations. The new results presented in this work are contained in 3 papers (see [46]) and in two chapters of the book [13].