Invariance properties in the theory of stability for ordinary differential systems and applications ^†

^†This research was supported by the Fundação de Amparo à Pesquisa do Estado de São Paulo and the Conselho Nacional de pesquisas, Brazil

Abstract

From Yoshizawa's result [5-a- Theorem 5] we have that, under convenient conditions, the ω-limit set of every bounded solution of is contained in the set of all x such that W(x) = 0. G{t, x) is assumed to be integrable and F(t, x) bounded in a certain sense; W(x) is a non-negative continuous function such that V(t, x) ≦ - W(x) where V(t, x) is a non-negative continuous function, locally Lipschtzian with respect to x, where the derivative is considered with respect to the above system. In this paper, by using ideas closely related to the ones in Yoshizawa's mentioned theorem, we prove a more general result concerning the case in which W depends upon t and x and G satisfies a condition weaker than the one in [5-a]. By using the above mentioned extend and a modified version of Yoshizawa's invariance principle [5-a-Theorem 3]we obtain sufficient conditions under which, for every solution x(t) of a second order differential equation

we can guarantee that .

^†This research was supported by the Fundação de Amparo à Pesquisa do Estado de São Paulo and the Conselho Nacional de pesquisas, Brazil

^†This research was supported by the Fundação de Amparo à Pesquisa do Estado de São Paulo and the Conselho Nacional de pesquisas, Brazil

Notes

^†This research was supported by the Fundação de Amparo à Pesquisa do Estado de São Paulo and the Conselho Nacional de pesquisas, Brazil