Abstract
In this paper, we use the Lyapunov's second method to obtain new sufficient conditions for many types of stability like exponential stability, uniform exponential stability, h-stability, and uniform h-stability of the nonlinear dynamic equation
x(Delta)(t) = A(t)x(t) + f (t, x), t is an element of T-tau(+) := [tau, infinity)(T),
on a time scale T, where A is an element of C-rd(T, L(X)) and f : T x X -> X is rd-continuous in the first argument with f(t, 0) = 0. Here X is a Banach space. We also establish sufficient conditions for the nonhomogeneous particular dynamic equation
x(Delta)(t) = A(t)x(t) + f (t), t is an element of T-tau(+),
to be uniformly exponentially stable or uniformly h-stable, where f is an element of C-rd(T, X), the space of rd-continuous functions from T to X. We construct a Lyapunov function and we make use of this function to obtain our stability results. Finally, we give illustrative examples to show the applicability of the theoretical results.