Stability of abstract dynamic equations on time scales by Lyapunov's second method

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.