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

Abstract

In this paper, we use the Lyapunov’s second method to obtain new sufficient conditions for many types ofstability like exponential stability, uniform exponential stability, h-stability, and uniform h-stability of the nonlineardynamic equation$x^triangle(t);=;A(t)x(t);+;f(t,;x),;t;in;T_tau^+;:= lbracktau,;infty)_T$ on a time scale T, where A ∈ $C_{rd}$ (T, L(X)) and f : T × 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 $x^triangle(t);=;A(t)x(t);+;f(t),;t;in;T_tau^+,$ to be uniformly exponentially stable or uniformly h-stable, where f ∈ $C_{rd}$ (T, X), the space of rd-continuous functionsfrom 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.