Stability Of Dynamic Equation On Time Scale

We study the sufficient and necessary conditions of stability,asymptotical stabil-ity,uniform stability and instability of the equilibrium solution x = 0 to dynamic equation on time scale x~△ = f(t,x), (t,x) ∈ T x R~n.This paper firstly introduce the based knowledge of Time Scale,stability and several lemmas.In the proof,sufficient conditions are proved mainly by properties of K function and continuity of Lyapunov function:in necessary proof,we separately structure Lyapunov functions which satisfy the conditions of the theorems, such that the results establish.Finally we give an example to utilize the theorems to ensure stability of the equilibrium solution x = 0 to dynamic equation on time scale x~△ = f(t, x), (t, i) ∈ T ×R~n. On the other hand, we have known the stability of the equilibrium solution,and structure appropriate Lyapunov function V, such that relevant results establish.