In this paper, the bounded Φ-variation function and the generalized ordinary differen- tial equation are unified, the concept of Φ-variational stability is established and the stability of the bounded Φ-variation solutions to Kurzweil equations is discussed.

By using the Henstock integral and Ljapunov function,the variational stability of bounded variation solutions for a class of discontinuous system is discussed.

In this paper,by using the Henstock integral and Lyapunov function, the variational stability for bounded variation solutions of a class of discontinuous systems is considered.

At the same time, the variational stability of bounded variation solutions at the effect of perturbation and non-perturbation for the impulsive differential system are discussed, the Ljapunov type theorems for variational stability and asymptotically variational stability for impulsive differential systems are established.

In this paper,the variational stability of bounded variation solutions to homogeneous linear ordinary differential equations are disscussed by using Henstock integral and Lyapunov function,the Lyapunov type theories for variation stability and variational-asymptotically stability of bounded variation solutions are established.

For this kind of discontinuous systems ,we introduce some conceptions of the variational stability for bounded variation solutions,the Lyapunov type theorems for variational stability and variationally-asymptotically stability of bounded variation solutions are established.

By using probabilistic methods,the rate of convergence in terms of function f∈C(R)∩L∞(R)and the convergence rate in φ-variation function for the Gauss-Weierstrass operators were obtained.

If several weights have been calculated by the multiple combination of measurement and stability errors have been calculated by least square method, we may draw the graph of variation of stability errors.