The existence theorem for bounded Φ-variation solutions to the Caratheodory system is established by using the bounded Φ-variation function theory, which was introduced by Musielak and Orlicz.

In this paper,we discuss the bounded Φ-variation solutions of the Caratheodory system by combining the theories of function of bounded Φ-variation and the Caratheodory system.

Bounded Φ-Variation Solutions of Kurzweil Equations;

The local existence theorem of bounded Φ-variation solutions for impulsive differential equations at fixed times is established.

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.

The main aim of this dissertation is to investigate estimates of Lp modulus of continuity w(f; t)_p (1 ≤ p < ∞) of the class of functions of Λφ-bounded variation and its special cases.

The relation between a class of ordinary differential equations and Kurzweil generalized ordinary differential equations is discussed on weak conditions by using Kurzweil integral,and the theorem of continuous dependence on a parameter of bounded variation solutions for the class of ordinary differential equations is established.

With the aid of Henstock-Kurzweil integral which encompasses the Lebesgue integral,generalized Carathéodory system x =f(t,x)is investigated and the existence theorem of the bounded variation solution for this system is obtained.

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.