On Stability Oscillating and Asymptotic Behavior of Solutions of Nonlinear Differential Equations

Abstract

The variety of problems of qualitative properties of a nonlinear system of differential equations of the type

x'=f(t,x), (1)

and the perturbed system

x' = f(t,x) + R(t,x), (2)

where j,R E C[ + X n, n], and f(t,O) = R(t,O) =: 0, has been successfully studied by different approaches based on Liapunov's direct method, such as, cone­ valued Liapunov function and comparison technique. Liapunov's second method is an important ingredient, as it is a powerful tool in stability theory of nonlinear systems of differential equations. Theoretically this method is very appealing and has many applications. Liapunov function and related differential inequalities in + Liapunov's second method play essential roles to determine the stability behavior of solutions of linear and nonlinear systems. The major advantage of this method is that the stability in the large can be obtained without any period knowledge of solutions.

Lakshmikantham and Leela [23-25] initiated the development of the theory of differential inequalities through cone and cone-valued Liapunov function. They used the comparison principal to improve and extend different notions of stability such as, event.ual stability and £P-stability for the nonlinear system (1), integral stability and total stability for both nonlinear systems (1) and (2). They used the comparison technique to improve relative stability for the two differential systems

x' = JI(t,x), y' = h(t,y),

x(to) = xo, y(to) =Yo, (3)

where fi,h E C[+ x n], and f1(t, 0) = h(t,0) = 0. Moreover they used the same technique to improve partial stability of a nonlinear system of differential equations of the form

x' = F(t, x,y), y' = H(t,x,y),

x(to) = xo, y(to) =Yo, (4)