On the @,-Stability of Comparison Differential Systems

For many years now, the variety of problems of the qualitative properties of differential equations in the context of Lyaponov second method has been successfully investigated in a unified way using the comparison principle [S]. In this method, the qualitative properties of the system of differential equations are inferred from the corresponding properties of the solutions of the system of comparison equations. In order to successfully employ this method, it is generally required that the comparison systems possess some special properties. The conditions under which the scalar comparison equations possess positivity and stability behaviour have been investigated using scalar Lyapunov function method by Brauer [l], Siljak and Grujic [9], and others. Ladde [3] gave suficient conditions for nonnegativity and stability of solutions of systems of comparison equations. In his investigation of stability of comparison differential systems, Ladde [3] considered the asymptotic stability and exponential asymptotic stability of the comparison differential system using the method of vector Lyapunov functions. However, imbedded in the method of vector Lyapunov function is the requirement of quasimonotone nondecreasing property of the comparison system. But quasimonotonicity of the comparison system is not a necessary condition for the system to be stable. Thus the limitation of the application potential of this general and effective method is due to the fact that 307