The Analysis On Practical Stability Of A Kind Of Differential Equations

Theory of Lyapunov stability is now well known and is widely used in concrete problem of the real world. But the range of initial value is too small to apply in actual world. In this situation practical stability generates. Leela firstly put forward the definition of practical stable and gives a series of conditions which satisfied normal differential equations. Mohapatra advances the concept of strict practical stable for different systems. V.L.akshmilantham develops the concepts of strict practical stable to delay differential equations.The main work follow below:firstly, this paper introduces related concepts of practical stability of the systems, and it illustrates the research about the practical stability of the systems has very important significance by comparing the concepts. Secondly, the practical stability of the given systems is researched by Lyapunov function defined on ring neighborhood. It allows that the derivative of Lyapunov function along a solution of the given differential equations is variable number outside of ring neighborhood. Thirdly, differential equation with variation in the initial time is studied by Lyapunov function. We give some efficient conditions by monotonous criterion, k-functions, inequality principle and compared theory. Fourthly, differential equation with finite delay is researched by using auxiliary function. Further, we give sufficient condition which guarantees the strict practical stability, practical asymptotical stability and strong practical stability. Lastly, uniform extremely stability and uniform extremely asymptotic stability of a first-order dynamical system has been discusses through construction of specific Lyapunov function.