Stability And Oscillation Of Functional Differential Equations

Following the development of science and technology, more and more differential difference equations and differential integral equations are researched in many fields, for instance mechanics, physics,biomathematics, economics, autocontrol and communication, because these equations are used to make a study of problems more accurate than ordinary differential equations. Then these equations are called functional differential equations.The study on functional differential equations is developed rapidly after we consider the equations with functional analysis. Every year we get a lot of papers on basic theory, stability, oscillation, periodic solutions, solution operator theory, branch theory, etc.The first part of this paper discusses the stability theorems for P- retarded functional differential equations which is first defined in [1] and called extended functional differential equations. This kind of equations includes retarded functional differential equations with bounded delay and some retarded functional differential equations with unbounded delay. Sufficient conditions are given to insure thatthe zero solution of P- retarded functional differential equations is uniform stable. The upper bound of V is allowed to be positive function under some conditions. For the study of asymptotic stability, the assumption of boundedness for / has been removed, and the upper-bound of V is allowed to be negative semi-definite under some conditions. The second part of the paper considers some kind of the first order neutral differential equation with multiple delays. Sufficient conditions are given to insure that all solutions of this equation are oscillation. The conclusions in [3] are extended.Ma Liwei (Applied Mathematics) Supervised by Gao Guozhu...