The Oscillation, Stability And Solution Of Impulsive Differential System Of Alternate Type

This paper mainly studies a class of impulsive differential equations of alternately advanced and retarded type:where x(t) ∈ R^m and x expresses state variable ; A , B and D_n are m × m constant matrices; A is nonsigular matrices , C₀ is constant m-vector ,Z denotes integer set;[·] express the greatest integer function .The argument deviation:The argument deviation ι(t) is negative in [2n — 1,2n) , so (1) is said to be of advanced type ; and it is positive in (2n, 2n + 1) , so (1) is said to be of retarded type.Therefore (1) is said to be of alternately type in [2n — 1,2n +1). In paper [1] and [2] ,the authors investigated respectivly RFDE(l) andwhere a, b ∈ R, a ≠ 0. The authors have established their theorems.In paper [3],the delay equation with impulsive perturbations have been discussed.The above papers affect this paper.This paper imposes the impulsive equation on delay equation (1) and (1) and generalizes some known results of [1] and [2]. In chapter 2 we establish the existence uniqueness theorems of solutions of systems (1)-(2) , prove these theorems and obtain the solutions .The equation (1) with variable coefficient is also discussed.In chapter 3,we get sometheorems of stability of solutions of systems (l)-(2) and (l') — (2).Suffient condition is obtained for stability of ordinary solution of system with constant coefficient.In chaper 4,we have studied the problem of oscillation of system.We mainly discuss the system with variable cofficient and obtain theorems of os-cillationjsufficient and necessary condition is given for oscillation of solution of system (l)-(2).