Boundedness and periodicity in impulsive ordinary and functional differential equations

Abstract In this paper, we study the boundedness and periodicity of solutions for impulsive ordinary differential equations (ODEs) and functional differential equations (FDEs). By using Horn’s fixed point theorem, Hale–Yoshizawa type criteria for the existence of T -periodic solutions are established. To use these criteria, we also give two new boundedness theorems, and establish new existence results for T -periodic solutions which show that the impulsive perturbations do contribute to yielding periodic solutions even when the underlying systems do not enjoy periodic solutions.

[1]  Shui-Nee Chow,et al.  Remarks on one-dimensional delay-differential equations , 1973 .

[2]  T. A. Burton,et al.  Stability and Periodic Solutions of Ordinary and Functional Differential Equations , 1986 .

[3]  J. L. Massera.  The existence of periodic solutions of systems of differential equations , 1950 .

[4]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[5]  D. Bainov,et al.  Impulsive Differential Equations: Periodic Solutions and Applications , 1993 .

[6]  D. R. Smart.  Fixed Point Theorems , 1974 .

[7]  Zhiguo Luo,et al.  New Razumikhin type theorems for impulsive functional differential equations , 2002, Appl. Math. Comput..

[8]  James H. Liu.  Bounded and periodic solutions of differential equations in Banach space , 1994 .

[9]  James H. Liu,et al.  Bounded and periodic solutions of finite delay evolution equations , 1998 .

[10]  T. A. Burton,et al.  Unified boundedness, periodicity, and stability in ordinary and functional differential equations , 1986 .

[11]  Jianhua Shen,et al.  Impulsive stabilization of functional differential equations by Lyapunov-Razumikhin functions , 1999 .

[12]  G. Makay.  Periodic solutions of linear differential and integral equations , 1995 .

[13]  Jack K. Hale,et al.  Fixed point theorems and dissipative processes. , 1973 .

[14]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[15]  Jianhua Shen,et al.  Razumikhin type stability theorems for impulsive functional differential equations 1 1 Research was , 1998 .