ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

9. Ordinary Differential Equations and Dynamical Systems

1991 MS Classification: 34, 39 Zhang, Shunian, Shanghai Jiaotong University, China: Stability of neutral delay difference systems Consider the neutral delay difference systems of the general form: Δ(D(n, x_{n})) = f(n, x_{n}), n ∈ Z^{+}, where Z^{+} denotes the set of non-negative integers, x ∈ R^{k} with some positive integer k, Δ is the forward difference operator, let C = {φ : {−r, −r + 1, …, −1, 0} → R^{k}} with some r ∈ Z^{+}; de-

fine the norm of φ ∈ C as ∥φ∥ = max_{s=−r,−r+1,...,0} |φ(s)| with |·| a norm in R^{k},

denote C_{H} = {φ ∈ C : ∥φ∥ < H} for some constant H > 0, and D, f : Z^{+} × C_{H} → R^{k}, while x_{n} ∈ C is defined as x_{n}(s) = x(n + s) for s = −r, −r+1, …, 0. Two kinds of stability criteria are established in this work. One is by means of the discrete Liapunov functionals; while the other is with the discrete Liapunov functions. Both make use of Razumikhin techniques. References: S. Elaydi, and S. Zhang, Stability and periodicity of difference equations with finite delay, Funkcial. Ekvac. , 37(1994), 401-413; S. Zhang, Razumikhin techniques in delay difference systems, PanAmerican Math. J. 3 (1993), 1–16.