Vol.15 No.1 ACTA MATHEMATICAE APPLICATAE SINICA Jan., 1999 EXPONENTIAL STABILITY OF A PARTIAL DIFFERENCE EQUATION WITH NONLINEAR PERTURBATION CHENG SuI-SuN (~~:~k) (Department of Mathematics, Tsing Hua University, Hsinchu 30043, Chinese Taiwan) LIN YIZHONG (~-~) (Department of Mathematics, Fujian Normal University, Fuzhou 350007, China) Abstract Stability criteria are established for partial difference equations with nonlinear pertubations. Key words. Partial difference equation, stability criteria, exponential stability 1. Introduction Partial difference equations have appeared in many branches of mathematics. Indeed, Lagrange and Laplace have discussed such equations in relation to probabihty [11, Courant et al. [2] have considered them in relation to differential equations of mathematical physics. In recent years, signal and image processing theory (see e.g. [3]) also makes use of the theory of partial difference equations. Oscillation theory for these equations has been investigated by a number of authors recently[ 41. However, a systematic investigation of the stability theory of partial difference equations is relatively unknown. In [5], stability criteria are obtained for the trivial solutions of partial difference equations of the form O~Ui+1,j+ 1 ~- ~Ui+l,j ~- ~[Ui,j+ 1 "Jr ~Ui] -- 0, (i, j) e z ~, where a,/3, 7 and 6 are real constants, Z is the set of nonnegative integers and z: = {(~,~) I ~,~ = 0, ~,.. }. When the trivial solution of a linear homogeneous evolutionary equation is stable, it is of great interest to know that it remains stable when nonlinear perturbation is introduced. This is due to the fact that an equation with nonlinear perturbation is usually a better model for realistic problems. In this paper, we will be concerned with a class of perturbed partial difference equations of the form O~U~+I,j+I -~-~Zti+l,j -~" "YUi,j+l -~- 6Uij = F (u~j ), (i,j) e z ~, (1) Received October 10, 1996. Revised May 26, 1998.