Hyers–Ulam stability for a partial difference equation

Abstract

Under the exponential trichotomy condition we study the Hyers–Ulam stability for the linear partial difference equation: \[ x_{n+1,m}=A_nx_{n,m}+B_{n,m}x_{n,m+1}+f(x_{n,m}),\qquad n,m\in \mathbb{Z} \] where $A_n$ is a $k\times k$ matrix whose elements are sequences of $n$, $B_{n,m}$ is a $k\times k$ matrix whose elements are double sequences of $m,n$ and $f:\mathbb{R}^k\rightarrow \mathbb{R}^k$ is a vector function. We also investigate the Hyers–Ulam stability in the case where the matrices $A_n, B_{n,m}$ and the vector function $f=f_{n,m}$ are constant.

Keywords

• partial difference equations
• hyers–ulam stability
• exponential trichotomy.