^{1}

^{2}

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We study the Hyers-Ulam stability in a Banach space

The issue of stability of a functional equation can be expressed in the following way.

Let

Let

Let

if

if

Let

We somehow complement those results in this paper by the study of the Hyers-Ulam stability of the following system of first order linear difference equations in

Our results correspond, in particular, not only to the outcomes in [

By an elementary induction on

If a sequence

In this paper,

Further, write

Let

Let

Since, in view of (

Let

The next theorem is the main result of this paper.

Assume that

Let

First, consider the case where

Define projections

Let

Now, consider the case

Define

The assumption that

It seems that our method cannot be easily applied to the systems of linear difference equations of higher orders, because it is difficult in such cases to obtain a formula analogous as (

The authors declare that there is no conflict of interests regarding the publication of this paper.

Bing Xu is supported by NSFC (China) Grant no. 11101295.