It seems that the first result on the stability of functional equations appeared in the famous book of Pólya and Szegö [13] and concerns the Cauchy functional equation on the set of positive integers. But the stability problem of functional equations was originally raised by S.M. Ulam [18] in 1940 who formulated a question concerning the perturbation of homomorphisms of metric groups. In 1941, this problem was solved by D. H. Hyers [9] for the Cauchy functional equation in Banach spaces. After Hyers’ result many papers dedicated to this topic have been published (see e.g. [3], [6], [7], [11], [14]).

Recall the result on stability for the Cauchy’s functional equation (see [4], [18]):

”Let $(G,+)$ be an abelian group, $X$ a Banach space and $\varepsilon>0$. Then for every $f:G\rightarrow X$ satisfying

there exists a unique additive function $a:G\rightarrow X$ such that

Remark that conditions (1.1) and (1.2) can be rewritten in the form

respectively

where $B(0,1)$ is the unit ball of the Banach space $X$.

This new form of the relations (1.1) and (1.2) suggests the possibility to formulate the problem of Ulam stability in topological vector spaces for various equations. The goal of this paper is to give a result on Ulam stability for the linear difference equation with constant coefficients in locally convex spaces and to obtain in some cases the best Ulam constant.

Throughout this paper, it will be assumed that $X$ is a sequentially complete locally convex space over the field $\mathbb{C}$ of complex numbers.

Let $A$ be a nonempty subset of $X$.

The following relation holds:

If $A$ is a convex set and $\lambda,\mu$ are positive numbers, then

For more details and terminology, see [17].

By a convex series of elements of $A$ we mean a series of the form $\sum\limits_{n=1}^{\infty}\lambda_{n}a_{n}$, where $a_{n}\in A$ and $\lambda_{n}\geq 0,\ n\geq 1$, and $\sum\limits_{n=1}^{\infty}\lambda_{n}=1$. The set $A$ is called:

• (i)

  CS-closed if the sum of every convergent series of elements of $A$ belongs to $A$;

  Report issue for preceding element
• (ii)

  CS-compact if every convex series of elements of $A$ is convergent to an element of $A$.

  Report issue for preceding element

It can be proved that every CS-compact set is a CS-closed set and bounded, and every CS-closed set is a convex set (for more details, see [10]).

As a consequence of the previous assertions it follows that if $\sum\limits_{n=1}^{\infty}p_{n}$ is a convergent series of positive terms, $A$ is a CS-compact set and $a_{n}\in A,\ n\geq 1$, then

All over this paper by $V$ we denote a CS-compact and balanced neighbourhood of $0_{X}$.

We consider the linear difference equation of order $p$

where $a_{1},a_{2},\ldots,a_{p}\in\mathbb{C}$, $x_{0},x_{1},\ldots,x_{p-1}\in X$, and $p\in\mathbb{N}=\{1,2,\ldots\}.$

The equation (1.3) is called Ulam stable (or $V$-Ulam stable) if there exists a positive constant $K$ such that for every $\varepsilon>0$ and every $(x_{n})_{n\geq 0}$ satisfying

there exists a sequence $(y_{n})_{n\geq 0}$ in $X$ such that

A positive number $K$ satisfying (1.5) is called an Ulam constant of (1.3). Denote by $K_{R}$ the infimum of all Ulam constants of (1.3). Generally $K_{R}$ is not necessary an Ulam constant of (1.3) (see [16]), but if it is an Ulam constant we call it the best Ulam constant or the Ulam constant of (1.3). If in (1.4) the number $\varepsilon$ is replaced by a sequence of positive numbers $(\varepsilon_{n})_{n\geq 0}$ and $K\varepsilon$ in (1.5) by a sequence of positive numbers $(\delta_{n})_{n\geq 0}$ depending on $(\varepsilon_{n})_{n\geq 0}$ then the linear difference equation is called generalized Ulam stable.

Let

be the characteristic equation of (1.4) and denote by $r_{1},\ldots,r_{p}$ its complex roots.

A first result on Ulam stability for the equation (1.3) in Banach spaces was given by D. Popa in [14] who proved that if all the roots of (1.6) are not on the unit circle then the equation (1.3) is stable with the Ulam constant

Later J. Brzdȩk, D. Popa and B. Xu proved that if the characteristic equation (1.6) has at least a root on the unit circle, then the equation (1.3) is not stable in Ulam sense ([4], [7], [8]).

A first approach of Ulam stability for a first order linear difference equation in topological vector spaces was considered by S. Moslehian and D. Popa in [11]. The results of the present paper extend and complement the results obtained in [11]. For more details on Ulam stability for linear difference equations we refer the reader to [1], [2], [5], [12], [15].

First we give a result on Ulam stability for a first order linear difference equation of the form

where $a\in\mathbb{C},\ a\neq 0$, and$\ (b_{n})_{n\geq 0}$ is a sequence in $X,\ x_{0}\in X$.

A result on classical Ulam stability for the linear difference equation $x_{n+1}=a_{n}x_{n}+b_{n},\ n\geq 0$ was given in [11] in the case when $(a_{n})_{n\geq 0}$ is a sequence in $K\backslash\{0\}$, and the field $K$ is $\mathbb{R}$ or $\mathbb{C}$. In the next theorem we present a result on generalized Ulam stability for the case where $\left(a_{n}\right)_{n\geq 0}$ is a constant sequence in $K$.

As a consequence it follows a result on classical Ulam stability for the equation (2.1)(see [11]).

In the next theorem we obtain the best Ulam constant for the equation (2.1).

In this section we give stability and nonstability results for the linear difference equation (1.3).

To prove the nonstability result for the equation (1.3) we need the following Lemma, concerning the nonstability of the linear difference equation of order one.