Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant Coefficients

Abstract

We study the Hyers-Ulam stability in a Banach space of the system of first order linear difference equations of the form for (nonnegative integers), where is a given matrix with real or complex coefficients, respectively, and is a fixed sequence in . That is, we investigate the sequences in such that (with the maximum norm in ) and show that, in the case where all the eigenvalues of are not of modulus 1, there is a positive real constant (dependent only on ) such that, for each such a sequence , there is a solution of the system with .

1. Introduction

The issue of stability of a functional equation can be expressed in the following way. When must a function satisfying an equation approximately (in some sense) be near an exact solution to the equation? It has been motivated by a question raised in 1940 by Ulam, concerning approximate homomorphisms of groups (see [1, 2]). The first partial answer to Ulam’s question (in the case of Cauchy’s functional equation in Banach spaces) was given by Hyers in [1]. After that result, a great number of papers on the subject have been published (see, e.g., monographs [3–5], survey articles [6–11], and the references given there), generalizing Ulam’s problem and Hyers’s theorem in various directions and to other equations (not necessarily functional) (see [12]). In particular, some results have been proved in [13], which concern the stability of linear difference equations of higher order of form (1). We describe them as follows.

Let be either (the set of nonnegative integers) or (the set of integers), let be either the field of reals or the field of complex numbers , let (the set of positive integers), let be fixed, and let be a given sequence in a Banach space over . The investigation of the Hyers-Ulam stability in of the difference equation actually means a study of the sequences in , satisfying the condition

Let and denote the roots of the characteristic equation of (1), which has the following form: . The following two theorems have been proved in [13] (see also [14, 15]).

Theorem 1. Let and . Suppose that is a sequence in such that (2) holds. Then, there exists a sequence in satisfying (1) such that Moreover,(a) is unique if and only if for or ;(b)if for or , then is the unique sequence in such that (1) holds and ;(c)if and for some , then the cardinality of the set of all sequences in , satisfying (1) and (3), equals the cardinality of .

Theorem 2. Let for some . Then, for any , there exists a sequence in , satisfying inequality (2), such that, for every sequence in , fulfilling recurrence (1), Moreover, if or there is a bounded sequence in fulfilling (1), then can be chosen unbounded.

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 with constant coefficients , ( is fixed): for all , where for are given. If we write then (5) can be expressed in the following simple form: To simplify the notations, we consider and to be elements of , when it is convenient (and when this makes no confusion); that is, we identify with and with .

Our results correspond, in particular, not only to the outcomes in [13], but also to those in [14, 15], where similar problems have been studied for .

2. Some Auxiliary Results

By an elementary induction on , we obtain the following simple observation.

Lemma 3. If a sequence in satisfies (7), then

In this paper, denote the binomial coefficients and for . The subsequent formula is well known: Also, replacing by , we easily obtain that

Further, write Then, is a Banach space and we have the following result, which will be useful in the proof of the main theorem.

Theorem 4. Let be a Jordan matrix of the form with some . If a sequence in satisfies then there exists a sequence in such that

Proof. Let for . Then, by (14), for and (see Lemma 3)
Case 1 (). Define the sequence by (15) with . Then, It is easy to show by induction on that is an upper (right) triangular matrix of the form (it is enough to use the well-known formula ) whence we derive that for .
Since, in view of (10), we have and, by (18), That is, inequality (16) holds.
Case 2 (). Since is an upper triangular matrix of the form it is easy to check that is also an upper triangular matrix for each and has the form Hence, Consequently, in view of (11), Taking into account that for , we deduce that the series is convergent. Take and define by (15) with . Then, (see Lemma 3) Next, by (26), for every , we have whence

3. The Main Result

Let be the eigenvalues of with multiplicities , respectively. There exists a nonsingular matrix in with , where

The next theorem is the main result of this paper.

Theorem 5. Assume that for . For any sequence in , satisfying there exists a sequence in such that

Proof. Let be a sequence in such that (32) holds.
First, consider the case where . Write and for . Then,
Define projections (for ) by for . For simplicity, we write and for . It is easily seen that (in analogous notation) According to Theorem 4 (applied for each , separately), there exists a sequence in such that Clearly,
Let for . Then,
Now, consider the case . Define the linear structure in by and for . Then, is a complex Banach space (see, e.g., [16, page 39], [17], or [18, 1.9.6, page 66]), when endowed with the Taylor norm given by Note that
Define by for . Let and for . Then, is a sequence in and So, by the first part of the proof, there is a sequence in such that Write for . Then, it is easily seen that (33) and (34) are valid (in view of (42)).

Remark 6. The assumption that for cannot be omitted in the general case (at least when ), in view of Theorem 2.
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 (8).

Open Problems. There arises a natural question if some results similar to Theorem 2 and statements (a)–(c) of Theorem 1 can be obtained for difference equation (33) with (also with replaced by ).

Conflict of Interests

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

Acknowledgment

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