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

Bing Xu,Janusz Brzdęk

doi:10.1155/2015/269356

Abstract

We study the Hyers-Ulam stability in a Banach spaceXof the system of first order linear difference equations of the formxn+1=Axn+dnforn∈N0(nonnegative integers), whereAis a givenr×rmatrix with real or complex coefficients, respectively, and(dn)n∈N0is a fixed sequence inXr. That is, we investigate the sequences(yn)n∈N0inXrsuch thatδ∶=supn∈N0yn+1-Ayn-dn<∞(with the maximum norm inXr) and show that, in the case where all the eigenvalues ofAare not of modulus 1, there is a positive real constantc(dependent only onA) such that, for each such a sequence(yn)n∈N0, there is a solution(xn)n∈N0of the system withsupn∈N0yn-xn≤cδ.

Highlights

The issue of stability of a functional equation can be expressed in the following way
If t1, . . . , tp ∈ K or there is a bounded sequencen∈T in X fulfilling (1),n∈T 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 X with constant coefficients aij ∈ K, i, j = 1, . . . , r (r ∈ N is fixed): xn1+1 = a11xn1 + a12xn2 + ⋅ ⋅ ⋅ + a1rxnr + dn1, xn2+1 = a21xn1 + a22xn2 + ⋅ ⋅ ⋅ + a2rxnr + dn2
We consider ments of Xr, when it is convenient; that is, we identify xn with and dn with

Summary

Introduction

The issue of stability of a functional equation can be expressed in the following way. Some results have been proved in [13], which concern the stability of linear difference equations of higher order of form (1). Ap ∈ K be fixed, and let (bn)n∈T be a given sequence in a Banach space X over K. There exists a sequence (xn)n∈T in X satisfying (1) such that

Results

Conclusion