On approximate solutions of the linear functional equation of higher order

Abstract

We show that, under some assumptions, every approximate solution of the linear functional equation of higher order, in single variable, generates a solution of the equation that is close to it. We also give a description of a procedure that yields such a solution, estimate the distance between those approximate and exact solutions to the equation, and discuss the problem of uniqueness. Moreover, as a consequence we obtain some results concerning the Hyers-Ulam stability of the equation.

On the stability of the linear functional equation in a single variable on complete metric groups

“…Some recent results on the stability and nonstability of the equation (1.1) and the linear functional equation of higher order in a single variable were obtained by J. Brzdek, D. Popa, B. Xu (see [6,7,8,9,10]). …”

Hyers-Ulam-Rassias Stability of Functional Differential Systems with Point and Distributed Delays

“…See, for instance, [10,20,24,[26][27][28] and references therein. On the other hand, Hyers-Ulam stability has been also invoked in difference-type linear and nonlinear equations as, for instance, in [25] and several background references therein. Note that difference equations are sometimes got from the discretization of continuous-time system either via the use of numerical tools or by the use of physical sampling and hold devices and that the stability of such discretized systems can be, in general, either studied independently of that of their continuous-time counterparts, via "ad hoc" discrete analysis methods, or based with the stability properties of the continuous-time version with extra conditions on the sequence of sampling instants (see, e.g., [22]).…”

“…See, for instance, [14][15][16][17][18][19][20][21][22][23] and references therein. The stability and the derivation of approximate solutions of some kinds of functional equations have been also investigated in [24,25] and references therein, under close analysis methods.…”

On approximate isometries and application to stability of a functional equation

“…We refer to the survey [28] and the celebrated book [18] for this topic and related matters. Some of the latest developments of the stability of functional equations can be found in the papers [2,[4][5][6]15,[19][20][21]25].…”