A Linear Functional Equation of Third Order Associated with the Fibonacci Numbers

Abstract

Given a vector spaceX, we investigate the solutionsf:R→Xof the linear functional equation of third orderfx=pfx-1+qfx-2+rf(x-3), which is strongly associated with a well-known identity for the Fibonacci numbers. Moreover, we prove the Hyers-Ulam stability of that equation.

Ulam–Hyers Stability for Fractional Differential Equations in Quaternionic Analysis

“…Rassias [24][25][26][27] and Xu [35][36][37] generalized the Hyers stability result by introducing two weaker conditions controlled by a product of different powers of norms and a mixed product-sum of powers of norms, respectively. Furthermore, Jung [15,17,18] has also proved the Ulam-Hyers stability of linear functional equations. Castro et al [6] and Jung [16] considered the Ulam-Hyers stability for a class of Volterra integral equations.…”

On fixed points of a linear operator of polynomial form of order 3

“…There are several papers showing how to deal with the problem of stability of various linear equations of higher orders (see, e.g., [7,8,12,21,22,24,[28][29][30]) of the form:…”

Hyers-Ulam stability of Pielou logistic difference equation

“…More recently, Hyers-Ulam stability of difference equations has been given attention. For instance, see [2,[5][6][7][8][9]. However, this stability for difference equations is not yet studied far beyond the linear difference equation as far as we know.…”