Note on nonstability of the linear functional equation of higher order

AbstractWe provide a complete solution of the problem of Hyers–Ulam stability for a large class of higher order linear functional equations in single variable, with constant coefficients. We obtain this by showing that such an equation is nonstable in the case where at least one of the roots of the characteristic equation is of module 1. Our results are related to the notions of shadowing (in dynamical systems and computer science) and controlled chaos. They also correspond to some earlier results on approximate solutions of functional equations in single variable

A Note on Stability of an Operator Linear Equation of the Second Order

We prove some Hyers-Ulam stability results for an operator linear equation of the second order that is patterned on the difference equation, which defines the Lucas sequences (and in particular the Fibonacci numbers). In this way, we obtain several results on stability of some linear functional and differential and integral equations of the second order and some fixed point results for a particular (not necessarily linear) operator

HYERS–ULAM STABILITY OF the delay equation . . .

We investigate the approximate solutions y : −τ, ∞ → R of the delay differential equation y t λy t − τ t ∈ 0, ∞ with an initial condition, where λ > 0 and τ > 0 are real constants. We show that they can be "approximated" by solutions of the equation that are constant on the interval −τ, 0 and, therefore, have quite simple forms. Our results correspond to the notion of stability introduced by Ulam and Hyers

On the Stability of a Generalized Fréchet Functional Equation with Respect to Hyperplanes in the Parameter Space

We study the Ulam-type stability of a generalization of the Fréchet functional equation. Our aim is to present a method that gives an estimate of the difference between approximate and exact solutions of this equation. The obtained estimate depends on the values of the coefficients of the equation and the form of the control function. In the proofs of the main results, we use a fixed point theorem to get an exact solution of the equation close to a given approximate solution

On the Stability of a Generalized Fréchet Functional Equation with Respect to Hyperplanes in the Parameter Space

We study the Ulam-type stability of a generalization of the Fréchet functional equation. Our aim is to present a method that gives an estimate of the difference between approximate and exact solutions of this equation. The obtained estimate depends on the values of the coefficients of the equation and the form of the control function. In the proofs of the main results, we use a fixed point theorem to get an exact solution of the equation close to a given approximate solution