Hyers–Ulam Stability for Quantum Equations of Euler Type

Douglas R Anderson,Masakazu Onitsuka

doi:10.1155/2020/5626481

Abstract

Many applications using discrete dynamics employ either q-difference equations or h-difference equations. In this work, we introduce and study the Hyers–Ulam stability (HUS) of a quantum (q-difference) equation of Euler type. In particular, we show a direct connection between quantum equations of Euler type and h-difference equations of constant step size h with constant coefficients and an arbitrary integer order. For equation orders greater than two, the h-difference results extend first-order and second-order results found in the literature, and the Euler-type q-difference results are completely novel for any order. In many cases, the best HUS constant is found.

Highlights

There has been much interest in questions of Hyers–Ulam stability for differential equations and h-difference equations, but little has been published on q-difference equations [1], in particular, on quantum equations of Euler type
We introduce a new and direct connection between Hyers–Ulam stability results for h-difference equations with constant coefficients, of first, second, and all higher orders, with Hyers–Ulam stability results for quantum equations of Euler type, of all integer orders, through a change of variables
ConclusionNew results connecting h-difference equations with complex constant coefficients and q-difference equations of the Euler type are presented, for equations of all integer orders

Summary

Introduction

There has been much interest in questions of Hyers–Ulam stability for differential equations and h-difference equations, but little has been published on q-difference (quantum) equations [1], in particular, on quantum equations of Euler type. We introduce a new and direct connection between Hyers–Ulam stability results for h-difference equations with constant coefficients, of first, second, and all higher orders, with Hyers–Ulam stability results for quantum equations of Euler type, of all integer orders, through a change of variables.

Connections and First-Order Stability

Second-Order Quantum Equations of Euler Type

Higher-Order Quantum Equations of Euler Type

Perturbed Quantum Equations of Euler Type

Conclusion