Hyers–Ulam Stability for Quantum Equations of Euler Type

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.

Hyers–Ulam Stability for Cayley Quantum Equations and Its Application to h-Difference Equations

“…In [4], it has been shown that there is a suitable transformation between the quantum (q and h difference) equations on two different time scales to guarantee stability for both equations. More specifically, it turns out that if the h-difference equation has HUS, then the corresponding quantum equation of Euler type also has HUS.…”

“…Under assumption (1.2), they proved the following facts: if λ = 0, then (1. In 2020, the authors [4] introduced a new, direct connection between HUS for h-difference equations and HUS for quantum equations of Euler type. The second purpose of this study is to establish a novel connection between HUS for Cayley quantum equations and HUS for the h-difference equations with a specific variable coefficient, based on the ideas in this paper.…”

Nonlocal Problems for Hilfer Fractional q-Difference Equations

“…With the attention of many experts and scholars on fractional q-difference, rich results have been achieved on fractional q-difference equations via q-Gronwall equality (see [18]), the existence and stability of the solutions for Riemann-Liouville fractional q-difference equations (see [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]), Caputo fractional q-difference initial boundary value problems (see [34][35][36][37][38][39]). In [40], Boutiara explored the mixed multi-term fractional q-difference equations with q-integral boundary conditions by using topological degree theory.…”

Existence and stability of solutions for coupled fractional delay q-difference systems

“…But the stability of coupled fractional q-difference equations were relatively less explored. In [7,23], the stability of fractional difference about non-autonomous systems and quantum equations of Euler type was discussed, respectively. In [1,2,4,9,13], the existence and finite-time stability of a class of fractional q-difference time-delay systems by the q-Gronwall inequality and the fixed point theorem were studied.…”