$(L^p, L^q)$ Hyers-Ulam stability
Davor Dragicevic, Masakazu Onitsuka
TL;DR
This work generalizes Hyers-Ulam stability to a mixed $L^p$-$L^q$ setting for ordinary differential equations by measuring the residual error in $L^q$ and the deviation from an exact solution in $L^p$. For semilinear dynamics $x'=A(t)x+f(t,x)$, it proves that if the linear part admits an exponential dichotomy and the nonlinearity is Lipschitz with a small constant, the system exhibits $(L^p,L^q)$ Hyers-Ulam stability with an explicit constant $L=\dfrac{2D\left(\dfrac{1}{\lambda r}\right)^{\frac{1}{r}}}{1-\dfrac{2cD}{\lambda}}$, where $r$ satisfies $\dfrac{1}{p}+1=\dfrac{1}{q}+\dfrac{1}{r}$. This generalizes prior BDOP results (the case $p=q=\infty$) and provides concrete bounds and sharpness via examples. The paper also furnishes a lower-bound analysis for the HUS constant in 2D autonomous linear systems, establishing fundamental limits and demonstrating sharpness (e.g., minimal $L$ equals $1/\min\{\Re(\mu_1),\Re(\mu_2)\}$ in certain diagonalizable cases). Overall, the results extend HUS theory to broader function spaces and yield practical stability constants for semilinear dynamics.
Abstract
We introduce a new concept of Hyers-Ulam stability, in which in the size of a pseudosolution of a given ordinary differential equation and its deviation from an exact solution are measured with respect to different norms. These norms are associated to $L^p$-spaces for $p\in [1, \infty]$. Our main objective is to formulate sufficient conditions under which semilinear ordinary differential equations exhibit such property. In addition, in certain special cases we obtain explicit formulas for the best Hyers-Ulam constant.