$(L^p, L^q)$ Hyers-Ulam stability

TL;DR

This paper introduces a novel Hyers-Ulam stability concept for differential equations using different $L^p$-space norms, providing conditions for stability and explicit constants in special cases.

Contribution

It develops a new stability framework involving mixed $L^p$-norms and derives explicit stability constants for certain classes of differential equations.

Findings

Established sufficient conditions for $(L^p, L^q)$ Hyers-Ulam stability.

Derived explicit formulas for the best stability constants in specific cases.

Extended the stability analysis to semilinear ordinary differential equations.

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.