Stability of quadratic functional equation in modular spaces
Abderrahman Baza, Mohamed Rossafi, Arul Joseph Gnanaprakasam
TL;DR
This work develops Hyers-Ulam stability results for the quadratic functional equation $φ(x+y-z)+φ(x+z-y)+φ(y+z-x)=φ(x-y)+φ(x-z)+φ(z-y)+φ(x)+φ(y)+φ(z)$ within modular spaces and $β$-homogeneous Banach spaces. Using a direct-method approach, it establishes the existence and uniqueness of a quadratic mapping $h$ that approximates any $φ$ under explicit modular- or norm-based bounds, with $h(x)$ characterized as a $ρ$-limit or limit in the respective space. The results cover cases without the Δ2-condition, with Δ2-condition, and in β-homogeneous settings, yielding concrete stability estimates and conditions for uniqueness; classical Hyers-Ulam stability is recovered as a special case when the perturbation bound is constant. Collectively, the paper extends stability theory for quadratic equations to modular and β-homogeneous contexts, providing explicit bounds and constructive limits for the stabilizing quadratic map.
Abstract
In this paper, we study the Hyers-Ulam stability of the following equation \begin{multline*} φ(x+y-z)+φ(x+z-y)+φ(y+z-x)=φ(x-y)+φ(x-z)+φ(z-y) +φ(x)+φ(y) +φ(z) \end{multline*} in modular space, with or without $Δ_2$-condition, and in $β$-homogeneous Banach space.