On a Measure Zero Stability Problem of A cyclic equation

Abstract

Let $G$ be a commutative group, $Y$ a real Banach space and $f:G\rightarrow Y$. We prove the Ulam–Hyers stability theorem for the cyclic functional equation $$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$ for all $x,y\in {\rm\Omega}$, where $H$ is a finite cyclic subgroup of $\text{Aut}(G)$ and ${\rm\Omega}\subset G\times G$ satisfies a certain condition. As a consequence, we consider a measure zero stability problem of the functional equation $$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$ for all $(z,{\it\zeta})\in {\rm\Omega}$, where $f:\mathbb{C}\rightarrow Y,\,{\it\omega}=e^{2{\it\pi}i/N}$ and ${\rm\Omega}\subset \mathbb{C}^{2}$ has four-dimensional Lebesgue measure $0$.

Stability of generalized Jensen functional equation on a set of measure zero

“…Our method in this proof is inspired by [10,Theorem 3.2]. Let Then Ω satisfies (3.3) if and only if, for every z = x+iy and ξ = u+iv ∈ C, there exists η = t + is ∈ C such that…”

“…The stability problems of several functional equations on a restricted domain have been extensively investigated by a number of authors, for example [[4], [9], [8], [10] [11], [17], [20], [26], [23] ].…”

Stability of a Quartic Functional Equation in Restricted Domains

“…Among the results, Jung [14] and Rassias [17] proved the Ulam-Hyers stability of the quadratic functional equations in a restricted domain. As a refined version of the results in [14,17] we state the result in [20]. Theorem 2.…”