A fixed points approach to stability of the Pexider equation

Bouikhalene, Belaid; Elqorachi, Elhoucien; Rassias, John Michael

doi:10.2478/tmj-2014-0021

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2015

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“…Generalising the above result, Sibaha et al [14] proved the Ulam-Hyers stability of the functional equation for all x, y ∈ G. We call f : G → Y satisfying (1.2) an H-cyclic mapping. (See [5] for more general results.) Theorem 1.2 [14].…”

Section: Introductionmentioning

confidence: 96%

On a Measure Zero Stability Problem of A cyclic equation

Chung

Rassias

2015

Bull. Aust. Math. Soc.

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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 $0$.

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Section: Introductionmentioning

confidence: 96%