4 results
ON THE HYPERSTABILITY OF A PEXIDERISED
$\unicode[STIX]{x1D70E}$-QUADRATIC FUNCTIONAL EQUATION ON SEMIGROUPS
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 3 / June 2018
- Published online by Cambridge University Press:
- 07 March 2018, pp. 459-470
- Print publication:
- June 2018
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Motivated by the notion of Ulam stability, we investigate some inequalities connected with the functional equation
for functions
$$\begin{eqnarray}f(xy)+f(x\unicode[STIX]{x1D70E}(y))=2f(x)+h(y),\quad x,y\in G,\end{eqnarray}$$
$f$ and 
$h$ mapping a semigroup 
$(G,\cdot )$ into a commutative semigroup 
$(E,+)$, where the map 
$\unicode[STIX]{x1D70E}:G\rightarrow G$ is an endomorphism of 
$G$ with 
$\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D70E}(x))=x$ for all 
$x\in G$. We derive from these results some characterisations of inner product spaces. We also obtain a description of solutions to the equation and hyperstability results for the 
$\unicode[STIX]{x1D70E}$-quadratic and 
$\unicode[STIX]{x1D70E}$-Drygas equations.
ON A MEASURE ZERO STABILITY PROBLEM OF A CYCLIC EQUATION
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 93 / Issue 2 / April 2016
- Published online by Cambridge University Press:
- 27 October 2015, pp. 272-282
- Print publication:
- April 2016
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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 for all
$$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$
$x,y\in {\rm\Omega}$, 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 for all
$$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$
$(z,{\it\zeta})\in {\rm\Omega}$, 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 
$0$.
ON HYPERSTABILITY OF GENERALISED LINEAR FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 92 / Issue 2 / October 2015
- Published online by Cambridge University Press:
- 02 June 2015, pp. 259-267
- Print publication:
- October 2015
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We obtain some results on approximate solutions of the generalised linear functional equation
$\sum _{i=1}^{m}L_{i}f(\sum _{j=1}^{n}a_{ij}x_{j})=0$ for functions mapping a normed space into a normed space. We show that, under suitable assumptions, the approximate solutions are in fact exact solutions. The theorems correspond to and complement recent results on the hyperstability of generalised linear functional equations.
STABILITY OF MAPPINGS ON MULTI-NORMED SPACES
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- Journal:
- Glasgow Mathematical Journal / Volume 49 / Issue 2 / May 2007
- Published online by Cambridge University Press:
- 09 August 2007, pp. 321-332
- Print publication:
- May 2007
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