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ON THE HYPERSTABILITY OF A PEXIDERISED $\unicode[STIX]{x1D70E}$-QUADRATIC FUNCTIONAL EQUATION ON SEMIGROUPS

Published online by Cambridge University Press:  07 March 2018

IZ-IDDINE EL-FASSI  and
JANUSZ BRZDĘK
IZ-IDDINE EL-FASSI*
Affiliation:
Department of Mathematics, Faculty of Sciences, Ibn Tofail University, BP 133, Kenitra, Morocco email [email protected]
JANUSZ BRZDĘK
Affiliation:
Department of Mathematics, Pedagogical University, Podchorążych 2, 30-084 Kraków, Poland email [email protected]
*
✉[email protected]
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Abstract

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Motivated by the notion of Ulam stability, we investigate some inequalities connected with the functional equation

$$\begin{eqnarray}f(xy)+f(x\unicode[STIX]{x1D70E}(y))=2f(x)+h(y),\quad x,y\in G,\end{eqnarray}$$
for functions $f$ 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.

Keywords

hyperstability𝜎-quadratic functional equationsemigroupendomorphism of semigroup

MSC classification

Primary: 39B82: Stability, separation, extension, and related topics
Secondary: 39B52: Equations for functions with more general domains and/or ranges
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 3 , June 2018 , pp. 459 - 470
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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