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TWO NEW GENERALISED HYPERSTABILITY RESULTS FOR THE DRYGAS FUNCTIONAL EQUATION

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  09 January 2017

LADDAWAN AIEMSOMBOON  and
WUTIPHOL SINTUNAVARAT
LADDAWAN AIEMSOMBOON
Affiliation:
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand email [email protected]
WUTIPHOL SINTUNAVARAT*
Affiliation:
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand email [email protected]
*
✉[email protected]
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Abstract

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Let $X$ be a nonempty subset of a normed space such that $0\notin X$ and $X$ is symmetric with respect to $0$ and let $Y$ be a Banach space. We study the generalised hyperstability of the Drygas functional equation

$$\begin{eqnarray}f(x+y)+f(x-y)=2f(x)+f(y)+f(-y),\end{eqnarray}$$
where $f$ maps $X$ into $Y$ and $x,y\in X$ with $x+y,x-y\in X$. Our first main result improves the results of Piszczek and Szczawińska [‘Hyperstability of the Drygas functional equation’, J. Funct. Space Appl.2013 (2013), Article ID 912718, 4 pages]. Hyperstability results for the inhomogeneous Drygas functional equation can be derived from our results.

Keywords

generalised hyperstabilityDrygas functional equationfixed point theorem

MSC classification

Primary: 39B82: Stability, separation, extension, and related topics
Secondary: 39B62: Functional inequalities, including subadditivity, convexity, etc. 47H14: Perturbations of nonlinear operators 47H10: Fixed-point theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 2 , April 2017 , pp. 269 - 280
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

This work was supported by the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST). The second author was supported by the Thailand Research Fund and Office of the Higher Education Commission under grant no. MRG5980242.

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