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A NOTE ON THE GENERALISED HYPERSTABILITY OF THE GENERAL LINEAR EQUATION

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  29 August 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 Laddawan_Aiemsomboon@hotmail.com
WUTIPHOL SINTUNAVARAT*
Affiliation:
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand email wutiphol@mathstat.sci.tu.ac.th
*
✉wutiphol@mathstat.sci.tu.ac.th
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Abstract

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Let $X$ and $Y$ be two normed spaces over fields $\mathbb{F}$ and $\mathbb{K}$, respectively. We prove new generalised hyperstability results for the general linear equation of the form $g(ax+by)=Ag(x)+Bg(y)$, where $g:X\rightarrow Y$ is a mapping and $a,b\in \mathbb{F}$, $A,B\in \mathbb{K}\backslash \{0\}$, using a modification of the method of Brzdęk [‘Stability of additivity and fixed point methods’, Fixed Point Theory Appl.2013 (2013), Art. ID 285, 9 pages]. The hyperstability results of Piszczek [‘Hyperstability of the general linear functional equation’, Bull. Korean Math. Soc.52 (2015), 1827–1838] can be derived from our main result.

Keywords

fixed pointgeneralised hyperstabilitygeneral linear equation

MSC classification

Primary: 39B82: Stability, separation, extension, and related topics
Secondary: 39B62: Functional inequalities, including subadditivity, convexity, etc. 47H10: Fixed-point theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 2 , October 2017 , pp. 263 - 273
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

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

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