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ON THE HYPERSTABILITY OF THE DRYGAS FUNCTIONAL EQUATION ON A RESTRICTED DOMAIN

Published online by Cambridge University Press:  29 October 2019

JEDSADA SENASUKH Open the ORCID record for JEDSADA SENASUKH [Opens in a new window]  and
SATIT SAEJUNG Open the ORCID record for SATIT SAEJUNG [Opens in a new window]
JEDSADA SENASUKH
Affiliation:
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand email senasukh@kkumail.com
SATIT SAEJUNG*
Affiliation:
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand Research Center for Environmental and Hazardous Substance Management, Khon Kaen University, Khon Kaen, 40002, Thailand email saejung@kku.ac.th
*
✉saejung@kku.ac.th
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Abstract

We prove hyperstability results for the Drygas functional equation on a restricted domain (a certain subset of a normed space). Our results are more general than the ones proposed by Aiemsomboon and Sintunavarat [‘Two new generalised hyperstability results for the Drygas functional equation’, Bull. Aust. Math. Soc.95 (2017), 269–280] and our proof does not rely on the fixed point theorem of Brzdęk as was the case there. A characterisation of the Drygas functional equation in terms of its asymptotic behaviour is given. Several examples are given to illustrate our generalisations. Finally, we point out a misleading statement in the proof of the second result in the paper by Aiemsomboon and Sintunavarat and propose its correction.

Keywords

hyperstabilityrestricted domainDrygas functional equation

MSC classification

Primary: 39B82: Stability, separation, extension, and related topics
Secondary: 39B62: Functional inequalities, including subadditivity, convexity, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 1 , August 2020 , pp. 126 - 137
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author would like to thank the Development and Promotion for Science and Technology talents project (DPST) for the financial support to this paper. The second author is supported by the Thailand Research Fund and Khon Kaen University under grant RSA6280002.

References

Aiemsomboon, L. and Sintunavarat, W., ‘Two new generalised hyperstability results for the Drygas functional equation’, Bull. Aust. Math. Soc. 95 (2017), 269–280.10.1017/S000497271600126XCrossRefGoogle Scholar
Aoki, T., ‘On the stability of the linear transformation in Banach spaces’, J. Math. Soc. Japan 2 (1950), 64–66.10.2969/jmsj/00210064CrossRefGoogle Scholar
Bourgin, D. G., ‘Approximately isometric and multiplicative transformations on continuous function rings’, Duke Math. J. 16 (1949), 385–397.10.1215/S0012-7094-49-01639-7CrossRefGoogle Scholar
Brillouë-Belluot, N., Brzdęk, J. and Ciepliński, K., ‘On some recent developments in Ulam’s type stability’, Abstr. Appl. Anal. 2012 (2012), Article ID 716936, 41 pages.Google Scholar
Brzdęk, J., ‘Hyperstability of the Cauchy equation on restricted domains’, Acta Math. Hungar. 141(1–2) (2013), 58–67.10.1007/s10474-013-0302-3CrossRefGoogle Scholar
Brzdęk, J., ‘Remarks on hyperstability of the Cauchy functional equation’, Aequationes Math. 86(3) (2013), 255–267.10.1007/s00010-012-0168-4CrossRefGoogle Scholar
Brzdęk, J., ‘Stability of additivity and fixed point methods’, Fixed Point Theory Appl. 2013 (2013), Article ID 401756, 9 pages.10.1186/1687-1812-2013-285CrossRefGoogle Scholar
Drygas, H., ‘Quasi-inner products and their applications’, in: Advances in Multivariate Statistical Analysis (ed. Gupta, K.) (Springer, Dordrecht, 1987), 13–30.10.1007/978-94-017-0653-7_2CrossRefGoogle Scholar
Ebanks, B. R., Kannappan, Pl. and Sahoo, P. K., ‘A common generalization of functional equations characterizing normed and quasi-inner-product spaces’, Canad. Math. Bull. 35(3) (1992), 321–327.10.4153/CMB-1992-044-6CrossRefGoogle Scholar
Hyers, D. H., ‘On the stability of the linear functional equation’, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224.10.1073/pnas.27.4.222CrossRefGoogle ScholarPubMed
Piszczek, M. and Szczawińska, J., ‘Hyperstability of the Drygas functional equation’, J. Funct. Spaces Appl. 2013 (2013), Article ID 912718, 4 pages.10.1155/2013/912718CrossRefGoogle Scholar
Rassias, Th. M., ‘On the stability of the linear mapping in Banach spaces’, Proc. Amer. Math. Soc. 72(2) (1978), 297–300.10.1090/S0002-9939-1978-0507327-1CrossRefGoogle Scholar
Skof, F., ‘Proprietá locali e approssimazione di operatori’, Rend. Semin. Mat. Fis. Milano 53(1) (1983), 113–129.CrossRefGoogle Scholar
Ulam, S. M., A Collection of Mathematical Problems (Interscience Publishers, New York–London, 1960).Google Scholar