On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey

The theory of Ulam stability was initiated by a problem raised in 1940 by S. Ulam and concerning approximate solutions to the equation of homomorphism in groups. It is somehow connected to various other areas of investigation such as, e.g., optimization and approximation theory. Its main issue is th...

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Autores principales:	Anna Bahyrycz, Janusz Brzdęk, El-sayed El-hady, Zbigniew Leśniak
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	Ulam stability functional equation 2-norm <i>n</i>-norm Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/3680c80f6fec4e16ad02d7e80720cc10
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id	oai:doaj.org-article:3680c80f6fec4e16ad02d7e80720cc10
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spelling	oai:doaj.org-article:3680c80f6fec4e16ad02d7e80720cc102021-11-25T19:07:31ZOn Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey10.3390/sym131122002073-8994https://doaj.org/article/3680c80f6fec4e16ad02d7e80720cc102021-11-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2200https://doaj.org/toc/2073-8994The theory of Ulam stability was initiated by a problem raised in 1940 by S. Ulam and concerning approximate solutions to the equation of homomorphism in groups. It is somehow connected to various other areas of investigation such as, e.g., optimization and approximation theory. Its main issue is the error that we make when replacing functions satisfying the equation approximately with exact solutions of the equation. This article is a survey of the published so far results on Ulam stability for functional equations in 2-normed spaces. We present and discuss them, pointing to the various pitfalls they contain and showing possible simple generalizations. In this way, in particular, we demonstrate that the easily noticeable symmetry between them and the analogous results obtained for the classical metric or normed spaces is in fact only apparent.Anna BahyryczJanusz BrzdękEl-sayed El-hadyZbigniew LeśniakMDPI AGarticleUlam stabilityfunctional equation2-norm<i>n</i>-normMathematicsQA1-939ENSymmetry, Vol 13, Iss 2200, p 2200 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Ulam stability functional equation 2-norm <i>n</i>-norm Mathematics QA1-939
spellingShingle	Ulam stability functional equation 2-norm <i>n</i>-norm Mathematics QA1-939 Anna Bahyrycz Janusz Brzdęk El-sayed El-hady Zbigniew Leśniak On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey
description	The theory of Ulam stability was initiated by a problem raised in 1940 by S. Ulam and concerning approximate solutions to the equation of homomorphism in groups. It is somehow connected to various other areas of investigation such as, e.g., optimization and approximation theory. Its main issue is the error that we make when replacing functions satisfying the equation approximately with exact solutions of the equation. This article is a survey of the published so far results on Ulam stability for functional equations in 2-normed spaces. We present and discuss them, pointing to the various pitfalls they contain and showing possible simple generalizations. In this way, in particular, we demonstrate that the easily noticeable symmetry between them and the analogous results obtained for the classical metric or normed spaces is in fact only apparent.
format	article
author	Anna Bahyrycz Janusz Brzdęk El-sayed El-hady Zbigniew Leśniak
author_facet	Anna Bahyrycz Janusz Brzdęk El-sayed El-hady Zbigniew Leśniak
author_sort	Anna Bahyrycz
title	On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey
title_short	On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey
title_full	On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey
title_fullStr	On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey
title_full_unstemmed	On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey
title_sort	on ulam stability of functional equations in 2-normed spaces—a survey
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/3680c80f6fec4e16ad02d7e80720cc10
work_keys_str_mv	AT annabahyrycz onulamstabilityoffunctionalequationsin2normedspacesasurvey AT januszbrzdek onulamstabilityoffunctionalequationsin2normedspacesasurvey AT elsayedelhady onulamstabilityoffunctionalequationsin2normedspacesasurvey AT zbigniewlesniak onulamstabilityoffunctionalequationsin2normedspacesasurvey
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On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey

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