Stability of generalized quadratic functional equation on a set of measure zero

Youssef Aribou; Hajira Dimou; Abdellatif Chahbi; Samir Kabbaj

doi:10.1515/aupcsm-2015-0011

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Stability of generalized quadratic functional equation on a set of measure zero

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

Volume 14 (2015): Issue 1 (December 2015)

By: Youssef Aribou, Hajira Dimou, Abdellatif Chahbi and Samir Kabbaj

Open Access

|Dec 2015

Abstract

In this paper we prove the Hyers-Ulam stability of the following K-quadratic functional equation

where E is a real (or complex) vector space. This result was used to demonstrate the Hyers-Ulam stability on a set of Lebesgue measure zero for the same functional equation.

References

[1] M. Ait Sibaha, B. Bouikhalene, E. Elqorachi, Hyers-Ulam-Rassias stability of the K-quadratic functional equation, JIPAM. J. Inequal. Pure Appl. Math. 8 (2007), no. 3, Article 89, 13pp. Cited on 150 and 151.
Search in Google Scholar
[2] M. Almahalebi, Approximate Drygas mappings on a set of measure zero, (submitted). Cited on 150.
Search in Google Scholar
[3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66. Cited on 149.
Search in Google Scholar
[4] N. Brillouët-Belluot, J. Brzdęk, K. Ciepliński, On some recent developments in Ulam’s type stability, Abstr. Appl. Anal. 2012, Art. ID 716936, 41pp. Cited on 150.
Search in Google Scholar
[5] J. Brzdęk, On a method of proving the Hyers-Ulam stability of functional equations on restricted domains, Austr. J. Math. Anal. Appl. 6 (2009), 1-10. Cited on 150.
Search in Google Scholar
[6] J. Brzdęk, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar. 141 (2013), no. 1-2, 58-67. Cited on 150.
Search in Google Scholar
[7] A. Charifi, B. Bouikhalene, E. Elqorachi, Hyers-Ulam-Rassias stability of a generalized Pexider functional equation, Banach J. Math. Anal. 1 (2007), no. 2, 176-185. Cited on 150 and 151.
Search in Google Scholar
[8] A.B. Chahbi, A. Charifi, B. Bouikhalene, S. Kabbaj, Operatorial approach to the non-Archimedean stability of a Pexider K-quadratic functional equation, Arab J. Math. Sci. 21 (2015), no. 1, 67-83. Cited on 150.
Search in Google Scholar
[9] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76-86. Cited on 150.
Search in Google Scholar
[10] Jaeyoung Chung, Stability of a conditional Cauchy equation on a set of measure zero, Aequationes Math. 87 (2014), no. 3, 391-400. Cited on 150 and 158.
Search in Google Scholar
[11] Jaeyoung Chung, J.M. Rassias, Quadratic functional equations in a set of Lebesgue measure zero, J. Math. Anal. Appl. 419 (2014), no. 2, 1065-1075. Cited on 150 and 158.
Search in Google Scholar
[12] K. Ciepliński, Applications of fixed point theorems to the Hyers-Ulam stability of functional equations - a survey, Ann. Funct. Anal. 3 (2012), no. 1, 151-164. Cited on 150.
Search in Google Scholar
[13] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64. Cited on 150.
Search in Google Scholar
[14] D.Ž. Djoković, A representation theorem for (X1 −1)(X2 −1) · · · (Xn −1) and its applications, Ann. Polon. Math. 22 (1969/1970) 189-198. Cited on 151.
Search in Google Scholar
[15] P. Gǎvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436. Cited on 150.
Search in Google Scholar
[16] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. Cited on 149 and 150.
Search in Google Scholar
[17] D.H. Hyers, Th.M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125-153. Cited on 150.
Search in Google Scholar
[18] D.H. Hyers, G.I. Isac, Th.M. Rassias, Stability of functional equations in several variables, (English summary) Progress in Nonlinear Differential Equations and their Applications 34, Birkhäuser Boston, Inc., Boston, MA, 1998. Cited on 150.
Search in Google Scholar
[19] D.H. Hyers, Transformations with bounded m-th differences, Pacific J. Math. 11 (1961), 591-602. Cited on 150, 153 and 155.
Search in Google Scholar
[20] S.-M. Jung, Stability of the quadratic equation of Pexider type, Abh. Math. Sem. Univ. Hamburg 70 (2000), 175-190. Cited on 150.
Search in Google Scholar
[21] S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), no. 1, 126-137. Cited on 150.
Search in Google Scholar
[22] S.-M. Jung, P.K. Sahoo, Hyers-Ulam stability of the quadratic equation of Pexider type, J. Korean Math. Soc. 38 (2001), no. 3, 645-656. Cited on 150.
Search in Google Scholar
[23] C.F.K. Jung, On generalized complete metric spaces, Bull. Amer. Math. Soc. 75 (1969), 113-116. Cited on 150.
Search in Google Scholar
[24] Yongjin Li, Liubin Hua, Hyers-Ulam stability of a polynomial equation, Banach J. Math. Anal. 3 (2009), no. 2, 86-90. Cited on 150.
Search in Google Scholar
[25] R. Łukasik, Some generalization of Cauchy’s and the quadratic functional equations, Aequationes Math. 83 (2012), no. 1-2, 75-86. Cited on 150 and 157.
Search in Google Scholar
[26] R. Łukasik, The stability of some generalization of the quadratic functional equation, Georgian Math. J. 21 (2014), no. 4, 463-474. Cited on 151.
Search in Google Scholar
[27] S. Mazur, W. Orlicz, Grundlegende Eigenschaften der polynomischen Operationen. Erst Mitteilung, Studia Math. 5 (1934), 50-68. Cited on 151.
Search in Google Scholar
[28] J.C. Oxtoby, Measure and category. A survey of the analogies between topological and measure spaces, Second edition, Graduate Texts in Mathematics 2, Springer- Verlag, New York-Berlin, 1980. Cited on 157.
Search in Google Scholar
[29] J.M. Rassias, On the Ulam stability of mixed type mappings on restricted domains, J. Math. Anal. Appl. 276 (2002), 747-762. Cited on 150.
Search in Google Scholar
[30] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300. Cited on 149 and 150.
Search in Google Scholar
[31] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23-130. Cited on 150.
Search in Google Scholar
[32] F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. Cited on 150.
Search in Google Scholar
[33] S.M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics 8, Interscience Publishers, New York-London 1960. Cited on 149.
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.1515/aupcsm-2015-0011 | Journal eISSN: 2300-133X | Journal ISSN: 2081-545X

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Language: English

Page range: 149 - 162

Submitted on: Jun 14, 2015

Published on: Dec 30, 2015

Published by: Pedagogical University of Cracow

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

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© 2015 Youssef Aribou, Hajira Dimou, Abdellatif Chahbi, Samir Kabbaj, published by Pedagogical University of Cracow
This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 License.

Volume 14 (2015): Issue 1 (December 2015)