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

Youssef Aribou, Hajira Dimou, Abdellatif Chahbi, Samir Kabbaj

Abstract


In this paper we prove the Hyers-Ulam stability of the following K-quadratic functional equation ∑ k ∈ K f(x+ k ⋅ y)= Lf(x)+ Lf(y), x,y ∈ E, 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.

Keywords


group automorphisms; Jensen functional equation; quadratic functional equation; K -quadratic functional equation; Pexider functional equation; Hyers-Ulam stability

Mathematics Subject Classification (2010)


Primary 39B82; Secondary 39B52;

References


Ait Sibaha, M. and B. Bouikhalene and E. Elqorachi. "Hyers-Ulam-Rassias stability of the K-quadratic functional equation." JIPAM. J. Inequal. Pure Appl. Math. 8.3 (2007): Article 89.

Almahalebi, M. "Approximate Drygas mappings on a set of measure zero." (submitted).

Aoki, T. "On the stability of the linear transformation in Banach spaces." J. Math. Soc. Japan 2 (1950): 64–66.

Brillouët-Belluot, N. and J. Brzdęk and K. Ciepliński. "On some recent developments in Ulam’s type stability." Abstr. Appl. Anal. 2012, Art. ID 716936

Brzdęk, J. "On a method of proving the Hyers-Ulam stability of functional equations on restricted domains." Austr. J. Math. Anal. Appl. 6 (2009): 1–10.

Brzdek, J. "Hyperstability of the Cauchy equation on restricted domains." Acta Math. Hungar. 141.1-2 (2013): 58–67.

Charifi, A. and B. Bouikhalene and E. Elqorachi. "Hyers-Ulam-Rassias stability of a generalized Pexider functional equation." Banach J. Math. Anal. 1.2 (2007): 176–185.

Chahbi, A.B. and A. Charifi and B. Bouikhalene and S. Kabbaj. "Operatorial approach to the non-Archimedean stability of a Pexider K-quadratic functional equation." Arab J. Math. Sci. 21.1 (2015):67–83.

Cholewa, P.W. "Remarks on the stability of functional equations." Aequationes Math. 27.1-2 (1984): 76–86.

Chung, Jaeyoung . "Stability of a conditional Cauchy equation on a set of measure zero." Aequationes Math. 87.3 (2014): 391–400.

Chung, Jaeyoung and J.M. Rassias. "Quadratic functional equations in a set of Lebesgue measure zero." J. Math. Anal. Appl. 419.2 (2014): 1065–1075.

Ciepliński, K. "Applications of fixed point theorems to the Hyers-Ulam stability of functional equations – a survey." Ann. Funct. Anal. 3.1 (2012): 151–164.

Czerwik, S. "On the stability of the quadratic mapping in normed spaces." Abh. Math. Sem. Univ. Hamburg 62 (1992): 59–64.

Djoković, D.Ž. "A representation theorem for (X1 −1)(X2 −1) · · · (Xn −1) and its applications." Ann. Polon. Math. 22 (1969/1970): 189–198.

Gâvruta, P. "A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings." J. Math. Anal. Appl. 184 (1994): 431–436.

Hyers, D.H. "On the stability of the linear functional equation." Proc. Nat. Acad. Sci. U.S.A. 27 (1941): 222–224.

Hyers, D.H. and Th.M. Rassias. "Approximate homomorphisms." Aequationes Math. 44.2-3 (1992): 125–153.

Hyers, D.H. and G.I. Isac and Th.M. Rassias. "Stability of functional equations in several variables." Progress in Nonlinear Differential Equations and their Applications 34. Boston, MA: Birkhäuser Boston, Inc., 1998.

Hyers, D.H. "Transformations with bounded m-th differences." Pacific J. Math. 11 (1961): 591–602.

Jung, S.-M. "Stability of the quadratic equation of Pexider type." Abh. Math. Sem. Univ. Hamburg 70 (2000): 175–190.

Jung, S.-M. "On the Hyers-Ulam stability of the functional equations that have the quadratic property." J. Math. Anal. Appl. 222.1 (1998): 126–137.

Jung, S.-M. and P.K. Sahoo. "Hyers-Ulam stability of the quadratic equation of Pexider type." J. Korean Math. Soc. 38.3 (2001): 645–656.

Jung, C.F.K. "On generalized complete metric spaces." Bull. Amer. Math. Soc. 75 (1969): 113–116.

Li, Yongjin and Liubin Hua. "Hyers-Ulam stability of a polynomial equation." Banach J. Math. Anal. 3.2 (2009): 86–90.

Łukasik, R. "Some generalization of Cauchy’s and the quadratic functional equations." Aequationes Math. 83.1-2 (2012): 75–86.

Łukasik, R. "The stability of some generalization of the quadratic functional equation." Georgian Math. J. 21.4 (2014): 463–474.

Mazur, S. and W. Orlicz. "Grundlegende Eigenschaften der polynomischen Operationen." Erst Mitteilung, Studia Math. 5 (1934): 50–68.

Oxtoby, J.C. "Measure and category. A survey of the analogies between topological and measure spaces," Second edition. Graduate Texts in Mathematics 2. New York-Berlin: Springer-Verlag, 1980.

Rassias, J.M. "On the Ulam stability of mixed type mappings on restricted domains." J. Math. Anal. Appl. 276 (2002): 747–762.

Rassias, Th.M. "On the stability of the linear mapping in Banach spaces." Proc. Amer. Math. Soc. 72.2 (1978): 297–300.

Rassias, Th.M. "On the stability of functional equations and a problem of Ulam." Acta Appl. Math. 62.1 (2000): 23–130.

Skof, F. "Local properties and approximation of operators." Rend. Sem. Mat. Fis. Milano 53 (1983): 113–129.

Ulam, S.M. "A collection of mathematical problems." Interscience Tracts in Pure and Applied Mathematics 8. New York-London: Interscience Publishers, 1960.


Full Text: PDF

Download statistics: 1814

Licencja Creative Commons
This article by Youssef Aribou, Hajira Dimou, Abdellatif Chahbi, Samir Kabbaj is governed by the Creative Commons Attribution-ShareAlike 4.0 International(CC BY-SA 4.0) licence.


e-ISSN: 2300-133X, ISSN: 2081-545X

Since 2017 Open Access in De Gruyter and CrossCheck access cofinanced by The Ministry of Science and Higher Education - Republic of Poland - DUN 775/P-DUN/2017 see more

The Journal is indexed in:
and others see Abstracting and Indexing list

AUPC SM is on the List of the Ministry’s scored journals with 20 points for 2019

Deklaracja dostępności cyfrowej