SciELO - Scientific Electronic Library Online

 
vol.41 issue3On fuzzy γµ-open sets in generalized fuzzy topological spacesNear-Zumkeller numbers author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Share


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.41 no.3 Antofagasta June 2022

http://dx.doi.org/10.22199/issn.0717-6279-4691 

Artículos

Stability problem in a set of Lebesgue measure zero of bi-additive functional equation

Rachid El Ghali1 

Samir Kabbaj2 

1Department of Mathematics, Faculty of Sciences University of Ibn Tofail, Kenitra, Morocco e-mail: rachid2810@gmail.com

2 Department of Mathematics, Faculty of Sciences University of Ibn Tofail, Kenitra, Morocco e-mail: samkabbaj@yahoo.fr

Abstract

Let X be a vector space and Y be a Banach space. Our aim in this paper is to investigate the Hyers-Ulam stability problem of the following bi-additive functional equation

f(x + y, s − t) + f(x − y, s + t)=2f(x, s) − 2f(y, t), x, y, s, tX,

where f : X × X → Y . As a consequence, we discuss the stability of the considered functional equation in a restricted domain and in the set of Lebesgue measure zero.

Keyword: bi-additive functional equation; Hyers-Ulam stability; functional equation; Baire category theorem; first category; Lebesgue measure

Texto completo disponible sólo en PDF

Full text available only in PDF format.

References

[1] J. Aczél and J. Dhombres, Functional Equations in Several Variables. Cambridge: Cambridge University Press, 1989. [ Links ]

[2] T. Aoki, “On the stability of the linear transformation in Banach spaces”, Journal of the Mathematical Society of Japan, vol. 2, pp. 64-66, 1950. [ Links ]

[3] A. Bahyrycz and J. Brzdęk, “On solutions of the d’Alembert equation on a restricted domain”, Aequationes mathematicae, vol. 85, pp. 169-183, 2013. [ Links ]

[4] D. G. Bourgin, “Classes of transformations and bordering transformations”, Bulletin of the American Mathematical Society, vol. 57, pp. 223-237, 1951. [ Links ]

[5] J. Brzdęk, “On the quotient stability of a family of functional equations”, Nonlinear Analysis: Theory, Methods & Applications, vol. 71, pp. 4396-4404, 2009. [ Links ]

[6] J. Brzdęk, “On a method of proving the Hyers-Ulam stability of functional equations on restricted domains”, Australian Journal of Mathematical Analysis and Applications, vol. 6, pp. 1-10, 2009. [ Links ]

[7] J. Brzdęk and J. Sikorska, “A conditional exponential functional equation and its stability”, Nonlinear Analysis: Theory, Methods & Applications , vol. 72, pp. 2929-2934, 2010. [ Links ]

[8] J. Chung, “Stability of functional equations on restricted domains in a group and their asymptotic behaviors”, Computers & Mathematics with Applications, vol. 60, pp. 2653-2665, 2010. [ Links ]

[9] J. Chung, “Stability of a conditional Cauchy equation on a set of measure zero”, Aequationes mathematicae , vol. 87, pp. 391-400, 2014. doi: 10.1007/s00010-013-0235-5 [ Links ]

[10] J. Chung and J. M. Rassias, “Quadratic functional equations in a set of Lebesgue measure zero”, Journal of Mathematical Analysis and Applications, vol. 419, no. 2, pp. 1065-1075, 2014. [ Links ]

[11] S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type. Palm Harbor (FL): Hadronic, 2003. [ Links ]

[12] I. EL-Fassi, J. Brzdęk, A. Chahbi, and S. Kabbaj, “On the Hyperstability of the biadditive functional equation”, Acta Mathematica Scientia, vol. 37, no. 6, pp. 1727-1739, 2017. [ Links ]

[13] M. Fochi, “An alternative functional equation on restricted domain”, Aequationes mathematicae , vol. 70, pp. 201-212, 2005. [ Links ]

[14] Z. Gajda, “On stability of additive mappings”, International Journal of Mathematics and Mathematical Sciences, vol. 14, pp. 431-434, 1991. [ Links ]

[15] P. Gávrutá, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings”, Journal of Mathematical Analysis and Applications , vol. 184, no. 1, pp. 431-436, 1984. [ Links ]

[16] R. Ger, and J. Sikorska, “On the Cauchy equation on spheres”, Annales Mathematicae Silesianae, vol. 11, pp. 89-99, 1997. [ Links ]

[17] D. H. Hyers, “On the stability of the linear functional equation”, Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222-224, 1941. [ Links ]

[18] S. -M. Jung, “On the Hyers-Ulam stability of the functional equations that have the quadratic property”, Journal of Mathematical Analysis and Applications , vol. 222, pp. 126-137, 1998. [ Links ]

[19] M. Kuczma, “Functional equations on restricted domains”, Aequationes mathematicae , vol. 18, pp. 1-34, 1978. [ Links ]

[20] Y.-H. Lee, “Hyers-Ulam-Rassias stability of a quadratic-additive type functional equation on a restricted domain”, International Journal of Mathematical Analysis, vol. 7, no. 55, pp. 2745-2752, 2013. [ Links ]

[21] A. Nuino, M. Almahalebi, and A. Charif, “Measure Zero Stability Problem for Drygas Functional Equation with Complex Involution”, in Frontiers in Functional Equations and Analytic Inequalities, G. A. Anastassiou and J. M. Rassias, Eds. Cham: Springer, 2019, pp. 183-193. [ Links ]

[22] J. C. Oxtoby, Measure and Category. New York: Springer, 1980. [ Links ]

[23] W.G. Park, and J.H. Bae, “Stability of a bi-additive functional equation in Banach modules over a C*-Algebra”, Discrete Dynamics in Nature and Society, 2012, Art. ID 835893. [ Links ]

[24] Th. M. Rassias, “On the stability of the linear mapping in Banach spaces”, Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978. [ Links ]

[25] Th.M. Rassias, “On a modified Hyers-Ulam sequence”, Journal of Mathematical Analysis and Applications , vol. 158, pp. 106-113, 1991. [ Links ]

[26] J. M. Rassias, “On the Ulam stability of mixed type mappings on restricted domains”, Journal of Mathematical Analysis and Applications , vol. 281, pp. 747-762, 2002. [ Links ]

[27] J. M. Rassias, “On the Ulam stability of Jensen and Jensen type mappings on restricted domains”, Journal of Mathematical Analysis and Applications , vol. 281, pp. 516- 524, 2003. [ Links ]

[28] S. M. Ulam, A Collection of Mathematical Problems. New York: Wiley, 1964 [ Links ]

Received: January 30, 2021; Accepted: November 30, 2021

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License