2-Banach stability results for the radical cubic functional equation related to quadratic mapping
الموضوعات :
R. E. Ghali
1
(Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, BP 133 Kenitra, Morocco)
S. Kabbaj
2
(Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, BP 133 Kenitra, Morocco)
الکلمات المفتاحية: Stability, hyperstability, 2-Banach spaces, radical functional equation,
ملخص المقالة :
The aim of this paper is to introduce and solve the generalized radical cubic functional equation related to quadraticfunctional equation$$f\left(\sqrt[3]{ax^{3}+by^{3}}\right)+f\left(\sqrt[3]{ax^{3}-by^{3}}\right)=2a^{2}f(x)+2b^{2}f(y),\;\; x,y\in\mathbb{R},$$for a mapping $f$ from $\mathbb{R}$ into a vector space.We also investigate some stability and hyperstability results forthe considered equation in 2-Banach spaces by using an analogue theorem of Brzd\c{e}k in [17].
[1] M. Almahalebi, On the stability of a generalization of Jensen functional equation, Acta Math. Hungar. 154 (1) (2018), 187-198.
[2] M. Almahalebi, Stability of a generalization of Cauchy’s and the quadratic functional equations, J. Fixed Point Theory Appl. 2018, 20:12.
[3] M. Almahalebi, A. Chahbi, Approximate solution of p-radical functional equation in 2-Banach spaces, (preprint).
[4] M. Almahalebi, A. Chahbi, Hyperstability of the Jensen functional equation in ultrametric spaces, Aequat. Math. 91 (4) (2017), 647-661.
[5] M. Almahalebi, A. Charifi, S. Kabbaj, Hyperstability of a Cauchy functional equation, J. Nonlinear Anal. Optim. Theor & Appl. 6 (2) (2015), 127-137.
[6] M. Almahalebi, C. Park, On the hyperstability of a functional equation in commutative groups, J. Comput Anal. Appl. 20 (1) (2016), 826-833.
[7] L. Aiemsomboon, W. Sintunavarat, On a new type of stability of a radical quadratic functional equation using Brzdek’s fixed point theorem, Acta Math. Hungar. 2017, 151: 35.
[8] L. Aiemsomboon, W. Sintunavarat, On generalized hyperstability of a general linear equation, Acta Math. Hungar. 149 (2016), 413-422.
[9] Z. Alizadeh, A. G. Ghazanfari, On the stability of a radical cubic functional equation in quasi-β-spaces, J. Fixed Point Theory Appl. 2016, 18:843.
[10] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan. 2 (1950), 64-66.
[11] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223-237.
[12] J. Brzdek, A note on stability of additive mappings, in: Stability of Mappings of Hyers-Ulam Type, Rassias, T.M., Tabor, J. (eds.), Hadronic Press (Palm Harbor, 1994), pp. 19-22.
[13] J. Brzdek, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar. 141 (2013), 58-67.
[14] J. Brzdek, Remark 3, In: Report of Meeting of 16th International Conference on Functional Equations and Inequalities (B edlewo, Poland, May 17-23,(2015), p. 196, Ann. Univ. Paedagog. Crac. Stud. Math. 14 (2015), 163-202.
[15] J. Brzdek, Stability of additivity and fixed point methods, Fixed Point Theory Appl. 2013, 2013:265.
[16] J. Brzdek, L. Cadariu, K. Cieplinski, Fixed point theory and the Ulam stability, J. Funct. Spaces. 2014, 2014:829419.
[17] J. Brzdek, J. Chudziak, Z. Páles, A fixed point approach to stability of functional equations, Nonlinear Anal. 74 (2011), 6728-6732.
[18] J. Brzdek, W. Fechner, M. S. Moslehian and J. Sikorska, Recent developments of the conditional stability of the homomorphism equation, Banach J. Math. Anal. 9 (2015), 278-327.
[19] M. Eshaghi Gordji, H. Khodaei, A. Ebadian and G. H. Kim, Nearly radical quadratic functional equations in p-2-normed spaces, Abstr. Appl. Anal. 2012, 2012:896032.
[20] M. Eshaghi Gordji, M. Parviz, On the HyersUlam stability of the functional equation f(2√x2+ y2)
=f(x) + f(y), Nonlinear Funct. Anal. Appl. 14 (2009), 413-420.
[21] S. Gahler, 2-metrische Raume und ihre topologische Struktur, Math. Nachr. 26 (1963), 115-148.
[22] S. Gahler, Linear 2-normiete R¨ aumen, Math. Nachr. 28 (1964), 1-43.
[23] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222-224.
[24] H. Khodaei, M. Eshaghi Gordji, S. S. Kim, Y. J. Cho, Approximation of radical functional equations related to quadratic and quartic mappings, J. Math. Anal. Appl. 395 (2012), 284-297.
[25] S. S. Kim, Y. J. Cho, M. Eshaghi Gordji, On the generalized Hyers-Ulam-Rassias stability problem of radical functional equations, J. Inequal. Appl. 2012, 2012:186.
[26] G. Maksa, Z. Pales, Hyperstability of a class of linear functional equations, Acta. Math. 17 (2001), 107-112.
[27] W. G. Park, Approximate additive mappings in 2-Banach spaces and related topics, J. Math. Anal. Appl. 376 (2011), 193-202.
[28] Th. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), 106-113.
[29] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
[30] Th. M. Rassias, Problem 16; 2. Report of the 27th international symposium on functional equations, Aequationes Math. 39 (1990), 292-293.
[31] S. M. Ulam, Problems in Modern Mathematics, Science Editions, John-Wiley & Sons Inc., New York, 1964.
