On an equation characterizing multi-quartic mappings and its stability

Document Type : Research Paper

Authors

1 Department of Mathematics, Tehran North Branch, Islamic Azad University, Tehran, Iran

2 Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran

Abstract

In this paper, we define and investigate the mappings of several variables which are quartic in each variable. We show that such mappings can be unified as an equation, say the multi-quartic functional equation. We also establish the Hyers-Ulam stability of a such functional equation by a fixed point theorem in non-Archimedean normed spaces. Moreover, we generalize some known stability and hyperstability results.

Keywords

[1] A. Bahyrycz, K. Ciepli´nski and J. Olko, On Hyers-Ulam stability of two functional equations in non-Archimedean
spaces, J. Fixed Point Theory Appl. 18 (2016) 433–444.
[2] A. Bodaghi, Functional inequalities for generalized multi-quadratic mappings, J. Inequal. Appl. 2021 (2021) Paper
No. 145.
[3] A. Bodaghi, Intuitionistic fuzzy stability of the generalized forms of cubic and quartic functional equations, J.
Intel. Fuzzy Syst. 30 (2016) 2309–2317.
[4] A. Bodaghi, Stability of a quartic functional equation, Sci. World J. 2014 Art. ID 752146, 9 pages.
[5] A. Bodaghi, C. Park and O. T. Mewomo, Multiquartic functional equations, Adv. Diff. Equa. 2019 (2019) Paper
No. 312.
[6] A. Bodaghi, C. Park and S. Yun, Almost multi-quadratic mappings in non-Archimedean spaces, AIMS Math. 5(5)
(2020) 5230–5239.
[7] A. Bodaghi, S. Salimi and G. Abbasi, Characterization and stability of multi-quadratic functional equations in
non-Archimedean spaces, Ann. Uni. Craiova-Math. Comp. Sci. Ser. 48(1) (2021) 88–97.
[8] A. Bodaghi and B. Shojaee, On an equation characterizing multi-cubic mappings and its stability and hyperstability, Fixed Point Theory 22(1) (2021) 83–92.
[9] J. Brzd¸ek and K. Ciepli´nski, A fixed point approach to the stability of functional equations in non-Archimedean
metric spaces, Nonlinear Anal. 74 (2011) 6861–6867.
[10] K. Ciepli´nski, Ulam stability of functional equations in 2-Banach spaces via the fixed point method, J. Fixed Point
Theory Appl. 23 (2021) Paper No. 33.
[11] K. Ciepli´nski, On the generalized Hyers-Ulam stability of multi-quadratic mappings, Comput. Math. Appl. 62
(2011) 3418–3426.
[12] K. Ciepli´nski, Generalized stability of multi-additive mappings, Appl. Math. Lett. 23 (2010) 1291–1294.
[13] M. Dashti and H. Khodaei, Stability of generalized multi-quadratic mappings in Lipschitz spaces, Results Math.
74 (2019) Paper No. 163.
[14] N. Ebrahimi Hoseinzadeh, A. Bodaghi and M.R. Mardanbeigi, Almost multi-cubic mappings and a fixed point
application, Sahand Commun. Math. Anal. 17(3) (2020) 131–143.
[15] M.B. Ghaemi, M. Majani and M.E. Gordji, General system of cubic functional equations in non-Archimedean
spaces, Tamsui Oxford J. Inf. Math. Sci. 28(4) (2012) 407–423.
[16] K. Hensel, Uber eine neue Begrndung der Theorie der algebraischen Zahlen, Jahresber, Deutsche MathematikerVereinigung. 6 (1897) 83–88.
[17] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941) 222–224.
[18] K.W. Jun and H.M. Kim, On the Hyers-Ulam-Rassias stability of a general cubic functional equation, Math.
Inequ. Appl. 6(2) (2003) 289–302.
[19] K.W. Jun and H.M. Kim, The generalized Hyers-Ulam-Russias stability of a cubic functional equation, J. Math.
Anal. Appl. 274(2) (2002) 267–278.
[20] A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models,
Mathematics and its Applications, vol. 427, Kluwer Academic Publishers, Dordrecht, 1997.
[21] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and
Jensen’s Inequality, Birkhauser Verlag, Basel, 2009.
[22] Y. Lee and S. Chung, Stability of quartic functional equations in the spaces of generalized functions, Adv. Diff.
Equ. 2009 (2009) Art. ID. 838347.
[23] S. Lee, S. Im and I. Hwang, Quartic functional equations, J. Math. Anal. Appl. 307 (2005) 387–394.
[24] C.-G. Park, Multi-quadratic mappings in Banach spaces, Proc. Amer. Math. Soc. 131 (2002) 2501–2504.
[25] C. Park and A. Bodaghi, Two multi-cubic functional equations and some results on the stability in modular spaces,
J. Inequ. Appl. 2020 (2020), Paper No. 6.
[26] C. Park, A. Bodaghi and T.-Z. Xu, On an equation characterizing multi-Jensen-quartic mappings and its stability,
J. Math. Inequa. 15(1) (2021) 333–347.
[27] J.M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnik Matematicki Series III. 34(2)
(1999) 243–252.[28] J.M. Rassias, Solution of the Ulam stability problem for cubic mappings, Glasnik Matematicki. Serija III. 36(1)
(2001) 63–72.
[29] S. Salimi and A. Bodaghi, A fixed point application for the stability and hyperstability of multi-Jensen-quadratic
mappings, J. Fixed Point Theory Appl. 22 (2020) Paper No. 9.
[30] S.M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, New York, 1964.
[31] T.-Z. Xu, Stability of multi-Jensen mappings in non-Archimedean normed spaces, J. Math. Phys. 53 (2012) Art.
ID. 023507.
[32] T.-Z. Xu, Ch. Wang and Th. M. Rassias, On the stability of multi-additive mappings in non-Archimedean normed
spaces. J. Comput. Anal. Appl. 18 (2015), 1102–1110.
[33] X. Zhao, X. Yang and C.-T. Pang, Solution and stability of the multiquadratic functional equation, Abstr. Appl.
Anal. 2013 (2013), Art. ID 415053, 8 pp.
Volume 13, Issue 1
March 2022
Pages 991-1002
  • Receive Date: 18 May 2021
  • Revise Date: 22 July 2021
  • Accept Date: 07 September 2021