Fixed point approach to the stability of a cubic and quartic mixed type functional equation in non-archimedean spaces

Volume 33, Issue 2, pp 124--136 http://dx.doi.org/10.22436/jmcs.033.02.01

Publication Date: December 05, 2023 Submission Date: April 27, 2023 Revision Date: August 18, 2023 Accteptance Date: November 01, 2023

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 306 Downloads
• 877 Views

Authors

P. Elumalai - Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur - 603 203, Tamil Nadu, India. S. Sangeetha - Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur - 603 203, Tamil Nadu, India. A. P. Selvan - Department of Mathematics, Rajalakshmi Engineering College, Thandalam, Chennai - 602 105, Tamil Nadu, India.

Abstract

The objective of this article is to establish the generalized Hyers-Ulam stability of the following cubic-quartic(\(C_3 Q_4\)) functional equation \[ \mathfrak{g}(\upsilon + 2\nu) + \mathfrak{g}(\upsilon - 2\nu) - 4 [\mathfrak{g}(\upsilon+\nu) + \mathfrak{g}(\upsilon-\nu)] = 3 \mathfrak{g}(2\nu) - 24 \mathfrak{g}(\nu) - 6 \mathfrak{g}(\upsilon) \] in non-Archimedean normed spaces by using alternative fixed point theorem.

Share and Cite

Keywords

• Generalized Hyers-Ulam (HU) stability
• cubic-quartic (\(C_3 Q_4\)) functional equation (FE)
• non-archimedean (NA) normed spaces
• fixed point theorem

MSC

•  39B52
•  47H10
•  26E30
•  46S10
•  47S10

References

• [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64–66
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] A. Bahyrycz, J. Brzdek, E. Jabło ´ nska, J. Olko, On functions that are approximate fixed points almost everywhere and Ulam’s type stability, J. Fixed Point Theory Appl., 17 (2015), 659–668
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] J. Brzde¸k, K. Ciepli ´ nski, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal., 74 (2011), 6861–6867
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] J. Brzde¸k, K. Ciepli ´ nski, A fixed point theorem and the Hyers-Ulam stability in non-Archimedean spaces, J. Math. Anal. Appl., 400 (2013), 68–75
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] L. C˘adariu, V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math., 4 (2003), 7 pages
  • View Article
  • Google Scholar
  • MATH


• [6] J. B. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305–309
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci., 14 (1991), 431–434
  • View Article
  • MathSciNet
  • Google Scholar


• [8] P. G˘avrut¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431–436
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] P. G˘avrut¸a, On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings, J. Math. Anal. Appl., 261 (2001), 543–553
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] H. Gharib, M. B. Moghimi, A. Najati, J. H. Bae, Asymptotic stability of the Pexider-Cauchy functional equation in non-Archimedean spaces, Mathematics, 9 (2021), 12 pages
  • View Article
  • Google Scholar


• [11] M. E. Gordji, A. Ebadian, S. Zolfaghari, Stability of a functional equation deriving from cubic and quartic functions, Abstr. Appl. Anal., 2008 (2008), 17 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] M. E. Gordji, M. B. Savadkouhi, Stability of cubic and quartic functional equations in non-Archimedean spaces, Acta Appl. Math., 110 (2010), 1321–1329
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] N. B. Huy, T. D. Thanh, Fixed point theorems and the Ulam-Hyers stability in non-Archimedean cone metric spaces, J. Math. Anal. Appl., 414 (2014), 10–20
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222–224
  • View Article
  • MathSciNet
  • Google Scholar


• [15] G. Isac, Th. M. Rassias, Stability of -additive mappings: applications to nonlinear analysis, nternat. J. Math. Math. Sci., 19 (1996), 219–228
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] K.-W. Jun, H.-M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl., 247 (2002), 267–278
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] M. S. Moslehian, T. M. Rassias, Stability of functional equations in non-Archimedean spaces, Appl. Anal. Discrete Math., 1 (2007), 325–334
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] R. Murali, M. J. Rassias, V. Vithya, The general solution and stability of nonadecic functional equation in matrix normed spaces, Malaya J. Mat., 5 (2017), 416–427
  • View Article
  • Google Scholar


• [19] A. Najati, M. B. Moghimi, Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces, J. Math. Anal. Appl., 337 (2008), 399–415
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] A. Najati, B. Noori, M. B. Moghimi, On Approximation of Some Mixed Functional Equations, Sahand Commun. Math. Anal., 18 (2021), 35–46
  • View Article
  • Google Scholar


• [21] A. Najati, P. K. Sahoo, On some functional equations and their stability, J. Interdiscip. Math., 23 (2020), 755–765
  • View Article
  • Google Scholar


• [22] N. Noori, M. B. Moghimi, A. Najati, C. Park, J. R. Lee, On superstability of exponential functional equations, J. Inequal. Appl., 2021 (2021), 17 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] C. Park, Fixed points and the stability of an AQCQ-functional equation in non-Archimedean normed spaces, Abstr. Appl. Anal., 2010 (2010), 15 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] W.-G. Park, J.-H. Bae, On a bi-quadratic functional equation and its stability, Nonlinear Anal., 62 (2005), 643–654
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] A. Ponmana Selvan, A. Najati, Hyers–Ulam stability and Hyper-stability of a Jensen-type functional equation on 2-Banach spaces, J. Inequal. Appl., 2022 (2022), 1–11
  • View Article
  • Google Scholar


• [26] T. M. Rassias, On the stability of the linear mappings in Banach Spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300
  • View Article
  • MathSciNet
  • Google Scholar


• [27] T. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta. Appl. Math., 62 (2000), 23–130
  • View Article
  • MathSciNet
  • Google Scholar


• [28] Y. Shehu, G. Cai, O. S. Iyiola, Iterative approximation of solutions for proximal split feasibility problems, Fixed Point Theory Appl., 2015 (2015), 18 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] K. Tamilvanan, A. H. Alkhaldi, R. P. Agarwal, A. M. Alanazi, Fixed point approach: Ulam stability results of functional equation in non-Archimedean fuzzy '-2-normed spaces and non-Archimedean Banach spaces, Mathematics, 11 (2023), 1–24
  • View Article
  • Google Scholar


• [30] S. M. Ulam, Problem in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York (1964)
  • View Article
  • Google Scholar
  • MATH