Hyers-Ulam Stability of Quartic Functional Equation in IFN-Spaces and 2-Banach Spaces by Classical Methods

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-23-2025-68
Published: Mar 17, 2025
Cite as:
Gowri Senthil, N. Vijaya, S. Gayathri, P. Vijayalakshmi, M. Balamurugan, S. Karthikeyan, Hyers-Ulam Stability of Quartic Functional Equation in IFN-Spaces and 2-Banach Spaces by Classical Methods, Int. J. Anal. Appl., 23 (2025), 68.

Main Article Content

Gowri Senthil, N. Vijaya, S. Gayathri, P. Vijayalakshmi, M. Balamurugan, S. Karthikeyan

Abstract

There are many practical applications of functional equations that depend on the Ulam stability. Important for real-world applications, this stability idea makes sure that slight modifications to the functional equation don’t cause significant modifications to the solutions. The purpose of this work is to examine the Hyers-Ulam stability of a finite-dimensional quartic functional equation in 2-Banach spaces and IFN-spaces (Intuitionistic Fuzzy Normed spaces) by utilizing fixed point and direct approaches. Within the context of this quartic functional equation, as an illustration of the stability of the equation can be regulated by sums and products of powers of norms, we present several instances.

Article Details

References

  1. A.M. Alanazi, G. Muhiuddin, K. Tamilvanan, E.N. Alenze, A. Ebaid, K. Loganathan, Fuzzy Stability Results of Finite Variable Additive Functional Equation: Direct and Fixed Point Methods, Mathematics 8 (2020), 1050. https://doi.org/10.3390/math8071050.
  2. T. Aoki, On the Stability of the Linear Transformation in Banach Spaces, J. Math. Soc. Japan 2 (1950), 64–66. https://doi.org/10.2969/jmsj/00210064.
  3. T. Bag, S.K. Samanta, Fuzzy Bounded Linear Operators, Fuzzy Sets Syst. 151 (2005), 513–547. https://doi.org/10.1016/j.fss.2004.05.004.
  4. A. Bodaghi, C. Park, J.M. Rassias, Fundamental Stabilities of the Nonic Functional Equation in Intuitionistic Fuzzy Normed Spaces, Commun. Korean Math. Soc. 31 (2016), 729–743. https://doi.org/10.4134/CKMS.C150147.
  5. J. Fang, On I-Topology Generated by Fuzzy Norm, Fuzzy Sets Syst. 157 (2006), 2739–2750. https://doi.org/10.1016/j.fss.2006.03.024.
  6. S. Gähler, Lineare 2-normierte Räume, Math. Nachr. 28 (1964), 1–43. https://doi.org/10.1002/mana.19640280102.
  7. P. Gavruta, A Generalization of the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings, J. Math. Anal. Appl. 184 (1994), 431–436. https://doi.org/10.1006/jmaa.1994.1211.
  8. D.H. Hyers, On the Stability of the Linear Functional Equation, Proc. Nat. Acad. Sci. 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222.
  9. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer New York, 2006. https://doi.org/10.1007/978-1-4419-9637-4.
  10. S.A. Mohiuddine, H. ¸Sevli, Stability of Pexiderized Quadratic Functional Equation in Intuitionistic Fuzzy Normed Space, J. Comput. Appl. Math. 235 (2011), 2137–2146. https://doi.org/10.1016/j.cam.2010.10.010.
  11. M. Mursaleen, Q.M. Danish Lohani, Intuitionistic Fuzzy 2-Normed Space and Some Related Concepts, Chaos Solitons Fractals 42 (2009), 224–234. https://doi.org/10.1016/j.chaos.2008.11.006.
  12. C. Park, K. Tamilvanan, G. Balasubramanian, B. Noori, A. Najati, On a Functional Equation That Has the QuadraticMultiplicative Property, Open Math. 18 (2020), 837–845. https://doi.org/10.1515/math-2020-0032.
  13. W.-G. Park, Approximate Additive Mappings in 2-Banach Spaces and Related Topics, J. Math. Anal. Appl. 376 (2011), 193–202. https://doi.org/10.1016/j.jmaa.2010.10.004.
  14. V. Radu, The Fixed Point Alternative and the Stability of Functional Equations, Fixed Point Theory 4 (2003), 91–96. https://api.semanticscholar.org/CorpusID:56040530.
  15. T.M. Rassias, On the Stability of the Linear Mapping in Banach Spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. https://doi.org/10.1090/S0002-9939-1978-0507327-1.
  16. R. Saadati, Y.J. Cho, J. Vahidi, The Stability of the Quartic Functional Equation in Various Spaces, Comput. Math. Appl. 60 (2010), 1994–2002. https://doi.org/10.1016/j.camwa.2010.07.034.
  17. R. Saadati, C. Park, Non-Archimedean L -Fuzzy Normed Spaces and Stability of Functional Equations, Comput. Math. Appl. 60 (2010), 2488–2496. https://doi.org/10.1016/j.camwa.2010.08.055.
  18. K. Tamilvanan, A.M. Alanazi, M.G. Alshehri, J. Kafle, Hyers-Ulam Stability of Quadratic Functional Equation Based on Fixed Point Technique in Banach Spaces and Non-Archimedean Banach Spaces, Mathematics 9 (2021), 2575. https://doi.org/10.3390/math9202575.
  19. K. Tamilvanan, J. Rye Lee, C. Park, Hyers-Ulam Stability of a Finite Variable Mixed Type Quadratic-Additive Functional Equation in Quasi-Banach Spaces, AIMS Math. 5 (2020), 5993–6005. https://doi.org/10.3934/math.2020383.
  20. S.M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1960.
  21. T.Z. Xu, J.M. Rassias, W.X. Xu, Stability of a General Mixed Additive-Cubic Functional Equation in Non-Archimedean Fuzzy Normed Spaces, J. Math. Phys. 51 (2010), 093508. https://doi.org/10.1063/1.3482073.