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Hyperstability of the General Linear Functional Equation in Non-Archimedean Banach Spaces

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Abstract

Let \( X \) be a normed space over \( \mathbb{F} \in\{ \mathbb{R}, \mathbb{C}\} \), \( Y \) be a non-Archimedean Banach space over a non-Archimedean non-trivial field \(\mathbb{K}\) and \(c,d,C,D\) be constants such that, \( c, d \in \mathbb{F}\setminus\{0\} \) and \( C, D \in \mathbb{K}\setminus\{0\} \). In this paper, some preliminaries on non-Archimedean Banach spaces and the concept of hyperstability are presented. Next, the well-known fixed point method [7, Theorem1] is reformulated in non-Archimedean Banach spaces. Using this method, we prove that the general linear functional equation \( h(cx+dy)= Ch(x)+Dh(y) \) is hyperstable in the class of functions \( h:X\rightarrow Y \). In fact, by exerting some natural assumptions on control function \( \gamma:X^{2}\setminus\{0\}\rightarrow \mathbb{R}_{+} \), we show that the map \( h:X\rightarrow Y \) that satisfies the inequality \( \lVert h(cx+dy)- Ch(x)-Dh(y)\rVert_{\ast}\leq \gamma(x,y) \), is a solution to general linear functional equation for every \( x, y \in X\setminus\{0\} \). Finally, this paper concludes with some consequences of the results.

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References

  1. A. El Amrani, A. Razouki, R. A. Hassani and M. Babahmed, “On the \(K\)-vector sequential topology on a non-Archimedean valued field,” \(p\)-Adic Num. Ultrametr. Anal. Appl. 12, 177–184 (2020).

    Article  MathSciNet  Google Scholar 

  2. T. Aoki, “On the stability of the linear transformation in Banach spaces,” J. Math. Soc. Japan 2 (1-2), 64–66 (1950).

    Article  MathSciNet  Google Scholar 

  3. M. Almahalebi and A. Chahbi, “Hyperstability of the Jensen functional equation in ultrametric spaces,” Aequationes Math. 91 (4), 647–661 (2017).

    Article  MathSciNet  Google Scholar 

  4. A. Bahyrycz and M. Piszczek, “Hyperstability of the Jensen functional equation,” Acta Math. Hung. 142 (2), 353–365 (2014).

    Article  MathSciNet  Google Scholar 

  5. D. G. Bourgin, “Approximately isometric and multiplicative transformations on continuous function rings,” Duke Math. J. 16 (2), 385–397 (1949).

    Article  MathSciNet  Google Scholar 

  6. D. G. Bourgin, “Classes of transformations and bordering transformations,” Bull. Amer. Math. Soc. 57 (4), 223–237 (1951).

    Article  MathSciNet  Google Scholar 

  7. J. Brzdek and K. Ciepliński, “A fixed point approach to the stability of functional equations in non-Archimedean metric spaces,” Nonlin. Anal.: Theory Meth. Appl. 74 (18), 6861–6867 (2011).

    Article  MathSciNet  Google Scholar 

  8. J. Brzdek and K. Ciepliński, “Hyperstability and superstability,” Abstr. Appl. Anal. 2013 (2013).

    Article  MathSciNet  Google Scholar 

  9. J. Brzdek, “Remarks on hyperstability of the Cauchy functional equation,” Aequationes Math. 86 (3), 255–267 (2013).

    Article  MathSciNet  Google Scholar 

  10. J. Brzdek, “Stability of additivity and fixed point methods,” Fixed Point Theo. Appl. 2013 (1), 1–9 (2013).

    MathSciNet  Google Scholar 

  11. J. Brzdek, “A hyperstability result for the Cauchy equation,” Bull. Austral. Math. Soc. 89 (1), 33–40 (2014).

    Article  MathSciNet  Google Scholar 

  12. J. Brzdek, “Remarks on stability of some inhomogeneous functional equations,” Aequationes Math. 89 (1), 83–96 (2015).

    Article  MathSciNet  Google Scholar 

  13. J. Brzdek, “Hyperstability of the Cauchy equation on restricted domains,” Acta Math. Hung. 141 (1-2), 58–67 (2013).

    Article  MathSciNet  Google Scholar 

  14. K. Ciepliński, “Stability of multi-additive mappings in non-Archimedean normed spaces,” J. Math. Anal. Appl. 373 (2), 376–383 (2011).

    Article  MathSciNet  Google Scholar 

  15. T. Diagana and F. Ramaroson, Non-Archimedean Operator Theory (Springer, 2016).

    Book  Google Scholar 

  16. Iz. El-Fassi, “A new type of approximation for the radical quintic functional equation in non-Archimedean (2, \(\beta\))-Banach spaces,” J. Math. Anal. Appl. 457 (1), 322–335 (2018).

    Article  MathSciNet  Google Scholar 

  17. Iz. El-Fassi and S. Kabbaj, “Non-Archimedean random stability of \(\sigma\)-quadratic functional equation,” Thai J. Math. 14 (1), 151–165 (2015).

    MathSciNet  Google Scholar 

  18. Iz. El-Fassi and S. Kabbaj, “On the hyperstability of a Cauchy-Jensen type functional equation in Banach spaces,” Proyecciones (Antofagasta) 34 (4), 359–375 (2015).

    Article  MathSciNet  Google Scholar 

  19. Iz. El-Fassi, S. Kabbaj and A. Charifi, “Hyperstability of Cauchy-Jensen functional equations,” Indagationes Math. 27 (3), 855–867 (2016).

    Article  MathSciNet  Google Scholar 

  20. Iz. El-Fassi, “On a new type of hyperstability for radical cubic functional equation in non-Archimedean metric spaces,” Results Math. 72 (1), 991–1005 (2017).

    Article  MathSciNet  Google Scholar 

  21. Iz. El-Fassi, S. Kabbaj and A. Chahbi, “A fixed point approach to the hyperstability of the general linear equation in \(\beta\)-Banach spaces,” Analysis (0174-4747) 38 (3), (2018).

    Article  MathSciNet  Google Scholar 

  22. Iz. El-Fassi, “Non-Archimedean hyperstability of Cauchy-Jensen functional equations on a restricted domain,” J. Appl. Anal. 24 (2), 155–165 (2018).

    Article  MathSciNet  Google Scholar 

  23. Iz. El-Fassi, E. Elqorachi and H. Khodaei, “A fixed point approach to stability of \(k\)-th radical functional equation in Non-Archimedean \((n, \beta)\)-Banach spaces,” Bull. Iran. Math. Soc. 1–18 (2020).

    Google Scholar 

  24. A. El Amrani, A. Razouki, R. A. Hassani and M. Babahmed, “On the K-vector sequential topology on a non-Archimedean valued field,” \(p\)-Adic Num. Ultrametr. Anal. Appl. 12, 177–184 (2020).

    Article  MathSciNet  Google Scholar 

  25. Z. Gajda, “On stability of additive mappings,” Int. J. Math. Math. Sci. 14 (3), 431–434 (1991).

    Article  MathSciNet  Google Scholar 

  26. M. Gordji and M. Savadkouhi, “Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces,” Appl. Math. Lett. 23 (10), 1198–1202 (2010).

    Article  MathSciNet  Google Scholar 

  27. E. Gselmann, “Hyperstability of a functional equation,” Acta Math. Hung. 124 (1-2), 179–188 (2009).

    Article  MathSciNet  Google Scholar 

  28. K. Hensel, “Über eine neue Begründung der Theorie der algebraischen Zahlen,” Jahresbericht der Deutschen Mathematiker-Vereinigung 6, 83–88 (1897).

    Google Scholar 

  29. D. H. Hyers, “On the stability of the linear functional equation,” Proc. Nat. Acad. Sci. U. S, A. 27 (4), 222 (1941).

    Article  MathSciNet  CAS  ADS  Google Scholar 

  30. A. K. Katsaras and A. Beloyiannis, “Tensor products of non-Archimedean weighted spaces of continuous functions,” Georgian Math. J. 6 (1), 33–44 (1999).

    Article  MathSciNet  Google Scholar 

  31. A. Y. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer Academic Publishers, Dordrecht, 1997).

    Book  Google Scholar 

  32. H. Khodaei, M. E Gordji, S. Kim and Y. J. Cho, “Approximation of radical functional equations related to quadratic and quartic mappings,” J. Math. Anal. Appl. 395 (1), 284–297 (2012).

    Article  MathSciNet  Google Scholar 

  33. Y. Lee, “On the stability of the monomial functional equation,” Bull. Korean Math. Soc. 45 (2), 397–403 (2008).

    Article  MathSciNet  Google Scholar 

  34. G. Maksa and Z. Páles, “Hyperstability of a class of linear functional equations,” Acta Math. Acad/ Paedag. Nyí Regyháziensis. New Series [electronic only] 17, 107–112 (2001).

    MathSciNet  Google Scholar 

  35. A. K. Mirmostafaee, “Hyers-Ulam stability of cubic mappings in non-Archimedean normed spaces,” Kyungpook Math. J. 50 (2), 315–327 (2010).

    Article  MathSciNet  Google Scholar 

  36. M. S. Moslehian and Th. M. Rassias, “Stability of functional equations in non-Archimedean spaces,” Appl. Anal. Discr. Math. 1 (2), 325–334 (2007).

    Article  MathSciNet  Google Scholar 

  37. M. S. Moslehian and Gh. Sadeghi, “Stability of two types of cubic functional equations in non-Archimedean spaces,” Real Anal. Exch. 33 (2), 375–384 (2008).

    Article  MathSciNet  Google Scholar 

  38. F. Mukhamedov and O. Khakimov, “Hypercyclic and supercyclic linear operators on non-Archimedean vector spaces,” Bull. Belgian Math. Soc. 25 (1), 85–105 (2018).

    MathSciNet  Google Scholar 

  39. F. Mukhamedov, “Recurrence equations over trees in a non-Archimedean context,” \(p\)-Adic Num. Ultrametr. Anal. Appl. 6, 310–317 (2014).

    Article  MathSciNet  Google Scholar 

  40. F. Mukhamedov, “On \(p\)-adic quasi Gibbs measures for q+1-state Potts model on Cayley tree,” \(p\)-Adic Num. Ultrametr. Anal. Appl. 2, 241–251 (2010).

    Article  MathSciNet  Google Scholar 

  41. F. Mukhamedove and H. Akin, “On non-Archimedean recurrence equations and their applications,” J. Math. Anal. Appl. 423, 1203–1218 (2015).

    Article  MathSciNet  Google Scholar 

  42. F. Mukhamedove, O. Khakimov and A. Souissi, “A few remarks on supercyclicity of non-Archimedean linear operators on c0(N),” \(p\)-Adic Num. Ultra. Anal. Appl. 14 (1), 63–75 (2022).

    Google Scholar 

  43. P. J. Nyikos, “On some non-Archimedean spaces of Alexandroff and Urysohn,” Topol. Appl. 91 (1), 1–23 (1999).

    Article  MathSciNet  Google Scholar 

  44. M. Piszczek, “Hyperstability of the general linear functional equation,” Bull. Korean Math. Soc. 52 (6), 1827–1838 (2015).

    Article  MathSciNet  Google Scholar 

  45. Th. M. Rassias, “On a modified Hyers-Ulam sequence,” J. Math. Anal. Appl. 158 (1), 106–113 (1991).

    Article  MathSciNet  Google Scholar 

  46. Th. M. Rassias and P. Šemrl, “On the behavior of mappings which do not satisfy Hyers-Ulam stability,” Proc. Amer. Math. Soc. 989–993 (1992).

    Article  MathSciNet  Google Scholar 

  47. A. M. Robert, A Course in \(p\)-Adic Analysis (Springer, New York, 2000).

    Book  Google Scholar 

  48. A. van Rooij, Non-Archimedean Functional Analysis (M. Dekker, New York, 1978).

    Google Scholar 

  49. S. M. Ulam, Problems in Modern Mathematics (Courier Corporation, 2004).

    Google Scholar 

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Acknowledgments

The authors are grateful to an anonymous referee for his/her useful suggestions which improved the presentation of this paper.

Funding

This work was supported by the Ministry of Higher Education under Fundamental Research Grant Scheme (FRGS/1/2021/STG06/UTM/02/5). The first author expresses gratitude to the Ministry of higher education of Afghanistan for supporting this research under the program of HEDP.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Malaysia

    Shujauddin Shuja, Ahmad F. Embong & Nor M. M. Ali

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  1. Shujauddin Shuja
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Correspondence to Shujauddin Shuja, Ahmad F. Embong or Nor M. M. Ali.

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Shuja, S., Embong, A.F. & Ali, N.M.M. Hyperstability of the General Linear Functional Equation in Non-Archimedean Banach Spaces. P-Adic Num Ultrametr Anal Appl 16, 70–81 (2024). https://doi.org/10.1134/S2070046624010060

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