The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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ON THE SUPERSTABILITY OF THE p-RADICAL SINE TYPE FUNCTIONAL EQUATIONS

  • Kim, Gwang Hui (Department of Applied Mathematics, Kangnam University)
  • Received : 2021.08.30
  • Accepted : 2021.10.14
  • Published : 2021.11.30
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Abstract

In this paper, we will find solutions and investigate the superstability bounded by constant for the p-radical functional equations as follows: $f\(\sqrt[p]{\frac{x^p+y^p}{2}}\)^2-f\(\sqrt[p]{\frac{x^p-y^p}{2}}\)^2=\;\{(i)\;f(x)f(y),\\(ii)\;g(x)f(y),\\(iii)\;f(x)g(y),\\(iv)\;g(x)g(y).$ with respect to the sine functional equation, where p is an odd positive integer and f is a complex valued function. Furthermore, the results are extended to Banach algebra.

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References

  1. M. Almahalebi, R. El Ghali, S. Kabbaj & C. Park: Superstability of p-radical functional equations related to WilsonKannappanKim functional equations. Results Math. 76 (2021), Paper No. 97.
  2. R. Badora: On the stability of cosine functional equation. Rocznik Naukowo-Dydak., Prace Mat. 15 (1998), 1-14.
  3. R. Badora & R. Ger: On some trigonometric functional inequalities. Functional Equations- Results and Advances, 3-15 (2002).
  4. J.A. Baker: The stability of the cosine equation. Proc. Amer. Math. Soc. 80 (1980), 411-416. https://doi.org/10.1090/S0002-9939-1980-0580995-3
  5. J. Baker, J. Lawrence & F. Zorzitto: The stability of the equation f(x+y) = f(x)f(y). Proc. Amer. Math. Soc. 74 (1979), 242-246. https://doi.org/10.1090/S0002-9939-1979-0524294-6
  6. B. Bouikhalene, E. Elquorachi & J.M. Rassias: The superstability of d'Alembert's functional equation on the Heisenberg group. Appl. Math. Lett. 23 (2010), 105-109. https://doi.org/10.1016/j.aml.2009.08.013
  7. P.W. Cholewa: The stability of the sine equation. Proc. Amer. Math. Soc. 88 (1983), 631-634. https://doi.org/10.1090/S0002-9939-1983-0702289-8
  8. E. Elqorachi & M. Akkouchi: On Hyers-Ulam stability of the generalized Cauchy and Wilson equations. Publ. Math. Debrecen 66 (2005), 283-301. https://doi.org/10.5486/PMD.2005.2956
  9. P. de P. Friis: d'Alembert's and Wilson's equations on Lie groups. Aequationes Math. 67 (2004), 12-25. https://doi.org/10.1007/s00010-002-2665-3
  10. M. Eshaghi Gordji & M. Parviz: On the Hyers-Ulam-Rassias stability of the functional equation $f(\sqrt{x^2+y^2})=f(x)+f(y)$. Nonlinear Funct. Anal. Appl. 14 (2009), 413-420.
  11. D.H. Hyers: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  12. Pl. Kannappan & G.H. Kim: On the stability of the generalized cosine functional equations. Annales Acadedmiae Paedagogicae Cracoviensis - Studia Mathematica 1 (2001), 49-58.
  13. G.H. Kim: Superstability of the p-radical functional equations related to Wilson's and Kim's equation. Int. J. Nonlinear. Anal. Appl. 12 (2021), 571-582.
  14. G.H. Kim: On the stability of the generalized sine functional equations. Acta Math. Sin., Engl. Ser. 25 (2009), 965-972
  15. G.H. Kim: A stability of the generalized sine functional equations. Jour. Math. Anal & Appl. 331 (2007), 886-894. https://doi.org/10.1016/j.jmaa.2006.09.037
  16. G.H. Kim: On the superstability of the pexider type trigonometric functional equation. Journal of Inequalities Applications 2010 (2010), Doi:10.1155/2010/897123.
  17. G.H. Kim: The stability of the d'Alembert and Jensen type functional equations. J. Math. Anal & Appl. 325 (2007), 237-248. https://doi.org/10.1016/j.jmaa.2006.01.062
  18. G.H. Kim: The stability of pexiderized cosine functional equations. Korean J. Math. 16 (2008), no. 1, 103-114. https://doi.org/10.3740/MRSK.2008.18.2.103
  19. G.H. Kim: A stability of the generalized sine functional equations. J. Math. Anal & Appl. preprint (2007).
  20. G.H. Kim: On the stability of mixed trigonometric functional equations. Banach J. Math. Anal. 2 (2007), 227-236. https://doi.org/10.15352/bjma/1240336221
  21. G.H. Kim & Sever S. Dragomir: On the the Stability of generalized d'Alembert and Jensen functional equation. Intern. Jour. Math. & Math. Sci. preprint (2006).
  22. G.H. Kim & S.H. Lee: Stability of the d'Alembert type functional equations. Nonlinear Funct. Anal. Appl. 9 (2004), 593-604.
  23. P. Sinopoulos: Generalized sine equations. III. Aequationes Math. 51 (1996), 311-327. https://doi.org/10.1007/BF01833286
  24. S.M. Ulam: 'Problems in Modern Mathematics' Chap. VI, Science editions. Wiley, New York, (1964)