On the superstability of the \(p\)-power-radical sine functional equation related to Pexider type

Volume 34, Issue 1, pp 52--64 https://dx.doi.org/10.22436/jmcs.034.01.05

Publication Date: February 14, 2024 Submission Date: December 15, 2023 Revision Date: December 23, 2023 Accteptance Date: January 03, 2024

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)

• 136 Downloads
• 680 Views

Authors

W.-G. Park - Department of Mathematics Education, College of Education, Mokwon University, Daejeon 35349, Korea. M. Donganont - School of Science, University of Phayao, Phayao 56000, Thailand. G. H. Kim - Department of Mathematics, Kangnam University, Yongin, Gyeonggi 16979, Korea.

Abstract

In this paper, we investigate the superstability bounded by a function (G{\v a}vruta sense) for the \(p\)-power-radical sine functional equation from the \(p\)-power-radical Pexider type\rq{}s functional equation: \[ f\left(\sqrt[p]{\frac{x+y}{2}}\right)^{2}-g\left(\sqrt[p]{\frac{x-y}{2}}\right)^{2}= h(\sqrt[p]{x})k(\sqrt[p]{y}), \] where \(p\) is an odd positive integer and \(f, g, h, k\) are complex valued functions on \(\mathbb{R}\). Furthermore, the obtained results are extended to Banach algebras.

Share and Cite

Keywords

• Stability
• superstability
• sine functional equation
• \(p\)-radical functional equation
• \(p\)-power-radical functional equation

MSC

•  39B82
•  39B52

References

• [1] M. R. Abdollahpour, R. Aghayari, M. Th. Rassias, Hyers-Ulam stability of associated Laguerre differential equations in a subclass of analytic functions, J. Math. Anal. Appl., 437 (2016), 605–612
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] M. R. Abdollahpour, M. Th. Rassias, Hyers-Ulam stability of hypergeometric differential equations, Aequationes Math., 93 (2019), 691–698
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] M. Almahalebi, R. El Ghali, S. Kabbaj, C. Park, Superstability of p-radical functional equations related to Wilson- Kannappan-Kim functional equations, Results Math., 76 (2021), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] 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


• [5] R. Badora, On the stability of cosine functional equation, Rocznik Nauk.-Dydakt. Prace Mat., 15 (1998), 5–14
  • View Article
  • Google Scholar
  • MATH


• [6] R. Badora, R. Ger, On some trigonometric functional inequalities, Kluwer Acad. Publ., Dordrecht, 3 (2002), 3–15
  • View Article
  • Google Scholar
  • MATH


• [7] J. A. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc., 80 (1980), 411–416
  • View Article
  • MathSciNet
  • Google Scholar


• [8] J. A. 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
  • View Article
  • MathSciNet
  • Google Scholar


• [9] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16 (1949), 385–397
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] P. W. Cholewa, The stability of the sine equation, Proc. Amer. Math. Soc., 88 (1983), 631–634
  • View Article
  • MathSciNet
  • Google Scholar


• [11] P. Gˇavruta, On the stability of some functional equations, In: Stability of mappings of Hyers-Ulam type, Hadronic Press, Palm Harbor, FL (1994)
  • View Article
  • Google Scholar


• [12] M. E. Gordji, M. Parviz, On the Hyers-Ulam-Rassias stability of the functional equation f( p x2 + y2) = f(x) + f(y), Nonlinear Funct. Anal. Appl., 14 (2009), 413–420
  • View Article
  • Google Scholar


• [13] H. J. Hwang, G. H. Kim, Superstability of the p-power-radical functional equation related to sine function equation, Electron. Res. Arch., 31 (2023), 6347–6362
  • View Article
  • MathSciNet
  • Google Scholar


• [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
  • MATH


• [15] D. H. Hyers, G. Isac, Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨auser Boston, Boston, MA (1998)
  • View Article
  • MathSciNet
  • Google Scholar


• [16] D. H. Hyers, Th. M. Rassias, Approximate homomorphisms, Aequationes Math., 42 (1992), 125–153
  • View Article
  • MathSciNet
  • Google Scholar


• [17] S.-M. Jung, M. Th. Rassias, C. Mortici, On a functional equation of trigonometric type, Appl. Math. Comput., 252 (2015), 294–303
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] Pl. Kannappan, G. H. Kim, On the stability of the generalized cosine functional equations, Ann. Acad. Paedag. Crac.- Studia Math., 1 (2001), 49–58
  • View Article
  • Google Scholar


• [19] Nonlinear Anal., Addendum to ‘On the stability of functional equations on square-symmetric groupoid’, Nonlinear Anal., 62 (2005), 365–381
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] G. H. Kim, The stability of d’Alembert and Jensen type functional equations, J. Math. Anal. Appl., 325 (2007), 237–248
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] G. H. Kim, A stability of the generalized sine functional equations, J. Math. Anal. Appl., 331 (2007), 886–894
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] G. H. Kim, On the stability of mixed trigonometric functional equations, Banach J. Math. Anal., 1 (2007), 227–236
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] G. H. Kim, On the stability of the Pexiderized trigonometric functional equation, Appl. Math. Comput., 203 (2008), 99–105
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] G. H. Kim, On the stability of the generalized sine functional equations, Acta Math. Sin. (Engl. Ser.), 25 (2009), 29–38
  • View Article
  • MathSciNet
  • MATH


• [25] G. H. Kim, On the superstability of the Pexider type trigonometric functional equation, J. Inequal. Appl., 2010 (2010), 14 pages
  • View Article
  • MathSciNet
  • MATH


• [26] G. H. Kim, The stability of the p-radical sine type functional equations, J. Korean Soc. Math. Educ. Ser. B: Pure. Appl. Math., 28 (2021), 387–398
  • View Article


• [27] 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
  • View Article
  • Google Scholar


• [28] G. H. Kim, On the superstability for the p-power-radical sine functional equation, Nonlinear Funct. Anal. Appl., 28 (2023), 801–812
  • View Article
  • Google Scholar
  • MATH


• [29] G. H. Kim, S. S. Dragomir, On the stability of generalized d’Alembert and Jensen functional equations, Int. J. Math. Math. Sci., 2006 (2006), 12 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] G. H. Kim, M. Eshaghi Gordji, Superstability of the p-radical sine functional equation, Int. J. Nonlinear Anal. Appl., 14 (2022), 161–170
  • View Article
  • Google Scholar


• [31] G. H. Kim, Y. W. Lee, Boundedness of approximate trigonometric functions, Appl. Math. Lett., 22 (2009), 439–443
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] G. H. Kim, J. Roh, Transferred superstability of the p-radical sine functional equation, J. Chungcheong Math. Soc., 35 (2022), 315–327
  • View Article
  • Google Scholar


• [33] Y.-H. Lee, S.-M. Jung, M. Th. Rassias, On an n-dimensional mixed type additive and quadratic functional equation, Appl. Math. Comput., 228 (2014), 13–16
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] Y.-H. Lee, S.-M. Jung, M. Th. Rassias, Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation, J. Math. Inequal., 12 (2018), 43–61
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [35] G. Lu, J. Xin, Y. Jin, C. Park, Approximation of general Pexider functional inequalities in fuzzy Banach spaces, J. Nonlinear Sci. Appl., 12 (2019), 206–216
  • View Article
  • MathSciNet
  • Google Scholar


• [36] C. Park, M. Th. Rassias, Additive functional equations and partial multipliers in C-algebras, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 113 (2019), 2261–2275
  • View Article
  • MathSciNet
  • Google Scholar


• [37] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300
  • View Article
  • MathSciNet
  • Google Scholar


• [38] P. K. Sahoo, P. Kannappan, Introduction to functional equations, CRC Press, Boca Raton, FL (2011)
  • View Article
  • Google Scholar


• [39] D. K. Singh, S. Grover, On the stability of a sum form functional equation related to entropies of type (, ), J. Nonlinear Sci. Appl., 14 (2021), 168–180
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [40] S. M. Ulam, Problems in modern mathematics, Science Editions John Wiley & Sons, New York (1964)
  • View Article
  • Google Scholar