The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

ADDITIVE ρ-FUNCTIONAL EQUATIONS IN β-HOMOGENEOUS F-SPACES

  • Shim, EunHwa (Department of Mathematics, Hanyang University)
  • Received : 2017.11.07
  • Accepted : 2017.11.20
  • Published : 2017.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we solve the additive ${\rho}-functional$ equations (0.1) $f(x+y)+f(x-y)-2f(x)={\rho}(2f(\frac{x+y}{2})+f(x-y)-2f(x))$, and (0.2) $2f(\frac{x+y}{2})+f(x-y)-2f(x)={\rho}(f(x+y)+f(x-y)-2f(x))$, where ${\rho}$ is a fixed (complex) number with ${\rho}{\neq}1$, Using the direct method, we prove the Hyers-Ulam stability of the additive ${\rho}-functional$ equations (0.1) and (0.2) in ${\beta}-homogeneous$ (complex) F-spaces.

Keywords

References

  1. M. Adam: On the stability of some quadratic functional equation. J. Nonlinear Sci. Appl. 4 (2011), 50-59. https://doi.org/10.22436/jnsa.004.01.05
  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. L. Cadariu, L. Gavruta & P. Gavruta: On the stability of an affine functional equation. J. Nonlinear Sci. Appl. 6 (2013), 60-67. https://doi.org/10.22436/jnsa.006.02.01
  4. A. Chahbi & N. Bounader: On the generalized stability of d'Alembert functional equation. J. Nonlinear Sci. Appl. 6 (2013), 198-204. https://doi.org/10.22436/jnsa.006.03.05
  5. P.W. Cholewa: Remarks on the stability of functional equations. Aequationes Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  6. G.Z. Eskandani & P. Gavruta: Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2-Banach spaces. J. Nonlinear Sci. Appl. 5 (2012), 459-465. https://doi.org/10.22436/jnsa.005.06.06
  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. Natl. Acad. Sci. U.S.A. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  9. C. Park: Orthogonal stability of a cubic-quartic functional equation. J. Nonlinear Sci. Appl. 5 (2012), 28-36. https://doi.org/10.22436/jnsa.005.01.04
  10. C. Park: Additive $\rho$-functional inequalities and equations. J. Math. Inequal. 9 (2015), 17-26.
  11. C. Park: Additive $\rho$-functional inequalities in non-Archimedean normed spaces. J. Math. Inequal. 9 (2015), 397-407.
  12. C. Park, K. Ghasemi, S.G. Ghaleh & S. Jang: Approximate n-Jordan *- homomorphisms in C*-algebras. J. Comput. Anal. Appl. 15 (2013), 365-368.
  13. C. Park, A. Najati & S. Jang: Fixed points and fuzzy stability of an additive-quadratic functional equation. J. Comput. Anal. Appl. 15 (2013), 452-462.
  14. Th.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
  15. K. Ravi, E. Thandapani & B.V. Senthil Kumar: Solution and stability of a reciprocal type functional equation in several variables. J. Nonlinear Sci. Appl. 7 (2014), 18-27.
  16. S. Rolewicz: Metric Linear Spaces. PWN-Polish Scientific Publishers, Warsaw, 1972.
  17. S. Schin, D. Ki, J. Chang & M. Kim: Random stability of quadratic functional equations: a fixed point approach. J. Nonlinear Sci. Appl. 4 (2011), 37-49. https://doi.org/10.22436/jnsa.004.01.04
  18. S. Shagholi, M. Bavand Savadkouhi & M. Eshaghi Gordji: Nearly ternary cubic homomorphism in ternary Frechet algebras. J. Comput. Anal. Appl. 13 (2011), 1106-1114.
  19. S. Shagholi, M. Eshaghi Gordji & M. Bavand Savadkouhi: Stability of ternary quadratic derivation on ternary Banach algebras. J. Comput. Anal. Appl. 13 (2011), 1097-1105.
  20. D. Shin, C. Park & Sh. Farhadabadi: On the superstability of ternary Jordan C*-homomorphisms. J. Comput. Anal. Appl. 16 (2014), 964-973.
  21. D. Shin, C. Park & Sh. Farhadabadi: Stability and superstability of J*-homomorphisms and J*-derivations for a generalized Cauchy-Jensen equation. J. Comput. Anal. Appl. 17 (2014), 125-134.
  22. F. Skof: Propriet locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  23. S.M. Ulam: A Collection of the Mathematical Problems. Interscience Publ. New York, 1960.
  24. C. Zaharia: On the probabilistic stability of the monomial functional equation. J. Nonlinear Sci. Appl. 6 (2013), 51-59. https://doi.org/10.22436/jnsa.006.01.08
  25. S. Zolfaghari: Approximation of mixed type functional equations in p-Banach spaces. J. Nonlinear Sci. Appl. 3 (2010), 110-122. https://doi.org/10.22436/jnsa.003.02.04