Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN BANACH SPACES: A FIXED POINT APPROACH

  • Received : 2015.03.15
  • Accepted : 2015.06.03
  • Published : 2015.06.30
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Abstract

In this paper, we solve the following quadratic $\rho$-functional inequalities ${\parallel}f(\frac{x+y+z}{2})+f(\frac{x-y-z}{2})+f(\frac{y-x-z}{2})+f(\frac{z-x-y}{2})-f(x)-x(y)-f(z){\parallel}\;(0.1)\\{\leq}{\parallel}{\rho}(f(x+y+z+)+f(x-y-z)+f(y-x-z)+f(z-x-y)-4f(x)-4f(y)-f(z)){\parallel}$ where $\rho$ is a fixed complex number with ${\left|\rho\right|}<\frac{1}{8}$, and ${\parallel}f(x+y+z)+f(x-y-z)+f(y-x-z)+f(z-x-y)-4f(x)-4f(y)-4f(z){\parallel}\;(0.2)\\{\leq}{\parallel}{\rho}(f(\frac{x+y+z}{2})+f(\frac{x-y-z}{2})+f(\frac{y-x-z}{2})+f(\frac{z-x-y}{2})-f(x)-f(y)-f(z)){\parallel}$ where $\rho$ is a fixed complex number with ${\left|\rho\right|}$ < 4. Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic $\rho$-functional inequalities (0.1) and (0.2) in complex Banach spaces.

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References

  1. 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
  2. L. Cadariu, V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003).
  3. L. Cadariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43-52.
  4. L. Cadariu, V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008).
  5. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  6. J. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309. https://doi.org/10.1090/S0002-9904-1968-11933-0
  7. W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149-161. https://doi.org/10.1007/s00010-005-2775-9
  8. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-43. https://doi.org/10.1006/jmaa.1994.1211
  9. A. Gilanyi, Eine zur Parallelogrammgleichung aquivalente Ungleichung, Aequationes Math. 62 (2001), 303-309. https://doi.org/10.1007/PL00000156
  10. A. Gilanyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707-710.
  11. 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
  12. G. Isac, Th. M. Rassias, Stability of $\psi$-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219-228. https://doi.org/10.1155/S0161171296000324
  13. D. Mihet, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567-572. https://doi.org/10.1016/j.jmaa.2008.01.100
  14. C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007).
  15. C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008).
  16. C. Park, Y. Cho, M. Han, Functional inequalities associated with Jordan-von Neumann-type additive functional equations, J. Inequal. Appl. 2007 (2007), Article ID 41820, 13 pages.
  17. 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
  18. V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91-96.
  19. J. Ratz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191-200. https://doi.org/10.1007/s00010-003-2684-8
  20. F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  21. S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ., New York, 1960.