The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

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

DOI QR Code

QUADRATIC (ρ1, ρ2)-FUNCTIONAL INEQUALITY IN FUZZY BANACH SPACES

  • Park, Junha (Mathematics Branch, Seoul Science High School) ;
  • Jo, Younghun (Mathematics Branch, Seoul Science High School) ;
  • Kim, Jaemin (Mathematics Branch, Seoul Science High School) ;
  • Kim, Taekseung (Mathematics Branch, Seoul Science High School)
  • Received : 2017.08.08
  • Accepted : 2017.08.21
  • Published : 2017.08.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce and solve the following quadratic (${\rho}_1$, ${\rho}_2$)-functional inequality (0.1) $$N\left(2f({\frac{x+y}{2}})+2f({\frac{x-y}{2}})-f(x)-f(y),t\right){\leq}min\left(N({\rho}_1(f(x+y)+f(x-y)-2f(x)-2f(y)),t),\;N({\rho}_2(4f(\frac{x+y}{2})+f(x-y)-2f(x)-2f(y)),t)\right)$$ in fuzzy normed spaces, where ${\rho}_1$ and ${\rho}_2$ are fixed nonzero real numbers with ${{\frac{1}{{4\left|{\rho}_1\right|}}+{{\frac{1}{{4\left|{\rho}_2\right|}}$ < 1, and f(0) = 0. Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic (${\rho}_1$, ${\rho}_2$)-functional inequality (0.1) in fuzzy Banach spaces.

Keywords

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. T. Bag & S.K. Samanta: Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 11 (2003), 687-705.
  3. T. Bag & S.K. Samanta: Fuzzy bounded linear operators. Fuzzy Sets and Systems 151 (2005), 513-547. https://doi.org/10.1016/j.fss.2004.05.004
  4. 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).
  5. L. Cadariu & V. Radu: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346 (2004), 43-52.
  6. 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).
  7. I. Chang & Y. Lee: Additive and quadratic type functional equation and its fuzzy stability. Results Math. 63 (2013), 717-730. https://doi.org/10.1007/s00025-012-0229-y
  8. S.C. Cheng & J.M. Mordeson: Fuzzy linear operators and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 86 (1994), 429-436.
  9. 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
  10. W. Fechner: Stability of a functional inequalities associated with the Jordan-von Neu-mann functional equation. Aequationes Math. 71 (2006), 149-161. https://doi.org/10.1007/s00010-005-2775-9
  11. C. Felbin: Finite dimensional fuzzy normed linear spaces. Fuzzy Sets and Systems 48 (1992), 239-248. https://doi.org/10.1016/0165-0114(92)90338-5
  12. 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
  13. A. Gilanyi: Eine zur Parallelogrammgleichung Äaquivalente Ungleichung. Aequationes Math. 62 (2001), 303-309. https://doi.org/10.1007/PL00000156
  14. A. Gilanyi: On a problem by K. Nikodem. Math. Inequal. Appl. 5 (2002), 707-710.
  15. 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
  16. D.H. Hyers, G. Isac & Th.M. Rassias: Stability of Functional Equations in Several Variables. Birkhauser, Basel, 1998.
  17. 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
  18. K. Jun & H. Kim: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 274 (2002), 867-878. https://doi.org/10.1016/S0022-247X(02)00415-8
  19. S. Jung: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press lnc., Palm Harbor, Florida, 2001.
  20. A.K. Katsaras: Fuzzy topological vector spaces II. Fuzzy Sets and Systems 12 (1984), 143-154. https://doi.org/10.1016/0165-0114(84)90034-4
  21. H. Kim, M. Eshaghi Gordji, A. Javadian & I. Chang: Homomorphisms and derivations on unital C*-algebras related to Cauchy-Jensen functional inequality. J. Math. Inequal. 6 (2012), 557-565.
  22. H. Kim, J. Lee & E. Son: Approximate functional inequalities by additive mappings. J. Math. Inequal. 6 (2012), 461-471.
  23. I. Kramosil & J. Michalek: Fuzzy metric and statistical metric spaces. Kybernetica 11 (1975), 326-334.
  24. S.V. Krishna & K.K.M. Sarma: Separation of fuzzy normed linear spaces. Fuzzy Sets and Systems 63 (1994), 207-217. https://doi.org/10.1016/0165-0114(94)90351-4
  25. J. Lee, C. Park & D. Shin: An AQCQ-functional equation in matrix normed spaces. Results Math. 27 (2013), 305-318.
  26. S. Lee, S. Im & I. Hwang: Quartic functional equations. J. Math. Anal. Appl. 307 (2005), 387-394. https://doi.org/10.1016/j.jmaa.2004.12.062
  27. 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
  28. M. Mirzavaziri & M.S. Moslehian: A fixed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc. 37 (2006), 361-376. https://doi.org/10.1007/s00574-006-0016-z
  29. A.K. Mirmostafaee, M. Mirzavaziri & M.S. Moslehian: Fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems 159 (2008), 730-738. https://doi.org/10.1016/j.fss.2007.07.011
  30. A.K. Mirmostafaee & M.S. Moslehian: Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets and Systems 159 (2008), 720-729. https://doi.org/10.1016/j.fss.2007.09.016
  31. A.K. Mirmostafaee & M.S. Moslehian: Fuzzy approximately cubic mappings. Inform. Sci. 178 (2008), 3791-3798. https://doi.org/10.1016/j.ins.2008.05.032
  32. 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).
  33. 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).
  34. C. Park: Additive $\rho$-functional inequalities and equations. J. Math. Inequal. 9 (2015), 17-26.
  35. C. Park: Additive $\rho$-functional inequalities in non-Archimedean normed spaces. J. Math. Inequal. 9 (2015), 397-407.
  36. C. Park, Y. Cho & M. Han: Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations. J. Inequal. Appl. 2007, Art. ID 41820 (2007).
  37. C. Park, K. Ghasemi, S.G. Ghaleh & S. Jang: Approximate n-Jordan *-homo-morphisms in C*-algebras. J. Comput. Anal. Appl. 15 (2013), 365-368.
  38. 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.
  39. C. Park & Th.M. Rassias: Fixed points and generalized Hyers-Ulam stability of quadratic functional equations. J. Math. Inequal. 1 (2007), 515-528.
  40. V. Radu: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4 (2003), 91-96.
  41. 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
  42. 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
  43. L. Reich, J. Smital & M. Stefankova: Singular solutions of the generalized Dhombres functional equation. Results Math. 65 (2014), 251-261. https://doi.org/10.1007/s00025-013-0345-3
  44. 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
  45. S. Shagholi, M. Bavand Savadkouhi & M. Eshaghi Gordji: Nearly ternary cubic homo-morphism in ternary Frechet algebras. J. Comput. Anal. Appl. 13 (2011), 1106-1114.
  46. 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.
  47. D. Shin, C. Park & Sh. Farhadabadi: On the superstability of ternary Jordan C*-homomorphisms. J. Comput. Anal. Appl. 16 (2014), 964-973.
  48. 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.
  49. S.M. Ulam: A Collection of the Mathematical Problems. Interscience Publ. New York, 1960.
  50. J.Z. Xiao & X.H. Zhu: Fuzzy normed spaces of operators and its completeness. Fuzzy Sets and Systems 133 (2003), 389-399. https://doi.org/10.1016/S0165-0114(02)00274-9