Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

THE STABILITY OF GENERALIZED RECIPROCAL-NEGATIVE FERMAT'S EQUATIONS IN QUASI-β-NORMED SPACES

  • Received : 2018.09.12
  • Accepted : 2018.12.31
  • Published : 2019.03.30
Download PDF
⟨ Previous Next ⟩

Abstract

We introduce a reciprocal-negative Fermat's equation generalized with constants coefficients and investigate its stability in a quasi-${\beta}$-normed space.

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. J.-H. Bae and W.-G. Park, On the generalized Hyers-Ulam-Rassias stability in Banach modules over a $C^{\ast}$-algebra, J. Math. Anal. Appl. 294 (2004), 196-205. https://doi.org/10.1016/j.jmaa.2004.02.009
  3. Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, Colloq. Publ., vol. 48, Amer. Math. Soc., Providence, (2000).
  4. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes. Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  5. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64. https://doi.org/10.1007/BF02941618
  6. Z. Gajda, On the stability of additive mappings, Internat. J. Math. Math. Sci., 14 (1991), 431-434. https://doi.org/10.1155/S016117129100056X
  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 equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  9. J. K. Chung and P. K. Sahoo, On the general solution of a quartic functional equation, Bulletin of the Korean Mathematical Society, 40 (4) (2003), 565-576. https://doi.org/10.4134/BKMS.2003.40.4.565
  10. R. Ger, Tatra Mt. Math. Publ. 55 (2013), 67-75.
  11. S.M. Jung, A Fixed Point Approach to the Stability of the Equation f(x + y)=${\frac}{f(x)f(y)}{f(x)+f(y)}$, The Australian Journal of Math. Anal. and Appl. Vol. 6 (1) (2009), 1-6
  12. Y.-S. Jung and I.-S. Chang, The stability of a cubic type functional equation with the fixed point alternative, J. Math. Anal. Appl. (2005), 264-284.
  13. K.-W. Jun and H.-M. Kim, On the stability of Euler-Lagrange type cubic functional equations in quasi-Banach spaces, J. Math. Anal. Appl. 332 (2007), 1335-1350. https://doi.org/10.1016/j.jmaa.2006.11.024
  14. K. Jun and H. Kim, Solution of Ulam stability problem for approximately biquadratic mappings and functional inequalities, J. Inequal. Appl. 10 (4) (2007), 895-908
  15. Y.-S. Lee and S.-Y. Chung, Stability of quartic functional equations in the spaces of generalized functions, Adv. Diff. Equa. (2009), 2009: 838347 https://doi.org/10.1155/2009/838347
  16. R. Kadisona and G. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985), 249-266. https://doi.org/10.7146/math.scand.a-12116
  17. H.-M. Kim,On the stability problem for a mixed type of quartic and quadratic functional equation, J. Math. Anal. Appl. 324 (2006), 358-372. https://doi.org/10.1016/j.jmaa.2005.11.053
  18. D. Kang and H.B. Kim, On the stability of reciprocal-negative Fermat's Equations in quasi-${\beta}$-normed spaces, preprint
  19. B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 126, 74 (1968), 305-309. https://doi.org/10.1090/S0002-9904-1968-11933-0
  20. P. Narasimman, K. Ravi and Sandra Pinelas, Stability of Pythagorean Mean Functional Equation, Global Journal of Mathematics 4 (1) (2015), 398-411
  21. 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
  22. Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264-284. https://doi.org/10.1006/jmaa.2000.7046
  23. Th. M. Rassias, P. Semrl On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325-338. https://doi.org/10.1006/jmaa.1993.1070
  24. Th. M. Rassias, K. Shibata, Variational problem of some quadratic functions in complex analysis, J. Math. Anal. Appl. 228 (1998), 234-253. https://doi.org/10.1006/jmaa.1998.6129
  25. J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnik Matematicki Series III, 34 (2) (1999) 243-252.
  26. J. M. Rassias, On the stability of the Euler-Lagrange functional equation, Chinese J. Math. 20 (1992) 185-190.
  27. J. M. Rassias, H.-M. Kim Generalized Hyers.Ulam stability for general additive functional equations in quasi-${\beta}$-normed spaces, J. Math. Anal. Appl. 356 (2009), 302-309. https://doi.org/10.1016/j.jmaa.2009.03.005
  28. K. Ravi and B.V. Senthil Kumar Ulam-Gavruta-Rassias stability of Rassias Reciprocal functional equation, Global Journal of App. Math. and Math. Sci. 3(1-2), Jan-Dec 2010, 57-79.
  29. S. Rolewicz, Metric Linear Spaces, Reidel/PWN-Polish Sci. Publ., Dordrecht, (1984).
  30. I.A. Rus, Principles and Appications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, 1979 (in Romanian).
  31. F. Skof, Proprieta locali e approssimazione di operatori, Rend. Semin. Mat. Fis. Milano 53 (1983) 113-129. https://doi.org/10.1007/BF02924890
  32. S. M. Ulam, Problems in Morden Mathematics, Wiley, New York (1960).