한국수학교육학회지시리즈B:순수및응용수학 (The Pure and Applied Mathematics)

  • 제26권2호
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  • Pages.85-97
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  • 2019
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  • 1226-0657(pISSN)
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  • 2287-6081(eISSN)
한국수학교육학회 (Korean Society of Mathematical Education)
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ON THE STABILITY OF RECIPROCAL-NEGATIVE FERMAT'S EQUATION IN QUASI-β-NORMED SPACES

  • Kang, Dongseung (Mathematics Education, Dankook University) ;
  • Kim, Hoewoon B. (Department of Mathematics, Oregon State University)
  • 투고 : 2018.07.25
  • 심사 : 2019.04.03
  • 발행 : 2019.05.31
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초록

In this paper we introduce the reciprocal-negative Fermat's equation induced by the famous equation in the Fermat's Last Theorem, establish the general solution in the simplest cases and the differential solution to the equation, and investigate, then, the generalized Hyers-Ulam stability in a $quasi-{\beta}-normed$ space with both the direct estimation method and the fixed point approach.

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참고문헌

  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. Y. Benyamini & J. Lindenstrauss: Geometric Nonlinear Functional Analysis. vol. 1 Colloq. Publ., vol. 48, Amer. Math. Soc., Providence, (2000).
  3. P.W. Cholewa: Remarks on the stability of functional equations. Aequationes. Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  4. 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
  5. J.B. 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
  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. R. Ger: Abstract Pythagorean theorem and corresponding functional equations. Tatra Mt. Math. Publ. 55 (2013), 67-75.
  9. 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
  10. K. Jun & H. Kim: Solution of Ulam stability problem for approximately biquadratic mappings and functional inequalities. J. Inequal. Appl. (2008), 109-124.
  11. S.-M. Jung: A fixed point approach to the stability of the equation f(x + y) = ${\frac{f(x)f(y)}{f(x)+f(y)}}$. Austral. J. Math. Anal. Appl. 6 (2009), no. 1, 1-6
  12. 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
  13. Y.-S. Lee and S.-Y. Chung: Stability of quartic functional equations in the spaces of generalized functions. Adv. Diff. Equa. (2009).
  14. D. Mihett & V. Radu: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343 (2008), no. 1, 567-572 https://doi.org/10.1016/j.jmaa.2008.01.100
  15. P. Narasimman, K. Ravi & Sandra Pinelas: Stability of Pythagorean mean functional equation. Global J. Math. 4 (2015), 398-411.
  16. 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
  17. K. Ravi & B.V. Senthil Kumar: Ulam-Gavruta-Rassias stability of Rassias reciprocal functional equation. Global J. Appl. Math. Math. Sci. 3 (2010), 57-79.
  18. S. Rolewicz: Metric Linear Spaces. Reidel/PWN-Polish Sci. Publ., Dordrecht, 1984.
  19. S.M. Ulam: Problems in Morden Mathematics. Wiley, New York, 1960.