Korean J. Math. Vol. 27 No. 1 (2019) pp.81-92
DOI: https://doi.org/10.11568/kjm.2019.27.1.81

The stability of generalized reciprocal-negative Fermat's equations in quasi-$\beta$-normed spaces

Main Article Content

DognSeung Kang
Hoewoon Kim

Abstract

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



Article Details

References

[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950) 64–66. Google Scholar

[2] J.-H. Bae and W.-G. Park, On the generalized Hyers-Ulam-Rassias stability in Banach modules over a C∗−algebra, J. Math. Anal. Appl. 294 (2004), 196–205. Google Scholar

[3] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, Colloq. Publ., vol. 48, Amer. Math. Soc., Providence, (2000). Google Scholar

[4] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes. Math. 27 (1984), 76–86. Google Scholar

[5] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. Google Scholar

[6] Z. Gajda, On the stability of additive mappings, Internat. J. Math. Math. Sci., 14 (1991), 431–434. Google Scholar

[7] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. Google Scholar

[8] D. H. Hyers, On the stability of the linear equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. Google Scholar

[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. Google Scholar

[10] R. Ger, Tatra Mt. Math. Publ. 55 (2013), 67–75. Google Scholar

[11] S.M. Jung, A Fixed Point Approach to the Stability of the Equation f(x+y)=f(x)f(y))/((f(x)+f(y)), The Australian Journal of Math. Anal. and Appl. Vol. 6 (1) (2009), 1–6 Google Scholar

[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. Google Scholar

[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. Google Scholar

[14] K. Jun and H. Kim, Solution of Ulam stability problem for approximately bi-quadratic mappings and functional inequalities, J. Inequal. Appl. 10 (4) (2007), 895–908 Google Scholar

[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 Google Scholar

[16] R. Kadisona and G. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985), 249–266. Google Scholar

[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. Google Scholar

[18] D. Kang and H.B. Kim, On the stability of reciprocal-negative Fermat’s Equations in quasi-β-normed spaces, preprint Google Scholar

[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. Google Scholar

[20] P. Narasimman, K. Ravi and Sandra Pinelas, Stability of Pythagorean Mean Functional Equation, Global Journal of Mathematics 4 (1) (2015), 398–411 Google Scholar

[21] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. Google Scholar

[22] Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. Google Scholar

[23] Th. M. Rassias, P. Sˇemrl On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325–338. Google Scholar

[24] Th. M. Rassias, K. Shibata, Variational problem of some quadratic functions in complex analysis, J. Math. Anal. Appl. 228 (1998), 234–253. Google Scholar

[25] J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnik Matematicki Series III, 34 (2) (1999) 243–252. Google Scholar

[26] J. M. Rassias, On the stability of the Euler-Lagrange functional equation, Chinese J. Math. 20 (1992) 185–190. Google Scholar

[27] J. M. Rassias, H.-M. Kim Generalized Hyers.Ulam stability for general additive functional equations in quasi-β-normed spaces, J. Math. Anal. Appl. 356 (2009), 302–309. Google Scholar

[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. Google Scholar

[29] S. Rolewicz, Metric Linear Spaces, Reidel/PWN-Polish Sci. Publ., Dordrecht, (1984). Google Scholar

[30] I.A. Rus, Principles and Appications of Fixed Point Theory, Ed. Dacia, ClujNapoca, 1979 (in Romanian). Google Scholar

[31] F. Skof, Propriet`a locali e approssimazione di operatori,Rend. Semin. Mat. Fis. Milano 53 (1983) 113–129. Google Scholar

[32] S. M. Ulam, Problems in Morden Mathematics, Wiley, New York (1960). Google Scholar