Ulam stability of the reciprocal functional equation in non-Archimedean fields
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Abstract
In this paper, we introduce a new generalized reciprocal functional equation and study its Hyers-Ulam-Rassias stability. We also provide the counter examples for some cases, Ulam-Gavruta-Rassias stability and Hyers-Ulam-Rassias stability in non-Archimedean fields.
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How to Cite
Bodaghi, A., Narasimman, P., Rassia, J., & Ravi, K.
(2016).
Ulam stability of the reciprocal functional equation in non-Archimedean fields.
Acta Mathematica Universitatis Comenianae, 85(1), 113-124.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/213/292
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References
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[2] A. Bodaghi, Cubic derivations on Banach algebras, Acta Math. Vietnamica 38 (2013), 517{528.
[3] A. Bodaghi and I. A. Alias, Approximate ternary quadratic derivations on ternary Banach algebras and C*-ternary rings, Adv. Difference Equ. 2012, Art. No. 11 (2012), doi:10.1186/1687-1847-2012-11.
[4] A. Bodaghi and S. O. Kim, Approximation on the quadratic reciprocal functional equation, Journal of Function Spaces. Volume 2014, Article ID 532463,5 pages.
[5] P. Gavruta, A generalization of the Hyers-Ulam-Rassias Stability of approximately additive mapping, J. Math. Anal. Appl., 184 (1994),431-436.
[6] P. M. Gruber, Stability of isometries, Trans. Amer. Math. Soc., 245 (1978), 263-277.
[7] K. Hensel, Uber eine neue Begrndung der Theorie der algebraischen Zahlen, Jahresber, Deutsche Mathematiker-Vereinigung, 6 (1897), 83-88.
[8] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.
[9] G. Isac and Th. M. Rassias, Stability of $\psi$-additive mappings: Applications to nonlinear analysis, Int. J. Math. Math. Sci., 19 (2)(1996), 219-228.
[10] 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 Mathematical Analysis and Applications, 6 (8)(2009), 1-6.
[11] P. Kannappan, Quadratic functional equation and inner product spaces, Results Math., 27 (1995), 368{372.
[12] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126-130.
[13] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), 445-446.
[14] J.M. Rassias, Solution of a problem of Ulam, J. Approx. Theory, 57 (1989), 268-273.
[15] K. Ravi, M. Arunkumar, J. M. Rassias, On the Ulam stability for a orthogonally general Euler-Lagrange type functional equation, Int. J. Math. Stat., 3, No. A08 (2008),
36-46.
[16] K. Ravi and P. Narasimman, Stability of Generalized Quadratic Functional Equation in Non-Archmedean L-Fuzzy Normed Space, Int. Journal of Math. Analysis, 6, (2012)
no. 24, 1193-1203.
[17] K. Ravi and P. Narasimman, Generalized Hyers-Ulam Stability of Radical Reciprocal-Quadratic Functional Equation, Indian Journal of Science and Technology, Vol. 6,
Issue: 2S3, March 2013, pp 121-125.
[18] 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) (2010),
57-79.
[19] K. Ravi, J. M. Rassias and B. V. Senthil Kumar, Ulam stability of Reciprocal Difference and Adjoint Funtional Equations, The Australian Journal of Mathematical
Analysis and Applications, 8 (1), Art.13 (2011), 1-18.
[20] K. Ravi, E. Thandapani and B. V. Senthil Kumar, Stability of reciprocal type functional equations, Pan American Math. Journal, 21 (2012) No. 1, 59-70.
[21] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
[22] S. M. Ulam, A collection of mathematical problems, Interscience Publishers, Inc. New York, 1960.
[23] S. Y. Yang, A. Bodaghi and K. A. Mohd Atan, Approximate cubic *-derivations on Banach *-algebras, Abst. Appl. Anal. Volume 2012, Article ID 684179, 12 pages,
doi:10.1155/2012/684179.
[2] A. Bodaghi, Cubic derivations on Banach algebras, Acta Math. Vietnamica 38 (2013), 517{528.
[3] A. Bodaghi and I. A. Alias, Approximate ternary quadratic derivations on ternary Banach algebras and C*-ternary rings, Adv. Difference Equ. 2012, Art. No. 11 (2012), doi:10.1186/1687-1847-2012-11.
[4] A. Bodaghi and S. O. Kim, Approximation on the quadratic reciprocal functional equation, Journal of Function Spaces. Volume 2014, Article ID 532463,5 pages.
[5] P. Gavruta, A generalization of the Hyers-Ulam-Rassias Stability of approximately additive mapping, J. Math. Anal. Appl., 184 (1994),431-436.
[6] P. M. Gruber, Stability of isometries, Trans. Amer. Math. Soc., 245 (1978), 263-277.
[7] K. Hensel, Uber eine neue Begrndung der Theorie der algebraischen Zahlen, Jahresber, Deutsche Mathematiker-Vereinigung, 6 (1897), 83-88.
[8] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.
[9] G. Isac and Th. M. Rassias, Stability of $\psi$-additive mappings: Applications to nonlinear analysis, Int. J. Math. Math. Sci., 19 (2)(1996), 219-228.
[10] 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 Mathematical Analysis and Applications, 6 (8)(2009), 1-6.
[11] P. Kannappan, Quadratic functional equation and inner product spaces, Results Math., 27 (1995), 368{372.
[12] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126-130.
[13] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), 445-446.
[14] J.M. Rassias, Solution of a problem of Ulam, J. Approx. Theory, 57 (1989), 268-273.
[15] K. Ravi, M. Arunkumar, J. M. Rassias, On the Ulam stability for a orthogonally general Euler-Lagrange type functional equation, Int. J. Math. Stat., 3, No. A08 (2008),
36-46.
[16] K. Ravi and P. Narasimman, Stability of Generalized Quadratic Functional Equation in Non-Archmedean L-Fuzzy Normed Space, Int. Journal of Math. Analysis, 6, (2012)
no. 24, 1193-1203.
[17] K. Ravi and P. Narasimman, Generalized Hyers-Ulam Stability of Radical Reciprocal-Quadratic Functional Equation, Indian Journal of Science and Technology, Vol. 6,
Issue: 2S3, March 2013, pp 121-125.
[18] 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) (2010),
57-79.
[19] K. Ravi, J. M. Rassias and B. V. Senthil Kumar, Ulam stability of Reciprocal Difference and Adjoint Funtional Equations, The Australian Journal of Mathematical
Analysis and Applications, 8 (1), Art.13 (2011), 1-18.
[20] K. Ravi, E. Thandapani and B. V. Senthil Kumar, Stability of reciprocal type functional equations, Pan American Math. Journal, 21 (2012) No. 1, 59-70.
[21] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
[22] S. M. Ulam, A collection of mathematical problems, Interscience Publishers, Inc. New York, 1960.
[23] S. Y. Yang, A. Bodaghi and K. A. Mohd Atan, Approximate cubic *-derivations on Banach *-algebras, Abst. Appl. Anal. Volume 2012, Article ID 684179, 12 pages,
doi:10.1155/2012/684179.