A Functional Equation Originating from Elliptic Curves
Won-Gil Park and Jae-Hyeong Bae
Source: Abstr. Appl. Anal. Volume 2008 (2008), 10 pages.
Abstract
We obtain the general solution and the stability of the functional equation $f(x+y+z,u+v+w)+f(x+y-z,u+v+w)+2f(x,u-w)+2f(y,v-w)=f(x+y,u+w)+f(x+y,v+w)+f(x+z,u+w)+f(x-z,u+v-w)+f(y+z,v+w)+f(y-z,u+v-w)$. The function $f(x,y)={x}^{3}+ax+b-{y}^{2}$ having level curves as elliptic curves is a solution of the above functional equation.
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aaa/1220969161
Digital Object Identifier: doi:10.1155/2008/135237
Mathematical Reviews number (MathSciNet):
MR2407274
Zentralblatt MATH identifier:
1146.39037
References
L. C. Washington, Elliptic Curves: Number Theory and Cryptography, Discrete Mathematics and Its Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2003.
Mathematical Reviews (MathSciNet):
MR1989729
Zentralblatt MATH:
1034.11037
K.-W. Jun and H.-M. Kim, ``Ulam stability problem for a mixed type of cubic and additive functional equation,'' Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 13, no. 2, pp. 271--285, 2006.
Mathematical Reviews (MathSciNet):
MR2259906
Zentralblatt MATH:
1132.39022
Project Euclid: euclid.bbms/1148059462
T. Aoki, ``On the stability of the linear transformation in Banach spaces,'' Journal of the Mathematical Society of Japan, vol. 2, pp. 64--66, 1950.
Mathematical Reviews (MathSciNet):
MR0040580
Zentralblatt MATH:
0040.35501
J.-H. Bae and K.-W. Jun, ``On the generalized Hyers-Ulam-Rassias stability of an $n$-dimensional quadratic functional equation,'' Journal of Mathematical Analysis and Applications, vol. 258, no. 1, pp. 183--193, 2001.
Mathematical Reviews (MathSciNet):
MR1828099
Zentralblatt MATH:
0983.39013
Digital Object Identifier: doi:10.1006/jmaa.2000.7372
Y.-S. Cho and H.-M. Kim, ``Stability of functional inequalities with Cauchy-Jensen additive mappings,'' Abstract and Applied Analysis, vol. 2007, Article ID 89180, 13 pages, 2007.
Mathematical Reviews (MathSciNet):
MR2320798
Zentralblatt MATH:
1149.39020
Digital Object Identifier: doi:10.1155/2007/89180
S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002.
Mathematical Reviews (MathSciNet):
MR1904790
Zentralblatt MATH:
1011.39019
S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001.
Mathematical Reviews (MathSciNet):
MR1841182
Zentralblatt MATH:
0980.39024
M. S. Moslehian, ``Ternary derivations, stability and physical aspects,'' Acta Applicandae Mathematicae, vol. 100, no. 2, pp. 187--199, 2008.
Mathematical Reviews (MathSciNet):
MR2373684
Digital Object Identifier: doi:10.1007/s10440-007-9179-x
C. Park and J. Cui, ``Generalized stability of $C^\ast$-ternary quadratic mappings,'' Abstract and Applied Analysis, vol. 2007, Article ID 23282, 6 pages, 2007.
Mathematical Reviews (MathSciNet):
MR2302189
Zentralblatt MATH:
1158.39020
W.-G. Park and J.-H. Bae, ``On a bi-quadratic functional equation and its stability,'' Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 4, pp. 643--654, 2005.
Mathematical Reviews (MathSciNet):
MR2149907
Zentralblatt MATH:
1076.39027
W.-G. Park and J.-H. Bae, ``A multidimensional functional equation having quadratic forms as solutions,'' Journal of Inequalities and Applications, vol. 2007, Article ID 24716, 8 pages, 2007.
Mathematical Reviews (MathSciNet):
MR2366340
Zentralblatt MATH:
1133.39032
Th. M. Rassias, ``On the stability of functional equations and a problem of Ulam,'' Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23--130, 2000.
Mathematical Reviews (MathSciNet):
MR1778016
Zentralblatt MATH:
0981.39014
Digital Object Identifier: doi:10.1023/A:1006499223572
J. Aczél and J. Dhombres, Functional Equations in Several Variables, vol. 31 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1989.
Mathematical Reviews (MathSciNet):
MR1004465
Zentralblatt MATH:
0731.39010
Th. M. Rassias, ``On the stability of the linear mapping in Banach spaces,'' Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297--300, 1978.
Mathematical Reviews (MathSciNet):
MR507327
Zentralblatt MATH:
0398.47040
Digital Object Identifier: doi:10.2307/2042795
Abstract and Applied Analysis
