×

On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations. (English) Zbl 1167.39014

Summary: We achieve the general solution and the generalized Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities for quadratic functional equations
\[ \begin{split} f(ax+by)+f(ax - by)\\ =(b(a+b)/2)f(x+y)+(b(a+b)/2)f(x - y)+(2a^{2} - ab - b^{2})f(x)+(b^{2} - ab)f(y)\end{split} \]
where \(a, b\) are nonzero fixed integers with \(b\neq \pm a, - 3a\), and \[ f(ax+by)+f(ax - by)=2a^{2}f(x)+2b^{2}f(y) \]
for fixed integers \(a, b\) with \(a,b\neq 0\) and \(a\pm b\neq 0\).

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B22 Functional equations for real functions
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] S. M. Ulam, Problems in Modern Mathematics, chapter 6, John Wiley & Sons, New York, NY, USA, 1940. · Zbl 0137.24201
[2] D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222-224, 1941. · Zbl 0061.26403
[3] 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. · Zbl 0040.35501
[4] 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. · Zbl 0398.47040
[5] J. M. Rassias, “On approximation of approximately quadratic mappings by quadratic mappings,” Annales Mathematicae Silesianae, no. 15, pp. 67-78, 2001. · Zbl 1087.39518
[6] J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Journal of Functional Analysis, vol. 46, no. 1, pp. 126-130, 1982. · Zbl 0482.47033
[7] J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Bulletin des Sciences Mathématiques, vol. 108, no. 4, pp. 445-446, 1984. · Zbl 0599.47106
[8] J. M. Rassias, “On a new approximation of approximately linear mappings by linear mappings,” Discussiones Mathematicae, vol. 7, pp. 193-196, 1985. · Zbl 0592.46004
[9] J. M. Rassias, “Solution of a problem of Ulam,” Journal of Approximation Theory, vol. 57, no. 3, pp. 268-273, 1989. · Zbl 0672.41027
[10] J. M. Rassias, “Solution of a stability problem of Ulam,” Discussiones Mathematicae, vol. 12, pp. 95-103, 1992. · Zbl 0779.47005
[11] P. G\uavru\cta, “An answer to a question of John M. Rassias concerning the stability of Cauchy equation,” in Advances in Equations and Inequalities, Hadronic Mathematics, pp. 67-71, Hadronic Press, Palm Harbor, Fla, USA, 1999.
[12] Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431-434, 1991. · Zbl 0739.39013
[13] P. G\uavru\cta, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431-436, 1994. · Zbl 0818.46043
[14] K. Ravi, M. Arunkumar, and J. M. Rassias, “Ulam stability for the orthogonally general Euler-Lagrange type functional equation,” International Journal of Mathematics and Statistics, vol. 3, no. A08, pp. 36-46, 2008. · Zbl 1144.39029
[15] K.-W. Jun, H.-M. Kim, and J. M. Rassias, “Extended Hyers-Ulam stability for Cauchy-Jensen mappings,” Journal of Difference Equations and Applications, vol. 13, no. 12, pp. 1139-1153, 2007. · Zbl 1135.39013
[16] H.-M. Kim, K.-W. Jun, and J. M. Rassias, “Extended stability problem for alternative Cauchy-Jensen mappings,” Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 4, article 120, pp. 1-17, 2007. · Zbl 1141.39028
[17] C.-G. Park and J. M. Rassias, “Hyers-Ulam stability of an Euler-Lagrange type additive mapping,” International Journal of Applied Mathematics & Statistics, vol. 7, no. Fe07, pp. 112-125, 2007.
[18] J. M. Rassias, “Solution of a quadratic stability Hyers-Ulam type problem,” Ricerche di Matematica, vol. 50, no. 1, pp. 9-17, 2001. · Zbl 1221.39039
[19] J. M. Rassias, “On the stability of a multi-dimensional Cauchy type functional equation,” in Geometry, Analysis and Mechanics, pp. 365-376, World Scientific, River Edge, NJ, USA, 1994. · Zbl 0842.39014
[20] J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,” Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185-190, 1992. · Zbl 0789.46036
[21] J. M. Rassias, “Refined Hyers-Ulam approximation of approximately Jensen type mappings,” Bulletin des Sciences Mathématiques, vol. 131, no. 1, pp. 89-98, 2007. · Zbl 1112.39025
[22] J. M. Rassias, “Solution of a Cauchy-Jensen stability Ulam type problem,” Archivum Mathematicum, vol. 37, no. 3, pp. 161-177, 2001. · Zbl 1090.39014
[23] 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. · Zbl 0685.39006
[24] Pl. Kannappan, “Quadratic functional equation and inner product spaces,” Results in Mathematics, vol. 27, no. 3-4, pp. 368-372, 1995. · Zbl 0836.39006
[25] F. Skof, “Proprieta’ locali e approssimazione di operatori,” Rendiconti del Seminario Matemàtico e Fisico di Milano, vol. 53, no. 1, pp. 113-129, 1983. · Zbl 0599.39007
[26] P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76-86, 1984. · Zbl 0549.39006
[27] St. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, no. 1, pp. 59-64, 1992. · Zbl 0779.39003
[28] A. Grabiec, “The generalized Hyers-Ulam stability of a class of functional equations,” Publicationes Mathematicae Debrecen, vol. 48, no. 3-4, pp. 217-235, 1996. · Zbl 1274.39058
[29] J. R. Lee, J. S. An, and C. Park, “On the stability of quadratic functional equations,” Abstract and Applied Analysis, vol. 2008, Article ID 628178, 8 pages, 2008. · Zbl 1146.39045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.