zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Generalized stability of Euler-Lagrange quadratic functional equation. (English) Zbl 1246.39026
Summary: The main goal of this paper is the investigation of the general solution and the generalized Hyers-Ulam stability theorem of the following Euler-Lagrange type quadratic functional equation $f(ax + by) + af(x - by) = (a + 1)b^2 f(y) + a(a + 1)f(x)$, in $(\beta, p)$-Banach space, where $a, b$ are fixed rational numbers such that $a \neq - 1, 0$ and $b \neq 0$.
MSC:
39B82Stability, separation, extension, and related topics
WorldCat.org
Full Text: DOI
References:
[1] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics No. 8, Interscience Publishers, New York, NY, USA, 1960. · Zbl 0086.24101
[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, pp. 222-224, 1941. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[3] J. Aczél and J. Dhombres, Functional Equations in Several Variables, vol. 31 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, Mass, USA, 1989. · Zbl 0731.39010 · doi:10.1017/CBO9781139086578
[4] D. H. Hyers and Th. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125-153, 1992. · Zbl 0806.47056 · doi:10.1007/BF01830975 · eudml:137488
[5] F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113-129, 1983. · Zbl 0599.39007 · doi:10.1007/BF02924890
[6] P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76-86, 1984. · Zbl 0549.39006 · doi:10.1007/BF02192660 · eudml:137013
[7] S. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, pp. 59-64, 1992. · Zbl 0779.39003 · doi:10.1007/BF02941618
[8] C. Borelli and G. L. Forti, “On a general Hyers-Ulam stability result,” International Journal of Mathematics and Mathematical Sciences, vol. 18, no. 2, pp. 229-236, 1995. · Zbl 0826.39009 · doi:10.1155/S0161171295000287 · http://www.hindawi.com/journals/ijmms/volume-18/issue-2.html · eudml:47292
[9] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002. · Zbl 1011.39019 · doi:10.1142/9789812778116
[10] G. L. Forti, “Hyers-Ulam stability of functional equations in several variables,” Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143-190, 1995. · Zbl 0836.39007 · doi:10.1007/BF01831117 · eudml:137624
[11] D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and their Applications 34, Birkhäuser, Boston, Mass, USA, 1998. · Zbl 0907.39025 · doi:10.1007/978-1-4612-1790-9
[12] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001. · Zbl 0980.39024
[13] 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. · Zbl 0981.39014 · doi:10.1023/A:1006499223572
[14] J. M. Rassias and H.-M. Kim, “Generalized Hyers-Ulam stability for general additive functional equations in quasi-\beta -normed spaces,” Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp. 302-309, 2009. · Zbl 1168.39015 · doi:10.1016/j.jmaa.2009.03.005
[15] A. Najati and M. B. Moghimi, “Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 399-415, 2008. · Zbl 1127.39055 · doi:10.1016/j.jmaa.2007.03.104
[16] 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
[17] M. E. Gordji and H. Khodaei, “On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations,” Abstract and Applied Analysis, vol. 2009, Article ID 923476, 11 pages, 2009. · Zbl 1167.39014 · doi:10.1155/2009/923476 · eudml:55826
[18] K. Jun, H. Kim, and J. Son, “Generalized Hyers-Ulam stability of a quadratic functional equation,” in Functional Equations in Mathematical Analysis, Th. M. Rassias and J. Brzdek, Eds., chapter 12, pp. 153-164, 2011. · Zbl 1248.39037
[19] K.-W. Jun and H.-M. Kim, “Ulam stability problem for generalized A-quadratic mappings,” Journal of Mathematical Analysis and Applications, vol. 305, no. 2, pp. 466-476, 2005. · Zbl 1069.39030 · doi:10.1016/j.jmaa.2004.10.058
[20] J.-H. Bae and W.-G. Park, “Stability of a cauchy-jensen functional equation in quasi-banach spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 151547, 9 pages, 2010. · Zbl 1187.39031 · doi:10.1155/2010/151547 · eudml:224777
[21] M. E. Gordji and H. Khodaei, “Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5629-5643, 2009. · Zbl 1179.39034 · doi:10.1016/j.na.2009.04.052
[22] A. Najati and G. Z. Eskandani, “Stability of a mixed additive and cubic functional equation in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 1318-1331, 2008. · Zbl 1228.39028 · doi:10.1016/j.jmaa.2007.12.039
[23] A. Najati and F. Moradlou, “Stability of a quadratic functional equation in quasi-Banach spaces,” Bulletin of the Korean Mathematical Society, vol. 45, no. 3, pp. 587-600, 2008. · Zbl 1168.39013 · doi:10.4134/BKMS.2008.45.3.587 · http://ajmaa.org/cgi-bin/paper.pl?string=v5n2/V5I2P10.tex
[24] T. Z. Xu, J. M. Rassias, M. J. Rassias, and W. X. Xu, “A fixed point approach to the stability of quintic and sextic functional equations in quasi-\beta -normed spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 423231, 23 pages, 2010. · Zbl 1219.39020 · doi:10.1155/2010/423231 · eudml:233500
[25] L. G. Wang and B. Liu, “The Hyers-Ulam stability of a functional equation deriving from quadratic and cubic functions in quasi-\beta -normed spaces,” Acta Mathematica Sinica (English Series), vol. 26, no. 12, pp. 2335-2348, 2010. · Zbl 1222.39024 · doi:10.1007/s10114-010-9330-x