×

Stability of a functional equation deriving from cubic and quartic functions. (English) Zbl 1160.39334

Summary: We obtain the general solution and the generalized Ulam-Hyers stability of the cubic and quartic functional equation \(4(f(3x+y)+f(3x - y))= - 12(f(x+y)+f(x - y))+12(f(2x+y)+f(2x - y)) - 8f(y) - 192f(x)+f(2y)+30f(2x).\)

MSC:

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

References:

[1] S. M. Ulam, Problems in Modern Mathematics, 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, pp. 222-224, 1941. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[3] 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 · doi:10.2307/2042795
[4] 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 · doi:10.1155/S016117129100056X
[5] 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 · doi:10.2969/jmsj/00210064
[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
[7] 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 · doi:10.1006/jmaa.1994.1211
[8] 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
[9] D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, vol. 34 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 1998. · Zbl 0907.39025
[10] G. Isac and Th. M. Rassias, “On the Hyers-Ulam stability of \psi -additive mappings,” Journal of Approximation Theory, vol. 72, no. 2, pp. 131-137, 1993. · Zbl 0770.41018 · doi:10.1006/jath.1993.1010
[11] Th. M. Rassias, Ed., Functional Equations and Inequalities, Th. M. Rassias, Ed., vol. 518 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. · Zbl 0945.00010
[12] 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
[13] Th. M. Rassias, “On the stability of functional equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 264-284, 2000. · Zbl 0964.39026 · doi:10.1006/jmaa.2000.7046
[14] 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
[15] 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 · doi:10.1016/0022-1236(82)90048-9
[16] J. M. Rassias, “Solution of the Ulam stability problem for quartic mappings,” Glasnik Matemati\vcki, vol. 34(54), no. 2, pp. 243-252, 1999. · Zbl 0951.39008
[17] J. M. Rassias, “Complete solution of the multi-dimensional problem of Ulam,” Discussiones Mathematicae, vol. 14, pp. 101-107, 1994. · Zbl 0819.39012
[18] 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
[19] J. M. Rassias, “Solution of a stability problem of Ulam,” in Functional Analysis, Approximation Theory and Numerical Analysis, pp. 241-249, World Scientific, River Edge, NJ, USA, 1994. · Zbl 0878.46032
[20] J. M. Rassias, “Solution of a stability problem of Ulam,” Discussiones Mathematicae, vol. 12, pp. 95-103, 1992. · Zbl 0779.47005
[21] J. M. Rassias, “Solution of a problem of Ulam,” Journal of Approximation Theory, vol. 57, no. 3, pp. 268-273, 1989. · Zbl 0672.41027 · doi:10.1016/0021-9045(89)90041-5
[22] 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
[23] J. M. Rassias, “Solution of the Ulam stability problem for cubic mappings,” Glasnik Matemati\vcki, vol. 36(56), no. 1, pp. 63-72, 2001. · Zbl 0984.39014
[24] J. M. Rassias, “Solution of the Ulam problem for cubic mappings,” Analele Universit\ua\ctii din Timi\csoara. Seria Matematic\ua-Informatic\ua, vol. 38, no. 1, pp. 121-132, 2000. · Zbl 1004.39025
[25] K.-W. Jun and H.-M. Kim, “The generalized Hyers-Ulam-Rassias stability of a cubic functional equation,” Journal of Mathematical Analysis and Applications, vol. 274, no. 2, pp. 267-278, 2002. · Zbl 1021.39014 · doi:10.1016/S0022-247X(02)00415-8
[26] J. M. Rassias, “Solution of the Ulam stability problem for quartic mappings,” The Journal of the Indian Mathematical Society, vol. 67, no. 1-4, pp. 169-178, 2000. · Zbl 1083.39503
[27] 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. · Zbl 1076.39027 · doi:10.1016/j.na.2005.03.075
[28] J. K. Chung and P. K. Sahoo, “On the general solution of a quartic functional equation,” Bulletin of the Korean Mathematical Society, vol. 40, no. 4, pp. 565-576, 2003. · Zbl 1048.39017 · doi:10.4134/BKMS.2003.40.4.565
[29] L. C\uadariu, “Fixed points in generalized metric space and the stability of a quartic functional equation,” Buletinul \vStiin\ctific al Universit\ua\ctii Politehnica din Timi\csoara. Seria Matematic\ua-Fizic\ua, vol. 50(64), no. 2, pp. 25-34, 2005. · Zbl 1135.39304
[30] S. H. Lee, S. M. Im, and I. S. Hwang, “Quartic functional equations,” Journal of Mathematical Analysis and Applications, vol. 307, no. 2, pp. 387-394, 2005. · Zbl 1072.39024 · doi:10.1016/j.jmaa.2004.12.062
[31] A. Najati, “On the stability of a quartic functional equation,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 569-574, 2008. · Zbl 1133.39030 · doi:10.1016/j.jmaa.2007.08.048
[32] C.-G. Park, “On the stability of the orthogonally quartic functional equation,” Bulletin of the Iranian Mathematical Society, vol. 31, no. 1, pp. 63-70, 2005. · Zbl 1117.39020
[33] E. Thandapani, K. Ravi, and M. Arunkumar, “On the solution of the generalized quartic functional equation,” Far East Journal of Applied Mathematics, vol. 24, no. 3, pp. 297-312, 2006. · Zbl 1123.39016
[34] 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
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.