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Approximate tri-quadratic functional equations via Lipschitz conditions

Ismail Nikoufar

Received February 17, 2016.   First published January 26, 2017.

Abstract: In this paper, we consider Lipschitz conditions for tri-quadratic functional equations. We introduce a new notion similar to that of the left invariant mean and prove that a family of functions with this property can be approximated by tri-quadratic functions via a Lipschitz norm.

Keywords: tri-quadratic functional equation; Lipschitz space; stability

Classification (MSC 2010): 39B82, 39B52

DOI: 10.21136/MB.2017.0028-16

Full text available as PDF.

References:
  [1] S. Czerwik, K. Dłutek: Stability of the quadratic functional equation in Lipschitz spaces. J. Math. Anal. Appl. 293 (2004), 79-88. DOI 10.1016/j.jmaa.2003.12.034 | MR 2052533 | Zbl 1052.39030
  [2] A. Ebadian, N. Ghobadipour, I. Nikoufar, M. Eshaghi Gordji: Approximation of the cubic functional equations in Lipschitz spaces. Anal. Theory Appl. 30 (2014), 354-362. DOI 10.4208/ata.2014.v30.n4.2 | MR 3303361 | Zbl 1340.39040
  [3] S.-M. Jung, P. K. Sahoo: Hyers-Ulam stability of the quadratic equation of Pexider type. J. Korean Math. Soc. 38 (2001), 645-656. MR 1826928 | Zbl 0980.39023
  [4] J. R. Lee, S.-Y. Jang, C. Park, D. Y. Shin: Fuzzy stability of quadratic functional equations. Adv. Difference Equ. 2010 (2010), Article ID 412160, 16 pages. DOI 10.1155/2010/412160 | MR 2652450 | Zbl 1192.39021
  [5] I. Nikoufar: Lipschitz approximation of the $n$-quadratic functional equations. Mathematica (Cluj) - Tome 57 (2015), 67-74.
  [6] I. Nikoufar: Quartic functional equations in Lipschitz spaces. Rend. Circ. Mat. Palermo, Ser. 2 64 (2015), 171-176. DOI 10.1007/s12215-014-0187-1 | MR 3371402 | Zbl 1328.39046
  [7] I. Nikoufar: Erratum to: Quartic functional equations in Lipschitz spaces. Rend. Circ. Mat. Palermo, Ser. 2 65 (2016), 345-350. DOI 10.1007/s12215-015-0222-x | MR 3535460 | Zbl 06643403
  [8] I. Nikoufar: Lipschitz criteria for bi-quadratic functional equations. Commun. Korean Math. Soc 31 (2016), 819-825. DOI 10.4134/CKMS.c150249
  [9] C.-G. Park: On the stability of the quadratic mapping in Banach modules. J. Math. Anal. Appl. 276 (2002), 135-144. DOI 10.1016/S0022-247X(02)00387-6 | MR 1944341 | Zbl 1017.39010
  [10] W.-G. Park, J.-H. Bae: Approximate behavior of bi-quadratic mappings in quasinormed spaces. J. Inequal. Appl. (2010), Article ID 472721, 8 pages. DOI 10.1155/2010/472721 | MR 2665495 | Zbl 1216.39039
  [11] F. Skof: Local properties and approximations of operators. Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129 (In Italian.). DOI 10.1007/BF02924890 | MR 0858541 | Zbl 0599.39007
  [12] J. Tabor: Lipschitz stability of the Cauchy and Jensen equations. Result. Math. 32 (1997), 133-144. DOI 10.1007/BF03322533 | MR 1464682 | Zbl 0890.39023
  [13] J. Tabor: Superstability of the Cauchy, Jensen and isometry equations. Result. Math. 35 (1999), 355-379. DOI 10.1007/BF03322824 | MR 1694913 | Zbl 0929.39015