International Journal of Nonlinear Analysis and Applications

Asymptotic behavior of generalized quadratic mappings

Document Type : Research Paper

Authors

1 Department of Mathematics, Chungnam National University 99 Daehangno, Yuseong-gu, Daejeon 305-764, Korea

2 Department of Mathematics, Chungnam National University, 99 Daehangno, Yuseong-gu, Daejeon 34134, Korea

Abstract

We show in this paper that a mapping $f$ satisfies the following functional equation
\begin{eqnarray*}
\biguplus_{x_2,\cdots,x_{d+1}}^{d}f(x_1) = 2^{d} \sum_{i=1}^{d+1}f(x_i),
\end{eqnarray*}
if and only if it is quadratic. In addition, we investigate generalized Hyers-Ulam stability problem for the equation, and thus obtain an asymptotic property of quadratic mappings as applications.

Keywords

[1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, 1989.
[2] Dan Amir, Characterizations of inner product spaces, Birkh¨auser-Verlag, Basel, 1986.
[3] J. Bae, K. Jun and S. Jung, On the stability of a quadratic functional equation, Kyungpook Math. J. 43 (2003) 415-423.
[4] C. Borelli and G.L. Forti, On a general Hyers-Ulam stability result, Internat. J. Math. Math. Sci. 18 (1995) 229-236.
[5] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992) 59-64.
[6] G.L. Forti, Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations, J. Math. Anal. Appl. 295 (2004) 127-133.
[7] P.M. Gruber, Stability of isometries, Trans. Amer. Math. Soc. 245 (1978) 263-277.
[8] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
[9] S. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998) 126-137.
[10] S. Jung, Quadratic functional equations of Pexider type, Int. J. Math. Math. Sci. 24 (2000) 351-359.
[11] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
[12] H. Kim, On the stability problem for a mixed type of quartic and quadratic functional equation, J. Math. Anal. Appl. 324(1) (2006) 358-372.
[13] H. Kim, On the stability and asymptotic behavior of generalized quadratic mappings, J. Comput. Anal. Appl. 10(3) (2008), 319-331.
[14] J.M. Rassias, Asymptotic behavior of mixed type functional equations, Austral. J. Math. Anal. Appl. 1(1) (2004) 1-21.
[15] Th.M. Rassias and J. Tabor, Stability of mapping of Hyers-Ulam Type, Hadronic Press, Inc., Florida, 1994.
[16] Th.M. Rassias, Inner product spaces and applications, Longman, 1997.
[17] Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia, Univ. Babes Bolyai, XLIII (3) (1998) 89-124.
[18] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000) 264-284.
[19] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000) 23-130.
[20] D.A. Senechalle, A characterization of inner product spaces, Proc. Amer. Math. Soc. 19 (1968) 1306-1312.
[21] F. Skof, Sull’ approssimazione delle applicazioni localmente δ-additive, Atti Accad. Sci. Torino Cl Sci. Fis. Mat. Natur. 117 (1983) 377-389.
[22] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.
[23] Ding-Xuan Zhou, On a conjecture of Z. Ditzian, J. Approx. Theory 69 (1992) 167-172.
International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 1
May 2021
Pages 1153-1165
  • Receive Date: 09 February 2021
  • Revise Date: 05 March 2021
  • Accept Date: 13 March 2021