Browse > Article
http://dx.doi.org/10.14403/jcms.2020.33.3.319

ON STABILITY OF A GENERALIZED QUADRATIC FUNCTIONAL EQUATION WITH n-VARIABLES AND m-COMBINATIONS IN QUASI-𝛽-NORMED SPACES  

Koh, Heejeong (College of Liberal Arts Dankook University)
Lee, Yonghoon (Department of Mathematics Dankook University)
Publication Information
Journal of the Chungcheong Mathematical Society / v.33, no.3, 2020 , pp. 319-326 More about this Journal
Abstract
In this paper, we establish a general solution of the following functional equation $$mf\({\sum\limits_{k=1}^{n}}x_k\)+{\sum\limits_{t=1}^{m}}f\({\sum\limits_{k=1}^{n-i_t}}x_k-{\sum\limits_{k=n-i_t+1}^{n}}x_k\)=2{\sum\limits_{t=1}^{m}}\(f\({\sum\limits_{k=1}^{n-i_t}}x_k\)+f\({\sum\limits_{k=n-i_t+1}^{n}}x_k\)\)$$ where m, n, t, it ∈ ℕ such that 1 ≤ t ≤ m < n. Also, we study Hyers-Ulam-Rassias stability for the generalized quadratic functional equation with n-variables and m-combinations form in quasi-𝛽-normed spaces and then we investigate its application.
Keywords
generalized quadratic functional equation with n-variable; Hyers-Ulam-Rassias stability; quasi-${\beta}$-normed space;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Z. Alizadeh and A. G. Ghazanfari, On the stability of a radical cubic functional equation in quasi-β-spaces, Journal of Fixed Point Theory and Applications, 18 (2016), no. 4, 843-853.   DOI
2 T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.   DOI
3 St. Czerwik, On the stability of the quadratic mapping in normed spaces, Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 62 (1992), 59-64.
4 D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci, 27 (1941), 222-224.   DOI
5 S.M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, Journal of Mathematical Analysis and Applications, 222 (1998), no. 1, 126-137.   DOI
6 K.W. Jun and Y.H. Lee, On the Hyers-Ulam-Rassias stability of a pexiderized quadratic inequality, Mathematical Inequalities and Applications, 4 (2001), no. 1, pp. 93-118.
7 K.W. Jun and H. M. Kim, On the stability of an n-dimensional quadratic and additive functional equation, Mathematical Inequalities and Applications, 9 (2006), no. 1, 153-165.
8 D. S. Kang, On the Stability of Generalized Quartic Mappings in Quasi-β-Normed Spaces, Journal of Inequalities and Applications, 2010 (2010), no. 2.
9 R. Krishnan, E. Thandapani, and B.V. Senthil Kumar, Solution and stability of a reciprocal type functional equation in several variables, Journal of Nonlinear Science and Applications, 7 (2014), no. 1.
10 Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc, 72 (1978), 297-300.   DOI
11 Th. M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Universitatis Babe-Bolyai, 43 (1998), no. 3, 89-124.
12 B. V. Senthil Kumar and K. Ravi, Ulam stability of a reciprocal functional equation in quasi-beta-normed spaces, Global Journal of Pure and Applied Mathematics, 12 (2016), no. 1, 125-128.
13 S. M. Ulam, A Collection of Mathematical Problems, Interscience Publ. New York, 1960.
14 D. Y. Shin, C.K. Park, and R. A. Aghjeubeh, S. Farhadabadi, Fuzzy stability of functional equations in n-variable fuzzy Banach spaces, Journal of Computational Analysis and Applications, 19 (2015), no. 1, 186-196.
15 F. Skof, Proprieta' locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129.   DOI