Browse > Article
http://dx.doi.org/10.4134/BKMS.2008.45.3.587

STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN QUASI-BANACH SPACES  

Najati, Abbas (DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCES UNIVERSITY OF MOHAGHEGH ARDABILI)
Moradlou, Fridoun (FACULTY OF MATHEMATICAL SCIENCES UNIVERSITY OF TABRIZ)
Publication Information
Bulletin of the Korean Mathematical Society / v.45, no.3, 2008 , pp. 587-600 More about this Journal
Abstract
In this paper we establish the general solution and investigate the Hyers-Ulam-Rassias stability of the following functional equation in quasi-Banach spaces. $${\sum\limits_{{{1{\leq}i<j{\leq}4}\limits_{1{\leq}k<l{\leq}4}}\limits_{k,l{\in}I_{ij}}}\;f(x_i+x_j-x_k-x_l)=2\;\sum\limits_{1{\leq}i<j{\leq}4}}\;f(x_i-x_j)$$ where $I_{ij}$={1, 2, 3, 4}\backslash${i, j} for all $1{\leq}i<j{\leq}4$. The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc.
Keywords
Hyers-Ulam-Rassias stability; quadratic function; quasi-Banach space; p-Banach space;
Citations & Related Records

Times Cited By Web Of Science : 4  (Related Records In Web of Science)
Times Cited By SCOPUS : 3
연도 인용수 순위
1 A. Najati and M. B. Moghimi, Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces, J. Math. Anal. Appl. 337 (2008), no. 1, 399-415   DOI   ScienceOn
2 P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436   DOI   ScienceOn
3 P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76-86   DOI
4 S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64   DOI
5 A. Grabiec, The generalized Hyers-Ulam stability of a class of functional equations, Publ. Math. Debrecen 48 (1996), no. 3-4, 217-235
6 D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222-224
7 K. Jun and Y. Lee, On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality, Math. Inequal. Appl. 4 (2001), no. 1, 93-118
8 D. Amir, Characterizations of inner product spaces, Operator Theory: Advances and Applications, 20. Birkhauser Verlag, Basel, 1986
9 P. Jordan and J. von Neumann, On inner products in linear, metric spaces, Ann. of Math. (2) 36 (1935), no. 3, 719-723   DOI
10 J. Aczel and J. Dhombres, Functional Equations in Several Variables, With applications to mathematics, information theory and to the natural and social sciences. Encyclopedia of Mathematics and its Applications, 31. Cambridge University Press, Cambridge, 1989
11 Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, 2000
12 Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), no. 3-4, 368-372   DOI
13 Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300
14 S. Rolewicz, Metric Linear Spaces, PWN?Polish Scientific Publishers, Warsaw; D. Reidel Publishing Co., Dordrecht, 1984
15 S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8 Interscience Publishers, New York-London, 1960
16 F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129   DOI