STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN QUASI-BANACH SPACES

Najati, Abbas; Moradlou, Fridoun

doi:10.4134/BKMS.2008.45.3.587

HOME > Journal Browse > About Journal > Journal Vol & Issue > Full Record

STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN QUASI-BANACH SPACES

Journal title : Bulletin of the Korean Mathematical Society
Volume 45, Issue 3, 2008, pp.587-600
Publisher : The Korean Mathematical Society
DOI : 10.4134/BKMS.2008.45.3.587

Title & Authors

STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN QUASI-BANACH SPACES
Najati, Abbas; Moradlou, Fridoun;

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;

Language

English

Cited by

5time(s) in CrossRef

Homomorphisms in quasi-Banach algebras associated with a Pexiderized Cauchy-Jensen functional equation, Acta Mathematica Sinica, English Series, 2009, 25, 9, 1529

On Approximate Additive–Quartic and Quadratic–Cubic Functional Equations in Two Variables on Abelian Groups, Results in Mathematics, 2010, 58, 1-2, 39

Approximate Behavior of Bi-Quadratic Mappings in Quasinormed Spaces, Journal of Inequalities and Applications, 2010, 2010, 1, 472721

Generalized Stability of Euler-Lagrange Quadratic Functional Equation, Abstract and Applied Analysis, 2012, 2012, 1

On the stability of a functional equation deriving from additive and quadratic functions, Advances in Difference Equations, 2012, 2012, 1, 98

References

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

D. Amir, Characterizations of inner product spaces, Operator Theory: Advances and Applications, 20. Birkhauser Verlag, Basel, 1986

Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, 2000

P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76-86

S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64

P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436

A. Grabiec, The generalized Hyers-Ulam stability of a class of functional equations, Publ. Math. Debrecen 48 (1996), no. 3-4, 217-235

D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222-224

P. Jordan and J. von Neumann, On inner products in linear, metric spaces, Ann. of Math. (2) 36 (1935), no. 3, 719-723

10.

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

11.

Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), no. 3-4, 368-372

12.

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

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.

F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129

16.

S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8 Interscience Publishers, New York-London, 1960