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http://dx.doi.org/10.4134/CKMS.2010.25.3.405

FUZZY STABILITY OF A GENERALIZED QUADRATIC FUNCTIONAL EQUATION  

Najati, Abbas (DEPARTMENT OF MATHEMATICS UNIVERSITY OF MOHAGHEGH ARDABILI)
Publication Information
Communications of the Korean Mathematical Society / v.25, no.3, 2010 , pp. 405-417 More about this Journal
Abstract
We prove the generalized Hyers-Ulam stability of the generalized quadratic functional equation $$f(rx\;+\;sy)\;=\;r^2f(x)\;+\;s^2f(y)\;+\;\frac{rs}{2}[f(x\;+\;y)\;-\;f(x\;-\;y)]$$ in fuzzy Banach spaces, where r, s are non-zero rational numbers with $r^2\;+\;s^2\;{\neq}\;1$.
Keywords
stability; fuzzy Banach spaces; quadratic mapping;
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1 S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.
2 F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129.   DOI
3 A. K. Mirmostafee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), no. 6, 720–729.   DOI   ScienceOn
4 A. K. Mirmostafee and M. S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), no. 19, 3791–3798.   DOI   ScienceOn
5 A. Najati and C. Park, Fixed points and stability of a generalized quadratic functional equation, J. Inequal. Appl. 2009 (2009), Article ID 193035, 19 pages.
6 C. Park, Fuzzy stability of a functional equation associated with inner product spaces, Fuzzy Sets and Systems 160 (2009), no. 11, 1632–1642.   DOI   ScienceOn
7 C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), no. 2, 711–720.   DOI   ScienceOn
8 Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297–300.   DOI   ScienceOn
9 Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers Co., Dordrecht, Boston, London, 2003.
10 D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
11 P. Jordan and J. von Neumann, On inner products in linear, metric spaces, Ann. of Math. (2) 36 (1935), no. 3, 719–723.   DOI
12 D. Mihet and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), no. 1, 567–572.   DOI   ScienceOn
13 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.
14 S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Inc., Palm Harbor, FL, 2001.
15 Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), no. 3-4, 368–372.   DOI
16 A. K. Mirmostafaee, M. Mirzavaziri, and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), no. 6, 730–738.   DOI   ScienceOn
17 S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002.
18 D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.   DOI   ScienceOn
19 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
20 A. Grabiec, The generalized Hyers-Ulam stability of a class of functional equations, Publ. Math. Debrecen 48 (1996), no. 3-4, 217–235.
21 P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76–86.   DOI
22 T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.   DOI
23 T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), no. 3, 687–705.
24 T. Bag and S. K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), no. 3, 513–547.   DOI   ScienceOn
25 S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64.   DOI
26 J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989.
27 D. Amir, Characterizations of Inner Product Spaces, Birkhauser, Basel, 1986.