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http://dx.doi.org/10.4134/BKMS.2010.47.5.987

A FIXED POINT APPROACH TO THE CAUCHY-RASSIAS STABILITY OF GENERAL JENSEN TYPE QUADRATIC-QUADRATIC MAPPINGS  

Park, Choon-Kil (DEPARTMENT OF MATHEMATICS RESEARCH INSTITUTE FOR NATURAL SCIENCES HANYANG UNIVERSITY)
Gordji, M. Eshaghi (DEPARTMENT OF MATHEMATICS SEMNAN UNIVERSITY)
Khodaei, H. (DEPARTMENT OF MATHEMATICS SEMNAN UNIVERSITY)
Publication Information
Bulletin of the Korean Mathematical Society / v.47, no.5, 2010 , pp. 987-996 More about this Journal
Abstract
In this paper, we investigate the Cauchy-Rassias stability in Banach spaces and also the Cauchy-Rassias stability using the alternative fixed point for the functional equation: $$f(\frac{sx+ty}{2}+rz)+f(\frac{sx+ty}{2}-rz)+f(\frac{sx-ty}{2}+rz)+f(\frac{sx-ty}{2}-rz)=s^2f(x)+t^2f(y)+4r^2f(z)$$ for any fixed nonzero integers s, t, r with $r\;{\neq}\;{\pm}1$.
Keywords
Cauchy-Rassias stability; quadratic mapping; fixed point method;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
Times Cited By Web Of Science : 1  (Related Records In Web of Science)
Times Cited By SCOPUS : 1
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1 T. Trif, On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions, J. Math. Anal. Appl. 272 (2002), no. 2, 604-616.   DOI   ScienceOn
2 S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, New York, 1964.
3 C. Park, Modified Trifs functional equations in Banach modules over a $C^{\ast}$-algebra and approximate algebra homomorphisms, J. Math. Anal. Appl. 278 (2003), no. 1, 93-108.   DOI   ScienceOn
4 V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), no. 1, 91-96.
5 C. Park, Generalized quadratic mappings in several variables, Nonlinear Anal. 57 (2004), no. 5-6, 713-722.   DOI   ScienceOn
6 C. Park, S. Hong, and M. Kim, Jensen type quadratic-quadratic mapping in Banach spaces, Bull. Korean Math. Soc. 43 (2006), no. 4, 703-709.   과학기술학회마을   DOI   ScienceOn
7 C. Park, J. Park, and J. Shin, Hyers-Ulam-Rassias stability of quadratic functional equations in Banach modules over a $C^*$-algebra, Chinese Ann. Math. Ser. B 24 (2003), no. 2, 261-266.   DOI   ScienceOn
8 J. M. Rassias and M. J. Rassias, Asymptotic behavior of Jensen and Jensen type functional equations, Panamer. Math. J. 15 (2005), no. 4, 21-35.
9 J. M. Rassias and M. J. Rassias, Asymptotic behavior of alternative Jensen and Jensen type functional equations, Bull. Sci. Math. 129 (2005), no. 7, 545-558.   DOI   ScienceOn
10 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
11 Th. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), no. 4, 989-993.   DOI   ScienceOn
12 Th. M. Rassias and K. Shibata, Variational problem of some quadratic functionals in complex analysis, J. Math. Anal. Appl. 228 (1998), 234-253.   DOI   ScienceOn
13 F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129.   DOI
14 Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434.   DOI   ScienceOn
15 L. Cadariu and V. Radu, Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara Ser. Mat.-Inform. 41 (2003), no. 1, 25-48.
16 St. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64.   DOI
17 J. B. Diaz and B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309.   DOI
18 D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222-224.   DOI   ScienceOn
19 S. Jung, Hyers-Ulam-Rassias stability of Jensen's equation and its application, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3137-3143.   DOI   ScienceOn
20 Y. Jung and I. Chang, The stability of a cubic type functional equation with the fixed point alternative, J. Math. Anal. Appl. 306 (2005), no. 2, 752-760.   DOI   ScienceOn
21 Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), no. 3-4, 368-372.   DOI
22 Y. Lee and K. Jun, A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation, J. Math. Anal. Appl. 238 (1999), no. 1, 305-315.   DOI   ScienceOn
23 M. Mirzavaziri and M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. (N.S.) 37 (2006), no. 3, 361-376.   DOI   ScienceOn
24 D. Boo, S. Oh, C. Park, and J. Park, Generalized Jensen’s equations in Banach modules over a $C^*$-algebra and its unitary group, Taiwanese J. Math. 7 (2003), no. 4, 641-655.   DOI
25 M. S. Moslehian and L. Szekelyhidi, Stability of ternary homomorphisms via generalized Jensen equation, Results Math. 49 (2006), no. 3-4, 289-300.   DOI
26 J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989.
27 J. Bae and W. Park, Generalized Jensen’s functional equations and approximate algebra homomorphisms, Bull. Korean Math. Soc. 39 (2002), no. 3, 401-410.   과학기술학회마을   DOI   ScienceOn
28 L. Cadariu and V. Radu, The fixed points method for the stability of some functional equations, Carpathian J. Math. 23 (2007), no. 1-2, 63-72.
29 L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Iteration theory (ECIT ’02), 43-52, Grazer Math. Ber., 346, Karl-Franzens-Univ. Graz, Graz, 2004.