# 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)
• Published : 2010.09.30

#### 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 $r\;{\neq}\;{\pm}1$.

#### Acknowledgement

Supported by : National Research Foundation of Korea

#### References

1. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989.
2. J. Bae and W. Park, Generalized Jensen’s functional equations and approximate algebra homomorphisms, Bull. Korean Math. Soc. 39 (2002), no. 3, 401-410. https://doi.org/10.4134/BKMS.2002.39.3.401
3. 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. https://doi.org/10.11650/twjm/1500407583
4. 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.
5. 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.
6. 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.
7. St. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64. https://doi.org/10.1007/BF02941618
8. 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. https://doi.org/10.1090/S0002-9904-1968-11933-0
9. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434. https://doi.org/10.1155/S016117129100056X
10. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
11. S. Jung, Hyers-Ulam-Rassias stability of Jensen's equation and its application, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3137-3143. https://doi.org/10.1090/S0002-9939-98-04680-2
12. 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. https://doi.org/10.1016/j.jmaa.2004.10.017
13. Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), no. 3-4, 368-372. https://doi.org/10.1007/BF03322841
14. 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. https://doi.org/10.1006/jmaa.1999.6546
15. 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. https://doi.org/10.1007/s00574-006-0016-z
16. M. S. Moslehian and L. Szekelyhidi, Stability of ternary homomorphisms via generalized Jensen equation, Results Math. 49 (2006), no. 3-4, 289-300. https://doi.org/10.1007/s00025-006-0225-1
17. 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. https://doi.org/10.1016/S0022-247X(02)00573-5
18. C. Park, Generalized quadratic mappings in several variables, Nonlinear Anal. 57 (2004), no. 5-6, 713-722. https://doi.org/10.1016/j.na.2004.03.013
19. 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. https://doi.org/10.4134/BKMS.2006.43.4.703
20. 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. https://doi.org/10.1142/S0252959903000244
21. V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), no. 1, 91-96.
22. 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.
23. 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. https://doi.org/10.1016/j.bulsci.2005.02.001
24. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
25. 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. https://doi.org/10.1090/S0002-9939-1992-1059634-1
26. Th. M. Rassias and K. Shibata, Variational problem of some quadratic functionals in complex analysis, J. Math. Anal. Appl. 228 (1998), 234-253. https://doi.org/10.1006/jmaa.1998.6129
27. F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
28. 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. https://doi.org/10.1016/S0022-247X(02)00181-6
29. S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, New York, 1964.

#### Cited by

1. Approximate homomorphisms and derivations on random Banach algebras vol.2012, pp.1, 2012, https://doi.org/10.1186/1029-242X-2012-157
2. On the Stability of an -Variables Functional Equation in Random Normed Spaces via Fixed Point Method vol.2012, 2012, https://doi.org/10.1155/2012/346561