Abstract

In this paper, using the fixed point approach, we proved the Hyers-Ulam-Rassias stability of a Jensen-type quadratic functional equations f(ax±ay)-a2[f(x)+f(y)] and f((x±y)/2)-f(x)-f(y)in Multi-Banach Spaces using the ideas from Dales and Polyakov [4].

References

• C. Borelli and G. L. Forti, On a general Hyers-Ulam stability, Int. J. Math. Math. Sci. (18)(1995), pp. 229-236.
• D. H. Hyers, On the Stability of the Linear Functional Equation, Proc. Nat. Acad. Sci. U. S. A. (27)(1941), pp. 222–224.
• F. Skof, Proprietà locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, (53)(1983), pp. 113–129.
• H. G. Dales and M. E. Polyakov, Multi-normed spaces and multi-Banach algebras, preprint.
• H. G. Dales and M. S. Moslehian, Stability of mappings on multi- normed spaces, Glasgow Mathematical Journal, (49)(2)(2007), pp. 321– 332.
• J. B. Diaz and B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bulletin of the American Mathematical Society, (74)(1968), pp. 305–309.
• L. Wang, B. Liu and R. Bai, Stability of a Mixed Type Functional Equation on Multi-Banach Spaces: A Fixed Point Approach, Fixed Point Theory and Applications, vol. 2010, Art. ID 283827, 9 pages
• M. S. Moslehian, Superstability of higher derivations in multi-Banach algebras, Tamsui Oxford Journal of Mathematical Sciences, (24)(4) (2008), pp. 417–427.
• M. S. Moslehian, K. Nikodem, and D. Popa, Asymptotic aspect of the quadratic functional equation in multi-normed spaces, Journal of Mathematical Analysis and Applications, (355)(2)(2009), pp. 717– 724.
• M. S. Moslehian and H. M. Srivastava, Jensens functional equations in multi-normed spaces, Taiwnese J. of Math. , (14)(2)(2010), pp. 453-462.
• O. Had?zi´c, E. Pap, and V. Radu, Generalized contraction mapping principles in probabilistic metric spaces, Acta Mathematica Hungarica, (101)(1-2)(2003), pp. 131–148.
• P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. , (27)(1984), pp. 76–86.
• P. Gavruta, A Generalization of the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings, J. Math. Anal. Appl. (184)(1994), pp. 431–436.
• S. Y. Jang, Rye Lee, C. Park and Dong Yun Shin, Fuzzy stability of Jensen – Type Quadratic functional equations, Abstract and Applied Analysis, vol. 2009, Article ID 535678
• S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.
• S. Czerwik, On the stability of the quadratic mappings in normed spaces, Abh. Math. Sem. Univ. Hamburg, (62)(1992), pp. 59–64.
• Th. M. Rassias, On the stability of the Linear mapping in Banach spaces, Procc. Of the American Mathematical Society, (72)(2)(1978), pp. 297-300.
• Th. M. Rassias, On the stability of the functional equations in Banach spaces, J. Math. Anal. Appl. , (251)(2000), pp. 264–284.
• T. Aoki, On the stability of the linear transformation in Banach spaces, Journal of the Mathematical Society, (2)(1-2)(1950), pp. 64-66.
• V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, (4)(1)(2003), pp. 91–96.
• Z. Wang, X. Li, and Th. M. Rassias, Stability of an Additive-Cubic- Quartic Functional Equation in Multi-Banach Spaces, Abstract and Applied Analysis, vol. 2011, Article ID 536520, 11 pages