Generalized random stability of Jensen type mapping by ProQuest

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									   Scientia Magna
  Vol. 5 (2009), No. 3, 19-24




    Generalized random stability of Jensen type
                     mapping
                      Saleh Shakeri ‡ , Yeol Je Cho   †
                                                          and Reza Saadati


 ‡ Department of Mathematics, Islamic Azad University-Ayatollah Amoli branch, Amol P.O.
                                      Box 678, Iran

 † Department of Mathematics Education and the RINS College of Education, Gyeongsang
                      National University Chinju 660-701, Korea

             Faculty of Science, University of Shomal, Amol, P.O. Box 731, Iran
         E-mail: salehshakeri@yahoo.com yjcho@nongae.gsnu.ac.kr rsaadati@eml.cc

    Abstract In this paper, we consider Jensen type mapping in the setting of generalized random
    normed spaces. We generalize a Hyers-Ulam stability result in the framework of classical
    normed spaces.
    Keywords Jensen type mapping, generalized random normed space.



§1. Introduction
     In 1941 D.H. Hyers [5] solved this stability problem for additive mappings subject to the
Hyers condition on approximately additive mappings. In 1951 D.G. Bourgin [2] was the second
author to treat the Ulam stability problem for additive mappings. In 1978 P.M. Gruber [4]
remarked that Ulam’s problem is of particular interest in probability theory and in the case of
functional equations of different types.
     We wish to note that stability properties of different functional equations can have appli-
cations to unrelated fields. For instance, Zhou [10] used a stability property of the functional
equation
                                 f (x − y) + f (x + y) = 2f (x)                           (0.1)

to prove a conjecture of Z. Ditzian about the relationship between the smoothness of a mapping
and the degree of its approximation by the associated Bernstein polynomials. In 2003–2006 J.M.
Rassias and M.J. Rassias [6] and J.M. Rassias [7] solved the above Ulam problem for Jensen
and Jensen type mappings. In this paper we consider the stability of Jensen type mapping in
the setting of intuitionistic fuzzy normed spaces.


§2. Preliminaries
     In the sequel, we shall adopt the usual terminology, notations and conventions of the theory
of intuitionistic random normed spaces as in [8].
20                           Saleh Shakeri, Yeol Je Cho and Reza Saadati                         No. 3


    Definition 1. A measure distribution function is a function µ : R → [0, 1]
								
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