Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 868193, 14 pages
doi:10.1155/2010/868193
Research Article

Generalized Ulam-Hyers Stability of Jensen Functional Equation in Šerstnev PN Spaces

M. Eshaghi Gordji,1 M. B. Ghaemi,2 H. Majani,2 and C. Park3

1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
2Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
3Department of Mathematics, Hanyang University, Seoul 133-791, South Korea

Received 17 November 2009; Revised 31 January 2010; Accepted 1 March 2010

Academic Editor: Sin-Ei Takahasi

Copyright © 2010 M. Eshaghi Gordji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We establish a generalized Ulam-Hyers stability theorem in a Šerstnev probabilistic normed space (briefly, Šerstnev PN-space) endowed with ΠM. In particular, we introduce the notion of approximate Jensen mapping in PN-spaces and prove that if an approximate Jensen mapping in a Šerstnev PN-space is continuous at a point then we can approximate it by an everywhere continuous Jensen mapping. As a version of a theorem of Schwaiger, we also show that if every approximate Jensen type mapping from the natural numbers into a Šerstnev PN-space can be approximated by an additive mapping, then the norm of Šerstnev PN-space is complete.