Abstract and Applied Analysis
Volume 2011 (2011), Article ID 370104, 10 pages
http://dx.doi.org/10.1155/2011/370104
Research Article

On Pexider Differences in Topological Vector Spaces

Abbas Najati,1 M. R. Abdollahpour,1 and Gwang Hui Kim2

1Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran
2Department of Applied Mathematics, Kangnam University, Gyeonggi, Giheung-gu, Yongin 446-702, Republic of Korea

Received 31 May 2011; Accepted 30 August 2011

Academic Editor: Alberto d'Onofrio

Copyright © 2011 Abbas Najati 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

Let 𝑋 be a normed space and 𝑌 a sequentially complete Hausdorff topological vector space over the field ℚ of rational numbers. Let 𝐷 1 = { ( 𝑥 , 𝑦 ) ∈ 𝑋 × 𝑋 ∶ ‖ 𝑥 ‖ + ‖ 𝑦 ‖ ≥ 𝑑 } , and 𝐷 2 = { ( 𝑥 , 𝑦 ) ∈ 𝑋 × 𝑋 ∶ ‖ 𝑥 ‖ + ‖ 𝑦 ‖ < 𝑑 } where 𝑑 > 0 . We prove that the Pexiderized Jensen functional equation is stable for functions defined on 𝐷 1 ( 𝐷 2 ) , and taking values in 𝑌 . We consider also the Pexiderized Cauchy functional equation.