Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 156167, 12 pages
doi:10.1155/2009/156167
Research Article
Investigation of the Stability via Shadowing Property
1School of Mecatronics Engineering, Changwon National University, Sarim 9, Changwon, Gyeongnam 641-773, South Korea
2Department of Mathematics Education, College of Education, Dankook University, 126, Jukjeon, Suji, Yongin, Gyeongi 448-701, South Korea
3Department of Mathematics, Chungnam National University, 79, Daehangno, Yuseong-Gu, Daejeon 305-764, South Korea
Received 25 November 2008; Revised 16 February 2009; Accepted 19 May 2009
Academic Editor: Ulrich Abel
Copyright © 2009 Sang-Hyuk Lee 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
The shadowing property is to find an exact solution to an iterated map that remains close to an approximate solution.
In this article, using shadowing property, we show the stability of the following equation in normed group:
4n−2Cn/2−1r2f(∑j=1n(xj/r))+n∑ik∈{0,1}, ∑k=1nik=n/2r2f(∑i=1n(−1)ik(xi/r))=4n⋅n−2Cn/2−1∑i=1nf(xi), where n≥2, r∈R (r2≠n) and f is a mapping. And we prove that the even mapping which satisfies the above equation is quadratic and also the Hyers-Ulam stability of the functional equation in Banach spaces.