Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 12404, 23 pages
doi:10.1155/JIA/2006/12404
Generalized orthogonal stability of some functional equations
Institute of Mathematics, University of Silesian, Bankowa 14, Katowice 40-007, Poland
Received 19 November 2005; Accepted 2 July 2006
Copyright © 2006 Justyna Sikorska. 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 deal with a conditional functional inequality x⊥y⇒‖f(x+y)−f(x)−f(y)‖≤ε(‖x‖p+‖y‖p), where ⊥ is a given orthogonality relation, ε is a given nonnegative number, and p is a given real number. Under suitable assumptions, we prove that any solution f of the above inequality has to be uniformly close to an orthogonally additive mapping g, that is, satisfying the condition x⊥y⇒g(x+y)=g(x)+g(y). In the sequel, we deal with some other functional inequalities and we also present some applications and generalizations of the first result.