Abstract and Applied Analysis
Volume 2011 (2011), Article ID 502903, 15 pages
http://dx.doi.org/10.1155/2011/502903
Research Article

Stability in Generalized Functions

Young-Su Lee

Department of Mathematics, Sogang University, Seoul 121-741, Republic of Korea

Received 31 July 2011; Revised 17 September 2011; Accepted 21 September 2011

Academic Editor: Dumitru Baleanu

Copyright © 2011 Young-Su Lee. 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 consider the following additive functional equation with 𝑛 -independent variables: ∑ 𝑓 ( 𝑛 𝑖 = 1 𝑥 𝑖 ∑ ) = 𝑛 𝑖 = 1 𝑓 ( 𝑥 𝑖 ∑ ) + 𝑛 𝑖 = 1 𝑓 ( 𝑥 𝑖 − 𝑥 𝑖 − 1 ) in the spaces of generalized functions. Making use of the heat kernels, we solve the general solutions and the stability problems of the above equation in the spaces of tempered distributions and Fourier hyperfunctions. Moreover, using the mollifiers, we extend these results to the space of distributions.