Journal of Applied Mathematics
Volume 2012 (2012), Article ID 610714, 14 pages
http://dx.doi.org/10.1155/2012/610714
Research Article

On a Generalized Hyers-Ulam Stability of Trigonometric Functional Equations

Jaeyoung Chung1 and Jeongwook Chang2

1Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea
2Department of Mathematics Education, Dankook University, Yongin 448-701, Republic of Korea

Received 4 July 2011; Revised 3 January 2012; Accepted 3 January 2012

Academic Editor: Yongkun Li

Copyright © 2012 Jaeyoung Chung and Jeongwook Chang. 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 𝐺 be an Abelian group, let ℂ be the field of complex numbers, and let 𝑓 , 𝑔 ∶ 𝐺 → ℂ . We consider the generalized Hyers-Ulam stability for a class of trigonometric functional inequalities, | 𝑓 ( 𝑥 − 𝑦 ) − 𝑓 ( 𝑥 ) 𝑔 ( 𝑦 ) + 𝑔 ( 𝑥 ) 𝑓 ( 𝑦 ) | ≤ 𝜓 ( 𝑦 ) , | 𝑔 ( 𝑥 − 𝑦 ) − 𝑔 ( 𝑥 ) 𝑔 ( 𝑦 ) − 𝑓 ( 𝑥 ) 𝑓 ( 𝑦 ) | ≤ 𝜓 ( 𝑦 ) , where 𝜓 ∶ 𝐺 → ℝ is an arbitrary nonnegative function.