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Journal of Function Spaces
Volume 2016 (2016), Article ID 1036094, 7 pages
http://dx.doi.org/10.1155/2016/1036094
Research Article

On the Superstability of Lobačevskiǐ’s Functional Equations with Involution

1Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea
2Department of Mathematics, Jeonbuk National University, Jeonju 561-756, Republic of Korea

Received 29 November 2015; Accepted 22 February 2016

Academic Editor: Alberto Fiorenza

Copyright © 2016 Jaeyoung Chung 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 be a uniquely -divisible commutative group and let and be an involution. In this paper, generalizing the superstability of Lobačevskiǐ’s functional equation, we consider or for all , where . As a direct consequence, we find a weaker condition for the functions satisfying the Lobačevskiǐ functional inequality to be unbounded, which refines the result of Găvrută and shows the behaviors of bounded functions satisfying the inequality. We also give various examples with explicit involutions on Euclidean space.