On the Superstability of Lobačevskiǐ's Functional Equations with Involution. (3rd April 2016)

• Record Type: Journal Article
• Authors: Chung, Jaeyoung; Lee, Bogeun; Ha, Misuk
• Other Names: Fiorenza Alberto Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2016(2016)
• Publication Date: 2016-04-03
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2016/1036094
• Languages: English
• ISSNs: 2314-8896

Abstract

Abstract : Let G be a uniquely 2 -divisible commutative group and let f, g : G → C and σ : G → G be an involution. In this paper, generalizing the superstability of Lobačevskiǐ's functional equation, we consider f ( x + σ y ) / 2 2 - g ( x ) f ( y ) ≤ ψ ( x ) or ψ ( y ) for all x, y ∈ G, where ψ : G → R + . As a direct consequence, we find a weaker condition for the functions f 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.