Functional Inequalities Associated with Additive Mappings. (4th August 2008)

• Record Type: Journal Article
• Authors: Roh, Jaiok; Chang, Ick-Soon
• Other Names: Rassias John Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2008(2008)
• Publication Date: 2008-08-04
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2008/136592
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : The functional inequality∥ f ( x ) + 2 f ( y ) + 2 f ( z ) ∥ ≤ ∥ 2 f ( x / 2 + y + z ) ∥ + ϕ ( x, y, z ) ( x, y, z ∈ G ) is investigated, whereG is a group divisible by2, f : G → X andϕ : G 3 → [ 0, ∞ ) are mappings, andX is a Banach space. The main result of the paper states that the assumptions above together with (1)ϕ ( 2 x, − x, 0 ) = 0 = ϕ ( 0, x, − x ) ( x ∈ G ) and (2)lim n → ∞ ( 1 / 2 n ) ϕ ( 2 n + 1 x, 2 n y, 2 n z ) = 0, orlim n → ∞ 2 n ϕ ( x / 2 n − 1, y / 2 n, z / 2 n ) = 0 ( x, y, z ∈ G ), imply thatf is additive. In addition, some stability theorems are proved.