On approximate homomorphisms: a fixed point approach

• Published in: Mathematical sciences (Karaj, Iran) Vol. 6; no. 1; pp. 59 - 8
• Main Authors: Gordji, Madjid Eshaghi, Alizadeh, Zahra, Khodaei, Hamid, Park, Choonkil
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2012; Springer Nature B.V; BioMed Central Ltd
• Subjects: Algebra; Applications of Mathematics; Approximation; Classification; Exact solutions; Fixed points (mathematics); Functions (mathematics); Homomorphisms; Mathematical analysis; Mathematics; Mathematics and Statistics; Original Research; Stability; Fixed point approach; Additive; Approximate homomorphism; Banach algebra; Quadratic; Cubic and quartic functional equation
• ISSN: 2008-1359; 2251-7456; 2251-7456
• DOI: 10.1186/2251-7456-6-59

Consider the functional equation ℑ 1 ( f ) = ℑ 2 ( f ) ( ℑ )in a certain general setting. A function g is an approximate solution of ( ℑ )if ℑ 1 ( g )and ℑ 2 ( g )are close in some sense. The Ulam stability problem asks whether or not there is a true solution of ( ℑ )near g . A functional equation is superstable if every approximate solution of the functional equation is an exact solution of it. In this paper, for each m = 1,2,3,4, we will find out the general solution of the functional equation f ( ax + y ) + f ( ax - y ) = a m - 2 [ f ( x + y ) + f ( x - y ) ] + 2 ( a 2 - 1 ) [ a m - 2 f ( x ) + ( m - 2 ) ( 1 - ( m - 2 ) 2 ) 6 f ( y ) ] for any fixed integer a with a ≠ 0, ± 1. Using a fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in real Banach algebras for this functional equation. Moreover, we establish the superstability of this functional equation by suitable control functions. 2010 Mathematics Subject Classification 39B52, 47H10, 39B82