International Journal of Research and Scientific Innovation (IJRSI) | Volume V, Issue VII, July 2018 | ISSN 2321–2705
Stability of a Quadratic-Additive Functional Equation: A Fixed Point Approach
Rajkumar.V
Hindusthan Institute of Technology, Coimbatore, Tamil Nadu, India
Abstract: In this paper, we investigate the stability of a functional equation
f(x+y+z)+f(x-y)+f(y-z)+f(z-x) = 2f(x)+2f(y)+2f(z)+f(-x)+f(-y)+f(-z)
by using the fixed point theory in the sense of L. Cadariu and V. Radu.
Key words and phrases: Hyers-Ulam stability, fixed point method, a mixed type functional equation.
I. INTRODUCTION
In d(f(xy), f(x)f(y)) < δn 1940, S. M. Ulam [14] raised a question concerning the stability of homomorphisms: Given a group G1, a metric group G2 with the metric d(.,.), and a positive number ε, does there exist a δ> 0 such that if a mapping f : G1→G2 satisfies the inequality
for all x, y є G1 then there exists a homomorphism F : G1→ G2 with
d(f(x), F(x)) <ε
for all x є G1?
When this problem has a solution, we say that the homomorphisms from G1 to G2 are stable. In the next year, D. H. Hyers[6] gave a partial solution of Ulam’s problem for the case of approximate additive mappings under the assumption that G1 and G2 are Banach spaces. Hyers’ result was generalized by T. Aoki [1] for additive mappings, and by Th. M. Rassias [12] for linear mappings, to considering the stability problem with unbounded Cauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians, see [5], [8]-[10].
Almost all subsequent proofs, in this very active area, have used Hyers’ method. Namely, the function F, which is the solution of a functional equation, is explicitly constructed, starting from the given function f, by the formulae