Abstract
Cădariu and Radu applied the fixed point theorem to prove the stability theorem of Cauchy and Jensen functional equations. In this paper, we prove the generalized Hyers-Ulam stability via the fixed point method and investigate new theorems via direct method concerning the stability of a general quadratic functional equation.
Highlights
In 1940, Ulam 1 gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems
Given > 0, does there exist a δ > 0 such that if f : G → G satisfies ρ f xy, f x f y < δ for all x, y ∈ G, a homomorphism h : G → G exists with ρ f x, h x < for all x ∈ G?
We say that a functional equation E1 f E2 f is stable if any mapping g approximately satisfying the equation d E1 g, E2 g ≤ φ x is near to a true solution f such that E1 f E2 f and d f x, g x ≤ Φ x for some function Φ depending on the given function φ
Summary
In 1940, Ulam 1 gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. In 1941, the first result concerning the stability of functional equations for the case where G1 and G2 are Banach spaces was presented by Hyers 2 .
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