Abstract
A symmetric functional equation is one whose form is the same regardless of the order of the arguments. A remarkable example is the Cauchy functional equation: f ( x + y ) = f ( x ) + f ( y ) . Interesting results in the study of the rigidity of quasi-isometries for symmetric spaces were obtained by B. Kleiner and B. Leeb, using the Hyers-Ulam stability of a Cauchy equation. In this paper, some results on the Ulam’s type stability of the Cauchy functional equation are provided by extending the traditional norm estimations to ther measurements called generalized norm of convex type (v-norm) and generalized norm of subadditive type (s-norm).
Highlights
Ulam [1] posed the following problem concerning group homomorphism when presenting a talk at the University of Wisconsin: Let ( G, ◦) be a group, ( G2, ∗) be a metric group with the metric d(·, ·) and ε > 0
The first affirmative answer to this problem was the one provided by Hyers [2], who solved the problem for additive mappings in Banach spaces
A further generalization was obtained by Găvruţa [4], where he introduced the concept of generalized Hyers–Ulam–Rassias stability in the spirit of Rassias’s approach
Summary
Ulam [1] posed the following problem concerning group homomorphism when presenting a talk at the University of Wisconsin: Let ( G, ◦) be a group, ( G2 , ∗) be a metric group with the metric d(·, ·) and ε > 0. In [13], we proved a fixed-point theorem for a class of operators with suitable properties in very general conditions and some corollaries, which showed that our main result is a useful tool for proving properties of generalized Hyers–Ulam stability for some functional equations in a single variable. A mapping k · kv : X → R+ = [0, ∞) is called a generalized norm of convex type or a v-norm if (V2)
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