Showing papers by "Prasanna K. Sahoo published in 2004"

Superstability of the generalized orthogonality equation on restricted domains

Abstract: Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation ¦〈f(x), f(y)〉¦ = ¦〈x,y〉¦. In this paper, we will extend the result of Chmielinski by proving a theorem: LetDn be a suitable subset of ℝn. If a function f:Dn → ℝn satisfies the inequality ∥〈f(x), f(y)〉¦ ¦〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ Dn, thenf satisfies the generalized orthogonality equation for anyx, y ∈ Dn.

Stability of a Functional Equation that Arises in the Theory of Conditionally Specified Distributions

Abstract: The functional equation arises in the theory of conditionally specified distributions In this paper, we investigate the Hyers-Ulam stability of this functional equation

On quadratic differences that depend on the product of arguments - revisited

Abstract: Let $$ \mathbb{K} $$ be a field of real or complex numbers and $$ \mathbb{K}_0 $$ denote the set of nonzero elements of $$ \mathbb{K} $$ . Let $$ \mathbb{G} $$ be an abelian group. In this paper, we solve the functional equation f 1 (x + y) + f 2 (x - y) = f 3 (x) + f 4 (y) + g(xy) by modifying the domain of the unknown functions f 3, f 4, and g from $$ \mathbb{K} $$ to $$ \mathbb{K}_0 $$ and using a method different from [3]. Using this result, we determine all functions f defined on $$ \mathbb{K}_0 $$ and taking values on $$ \mathbb{G} $$ such that the difference f(x + y) + f (x - y) - 2 f(x) - 2 f(y) depends only on the product xy for all x and y in $$ \mathbb{K}_0 $$