Generalized Hyers-Ulam stability of a 3-dimensional quadratic functional equation

Abstract

In this paper, we investigate the stability of a functional equationby applying the direct method in the sense of Hyers. Mathematics Subject Classification: 39B82, 39B52

Stability of a functional equation related to quadratic mappings

“…is called a quadratic functional equation, and every solution of a quadratic functional equation is said to be a quadratic mapping. F. Skof [9], P. W. Cholewa [1], and S. Czerwik [2] proved the stability of quadratic functional equations, and the authors have investigated the stability problems of quadratic functional equations, see [3] and [4]. Moreover, Najati and Jung [8] have observed the Hyers-Ulam stability of a generalized quadratic functional equation f (ax + by) + abf (x − y) = af (x) + bf (y) (1.1) where a, b are non-zero rational numbers with a + b = 1.…”

Generalized Hyers-Ulam stability of a 3-dimensional quadratic functional equation in modular spaces

“…is called a quadratic mapping ( [2,13]). A functional equation is called a quadratic functional equation if every solution of that equation is a quadratic mapping and any quadratic mapping is a solution of the equation( [5,6,7]).…”

“…for all nonzero real numbers k, i.e., f (0) = 0 when p = 0. If we put ϕ(x, y, z) := θ( x p + y p + z p ) for all x, y, z ∈ X\{0}, then ϕ satisfies (5). Therefore, by Theorems 2.3, there exists a unique quadratic mapping F satisfying the inequality (10) for all x ∈ X\{0}.…”