A Common Generalization of Functional Equations Characterizing Normed and Quasi-Inner-Product Spaces

Abstract

We determine the general solutions of the functional equation for ƒi: G → F (i = 1,2,3,4), where G is a 2-divisible group and F is a commutative field of characteristic different from 2. The motivation for studying this equation came from a result due to Dry gas [4] where he proved a Jordan and von Neumann type characterization theorem for quasi-inner products. Also, this equation is a generalization of the quadratic functional equation investigated by several authors in connection with inner product spaces and their generalizations. Special cases of this equation include the Cauchy equation, the Jensen equation, the Pexider equation and many more. Here, we determine the general solution of this equation without any regularity assumptions on ƒi.

Remarks on the stability of Drygas’ equation and the Pexider-quadratic equation

“…for all x, y ∈ G. Theorem 4 and Corollary 6 in [1] (or Theorem 6 in [13]) imply the following directly. …”

On a Quadratic-Trigonometric Functional Equation and Some Applications

“…Lemma 2.7 [1,5]. The general solution f, g : G -» C of (2.34) f(xy) + f(xy~x) = 2fax) + g(y) (x, y £ G) with f satisfying (FC) is given by…”

On a Method of Solving Some Functional Equations for Set-Valued Functions

“…Equation (5) is now known in the literature as the Drygas equation. For solving the equation for single-valued functions from an abelian group into a uniquely 2-divisible abelian group one may consult, e.g., [5].…”