General Solutions of Two Quadratic Functional Equations of Pexider Type on Orthogonal Vectors

Abstract

Based on the studies on the Hyers-Ulam stability and the orthogonal stability of some Pexider-quadratic functional equations, in this paper we find the general solutions of two quadratic functional equations of Pexider type. Both equations are studied in restricted domains: the first equation is studied on the restricted domain of the orthogonal vectors in the sense of Rätz, and the second equation is considered on the orthogonal vectors in the inner product spaces with the usual orthogonality.

Orthogonalities and functional equations

“…We cite here the main theorems presented then. In [76], Fochi was looking for the solutions of some pexiderized forms of an orthogonally quadratic equation, namely f (x + y) + f (x − y) = 2g(x) + 2h(y) for all x, y ∈ X with x ⊥ y (2.9) and…”

Approximate Euler-Lagrange Quadratic Mappings in Fuzzy Banach Spaces

“…for all , ∈ . It easily follows from (14) that is additive; that is, ( + ) = ( ) + ( ) for all , ∈ . Since is a rational number, ( ) = ( ) for all ∈ .…”

“…in [8][9][10]. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [11][12][13][14]). The theory of fuzzy space has much progressed as the theory of randomness has developed.…”