A functional equation originating from quadratic forms

Abstract

In this paper, we obtain the general solution and the stability of the 2-variable quadratic functional equationThe quadratic form f (x, y) = ax 2 + bxy + cy 2 is a solution of the above functional equation.

2 - Variable AQCQ - Functional equation

“…f (x + 2y) + f (x − 2y) = 4f (x + y) + 4f (x − y) − 6f (x) + f (2y) + f (−2y) − 4f (y) − 4f (−y) (4) in paranormed spaces. In this paper, the authors obtain the general solution and generalized Ulam -Hyers stability of a 2 -variable AQCQ functional equation g(x + 2y, u + 2v) + g(x − 2y, u − 2v) = 4[g(x + y, u + v) + g(x − y, u − v)] − 6g(x, u) + g(2y, 2v) + g(−2y, −2v) − 4g(y, v) − 4g(−y, −v)…”

Quadratic Functional Equations Associated With Borel Functions and Module Actions

“…The result also holds when X and Y be arbitrary real or complex vector spaces. For a mapping f : X × X → Y , consider the 2-dimensional quadratic functional equation: (2) f (x + y, z + w) + f (x − y, z − w) = 2f (x, z) + 2f (y, w).…”

“…The quadratic form f : R × R → R given by f (x, y) := ax 2 + bxy + cy 2 is a solution of the equation (2). In 2007, The authors [2] acquired the general solution and proved the stability of the 2-dimensional quadratic functional equation (2) for the case that X and Y be real vector spaces as follows.…”

“…In 2007, The authors [2] acquired the general solution and proved the stability of the 2-dimensional quadratic functional equation (2) for the case that X and Y be real vector spaces as follows. for all t ∈ R. By the assumption that f (tx, ty) is continuous in t ∈ R for each fixed x, y ∈ A M, the function ψ n is continuous for all n ∈ N. Note that…”

“…0 ) for all n ∈ N and all t ∈ R. By [2], the sequence ψ n (t) is a Cauchy sequence for all t ∈ R. Define a function ψ : R → R by ψ(t) := lim n→∞ ψ n (t) for all t ∈ R. Note that…”