Stability of a 3-dimensional quadratic-additive type functional equation (Q2788841)

Let \(V\) and \(W\) be two real vector spaces. It is shown that a function \(f: V \to W\) satisfies the functional equation NEWLINE\[NEWLINE\begin{multlined} Df(x,y,z) := f(-x -y -z) - f(x + z) - f(y + z) - f(z + x) \\ + 2f(x) + 2f(y) + 2f(z) - f(-x) - f(-y) - f(-z) = 0 \end{multlined}\tag{1}NEWLINE\]NEWLINE if and only if it is the sum of a quadratic function and an additive function.NEWLINENEWLINEThree stability theorems for the functional equation (1) are proved. One of them is the following.NEWLINENEWLINELet \(V\) be a real vector space, and let \(Y\) be a real Banach space. If \(\varphi: (V \setminus \{0 \})^3 \to[0, \infty)\) is a function such that NEWLINE\[NEWLINE \Phi(x,y,z) := \sum_{i=0}^{\infty} 2^{-i}\varphi(2^ix,2^iy,2^iz) < \infty NEWLINE\]NEWLINE for all \(x, y, z \in V \setminus \{0 \}\) and a mapping \(f: V \to Y\) satisfies \(f(0) = 0\) as well as NEWLINE\[NEWLINE \|Df(x,y,z) \| \leq \varphi(x,y,z) NEWLINE\]NEWLINE for all \(x, y, z \in V \setminus \{0 \}\), then there exists a unique function \(F: V \to Y\) fulfilling (1) for which NEWLINE\[NEWLINE \|f(x) - F(x)\| \leq \frac{3}{8} \left( \Phi(x,x,-x) + \Phi(x,-x,-x) \right) NEWLINE\]NEWLINE for all \(x \in V \setminus \{0 \}\).