On the hyperstability of a Cauchy-Jensen type functional equation in Banach spaces (Q2812614)

Let \(X\) be a normed space and \(Y\) be a Banach space. The following two results are achieved.NEWLINENEWLINEIf \(f: X \to Y\) satisfies the condition NEWLINE\[NEWLINE \|f \left( \frac{x + y}{2} + z \right) + f \left( \frac{x - y}{2} + z \right) - 2f(z) - f(x) \| \leq c \|x\|^p \|y\|^q \|z\|^r, \quad x, y, z \in X \setminus \{0\}, NEWLINE\]NEWLINE where \(c \geq 0\) and \(p, q, r \in \mathbb{R}\) with \(p + q + r \notin \{0,1 \}\), then NEWLINE\[NEWLINEf \left( \frac{x + y}{2} + z \right) + f \left( \frac{x - y}{2} + z \right) = 2f(z) + f(x)\tag{a}NEWLINE\]NEWLINE on \(X \setminus \{0 \}\) (Theorems 2.1--2.3).NEWLINENEWLINEIf \(f: X \to Y\) satisfies the condition NEWLINE\[NEWLINE \|f \left( \frac{x + y}{2} + z \right) + f \left( \frac{x - y}{2} + z \right) - 2f(z) - f(x) \| \leq c(\|x\|^p + \|y\|^p + \|z\|^p), \;x, y, z \in X \setminus \{0\}, NEWLINE\]NEWLINE where \(c \geq 0\) and \(p< 0\), then \(f\) is a solution of (a) (Theorem 2.4).