A conditional exponential functional equation and its stability (Q847343)

The authors study the conditional functional equation \[ \gamma(x+y)=\gamma(x-y) \implies f(x+y)=f(x)f(y), \leqno(*) \] where \(f:X \to Y\), \(X\) real linear space with dim\(X \geq 2\) and \(Y\) is a semigroup with neutral element, \(\gamma:X \to Z\) (\(Z\) is a given non empty set) is a given even function satisfying a series of conditions generalizing various well-known orthogonality notions. In Section 2 of the paper a description of the solutions of equation (\(*\)) is provided. Then the stability of (\(*\)) is studied, namely the functions \(f:X \to \mathbb K\) (\(\mathbb K \in \{\mathbb R, \mathbb C\}\)) satisfying the two inequalities \[ [\gamma(x+y)=\gamma(x-y), f(x)f(y) \neq 0] \implies \left|\frac{f(x+y)}{f(x)f(y)}-1\right| \leq \varepsilon \] and \[ [\gamma(x+y)=\gamma(x-y), f(x+y) \neq 0] \implies \left|\frac{f(x)f(y)}{f(x+y)}-1\right| \leq \varepsilon \] are investigated. In the last section the stability of the following pexiderized functional equation \[ \gamma(x+y)=\gamma(x-y) \implies f(x+y)=f_1(x)f_2(y) \] is considered.