\(n\)-Jordan \(\ast\)-derivations on induced fuzzy \(C^*\)-algebras (Q2794919)

Let \((X, N)\) be an induced fuzzy \({C}^{\ast}\)-algebra and \((Z, N)\) be a fuzzy normed space. A \(\mathbb{C}\)-linear mapping \(D : (X, N) \to (X, N)\) is called a fuzzy \(n\)-Jordan \(\ast\)-derivation if NEWLINE\[NEWLINED(x^n) = D(x) x^{n-1} + x D(x) x^{n-2} + \cdots + x^{n-2} D(x) x + x^{n-1} D(x) , \quad D(x^{\ast} ) = D(x)^{\ast}NEWLINE\]NEWLINE for all \(x \in X\). The authors consider the functional equation NEWLINE\[NEWLINEf(y-x)+f(x-z)+f(3x-y+z) = f(3x), NEWLINE\]NEWLINE where \(f : X \to Z\) is a unknown function. First, they show that \(f\) is an additive function which is an easy exercise. Then, using a fixed point method, they prove a theorem concerning the Hyers-Ulam stability of \(n\)-Jordan \(\ast\)-derivations on induced fuzzy \({C}^{\ast}\)-algebras associated with the above functional equation.