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On approximate homomorphisms: A fixed point approach. (English) Zbl 1271.39023
Summary: Consider the functional equation \((I)~\mathfrak I_1(f) = \mathfrak I_2(f) \) in a certain general setting. A function \(g\) is an approximate solution of \((I)\) if \(\mathfrak I_1(g)\) and \(\mathfrak I_2(g)\) are close in some sense. The Ulam stability problem asks whether or not there is a true solution of \((I)\) near \(g\). A functional equation is superstable if every approximate solution of the functional equation is an exact solution of it. In this paper, for each \(m = 1, 2, 3, 4\), we will find out the general solution of the functional equation \[ \begin{split} f(ax+y)+f(ax-y)\\ =a^{m-2}\left[f(x+y)+f(x-y)\right]+2(a^2-1)\left[a^{m-2}f(x)+\frac{{(m-2)(1-(m-2)^2)}}{6}f(y)\right]\end{split} \] for any fixed integer a with \(a \neq 0, \;\pm 1\). Using a fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in real Banach algebras for this functional equation. Moreover, we establish the superstability of this functional equation by suitable control functions.

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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