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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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