Comments on the core of the direct method for proving Hyers–Ulam stability of functional equations

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Hyperstability of the Cauchy equation on restricted domains (2013, background): “2 corresponds to some observations in [18,19,40], but actually is based on a fixed point theorem in [7, Theorem 1] (see also [8, Theorem 2]).”
On Ulam’s Type Stability of the Linear Equation and Related Issues (2014, background): “Let us yet mention that the equation plays a significant role in the investigations of stability of the functional equations in several variables; for suitable examples, we refer the reader, for example, to [8, 27–31].”
The Generalized Hyers--Ulam--Rassias Stability of a Cubic Functional Equation (2007, background): “In recent decades, the stability problems of several functional equations have been extensively investigated by a number of authors [2, 5, 9, 13].”
On a direct method for proving the Hyers–Ulam stability of functional equations (2010, background): “1, we have a special case of the considerations contained in [5].”
Elementary remarks on Ulam–Hyers stability of linear functional equations (2007, methodology): “The conditions giving stability (see [10]) are the convergence of the series ∞ ∑ |h|j δ̄(gj z) j=0”