Applications of fixed point theorems to the Hyers-Ulam stability of functional equations–a survey

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Hyperstability of the Cauchy equation on restricted domains (2013, methodology): “The fixed point results have been already used for proving stability of functional equation for instance in [2,9,28–30,34,35] and our method can be considered to be an extension of those investigations (for a survey on this subject see [10]).”
On Ulam's type stability for a class of impulsive fractional differential equations with nonlinear integral boundary conditions (2017, background): “Further, about the stability of functional equations, ordinary differential equations and FDEs for some recent work, we recommend the readers to study [3, 3, 14, 30, 36].”
Hyers–Ulam stability of delay differential equations of first order (2016, background): “For some papers concerning the connection between Hyers–Ulam stability and fixed point theorem, we refer the interested readers to [6], [15], [25], [27], and [33].”
Recent developments of the conditional stability of the homomorphism equation (2015, methodology): “The foundation of the third method are fixed point techniques (see [28, 31, 35]) and the fourth technique uses the invariant means (see [113, 114]).”