Hyers–Ulam stability of a functional equation with several parameters

A fixed point theorem and Ulam stability of a general linear functional equation in random normed spaces

“…Further, in 2003, Radu [82] proposed a new method to retrieve the main result of Rassias [84], based on the fixed point alternative in [34]. The same fixed point method, using also Banach Contraction Principle, has subsequently been used by many other authors to study the stability of a large variety of functional equations (see for example [21,27,33,69,74] and the references therein). A modification of it was proposed in [74,75], where the author tied some set of functions to the given approximate solution of a given functional equation to make it a complete metric space, and then to apply the Banach theorem.…”

“…The same fixed point method, using also Banach Contraction Principle, has subsequently been used by many other authors to study the stability of a large variety of functional equations (see for example [21,27,33,69,74] and the references therein). A modification of it was proposed in [74,75], where the author tied some set of functions to the given approximate solution of a given functional equation to make it a complete metric space, and then to apply the Banach theorem. Many new fixed point theorems have been shown in the literature, to investigate Ulam stability in spaces endowed with some kind of generalized metrics, such as fuzzy metric, quasi-metric, partial metric, G-metric, D-metric, b-metric, 2-metric, ultrametric, modular metric, and dislocated metric; see for instance [5,9,46,56,64].…”