A fixed point approach to stability of functional equations

Stability of a generalization of the Fréchet functional equation

“…We need the subsequent fixed point theorem proved for function spaces in [6]; it will be the main tool in the proof of our main theorem (R + stands for the set of nonnegative reals and A B denotes the family of all functions mapping a set B = ∅ into a set A = ∅). For related outcomes we refer to [7,9]; a similar approach to stability of functional equations has been already applied in [5,24].…”

Hyperstability of general linear functional equation

“…Note that the operators T k and Λ k satisfy the assumptions of Theorem 1 in [5]. Applying this version of the fixed point theorem we obtain that there exists a unique fixed point…”

Hyperstability of a quadratic functional equation on abelian group and inner product spaces

“…Let R + := [0, ∞) be the set of nonnegative real numbers and Y X denotes the family of all mappings from a nonempty set X into a nonempty set Y . The proof's method of the main results is based on a fixed point theorem in [10,Theorem 1]. Our method can be considered to be an extension of the investigations in [2-4, 18, 21].…”

“…Our method can be considered to be an extension of the investigations in [2-4, 18, 21]. Now, we will take the following three hypotheses (all notations come from [10]). …”