We prove, using the fixed point approach, some stability results for the general linear functional equation. Namely we obtain sufficient conditions for the stability of a wide class of functional equations and control functions. Our results generalize a lot of the well known and recent outcomes concerning stability. In some examples we indicate how our method may be used to check if the particular functional equation is stable and we discuss the optimality of obtained bounding constants.

Our purpose is to investigate criteria for hyperstability of linear type functional equations. We prove that a function satisfying the equation approximately in some sense, must be a solution of it. We give some conditions on coefficients of the functional equation and a control function which guarantee hyperstability. Moreover, we show how our outcomes may be used to check whether the particular functional equation is hyperstable. Some relevant examples of applications are presented.

We prove some stability and hyperstability results for the well-known Fréchet equation stemming from one of the characterizations of the inner product spaces. As the main tool, we use a fixed point theorem for the function spaces. We finish the paper with some new inequalities characterizing the inner product spaces.

Let A be a subgroup of a commutative group (G, + ) and P be a commutative ring. We give the full description of functions \({g: G \rightarrow P}\) satisfying $$g(x + y) + g(x - y) = 2g(x)g(y) \quad (x, y) \in A \times G. \quad\quad\quad\quad (A)$$ Thus we obtain a family of functions depicting evolutions of quite arbitrary functions \({g_0 : G \to P}\) into cosine functions \({g: G \to P}\) , i.e., solutions of the d’Alembert (cosine) functional equation $$g(x + y) + g(x - y) = 2g(x)g(y) \quad x, y \in G. \quad\quad\quad\quad (B)$$ We also show that every function \({g: G \rightarrow P}\) , fulfilling (A), is a solution of (B) if and only if A = G.