Abstract
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.
Highlights
Let X, Y be linear spaces over the field F ∈ {R, C}
In the paper we prove, applying the fixed point approach, criteria for the θ-hyperstability of (1.1) under some natural assumptions on θ
We show how our outcomes may be used to check whether the particular functional equation is θ-hyperstable
Summary
The stability and hyperstability of the particular cases of the functional equation (1.1), among others those mentioned above were studied by many. We say that the equation (1.1) is θ-hyperstable in the class of functions g : X → Y (with a control function θ : (X\{0})n → Y ), if g : X → Y satisfying the inequality. In the paper we prove, applying the fixed point approach, criteria for the θ-hyperstability of (1.1) under some natural assumptions on θ. In this way we obtain sufficient conditions for the θ-hyperstability of a wide class of functional equations and control functions θ. We show how our outcomes may be used to check whether the particular functional equation is θ-hyperstable. The equation of the p-Wright affine function will be called shortly the p-Wright equation
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