Abstract
Stability of functional equations has recent applications in many fields. We show that the stability results obtained by J. Brzdęk and concerning the functional equation of the p-Wright affine function:f(px1+(1−p)x2)+f((1−p)x1+px2)=f(x1)+f(x2),can be proved also in (2, α)-Banach spaces, for some real number α∈(0,1). This is done using some fixed-point theorem.
Highlights
The rabid development of the theory of functional equations has been strongly promoted by its applications in various fields, e.g., networks and communication
We investigate the stability of the functional equation of the p-Wright affine functions investigated in [3] but in (2,α)-Banach spaces
The article is organized as follows: in the “Preliminaries” section, we recall some definitions and the functional equation of our interest; the “Fixed-point theorem” section introduces the fixed-point theorem used in the stability; in the “Stability” section, we investigate the stability of the functional equation of the p-Wright affine functions; the “An observation on superstability” section introduces a simple observation on superstability; and the “Conclusion” section concludes our work
Summary
The rabid development of the theory of functional equations has been strongly promoted by its applications in various fields, e.g., networks and communication (see, e.g., [4, 16, 17, 21, 31]). We investigate the stability of the functional equation of the p-Wright affine functions investigated in [3] but in (2,α)-Banach spaces. A generalized version of a linear 2-normed spaces is the (2, α)-normed space defined in the following manner: Definition 2 Let α be a fixed real number with 0 < α ≤ 1, and let X be a linear space over K with dim X > 1. Definition 3 A sequence (xn)n∈N of elements of a linear (2, α)-normed space X is called a Cauchy sequence if there are linearly independent y, z ∈ X such that lim n,m→∞. The main tool used in this article is the following fixed-point theorem. Fixed-point theorem Let us introduce the following three assumptions:. Suppose that M0 ∈ (0, ∞) and G1 : X → Y is a solution to (1) with g(x1) − G1(x1), y α ≤ M0 x1, y α, x1 ∈ E, y ∈ Y0
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