A fixed point theorem and the Hyers-Ulam stability in Riesz spaces

Bogdan Batko,Janusz Brzdęk

doi:10.1186/1687-1847-2013-138

Bogdan Batko, Janusz Brzdęk

Open Access

https://doi.org/10.1186/1687-1847-2013-138

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Abstract

We prove a fixed point theorem and show its applications in investigations of the Hyers-Ulam type stability of some functional equations (in single and many variables) in Riesz spaces.

Highlights

The Hyers-Ulam stability for functional, and for difference, differential and integral equations, is a very quickly growing area of investigations
It is related to the notions of shadowing as well as to the theories of perturbation and optimization
The first known result on such stability is due to Pólya and Szegö [ ] and reads as follows

Summary

Introduction

The Hyers-Ulam stability for functional, and for difference, differential and integral equations, is a very quickly growing area of investigations (for more details and further references, see, e.g., [ – ]; examples of some recent results can be found in [ – ]). The Hyers-Ulam stability in Riesz spaces have already been studied in [ ] (with a direct method) and in [ ] (with an application of the spectral representation theorem). The main motivation for this kind of investigations follows from the pretty natural concept to pose the stability problem for a given functional equation in the settings of an ordered structure as an alternative for the topological or metric ones. ]) that in a Riesz space L that is Archimedean, the euniform limit of a sequence in it, if exists, is unique and the fact that {fn}n∈N converges e-uniformly to f will be denoted by limen→∞ fn = f (in particular, if v ∈ L and fn ≥ v for n ∈ N, f ≥ v)

Riesz space

First we show by induction that

This means that