Inclusions in a Single Variable in Ultrametric Spaces and Hyers-Ulam Stability

Magdalena Piszczek

doi:10.1155/2013/129637

Abstract

We present some properties of set-valued inclusions in a single variable in ultrametric spaces. As a consequence, we obtain stability results for the corresponding functional equations.

Highlights

A metric space (X, d) is called an ultrametric space, if d, called an ultrametric, satisfies the strong triangle inequality d (x, z) ≤ max {d (x, y), d (y, z)} for x, y, z ∈ X. (1)One of the typical ultrametrics is a p-adic metric
We present some properties of set-valued inclusions in a single variable in ultrametric spaces
Where r is the largest nonnegative integer such that pr divides m−n. This example is the introduction to the p-adic numbers which play the essential role because of their connections with some problem coming from quantum physics, p-adic string or superstring

Summary

Introduction

A metric space (X, d) is called an ultrametric space (or non-Archimedean metric space), if d, called an ultrametric, satisfies the strong triangle inequality d (x, z) ≤ max {d (x, y) , d (y, z)} for x, y, z ∈ X. (1). Where r is the largest nonnegative integer such that pr divides m−n. This example is the introduction to the p-adic numbers which play the essential role because of their connections with some problem coming from quantum physics, p-adic string or superstring (see [1]). Let (X, d) be an ultrametric space. It is easy to see that (b cl(X), h) is an ultrametric space, that is, h satisfies the strong triangle inequality h (A, C) ≤ max {h (A, B) , h (B, C)} for A, B, C ∈ b cl (X) . Some results of the stability of functional equations in non-Archimedean spaces can be found in [3–8]

Main Results

Stability Results

Then there exists a unique function g such that