The Scientific World Journal

Volume 2013 (2013), Article ID 129637, 5 pages

http://dx.doi.org/10.1155/2013/129637

Research Article

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

Magdalena Piszczek

Pedagogical University, Podchorążych 2, 30-084 Kraków, Poland

Received 7 August 2013; Accepted 2 September 2013

Academic Editors: G. Dai and F. J. Garcia-Pacheco

Copyright © 2013 Magdalena Piszczek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

A metric space is called an ultrametric space (or non-Archimedean metric space), if , called an ultrametric, satisfies the strong triangle inequality

One of the typical ultrametrics is a -adic metric. Let be a fixed prime. For , we define

where is the largest nonnegative integer such that divides . This example is the introduction to the -adic numbers which play the essential role because of their connections with some problem coming from quantum physics, -adic string or superstring (see [1]).

The inequality

yields and implies the following lemma.

Lemma 1. A sequence in an ultrametric space is a Cauchy sequence if and only if .

Let be an ultrametric space. The number is called the diameter of . We will denote by the family of all nonempty subsets of . Moreover, let stand for the family of all bounded sets of let and denote the family of all closed sets of . We understand the convergence of sets with respect to the Hausdorff metric derived from the metric . It is easy to see that is also an ultrametric space, that is, satisfies the strong triangle inequality

We say that is a complete ultrametric commutative groupoid with 0, if is a commutative groupoid with a neutral element 0, is a complete ultrametric space and the operation + is continuous with respect to the metric .

From now on, we assume that is a nonempty set and is a complete ultrametric commutative groupoid with 0. For , we define

The aim of the paper is to obtain some results concerning the following inclusion:

where , , and and its generalization in an ultrametric space. In ultrametric spaces, it is possible to get better estimation with weaker assumptions, than in metric spaces. The ideas of proofs are based on the ideas in [2]. As a consequence we obtain stability results for the corresponding functional equation and its generalization. Some results of the stability of functional equations in non-Archimedean spaces can be found in [3–8].

2. Main Results

Theorem 2.  Let , for all , , , , and

Then there exists a unique function such that and

Proof. Let us fix . Replacing by in (8), we get

and as , we have

for . Thus,

for all . According to Lemma 1 and , the sequence is a Cauchy sequence. Since is complete, there exists the limit . Moreover,

and the right side of the last inequality converges to 0 with . Therefore,

is a singleton and as is continuous,

so . Notice that

Consequently,

where is a closed unit ball.

It remains to prove the uniqueness of . Let be such that , , , for . By induction we get and for . Hence,

Since , we have .

Theorem 3. Assume that , for all , , , , are such that

for , ,

Then there exists a unique function such that and for , where

Proof. Let us fix . Replacing by , , in (21), we obtain

Since , we have for . Hence,

We define a sequence by the following recurrence relation:

It is easy to see that

for . In virtue of (20), the sequence is a Cauchy sequence. As is a complete metric space, there exists the limit . Moreover,

and the right side of the last inequality converges to 0 as . Therefore,

is a singleton, and as satisfies (19),

By (26), we get

which, with , yields

It remains to prove the uniqueness of . Suppose that are such that , , , and . Replacing by , , in the penultimate equality, we get

Thus,

and we get a constant sequence

In the same way, we get a constant sequence

Hence,

Using the definition of , we get

It follows that with , and the proof is completed.

3. Stability Results

We present the applications of the above theorems to the results concerning the stability of functional equations.

Corollary 4.  Let , , , , satisfy

Then there exists a unique function such that and

Proof. Let for . Then

where is a closed unit ball and

According to Theorem 2 there exists a unique function such that and for .

Corollary 5. Let , , , , , satisfy (19) and

Then there exists a unique function such that

Proof. Let for . Then

where is a closed unit ball and

By Theorem 3, we get the assertion.

As it was observed in [9, 10] from the stability results concerning the equation , we can easily derive the stability of functional equations in several variables, for example, the Cauchy equation, the Jensen equation, or the quadratic equation. The equation is a generalization of the gamma-type equations or the linear equations (see [11, 12]).

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