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