Abstract
We investigate the superstability of generalized derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy functional equation.
1. Introduction
A functional equation is superstable if every approximately solution of is an exact solution of it.
The stability of functional equations was first introduced by Ulam [1] during his talk before a Mathematical Colloquium at the University of Wisconsin in 1940.
Given a metric group , a number , and a mapping which satisfies the inequality for all in , does there exist an automorphism of and a constant , depending only on such that for all
If the answer is affirmative, we would call the equation of automorphism is stable. In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Rassias [3] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences . In 1991, Gajda [4] answered the question for the case , which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias or generalized Hyers-Ulam stability of functional equations [5, 6].
In 1992, GΔvruΕ£a [7] generalized the Th. M. Rassias Theorem as follows.
Suppose that is an ablian group, is a Banach space which satisfies for all . If is a mapping with for all , then there exists a unique mapping such that and for all .
In 1949, Bourgin [8] proved the following result, which is sometimes called the superstability of ring homomorphisms: suppose that and are Banach algebras with unit. If is a surjective mapping such that for some and for all , then is a ring homomorphism.
Badora [9] and Miura et al. [10] proved the Ulam-Hyers stability and the Isac and Rassias-type stability of derivations [11] (see also [12, 13]); Savadkouhi et al. [14] have contributed works regarding the stability of ternary Jordan derivations. Jung and Chang [15] investigated the stability and superstability of higher derivations on rings. Recently, Ansari-Piri and Anjidani [16] discussed the superstability of generalized derivations on Banach algebras. In this paper, we investigate the superstability of generalized derivations on non-Archimedean Banach algebras by using the fixed point methods.
2. Preliminaries
In 1897, Hensel [17] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications [18, 19].
A non-Archimedean field is a field equipped with a function (valuation) from into such that if and only if , and for all (see [20, 21]).
Definition 2.1. Let be a vector space over a scalar field with a non-Archimedean nontrivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions:(NA1) if and only if ,(NA2) for all and ,(NA3) for all (the strong triangle inequality).
A sequence in a non-Archimedean space is Cauchy if and only if converges to zero. By a complete non-Archimedean space, we mean one in which every Cauchy sequence is convergent. A non-Archimedean normed algebra is a non-Archimedean normed space with a linear associative multiplication, satisfying for all . A non-Archimedean complete normed algebra is called a non-Archimedean Banach algebra (see [22]).
Example 2.2. Let be a prime number. For any nonzero rational number such that and are integers not divisible by , define the βadic absolute value . Then, is a non-Archimedean norm on . The completion of with respect to is denoted by which is called the -adic number field.
Definition 2.3. Let be a nonempty set and satisfy the following properties: (D1) if and only if ,(D2) (symmetry),(D3) (strong triangle in equality),for all . Then, is called a non-Archimedean generalized metric space. is called complete if every -Cauchy sequence in is -convergent.
Definition 2.4. Let be a non-Archimedean algebra. An additive mapping is said to be a ring derivation if for all . An additive mapping is said to be a generalized ring derivation if there exists a ring derivation such that
for all .
We need the following fixed point theorem (see [23, 24]).
Theorem 2.5 (non-Archimedean alternative Contraction Principle). Suppose that is a non-Archimedean generalized complete metric space and is a strictly contractive mapping; that is,
for some . If there exists a nonnegative integer such that for some , then the followings are true:
(a) the sequence converges to a fixed point of ,
(b)β is a unique fixed point of in
(c) if , then
3. Non-Archimedean Superstability of Generalized Derivations
Hereafter, we will assume that is a non-Archimedean Banach algebra with unit over a non-Archimedean field .
Theorem 3.1. Let be a function. Suppose that are mappings such that is additive and for all . If there exists a natural number and , for all . Then, is a generalized ring derivation and is a ring derivation.
Proof. By induction on , we prove that
for all and . Let in (3.1). Then,
This proves (3.4) for . Let (3.4) holds for . Replacing by and by in (3.1) for each , and for all , we get
Since
for all , it follows from induction hypothesis and (3.6) that
for all . This proves (3.4) for all . In particular,
for all where
Let be the set of all functions . We define as follows:
It is easy to see that defines a generalized complete metric on . Define by . Then, is strictly contractive on , in fact, if
then by (3.3),
It follows that
Hence, is a strictly contractive mapping with Lipschitz constant . By (3.9),
This means that . By Theoremβ2.5, has a unique fixed point in the set
and for each ,
Therefore,
for all . This shows that is additive.
Replacing by in (3.2) to get
and so
for all and all . By taking , we have
for all .
Fix . By (3.21), we have
for all . Then, for all and each , and so by taking , we have . Now, we obtain , since is with unit. Replacing by in (3.2), we obtain
and hence,
for all and each . Letting tends to infinite, we have
Now, we show that is a ring derivation. By (3.25), we get
for all . Therefore, we have .
The proof of following theorem is similar to that in Theorem 3.1, hence it is omitted.
Theorem 3.2. Let be a function. Suppose that are mappings such that is additive and for all . If there exists a natural number and , for all . Then, is a generalized ring derivation and is a ring derivation.
The following results are immediate corollaries of Theorems 3.1 and 3.2 and Exampleβ2.3.
Corollary 3.3. Let be a non-Archimedean Banach algebra over , , and . Suppose that are mappings such that is additive and for all . Then, is a generalized ring derivation and is a ring derivation.
Corollary 3.4. Let be a non-Archimedean Banach algebra over , and . Suppose that are mappings such that is additive and for all . Then, is a generalized ring derivation and is a ring derivation.