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Hyers-Ulam-Rassias stability of the additive-quadratic mappings in non-Archimedean Banach spaces
Journal of Inequalities and Applications volume 2012, Article number: 174 (2012)
Abstract
Using the fixed point and direct methods, we prove the generalized Hyers-Ulam stability of the following additive-quadratic functional equation in non-Archimedean normed spaces
where r, s, γ are positive real numbers.
MSC:39B55, 46S10.
1 Introduction and preliminaries
A classical question in the theory of functional equations is the following: ‘When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?’ If the problem accepts a solution, we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [44] in 1940. In the next year, Hyers [23] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In 1978, Rassias [39] proved a generalization of Hyers’ theorem for additive mappings. This new concept is known as generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. Furthermore, in 1994, a generalization of Rassias’ theorem was obtained by Gǎvruta [21] by replacing the bound by a general control function .
In 1983, a generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [43] for mappings , where X is a normed space and Y is a Banach space. In 1984, Cholewa [11] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group and, in 2002, Czerwik [13] proved the generalized Hyers-Ulam stability of the quadratic functional equation. The reader is referred to [1–42] and references therein for detailed information on stability of functional equations.
In 1897, Hensel [22] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications (see [16, 25–27, 33]).
Definition 1.1 By a non-Archimedean field, we mean a field equipped with a function (valuation) such that for all , the following conditions hold:
-
(1)
if and only if ;
-
(2)
;
-
(3)
.
Definition 1.2 Let X be a vector space over a scalar field with a non-Archimedean non-trivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions:
-
(1)
if and only if ;
-
(2)
(, );
-
(3)
The strong triangle inequality (ultrametric); namely,
Then is called a non-Archimedean space.
Due to the fact that
Definition 1.3 A sequence is Cauchy if and only if converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.
Definition 1.4 Let X be a set. A function is called a generalized metric on X if d satisfies
-
(1)
if and only if ;
-
(2)
for all ;
-
(3)
for all .
We recall a fundamental result in fixed point theory.
Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either
for all nonnegative integers n or there exists a positive integer such that
-
(1)
, ;
-
(2)
the sequence converges to a fixed point of J;
-
(3)
is the unique fixed point of J in the set ;
-
(4)
for all .
In 1996, G. Isac and Th. M. Rassias [24] were the first to provide applications of stability theory of functional equations for the proof of new fixed-point theorems with applications. By using fixed-point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [14, 15, 35, 36, 40]).
This paper is organized as follows: In Section 2, using the fixed-point method, we prove the Hyers-Ulam stability of the following additive-quadratic functional equation:
where , in non-Archimedean normed space. In Section 3, using direct methods, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (1.1) in non-Archimedean normed spaces.
It is easy to see that a mapping f with is a solution of equation (1.1) if and only if f is of the form for all .
2 Stability of functional equation (1.1): a fixed point method
In this section, we deal with the stability problem for the additive-quadratic functional equation (1.1). In the rest of the present article, let .
Theorem 2.1 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let be a function such that there exists an with
for all . Let be an odd mapping satisfying
for all . Then there exists a unique additive mapping such that
for all .
Proof Putting in (2.2) and replacing x by 2x, we get
for all . Putting and in (2.2), we have
for all . By (2.4) and (2.5), we get
Consider the set and introduce the generalized metric on S:
where, as usual, . It is easy to show that is complete (see [30]). Now we consider the linear mapping such that for all . Let be given such that . Then
for all . Hence,
for all . So implies that . This means that for all .
It follows from (2.6) that . By Theorem 1.5, there exists a mapping satisfying the following:
-
(1)
A is a fixed point of J, i.e.,
(2.7)
for all . The mapping A is a unique fixed point of J in the set . This implies that A is a unique mapping satisfying (2.7) such that there exists a satisfying for all ;
-
(2)
as . This implies the equality
-
(3)
, which implies the inequality . This implies that the inequalities (2.3) holds.
It follows from (2.1) and (2.2) that
for all . So
for all . Hence, satisfying (1.1). This completes the proof. □
Corollary 2.2 Let θ be a positive real number and q is a real number with . Let be an odd mapping satisfying
for all . Then there exists a unique additive mapping such that
for all .
Proof The proof follows from Theorem 2.1 by taking for all . Then we can choose and we get the desired result. □
Theorem 2.3 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let be a function such that there exists an with
for all . Let be an odd mapping satisfying (2.2). Then there exists a unique additive mapping such that
for all .
Proof Let be the generalized metric space defined in the proof of Theorem 2.1.
Now we consider the linear mapping such that
for all .
Replacing x by in (2.6) and using (2.9), we have
So .
The rest of the proof is similar to the proof of Theorem 2.1. □
Corollary 2.4 Let θ be a positive real number and q is a real number with . Let be an odd mapping satisfying (2.8). Then there exists a unique additive mapping such that
for all .
Proof The proof follows from Theorem 2.3 by taking for all . Then we can choose and we get the desired result. □
Theorem 2.5 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let be a function such that there exists an with
for all . Let be an even mapping with and satisfying (2.2). Then there exists a unique quadratic mapping such that
for all .
Proof Consider the set and the generalized metric in defined by
where, as usual, . It is easy to show that is complete (see [30]). Now we consider the linear mapping such that
for all .
Putting and in (2.2), we have
for all .
Substituting and then replacing x by 2x in (2.2), we obtain
By (2.13) and (2.14), we get
The rest of the proof is similar to the proof of Theorem 2.1. □
Corollary 2.6 Let θ be a positive real number and q is a real number with . Let be an even mapping with and satisfying (2.8). Then there exists a unique quadratic mapping such that
for all .
Proof The proof follows from Theorem 2.5 by taking for all . Then we can choose and we get the desired result. □
Theorem 2.7 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let be a function such that there exists an with
for all . Let be an even mapping with and satisfying (2.2). Then there exists a unique quadratic mapping such that
for all .
Proof It follows from (2.15) that
The rest of the proof is similar to the proof of Theorems 2.1 and 2.5. □
Corollary 2.8 Let θ be a positive real number and q is a real number with . Let be an even mapping with and satisfying (2.8). Then there exists a unique quadratic mapping such that
for all .
Proof The proof follows from Theorem 2.7 by taking for all . Then we can choose and we get the desired result. □
Let be a mapping satisfying and (1.1). Let and . Then is an even mapping satisfying (1.1) and is an odd mapping satisfying (1.1) such that .
On the other hand
and
for all , where is the difference operator of the functional equation (1.1). So we obtain the following theorem.
Theorem 2.9 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let be a function such that there exists an with
for all . Let be a mapping with and satisfying (2.2). Then there exist a unique additive mapping and a unique quadratic mapping such that
for all .
Theorem 2.10 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let be a function such that there exists an with
for all . Let be a mapping with and satisfying (2.2). Then there exist a unique additive mapping and a unique quadratic mapping such that
for all .
3 Stability of functional equation (1.1): a direct method
In this section, using direct method, we prove the generalized Hyers-Ulam stability of the additive-quadratic functional equation (1.1) in non-Archimedean space.
Theorem 3.1 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that be a function such that
for all . Suppose that, for any , the limit
exists and be an odd mapping satisfying
Then the limit
exists for all and defines an additive mapping such that
Moreover, if
then A is the unique additive mapping satisfying (3.4).
Proof By (2.10), we know
for all . Replacing x by in (3.5), we obtain
Thus, it follows from (3.1) and (3.6) that the sequence is a Cauchy sequence. Since X is complete, it follows that is convergent. Set
By induction on n, one can show that
for all and . By taking in (3.7) and using (3.2), one obtains (3.4). By (3.1) and (3.3), we get
for all . Therefore, the mapping satisfies (1.1).
To prove the uniqueness property of A, let L be another mapping satisfying (3.4). Then we have
for all . Therefore, . This completes the proof. □
Corollary 3.2 Let be a function satisfying
for all . Assume that and be a mapping with such that
for all . Then there exists a unique additive mapping such that
Proof Defining by , then we have
for all . The last equality comes form the fact that . On the other hand, it follows that
exists for all . Also, we have
Thus, applying Theorem 3.1, we have the conclusion. This completes the proof. □
Theorem 3.3 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that be a function such that
for all . Suppose that, for any , the limit
exists and be an odd mapping satisfying (3.3). Then the limit exists for all and
for all . Moreover, if
then A is the unique mapping satisfying (3.11).
Proof By (2.6), we get
for all . Replacing x by in (3.12), we obtain
Thus, it follows from (3.9) and (3.13) that the sequence is convergent. Set
On the other hand, it follows from (3.13) that
for all and with . Letting , taking in the last inequality and using (3.10), we obtain (3.11).
The rest of the proof is similar to the proof of Theorem 3.1. This completes the proof. □
Theorem 3.4 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that be a function such that
for all . Suppose that, for any , the limit
exists and be an even mapping with and satisfying (3.3). Then the limit exists for all and defines a quadratic mapping such that
Moreover, if
then Q is the unique additive mapping satisfying (3.16).
Proof It follows from (2.15) that
Replacing x by in (3.18), we have
It follows from (3.14) and (3.18) that the sequence is Cauchy sequence. The rest of the proof is similar to the proof of Theorem 3.1. □
Similarly, we can obtain the followings. We will omit the proof.
Theorem 3.5 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that be a function such that
for all . Suppose that, for any , the limit
exists and be an even mapping with and satisfying (3.3). Then the limit exists for all and
for all . Moreover, if
then Q is the unique mapping satisfying (3.21).
Let be a mapping satisfying and (1.1). Let and . Then is an even mapping satisfying (1.1) and is an odd mapping satisfying (1.1) such that . On the other hand,
and
for all , where is the difference operator of the functional equation (1.1). So we obtain the following theorem.
Theorem 3.6 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that be a function such that
for all . Suppose that the limits
and
exist for all and be a mapping with and satisfying (3.3). Then there exist an additive mapping and a quadratic mapping such that
for all . Moreover, if
and
then A, Q are the unique mappings satisfying (3.22).
Theorem 3.7 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that be a function such that
for all . Suppose that the limits
and
exist for all and be a mapping with and satisfying (3.3). Then there exist an additive mapping and a quadratic mapping such that
for all . Moreover, if
and
then A, Q are the unique mappings satisfying (3.23).
4 Conclusion
We linked here two different disciplines, namely, the non-Archimedean normed spaces and functional equations. We established the generalized Hyers-Ulam stability of the functional equation (1.1) in non-Archimedean normed spaces.
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Park, C., Azadi Kenary, H. & Rassias, T. Hyers-Ulam-Rassias stability of the additive-quadratic mappings in non-Archimedean Banach spaces. J Inequal Appl 2012, 174 (2012). https://doi.org/10.1186/1029-242X-2012-174
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DOI: https://doi.org/10.1186/1029-242X-2012-174