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Ulam type stability problems for alternative homomorphisms
Journal of Inequalities and Applications volume 2014, Article number: 228 (2014)
Abstract
We introduce an alternative homomorphism with respect to binary operations and investigate the Ulam type stability problem for such a mapping. The obtained results apply to Ulam type stability problems for several important functional equations.
MSC: Primary 39B82; secondary 47H10.
1 Introduction
In 1940, SM Ulam proposed the following stability problem: Given an approximately additive mapping, can one find the strictly additive mapping near it? A year later, DH Hyers gave an affirmative answer to this problem for additive mappings between Banach spaces. Subsequently many mathematicians came to deal with this problem (cf. [1–5]).
We introduce an alternative homomorphism from a set X with two binary operations ∘ and ∗ to another set E with two binary operations ⋄ and ⋆ defined by
and we investigate the Ulam type stability problem for such a mapping when E is a complete metric space. In particular, if for all , then our results imply the stability results obtained in [6]. Also the method used in the paper have already applied for some other equations (cf. [7–15]).
One consequence of Banach’s fixed point theorem
A fixed point theorem has played an important role in the stability problem (cf. [16]). The authors used an easy consequence of Banach’s fixed point theorem in [6]. It will serve again in this paper. Here we review it.
Let X be a set and a complete metric space. Fix two mappings and , where denotes the set of all nonnegative real numbers. Denote by the set of all mappings such that there exists a finite constant satisfying
For any , we define
Then is a complete metric space which contains f.
Now, fix three mappings , and . For any mapping , we define the mapping by
Also, we consider three quantities:
If , and , then we have
respectively. We will use these inequalities throughout this paper.
We now state our fixed point theorem.
Lemma A ([[6], Proposition 2.1])
Let X be a set and a complete metric space. Suppose that four mappings , , and satisfy
Then and has a unique fixed point in . Moreover,
for all .
2 A stability of alternative homomorphisms
Let be a set X with two binary operations ∘ and ∗. Let be a complete metric space with two binary operations ⋄ and ⋆. Given , we consider the following commutative diagram:
This means that
In particular, if for all , then (1) and (2) become
and
In other words, f is a homomorphism from X to E. Thus, if a mapping satisfies (2), then we say that f is an alternative homomorphism.
In this section, we establish two general settings, on which we can give an affirmative answer to the Ulam type stability problem for the commutative diagram (1). These settings have a property such as duality, that is, each of them works as a complement of the other.
Let us describe the first setting. For and , we consider the following three conditions:
-
(i)
The square operator is an automorphism of X with respect to ∘ and ∗. We denote by σ the inverse mapping of this automorphism.
-
(ii)
The binary operations ⋄ and ⋆ on E are continuous. The square operator is an endomorphism of E with respect to ⋄ and ⋆.
-
(iii)
, , and .
Under the above conditions, we show the Ulam type stability for the commutative diagram (1), as follows.
Theorem 1 Let and be as above. Suppose that four mappings , , and satisfy (i), (ii), and (iii). If a mapping satisfies
then there exists a mapping such that
Moreover, if a mapping satisfies (5), (6), and
then .
Proof For simplicity, we write . We note that α, β, and γ are finite by (iii). Suppose that satisfies (3) and (4). Put for all . To apply Lemma A to f and φ, we first observe that . Fix . Replacing x and y in (3) by σx, we get
Since
and
it follows that
Using this and (4), we have
Hence and .
We next estimate the quantity . For , we have
Hence and by (iii).
Thus we can apply Lemma A. As a consequence, T has a unique fixed point . Moreover,
and
for all . Since and , (10) implies (7).
Here we show (5). If and , then we have
We will see that the right hand side of (11) tends to 0 as . The first and third terms on the right hand side tend to 0 as , because of (9) and the continuity of ⋆ and ⋄ in (ii). Moreover, the second term, say , is estimated as follows: By (i), (ii), and (3), we have
where and denote the n-fold compositions of endomorphisms τ and σ, respectively. Since by (iii), it follows that as . Thus the right hand side of (11) tends to 0, and we obtain (5).
Next, we show (6). For , we replace x and y in (5) by σx to get
Since and
we obtain (6).
Finally, we show the last statement. Since g satisfies (5) and (6), we have
for all . This says that g is a fixed point of T. Also, by (8), we have . Thus the uniqueness of a fixed point of T in implies that . □
The next corollary is obtained in [6].
Corollary 1 ([[6], Corollary 3.2])
Let X be a set with a binary operation ∘ such that the square operation is an automorphism of X with respect to ∘ and E a complete metric space with a continuous binary operation ⋄ such that the square operation is an endomorphism of E with respect to ⋄. Let and suppose that , and , where σ denotes the inverse mapping of the square operation . If a mapping satisfies
then there exists a unique mapping such that
for all .
Proof Consider the case that ∗ = ∘ and for , in Theorem 1. In this case, τ is clearly an endomorphism of E with respect to ⋆. Therefore the corollary follows immediately from Theorem 1 with . □
Now we turn to another setting. Let and be as in the first part of this section. For and , we consider the following three conditions:
-
(iv)
The square operator is an endomorphism of X with respect to ∘ and ∗.
-
(v)
The binary operations ⋄ and ⋆ on E are continuous. The square operator is an automorphism of E with respect to ⋄ and ⋆. We denote by the inverse mapping of this automorphism.
-
(vi)
, , , and .
Under the above conditions, we show the Ulam type stability for the commutative diagram (1), as follows.
Theorem 2 Let and be as above. Suppose that four mappings , , and satisfy (iv), (v), and (vi). If a mapping satisfies (3) and
then there exists a mapping satisfying (5)
Moreover, if a mapping satisfies (13), (14), and
then .
Proof For simplicity, we write , that is, for . We note that , and are finite by (vi). Suppose that satisfies (3) and (12). Put for all . To apply Lemma A to f and , we first observe that . Fix . Since , it follows from (3) and (12) that
Hence and .
We next estimate the quantity . For , we have
Hence and by (vi).
Thus we can apply Lemma A. As a consequence, has a unique fixed point . Moreover,
and
for all . Since and , (17) implies (14).
Here we show (5). If and , then we have
Letting , the first and third terms on the right hand side tend to 0, because of (16) and the continuity of ⋆ and ⋄ in (v). Moreover, the second term, say , is estimated as follows: By (iv), (v), and (3),
where and denote the n-fold compositions of endomorphisms and , respectively. Since by (vi), it follows that as . Thus we obtain (5).
Next, we show (13). Replacing y in (5) by x, we have
Also since
it follows that
Combining with (18), we obtain (13).
Finally, we show the last statement. Since g satisfies (14) and (13), we have
that is, for all . This says that g is a fixed point of . Also, by (15), we have . Hence the uniqueness of a fixed point of in implies that . □
The next corollary is obtained in [6].
Corollary 2 ([[6], Corollary 3.5])
Let X be a set with a binary operation ∘ such that the square operation is an endomorphism of X with respect to ∘ and E a complete metric space with a continuous binary operation ⋄ such that the square operation is an automorphism of E with respect to ⋄. Let and suppose that , and , where denotes the inverse mapping of the square operation . If a mapping satisfies
then there exists a unique mapping such that
for all .
Proof Consider the case that ∗ = ∘ and for , in Theorem 2. Then is clearly an endomorphism of E with respect to ⋆. Therefore the corollary follows immediately from Theorem 2 with . □
3 Application I
The Ulam type stability problem for Euler-Lagrange type additive mappings has been investigated in [17]. Here we take up the following Euler-Lagrange type mapping satisfying
where X is a complex normed space, E a complex Banach space and with . The following is an Ulam type stability result for this mapping.
Corollary 3 (cf. [[17], Theorem 2.1])
Let and suppose that
(vii) and ().
If a mapping satisfies
then there exists a unique mapping satisfying (19) and
Proof Put , for each . Under these transformations, (20) changes into the following estimate:
where ().
Now we define , for each . In this case, we can easily see that the square operator is an endomorphism of X with respect to ∘ and ∗. Also since , this endomorphism is bijective and so automorphic. We denote by σ the inverse mapping of this automorphism. Moreover, we define , for each . Then we can also see that the binary operations ⋄ and ⋆ on E are continuous and the square operator is an automorphism of E with respect to ⋄ and ⋆. Note that (22) changes into the following:
Since for all , it follows that for all . Also, since for all , it follows that for all and then (4) holds with . Moreover, must hold with . It is also obvious that from the definition of τ. We also note that from the second condition of (vii) and hence from the first condition of (vii). Therefore, by Theorem 1, there exists a unique mapping such that
namely, (19) holds and
and so (21) holds. □
The following is also an Ulam type stability result for the mapping satisfying (19).
Corollary 4 (cf. [[17], Theorem 2.2])
Let and suppose that
-
(viii)
and ().
If a mapping satisfies (20), then there exists a unique mapping satisfying (19) and
Proof As observed in the proof of Corollary 3, (20) changes into (22). Now we define , for each . In this case, we can easily see that the square operator is an endomorphism of X with respect to ∘ and ∗. Moreover, we define , for each . Then we can also see that the binary operations ⋄ and ⋆ on E are continuous and the square operator is an endomorphism of E with respect to ⋄ and ⋆. Also since , this endomorphism is bijective and so automorphic. We denote by the inverse mapping of this automorphism. Note that (22) changes into (23). Since () and (), it follows that for all and then (12) holds with .
Moreover, must hold with . It is also obvious that from the definition of . We also note that from the second condition of (viii) and hence from the first condition of (viii).
Therefore, by Theorem 2, there exists a unique mapping such that
namely, (19) holds and
and so (24) holds. □
Corollary 5 (cf. [[17], Corollary 2.3])
Suppose that , and . If a mapping satisfies
for all , then there exists a unique mapping satisfying (19) and
Proof Put for each .
-
(a)
The case where either
or
Put . Then K satisfies (vii). Note also that
for all . Then the desired result follows from Corollary 3.
-
(b)
The case where either
or
Put . Then K satisfies (viii). Note also that
for all . Then the desired result follows from Corollary 4. □
4 Application II
Let be an Abelian group. In [18], the following result has been shown by A. Simon and P. Volkmann.
Lemma B ([[18], Théorème 1)]
A mapping satisfies
if and only if () for some additive function .
In this section, we deal with the Ulam type stability problem for Equation (25). Put and for each . Moreover, put and for each . Then (25) changes into (2). Also we can easily see that the square operation is endomorphic with respect to ∘ and ∗ and that the square operator is automorphic with respect to ⋄ and ⋆. Denote by the inverse mapping of this automorphism. In this case, it is obvious that for each and hence .
Now let ε be a nonnegative constant and suppose that satisfies
Putting in (26), we obtain
Also, putting in (26), we obtain
Combining (27) and (28), we obtain
Put . By (27) and (28), we obtain
and hence (12) holds. Moreover, note that since ε and δ are constant. Then Lemma B and Theorem 2 easily imply the following.
Corollary 6 Let X be an Abelian group and ε a nonnegative constant. If satisfies (26), then there exists an additive mapping such that
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Acknowledgements
The authors are deeply grateful to the referees for careful reading of the paper and for the helpful suggestions and comments. All authors are partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science.
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Takahasi, SE., Tsukada, M., Miura, T. et al. Ulam type stability problems for alternative homomorphisms. J Inequal Appl 2014, 228 (2014). https://doi.org/10.1186/1029-242X-2014-228
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DOI: https://doi.org/10.1186/1029-242X-2014-228