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Approximation of a generalized additive mapping in multi-Banach modules and isomorphisms in multi--algebras: a fixed-point approach
Advances in Difference Equations volume 2012, Article number: 162 (2012)
Abstract
Let , be vector spaces. It is shown that if an odd mapping satisfies the functional equation
then the odd mapping is additive, and we use a fixed-point method to prove the Hyers-Ulam stability of the functional equation (0.1) in multi-Banach modules over a unital multi--algebra. As an application, we show that every almost linear bijection of a unital multi--algebra onto a unital multi--algebra is a -algebra isomorphism when for all unitaries , all , and .
MSC:39B52, 46L05, 47H10, 47B48.
1 Introduction
Throughout this paper we assume that r is a positive rational number and d, l are integers with .
Let X and Y be Banach spaces. Consider a mapping such that is continuous in for each fixed , and assume that there exist constants and with
Rassias [1] showed that there exists a unique -linear mapping such that
Găvruta [2] extended the above theorem as follows: let G be an Abelian group, Y be a Banach space and put
If is a mapping satisfying
then there exists a unique additive mapping such that
Park [3] applied Găvruta’s result to linear functional equations in Banach modules over a -algebra. Several functional equations have been investigated in [4, 5] and [6]. In 2006 Baak, Boo and Rassias [7] solved the following functional equation:
(any solution of (1.1) will be called a generalized additive mapping) and proved its Hyers-Ulam stability in Banach modules over a unital -algebra via the direct method. These results were applied to investigate -algebra isomorphisms in unital -algebras.
In this paper, we prove the Hyers-Ulam stability of the functional equation (1.1) in multi-Banach modules over a unital multi--algebra via the fixed-point method. These results are applied to investigate -algebra isomorphisms in unital multi--algebras.
2 Fixed-point theorems
We recall two fundamental results in the fixed-point theory.
Let be a complete metric space and let be strictly contractive, i.e.,
for a Lipschitz constant . Then
-
(1)
the mapping J has a unique fixed point ,
-
(2)
the fixed point is globally attractive, i.e.,
-
(3)
the following inequalities hold:
for all and nonnegative integers n.
Let X be a non-empty set. A function is called a generalized metric on X if for any , we have:
-
(1)
if and only if ,
-
(2)
,
-
(3)
.
Let be a complete generalized metric space and let be a strictly contractive mapping with a Lipschitz constant . Then, for each , either
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)
, .
3 Multi-normed spaces
The notion of a multi-normed space was introduced by Dales and Polyakov [11]. This concept is somewhat similar to an operator sequence space and has some connections with operator spaces and Banach lattices. Motivations for the study of multi-normed spaces and many examples are given in [11–13].
Let be a complex normed space and . We denote by the linear space consisting of k-tuples , where . The linear operations on are defined coordinate-wise. The zero element of either or is denoted by 0. Finally, we denote by the set and by the group of permutations on k symbols.
Definition 3.1 [11, 14] A multi-norm on is a sequence
such that is a norm on for , , and for any integer , we have
(A1) , , ,
(A2) , , ,
(A3) , ,
(A4) , .
A sequence is then said to be a multi-normed space.
Suppose that is a multi-normed space. Then for any , we have
-
(a)
, ,
-
(b)
, .
From Lemma 3.2(b), it follows that if is a Banach space, then is a Banach space for each (in this case we say that is a multi-Banach space).
Now, we recall two important examples of multi-norms (see [11, 12]).
Example 3.3 The sequence on defined by
is a multi-norm called the minimum multi-norm. The terminology ‘minimum’ is justified by property (b) from Lemma 3.2.
Example 3.4 Let be a (non-empty) family of all multi-norms on . For , set
Then is a multi-norm on called the maximum multi-norm.
Lemma 3.5 [14]
Suppose that and . For each , let be a sequence in such that . Then for each we have
Let be a multi-normed space. A sequence in is said to be a multi-null sequence if for each , there exists an such that
We say that the sequence is multi-convergent to and write if is a multi-null sequence.
Let be a normed algebra such that is a multi-normed space. Then is called a multi-normed algebra if
The multi-normed algebra is said to be a multi-Banach algebra if is a multi-Banach space.
Example 3.8 Let p, q be such that and . The algebra is a Banach sequence algebra with respect to coordinate-wise multiplication of sequences (see Example 4.1.42 of [15]). Let be the standard -multi-norm on (see [11]). Then is a multi-Banach algebra.
Definition 3.9 Let be a multi-Banach algebra and assume that is a (unital) -algebra. If the involution ∗ satisfies
then is called a (unital) multi--algebra.
Definition 3.10 Let be a multi-Banach algebra and be a multi-Banach space. Assume also that is a Banach left module over . We say that is a multi-Banach left module over if there is an such that
for all , , .
4 Stability of an odd functional equation in multi-Banach modules over a multi--algebra
Throughout this section, we assume that is a unital multi--algebra, and and are multi-Banach left modules over . Moreover, by we denote the unitary group of .
Lemma 4.1 [7]
Let X and Y be vector spaces. An odd mapping satisfies (1.1) for all if and only if f is additive.
Corollary 4.2 [7]
Let X and Y be vector spaces. An odd mapping satisfies
if and only if f is additive.
Given a mapping , we set
for all and .
Theorem 4.3 Let and be an odd mapping such that for every there is a function with
for all and . If there exists an such that
for all and , then there is a unique -linear generalized additive mapping with
for all and .
Proof Put
and
for all . It is easy to show that is a complete generalized metric space.
Define a mapping by
Analysis similar to that in the proof of Theorem 3.1 in [8] (see also the proof of Lemma 3.2 in [12]) shows that
Fix . Putting , , and for in (4.2), we have
because f is odd and . We thus get
and therefore,
Consequently, by Theorem 2.2, there exists a mapping such that
-
(1)
is a fixed point of J, i.e.,
(4.5)
and is unique in the set
This means that is a unique mapping satisfying (4.5) such that there exists a with
for all and .
-
(2)
as . This implies the equality
(4.6) -
(3)
, which together with (4.4) gives
and therefore, inequality (4.3) holds for all .
Next, note that the fact that the mapping f is odd and (4.6) imply that is odd. Moreover, by (4.1) and (4.2), we get
for all and , and therefore, is a generalized additive mapping.
Fix and . Using (4.1) and (4.2), we have
and consequently,
Since is a generalized additive mapping, from Lemma 4.1 it follows that is additive, and therefore,
As in the proof of Theorem 3.1, in [7] one can now show that is an -linear mapping. □
Corollary 4.4 Let and . Assume also that for , and for . If is an odd mapping such that
for all , , and , then there exists a unique -linear generalized additive mapping with
for all and .
Proof Putting and
for all and , in Theorem 4.3, we get the desired assertion. □
Theorem 4.5 Let . Let be an odd mapping for which there is a function such that
for all and all . If there exists an such that
for all . Then there exists a unique -linear generalized additive mapping such that
for all .
Proof Note that and for all since f is an odd mapping. Let . Putting and , in (4.7), we have
Letting , we get
for all .
The rest of the proof is similar to the proof of Theorem 4.3. □
Corollary 4.6 Let , and let θ and be positive real numbers, or let , and let θ and be positive real numbers. Let be an odd mapping such that
for all and all . Then there exists a unique -linear generalized additive mapping such that
for all .
Proof Define
Putting in Theorem 4.5, we get the desired result. □
Now we investigate the Hyers-Ulam stability of linear mappings for the case .
Theorem 4.7 Let . Let be an odd mapping for which there is a function such that
for all and all . If there exists an such that
for all . Then there exists a unique -linear generalized additive mapping such that
for all .
Proof Let . Putting in (4.8), we have
for all . So
for all .
The rest of the proof is the same as in the proof of Theorem 4.3. □
Corollary 4.8 Let , and let θ and be positive real numbers, or let , and let θ and be positive real numbers. Let be an odd mapping such that
for all and for all . Then there exists a unique -linear generalized additive mapping such that
for all .
Proof Define , and apply Theorem 4.7. Then we get the desired result. □
Theorem 4.9 Let . Let be an odd mapping for which there is a function such that
for all and all . If there exists an such that
for all . Then there exists a unique -linear generalized additive mapping such that
for all .
Proof Let . Putting in (4.9), we have
for all . So
for all .
The rest of the proof is similar to the proof of Theorem 4.3. □
Corollary 4.10 Let , and let θ and be positive real numbers. Or let , and let θ and be positive real numbers. Let be an odd mapping such that
for all and all . Then there exists a unique -linear generalized additive mapping such that
for all .
Proof Define , and apply Theorem 4.9. Then we get the desired result. □
5 Isomorphisms in unital multi--algebras
Throughout this section, assume that and are unital multi--algebras with unit e. Let be the set of unitary elements in .
We investigate -algebra isomorphisms in unital multi--algebras.
Theorem 5.1 Let . Let be an odd bijective mapping satisfying for all , all , and , for which there exists a function such that
for all , all , , and all . Assume that is invertible. Then the odd bijective mapping is a -algebra isomorphism.
Proof Consider the multi--algebras and as left Banach modules over the unital multi--algebra . By Theorem 4.3, there exists a unique -linear generalized additive mapping such that
for all in which is given by
for all .
The rest of the proof is similar to the proof of Theorem 4.1 of [7]. □
Corollary 5.2 Let , and let θ and be positive real numbers. Or let , and let θ and be positive real numbers. Let be an odd bijective mapping satisfying for all , all , and all , such that
for all , all , , and all . Assume that is invertible. Then the odd bijective mapping is a -algebra isomorphism.
Proof Define , and apply Theorem 5.1. Then we get the desired result. □
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The authors are grateful to the reviewers for their valuable comments and suggestions.
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All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
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Park, C., Saadati, R. Approximation of a generalized additive mapping in multi-Banach modules and isomorphisms in multi--algebras: a fixed-point approach. Adv Differ Equ 2012, 162 (2012). https://doi.org/10.1186/1687-1847-2012-162
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DOI: https://doi.org/10.1186/1687-1847-2012-162