1. Introduction
The issue of stability of functional equations has been motivated by a problem raised by S.M. Ulam (see [
1]), which at present can be understood in the following way:
When is it true that a function satisfying a certain property (e.g., equation) approximately must be close to a function satisfying the property exactly?
For more details and historical background, we refer to [
2], which is the first monograph on the subject (see also [
3]). An interesting discussion on various possible approaches to such stability has been presented in [
4] (see also [
5]); we use the one described in our Definitions 1 and 2.
Problems of that type are very natural for difference, differential, functional and integral equations and many examples of recent results concerning their stability as well as further references can be found in [
6,
7].
Roughly speaking, we say that a given functional equation is stable in some class of functions if any function from that class, satisfying the equation approximately, is near an exact solution of the equation. One of the classical outcomes in this area is the following theorem (see Theorems 3.1 and 3.4 of [
8]) concerning stability of the well-known Cauchy functional equation on a restricted domain (
denotes the set of positive integers).
Theorem 1. Let and be normed spaces, and be real numbers and . Let be a mapping such that Then the following two statements are valid.
- (i)
If , and there exists such that for every and every with , then f is additive on X: - (ii)
If , and is complete, then there exists a unique mappping that is additive on X and
Let us mention that the assumption
is necessary (see [
9]) and estimate (
3) is the best possible in the general case (see [
3] for more details and [
10] for a related result). If
and
, then we obtain from Theorem 1 the result of Hyers [
1].
Similar outcomes, but with (
1) replaced by the inequality
with some real numbers
p and
q, have been obtained in [
11,
12,
13] (see
Section 3 for more details).
Clearly, the concept of an approximate solution and the idea of nearness of two functions can be understood in many nonstandard ways, depending on the needs and tools available in a particular situation. One of such non-classical measures of distance can be created using the notion of
n-norm, introduced by A. Misiak [
14]. We refer to [
15] for several examples of investigations of stability of functional equations in the
n-normed spaces.
In this paper, we present two possible extensions of the results in [
13] to the case of
n-normed spaces.
2. Preliminaries
The notion of
n-normed space is an extension of those of the classical normed space and of the 2-normed space defined by Gähler [
16] (cf., e.g., [
17,
18,
19,
20]).
Let us now recall some basic definitions and facts concerning
n-normed spaces (for more details we refer the reader to [
15]; see also [
14,
21,
22,
23]).
Let , X be a real linear space, which is at least n-dimensional, and be a function mapping into (the set of non-negative reals) that satisfies the following conditions:
- (C1)
if and only if are linearly dependent,
- (C2)
is invariant under permutation of ,
- (C3)
,
- (C4)
for every (the set of reals) and . Then is called an n-norm on X and the pair is said to be an n-normed space.
If
and
is a real inner product space, which is at least
n-dimensional, then the formula
defines an
n-norm on
X, where
means the absolute value of a real number
x.
If
with the usual inner product, then in this way we obtain the Euclidean
n-norm on
, which also can be expressed by
where
If
is an
n-normed space and
, then (C4) implies that
and the function
is non-negative.
Let us recall that a sequence
of elements of an
n-normed space
is called a
Cauchy sequence if
whereas
is said to be
convergent if there exists an element
(called the limit of this sequence and denoted by
) with
An
-normed space in which every Cauchy sequence is convergent is called
n-Banach space. Moreover, we have the following property stated in [
23] (see also [
15]).
Lemma 1. Let be an n-normed space. If is a convergent sequence of elements of X, then Remark 1. It follows from (C1) that if is an n-normed space, are linearly independent, andthen . Finally, let us also mention that H. Gunawan and M. Mashadi [
21] showed that from every
n-norm one can derive an
-norm and thus, finally, a standard norm (cf. our Remark 2). More information on the
n-normed spaces and some problems investigated in them (among others in fixed-point theory) can be found for instance in [
15,
22,
23,
24,
25,
26,
27,
28].
3. Hyperstability Results
In the rest of the paper we assume that
and
is an
-normed space. For simplicity of the notation we write
Moreover, if A and B are nonempty sets, then denotes the family of all mappings from A into B.
The name of Ulam has been associated with various definitions of stability (see [
2,
4,
23,
29]). The following one, formulated for the
n-normed spaces, describes our considerations in this paper.
Definition 1. Let be a groupoid ( is a set endowed with a binary operation ), be a subset of P, be nonempty, and . The conditional functional equation for mappings is said to be -stable if for any and any withthere exists a solution of Equation (5) such that Let us mention that functional Equation (
5) is conditional, because it can be rewritten in the following conditional form:
In the very particular case, when
for every
,
and
, the
-stability is called hyperstability (see [
30] for more details). Specifically, we have the following definition.
Definition 2. Let be a groupoid, be a subset of P and be nonempty. The conditional Equation (5) is said to be E-hyperstable if every satisfying (6) with some , is a solution of Equation (5). In this paper, we investigate the hyperstabilty case for Equation (
5). An example of such hyperstability results for classical normed spaces, motivated by some earlier well-known outcomes of Th.M. Rassias [
29,
31] (see also [
9,
32]) and J.M. Rassias [
11,
12], is the following main theorem in [
13] (we write
for
).
Theorem 2. Let and be classic normed spaces, , , and be real numbers with . Suppose that there exists such that Then each mapping withis additive on X. The next two theorems present two possible ways to extend Theorem 2 to the case of n-normed spaces. They are the main results of this paper. Their proofs are provided in the next section.
Theorem 3. Let be a normed space, be nonempty, , , and , . Assume that there exists such that (7) holds. Then every mapping satisfying the inequalityis additive on X. Theorem 4. Let , be nonempty, , and . Then every mapping satisfying the inequalityfor every and every with and , is additive on X. Remark 2. If is a sequence of linearly independent vectors in Y, then it is easy to check that the formuladefines a norm in Y, where if and (cf. Remark 1). Thus, we see that the -norm in Y generates a large family of norms in Y. Let be norms in Y; they can be chosen from the norms generated by the -norm in the way described above. Let , and . Then we can use in (9) the following two natural examples of function : We end this section with two examples of simple corollaries that can be derived from Theorem 4.
Corollary 1. Assume that Y and are as in Theorem 4 and satisfies the inequalityfor every and every such that . Then the functional equationhas at least one solution if and only if Proof. Suppose that there exists a solution
of Equation (
12). Then
and consequently
for every
and every
such that
. Consequently, by Theorem 4,
g is additive, which means that
for every
.
The converse is trivial. Specifically, if
for every
, then the function
,
for
, is a solution of Equation (
12). □
Corollary 2. Assume that Y and are as in Theorem 4. Let be a solution to the cocycle functional equationand inequality (11) be valid for every and every such that . Then (13) holds. Proof. According to [
33] (Theorem 1), a mapping
is a solution to Equations (
14) and (
15) if and only if there exists a mapping
with
The rest of the proof is analogous as for Corollary 4. □
Similar results can be deduced from Theorem 3.
4. Proofs of Theorems 3 and 4
For the proof of Theorem 3 we need an auxiliary fixed-point theorem that can be easily derived from the main result in [
34]. To present it we introduce the following two hypotheses:
- (A1)
E is a nonempty set, , and .
- (A2)
is an operator defined by
We say that
is
contractive if
for any
and
with
For given set
and
we define
for
by:
The fixed-point theorem reads as follows.
Theorem 5. Let (A1)
and (A2)
be valid and be Λ-contractive. Let mappings and be such that Then, for each , there exists the limitand the function , defined in this way, is a unique fixed point of with Now, we are able to prove Theorem 3.
Proof of Theorem 3. Let
for
and
. From the above it follows that:
Please note that we must have or , because . We consider only the case where ; the case is analogous.
Let
be a mapping satisfying (
9). Fix
with
and take
such that
It is easy to see that (
9) with
gives
Define operators
and
by
Then
has the form as in (A2) with
,
for
and
and
is
-contractive. Please note that (
18) takes the form
Let
for
and
. Then, by (
17) (with
), we have
Next, from (
20) and (
21), by an easy induction, it follows that for each
,
Hence, from the geometric queue summation, it results that
Thus, we see that (
16) is valid.
Consequently, by Theorem 5, there is a solution
of the equation
such that
Now we show by induction that for every
with
and
The case
is just (
9). Next, fix
and assume that (
24) holds for
and for every
with
. Then, by (
22) and the triangle inequality, for every
with
we have
whence and by (
17) we finally get
This completes the proof of (
24).
Letting
in (
24), we obtain
for every
and every
with
. Thus, we have proved that (see Remark 1)
Next we show that
is the unique function mapping
X into
Y satisfying (
26) and such that there is
with
Suppose that
satisfies
and there exists
with
Then
where according to the summation of geometric queue
Next we prove that for each
Clearly, the case
is exactly (
28). Therefore, fix
and assume that (
29) holds for
. Then by (
21), (
26), (
27) and the triangle inequality, we get
This completes the inductive proof of (
29). Now, letting
in (
29), we get
.
In the same way, for each
we obtain a unique mapping
such that
and
Furthermore, the uniqueness of
implies that
for each
. Hence
for every
, every
and every
. Consequently, since
and
, letting
in (
31), we obtain
which means that
and consequently (
2) holds. This completes the proof. □
Remark 3. Please note that with only very small and obvious modification in the proof, we can replace condition (9) in Theorem 3 with the following one:where W is any function mapping into that fulfils the inequalitiesfor every , every and every . Finally, we prove Theorem 4.
Proof of Theorem 4. Let
be a mapping satisfying inequality (
10) for every
and every
with
and
. Take
with
. Since
, there exist
such that the sets of vectors
and
are linearly independent.
Fix
such that
for
, where
, and write
. Then, for each
, the sets of vectors
and
are linearly independent, which means that
(see (C1)). Consequently, by (
10),
This means that
whence
.
Thus, we have proved that for every (the case where for some is trivial, because then vectors are linearly dependent). Therefore, by Remark 1, . □
5. Conclusions
In this article, we presented two possible hyperstability results for the Cauchy functional equation on a restricted domain, for mappings that take values in an n-normed space. They correspond to some earlier classical results obtained for normed spaces. Moreover, the second one (Theorem 4), even if quite simple, could be described as somewhat unexpected, because the assumptions that Y is complete and are not necessary there.
Author Contributions
Conceptualization, J.B. and E.-s.E.-h.; methodology, J.B. and E.-s.E.-h.; software, J.B. and E.-s.E.-h.; validation, J.B. and E.-s.E.-h.; formal analysis, J.B. and E.-s.E.-h.; investigation, J.B. and E.-s.E.-h.; resources, J.B. and E.-s.E.-h.; data curation, J.B. and E.-s.E.-h.; writing—original draft preparation, E.-s.E.-h.; writing—review and editing, J.B. and E.-s.E.-h.; visualization, J.B. and E.-s.E.-h.; supervision, J.B.; project administration, E.-s.E.-h.; funding acquisition, J.B. and E.-s.E.-h. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
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