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Additive functional inequalities in 2-Banach spaces
Journal of Inequalities and Applications volume 2013, Article number: 447 (2013)
Abstract
We prove the Hyers-Ulam stability of the Cauchy functional inequality and theCauchy-Jensen functional inequality in 2-Banach spaces.
Moreover, we prove the superstability of the Cauchy functional inequality and theCauchy-Jensen functional inequality in 2-Banach spaces under someconditions.
MSC: 39B82, 39B52, 39B62, 46B99, 46A19.
1 Introduction and preliminaries
In 1940, Ulam [1] suggested the stability problem of functional equations concerning thestability of group homomorphisms as follows: Letbe a group and letbe a metric group with the metric. Given, does there exist asuch that if a mappingsatisfies the inequality
for all , then a homomorphism exists with
for all ?
In 1941, Hyers [2] gave a first (partial) affirmative answer to the question of Ulam forBanach spaces. Thereafter, we call that type the Hyers-Ulam stability.
Hyers’ theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. Ageneralization of the Rassias theorem was obtained by Gǎvruta [5] by replacing the unbounded Cauchy difference by a general controlfunction.
Gähler [6, 7] introduced the concept of linear 2-normed spaces.
Definition 1.1 Let be a real linear space with , and let be a function satisfying the following properties:
-
(a)
if and only if x and y are linearly dependent,
-
(b)
,
-
(c)
,
-
(d)
for all and . Then the function is called 2-norm on and the pair is called a linear 2-normed space.Sometimes condition (d) is called the triangle inequality.
See [8] for examples and properties of linear 2-normed spaces.
White [9, 10] introduced the concept of 2-Banach spaces. In order to definecompleteness, the concepts of Cauchy sequences and convergence are required.
Definition 1.2 A sequence in a linear 2-normed space is called a Cauchy sequence if
for all .
Definition 1.3 A sequence in a linear 2-normed space is called a convergent sequenceif there is an such that
for all . If converges to x, write as and call x the limit of. In this case, we also write .
The triangle inequality implies the following lemma.
Lemma 1.4[11]
For a convergent sequencein a linear 2-normed space,
for all.
Definition 1.5 A linear 2-normed space, in which every Cauchy sequence is aconvergent sequence, is called a 2-Banach space.
Eskandani and Gǎvruta [12] proved the Hyers-Ulam stability of a functional equation in 2-Banachspaces.
In [13], Gilányi showed that if f satisfies the functionalinequality
then f satisfies the Jordan-von Neumann functional equation
See also [14]. Gilányi [15] and Fechner [16] proved the Hyers-Ulam stability of functional inequality (1.1).
Park et al.[17] proved the Hyers-Ulam stability of the following functional inequalities:
In this paper, we prove the Hyers-Ulam stability of Cauchy functional inequality(1.2) and Cauchy-Jensen functional inequality (1.3) in 2-Banach spaces.
Moreover, we prove the superstability of Cauchy functional inequality (1.2) andCauchy-Jensen functional inequality (1.3) in 2-Banach spaces under someconditions.
Throughout this paper, let be a normed linear space, and let be a 2-Banach space.
2 Hyers-Ulam stability of Cauchy functional inequality (1.2) in 2-Banachspaces
In this section, we prove the Hyers-Ulam stability of Cauchy functional inequality(1.2) in 2-Banach spaces.
Proposition 2.1 Let be a mapping satisfying
for alland all. Then the mappingis additive.
Proof Letting in (2.1), we get and so for all . Hence .
Letting and in (2.1), we get and so for all and all . Hence for all .
Letting in (2.1), we get
and so
for all and all . Hence
for all . So, is additive. □
Theorem 2.2 Let, with, and letbe a mapping satisfying
for alland all. Then there is a unique additive mappingsuch that
for alland all.
Proof Letting in (2.2), we get and so for all . Hence .
Letting and in (2.2), we get and so for all and all . Hence for all .
Putting and in (2.2), we get
for all and all . So, we get
for all and all . Replacing x by in (2.5) and dividing by , we obtain
for all , all and all integers . For all integers l, m with, we get
for all and all . So, we get
for all and all . Thus the sequence is a Cauchy sequence in for each . Since is a 2-Banach space, the sequence converges for each . So, one can define the mapping by
for all . That is,
for all and all .
By (2.2), we get
for all and all . So,
for all and all . By Proposition 2.1, is additive.
By Lemma 1.4 and (2.6), we have
for all and all .
Now, let be another additive mapping satisfying (2.3). Then wehave
which tends to zero as for all and all . By Definition 1.1, we can conclude that for all . This proves the uniqueness ofA. □
Theorem 2.3 Let, with, and letbe a mapping satisfying (2.2). Then there is a unique additivemappingsuch that
for alland all.
Proof It follows from (2.4) that
for all and all . Replacing x by in (2.7) and multiplying by , we obtain
for all and all and all integers . For all integers l, m with, we get
for all and all . So, we get
for all and all . Thus the sequence is a Cauchy sequence in . Since is a 2-Banach space, the sequence converges. So, one can define the mapping by
for all . That is,
for all and all .
The further part of the proof is similar to the proof of Theorem2.2. □
Now we prove the superstability of the Cauchy functional inequality in 2-Banachspaces.
Theorem 2.4 Let, with, and letbe a mapping satisfying
for alland all. Thenis an additive mapping.
Proof Replacing w by sw in (2.8) for, we get
and so
for all , all and all .
If , then the right-hand side of (2.9) tends to as .
If , then the right-hand side of (2.9) tends to as .
Thus
for all and all . By Proposition 2.1, is additive. □
3 Hyers-Ulam stability of Cauchy-Jensen functional inequality (1.3) in 2-Banachspaces
In this section, we prove the Hyers-Ulam stability of Cauchy-Jensen functionalinequality (1.3) in 2-Banach spaces.
Proposition 3.1 Let be a mapping satisfying
for alland all. Then the mappingis additive.
Proof Letting in (3.1), we get and so for all . Hence .
Letting and in (3.1), we get and so for all and all . Hence for all .
Letting in (3.1), we get
and so
for all and all . Hence
for all . Since , is additive. □
Theorem 3.2 Let, with, and letbe a mapping satisfying
for alland all. Then there is a unique additive mappingsuch that
for alland all.
Proof Letting in (3.2), we get and so for all . Hence .
Letting and in (3.2), we get and so for all and all . Hence for all .
Letting and in (3.2), we get
for all and all . Replacing x by in (3.3) and dividing by , we obtain
for all and all and all integers . For all integers l, m with, we get
for all and all . So, we get
for all and all . Thus the sequence is a Cauchy sequence in for each . Since is a 2-Banach space, the sequence converges for each . So, one can define the mapping by
for all . That is,
for all and all .
The further part of the proof is similar to the proof of Theorem2.2. □
Theorem 3.3 Let, with, and letbe a mapping satisfying (3.2). Then there is a unique additivemappingsuch that
for alland all.
Proof It follows from (3.3) that
for all and all . Replacing x by in (3.4) and multiplying by , we obtain
for all and all and all integers . For all integers l, m with, we get
for all and all . So, we get
for all and all . Thus the sequence is a Cauchy sequence in for each . Since is a 2-Banach space, the sequence converges for each . So, one can define the mapping by
for all . That is,
for all and all .
The further part of the proof is similar to the proof of Theorem2.2. □
Now we prove the superstability of the Jensen functional equation in 2-Banachspaces.
Theorem 3.4 Let, with, and letbe a mapping satisfying
for alland all. Thenis an additive mapping.
Proof Replacing w by sw in (3.5) for, we get
and so
for all , all and all .
The rest of the proof is similar to the proof of Theorem 2.4. □
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CP conceived of the study, participated in its design and coordination, drafted themanuscript, participated in the sequence alignment, and read and approved the finalmanuscript.
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Park, C. Additive functional inequalities in 2-Banach spaces. J Inequal Appl 2013, 447 (2013). https://doi.org/10.1186/1029-242X-2013-447
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DOI: https://doi.org/10.1186/1029-242X-2013-447