1 Introduction and preliminaries
The concept of statistical convergence for sequences of real numbers was introduced by Fast [
1] and Steinhaus [
2] independently, and since then several generalizations and applications of this notion have been investigated by various authors (see [
3‐
7]). This notion was defined in normed spaces by Kolk [
8].
We recall some basic facts concerning Fréchet spaces.
Let
X be a vector space. A paranorm
is a function on
X such that
(3)
(triangle inequality)
(4)
If is a sequence of scalars with and with , then (continuity of multiplication).
The pair is called a paranormed space if P is a paranorm on X.
The paranorm is called
total if, in addition, we have
(5)
implies .
A Fréchet space is a total and complete paranormed space.
The stability problem of functional equations originated from a question of Ulam [
10] concerning the stability of group homomorphisms. Hyers [
11] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [
12] for additive mappings and by Rassias [
13] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta [
14] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.
In 1990, Rassias [
15] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for
. In 1991, Gajda [
16], following the same approach as in Rassias [
13], gave an affirmative solution to this question for
. It was shown by Gajda [
16], as well as by Rassias and Šemrl [
17], that one cannot prove a Rassias-type theorem when
(
cf. the books of Czerwik [
18], Hyers, Isac and Rassias [
19]).
The functional equation
(1.1)
is called a
quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a
quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [
20] for mappings
, where
X is a normed space and
Y is a Banach space. Cholewa [
21] noticed that the theorem of Skof is still true if the relevant domain
X is replaced by an Abelian group. Czerwik [
22] proved the Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [
23‐
29]).
In [
30], Jun and Kim considered the following cubic functional equation:
(1.2)
It is easy to show that the function
satisfies the functional equation (
1.2), which is called a
cubic functional equation, and every solution of the cubic functional equation is said to be a
cubic mapping.
In [
31], Lee
et al. considered the following quartic functional equation:
(1.3)
It is easy to show that the function
satisfies the functional equation (
1.3), which is called a
quartic functional equation, and every solution of the quartic functional equation is said to be a
quartic mapping.
In [
32], Gilányi showed that if
f satisfies the functional inequality
(1.4)
then
f satisfies the Jordan-von Neumann functional equation
See also [
33]. Fechner [
34] and Gilányi [
35] proved the Hyers-Ulam stability of the functional inequality (1.4).
Park, Cho and Han [
36] proved the Hyers-Ulam stability of the following functional inequalities:
Throughout this paper, assume that is a Fréchet space and that is a Banach space.
In this paper, we prove the Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation (
1.2), the cubic functional equation (
1.2) and the quartic functional equation (
1.3) in paranormed spaces by using the fixed point method and the direct method.
Furthermore, we prove the Hyers-Ulam stability of the functional inequalities (1.5), (1.6) and (1.7) in paranormed spaces by using the fixed point method and the direct method.
2 Hyers-Ulam stability of the Cauchy additive functional equation
Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the Cauchy additive functional equation in paranormed spaces.
Let
S be a set. A function
is called a
generalized metric on
S if
m 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 a 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, Isac and Rassias [
39] 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 [
40‐
44]).
Note that for all .
Theorem 2.2
Let
be a function such that there exists an
with
(2.1)
for all .
Let be a mapping such that (2.2)
for all .
Then there exists a unique Cauchy additive mapping such that (2.3)
for all .
Proof Letting
in (2.2), we get
and so
(2.4)
for all .
and introduce the generalized metric on
S:
where, as usual,
. It is easy to show that
is complete (see [[
45], Lemma 2.1]).
Now we consider the linear mapping
such that
for all .
Let
be given such that
. Since
for all
,
implies that
. This means that
for all .
It follows from (2.4) that .
By Theorem 2.1, there exists a mapping
satisfying the following:
(1)
A is a fixed point of
J,
i.e.,
(2.5)
for all
. The mapping
A is a unique fixed point of
J in the set
This implies that
A is a unique mapping satisfying (2.5) such that there exists a
satisfying
for all
;
(2)
as
. This implies the equality
for all
;
(3)
, which implies the inequality
This implies that the inequality (2.3) holds true.
It follows from (2.1) and (2.2) that
for all . So, for all . Thus is an additive mapping, as desired. □
Corollary 2.3 Let r be a positive real number with ,
and let be a mapping such that
for all .
Then there exists a unique Cauchy additive mapping such that (2.6)
for all .
Proof Taking for all and choosing in Theorem 2.2, we get the desired result. □
Theorem 2.4
Let
be a function such that
for all .
Let be a mapping satisfying (2.2).
Then there exists a unique Cauchy additive mapping such that
for all .
Proof The proof is similar to the proof of [[
46], Theorem 2.2]. □
Remark 2.5 Let
. Letting
for all
in Theorem 2.4, we obtain the inequality (2.6). The proof is given in [[
46], Theorem 2.2].
Theorem 2.6
Let
be a function such that there exists an
with
(2.7)
for all .
Let be a mapping such that (2.8)
for all .
Then there exists a unique additive mapping such that (2.9)
for all .
Proof Letting
in (2.8), we get
and so
(2.10)
for all .
and introduce the generalized metric on
S:
where, as usual,
. It is easy to show that
is complete (see [[
45], Lemma 2.1]).
Now we consider the linear mapping
such that
for all .
Let
be given such that
. Since
for all
,
implies that
. This means that
for all .
It follows from (2.10) that .
By Theorem 2.1, there exists a mapping
satisfying the following:
(1)
A is a fixed point of
J,
i.e.,
(2.11)
for all
. The mapping
A is a unique fixed point of
J in the set
This implies that
A is a unique mapping satisfying (2.11) such that there exists a
satisfying
for all
;
(2)
as
. This implies the equality
for all
;
(3)
, which implies the inequality
This implies that the inequality (2.9) holds true.
It follows from (2.7) and (2.8) that
for all . So, for all . Thus is an additive mapping, as desired. □
Corollary 2.7 Let r,
θ be positive real numbers with ,
and let be a mapping such that
for all .
Then there exists a unique Cauchy additive mapping such that (2.12)
for all .
Proof Taking for all and choosing in Theorem 2.6, we get the desired result. □
Theorem 2.8
Let
be a function such that
for all .
Let be a mapping satisfying (2.8).
Then there exists a unique Cauchy additive mapping such that
for all .
Proof The proof is similar to the proof of [[
46], Theorem 2.1]. □
Remark 2.9 Let
. Letting
for all
in Theorem 2.8, we obtain the inequality (2.12). The proof is given in [[
46], Theorem 2.1].
3 Hyers-Ulam stability of the quadratic functional equation (1.1)
Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the quadratic functional equation (
1.1) in paranormed spaces.
Theorem 3.1
Let
be a function such that there exists an
with
for all .
Let be a mapping satisfying and (3.1)
for all .
Then there exists a unique quadratic mapping such that
for all .
Proof Letting
in (3.1), we get
and so
for all .
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 3.2 Let r be a positive real number with ,
and let be a mapping satisfying and
for all .
Then there exists a unique quadratic mapping such that (3.2)
for all .
Proof Taking for all and choosing in Theorem 3.1, we get the desired result. □
Theorem 3.3
Let
be a function such that
for all .
Let be a mapping satisfying and (3.1).
Then there exists a unique quadratic mapping such that
for all .
Proof The proof is similar to the proof of [[
46], Theorem 3.2]. □
Remark 3.4 Let
. Letting
for all
in Theorem 3.3, we obtain the inequality (3.2). The proof is given in [[
46], Theorem 3.2].
Theorem 3.5
Let
be a function such that there exists an
with
for all .
Let be a mapping satisfying and (3.3)
for all .
Then there exists a unique quadratic mapping such that
for all .
Proof Letting
in (3.3), we get
and so
for all .
The rest of the proof is similar to the proof of Theorem 2.6. □
Corollary 3.6 Let r,
θ be positive real numbers with ,
and let be a mapping satisfying and
for all .
Then there exists a unique quadratic mapping such that (3.4)
for all .
Proof Taking for all and choosing in Theorem 3.5, we get the desired result. □
Theorem 3.7
Let
be a function such that
for all .
Let be a mapping satisfying and (3.3).
Then there exists a unique quadratic mapping such that
for all .
Proof The proof is similar to the proof of [[
46], Theorem 3.1]. □
Remark 3.8 Let
. Letting
for all
in Theorem 3.7, we obtain the inequality (3.4). The proof is given in [[
46], Theorem 3.1].
4 Hyers-Ulam stability of the cubic functional equation (1.2)
Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the cubic functional equation (
1.2) in paranormed spaces.
Theorem 4.1
Let
be a function such that there exists an
with
for all .
Let be a mapping such that (4.1)
for all .
Then there exists a unique cubic mapping such that
for all .
Proof Letting
in (4.1), we get
and so
for all .
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 4.2 Let r be a positive real number with ,
and let be a mapping such that
for all .
Then there exists a unique cubic mapping such that (4.2)
for all .
Proof Taking for all and choosing in Theorem 4.1, we get the desired result. □
Theorem 4.3
Let
be a function such that
for all .
Let be a mapping satisfying (4.1).
Then there exists a unique cubic mapping such that
for all .
Proof The proof is similar to the proof of [[
46], Theorem 4.2]. □
Remark 4.4 Let
. Letting
for all
in Theorem 4.3, we obtain the inequality (4.2). The proof is given in [[
46], Theorem 4.2].
Theorem 4.5
Let
be a function such that there exists an
with
for all .
Let be a mapping such that (4.3)
for all .
Then there exists a unique cubic mapping such that
for all .
Proof Letting
in (4.3), we get
and so
for all .
The rest of the proof is similar to the proof of Theorem 2.6. □
Corollary 4.6 Let r,
θ be positive real numbers with ,
and let be a mapping such that
for all .
Then there exists a unique cubic mapping such that (4.4)
for all .
Proof Taking for all and choosing in Theorem 4.5, we get the desired result. □
Theorem 4.7
Let
be a function such that
for all .
Let be a mapping satisfying (4.3).
Then there exists a unique cubic mapping such that
for all .
Proof The proof is similar to the proof of [[
46], Theorem 4.1]. □
Remark 4.8 Let
. Letting
for all
in Theorem 4.7, we obtain the inequality (4.4). The proof is given in [[
46], Theorem 4.1].
5 Hyers-Ulam stability of the quartic functional equation (1.3)
Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the quartic functional equation (
1.3) in paranormed spaces.
Theorem 5.1
Let
be a function such that there exists an
with
for all .
Let be a mapping satisfying and (5.1)
for all .
Then there exists a unique quartic mapping such that
for all .
Proof Letting
in (5.1), we get
and so
for all .
The rest of the proof is similar to the proof of Theorem 2.2. □
Corollary 5.2 Let r be a positive real number with ,
and let be a mapping satisfying and
for all .
Then there exists a unique quartic mapping such that (5.2)
for all .
Proof Taking for all and choosing in Theorem 5.1, we get the desired result. □
Theorem 5.3
Let
be a function such that
for all .
Let be a mapping satisfying and (5.1).
Then there exists a unique quartic mapping such that
for all .
Proof The proof is similar to the proof of [[
46], Theorem 5.2]. □
Remark 5.4 Let
. Letting
for all
in Theorem 5.3, we obtain the inequality (5.2). The proof is given in [[
46], Theorem 5.2].
Theorem 5.5
Let
be a function such that there exists an
with
for all .
Let be a mapping satisfying and (5.3)
for all .
Then there exists a unique quartic mapping such that
for all .
Proof Letting
in (5.3), we get
and so
for all .
The rest of the proof is similar to the proof of Theorem 2.6. □
Corollary 5.6 Let r,
θ be positive real numbers with ,
and let be a mapping satisfying and
for all .
Then there exists a unique quartic mapping such that (5.4)
for all .
Proof Taking for all and choosing in Theorem 5.5, we get the desired result. □
Theorem 5.7
Let
be a function such that
for all .
Let be a mapping satisfying and (5.3).
Then there exists a unique quartic mapping such that
for all .
Proof The proof is similar to the proof of [[
46], Theorem 5.1]. □
Remark 5.8 Let
. Letting
for all
in Theorem 5.7, we obtain the inequality (5.4). The proof is given in [[
46], Theorem 5.1].
6 Stability of a functional inequality associated with a three-variable Jensen additive functional equation
Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of a functional inequality associated with a Jordan-von Neumann type three-variable Jensen additive functional equation in paranormed spaces.
Proposition 6.1 [[
36], Proposition 2.1]
Let
be a mapping such that
for all . Then f is Cauchy additive.
Theorem 6.2
Let
be a function such that there exists an
with
(6.1)
for all .
Let be an odd mapping such that (6.2)
for all .
Then there exists a unique Cauchy additive mapping such that (6.3)
for all .
Proof Letting
and
in (6.2), we get
and so
(6.4)
for all .
and introduce the generalized metric on
S
where, as usual,
. It is easy to show that
is complete (see [[
45], Lemma 2.1]).
Now we consider the linear mapping
such that
for all .
Let
be given such that
. Since
for all
,
implies that
. This means that
for all .
It follows from (6.4) that .
By Theorem 2.1, there exists a mapping
satisfying the following:
(1)
A is a fixed point of
J,
i.e.,
(6.5)
for all
. The mapping
A is a unique fixed point of
J in the set
This implies that
A is a unique mapping satisfying (6.5) such that there exists a
satisfying
for all
;
(2)
as
. This implies the equality
for all
;
(3)
, which implies the inequality
This implies that the inequality (6.3) holds true.
It follows from (6.1) and (6.2) that
(6.6)
for all
. Letting
in (6.6), we get
for all . By Proposition 6.1, is Cauchy additive, as desired. □
Corollary 6.3 [[
47], Theorem 2.2]
Let r be a positive real number with ,
and let be an odd mapping such that
for all .
Then there exists a unique Cauchy additive mapping such that (6.7)
for all .
Proof Taking for all and choosing in Theorem 6.2, we get the desired result. □
Theorem 6.4
Let
be a function such that
for all .
Let be an odd mapping satisfying (6.2).
Then there exists a unique Cauchy additive mapping such that
for all .
Proof The proof is similar to the proof of [[
47], Theorem 2.2]. □
Remark 6.5 Let
. Letting
for all
in Theorem 6.4, we obtain the inequality (6.7). The proof is given in [[
47], Theorem 2.2].
7 Stability of a functional inequality associated with a three-variable Cauchy additive functional equation
Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of a functional inequality associated with a Jordan-von Neumann type three-variable Cauchy additive functional equation in paranormed spaces.
Proposition 7.1 [[
36], Proposition 2.2]
Let
be a mapping such that
for all . Then f is Cauchy additive.
Theorem 7.2
Let
be a function such that there exists an
with
for all .
Let be an odd mapping such that (7.1)
for all .
Then there exists a unique Cauchy additive mapping such that
for all .
Proof Letting
and
in (7.1), we get
and so
(7.2)
for all .
The rest of the proof is similar to the proof of Theorem 6.2. □
Corollary 7.3 [[
47], Theorem 3.2]
Let r be a positive real number with ,
and let be an odd mapping such that
for all .
Then there exists a unique Cauchy additive mapping such that (7.3)
for all .
Proof Taking for all and choosing in Theorem 7.2, we get the desired result. □
Theorem 7.4
Let
be a function such that
for all .
Let be an odd mapping satisfying (7.1).
Then there exists a unique Cauchy additive mapping such that
for all .
Proof The proof is similar to the proof of [[
47], Theorem 3.2]. □
Remark 7.5 Let
. Letting
for all
in Theorem 7.4, we obtain the inequality (7.3). The proof is given in [[
47], Theorem 3.2].
8 Stability of a functional inequality associated with the Cauchy-Jensen functional equation
Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of a functional inequality associated with a Jordan-von Neumann type Cauchy-Jensen additive functional equation in paranormed spaces.
Proposition 8.1 [[
36], Proposition 2.3]
Let
be a mapping such that
for all . Then f is Cauchy additive.
Theorem 8.2
Let
be a function such that there exists an
with
for all .
Let be an odd mapping such that (8.1)
for all .
Then there exists a unique Cauchy additive mapping such that
for all .
Proof Replacing
x by 2
x and letting
and
in (8.1), we get
for all .
The rest of the proof is the same as in the proof of Theorem 6.2. □
Corollary 8.3 [[
47], Theorem 4.2]
Let r be a positive real number with ,
and let be an odd mapping such that
for all .
Then there exists a unique Cauchy additive mapping such that (8.2)
for all .
Proof Taking for all and choosing in Theorem 8.2, we get the desired result. □
Theorem 8.4
Let
be a function such that
for all .
Let be an odd mapping satisfying (8.1).
Then there exists a unique Cauchy additive mapping such that (8.3)
for all .
Proof The proof is similar to the proof of [[
47], Theorem 4.2]. □
Remark 8.5 Let
. Letting
for all
in Theorem 8.4, we obtain the inequality (8.3). The proof is given in [[
47], Theorem 4.2].
Acknowledgements
This work was supported by the Daejin University Research Grant in 2013.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (
https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.