Abstract
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-quartic functional equation 
in complete random normed spaces.
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1. Introduction
The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability or as Hyers-Ulam-Rassias stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by G
vru
a [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.
The functional equation
is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [6] for mappings 
, where 
is a normed space and 
is a Banach space. Cholewa [7] noticed that the theorem of Skof is still true if the relevant domain 
is replaced by an Abelian group. Czerwik [8] proved the generalized 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 [4, 9–26]).
In [27], Lee et al. considered the following quartic functional equation
It is easy to show that the function 
satisfies the functional equation (1.2), which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping.
Let 
be a set. A function 
is called a generalized metric on 
if 
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 Lipschitz constant 
. Then for each given element 
, either
for all nonnegative integers 
or there exists a positive integer 
such that
(1)
(2)the sequence 
converges to a fixed point 
of 
,
(3)
is the unique fixed point of 
in the set 
,
(4)
for all 
.
In 1996, Isac and Th. M. Rassias [30] 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 [31–36]).
2. Preliminaries
In the sequel we adopt the usual terminology, notations and conventions of the theory of random normed spaces, as in [37–41]. Throughout this paper, 
is the space of all probability distribution functions that is, the space of all mappings 
, such that 
is left-continuous, non-decreasing on 
, 
and 
. 
is a subset of 
consising of all functions 
for which 
, where 
denotes the left limit of the function 
at the point 
, that is, 
. The space 
is partially ordered by the usual point-wise ordering of functions, that is, 
if and only if 
for all 
in 
. The maximal element for 
in this order is the distribution function 
given by
Definition 2.1 ([40]).
A mapping 
is a continuous triangular norm (briefly, a 
-norm) if 
satisfies the following conditions:
(a)
is commutative and associative;
(b)
is continuous;
(c)
for all 
;
(d)
whenever 
and 
for all 
.
Typical examples of continuous 
-norms are 
, 
and 
(the 
ukasiewicz 
-norm).
Recall (see [42, 43]) that if 
is a 
-norm and 
is a given sequence of numbers in 
, 
is defined recurrently by 
and 
for 
. 
is defined as 
.
It is known ([43]) that for the 
ukasiewicz 
-norm the following implication holds:
Definition 2.2 ([41]).
A Random Normed space (briefly, RN-space) is a triple 
, where 
is a vector space, 
is a continuous 
-norm, and 
is a mapping from 
into 
such that, the following conditions hold:
(RN1)
for all 
if and only if 
;
(RN2)
for all 
, 
;
(RN3)
for all 
and 
.
Definition 2.3.
Let 
be a RN-space.
(1)A sequence 
in 
is said to be convergent to 
in 
if, for every 
and 
, there exists positive integer 
such that 
whenever 
.
(2)A sequence 
in 
is called Cauchy if, for every 
and 
, there exists positive integer 
such that 
whenever 
.
(3)A RN-space 
is said to be complete if and only if every Cauchy sequence in 
is convergent to a point in 
. A complete RN-space is said to be random Banach space.
Theorem 2.4 ([40]).
If 
is a RN-space and 
is a sequence such that 
, then 
almost everywhere.
The theory of random normed spaces (RN-spaces) is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations. The RN-spaces may also provide us the appropriate tools to study the geometry of nuclear physics and have important application in quantum particle physics. The generalized Hyers-Ulam stability of different functional equations in random normed spaces, RN-spaces and fuzzy normed spaces has been recently studied in, Alsina [44], Mirmostafaee, Mirzavaziri and Moslehian [33, 45–47], Mihe
and Radu [38, 39, 48, 49], Mihe
, Saadati and Vaezpour [50, 51], Baktash et al. [52] and Saadati et al. [53].
3. Generalized Hyers-Ulam Stability of the Functional Equation 
: An Odd Case
: An Odd CaseOne can easily show that an odd mapping 
satisfies 
if and only if the odd mapping mapping 
is an additive mapping, that is,
One can easily show that an even mapping 
satisfies 
if and only if the even mapping 
is a quadratic-quartic mapping, that is,
It was shown in [54, Lemma 
] that 
and 
are quartic and quadratic, respectively, and that 
.
For a given mapping 
, we define
for all 
.
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation 
in complete RN-spaces: an odd case.
Theorem 3.1.
Let 
be a linear space, 
be a complete RN-space and 
be a mapping from 
to 
such that, for some 
,
Let 
be an odd mapping satisfying
for all 
and all 
. Then
exists for each 
and defines a unique additive mapping 
such that
for all 
and all 
.
Proof.
Letting 
in (3.5), we get
for all 
and all 
.
Consider the set
and introduce the generalized metric on 
:
where, as usual, 
. It is easy to show that 
is complete. (See the proof of Lemma 
in [38].)
Now we consider the linear mapping 
such that
for all 
and we prove that 
is a strictly contractive mapping with the Lipschitz constant 
.
Let 
be given such that 
. Then
for all 
and all 
. Hence
for all 
and all 
. So 
implies that 
. This means that
for all 
.
It follows from (3.8) that
for all 
and all 
. So
By Theorem 1.1, there exists a mapping 
satisfying the following:
-
(1)
is a fixed point of 
, that is, 
(3.17)
is a fixed point of 
, that is, 
for all 
. The mapping 
is a unique fixed point of 
in the set
This implies that 
is a unique mapping satisfying (3.17) such that there exists a 
satisfying
for all 
and all 
;
-
(2)
as 
. This implies the equality 
(3.20)
as 
. This implies the equality 
for all 
. Since 
is odd, 
is an odd mapping;
-
(3)
with 
, which implies the inequality 
(3.21)
with 
, which implies the inequality 
from which it follows
This implies that the inequality (3.7) holds.
Now, we have,
for all 
, all 
and all 
.
So, we obtain by (3.4)
for all 
, all 
and all 
.
Since 
for all 
and all 
, by Theorem 2.4, we deduce that
for all 
and all 
. Thus the mapping 
is additive, as desired.
Corollary 3.2.
Let 
and let 
be a real number with 
. Let 
be a normed vector space with norm 
. Let 
be an odd mapping satisfying
for all 
and all 
. Then
exists for each 
and defines an additive mapping 
such that
for all 
and all 
.
Proof.
The proof follows from Theorem 3.1 by taking
for all 
. Then we can choose 
and we get the desired result.
Similarly, we can obtain the following. We will omit the proof.
Theorem 3.3.
Let 
be a linear space, 
be a complete RN-space and 
be a mapping from 
to 
(
is denoted by 
)such that, for some 
,
Let 
be an odd mapping satisfying (3.5). Then
exists for each 
and defines a unique additive mapping 
such that
for all 
and all 
.
Corollary 3.4.
Let 
and let 
be a real number with 
. Let 
be a normed vector space with norm 
. Let 
be an odd mapping satisfying (3.26). Then
exists for each 
and defines a unique additive mapping 
such that
for all 
and all 
.
Proof.
The proof follows from Theorem 3.3 by taking
for all 
. Then we can choose 
and we get the desired result.
4. Generalized Hyers-Ulam Stability of the Functional Equation 
: An Even Case
: An Even CaseUsing the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation 
in random Banach spaces: an even case.
Theorem 4.1.
Let 
be a linear space, let 
be a complete RN-space and 
be a mapping from 
to 
(
is denoted by 
)such that, for some 
,
Let 
be an even mapping satisfying 
and (3.5). Then
exists for each 
and defines a quartic mapping 
such that
for all 
and all 
.
Proof.
Letting 
in (3.5), we get
for all 
and all 
.
Replacing 
by 
in (3.5), we get
for all 
and all 
.
By (4.4) and (4.5),
for all 
and all 
. Letting 
for all 
, we get
for all 
and all 
.
Let 
be the generalized metric space defined in the proof of Theorem 3.1.
Now we consider the linear mapping 
such that
for all 
. It is easy to see that 
is a strictly contractive self-mapping on 
with the Lipschitz constant 
.
It follows from (4.7) that
for all 
and all 
. So
By Theorem 1.1, there exists a mapping 
satisfying the following:
-
(1)
is a fixed point of 
, that is, 
(4.11)
is a fixed point of 
, that is, 
for all 
. Since 
is even with 
, 
is an even mapping with 
. The mapping 
is a unique fixed point of 
in the set
This implies that 
is a unique mapping satisfying (4.11) such that there exists a 
satisfying
for all 
and all 
;
-
(2)
as 
. This implies the equality 
(4.14)
as 
. This implies the equality 
for all 
;
-
(3)
for every 
, which implies the inequality 
(4.15)
for every 
, which implies the inequality 
This implies that the inequality (4.3) holds.
Proceeding as in the proof of Theorem 3.1, we obtain that the mapping 
satisfies 
.
Now, we have
for every 
. Since the mapping 
is quartic (see [54, Lemma 
]), we get that the mapping 
is quartic.
Corollary 4.2.
Let 
and let 
be a real number with 
. Let 
be a normed vector space with norm 
. Let 
be an even mapping satisfying 
and (3.26). Then
exists for each 
and defines a quartic mapping 
such that
for all 
and all 
.
Proof.
The proof follows from Theorem 3.1 by taking
for all 
. Then we can choose 
and we get the desired result.
Similarly, we can obtain the following. We will omit the proof.
Theorem 4.3.
Let 
be a linear space, 
be a complete RN-space and 
be a mapping from 
to 
(
is denoted by 
)such that, for some 
,
Let 
be an even mapping satisfying 
and (3.5). Then
exists for each 
and defines a quartic mapping 
such that
for all 
and all 
.
Corollary 4.4.
Let 
and let 
be a real number with 
. Let 
be a normed vector space with norm 
. Let 
be an even mapping satisfying 
and (3.26). Then
exists for each 
and defines a quartic mapping 
such that
for all 
and all 
.
Proof.
The proof follows from Theorem 3.3 by taking
for all 
. Then we can choose 
and we get the desired result.
Theorem 4.5.
Let 
be a linear space, 
be a complete RN-space and 
be a mapping from 
to 
(
is denoted by 
)such that, for some 
,
Let 
be an even mapping satisfying 
and (3.5). Then
exists for each 
and defines a quadratic mapping 
such that
for all 
and all 
.
Proof.
Let 
be the generalized metric space defined in the proof of Theorem 4.1.
Letting 
for all 
in (4.6), we get
for all 
and all 
.
It is easy to see that the linear mapping 
such that
for all 
, is a strictly contractive self-mapping with the Lipschitz constant 
.
It follows from (4.29) that
for all 
and all 
. So
By Theorem 1.1, there exists a mapping 
satisfying the following.
-
(1)
is a fixed point of 
, that is, 
(4.33)
is a fixed point of 
, that is, 
for all 
. Since 
is even with 
, 
is an even mapping with 
. The mapping 
is a unique fixed point of 
in the set
This implies that 
is a unique mapping satisfying (4.33) such that there exists a 
satisfying
for all 
and all 
;
-
(2)
as 
. This implies the equality 
(4.36)
as 
. This implies the equality 
for all 
;
-
(3)
for each 
, which implies the inequality 
(4.37)
for each 
, which implies the inequality 
This implies that the inequality (4.28) holds.
Proceeding as in the proof of Theorem 4.1, we obtain that the mapping 
satisfies 
.
Now, we have
for every 
. Since the mapping 
is quadratic (see [54, Lemma 
]), we get that the mapping 
is quadratic.
Corollary 4.6.
Let 
and let 
be a real number with 
. Let 
be a normed vector space with norm 
. Let 
be an even mapping satisfying 
and (3.26). Then
exists for each 
and defines a quadratic mapping 
such that
for all 
and all 
.
Proof.
The proof follows from Theorem 4.5 by taking
for all 
. Then we can choose 
and we get the desired result.
Similarly, we can obtain the following. We will omit the proof.
Theorem 4.7.
Let 
be a linear space, 
be a complete RN-space and 
be a mapping from 
to 
(
is denoted by 
) such that, for some 
,
Let 
be an even mapping satisfying 
and (3.5). Then
exists for each 
and defines a quadratic mapping 
such that
for all 
and all 
.
Corollary 4.8.
Let 
and let 
be a real number with 
. Let 
be a normed vector space with norm 
. Let 
be an even mapping satisfying 
and (3.26). Then
exists for each 
and defines a quadratic mapping 
such that
for all 
and all 
.
Proof.
The proof follows from Theorem 4.7 by taking
for all 
. Then we can choose 
and we get the desired result.
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The authors would like to thank referees for giving useful suggestions for the improvement of this paper. The first author is supported by Islamic Azad University-Ayatollah Amoli Branch, Amol, Iran. The second author was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050) and the third author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788). The fourth author is supported by Università degli Studi di Palermo, R.S. ex 60%.
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Mohamadi, M., Cho, Y., Park, C. et al. Random Stability of an Additive-Quadratic-Quartic Functional Equation. J Inequal Appl 2010, 754210 (2010). https://doi.org/10.1155/2010/754210
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DOI: https://doi.org/10.1155/2010/754210