Research Article  Open  Published:
Random Stability of an AdditiveQuadraticQuartic Functional Equation
Journal of Inequalities and Applicationsvolume 2010, Article number: 754210 (2010)
Abstract
Using the fixed point method, we prove the generalized HyersUlam stability of the following additivequadraticquartic functional equation in complete random normed spaces.
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 HyersUlam stability or as HyersUlamRassias stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Gvrua [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 HyersUlam 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 HyersUlam 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 leftcontinuous, nondecreasing 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 pointwise 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, RNspace) 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 RNspace.
(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 RNspace is said to be complete if and only if every Cauchy sequence in is convergent to a point in . A complete RNspace is said to be random Banach space.
Theorem 2.4 ([40]).
If is a RNspace and is a sequence such that , then almost everywhere.
The theory of random normed spaces (RNspaces) is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations. The RNspaces may also provide us the appropriate tools to study the geometry of nuclear physics and have important application in quantum particle physics. The generalized HyersUlam stability of different functional equations in random normed spaces, RNspaces 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 HyersUlam Stability of the Functional Equation : An Odd Case
One 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 quadraticquartic 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 HyersUlam stability of the functional equation in complete RNspaces: an odd case.
Theorem 3.1.
Let be a linear space, be a complete RNspace 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)
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)
for all . Since is odd, is an odd mapping;

(3)
with , which implies the inequality
(3.21)
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 RNspace 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 HyersUlam Stability of the Functional Equation : An Even Case
Using the fixed point method, we prove the generalized HyersUlam stability of the functional equation in random Banach spaces: an even case.
Theorem 4.1.
Let be a linear space, let be a complete RNspace 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 selfmapping 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)
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)
for all ;

(3)
for every , which implies the inequality
(4.15)
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 RNspace 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 RNspace 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 selfmapping 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)
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)
for all ;

(3)
for each , which implies the inequality
(4.37)
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 RNspace 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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Acknowledgments
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 UniversityAyatollah Amoli Branch, Amol, Iran. The second author was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF2008313C00050) 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 (NRF20090070788). The fourth author is supported by Università degli Studi di Palermo, R.S. ex 60%.
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Keywords
 Functional Equation
 Linear Space
 Quadratic Mapping
 Unique Fixed Point
 Normed Vector Space