 Research Article
 Open Access
 Published:
Random Stability of an AdditiveQuadraticQuartic Functional Equation
Journal of Inequalities and Applications volume 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.
References
 1.
Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.
 2.
Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222
 3.
Aoki T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan 1950, 2: 64–66. 10.2969/jmsj/00210064
 4.
Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978, 72(2):297–300. 10.1090/S00029939197805073271
 5.
Găvruţa P: A generalization of the HyersUlamRassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994, 184(3):431–436. 10.1006/jmaa.1994.1211
 6.
Skof F: Proprieta' locali e approssimazione di operatori. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113–129. 10.1007/BF02924890
 7.
Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984, 27(1–2):76–86.
 8.
Czerwik S: On the stability of the quadratic mapping in normed spaces. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1992, 62: 59–64. 10.1007/BF02941618
 9.
EshaghiGordji M, KaboliGharetapeh S, Park C, Zolfaghri S: Stability of an additivecubicquartic functional equation. Advances in Difference Equations 2009, 2009:20.
 10.
Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.
 11.
Jun KW, Kim HM: The generalized HyersUlamRassias stability of a cubic functional equation. Journal of Mathematical Analysis and Applications 2002, 274(2):267–278.
 12.
Jung S: HyersUlamRassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.
 13.
Park C: HyersUlamRassias stability of homomorphisms in quasiBanach algebras. Bulletin des Sciences Mathématiques 2008, 132(2):87–96.
 14.
Park C, Cui J: Generalized stability of ternary quadratic mappings. Abstract and Applied Analysis 2007, 2007:6.
 15.
Park C, Najati A: Homomorphisms and derivations in algebras. Abstract and Applied Analysis 2007, 2007:12.
 16.
Rassias JM: On approximation of approximately linear mappings by linear mappings. Bulletin des Sciences Mathématiques 1984, 108(4):445–446.
 17.
Rassias JM: Refined HyersUlam approximation of approximately Jensen type mappings. Bulletin des Sciences Mathématiques 2007, 131(1):89–98.
 18.
Rassias JM, Rassias MJ: Asymptotic behavior of alternative Jensen and Jensen type functional equations. Bulletin des Sciences Mathématiques 2005, 129(7):545–558.
 19.
Rassias ThM: Problem 16; 2, report of the 27th International Symposium on Functional Equations. Aequationes Mathematicae 39(2–3):292–293.
 20.
Rassias ThM: On the stability of the quadratic functional equation and its applications. Studia Universitatis BabeşBolyai. Mathematica 1998, 43(3):89–124.
 21.
Rassias ThM: The problem of S. M. Ulam for approximately multiplicative mappings. Journal of Mathematical Analysis and Applications 2000, 246(2):352–378. 10.1006/jmaa.2000.6788
 22.
Rassias ThM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications 2000, 251(1):264–284. 10.1006/jmaa.2000.7046
 23.
Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Applicandae Mathematicae 2000, 62(1):23–130. 10.1023/A:1006499223572
 24.
Rassias ThM, Šemrl P: On the behavior of mappings which do not satisfy HyersUlam stability. Proceedings of the American Mathematical Society 1992, 114(4):989–993. 10.1090/S00029939199210596341
 25.
Rassias ThM, Šemrl P: On the HyersUlam stability of linear mappings. Journal of Mathematical Analysis and Applications 1993, 173(2):325–338. 10.1006/jmaa.1993.1070
 26.
Rassias ThM, Shibata K: Variational problem of some quadratic functionals in complex analysis. Journal of Mathematical Analysis and Applications 1998, 228(1):234–253. 10.1006/jmaa.1998.6129
 27.
Lee SH, Im SM, Hwang IS: Quartic functional equations. Journal of Mathematical Analysis and Applications 2005, 307(2):387–394. 10.1016/j.jmaa.2004.12.062
 28.
Cădariu L, Radu V: Fixed points and the stability of Jensen's functional equation. Journal of Inequalities in Pure and Applied Mathematics 2003., 4(1, article 4):
 29.
Diaz JB, Margolis B: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bulletin of the American Mathematical Society 1968, 74: 305–309. 10.1090/S000299041968119330
 30.
Isac G, Rassias ThM: Stability of additive mappings: applications to nonlinear analysis. International Journal of Mathematics and Mathematical Sciences 1996, 19(2):219–228. 10.1155/S0161171296000324
 31.
Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. In Iteration Theory, Grazer Mathematische Berichte. Volume 346. KarlFranzensUniversitaet Graz, Graz, Austria; 2004:43–52.
 32.
Cădariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory and Applications 2008, 2008:15.
 33.
Mirzavaziri M, Moslehian MS: A fixed point approach to stability of a quadratic equation. Bulletin of the Brazilian Mathematical Society 2006, 37(3):361–376. 10.1007/s005740060016z
 34.
Park C: Fixed points and HyersUlamRassias stability of CauchyJensen functional equations in Banach algebras. Fixed Point Theory and Applications 2007, 2007:15.
 35.
Park C: Generalized HyersUlam stability of quadratic functional equations: a fixed point approach. Fixed Point Theory and Applications 2008, 2008:9.
 36.
Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4(1):91–96.
 37.
Chang S, Cho Y, Kang SM: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science, Huntington, NY, USA; 2001:x+338.
 38.
Miheţ D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces. Journal of Mathematical Analysis and Applications 2008, 343(1):567–572.
 39.
Miheţ D: Fuzzy stability of additive mappings in nonArchimedean fuzzy normed spaces. Fuzzy Sets and Systems. In press
 40.
Schweizer B, Sklar A: Probabilistic Metric Spaces, NorthHolland Series in Probability and Applied Mathematics. NorthHolland, New York, NY, USA; 1983:xvi+275.
 41.
Šerstnev AN: On the concept of a stochastic normalized space. Doklady Akademii Nauk SSSR 1963, 149: 280–283.
 42.
Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and Its Applications. Volume 536. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+273.
 43.
Hadžić O, Pap E, Budinčević M: Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces. Kybernetika 2002, 38(3):363–382.
 44.
Alsina C: On the stability of a functional equation arising in probabilistic normed spaces. In General Inequalities, Internationale Schriftenreihe zur Numerischen Mathematik. Volume 80. Birkhäuser, Basel, Switzerland; 1987:263–271.
 45.
Mirmostafaee AK, Mirzavaziri M, Moslehian MS: Fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems 2008, 159(6):730–738. 10.1016/j.fss.2007.07.011
 46.
Mirmostafaee AK, Moslehian MS: Fuzzy versions of HyersUlamRassias theorem. Fuzzy Sets and Systems 2008, 159(6):720–729. 10.1016/j.fss.2007.09.016
 47.
Mirmostafaee AK, Moslehian MS: Fuzzy approximately cubic mappings. Information Sciences 2008, 178(19):3791–3798. 10.1016/j.ins.2008.05.032
 48.
Miheţ D: The probabilistic stability for a functional equation in a single variable. Acta Mathematica Hungarica 2009, 123(3):249–256. 10.1007/s104740088101y
 49.
Miheţ D: The fixed point method for fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems 2009, 160(11):1663–1667. 10.1016/j.fss.2008.06.014
 50.
Miheţ D, Saadati R, Vaezpour SM: The stability of the quartic functional equation in random normed spaces. Acta Applicandae Mathematicae. In press
 51.
Miheţ D, Saadati R, Vaezpour SM: The stability of an additive functional equation in Menger probabilistic normed spaces. Mathematica Slovaca. In press
 52.
Baktash E, Cho YJ, Jalili M, Saadati R, Vaezpour SM: On the stability of cubic mappings and quadratic mappings in random normed spaces. Journal of Inequalities and Applications 2008, 2008:11.
 53.
Saadati R, Vaezpour SM, Cho YJ: A note to paper "On the stability of cubic mappings and quartic mappings in random normed spaces". Journal of Inequalities and Applications 2009, 2009:6.
 54.
EshaghiGordji M, Abbaszadeh S, Park C: On the stability of a generalized quadratic and quartic type functional equation in quasiBanach spaces. Journal of Inequalities and Applications 2009, 2009:26.
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%.
Author information
Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Mohamadi, M., Cho, Y., Park, C. et al. Random Stability of an AdditiveQuadraticQuartic Functional Equation. J Inequal Appl 2010, 754210 (2010). https://doi.org/10.1155/2010/754210
Received:
Accepted:
Published:
Keywords
 Functional Equation
 Linear Space
 Quadratic Mapping
 Unique Fixed Point
 Normed Vector Space