- Review Article
- Open Access

# Nonlinear -Random Stability of an ACQ Functional Equation

- Reza Saadati
^{1}Email author, - M M Zohdi
^{1}and - S M Vaezpour
^{1}

**2011**:194394

https://doi.org/10.1155/2011/194394

© Reza Saadati et al. 2011

**Received:**9 December 2010**Accepted:**6 February 2011**Published:**6 March 2011

## Abstract

We prove the generalized Hyers-Ulam stability of the following additive-cubic-quartic functional equation: in complete latticetic random normed spaces.

## Keywords

- Functional Equation
- Unique Mapping
- Additive Mapping
- Unique Fixed Point
- Generalize Uncertainty Principle

## 1. Introduction

Random theory is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. It has also very useful applications in various fields, for example, population dynamics, chaos control, computer programming, nonlinear dynamical systems, nonlinear operators, statistical convergence, and so forth. The random topology proves to be a very useful tool to deal with such situations where the use of classical theories breaks down. The usual uncertainty principle of Werner Heisenberg leads to a generalized uncertainty principle, which has been motivated by string theory and noncommutative geometry. In strong quantum gravity regime space-time points are determined in a random manner. Thus impossibility of determining the position of particles gives the space-time a random structure. Because of this random structure, position space representation of quantum mechanics breaks down, and therefore a generalized normed space of quasiposition eigenfunction is required. Hence, one needs to discuss on a new family of random norms. There are many situations where the norm of a vector is not possible to be found and the concept of random norm seems to be more suitable in such cases, that is, we can deal with such situations by modeling the inexactness through the random norm [1, 2].

*generalized Hyers-Ulam stability*or as

*Hyers-Ulam-Rassias stability*of functional equations

A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [7] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias approach.

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 [6, 8–24]).

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*.

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*.

The study of stability of functional equations is important problem in nonlinear sciences and application in solving integral equation via VIM [27–29] PDE and ODE [30–34]. 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.

for all nonnegative integers or there exists a positive integer such that

(1) , for all ;

(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 [37] 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 [38–43]).

## 2. Preliminaries

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 by Alsina [44], Mirmostafaee and Moslehian [45] and Mirzavaziri and Moslehian [40], Miheţ and Radu [46], Miheţ et al. [47, 48], Baktash et al. [49], and Saadati et al. [50].

Let be a complete lattice, that is, a partially ordered set in which every nonempty subset admits supremum and infimum, and , . The space of latticetic random distribution functions, denoted by , is defined as the set of all mappings such that is left continuous and nondecreasing on , .

Definition 2.1 (see [51]).

A *triangular norm* (
-norm) on
is a mapping
satisfying the following conditions:

(a) (boundary condition);

(b) (commutativity);

(c) (associativity);

(d) and (monotonicity).

*continuous*

*-norm*if

for all .

The limit on the right side of (2.4) exists since the sequence is nonincreasing and bounded from below.

Note that we put
whenever
. If
is a
-norm then
is defined for all
and
by 1, if
and
, if
. A
-norm
is said to be *of Hadžić-type* (we denote by
) if the family
is equicontinuous at
(cf. [52]).

Definition 2.2 (see [51]).

*continuous*

*-representable*if there exist a continuous -norm and a continuous -conorm on such that, for all , ,

for all , are continuous -representable.

Recall (see [52, 53]) that if is a given sequence in , is defined recurrently by and for .

A negation on is any decreasing mapping satisfying and . If , for all , then is called an involutive negation. In the following, is endowed with a (fixed) negation .

Definition 2.3.

A *latticetic random normed space* is a triple
, where
is a vector space and
is a mapping from
into
such that the following conditions hold:

(LRN1) for all if and only if ;

(LRN2) for all in , and ;

(LRN3) for all and .

We note that from (LPN2) it follows that .

Example 2.4.

Then is a latticetic random normed space.

is a complete system of neighborhoods of null vector for a linear topology on generated by the norm .

Definition 2.5.

Let be a latticetic random normed space.

(1)A sequence
in
is said to be *convergent* to
in
if, for every
and
, there exists a positive integer
such that
whenever
.

(2)A sequence
in
is called *Cauchy sequence* if, for every
and
, there exists a positive integer
such that
whenever
.

(3)A latticetic random normed spaces
is said to be *complete* if and only if every Cauchy sequence in
is convergent to a point in
.

Theorem 2.6.

If is a latticetic random normed space and is a sequence such that , then .

Proof.

The proof is the same as classical random normed spaces, see [54].

Lemma 2.7.

then and .

Proof.

Let for all . Since , we have , and by (LRN1) we conclude that .

## 3. Generalized Hyers-Ulam Stability of the Functional Equation (1.1): An Odd Case

It was shown in Lemma 2.2 of [55] that and are cubic and additive, respectively, and that .

for all .

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in complete LRN-spaces: an odd case.

Theorem 3.1.

for all and all .

Proof.

for all and all .

for all and all .

for all and all .

where, as usual, . It is easy to show that is complete. (See the proof of Lemma 2.1 of [46].)

for all .

for all .

By Theorem 1.1, there exists a mapping satisfying the following:

for all and all ;

for all ;

This implies that inequality (3.7) holds.

for all and all . Thus the mapping is cubic, as desired.

Corollary 3.2.

for all and all .

Proof.

for all . Then we can choose and we get the desired result.

Theorem 3.3.

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 3.1.

for all .

for all .

for all and all . So .

By Theorem 1.1, there exists a mapping satisfying the following:

for all and all ;

for all ;

This implies that inequality (3.37) holds.

The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.4.

for all and all .

Proof.

for all . Then we can choose , and we get the desired result.

Theorem 3.5.

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 3.1.

for all and all .

for all .

for all .

for all and all . So .

By Theorem 1.1, there exists a mapping satisfying the following:

for all and all ;

for all ;

This implies that inequality (3.54) holds.

The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.6.

for all and all .

Proof.

for all . Then we can choose and we get the desired result.

Theorem 3.7.

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 3.1.

for all .

for all .

for all and all . So .

By Theorem 1.1, there exists a mapping satisfying the following:

for all and all ;

for all ;

This implies that inequality (3.71) holds.

The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.8.

for all and all .

Proof.

for all . Then we can choose and we get the desired result.

## 4. Generalized Hyers-Ulam Stability of the Functional Equation (1.1): An Even Case

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in complete RN-spaces: an even case.

Theorem 4.1.

for all and all .

Proof.

for all and all .

for all and all .

for all and all .

where, as usual, . It is easy to show that is complete. (See the proof of Lemma 2.1 of [46].)

for all .

for all .

for all and all . So .

By Theorem 1.1, there exists a mapping satisfying the following:

for all and all ;

for all ;

This implies that inequality (4.3) holds.

The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 4.2.

for all and all .

Proof.

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.

for all and all .

Corollary 4.4.

for all and all .

Proof.

for all . Then we can choose , and we get the desired result.

## Authors’ Affiliations

## References

- El Naschie MS:
**On a fuzzy Kähler-like manifold which is consistent with the two slit experiment.***International Journal of Nonlinear Sciences and Numerical Simulation*2005,**6**(2):95–98. 10.1515/IJNSNS.2005.6.2.95View ArticleGoogle Scholar - Sigalotti LDG, Mejias A:
**On El Naschie's conjugate complex time, fractal space-time and faster-than-light particles.***International Journal of Nonlinear Sciences and Numerical Simulation*2006,**7**(4):467–472. 10.1515/IJNSNS.2006.7.4.467View ArticleGoogle Scholar - Ulam SM:
*A Collection of Mathematical Problems*. Interscience Publishers, New York, NY, USA; 1960:xiii+150.MATHGoogle Scholar - 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.27.4.222MathSciNetView ArticleGoogle Scholar - 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/00210064MATHMathSciNetView ArticleGoogle Scholar - 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/S0002-9939-1978-0507327-1MATHMathSciNetView ArticleGoogle Scholar - Găvruţa P:
**A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.***Journal of Mathematical Analysis and Applications*1994,**184**(3):431–436. 10.1006/jmaa.1994.1211MATHMathSciNetView ArticleGoogle Scholar - Hyers DH, Isac G, Rassias ThM:
*Stability of Functional Equations in Several Variables*. Birkhäuser, Boston, Mass, USA; 1998:vi+313.MATHView ArticleGoogle Scholar - Jung S-M:
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar - Park C:
**Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras.***Bulletin des Sciences Mathématiques*2008,**132**(2):87–96.MATHView ArticleGoogle Scholar - Park C:
**Hyers-Ulam-Rassias stability of a generalized Apollonius-Jensen type additive mapping and isomorphisms between -algebras.***Mathematische Nachrichten*2008,**281**(3):402–411. 10.1002/mana.200510611MATHMathSciNetView ArticleGoogle Scholar - Park C, Cui J:
**Generalized stability of -ternary quadratic mappings.***Abstract and Applied Analysis*2007,**2007:**-6.Google Scholar - Park C, Najati A:
**Homomorphisms and derivations in -algebras.***Abstract and Applied Analysis*2007,**2007:**-12.Google Scholar - Rassias JM:
**On approximation of approximately linear mappings by linear mappings.***Bulletin des Sciences Mathématiques*1984,**108**(4):445–446.MATHGoogle Scholar - Rassias JM:
**Refined Hyers-Ulam approximation of approximately Jensen type mappings.***Bulletin des Sciences Mathématiques*2007,**131**(1):89–98.MATHView ArticleGoogle Scholar - Rassias JM, Rassias MJ:
**Asymptotic behavior of alternative Jensen and Jensen type functional equations.***Bulletin des Sciences Mathématiques*2005,**129**(7):545–558.MATHView ArticleGoogle Scholar - Rassias ThM:
**Problem 16; 2, Report of the 27th International Symposium on Functional Equations.***Aequationes Mathematicae*1990,**39:**292–293.Google Scholar - Rassias ThM:
**On the stability of the quadratic functional equation and its applications.***Universitatis Babeş-Bolyai. Studia. Mathematica*1998,**43**(3):89–124.MATHGoogle Scholar - 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.2000.6788MATHMathSciNetView ArticleGoogle Scholar - 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.2000.7046MATHMathSciNetView ArticleGoogle Scholar - Rassias ThM:
**On the stability of functional equations and a problem of Ulam.***Acta Applicandae Mathematicae*2000,**62**(1):23–130. 10.1023/A:1006499223572MATHMathSciNetView ArticleGoogle Scholar - Rassias ThM, Šemrl P:
**On the behavior of mappings which do not satisfy Hyers-Ulam stability.***Proceedings of the American Mathematical Society*1992,**114**(4):989–993. 10.1090/S0002-9939-1992-1059634-1MATHMathSciNetView ArticleGoogle Scholar - Rassias ThM, Šemrl P:
**On the Hyers-Ulam stability of linear mappings.***Journal of Mathematical Analysis and Applications*1993,**173**(2):325–338. 10.1006/jmaa.1993.1070MATHMathSciNetView ArticleGoogle Scholar - 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.1998.6129MATHMathSciNetView ArticleGoogle Scholar - Jun K-W, Kim H-M:
**The generalized Hyers-Ulam-Rassias stability of a cubic functional equation.***Journal of Mathematical Analysis and Applications*2002,**274**(2):267–278.MathSciNetView ArticleGoogle Scholar - Lee SH, Im SM, Hwang IS:
**Quartic functional equations.***Journal of Mathematical Analysis and Applications*2005,**307**(2):387–394. 10.1016/j.jmaa.2004.12.062MATHMathSciNetView ArticleGoogle Scholar - Saadati R, Vaezpour SM, Rhoades BE:
**-stability approach to variational iteration method for solving integral equations.***Fixed Point Theory and Applications*2009,**2009:**-9.Google Scholar - He JH:
**Variational iteration method—a kind of nonlinear analytical technique: some examples.***International Journal of Non-Linear Mechanics*1999,**34:**699–708. 10.1016/S0020-7462(98)00048-1MATHView ArticleGoogle Scholar - He J-H:
**A review on some new recently developed nonlinear analytical techniques.***International Journal of Nonlinear Sciences and Numerical Simulation*2000,**1**(1):51–70. 10.1515/IJNSNS.2000.1.1.51MATHMathSciNetView ArticleGoogle Scholar - He J-H, Wu X-H:
**Variational iteration method: new development and applications.***Computers & Mathematics with Applications*2007,**54**(7–8):881–894. 10.1016/j.camwa.2006.12.083MATHMathSciNetView ArticleGoogle Scholar - He J-H:
**Variational iteration method—some recent results and new interpretations.***Journal of Computational and Applied Mathematics*2007,**207**(1):3–17. 10.1016/j.cam.2006.07.009MATHMathSciNetView ArticleGoogle Scholar - Ozer H:
**Application of the variational iteration method to the boundary value problems with jump discontinuities arising in solid mechanics.***International Journal of Nonlinear Sciences and Numerical Simulation*2007,**8**(4):513–518. 10.1515/IJNSNS.2007.8.4.513View ArticleGoogle Scholar - Biazar J, Ghazvini H:
**He's variational iteration method for solving hyperbolic differential equations.***International Journal of Nonlinear Sciences and Numerical Simulation*2007,**8:**311–314. 10.1515/IJNSNS.2007.8.3.311Google Scholar - Odibat ZM, Momani S:
**Application of variational iteration method to nonlinear differential equations of fractional order.***International Journal of Nonlinear Sciences and Numerical Simulation*2006,**7**(1):27–34. 10.1515/IJNSNS.2006.7.1.27MathSciNetView ArticleGoogle Scholar - 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):7.Google Scholar - 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/S0002-9904-1968-11933-0MATHMathSciNetView ArticleGoogle Scholar - 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/S0161171296000324MATHMathSciNetView ArticleGoogle Scholar - Cădariu L, Radu V:
**On the stability of the Cauchy functional equation: a fixed point approach.**In*Iteration Theory*.*Volume 346*. Karl-Franzens-Universität Graz, Graz, Austria; 2004:43–52.Google Scholar - 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.Google Scholar - 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/s00574-006-0016-zMATHMathSciNetView ArticleGoogle Scholar - Park C:
**Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras.***Fixed Point Theory and Applications*2007,**2007:**-15.Google Scholar - Park C:
**Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach.***Fixed Point Theory and Applications*2008,**2008:**-9.Google Scholar - Radu V:
**The fixed point alternative and the stability of functional equations.***Fixed Point Theory*2003,**4**(1):91–96.MATHMathSciNetGoogle Scholar - Alsina C:
**On the stability of a functional equation arising in probabilistic normed spaces.**In*General Inequalities*.*Volume 80*. Birkhäuser, Basel, Switzerland; 1987:263–271.Google Scholar - Mirmostafaee AK, Moslehian MS:
**Fuzzy approximately cubic mappings.***Information Sciences*2008,**178**(19):3791–3798. 10.1016/j.ins.2008.05.032MATHMathSciNetView ArticleGoogle Scholar - 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.MATHMathSciNetView ArticleGoogle Scholar - Miheţ D, Saadati R, Vaezpour SM:
**The stability of the quartic functional equation in random normed spaces.***Acta Applicandae Mathematicae*2011,**110**(2):797–803.Google Scholar - Miheţ D, Saadati R, Vaezpour SM: The stability of anadditive functional equation in Menger probabilistic -normed spaces. Mathematica Slovaca. In pressGoogle Scholar
- 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.Google Scholar - Saadati R, Vaezpour SM, Cho YJ:
**Erratumml: 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.Google Scholar - Deschrijver G, Kerre EE:
**On the relationship between some extensions of fuzzy set theory.***Fuzzy Sets and Systems*2003,**133**(2):227–235. 10.1016/S0165-0114(02)00127-6MATHMathSciNetView ArticleGoogle Scholar - 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.Google Scholar - 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.MATHMathSciNetGoogle Scholar - Schweizer B, Sklar A:
*Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics*. North-Holland Publishing, New York, NY, USA; 1983:xvi+275.Google Scholar - Eshaghi-Gordji M, Kaboli-Gharetapeh S, Park C, Zolfaghri S:
**Stability of an additive-cubic-quartic functional equation.***Advances in Difference Equations*2009,**2009:**20.Google Scholar

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