Discrete Dynamics in Nature and Society

VolumeΒ 2012Β (2012), Article IDΒ 346561, 13 pages

http://dx.doi.org/10.1155/2012/346561

## On the Stability of an -Variables Functional Equation in Random Normed Spaces via Fixed Point Method

^{1}Department of Mathematics, Payame Noor University, Tehran, Iran^{2}Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran^{3}Department of Mathematics, Iran University of Science and Technology, Behshahr, Iran^{4}Department of Mathematics and Computer Sciences, Tarbiat Moallem University Sabzevar, Sabzevar, P.O. Box 397, Iran

Received 17 September 2011; Revised 5 January 2012; Accepted 29 January 2012

Academic Editor: SeenithΒ Sivasundaram

Copyright Β© 2012 A. Ebadian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

At first we find the solution of the functional equation where is an integer number. Then, we obtain the generalized Hyers-Ulam-Rassias stability in random normed spaces via the fixed point method for the above functional equation.

#### 1. Introduction and Preliminaries

A basic question in the theory of functional equations is as follows: βwhen is it true that a function that approximately satisfies a functional equation must be close to an exact solution of the equation?β

If the problem accepts a solution, we say the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940 and affirmatively solved by Hyers [2]. The result of Hyers was generalized by Aoki [3] for approximate additive function and by Rassias [4] for approximate linear functions by allowing the difference Cauchy equation to be controlled by . Taking into consideration a lot of influence of Ulam, Hyers, and Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Rassias is called the Hyers-Ulam-Rassias stability. In 1994, a generalization of Rassias theorem was obtained by GΔvruΕ£a [5], who replaced by a general control function (see also [6β24]).

In the sequel we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [25β29]. Throughout this paper, let be the space of distribution functions, that is, and the subset is the set where denotes the left limit of function at the point . The space is partially ordered by the usual pointwise ordering of functions, that is, if and only if for all . The maximal element for in this order is the distribution function given by

*Definition 1.1 (see [28]). *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 Εukasiewicz -norm).

Recall (see [30, 31]) that if is a -norm and is a given sequence of numbers in , is defined recurrently by is defined as .

It is known [31] that for the Εukasiewicz -norm the following implication holds:

*Definition 1.2 (see [29]). *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 1.3. *Let be an 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)An 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 a random Banach space.

Theorem 1.4 (see [28]). *If is an RN space and is a sequence such that , then almost everywhere.*

Theorem 1.5 (see [32, 33]). *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)*, for all ,*(2)*the sequence converges to a fixed point of ,*(3)* is the unique fixed point of in the set ,*(4)* for all .*

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 notion of an RN space corresponds to the situations when we do not know exactly the norm of point and we know only probabilities of possible values of this norm. 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 [34], Mirmostafaee et al. [35β38], MiheΕ£ and Radu [26, 27, 39, 40], MiheΕ£ et al. [41, 42], Baktash et al. [43] and Saadati et al. [44].

In this paper, we consider the -dimensional additive functional equation where is an integer number. It is easy to see that the function is a solution of the functional equation (1.7).

As a special case, if in (1.7), then the functional equation (1.7) reduces to Also by putting in (1.7), we obtain that is,

The main purpose of this paper is to prove the stability of (1.7) in random normed spaces via the fixed point method.

#### 2. Results in RN spaces via Fixed Point Method

Lemma 2.1. *Let and be real vector spaces. A function with satisfies (1.7) if and only if is additive.*

*Proof. *Let satisfy the functional equation (1.7). Hence, according to (1.7), we get
for all . Setting in (2.1), we have
that is,
for all . On the other hand, we have the relation
for all . Hence, we obtain from (2.3) and (2.4) that
for all . Setting in (2.5) we get for all . Replacing and by and in (2.5), respectively, and then using , we obtain that
for all , which implies that is additive.

Conversely, suppose that is additive, and thus satisfies the equation . Hence we have and for all . Replacing and by and in the additive equation and then using lead to
for all .

Now, we are going to prove our assumption by induction on . It holds for ; see (2.7). Assume that (1.7) holds for the case, where ; that is, we have
for all . Replacing by in (2.8), we obtain
for all . Replacing by in (2.9), we obtain
for all . Adding (2.9) to (2.10), one gets
for all . Therefore, it follows from (2.7) and (2.11) that (1.7) holds for . This completes the proof of the theorem.

From now on, let be a linear space and a complete RN space. For convenience, we use the following abbreviation for a given function : for all , where is an integer number.

Theorem 2.2. *Let be a function ( is denoted by ) such that, for some ,
**
for all and all . Suppose that a function with satisfies the inequality
**
for all and all . Then, there exists a unique additive function such that
**
for all and all .*

*Proof. *Letting in (2.14), we get
for all and all . Setting in (2.16), we obtain from (2.4) and that
for all and all , or
for all and all . Let be the set of all functions with and introduce a generalized metric on as follows:
where, as usual, . It is easy to show that is a generalized complete metric space [26, 45].

Now we consider the function defined by
for all and .

Now let such that . Then,
that is, if , we have . This means that
for all , that is, is a strictly contractive self-function on with the Lipschitz constant .

It follows from (2.18) that
for all and all , which implies that .

Due to Theorem 1.5, there exists a function such that is a fixed point of , that is, for all .

Also, as , implies the equality
for all . If we replace with in (2.14), respectively, and divide by , then it follows from (2.13) that
for all and all . By letting in (2.25), we find that for all , which implies , and thus satisfies (1.7). Hence by Lemma 2.1, the function is additive.

According to the fixed point alterative, since is the unique fixed point of in the set , is the unique function such that
for all and all . Again using the fixed point alterative gives
which implies the inequality
for all and all . So,
for all and all . This completes the proof.

Now, we present a corollary that is an application of the last theorem in the classical case.

Corollary 2.3. *Let and , normed linear spaces, define
**
for and . Define
**
for all and all in which . Now, for , (2.13) holds for all and all . Suppose that an odd function satisfies (2.14) for all and all . Then, by the last theorem there exists a unique additive function such that
**
for all and all . Hence,
**
for all .*

Theorem 2.4. *Let be a function such that, for some ,
**
for all and all . Suppose that an odd function satisfies (2.14) for all and all . Then, there exists a unique additive function such that
**
for all and all .*

*Proof. *Letting in (2.14), we get
for all and all . Setting in the last inequality, we obtain by using oddness of and (2.4) that
for all and all , or
for all and all . Let be the set of all odd functions , and introduce a generalized metric on as follows:
It is easy to show that is a generalized complete metric space [26, 45]. Let be the function defined by
for all and . One can show that
for all , that is, is a strictly contractive self-function on with the Lipschitz constant .

It follows from (2.38) that
for all and all , which implies that .

Due to Theorem 1.5, the sequence converges to a fixed point of , that is,
and for all .

Also, is the unique fixed point of in the set , and
implies the inequality
for all and all . This implies that inequality (2.35) holds. Furthermore, we can obtain that the function satisfies (1.7). Hence by Lemma 2.1, we get that the function is additive.

Now, we present a corollary that is an application of the last theorem in the classical case.

Corollary 2.5. *Let and , normed linear spaces, define
**
for and . Define
**
for all and all in which . Now, for , (2.34) holds for all and all . Suppose that an odd function satisfies (2.14) for all and all . Then, by the last theorem there exists a unique additive function such that
**
for all and all . Hence,
**
for all .*

#### Acknowledgments

The authors would like to thank the referees and the Editor Professor Seenith Sivasundaram for giving useful suggestions for the improvement of this paper.

#### References

- S. M. Ulam,
*A Collection of Mathematical Problems*, Interscience Publishers, London, UK, 1960. - D. H. Hyers, βOn the stability of the linear functional equation,β
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 27, pp. 222β224, 1941. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. Aoki, βOn the stability of the linear transformation in Banach spaces,β
*Journal of the Mathematical Society of Japan*, vol. 2, pp. 64β66, 1950. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. M. Rassias, βOn the stability of the linear mapping in Banach spaces,β
*Proceedings of the American Mathematical Society*, vol. 72, no. 2, pp. 297β300, 1978. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - P. Găvruţa, βA generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,β
*Journal of Mathematical Analysis and Applications*, vol. 184, no. 3, pp. 431β436, 1994. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - P. W. Cholewa, βRemarks on the stability of functional equations,β
*Aequationes Mathematicae*, vol. 27, no. 1-2, pp. 76β86, 1984. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - M. Eshaghi Gordji and H. Khodaei, βSolution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 71, no. 11, pp. 5629β5643, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - M. Eshaghi Gordji and H. Khodaei, βOn the generalized Hyers-Ulam-Rassias stability of quadratic functional equations,β
*Abstract and Applied Analysis*, vol. 2009, Article ID 923476, 11 pages, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - C. Park, M. E. Gordji, and H. Khodaei, βA fixed point approach to the Cauchy-Rassias stability of general Jensen type quadratic-quadratic mappings,β
*Bulletin of the Korean Mathematical Society*, vol. 47, no. 5, pp. 987β996, 2010. View at Publisher Β· View at Google Scholar - Z. Gajda, βOn stability of additive mappings,β
*International Journal of Mathematics and Mathematical Sciences*, vol. 14, no. 3, pp. 431β434, 1991. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. H. Hyers, G. Isac, and T. M. Rassias,
*Stability of Functional Equations in Several Variables*, Birkhäuser Boston, Boston, Mass, USA, 1998. - G. Isac and T. M. Rassias, βStability of $\psi $-additive mappings: applications to nonlinear analysis,β
*International Journal of Mathematics and Mathematical Sciences*, vol. 19, no. 2, pp. 219β228, 1996. View at Publisher Β· View at Google Scholar - S.-M. Jung,
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*, Hadronic, Palm Harbor, Fla, USA, 2001. - C. Park, βFixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras,β
*Fixed Point Theory and Applications*, vol. 2007, Article ID 50175, 15 pages, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - C. Park, βHyers-Ulam-Rassias stability of a generalized Apollonius-Jensen type additive mapping and isomorphisms between ${C}^{\ast}$-algebras,β
*Mathematische Nachrichten*, vol. 281, no. 3, pp. 402β411, 2008. View at Publisher Β· View at Google Scholar - V. Radu, βThe fixed point alternative and the stability of functional equations,β
*Fixed Point Theory*, vol. 4, no. 1, pp. 91β96, 2003. View at Google Scholar Β· View at Zentralblatt MATH - T. M. Rassias, βProblem 16; 2, Report of the 27th International Symposium on Functional Equations,β
*Aequationes mathematicae*, vol. 39, article 309, pp. 292β293, 1990. View at Google Scholar - T. M. Rassias, βOn the stability of the quadratic functional equation and its applications,β
*Universitatis Babeş-Bolyai. Studia. Mathematica*, vol. 43, no. 3, pp. 89β124, 1998. View at Google Scholar Β· View at Zentralblatt MATH - T. M. Rassias, βThe problem of S. M. Ulam for approximately multiplicative mappings,β
*Journal of Mathematical Analysis and Applications*, vol. 246, no. 2, pp. 352β378, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. M. Rassias, βOn the stability of functional equations in Banach spaces,β
*Journal of Mathematical Analysis and Applications*, vol. 251, no. 1, pp. 264β284, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. M. Rassias, βOn the stability of functional equations and a problem of Ulam,β
*Acta Applicandae Mathematicae*, vol. 62, no. 1, pp. 23β130, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. M. Rassias and P. Šemrl, βOn the behavior of mappings which do not satisfy Hyers-Ulam stability,β
*Proceedings of the American Mathematical Society*, vol. 114, no. 4, pp. 989β993, 1992. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. M. Rassias and P. Šemrl, βOn the Hyers-Ulam stability of linear mappings,β
*Journal of Mathematical Analysis and Applications*, vol. 173, no. 2, pp. 325β338, 1993. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. M. Rassias and K. Shibata, βVariational problem of some quadratic functionals in complex analysis,β
*Journal of Mathematical Analysis and Applications*, vol. 228, no. 1, pp. 234β253, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - S.-S. Chang, Y. J. Cho, and S. M. Kang,
*Nonlinear Operator Theory in Probabilistic Metric Spaces*, Nova Science, Huntington, NY, USA, 2001. - D. Miheţ and V. Radu, βOn the stability of the additive Cauchy functional equation in random normed spaces,β
*Journal of Mathematical Analysis and Applications*, vol. 343, no. 1, pp. 567β572, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. Miheţ, βThe stability of the additive Cauchy functional equation in non-Archimedean fuzzy normed spaces,β
*Fuzzy Sets and Systems*, vol. 161, no. 16, pp. 2206β2212, 2010 (English). View at Google Scholar - B. Schweizer and A. Sklar,
*Probabilistic Metric Spaces*, North-Holland, New York, NY, USA, 1983. - A. N. Šerstnev, βOn the concept of a stochastic normalized space,β
*Doklady Akademii Nauk SSSR*, vol. 149, pp. 280β283, 1963 (Russian). View at Google Scholar - O. Hadžić and E. Pap,
*Fixed Point Theory in Probabilistic Metric Spaces*, vol. 536, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. - O. Hadžić, E. Pap, and M. Budinčević, βCountable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces,β
*Kybernetika*, vol. 38, no. 3, pp. 363β382, 2002. View at Google Scholar - L. Cădariu and V. Radu, βFixed points and the stability of Jensen's functional equation,β
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 4, no. 1, article 4, p. 7, 2003. View at Google Scholar Β· View at Zentralblatt MATH - J. B. Diaz and B. Margolis, βA fixed point theorem of the alternative, for contractions on a generalized complete metric space,β
*Bulletin of the American Mathematical Society*, vol. 74, pp. 305β309, 1968. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - C. Alsina, βOn the stability of a functional equation arising in probabilistic normed spaces,β in
*General Inequalities*, vol. 80, pp. 263β271, Birkhäuser, Basel, Switzerland, 1987. View at Google Scholar Β· View at Zentralblatt MATH - M. Mirzavaziri and M. S. Moslehian, βA fixed point approach to stability of a quadratic equation,β
*Bulletin of the Brazilian Mathematical Society*, vol. 37, no. 3, pp. 361β376, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. K. Mirmostafaee and M. S. Moslehian, βFuzzy approximately cubic mappings,β
*Information Sciences*, vol. 178, no. 19, pp. 3791β3798, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. K. Mirmostafaee, M. Mirzavaziri, and M. S. Moslehian, βFuzzy stability of the Jensen functional equation,β
*Fuzzy Sets and Systems*, vol. 159, no. 6, pp. 730β738, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. K. Mirmostafaee and M. S. Moslehian, βFuzzy versions of Hyers-Ulam-Rassias theorem,β
*Fuzzy Sets and Systems*, vol. 159, no. 6, pp. 720β729, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. Miheţ, βThe probabilistic stability for a functional equation in a single variable,β
*Acta Mathematica Hungarica*, vol. 123, no. 3, pp. 249β256, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. Miheţ, βThe fixed point method for fuzzy stability of the Jensen functional equation,β
*Fuzzy Sets and Systems*, vol. 160, no. 11, pp. 1663β1667, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. Miheţ, R. Saadati, and S. M. Vaezpour, βThe stability of the quartic functional equation in random normed spaces,β
*Acta Applicandae Mathematicae*, vol. 110, no. 2, pp. 797β803, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. Miheţ, R. Saadati, and S. M. Vaezpour, βThe stability of an additive functional equation in Menger probabilistic $\varphi $-normed spaces,β
*Mathematica Slovaca*, vol. 61, no. 5, pp. 817β826, 2011. View at Publisher Β· View at Google Scholar - E. Baktash, Y. J. Cho, M. Jalili, R. Saadati, and S. M. Vaezpour, βOn the stability of cubic mappings and quadratic mappings in random normed spaces,β
*Journal of Inequalities and Applications*, vol. 2008, Article ID 902187, 11 pages, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - R. Saadati, S. M. Vaezpour, and Y. J. Cho, βA note to paper “On the stability of cubic mappings and quartic mappings in random normed spaces”,β
*Journal of Inequalities and Applications*, vol. 2009, Article ID 214530, 6 pages, 2009. View at Publisher Β· View at Google Scholar - L. Cădariu and V. Radu, βOn the stability of the Cauchy functional equation: a fixed point approach,β in
*Iteration Theory*, vol. 346, pp. 43β52, Karl-Franzens-Universitaet Graz, Graz, Austria, 2004. View at Google Scholar Β· View at Zentralblatt MATH