- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2012 (2012), Article ID 763728, 14 pages

http://dx.doi.org/10.1155/2012/763728

## On the Stability Problem in Fuzzy Banach Space

^{1}Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran^{2}Department of Mathematics, University Politehnica of Timisoara, Piata Victoriei 2, 300006 Timisoara, Romania^{3}Department of Mathematics, Kangnam University, Suwon, Kyunggi 449-702, Republic of Korea

Received 6 February 2012; Accepted 17 May 2012

Academic Editor: Nicole Brillouet-Belluot

Copyright © 2012 G. Zamani Eskandani 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

We investigate the generalized Ulam-Hyers stability of the Cauchy functional equation and pose two open problems in fuzzy Banach space.

#### 1. Introduction and Preliminaries

In 1940, Ulam [1] asked the first question on the stability problem. In 1941, Hyers [2] solved the problem of Ulam. This result was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an *unbounded Cauchy difference*.

Theorem 1.1 (Th. M. Rassias). *Let be a mapping from a normed vector space into a Banach space subject to the inequality:
**
for all , where and are constants with and . Then, the limit exists for all and is the unique additive mapping which satisfies
**
for all . Also, if for each the function is continuous in , then is linear.*

In 1990, Th. M. Rassias [5] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for . In 1991, Gajda [6] gave an affirmative solution to this question for . It was shown by Gajda [6], as well as by Th. M. Rassias and Šemrl [7], that one cannot prove a Th. M. Rassias type theorem when . Găvruţa [8] proved that the function , if and satisfies (1.1) with but for any additive function . J. M. Rassias [9] replaced the factor by for with (see also [10, 11]) and has obtained the following theorem.

Theorem 1.2. *Let be a real normed linear space and a real complete normed linear space. Assume that is an approximately additive mapping for which there exist constants and such that satisfies the inequality:
**
for all . Then, there exists a unique additive mapping satisfying
**
for all . If, in addition, is a mapping such that the transformation is continuous in for each fixed , then is an -linear mapping.*

In the case , we do not have stability [12]. In 1994, a further generalization of Th. M. Rassias’ Theorem was obtained by Găvruţa [13], in which he replaced the bound by a general control function . Isac and Th. M. Rassias [14] replaced the factor by in Theorem 1.1 and solved stability problem when or , also they asked the question whether such a theorem can be proved for . Găvruţa [8] gave a negative answer to this question. Isac and Th. M. Rassias [15] applied the Ulam-Hyers-Rassias stability theory to prove fixed point theorems and study some new applications in nonlinear analysis. During the last two decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Ulam-Hyers stability to a number of functional equations and mappings (see [16–40]). We also refer the readers to the books of Czerwik [41] and Hyers et al. [42].

Th. M. Rassias [43] has obtained the following theorem and posed a problem.

Theorem 1.3. *Let and be two Banach spaces, and let be a mapping such that is continuous in for each fixed . Assume that there exist and such that
**
for all . Let be a positive integer . Then, there exists a unique linear mapping such that
**
for all , where
*

*Th. M. Rassias Problem*

What is the best possible value of in Theorem 1.3?

Găvruţa et al. have given a generalization of [13] and have answered to Th. M. Rassias problem [44].

In [45], J. M. Rassias et al. have investigated the generalized Ulam-Hyers “product-sum” stability of functional equations and have obtained the following theorem.

Theorem 1.4 (see [45]). *Let be a mapping which satisfies the inequality
**
for all with , where and p are constants with and either , or , with , , and . Then, the limit exists for all and is the unique orthogonally Euler-Lagrange quadratic mapping such that
**
for all .*

Note that the mixed “product-sum” function was introduced by J. M. Rassias in 2008-2009 [46–48].

We recall some basic facts concerning fuzzy normed space.

Let be a real linear space. A function (so-called fuzzy subset) is said to be a fuzzy norm on if for all and all ,

for ;

if and only if for all ;

if ;

;

is a nondecreasing function of and

The pair is called a fuzzy normed linear space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [49–51].

Let be a fuzzy normed space and let be a sequence in . Then, is said to be convergent if there exists such that for all . In that case, is called the limit of the sequence and we denote it by .

A sequence in a fuzzy normed space is called Cauchy if, for each and , one can find some such that

for all .

It is known that every convergent sequence in a fuzzy normed space is Cauchy. If, in a fuzzy-normed space, each Cauchy sequence is convergent, then the fuzzy-norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space.

Stability of Cauchy, Jensen, quadratic, and cubic function equation in fuzzy normed spaces have first been investigated in [50–53].

In this paper, we give a generalization of the results from [13] and pose two open problems in fuzzy Banach space. For convenience, we use the following abbreviation for a given mapping :

#### 2. Stability of the Cauchy Functional Equation

Hereafter, unless otherwise stated, we will assume that is real vector space, is a complete fuzzy norm space and is a fixed integer greater than 1.

Theorem 2.1. *Let be a fuzzy normed space and be a mapping such that, for some with . Suppose that be mapping such that
**
for all and all positive real number . Then, there is a unique additive mapping such that and
**
where .*

*Proof. *By induction on , we show that
for all and all positive real number . Letting in (2.1), we get
So we get (2.3) for .

Assume that (2.3) holds for with . Letting in (2.1), we get
for all . By using (2.3) and (2.5), we get (2.3) for and this completes the induction argument. Replacing by in (2.3), we get
Thus
for all and all positive real number . Hence,
Let and be given. Since , there is some such that . Since , there is some such that for all . It follows that

for all and all nonnegative integers and with . Therefore, the sequence is a Cauchy sequence in for all . Since is complete, the sequence converges in for all . So one can define the mapping by
for all . Now, we show that is an additive mapping. It follows from (2.1) and (2.10) that

for all and all positive real number . Therefore, the mapping is additive. Moreover, if we put in (2.8), we observe that
Therefore,
It follows from (2.13), for large enough , that
Now, we show that is unique. Let be another additive mapping from into , which satisfies the required inequality. Then, for each and , we have
So,
Hence, the right-hand side of the above inequality tends to as . It follows that for all .

Theorem 2.2. *Let be a fuzzy normed space and, be a mapping such that for some with . Suppose that be mapping such that
**
for all and all positive real number . Then, there is a unique additive mapping such that and
**
where .*

*Proof. *Similarly to the proof of Theorem 2.1, we have
for all and all positive real number . Replacing by in (2.19), we get
Thus,
for all and all positive real number . Hence,

Let and be given. Since , there is some such that . Since , there is some such that for all . It follows from (2.22) that
for all and all nonnegative integers and with . Therefore, the sequence is a Cauchy sequence in for all . Since is complete, the sequence converges in for all . So one can define the mapping by
for all . The rest of the proof is similar to the proof of Theorem 2.1

Theorem 2.3. *Let be a normed space, let be a fuzzy normed space, and let be a function such that **,
** for all .**
Suppose that a mapping satisfies the inequality:
**
for all and all positive real number , where is a fixed vector of . Then, there exists a unique additive mapping satisfying and
**
for all , where . Moreover, for all .*

*Proof. *Let
for all . So,
where . By using Theorem 2.1, we can get (2.26). Now, we show that . It follows from (1) that . Replacing by in (2.26), we get
for all . So we have
Using (2) and passing the limit in (2.30), we get .

Theorem 2.4. *Let be a normed space, let be a fuzzy normed space, and let be a function such that **,
** for all .**
Suppose that a mapping satisfies the inequality:
**
for all and all positive real number , where is a fixed vector of . Then, there exists a unique additive mapping satisfying and
**
for all , where
**
Moreover, for all .*

*Proof. *Let
for all . So, we have
where . It follows from (1) that . By using Theorem 2.2, we can get (2.32). Now, we show that . Replacing by in (2.32), we get
for all . So we have
Using (2) and passing the limit in (2.37), we get .

Theorem 2.5. *Let be a normed space, let be a nonnegative real number such that , and let be a homogeneous function of degree . Suppose that be a fuzzy normed space and let be mapping such that
**
for all and all positive real number , where is a fixed vector of . Then, there exists a unique additive mapping such that
**
where .*

*Proof. *The proof follows from Theorems 2.1 and 2.2.

For the particular cases , and , we have the following corollaries.

Corollary 2.6. *Let be a normed space, let be a nonnegative real number such that . Suppose that be a fuzzy normed space and be mapping such that
**
for all and all positive real number , where is a fixed vector of . Then, there exists a unique additive mapping such that
*

Corollary 2.7. *Let be a normed space, be non-negative real numbers such that . Suppose that be a fuzzy normed space and be mapping such that
**
for all and all positive real number , where is a fixed vector of . Then there exists a unique additive mapping such that
*

Corollary 2.8. *Let be a normed space, and let be nonnegative real numbers such that . Suppose that be a fuzzy normed space and let be mapping such that
**
for all and all positive real number , where is a fixed vector of . Then, there exists a unique additive mapping such that
*

Corollary 2.9. *Let be a normed space, let be a nonnegative real number such that . Suppose that be a fuzzy normed space and let be mapping such that
*

*Problem 1. *Whether Theorem 2.5 and/or such Corollaries can be proved for ?

*Problem 2. * What is the best possible value of in Corollaries 2.6 and 2.7?

#### Acknowledgment

G. H. Kim was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0005197).

#### References

- S. M. Ulam,
*A Collection of Mathematical Problems*, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York, NY, USA, 1960. - D. H. Hyers, “On the stability of the linear functional equation,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 27, pp. 222–224, 1941. - T. Aoki, “On the stability of the linear transformation in Banach spaces,”
*Journal of the Mathematical Society of Japan*, vol. 2, pp. 64–66, 1950. - Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 72, no. 2, pp. 297–300, 1978. View at Publisher · View at Google Scholar - Th. M. Rassias, “Problem 16; 2, report of the 27th International Symposium on functional equations,”
*Aequationes Mathematicae*, vol. 39, pp. 292–293, 1990. - Z. Gajda, “On stability of additive mappings,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 14, no. 3, pp. 431–434, 1991. View at Publisher · View at Google Scholar - Th. M. Rassias and P. Šemrl, “On the behavior of mappings which do not satisfy Hyers-Ulam stability,”
*Proceedings of the American Mathematical Society*, vol. 114, no. 4, pp. 989–993, 1992. View at Publisher · View at Google Scholar - P. Găvruţa, “On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 261, no. 2, pp. 543–553, 2001. View at Publisher · View at Google Scholar - J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”
*Bulletin des Sciences Mathématiques*, vol. 108, no. 4, pp. 445–446, 1984. - J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”
*Journal of Functional Analysis*, vol. 46, no. 1, pp. 126–130, 1982. View at Publisher · View at Google Scholar - J. M. Rassias, “Solution of a problem of Ulam,”
*Journal of Approximation Theory*, vol. 57, no. 3, pp. 268–273, 1989. View at Publisher · View at Google Scholar - P. Găvruţă, “An answer to question of John M. Rassias concerning the stability of Cauchy equation,”
*Advanced in Equation and Inequality*, pp. 67–71, 1999. - P. Găvruţa, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 184, no. 3, pp. 431–436, 1994. View at Publisher · View at Google Scholar - G. Isac and Th. M. Rassias, “Functional inequalities for approximately additive mappings,” in
*Stability of Mappings of Hyers-Ulam Type*, Hadronic Press Collection of Original Articles, pp. 117–125, Hadronic Press, Palm Harbor, Fla, USA, 1994. - G. Isac and Th. M. Rassias, “Stability of
*ψ*-additive mappings: applications to nonlinear analysis,”*International Journal of Mathematics and Mathematical Sciences*, vol. 19, no. 2, pp. 219–228, 1996. View at Publisher · View at Google Scholar - C. Baak and M. S. Moslehian, “On the stability of ${J}^{*}$-homomorphisms,”
*Nonlinear Analysis*, vol. 63, no. 1, pp. 42–48, 2005. View at Publisher · View at Google Scholar - B. Bouikhalene, E. Elqorachi, and J. M. Rassias, “The superstability of d'Alembert's functional equation on the Heisenberg group,”
*Applied Mathematics Letters*, vol. 23, no. 1, pp. 105–109, 2010. View at Publisher · View at Google Scholar - L. Cădariu and V. Radu, “The fixed points method for the stability of some functional equations,”
*Carpathian Journal of Mathematics*, vol. 23, no. 1-2, pp. 63–72, 2007. - G. Z. Eskandani, “On the Hyers-Ulam-Rassias stability of an additive functional equation in quasi-Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 345, no. 1, pp. 405–409, 2008. View at Publisher · View at Google Scholar - G. Z. Eskandani, H. Vaezi, and Y. N. Dehghan, “Stability of a mixed additive and quadratic functional equation in non-Archimedean Banach modules,”
*Taiwanese Journal of Mathematics*, vol. 14, no. 4, pp. 1309–1324, 2010. - G. Z. Eskandani, H. Vaezi, and F. Moradlou, “On the Hyers-Ulam-Rassias stability of functional equations in quasi-Banach spaces,”
*International Journal of Applied Mathematics & Statistics*, vol. 15, pp. 1–15, 2009. - P. Găvruţă, “On the Hyers-Ulam-Rassias stability of mappings,” in
*Recent Progress in Inequalities*, G. V. Milovanovic, Ed., vol. 430, pp. 465–469, Kluwer Academic, Dordrecht, The Netherlands, 1998. - P. Găvruţă and L. Cădariu, “General stability of the cubic functional equation,”
*Buletinul Stiintific al Universitătii “Politehnica” din Timişoara. Seria Matematica-Fizica*, vol. 47(61), no. 1, pp. 59–70, 2002. - K. W. Jun and Y. H. Lee, “On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality,”
*Mathematical Inequalities & Applications*, vol. 4, no. 1, pp. 93–118, 2001. View at Publisher · View at Google Scholar - K. Jun, H. Kim, and J. Rassias, “Extended Hyers-Ulam stability for Cauchy-Jensen mappings,”
*Journal of Difference Equations and Applications*, pp. 1–15, 2007. - S. M. Jung,
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*, Hadronic Press, Palm Harbor, Fla, USA, 2001. - S. M. Jung, “Asymptotic properties of isometries,”
*Journal of Mathematical Analysis and Applications*, vol. 276, no. 2, pp. 642–653, 2002. View at Publisher · View at Google Scholar - Pl. Kannappan, “Quadratic functional equation and inner product spaces,”
*Results in Mathematics*, vol. 27, no. 3-4, pp. 368–372, 1995. - H. M. Kim, J. M. Rassias, and Y. S. Cho, “Stability problem of Ulam for Euler-Lagrange quadratic mappings,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 10725, 15 pages, 2007. View at Publisher · View at Google Scholar - Y. S. Lee and S. Y. Chung, “Stability of an Euler-Lagrange-Rassias equation in the spaces of generalized functions,”
*Applied Mathematics Letters of Rapid Publication*, vol. 21, no. 7, pp. 694–700, 2008. View at Publisher · View at Google Scholar - D. Miheţ, “The fixed point method for fuzzy stability of the Jensen functional equation,”
*Fuzzy Sets and Systems*, vol. 160, no. 11, pp. 1663–1667, 2009. View at Publisher · View at Google Scholar - F. Moradlou, H. Vaezi, and G. Z. Eskandani, “Hyers-Ulam-Rassias stability of a quadratic and additive functional equation in quasi-Banach spaces,”
*Mediterranean Journal of Mathematics*, vol. 6, no. 2, pp. 233–248, 2009. View at Publisher · View at Google Scholar - M. S. Moslehian, “On the orthogonal stability of the Pexiderized quadratic equation,”
*Journal of Difference Equations and Applications*, vol. 11, no. 11, pp. 999–1004, 2005. View at Publisher · View at Google Scholar - P. Nakmahachalasint, “On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2007, Article ID 63239, 10 pages, 2007. View at Publisher · View at Google Scholar - C. Park and J. M. Rassias, “Stability of the Jensen-type functional equation in ${C}^{*}$-algebras: a fixed point approach,”
*Abstract and Applied Analysis*, vol. 2009, Article ID 360432, 17 pages, 2009. View at Publisher · View at Google Scholar - A. Pietrzyk, “Stability of the Euler-Lagrange-Rassias functional equation,”
*Demonstratio Mathematica*, vol. 39, no. 3, pp. 523–530, 2006. - J. M. Rassias, J. Lee, and H. M. Kim, “Refinned Hyers-Ulam stability for Jensen type mappings,”
*Journal of the Chungcheong Mathematical Society*, vol. 22, pp. 101–116, 2009. - J. M. Rassias, “Complete solution of the multi-dimensional problem of Ulam,”
*Discussiones Mathematicae*, vol. 14, pp. 101–107, 1994. - K. Ravi and M. Arunkumar, “On the Ulam-Gavruta-Rassias stability of the orthogonally Euler-Lagrange type functional equation,”
*International Journal of Applied Mathematics & Statistics*, vol. 7, pp. 143–156, 2007. - K. Ravi, J. M. Rassias, M. Arunkumar, and R. Kodandan, “Stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 10, no. 4, article 114, pp. 1–29, 2009. - S. Czerwik,
*Functional Equations and Inequalities in Several Variables*, World Scientific, River Edge, NJ, USA, 2002. View at Publisher · View at Google Scholar - D. H. Hyers, G. Isac, and Th. M. Rassias,
*Stability of Functional Equations in Several Variables*, Progress in Nonlinear Differential Equations and their Applications, 34, Birkhäuser, Basle, Switzerland, 1998. View at Publisher · View at Google Scholar - Th. M. Rassias, “On a modified Hyers-Ulam sequence,”
*Journal of Mathematical Analysis and Applications*, vol. 158, no. 1, pp. 106–113, 1991. View at Publisher · View at Google Scholar - P. Găvruţa, M. Hossu, D. Popescu, and C. Căprău, “On the stability of mappings and an answer to a problem of Th. M. Rassias,”
*Annales Mathématiques Blaise Pascal*, vol. 2, no. 2, pp. 55–60, 1995. - K. Ravi, M. Arunkumar, and J. M. Rassias, “Ulam stability for the orthogonally general Euler-Lagrange type functional equation,”
*International Journal of Mathematics and Statistics*, vol. 3, no. A08, pp. 36–46, 2008. - H. X. Cao, J. R. Lv, and J. M. Rassias, “Superstability for generalized module left derivations and generalized module derivations on a Banach module. I,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 718020, 10 pages, 2009. View at Publisher · View at Google Scholar - H. X. Cao, J. R. Lv, and J. M. Rassias, “Superstability for generalized module left derivations and generalized module derivations on a Banach module. II,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 10, no. 3, article 85, pp. 1–8, 2009. - M. B. Savadkouhi, M. E. Gordji, J. M. Rassias, and N. Ghobadipour, “Approximate ternary Jordan derivations on Banach ternary algebras,”
*Journal of Mathematical Physics*, vol. 50, no. 4, Article ID 042303, pp. 1–9, 2009. View at Publisher · View at Google Scholar - T. Bag and S. K. Samanta, “Finite dimensional fuzzy normed linear spaces,”
*Journal of Fuzzy Mathematics*, vol. 11, no. 3, pp. 687–705, 2003. - A. K. Mirmostafaee and M. S. Moslehian, “Fuzzy versions of Hyers-Ulam-Rassias theorem,”
*Fuzzy Sets and Systems*, vol. 159, no. 6, pp. 720–729, 2008. View at Publisher · View at Google Scholar - A. K. Mirmostafaee, M. Mirzavaziri, and M. S. Moslehian, “Fuzzy stability of the Jensen functional equation,”
*Fuzzy Sets and Systems*, vol. 159, no. 6, pp. 730–738, 2008. View at Publisher · View at Google Scholar - A. K. Mirmostafaee and M. S. Moslehian, “Fuzzy approximately cubic mappings,”
*Information Sciences*, vol. 178, no. 19, pp. 3791–3798, 2008. View at Publisher · View at Google Scholar - A. K. Mirmostafaee and M. S. Moslehian, “Fuzzy almost quadratic functions,”
*Results in Mathematics*, vol. 52, no. 1-2, pp. 161–177, 2008. View at Publisher · View at Google Scholar