Discrete Dynamics in Nature and Society

Volumeย 2010, Article IDย 140767, 15 pages

http://dx.doi.org/10.1155/2010/140767

## The Fixed Point Method for Fuzzy Approximation of a Functional Equation Associated with Inner Product Spaces

Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

Received 3 July 2010; Accepted 11 September 2010

Academic Editor: Johnย Rassias

Copyright ยฉ 2010 M. Eshaghi Gordji and H. Khodaei. 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

Th. M. Rassias (1984) proved that the norm defined over a real vector space is induced by an inner product if and only if for a fixed integer holds for all The aim of this paper is to extend the applications of the fixed point alternative method to provide a fuzzy stability for the functional equation which is said to be a functional equation associated with inner product spaces.

#### 1. Introduction

Studies on fuzzy normed linear spaces are relatively recent in the field of fuzzy functional analysis. In 1984, Katsaras [1] first introduced the notion of fuzzy norm on a linear space and at the same year Wu and Fang [2] also introduced a notion of fuzzy normed space and gave the generalization of the Kolmogoroff normalized theorem for a fuzzy topological linear space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [3โ6].

Nowadays, fixed point and operator theory play an important role in different areas of mathematics, and its applications, particularly in mathematics, physics, differential equation, game theory and dynamic programming. Since fuzzy mathematics and fuzzy physics along with the classical ones are constantly developing, the fuzzy type of the fixed point and operator theory can also play an important role in the new fuzzy area and fuzzy mathematical physics. Many authors [4, 7โ9] have also proved some different type of fixed point theorems in fuzzy (probabilistic) metric spaces and fuzzy normed linear spaces. In 2003, Bag and Samanta [10] modified the definition of Cheng and Mordeson [11] by removing a regular condition. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed linear spaces [12].

One of the most interesting questions in the theory of functional analysis concerning the Ulam stability problem of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to an exact solution of the given functional equation?

The first stability problem concerning group homomorphisms was raised by Ulam [13] in 1940 and affirmatively solved by Hyers [14]. The result of Hyers was generalized by Aoki [15] for approximate additive function and by Th. M. Rassias [16] 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 Th. M. Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Th. M. Rassias is called the generalized Hyers-Ulam stability. In 1994, a generalization of Th. M. Rassias theorem was obtained by Gฤvruลฃa [17], who replaced by a general control function .

On the other hand, J. M. Rassias [18โ25] considered the Cauchy difference controlled by a product of different powers of norm. However, there was a singular case; for this singularity a counterexample was given by Gฤvruลฃa [26]. This stability phenomen on is called the Ulam-Gฤvruลฃa-Rassias stability (see also [27]).

Theorem 1.1 (J. M. Rassias [18]). *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 and satisfies 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.*

Very recently, K. Ravi [28] in the inequality (1.1) replaced the bound by a mixed one involving the product and sum of powers of norms, that is, .

For more details about the results concerning such problems and mixed product-sum stability (J. M. Rassias Stability), the reader is referred to [29โ41].

Quadratic functional equations were used to characterize inner product spaces [42โ45]. A square norm on an inner product space satisfies the important parallelogram equality The functional equation is related to a symmetric biadditive function [46, 47]. It is natural that this equation is called a quadratic functional equation, and every solution of the quadratic equation (1.4) is said to be a quadratic function.

It was shown by Th. M. Rassias [48] that the norm defined over a real vector space is induced by an inner product if and only if for a fixed integer as follows: for all In [49], Park proved the generalized Hyers-Ulam stability of a functional equation associated with inner product spaces: in fuzzy normed spaces.

The main objective of this paper is to prove the the generalized Hyers-Ulam stability of the following functional equation associated with inner product spaces in fuzzy normed spaces, based on the fixed point method. Interesting new results concerning functional equations associated with inner product spaces have recently been obtained by Park et al. [50โ52] and Najati and Th. M. Rassias [53] as well as for the fuzzy stability of a functional equation associated with inner product spaces by Park [49].

The stability of different functional equations in fuzzy normed spaces and random normed spaces has been studied in [20, 21, 54โ77]. In this paper, we prove the generalized fuzzy stability of a functional equation associated with inner product spaces (1.7).

#### 2. Preliminaries

We start our work with the following notion of fixed point theory. For the proof, refer to [78]. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. [79].

Let be a generalized metric space. An operator satisfies a Lipschitz condition with Lipschitz constant if there exists a constant such that for all If the Lipschitz constant is less than , then the operator is called a strictly contractive operator. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity.

We recall the following theorem by Margolis and Diaz.

Theorem 2.1. *Suppose that one is given a complete generalized metric space and a strictly contractive function with Lipschitz constant , then for each given , either
**
or other exists a natural number such that*(i)* for all *(ii)*the sequence is convergent to a fixed point of ;*(iii)* is the unique fixed point of in *(iv)* for all *

Next, we define the notion of a fuzzy normed linear space.

Let be a real linear space. A function is said to be a fuzzy norm on [10] if and only if the following conditions are satisfied: for all and if and only if for all if ; for all and all is a nondecreasing function on and for all

In the following we will suppose that is left continuous for every

A fuzzy normed linear space is a pair , where is a real linear space and is a fuzzy norm on

Let be a normed linear space, then is a fuzzy norm on

Let be a fuzzy normed linear space. A sequence in is said to be convergent if there exists such that for all In that case, is called the limit of the sequence and we write .

A sequence in is called Cauchy if for each and each there exists such that . If 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.

From now on, let be a linear space, be a fuzzy normed space and be a fuzzy Banach space. For convenience, we use the following abbreviation for a given function : for all , where is a fixed integer.

#### 3. Fuzzy Approximation

In the following theorem, we prove the fuzzy stability of the functional equation (1.7) via fixed point method, for an even case.

Theorem 3.1. *Let be a function such that, for some real number with . Suppose that an even function with satisfies the inequality
**
for all and all then there exists a unique quadratic function such that and
**
for all and all , where
*

*Proof. *Letting , , and in (3.1) and using the evenness of , we obtain
for all and all Interchanging with in (3.4) and using the evenness of , we obtain
for all and all It follows from (3.4) and (3.5) that
for all and all Setting , and in (3.1) and using the evenness of , we obtain
for all and all So we obtain from (3.6) and (3.7) that
for all and all So
for all and all Putting , and in (3.1), we get
for all and all It follows from (3.9) and (3.10) that
for all and all Letting in (3.7) and replacing by in the obtained inequality, we get
for all and all It follows from (3.9), (3.10), (3.11) and (3.12) that
for all and all Applying (3.11) and (3.13), we obtain
for all and all Setting , and in (3.1), we obtain
for all and all It follows from (3.14) and (3.15) that
for all and all Therefore
for all and all which implies that
for all and all . Let be the set of all even 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 [80].

Without loss of generality, we consider . Let us now consider the function defined by for all and 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 (3.18) that
for all and all which implies that .

Due to Theorem 2.1, there exists a function such that is a fixed point of , that is, for all

Also, as implies the equality for all Setting and in (3.1), we obtain that
for all and all By letting in (3.22), we find that for all which implies = 0. Thus satisfies (1.7). Hence the function is quadratic (See Lemma of [53]).

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, we get
which implies the inequality
for all and all So
for all and all This completes the proof.

In the following theorem, we prove the fuzzy stability of the functional equation (1.7) via fixed point method, for an odd case.

Theorem 3.2. *Let be a function such that for some real number with . Suppose that an odd function satisfies the inequality (3.1) for all and all then there exists a unique additive function such that and
**
for all and all , where
*

*Proof. *Letting , and in (3.1) and using the oddness of , we obtain that
for all and all Interchanging with in (3.29) and using the oddness of , we get
for all and all It follows from (3.29) and (3.30) that
for all and all Setting , , and in (3.1) and using the oddness of , we get
for all and all So we obtain from (3.31) and (3.32) that
for all and all Putting , and in (3.1), we obtain
for all and all It follows from (3.33) and (3.34) that
for all and all Replacing and by and in (3.35), respectively, we obtain
for all and all Therefore
for all and all which implies that
for all and all . Let be the set of all odd functions and introduce a generalized metric on as follows:
where, as usual, . So is a generalized complete metric space. We consider the function defined by for all and 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 (3.38) that
for all and all which implies that .

Due to Theorem 2.1, there exists a function such that is a fixed point of , that is, for all

Also, as implies the equality for all Setting and in (3.1), we obtain that
for all and all By letting in (3.42), we find that for all which implies . Thus satisfies (1.7). Hence the function is additive (see Lemma of [53]).

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

The main result of the paper is the following.

Theorem 3.3. *Let be a function such that, for some real number with . Suppose that a function with satisfies (3.1) for all and all then there exist a unique quadratic function and a unique additive function such that
**
for all and all , where and are defined as in Theorems 3.1 and 3.2.*

*Proof. *Let for all then
for all and Hence, in view of Theorem 3.1, there exists a unique quadratic function such that
for all and On the other hand, let for all then, by using the above method from Theorem 3.2, there exists a unique additive function such that
for all and Hence, (3.43) follows from (3.45) and (3.46).

#### Acknowledgment

The second author would like to thank the Office of Gifted Students at Semnan University for its financial support.

#### References

- A. K. Katsaras, โFuzzy topological vector spaces. II,โ
*Fuzzy Sets and Systems*, vol. 12, no. 2, pp. 143โ154, 1984. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - C. Wu and J. Fang, โFuzzy generalization of Klomogoroffs theorem,โ
*Journal of Harbin Institute of Technology*, vol. 1, pp. 1โ7, 1984 (Chinese). View at Google Scholar - C. Felbin, โFinite-dimensional fuzzy normed linear space,โ
*Fuzzy Sets and Systems*, vol. 48, no. 2, pp. 239โ248, 1992. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - I. Kramosil and J. Michálek, โFuzzy metrics and statistical metric spaces,โ
*Kybernetika*, vol. 11, no. 5, pp. 336โ344, 1975. View at Google Scholar ยท View at Zentralblatt MATH - S. V. Krishna and K. K. M. Sarma, โSeparation of fuzzy normed linear spaces,โ
*Fuzzy Sets and Systems*, vol. 63, no. 2, pp. 207โ217, 1994. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - J.-z. Xiao and X.-h. Zhu, โFuzzy normed space of operators and its completeness,โ
*Fuzzy Sets and Systems*, vol. 133, no. 3, pp. 389โ399, 2003. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - T. Bag and S. K. Samanta, โFixed point theorems on fuzzy normed linear spaces,โ
*Information Sciences*, vol. 176, no. 19, pp. 2910โ2931, 2006. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - T. Bag and S. K. Samanta, โSome fixed point theorems in fuzzy normed linear spaces,โ
*Information Sciences*, vol. 177, no. 16, pp. 3271โ3289, 2007. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - V. Gregori and A. Sapena, โOn fixed-point theorems in fuzzy metric spaces,โ
*Fuzzy Sets and Systems*, vol. 125, no. 2, pp. 245โ252, 2002. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - T. Bag and S. K. Samanta, โFinite dimensional fuzzy normed linear spaces,โ
*Journal of Fuzzy Mathematics*, vol. 11, no. 3, pp. 687โ705, 2003. View at Google Scholar ยท View at Zentralblatt MATH - S. C. Cheng and J. N. Mordeson, โFuzzy linear operators and fuzzy normed linear spaces,โ
*Bulletin of the Calcutta Mathematical Society*, vol. 86, no. 5, pp. 429โ436, 1994. View at Google Scholar ยท View at Zentralblatt MATH - T. Bag and S. K. Samanta, โFuzzy bounded linear operators,โ
*Fuzzy Sets and Systems*, vol. 151, no. 3, pp. 513โ547, 2005. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - S. M. Ulam,
*A Collection of Mathematical Problems*, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, London, UK, 1960. - D. H. Hyers, โOn the stability of the linear functional equation,โ
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 27, pp. 222โ224, 1941. View at Google Scholar ยท View at Zentralblatt MATH - T. Aoki, โOn the stability of the linear transformation in Banach spaces,โ
*Journal of the Mathematical Society of Japan*, vol. 2, pp. 64โ66, 1950. View at Google Scholar ยท View at Zentralblatt MATH - Th. M. Rassias, โOn the stability of the linear mapping in Banach spaces,โ
*Proceedings of the American Mathematical Society*, vol. 72, no. 2, pp. 297โ300, 1978. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - P. Găvruţa, โA generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,โ
*Journal of Mathematical Analysis and Applications*, vol. 184, no. 3, pp. 431โ436, 1994. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - J. M. Rassias, โOn approximation of approximately linear mappings by linear mappings,โ
*Journal of Functional Analysis*, vol. 46, no. 1, pp. 126โ130, 1982. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - J. M. Rassias, โSolution of a problem of Ulam,โ
*Journal of Approximation Theory*, vol. 57, no. 3, pp. 268โ273, 1989. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - J. M. Rassias, โTwo new criteria on characterizations of inner products,โ
*Discussiones Mathematicae*, vol. 9, pp. 255โ267, 1988. View at Google Scholar - J. M. Rassias, โFour new criteria on characterizations of inner products,โ
*Discussiones Mathematicae*, vol. 10, pp. 139โ146, 1990. View at Google Scholar ยท View at Zentralblatt MATH - J. M. Rassias, โOn approximation of approximately linear mappings by linear mappings,โ
*Bulletin des Sciences Mathématiques*, vol. 108, no. 4, pp. 445โ446, 1984. View at Google Scholar ยท View at Zentralblatt MATH - J. M. Rassias, โOn the stability of the Euler-Lagrange functional equation,โ
*Chinese Journal of Mathematics*, vol. 20, no. 2, pp. 185โ190, 1992. View at Google Scholar ยท View at Zentralblatt MATH - J. M. Rassias, โSolution of a stability problem of Ulam,โ
*Discussiones Mathematicae*, vol. 12, pp. 95โ103, 1992. View at Google Scholar ยท View at Zentralblatt MATH - J. M. Rassias, โComplete solution of the multi-dimensional problem of Ulam,โ
*Discussiones Mathematicae*, vol. 14, pp. 101โ107, 1994. View at Google Scholar ยท View at Zentralblatt MATH - P. Găvruta, โAn answer to a question of John.M. Rassias concerning the stability of Cauchy equation,โ in
*Advances in Equations and Inequalities*, Hardronic Math. Ser., pp. 67โ71, Hadronic Press, 1999. View at Google Scholar - Z. Gajda, โOn stability of additive mappings,โ
*International Journal of Mathematics and Mathematical Sciences*, vol. 14, no. 3, pp. 431โ434, 1991. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - K. Ravi, M. Arunkumar, and J. M. Rassias, โUlam stability for the orthogonally general Euler-Lagrange type functional equation,โ
*International Journal of Mathematics and Statistics*, vol. 3, pp. 36โ46, 2008. View at Google Scholar - Gh. A. Tabadkan and A. Rahmani, โHyers-Ulam-Rassias and Ulam-Gavruta-Rassias stability of generalized quadratic functional equations,โ
*Advances in Applied Mathematical Analysis*, vol. 4, no. 1, pp. 31โ38, 2009. View at Google Scholar - B. Bouikhalene, E. Elqorachi, and J. M. Rassias, โThe superstability of d'Alembert's functional equation on the Heisenberg group,โ
*Applied Mathematics Letters*, vol. 23, no. 1, pp. 105โ109, 2010. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - H.-X. Cao, J.-R. Lv, and J. M. Rassias, โSuperstability for generalized module left derivations and generalized module derivations on a banach module (I),โ
*Journal of Inequalities and Applications*, vol. 2009, Article ID 718020, 10 pages, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - H.-X. Cao, J.-R. Lv, and J. M. Rassias, โSuperstability for generalized module left derivations and generalized module derivations on a Banach module. II,โ
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 10, no. 3, article 85, 2009. View at Google Scholar - M. Eshaghi Gordji and H. Khodaei, โOn the generalized Hyers-Ulam-Rassias stability of quadratic functional equations,โ
*Abstract and Applied Analysis*, vol. 2009, Article ID 923476, 11 pages, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - M. Eshaghi Gordji, J. M. Rassias, and N. Ghobadipour, โGeneralized Hyers-Ulam stability of generalized
*(N,K)*-derivations,โ*Abstract and Applied Analysis*, vol. 2009, Article ID 437931, 8 pages, 2009. View at Publisher ยท View at Google Scholar - M. Eshaghi Gordji, T. Karimi, and S. Kaboli Gharetapeh, โApproximately
*n*-Jordan homomorphisms on Banach algebras,โ*Journal of Inequalities and Applications*, vol. 2009, Article ID 870843, 8 pages, 2009. View at Publisher ยท View at Google Scholar ยท View at MathSciNet - M. Eshaghi Gordji, M. B. Ghaemi, S. Kaboli Gharetapeh, S. Shams, and A. Ebadian, โOn the stability of ${J}^{\ast}$-derivations,โ
*Journal of Geometry and Physics*, vol. 60, no. 3, pp. 454โ459, 2010. View at Publisher ยท View at Google Scholar - P. Găvruta and L. Găvruta, โA new method for the generalized Hyers-Ulam-Rassias stability,โ
*International Journal of Nonlinear Analysis and Applications*, vol. 1, no. 2, pp. 11โ18, 2010. View at Google Scholar - M. Eshaghi Gordji, S. Kaboli Gharetapeh, J. M. Rassias, and S. Zolfaghari, โSolution and stability of a mixed type additive, quadratic, and cubic functional equation,โ
*Advances in Difference Equations*, vol. 2009, Article ID 826130, 17 pages, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - M. Eshaghi Gordji and A. Najati, โApproximately ${J}^{\ast}$-homomorphisms: a fixed point approach,โ
*Journal of Geometry and Physics*, vol. 60, no. 5, pp. 809โ814, 2010. View at Publisher ยท View at Google Scholar - M. Eshaghi Gordji, S. Zolfaghari, J. M. Rassias, and M. B. Savadkouhi, โSolution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces,โ
*Abstract and Applied Analysis*, vol. 2009, Article ID 417473, 14 pages, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - A. Pietrzyk, โStability of the Euler-Lagrange-Rassias functional equation,โ
*Demonstratio Mathematica*, vol. 39, no. 3, pp. 523โ530, 2006. View at Google Scholar ยท View at Zentralblatt MATH - D. Amir,
*Characterizations of Inner Product Spaces*, vol. 20 of*Operator Theory: Advances and Applications*, Birkhäuser, Basel, Switzerland, 1986. - M. Eshaghi Gordji and H. Khodaei, โSolution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces,โ
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 11, pp. 5629โ5643, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - P. Jordan and J. Von Neumann, โOn inner products in linear, metric spaces,โ
*Annals of Mathematics*, vol. 36, no. 3, pp. 719โ723, 1935. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - Th. M. Rassias and K. Shibata, โVariational problem of some quadratic functionals in complex analysis,โ
*Journal of Mathematical Analysis and Applications*, vol. 228, no. 1, pp. 234โ253, 1998. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - J. Aczél and J. Dhombres,
*Functional Equations in Several Variables*, vol. 31 of*Encyclopedia of Mathematics and its Applications*, Cambridge University Press, Cambridge, Mass, USA, 1989. - Pl. Kannappan, โQuadratic functional equation and inner product spaces,โ
*Results in Mathematics*, vol. 27, no. 3-4, pp. 368โ372, 1995. View at Google Scholar - Th. M. Rassias, โNew characterizations of inner product spaces,โ
*Bulletin des Sciences Mathématiques*, vol. 108, no. 1, pp. 95โ99, 1984. View at Google Scholar ยท View at Zentralblatt MATH - C. Park, โFuzzy stability of a functional equation associated with inner product spaces,โ
*Fuzzy Sets and Systems*, vol. 160, no. 11, pp. 1632โ1642, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - C. Park, W.-G. Park, and A. Najati, โFunctional equations related to inner product spaces,โ
*Abstract and Applied Analysis*, vol. 2009, Article ID 907121, 11 pages, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - C. Park, J. S. Huh, W. J. Min, D. H. Nam, and S. H. Roh, โFunctional equations associated with inner product spaces,โ
*The Journal of Chungcheong Mathematical Society*, vol. 21, pp. 455โ466, 2008. View at Google Scholar - C. Park, J. Lee, and D. Shin, โQuadratic mappings associated with inner product spaces,โ preprint.
- A. Najati and Th. M. Rassias, โStability of a mixed functional equation in several variables on Banach modules,โ
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 3-4, pp. 1755โ1767, 2010. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - S. Abbaszadeh, โIntuitionistic fuzzy stability of a quadratic and quartic functional equation,โ
*International Journal of Nonlinear Analysis and Applications*, vol. 1, no. 2, pp. 100โ124, 2010. View at Google Scholar - C. Alsina, J. Sikorska, and M. S. Tomás,
*Norm Derivatives and Characterizations of Inner Product Spaces*, World Scientific, Hackensack, NJ, USA, 2010. - E. Baktash, Y. J. Cho, M. Jalili, R. Saadati, and S. M. Vaezpour, โOn the stability of cubic mappings and quadratic mappings in random normed spaces,โ
*Journal of Inequalities and Applications*, vol. 2008, Article ID 902187, 11 pages, 2008. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - M. Eshaghi Gordji, M. B. Ghaemi, and H. Majani, โGeneralized Hyers-Ulam-Rassias theorem in Menger probabilistic normed spaces,โ
*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 162371, 11 pages, 2010. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - M. Eshaghi Gordji, M. B. Ghaemi, H. Majani, and C. Park, โGeneralized Ulam-Hyers stability of Jensen functional equation in Šerstnev PN spaces,โ
*Journal of Inequalities and Applications*, vol. 2010, Article ID 868193, 14 pages, 2010. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - M. Eshaghi Gordji, J. M. Rassias, and M. Bavand Savadkouhi, โApproximation of the quadratic and cubic functional equations in RN-spaces,โ
*European Journal of Pure and Applied Mathematics*, vol. 2, no. 4, pp. 494โ507, 2009. View at Google Scholar - M. Eshaghi Gordji and M. B. Savadkouhi, โStability of mixed type cubic and quartic functional equations in random normed spaces,โ
*Journal of Inequalities and Applications*, vol. 2009, Article ID 527462, 9 pages, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH ยท View at MathSciNet - M. Eshaghi Gordji, M. B. Savadkouhi, and C. Park, โQuadratic-quartic functional equations in RN-spaces,โ
*Journal of Inequalities and Applications*, vol. 2009, Article ID 868423, 14 pages, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - H. Khodaei and M. Kamyar, โFuzzy approximately additive mappings,โ
*International Journal of Nonlinear Analysis and Applications*, vol. 1, no. 2, pp. 44โ53, 2010. View at Google Scholar - D. Miheţ, R. Saadati, and S. M. Vaezpour, โThe stability of the quartic functional equation in random normed spaces,โ
*Acta Applicandae Mathematicae*, vol. 110, no. 2, pp. 797โ803, 2010. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - D. Miheţ, R. Saadati, and S. M. Vaezpour, โThe stability of an additive functional equation in Menger
probabilistic $\phi $-normed spaces,โ
*Math. Slovak*. In press. - M. Mohamadi, Y. J. Cho, C. Park, F. Vetro, and R. Saadati, โRandom stability of an additive-quadratic-quartic functional equation,โ
*Journal of Inequalities and Applications*, vol. 2010, Article ID 754210, 18 pages, 2010. View at Publisher ยท View at Google Scholar - M. S. Moslehian and J. M. Rassias, โPower and Euler-Lagrange norms,โ
*The Australian Journal of Mathematical Analysis and Applications*, vol. 4, no. 1, article 17, 2007. View at Google Scholar ยท View at Zentralblatt MATH - M. S. Moslehian and J. M. Rassias, โA characterization of inner product spaces concerning an Euler-Lagrange identity,โ
*Communications in Mathematical Analysis*, vol. 8, no. 2, pp. 16โ21, 2010. View at Google Scholar - C. Park, โFuzzy stability of an additive-quadratic-quartic functional equation,โ
*Journal of Inequalities and Applications*, vol. 2010, Article ID 253040, 22 pages, 2010. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - C. Park, โFuzzy stability of additive functional inequalities with the fixed point alternative,โ
*Journal of Inequalities and Applications*, vol. 2009, Article ID 410576, 17 pages, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - C. Park, โA fixed point approach to the fuzzy stability of an additive-quadratic-cubic functional equation,โ
*Fixed Point Theory and Applications*, vol. 2009, Article ID 918785, 24 pages, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - C. Park and A. Najati, โGeneralized additive functional inequalities in Banach algebras,โ
*International Journal of Nonlinear Analysis and Applications*, vol. 1, no. 2, pp. 54โ62, 2010. View at Google Scholar - C. Park and J. M. Rassias, โStability of the Jensen-type functional equation in ${C}^{\ast}$-algebras: a fixed point approach,โ
*Abstract and Applied Analysis*, vol. 2009, Article ID 360432, 17 pages, 2009. View at Publisher ยท View at Google Scholar ยท View at MathSciNet - V. Radu, โThe fixed point alternative and the stability of functional equations,โ
*Fixed Point Theory*, vol. 4, no. 1, pp. 91โ96, 2003. View at Google Scholar ยท View at Zentralblatt MATH - Y. J. Cho, R. Saadati, and S. M. Vaezpour, โErratum: a note to paper on the stability of cubic mappings and quartic mappings in random normed spaces (Journal of Inequalities and Applications),โ
*Journal of Inequalities and Applications*, vol. 2009, Article ID 214530, 6 pages, 2009. View at Publisher ยท View at Google Scholar ยท View at MathSciNet - S. Shakeri, R. Saadati, and C. Park, โStability of the quadratic functional equation in non-Archimedean
*ℒ*-fuzzy normed spaces,โ*International Journal of Nonlinear Analysis and Applications*, vol. 1, no. 2, pp. 72โ83, 2010. View at Google Scholar - S.-s. Zhang, J. M. Rassias, and R. Saadati, โStability of a cubic functional equation in intuitionistic random normed spaces,โ
*Applied Mathematics and Mechanics*, vol. 31, no. 1, pp. 21โ26, 2010. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - S.-s. Zhang, M. Goudarzi, R. Saadati, and S. M. Vaezpour, โIntuitionistic Menger inner product spaces and applications to integral equations,โ
*Applied Mathematics and Mechanics*, vol. 31, no. 4, pp. 415โ424, 2010. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - J. B. Diaz and B. Margolis, โA fixed point theorem of the alternative, for contractions on a generalized complete metric space,โ
*Bulletin of the American Mathematical Society*, vol. 74, pp. 305โ309, 1968. View at Google Scholar ยท View at Zentralblatt MATH - 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, Boston, Mass, USA, 1998. - L. Cădariu and V. Radu, โOn the stability of the Cauchy functional equation: a fixed point approach,โ in
*Iteration Theory*, vol. 346 of*Grazer Math. Ber.*, pp. 43โ52, Karl-Franzens-Universitaet, Graz, Austria, 2004. View at Google Scholar ยท View at Zentralblatt MATH