The fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces have been investigated by Moslehian et al. In this paper, we prove the generalized HyersUlam stability of the following quadratic functional equations and in fuzzy Banach spaces.
1. Introduction and Preliminaries
Katsaras [1] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [2–4]. In particular, Bag and Samanta [5], following Cheng and Mordeson [6], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [7]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [8].
We use the definition of fuzzy normed spaces given in [5, 9, 10] to investigate a fuzzy version of the generalized HyersUlam stability for the quadratic functional equations
in the fuzzy normed vector space setting, where are nonzero real numbers with .
Definition 1.1 (see [5, 9, 10]).
Let be a real vector space. A function is called a fuzzy norm on if, for all and all ,
for ,
if and only if for all ,
if ,
,
is a nondecreasing function of and ,
for , is continuous on .
The pair is called a fuzzy normed vector space.
The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [9, 10].
Definition 1.2 (see [5, 9, 10]).
Let be a fuzzy normed vector space. A sequence in is said to be convergent or converges if there exists an such that for all . In this case, is called the limit of the sequence and we denote it by .
Definition 1.3 (see [5, 9, 10]).
Let be a fuzzy normed vector space. A sequence in is called Cauchyif for each and each there exists an such that, for all and all , we have .
It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.
We say that a mapping between fuzzy normed vector spaces and is continuous at a point if, for each sequence converging to in , the sequence converges to . If is continuous at each , then is said to be continuous on (see [8]).
The stability problem of functional equations is originated from a question of Ulam [11] concerning the stability of group homomorphisms. Hyers [12] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [13] for additive mappings and by Th. M. Rassias [14] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [14] has provided a lot of influence in the development of what we call generalized HyersUlam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [15] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.
A square norm on an inner product space satisfies the parallelogram equality
The functional equation
is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A generalized HyersUlam stability problem for the quadratic functional equation was proved by Skof [16] for mappings , where is a normed space and is a Banach space. Cholewa [17] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. In [18], Czerwik proved the generalized HyersUlam stability of the quadratic functional equation. During the last two decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized HyersUlam stability to a number of functional equations and mappings (see [19–31]).
This paper is organized as follows. In Section 2, we prove the generalized HyersUlam stability of the quadratic functional equation (1.1) in fuzzy Banach spaces. In Section 3, we prove the generalized HyersUlam stability of the quadratic functional equation (1.2) in fuzzy Banach spaces.
Throughout this paper, assume that is a vector space and that is a fuzzy Banach space. Let be nonzero real numbers with .
2. Generalized HyersUlam Stability of the Quadratic Functional Equation (1.1)
In this section, we prove the generalized HyersUlam stability of the quadratic functional equation (1.1) in fuzzy Banach spaces.
Theorem 2.1.
Let be a function such that
for all . Let be a mapping with such that
uniformly on . Then  exists for each and defines a quadratic mapping such that if for some
for all , then
for all .
Furthermore, the quadratic mapping is a unique mapping such that
uniformly on .
Proof.
For a given , by (2.2), we can find some such that
for all . By induction on , we show that
for all , all and all .
Letting in (2.6), we get
for all and all . So we get (2.7) for .
Assume that (2.7) holds for . Then
This completes the induction argument. Letting and replacing and by and in (2.7), respectively, we get
for all integers .
It follows from (2.1) and the equality
that for a given there is an such that
for all and . Now we deduce from (2.10) that
for all and all . Thus the sequence is Cauchy in . Since is a fuzzy Banach space, the sequence converges to some . So we can define a mapping by ; namely, for each and , .
Let . Fix and . Since , there is an such that for all . Hence for all , we have
The first four terms on the righthand side of the above inequality tend to 1 as , and the fifth term is greater than
which is greater than or equal to . Thus
for all . Since for all , by , for all . Thus the mapping is quadratic, that is, for all .
Now let, for some positive and , (2.3) hold. Let
for all . Let . By the same reasoning as in the beginning of the proof, one can deduce from (2.3) that
for all positive integers . Let . We have
Combining (2.18) and (2.19) and the fact that , we observe that
for large enough . Thanks to the continuity of the function , we see that . Letting , we conclude that
To end the proof, it remains to prove the uniqueness assertion. Let be another quadratic mapping satisfying (2.5). Fix . Given that , by (2.5) for and , we can find some such that
for all and all . Fix some and find some integer such that
for all . Since
we have
It follows that for all . Thus for all .
Corollary 2.2.
Let and let be a real number with . Let be a mapping with such that
uniformly on . Then  exists for each and defines a quadratic mapping such that if for some
for all , then
for all .
Furthermore, the quadratic mapping is a unique mapping such that
uniformly on .
Proof.
Define and apply Theorem 2.1 to get the result.
Similarly, we can obtain the following. We will omit the proof.
Theorem 2.3.
Let be a function such that
for all . Let be a mapping satisfying (2.2) and . Then  exists for each and defines a quadratic mapping such that if for some
for all , then
for all .
Furthermore, the quadratic mapping is a unique mapping such that
uniformly on .
Corollary 2.4.
Let and let be a real number with . Let be a mapping satisfying (2.26) and . Then  exists for each and defines a quadratic mapping such that if for some
for all , then
for all .
Furthermore, the quadratic mapping is a unique mapping such that
uniformly on .
Proof.
Define and apply Theorem 2.3 to get the result.
3. Generalized HyersUlam Stability of the Quadratic Functional Equation (1.2)
In this section, we prove the generalized HyersUlam stability of the quadratic functional equation (1.2) in fuzzy Banach spaces.
Lemma 3.1.
Let and be real vector spaces. If a mapping satisfies and
for all , then the mapping is quadratic, that is,
for all .
Proof.
Assume that satisfies (3.1).
Letting in (3.1), we get
for all .
Letting in (3.1), we get
for all . Replacing by in (3.4), we get
for all . It follows from (3.4) and (3.5) that for all . So
for all . Thus
for all . Replacing and by and in (3.7), respectively, we get
for all , as desired.
Theorem 3.2.
Let be a function such that
for all . Let be a mapping with such that
uniformly on . Then  exists for each and defines a quadratic mapping such that if for some
for all , then
for all .
Furthermore, the quadratic mapping is a unique mapping such that
uniformly on .
Proof.
For a given , by (3.10), we can find some such that
for all . By induction on , we show that
for all , all , and all .
Letting in (3.14), we get
for all and all . So we get (3.15) for .
Assume that (3.15) holds for . Then
This completes the induction argument. Letting and replacing and by and in (3.15), respectively, we get
for all integers .
It follows from (3.9) and the equality
that for a given there is an such that
for all and . Now we deduce from (3.18) that
for each and all . Thus the sequence is Cauchy in . Since is a fuzzy Banach space, the sequence converges to some . So we can define a mapping by ; namely, for each and , .
Let . Fix and . Since , there is an such that for all . Hence for each , we have
The first four terms on the righthand side of the above inequality tend to 1 as , and the fifth term is greater than
which is greater than or equal to . Thus
for all . Since for all , by , for all . By Lemma 3.1, the mapping is quadratic.
Now let, for some positive and , (3.18) hold. Let
for all . Let . By the same reasoning as in the beginning of the proof, one can deduce from (3.18) that
for all positive integers . Let . We have
Combining (3.26) and (3.27) and the fact that , we observe that
for large enough . Thanks to the continuity of the function , we see that . Letting , we conclude that
To end the proof, it remains to prove the uniqueness assertion. Let be another quadratic mapping satisfying (3.1) and (3.13). Fix . Given that , by (3.13) for and , we can find some such that
for all and all . Fix some and find some integer such that
for all . Since
we have
It follows that for all . Thus for all .
Corollary 3.3.
Let and let be a real number with if and with if . Let be a mapping with such that
uniformly on . Then  exists for each and defines a quadratic mapping such that if for some
for all , then
for all .
Furthermore, the quadratic mapping is a unique mapping such that
uniformly on .
Proof.
Define and apply Theorem 3.2 to get the result.
References

Katsaras, AK: Fuzzy topological vector spaces. II. Fuzzy Sets and Systems. 12(2), 143–154 (1984). Publisher Full Text

Felbin, C: Finitedimensional fuzzy normed linear space. Fuzzy Sets and Systems. 48(2), 239–248 (1992). Publisher Full Text

Krishna, SV, Sarma, KKM: Separation of fuzzy normed linear spaces. Fuzzy Sets and Systems. 63(2), 207–217 (1994). Publisher Full Text

Xiao, JZ, Zhu, XH: Fuzzy normed space of operators and its completeness. Fuzzy Sets and Systems. 133(3), 389–399 (2003). Publisher Full Text

Bag, T, Samanta, SK: Finite dimensional fuzzy normed linear spaces. Journal of Fuzzy Mathematics. 11(3), 687–705 (2003)

Cheng, SC, Mordeson, JN: Fuzzy linear operators and fuzzy normed linear spaces. Bulletin of the Calcutta Mathematical Society. 86(5), 429–436 (1994)

Kramosil, I, Michálek, J: Fuzzy metrics and statistical metric spaces. Kybernetika. 11(5), 336–344 (1975)

Bag, T, Samanta, SK: Fuzzy bounded linear operators. Fuzzy Sets and Systems. 151(3), 513–547 (2005). Publisher Full Text

Mirmostafaee, AK, Mirzavaziri, M, Moslehian, MS: Fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems. 159(6), 730–738 (2008). Publisher Full Text

Mirmostafaee, AK, Moslehian, MS: Fuzzy versions of HyersUlamRassias theorem. Fuzzy Sets and Systems. 159(6), 720–729 (2008). Publisher Full Text

Ulam, SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8,p. xiii+150. Interscience, New York, NY, USA (1960)

Hyers, DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America. 27, 222–224 (1941). PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Aoki, T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan. 2, 64–66 (1950). Publisher Full Text

Rassias, ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society. 72(2), 297–300 (1978). Publisher Full Text

Găvruţa, P: A generalization of the HyersUlamRassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications. 184(3), 431–436 (1994). Publisher Full Text

Skof, F: Proprietà locali e approssimazione di operatori. Rendiconti del Seminario Matematico e Fisico di Milano. 53(1), 113–129 (1983). Publisher Full Text

Cholewa, PW: Remarks on the stability of functional equations. Aequationes Mathematicae. 27(12), 76–86 (1984). Publisher Full Text

Czerwik, S: On the stability of the quadratic mapping in normed spaces. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 62, 59–64 (1992). Publisher Full Text

Czerwik, S: Functional Equations and Inequalities in Several Variables,p. x+410. World Scientific, River Edge, NJ, USA (2002)

Gajda, Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences. 14(3), 431–434 (1991). Publisher Full Text

Hyers, DH, Isac, G, Rassias, ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications,p. vi+313. Birkhäuser, Boston, Mass, USA (1998)

Jung, SM: HyersUlamRassias Stability of Functional Equations in Mathematical Analysis,p. ix+256. Hadronic Press, Palm Harbor, Fla, USA (2001)

Rassias, ThM: New characterizations of inner product spaces. Bulletin des Sciences Mathématiques. 108(1), 95–99 (1984)

Rassias, ThM: On the stability of the quadratic functional equation and its applications. Studia Universitatis BabeşBolyai. Mathematica. 43(3), 89–124 (1998)

Rassias, ThM: The problem of S. M. Ulam for approximately multiplicative mappings. Journal of Mathematical Analysis and Applications. 246(2), 352–378 (2000). Publisher Full Text

Rassias, ThM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications. 251(1), 264–284 (2000). Publisher Full Text

Rassias, ThM: On the stability of functional equations and a problem of Ulam. Acta Applicandae Mathematicae. 62(1), 23–130 (2000). Publisher Full Text

Rassias, ThM: Problem 16; 2, report of the 27th International Symposium on Functional Equations. Aequationes Mathematicae. 39, 292–293 (1990)

Rassias, ThM, Šemrl, P: On the behavior of mappings which do not satisfy HyersUlam stability. Proceedings of the American Mathematical Society. 114(4), 989–993 (1992). Publisher Full Text

Rassias, ThM, Šemrl, P: On the HyersUlam stability of linear mappings. Journal of Mathematical Analysis and Applications. 173(2), 325–338 (1993). Publisher Full Text

Rassias, ThM, Shibata, K: Variational problem of some quadratic functionals in complex analysis. Journal of Mathematical Analysis and Applications. 228(1), 234–253 (1998). Publisher Full Text