Fuzzy Stability of Quadratic Functional Equations
- JungRye Lee^{1},
- Sun-Young Jang^{2},
- Choonkil Park^{3} and
- DongYun Shin^{4}Email author
DOI: 10.1155/2010/412160
© Jung Rye Lee et al. 2010
Received: 10 February 2010
Accepted: 11 April 2010
Published: 17 May 2010
Abstract
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 Hyers-Ulam 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].
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 Cauchy if 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 Hyers-Ulam 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.
is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A generalized Hyers-Ulam 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 Hyers-Ulam 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 Hyers-Ulam 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 Hyers-Ulam stability of the quadratic functional equation (1.1) in fuzzy Banach spaces. In Section 3, we prove the generalized Hyers-Ulam 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 Hyers-Ulam Stability of the Quadratic Functional Equation (1.1)
In this section, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.1) in fuzzy Banach spaces.
Theorem 2.1.
for all .
uniformly on .
Proof.
for all , all and all .
for all and all . So we get (2.7) for .
for all integers .
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 , .
for all . Since for all , by , for all . Thus the mapping is quadratic, that is, for all .
It follows that for all . Thus for all .
Corollary 2.2.
for all .
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.
for all .
uniformly on .
Corollary 2.4.
for all .
uniformly on .
Proof.
Define and apply Theorem 2.3 to get the result.
3. Generalized Hyers-Ulam Stability of the Quadratic Functional Equation (1.2)
In this section, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.2) in fuzzy Banach spaces.
Lemma 3.1.
for all .
Proof.
Assume that satisfies (3.1).
for all .
for all , as desired.
Theorem 3.2.
for all .
uniformly on .
Proof.
for all , all , and all .
for all and all . So we get (3.15) for .
for all integers .
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 , .
for all . Since for all , by , for all . By Lemma 3.1, the mapping is quadratic.
It follows that for all . Thus for all .
Corollary 3.3.
for all .
uniformly on .
Proof.
Define and apply Theorem 3.2 to get the result.
Authors’ Affiliations
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