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Fuzzy Stability of Quadratic Functional Equations

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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 [24]. 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 Hyers-Ulam stability for the quadratic functional equations

(1.1)
(1.2)

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.

A square norm on an inner product space satisfies the parallelogram equality

(1.3)

The functional equation

(1.4)

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 [1931]).

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.

Let be a function such that

(2.1)

for all . Let be a mapping with such that

(2.2)

uniformly on . Then - exists for each and defines a quadratic mapping such that if for some

(2.3)

for all , then

(2.4)

for all .

Furthermore, the quadratic mapping is a unique mapping such that

(2.5)

uniformly on .

Proof.

For a given , by (2.2), we can find some such that

(2.6)

for all . By induction on , we show that

(2.7)

for all , all and all .

Letting in (2.6), we get

(2.8)

for all and all . So we get (2.7) for .

Assume that (2.7) holds for . Then

(2.9)

This completes the induction argument. Letting and replacing and by and in (2.7), respectively, we get

(2.10)

for all integers .

It follows from (2.1) and the equality

(2.11)

that for a given there is an such that

(2.12)

for all and . Now we deduce from (2.10) that

(2.13)

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

(2.14)

The first four terms on the right-hand side of the above inequality tend to 1 as , and the fifth term is greater than

(2.15)

which is greater than or equal to . Thus

(2.16)

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

(2.17)

for all . Let . By the same reasoning as in the beginning of the proof, one can deduce from (2.3) that

(2.18)

for all positive integers . Let . We have

(2.19)

Combining (2.18) and (2.19) and the fact that , we observe that

(2.20)

for large enough . Thanks to the continuity of the function , we see that . Letting , we conclude that

(2.21)

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

(2.22)

for all and all . Fix some and find some integer such that

(2.23)

for all . Since

(2.24)

we have

(2.25)

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

(2.26)

uniformly on . Then - exists for each and defines a quadratic mapping such that if for some

(2.27)

for all , then

(2.28)

for all .

Furthermore, the quadratic mapping is a unique mapping such that

(2.29)

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

(2.30)

for all . Let be a mapping satisfying (2.2) and . Then - exists for each and defines a quadratic mapping such that if for some

(2.31)

for all , then

(2.32)

for all .

Furthermore, the quadratic mapping is a unique mapping such that

(2.33)

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

(2.34)

for all , then

(2.35)

for all .

Furthermore, the quadratic mapping is a unique mapping such that

(2.36)

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.

Let and be real vector spaces. If a mapping satisfies and

(3.1)

for all , then the mapping is quadratic, that is,

(3.2)

for all .

Proof.

Assume that satisfies (3.1).

Letting in (3.1), we get

(3.3)

for all .

Letting in (3.1), we get

(3.4)

for all . Replacing by in (3.4), we get

(3.5)

for all . It follows from (3.4) and (3.5) that for all . So

(3.6)

for all . Thus

(3.7)

for all . Replacing and by and in (3.7), respectively, we get

(3.8)

for all , as desired.

Theorem 3.2.

Let be a function such that

(3.9)

for all . Let be a mapping with such that

(3.10)

uniformly on . Then - exists for each and defines a quadratic mapping such that if for some

(3.11)

for all , then

(3.12)

for all .

Furthermore, the quadratic mapping is a unique mapping such that

(3.13)

uniformly on .

Proof.

For a given , by (3.10), we can find some such that

(3.14)

for all . By induction on , we show that

(3.15)

for all , all , and all .

Letting in (3.14), we get

(3.16)

for all and all . So we get (3.15) for .

Assume that (3.15) holds for . Then

(3.17)

This completes the induction argument. Letting and replacing and by and in (3.15), respectively, we get

(3.18)

for all integers .

It follows from (3.9) and the equality

(3.19)

that for a given there is an such that

(3.20)

for all and . Now we deduce from (3.18) that

(3.21)

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

(3.22)

The first four terms on the right-hand side of the above inequality tend to 1 as , and the fifth term is greater than

(3.23)

which is greater than or equal to . Thus

(3.24)

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

(3.25)

for all . Let . By the same reasoning as in the beginning of the proof, one can deduce from (3.18) that

(3.26)

for all positive integers . Let . We have

(3.27)

Combining (3.26) and (3.27) and the fact that , we observe that

(3.28)

for large enough . Thanks to the continuity of the function , we see that . Letting , we conclude that

(3.29)

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

(3.30)

for all and all . Fix some and find some integer such that

(3.31)

for all . Since

(3.32)

we have

(3.33)

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

(3.34)

uniformly on . Then - exists for each and defines a quadratic mapping such that if for some

(3.35)

for all , then

(3.36)

for all .

Furthermore, the quadratic mapping is a unique mapping such that

(3.37)

uniformly on .

Proof.

Define and apply Theorem 3.2 to get the result.

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Correspondence to DongYun Shin.

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Keywords

  • Functional Equation
  • Quadratic Mapping
  • Unique Mapping
  • Quadratic Functional Equation
  • Fuzzy Normed Space