Advances in Fuzzy Systems

Volume 2013, Article ID 613604, 8 pages

http://dx.doi.org/10.1155/2013/613604

## Approximate Fixed Point Theorems in Fuzzy Norm Spaces for an Operator

^{1}Faculty of Mathematics, Vali-e-Asr University, Rafsanjan, Iran^{2}Faculty of Mathematics, Yazd University, Yazd, Iran

Received 31 December 2012; Accepted 22 March 2013

Academic Editor: Toly Chen

Copyright © 2013 S. A. M. Mohsenalhosseini and H. Mazaheri. 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 define approximate fixed point and fuzzy diameter in fuzzy norm spaces. We prove theorems for various types of well-known generalized contractions on fuzzy norm spaces with the use of two general lemmas that are given regarding approximate fixed points of operators on fuzzy norm spaces.

#### 1. Introduction

In this paper, starting from the article of Berinde [1], we study some well-known types of operators on fuzzy norm spaces, and we give some fuzzy approximate fixed points of such operators.

Fuzzy set was defined by Zadeh [2]. Katsaras [3], while studying fuzzy topological vector spaces, was the first to introduce in 1984 the idea of fuzzy norm on a linear space. In 1992, Felbin [4] defined a fuzzy norm on a linear space with an associated metric of the Kaleva and Seikkala type [5]. A further development along this line of inquiry took place when, in 1994, Cheng and Mordeson [6] evolved the definition of a further type of fuzzy norm having a corresponding metric of the Kramosil and Michálek type [7].

Chitra and Mordeson [8] introduce a definition of norm fuzzy, and thereafter the concept of fuzzy norm space has been introduced and generalized in different ways by Bag and Samanta in [9–11].

Throughout this paper, the symbols and mean the and the , respectively.

#### 2. Some Preliminary Results

We start our work with the following definitions.

*Definition 1. *Let be a linear space on . A function is called fuzzy norm if and only if for every and for every , the following properties are satisfied::
for every , : if and only if for every , : for every and , : for every , : the function is nondecreasing on , and .A pair is called a fuzzy norm space. Sometimes, we need two additional conditions as follows: : for all . : function is continuous for every and on subset
is strictly increasing.

Let be a fuzzy norm space. For all , we define norm on as follows:

Then is an ascending family of normed on and they are called -norm on corresponding to the fuzzy norm on . Some notation, lemmas, and examples which will be used in this paper are given in the following.

Lemma 2 (see [9]). *Let be a fuzzy norm space such that it satisfies conditions and . Define the function as follows:
**
Then*(a)* is a fuzzy norm on .*(b)*.
*

Lemma 3 (see [9]). *Let be a fuzzy norm space such that it satisfies conditions and , and . Then if and only if
**
for every .** Note that the sequence converges if there exists a such that
**
In this case, is called the limit of .*

*Example 4 (see [9]). *Let be the real or complex vector space and let be defined on as follows:
for all and . Then is a fuzzy norm space, and the function satisfies conditions and for every .

#### 3. Fuzzy Approximate Fixed Point

In the section, we begin with two lemmas which will be used in order to prove all the results given in the same section.

*Definition 5. *Let be a fuzzy norm space, , , and . Then is an -approximate fixed point (fuzzy approximate fixed point) of if for some ,

*Remark 6. *In the rest of the paper we will denote the set of all -approximate fixed points of , for a given , by

*Definition 7. *Let . Then has the fuzzy approximate fixed point property (f.a.f.p.p.) if

Lemma 8. *Let be a fuzzy norm space such that it satisfies conditions and , such that is asymptotic regular for some ; that is,
**
Then has the fuzzy approximate fixed point property.*

* Proof. *Suppose and . Since
s.t. for all for some .

If , then for some ,
Therefore, and . Hence there exists a fuzzy approximate fixed point in .

*Definition 9. *Let be a fuzzy norm space such that satisfies conditions and . We define fuzzy diameter of for some as

Lemma 10. *Let be a fuzzy norm space such that satisfy conditions and , , and . One assumes that for some , *(1)*,
*(2)*for all there exists such that
**Then
*

* Proof. *Let and . Then
We can write
Now by (44), it follows that
So
for some .

Lemma 11. *Let be a fuzzy norm space such that it satisfies conditions and , , and . One assumes that for some ,*(1)*, for all ,*(2)*for all there exists such that
**Then
*

*Definition 12. *A mapping is an -contraction if there exists such that

Proposition 13. *Let be a fuzzy norm space and an -contraction. Then
*

* Proof. *Let and . Consider
But . Therefore
Now by Lemma 8 it follows that , for all .

In 1968, Kannan (see [12, 13]) proved a fixed point theorem for operators which needs not be continuous. We apply it to fuzzy norm space for -approximate fixed points.

*Definition 14. *A mapping is an -Kannan operator if there exists such that
for all .

Proposition 15. *Let be a fuzzy norm space and a Kannan operator. Then
*

* Proof. *Let and . Consider
Therefore
Then
But ; hence . Therefore
Now by Lemma 8 it follows that , for all .

In 1972, Chatterjea (see [14]) considered another one which again does not impose the continuity of the operator. We apply it to fuzzy norm space for -approximate fixed points.

*Definition 16. *A mapping is an -Chatterjea operator if there exists such that
for all .

Proposition 17. *Let be a fuzzy norm space and a Chatterjea operator. Then
*

* Proof. *Let and . Consider
On the other hand,
Then
Hence
But ; hence . Therefore
Now by Lemma 8 it follows that , for all .

We, by combining the three independent contraction conditions above, obtain another -approximation fixed points result for operators which satisfies the following.

*Definition 18. *A mapping is an -Zamfirescu operator if there exist , , , such that for all , at least one of the following is true:

Proposition 19. *Let be a fuzzy norm space and an -Zamfirescu operator. Then
*

* Proof. *Let .

Supposing that holds, we have
Thus

Supposing holds, we have
Thus

Similarly,

Then

Now looking at , (42), (44), and (46), we can denote
and it is easy to see that .

For satisfying at least one of the conditions , , and , we have that
hold.

Using these conditions implied by ) and taking , we have
Then

Therefore
Now by Lemma 8 it follows that , for all .

*Definition 20. *A mapping is an -weak contraction operator if there exist and such that
for all .

Proposition 21. *Let be a fuzzy norm space and an -weak contraction operator. Then
*

* Proof. *Let . Consider
But . Therefore
Now by Lemma 8, it follows that , for all .

For the same operators, we will now use Lemma 10 in order to obtain new results.

Proposition 22. *Let be a fuzzy norm space and an -contraction operator.**Then for some ,
*

*Proof. *Let . Condition in Lemma 10 is satisfied, as one can see in the proof of Proposition 13.

We will show now that also holds for -contractions.

Let and and assume that
and aim to show that there exists an such that
We have

Therefore
which implies .

So for every , there exist such that
Therefore .

Now by Lemma 10 it follows that
for some , which means exactly that
for some .

Proposition 23. *Let be a fuzzy norm space and an -Kannan operator.**Then for some ,
*

* Proof. *Let . As in the proof of Proposition 22 we only verify that condition in Lemma 10 holds.

Let and and assume that
Then
As , we know that and .

Then . So for every there exist such that
Therefore . Now by Lemma 10 it follows that
for some , which means exactly that
for some .

Proposition 24. *Let be a fuzzy norm space and an -Chatterjea operator.**Then for some ,
*

* Proof. *Let . We will only verify that condition in Lemma 10 holds.

Let and and assume that
Then
As , it follows that
Then

So for every there exist such that
Then . Now by Lemma 10 it follows that
for some , which means exactly that
for some .

Proposition 25. *Let be a fuzzy norm space and an -Zamfirescu operator. Then for some ,
**
where , and , , as in Definition 18.*

* Proof. *In the proof of Proposition 19 we have already shown that if satisfies at least one of the conditions , , and from Definition 18, then
hold.

Let . We will only verify that condition in Lemma 10 is satisfied, as holds; see the Proof of Proposition 19.

Let and and assume that

Let and and assume that
Then

So for every there exist such that
Then . Now by Lemma 10 it follows that
for some , which means exactly that
for some .

*Example 26. *Let us consider the linear space , the space of all continuous real valued functions on with the usual linear operations.

Consider the two norms on defined by
Define two fuzzy norms and by
For the fuzzy norm , its -norm is given by for all , and for the fuzzy norm , its -norm is given by
Suppose that with for all . It can easily be seen that for and ,
However, has no any fixed point of course. But by the Proposition 13 for some , has a fuzzy approximate fixed point in . That is, there exists such that

#### 4. Conclusions

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.

We think that this paper could be of interest to the researchers working in the field fuzzy functional analysis in particular, fuzzy approximate fixed point theory. We proved results about fuzzy approximate fixed points and fuzzy diameter on fuzzy norm spaces, starting from a result presented in [1], but the study may go further to other classes of operators, which will be the subject of future papers.

#### Acknowledgment

The authors are extremely grateful to the referees for their helpful suggestions for the improvement of the paper.

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