common fixed point of weakly compatible

Network and Complex Systems www.iiste.org

ISSN 2224-610X (Paper) ISSN 2225-0603 (Online)

Vol 2, No.4, 2012

32

COMMON FIXED POINT OF WEAKLY COMPATIBLE

MAPPINGS UNDER A NEW PROPERTY IN FUZZY METRIC

SPACES

Rajinder Sharma

Department of General Requirements (Mathematics Section)

College of Applied Sciences, Sohar, Sultanate of Oman

E-mail : [email protected]

Abstract: In this paper, we prove some common fixed point theorems for weakly compatible mappings

under a new property in fuzzy metric spaces. We prove a new result under (S-B) property defined by

Sharma and Bamboria.

Keywords: Fixed point; Fuzzy metric space; (S-B) property.

Introduction:

The foundation of fuzzy mathematics was laid down by Zadeh [25] with the evolution of the concept of

fuzzy sets in 1965. The proven result becomes an asset for an applied mathematician due to its enormous

applications in various branches of mathematics which includes differential equations, integral equation

etc. and other areas of science involving mathematics especially in logic programming and electronic

engineering. It was developed extensively by many authors and used in various fields. Especially, Deng

[7], Erceg [8], and Kramosil and Michalek [17] have introduced the concepts of fuzzy metric spaces in

different ways. To use this concept in topology and analysis, several researchers have studied fixed point

theory in fuzzy metric spaces and fuzzy mappings [2],[3],[4],[5], [10],[14], [15] and many others.

Recently, George and Veeramani [12],[13] modified the concept of fuzzy metric spaces introduced by

Kramosil and Michalek and defined the Hausdoff topology of fuzzy metric spaces. They

showed also that every metric induces a fuzzy metric.Grabiec [11] extended the well known fixed point

theorem of Banach [1] and Edelstein [9] to fuzzy metric spaces in the sense of Kramosil and Michalek



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[15].Moreover, it appears that the study of Kramosil and Michalek [17] of fuzzy metric spaces

paves the way for developing a soothing machinery in the field of fixed point theorems in particular,

for the study of contractive type maps. In this paper we prove a new result under (S-B) property defined

by Sharma and Bamboria [22].

Preliminaries

Definition 1 : [20] A binary operation * is called a continuous t-norm if

* is an Abelian topological monoid with the unit 1 such that * * whenever

for all are in

Examples of are * *

Definition 2 : [17] The 3-tuple * is called a fuzzy metric space (shortly FM-space) if

is an arbitrary set, * is a continuous t-norm and M is a fuzzy set in satisfying the

following conditions for all

*

In what follows, will denote a fuzzy metric space. Note that

can be thought as the degree of nearness between and with



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respect to . We identify with for all and

with and we can find some topological properties and examples of fuzzy metric spaces in

(George and Veeramani [12]).

Example 1 : [12] Let be a metric space. Define * * and

for all ,

Then * is a fuzzy metric space. We call this fuzzy metric induced by the metric the

standard fuzzy metric.

In the following example, we know that every metric induces a fuzzy metric.

Lemma 1 : [11] For all is non-decreasing.

Definition 3 : [11] Let * be a fuzzy metric space :

(1) A sequence is said to be convergent to a point , if

,

for all

(2) A sequence called a Cauchy sequence if

, for all and .



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(3) A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.

Remark 1 : Since * is continuous, it follows from that the limit of the sequence in

space is uniquely determined.

Let * be a fuzzy metric space with the following condition:

for all .

Lemma 2 : [6],[18] If for all and for a number k ∈ (0,1),

then

Lemma 3 : [6],[18] Let be a sequence in a fuzzy metric space * with the condition

If there exists a number such that

for all and then is a Cauchy sequence in X.

Definition 4 : [18] Let and be mappings from a fuzzy metric space * into itself. The

mappings and are said to be compatible if

for all whenever is a sequence in such that



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Definition 5 : [6] Let and be mappings from a fuzzy metric space into itself. The

mappings and are said to be compatible of type if,

for all , whenever is a sequence in such that

Definition 6 : [16] A pair of mappings S and T is called weakly compatible pair in fuzzy metric

space if they commute at coincidence points; i.e. , if for some then

Definition 7 : Let and be two self mappings of a fuzzy metric space

* We say that and satisfy the property (S-B) if there exists a sequence in such

that

Example 2 : [22] Let by

Consider the sequence , clearly

Then and satisfy



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Example 3 : [22] Let

Suppose property holds; then there exists in a sequence satisfying

Therefore

Then which is a contradiction since . Hence and do not satisfy the property

Remark 2 : It is clear from the definition of Mishra et al. [18] and Sharma and Deshpande [23] that two

self mappings S and T of a fuzzy metric space * will be non-compatible if there exists at least

one sequence { } in X such that

but is either not equal to 1 or non-existent. Therefore

two noncompatible selfmappings of a fuzzy metric space * satisfy the property

It is easy to see that if and are compatible , then they are weakly compatible and the converse is not

true in general.

Example 2 : Let . Define and by :



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As

Therefore are weakly compatible.

Turkoglu, Kutukcu and Yildiz [24] prove the following :

Theorem A : Let * be a complete fuzzy metric space with * and let

be maps from into itself such that

(i)

(ii) there exists a constant such that

for all and , where , such that

(iii) the pairs and are compatible of type

(iv) and are continuous and .

Then have a unique common fixed point.

Sharma, Pathak and Tiwari [21] improved Theorem A and proved the following .



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Theorem B : * be a complete fuzzy metric space with and let

and be maps from into itself such that

(1.1)

(1.2) there exists a constant such that

for all where such that and

(1.3) the pairs are weak compatible.

Then have a unique common fixed point.

Rawal [19] proved the following :

Theorem C : Let * be a complete fuzzy metric space with . Let

be mappings of into itself such that

(1.1)

(1.2) there exists a constant k ∈ (0, 1) such that



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(1.3) If the pairs are weakly compatible, then

have a unique common fixed point in

Main Results

We prove Theorem C under a new property [22] in the following way.

Theorem 1 : Let * be a fuzzy metric space with for all and condition

Let be mappings of into itself such that

(1.1)

(1.2) satisfies the property




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(1.4) if the pairs are weakly compatible,

(1.5) one of is closed subset of then

have a unique common fixed point in

Proof : Suppose that satisfies the property Then there exists a sequence } in

such that for some

Since there exists in a sequence such that . Hence . Let

us show that . Indeed, in view of (1.3) for we have



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Thus it follows that

Since the t-norm * is continuous and is continuous, letting we have



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It follows that

and we deduce that

Suppose S(X) is a closed subset of X. Then z = Su for some u ∈ X. Subsequently, we have

= .

By (1.3) with we have

Taking the limn → ∞ , we have



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44

Thus .

This gives .

Therefore by Lemma 2, we have . The weak compatibility of and implies that

On the other hand, since

there exists a point such that . We claim that using (1.3) with

, we have

Thus



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45

It follows that

Therefore by Lemma 2, we have

Thus The weak compatibility of and implies

that and Let us show that is a common fixed

point of in view of (1.3) with we have

This gives



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46

Therefore by Lemma 2, we have and is a common fixed point of

Similarly, we can prove that is a common fixed point of and Since we conclude

that is a common fixed point of

If then by (1.3) with we

have

This gives

Thus



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47

By Lemma 2, we have and the common fixed point is a unique. This completes the proof of the

theorem.

If we take in Theorem 1, we have the following.

Corollary 1 : Let *) be a fuzzy metric space with and condition

Let be mappings of into itself such that

(1.3)

(1.4) satisfies the property


for all such that (1.4) if the

pairs and are weakly compatible,

(1.5) one of is closed subset of , then

have a unique common fixed point in X.

References:

[1] Banach, S. : Surles operations danseles ensembles abstracts at leur applications aux equations integrals. Fund.

Math., 3(1922), 133-181.

[2] Bose, B. K. and Sahani, D. : Fuzzy mappings and fixed point theorems, Fuzzy Sets and Systems,

21(1987),53-58.

[3] Butnariu, D.: Fixed points for fuzzy mappings , Fuzzy Sets and Systems, 7(1982),191-207.



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48

[4] Chang, S. S.: Fixed point theorem for fuzzy mappings, Fuzzy Sets and Systems, 17(1985),181-187.

[5] Chang, S.S., Cho, Y.J., Lee, B.S. and Lee, G.M. : Fixed degree and fixed point theorems for fuzzy mappings,

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[6] Cho, Y. J. : Fixed points in fuzzy metric spaces, J. Fuzzy Math., 5(4)(1997), 949- 962.

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[13] George, A. and Veeramani, P. : On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems,

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[17] Kramosil, I. and Michalek, J. : Fuzzy metric and statistical metric spaces, Kybernetika, 11(1975), 336-344.

[18] Mishra, S. N., Sharma, N. and Singh, S. L. : Common fixed points of maps in fuzzy metric spaces. Internat. J.

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[19] Rawal, M. : On common fixed point theorems in metric spaces and other spaces, Ph.D. thesis, Vikram Univ.,

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[20] Schweizer, B. and Sklar, A. : Statistical metric spaces, Pacific. J. Math., 10(1960), 313-334.

[21] Sharma, Sushil, Pathak, A. and Tiwari, R. : Common fixed point of weakly compatible maps without continuity

in fuzzy metric space, Internat. J. Appl. Math.Vol.20,No.4 (2007),495-506.

[22] Sharma, Sushil and Bamboria, D. : Some new common fixed point theorems in fuzzy

metric space under strict contractive conditions, J. Fuzzy Math., Vol. 14 No.2 (2006),1-11.

[23] Sharma, Sushil and Deshpande, B. : Common fixed point theorems for finite number of mappings without

continuity and compatibility on fuzzy metric spaces, Mathematica Moravica, Vol.12(2008),1-19.

[24] Turkoglu, D., Kutukcu, S. and Yildiz, C. : Common fixed point of compatible maps of type (α) on fuzzy metric

spaces, Internat. J. Appl. Math., 18(2)(2005),189-202.

[25] Zadeh, L.A. : Fuzzy Sets, Inform Contr., 8(1965), 338-353.

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