common fixed point of weakly compatible
TRANSCRIPT
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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
(1.3) there exists a constant such that
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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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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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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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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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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
(1.3) there exists a constant such that
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.
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