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FIXED POINT THEOREM IN COMPLEX VALUED METRIC SPACES USING SIX MAPS
Yogita R. Sharma*
Saffrony Institute Of Technology, Mehsana-384002, India.
(Received on: 30-11-13; Revised & Accepted on: 30-01-14)
ABSTRACT
R
ecently Rahul Tiwari [10] proved common fixed point theorem with six maps in complex valued metric spaces. In thispaper we obtain a common fixed point theorem for six maps in complex valued metric spaces having commuting and weakly compatible and satisfying different type of inequality. Our theorem generalizes and extends the result of R. Tiwari [10].
Mathematics Subject Classification: 47H10, 54H25.
Keywords: Weakly compatible maps, fixed points, common fixed points, complex valued metric spaces.
1. INTRODUCTION
The study of metric space expressed the most common important role to many fields both in pure and applied science [3]. Many authors generalized and extended the notion of a metric space such as vector valued metric space of Perov [2], a cone metric spaces of Huang and Zhang [8], a modular metric spaces of Chistyakov [13], etc.
Azam, Fisher and Khan [1] first introduced the complex valued metric spaces which is more general than well known metric spaces and also gave common fixed point theorems for maps satisfying generalized contraction condition.
2. PRELIMINARIES
Let β be the set of all complex numbers. For π§π§1,π§π§2β β, define partial order β€ on β by π§π§1β€ π§π§2 if and only if
π π π π (π§π§1)β€ π π π π (π§π§2) and πΌπΌπΌπΌ(π§π§1)β€ πΌπΌπΌπΌ(π§π§2).
That is π§π§1β€ π§π§2 if one of the following conditions holds (i) π π π π (π§π§1) =π π π π (π§π§2) and πΌπΌπΌπΌ(π§π§1) =πΌπΌπΌπΌ(π§π§2);
(ii) π π π π (π§π§1) <π π π π (π§π§2) and πΌπΌπΌπΌ(π§π§1) =πΌπΌπΌπΌ(π§π§2); (iii) π π π π (π§π§1) =π π π π (π§π§2) and πΌπΌπΌπΌ(π§π§1) <πΌπΌπΌπΌ(π§π§2); (iv) π π π π (π§π§1) <π π π π (π§π§2) and πΌπΌπΌπΌ(π§π§1) <πΌπΌπΌπΌ(π§π§2);
In particular, we will write π§π§1<π§π§2 if π§π§1β π§π§2 and one of (ii), (iii) and (iv) is satisfied and we will write π§π§1<π§π§2.
Definition: 2.1[1]Let X be a non-empty set and ππ:ππΓππ β β be a map, then d is said to be complex valued metric if (i) 0β€ ππ(π₯π₯,π¦π¦) for all π₯π₯,π¦π¦ β ππ and ππ(π₯π₯,π¦π¦) = 0 if and if only π₯π₯=π¦π¦;
(ii) ππ(π₯π₯,π¦π¦) =ππ(π¦π¦,π₯π₯) for all π₯π₯,π¦π¦ β ππ;
(iii) ππ(π₯π₯,π¦π¦)β€ ππ(π₯π₯,π§π§) +ππ(π§π§,π¦π¦) for all π₯π₯,π¦π¦,π§π§ β ππ.
Pair (ππ,ππ) is called a complex valued metric space.
Example: 2.2 Define a map ππ:βΓβ β β by ππ(π§π§1,π§π§2) =π π ππππ|π§π§1β π§π§2| where ππ β π π . Then (β,ππ) is a complex valued metric.
Corresponding author: Yogita R. Sharma*
Definition: 2.3 [1] Let (ππ,ππ) be a complex valued metric space then
(i) Any point π₯π₯ β ππ is said to be an interior point of π΄π΄ β ππ if there exists 0 <ππ β β such that π΅π΅(π₯π₯,ππ) = {π¦π¦ β ππ|ππ(π₯π₯,π¦π¦) <ππ}β π΄π΄.
(ii) Any point π₯π₯ β ππ is said to be a limit point of A if for every 0 <ππ β β, we have π΅π΅(π₯π₯,ππ)β©(π΄π΄ β ππ)β β . (iii) Any subset A of X is said to be an open if each element of A is an interior point of A.
(iv) Any subset A of X is said to be a closed if each limit point of A belongs to A.
(v) A sub-basis of a Hausdorff topology ππ on ππ is a family given by πΉπΉ= {π΅π΅(π₯π₯,ππ)|π₯π₯ β ππ ππππππ 0 <ππ}.
Definition: 2.4 [1] Let {π₯π₯ππ} be a sequence in complex valued metric space (ππ,ππ) and π₯π₯ β ππ. Then
(i) It is said to be a convergent sequence, {π₯π₯ππ} converges to π₯π₯ and π₯π₯ is the limit point of {π₯π₯ππ}, if for every ππ β β, with 0 <ππ there is a natural number N such that ππ(π₯π₯ππ,π₯π₯) <ππ, for all ππ>ππ. We denote it by limππββπ₯π₯ππ =π₯π₯
(ii) It is said to be a Cauchy sequence, if for every ππ β β, with 0 <ππ there is a natural number N such that ππ(π₯π₯ππ, π₯π₯ππ+πΌπΌ) <ππ , for all ππ>ππ and πΌπΌ β β.
(iii) (ππ,ππ) is said to be complete complex valued metric space if every Cauchy sequence in X is convergent .
Lemma: 2.5 [1] Any sequence {π₯π₯ππ} in complex valued metric space (ππ,ππ), converges to π₯π₯ if and only if
|ππ(π₯π₯ππ, π₯π₯)|β0 as ππ ββ.
Lemma: 2.6 [1] Any sequence {π₯π₯ππ} in complex valued metric space (ππ,ππ), Cauchy sequence if and only if
|ππ(π₯π₯ππ, π₯π₯ππ+πΌπΌ)|β0 as ππ ββ where πΌπΌ β β.
Definition: 2.7 Let S and T be self maps of a non-empty set X. Then (i) Any point π₯π₯ β ππ is said to be a fixed point T if πππ₯π₯=π₯π₯.
(ii) Any point π₯π₯ β ππ is said to be a coincidence point of S and T if πππ₯π₯=πππ₯π₯ and we shall called π€π€=πππ₯π₯=πππ₯π₯ that a point of coincidence of S and T.
(iii) Any point π₯π₯ β ππ is said to be a common fixed point of S and T if πππ₯π₯=πππ₯π₯=π₯π₯.
Definition: 2.8 [5] Two self maps S and T of a non-empty set X are commuting if
πππππ₯π₯=πππππ₯π₯, for all π₯π₯ β ππ.
Definition: 2.9 [12] Let S, T be self maps of metric space (ππ,ππ) then S, T are said to be weakly commuting if
ππ(πππππ₯π₯,πππππ₯π₯)β€ ππ(πππ₯π₯,πππ₯π₯), for all π₯π₯ β ππ.
Definition: 2.10 [6] Let S, T be self maps of metric space (ππ,ππ) then S, T are said to be compatible if
lim
ππββππ(πππππ₯π₯,πππππ₯π₯ππ) = 0
Whenever {π₯π₯ππ} is a sequence in X such that limππββπππ₯π₯ππ = limππββπππ₯π₯ππ =π§π§, for some π§π§ β ππ.
Remark: 2.11 In general, commuting maps are weakly commuting and weakly commuting maps are compatible, but the converse are not necessarily true and some examples can be found in [5-7, 9]
Definition: 2.12[7] Two self maps S, T of a non-empty set X are said to be weakly compatible if πππππ₯π₯=πππππ₯π₯ whenever πππ₯π₯=πππ₯π₯.
Lemma: 2.13 [9] Let ππ:ππ β ππ be a map, then there exists a subset E of X such that ππ(πΈπΈ) =ππ(ππ) and ππ:πΈπΈ β ππ is one to one.
3. MAIN RESULT
Theorem 3.1: Let (ππ,ππ) be a complex valued metric space and ππ,ππ,π π ,ππ,ππ,ππ be self maps of X satisfying the following conditions
ππππ(ππ)β ππ(ππ) and π π ππ(ππ)β ππ(ππ) (3.1)
ππ(π π πππ₯π₯,πππππ¦π¦)β€ ππππ(πππ₯π₯,πππ¦π¦) +πποΏ½ππ(πππ₯π₯,π π πππ₯π₯) +ππ(πππ¦π¦,πππππ¦π¦) +ππ(πππ₯π₯,πππππ¦π¦) +ππ(πππ¦π¦,π π πππ₯π₯)οΏ½ (3.2)
Assume that pairs (ππππ,ππ) and (π π ππ,ππ) are weakly compatible. Pairs (ππ,ππ), (ππ,ππ), (ππ,ππ), (π π ,ππ), (π π ,ππ) and (ππ,ππ) are commuting pairs of maps. Then ππ,ππ,π π ,ππ,ππ and ππ have a unique common fixed point in X.
Proof: Let π₯π₯0β ππ. By (3.1) we can define inductively a sequence {π¦π¦ππ} in ππ such that
π¦π¦2ππ =π π πππ₯π₯2ππ =πππ₯π₯2ππ and π¦π¦2ππ+1=πππππ₯π₯2ππ+1=πππ₯π₯2ππ+2 for all ππ= 1, 2, 3, β¦ (3.3)
By (3.2), we have
ππ(π¦π¦2ππ,π¦π¦2ππ+1) =ππ(π π πππ₯π₯2ππ,πππππ₯π₯2ππ+1)
β€ ππππ(πππ₯π₯2ππ,πππ₯π₯2ππ+1) +ππ οΏ½ππ(πππ₯π₯2ππ,π π πππ₯π₯2ππ) +ππ(πππ₯π₯2ππ+1,πππππ₯π₯2ππ+1) + ππ(πππ₯π₯2ππ,πππππ₯π₯2ππ+1) +ππ(πππ₯π₯2ππ+1,π π πππ₯π₯2ππ) οΏ½
=ππππ(π¦π¦2ππβ1,π¦π¦2ππ) +πποΏ½ππ(π¦π¦2ππβ1,π¦π¦2ππ) +ππ(π¦π¦2ππ,π¦π¦2ππ+1) +ππ(π¦π¦2ππβ1,π¦π¦2ππ+1) +ππ(π¦π¦2ππ,π¦π¦2ππ)οΏ½
β€(ππ+ 2ππ)ππ(π¦π¦2ππβ1,π¦π¦2ππ) + 2ππππ(π¦π¦2ππ,π¦π¦2ππ+1)
Which implies that
ππ(π¦π¦2ππ,π¦π¦2ππ+1)β€1ππβ+22ππππππ(π¦π¦2ππβ1,π¦π¦2ππ) =ππππ(π¦π¦2ππβ1,π¦π¦2ππ) Where ππ=ππ1+2β2ππππ< 1.
Similarly we obtain ππ(π¦π¦2ππ+1,π¦π¦2ππ+2)β€ ππππ(π¦π¦2ππ,π¦π¦2ππ+1)
Therefore,
ππ(π¦π¦ππ+1,π¦π¦ππ+2)β€ ππππ(π¦π¦ππ,π¦π¦ππβ1)β€ β― ππππ+1ππ(π¦π¦0,π¦π¦1) for ππ= 1, 2, 3, β¦
Now, for all πΌπΌ>ππ ,
ππ(π¦π¦ππ,π¦π¦πΌπΌ)β€ ππ(π¦π¦ππ,π¦π¦ππ+1) +ππ(π¦π¦ππ+1,π¦π¦ππ+2) +β― ππ(π¦π¦πΌπΌ β1,π¦π¦πΌπΌ)
β€(ππππ+ππππ+1+β―+πππΌπΌβ1)ππ(π¦π¦1,π¦π¦0)
β€ ππππ
ππβ1ππ(π¦π¦1,π¦π¦0)
ππ|(π¦π¦ππ,π¦π¦πΌπΌ)|β€ ππ ππ
ππ β1|ππ(π¦π¦1,π¦π¦0)|ππππ|ππ(π¦π¦1,π¦π¦0)|
Which implies that ππ|(π¦π¦ππ,π¦π¦πΌπΌ)|β0 as ππ,πΌπΌ ββ. Hence {π¦π¦ππ} is a Cauchy sequence.
Since X is complete, there exists a point π§π§ in ππ such that
lim
ππββπ π πππ₯π₯2ππ = limππββπππ₯π₯2ππ+1= limππββπππππ₯π₯2ππ+1= limππββπππ₯π₯2ππ+2 =π§π§
Since ππππ(ππ)β ππ(ππ), there exists a point π’π’ β ππ such that π§π§=πππ’π’ .
Then by (3.2), we have
ππ(π π πππ’π’,π§π§)β€ ππ(π π πππ’π’,πππππ₯π₯2ππβ1) +ππ(πππππ₯π₯2ππβ1,π§π§)
β€ ππππ(πππ’π’, πππ₯π₯2ππβ1) +ππ οΏ½ ππ(πππ’π’,π π πππ’π’) +ππ(πππ₯π₯2ππβ1,πππππ₯π₯2ππβ1)
+ππ(πππ’π’ ,πππππ₯π₯2ππβ1 ) +ππ(πππ₯π₯2ππβ1 ,π π πππ’π’) +ππ(πππππ₯π₯2ππβ1 ,π§π§)οΏ½
Taking the limit as ββ , we obtain
ππ(π π πππ’π’ ,π§π§)β€ ππππ(π§π§ ,π§π§) +πποΏ½ππ(π§π§ ,π π πππ’π’) +ππ(π§π§ ,π§π§) +ππ(π§π§ ,π§π§) +ππ(π§π§ ,π π πππ’π’)οΏ½+ππ(π§π§,π§π§)
= 2ππππ(π π π π π’π’,π§π§), a contradiction
Then by (3.2), we have
ππ(π§π§,ππππππ) =ππ(π π πππ’π’,ππππππ)
β€ ππππ(πππ’π’,ππππ) +πποΏ½ππ(πππ’π’,π π πππ’π’) +ππ(ππππ,ππππππ) +ππ(πππ’π’,ππππππ) +ππ(ππππ,π π πππ’π’)οΏ½
=ππππ(π§π§,π§π§) +πποΏ½ππ(π§π§,π§π§) +ππ(π§π§,ππππππ) +ππ(π§π§,ππππππ) +ππ(π§π§,π§π§)οΏ½
= 2ππππ(π§π§,ππππππ), which is a contradiction.
Therefore ππππππ=ππππ=π§π§ and so π π πππ’π’=πππ’π’=ππππππ=ππππ=π§π§.
Similarly, Q and TU are weakly compatible maps, we have πππππ§π§=πππ§π§
Now we claim that z is a fixed point of TU. If β π§π§, then by (3.2), we have
ππ(π§π§,πππππ§π§) =ππ(π π πππ§π§,πππππ§π§)
β€ ππππ(πππ§π§,πππ§π§) +πποΏ½ππ(πππ§π§,π π πππ§π§) +ππ(πππ§π§,πππππ§π§) +ππ(πππ§π§,πππππ§π§) +ππ(πππ§π§,π π πππ§π§)οΏ½
=ππππ(π§π§,πππππ§π§) +πποΏ½ππ(π§π§,π§π§) +ππ(πππππ§π§,πππππ§π§) +ππ(π§π§,πππππ§π§) +ππ(πππππ§π§,π§π§)οΏ½
= (ππ+ 2ππ)ππ(π§π§,πππππ§π§), a contradiction.
Therefore πππππ§π§=π§π§. Hence πππππ§π§=πππ§π§=π§π§. We have therefore proved that π π πππ§π§=πππππ§π§=πππ§π§=πππ§π§=π§π§. So z is common fixed point of ππ,ππ,π π ππ and ππ .
By commuting conditions of pairs we have
πππ§π§=ππ(πππππ§π§) =ππ(πππππ§π§) =ππππ(πππ§π§) .
πππ§π§=ππ(πππ§π§) =ππ(πππ§π§) and πππ§π§=ππ(πππππ§π§) = (ππππ)(πππ§π§) = (ππππ)(πππ§π§)
πππ§π§=ππ(πππ§π§) =ππ(πππ§π§), which follows that πππ§π§ and πππ§π§ are common fixed points of (ππππ,ππ)
Then πππ§π§=π§π§=πππ§π§=πππ§π§=πππππ§π§
Similarly π π π§π§=π§π§=πππ§π§=πππ§π§=π π πππ§π§
Therefore z is a common fixed point of ππ,ππ,π π ,ππ,ππ and ππ .
For uniqueness of z, let w be another common fixed point of ππ,ππ,π π ,ππ,ππ and ππ.
Then by (3.2), we have
ππ(π§π§,π€π€) =ππ(π π πππ§π§,πππππ€π€)
β€ ππππ(πππ§π§,πππ€π€) +πποΏ½ππ(πππ§π§,π π πππ§π§) +ππ(πππ€π€,πππππ€π€) +ππ(πππ§π§,πππππ€π€) +ππ(πππ€π€,π π πππ§π§)οΏ½
=ππππ(π§π§,π€π€) +πποΏ½ππ(π§π§,π§π§) +ππ(π€π€,π€π€) +ππ(π§π§,π€π€) +ππ(π€π€,π§π§)οΏ½
= (ππ+ 2ππ)ππ(π§π§,π€π€), a contradiction.
So, =π€π€.
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