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FIXED POINT THEOREM IN CONE
ππ
ππ-METRIC SPACE OVER BANACH ALGEBRA
M. RANGAMMA, P. RAMA BHADRA MURTHY*
Department of Mathematics, Osmania University, Telangana, India-500007.
(Received On: 21-08-16; Revised & Accepted On: 07-10-16)
ABSTRACT
T
he concept of cone ππ2-metric space is introduced as a generalization of cone 2-metric space and ππ2-metric space. We proved Hardy and Roger's type fixed point theorem in cone ππ2-metric space over banach algebra. Our results generalize many well known results in fixed point theory.Mathematics Subject Classification: 47H10.
Keywords: fixed point theorems, cone metric spaces, banach algebra, spectral radius.
1. INTRODUCTION
Czerwik introduced the concept of b-metric space as a generalization of metric spaces [12]. Many fixed point theorems were proved by several authors using b-metric [13]. The concept of 2-metric space was introduced by Gahler in 1960's keeping in view the area of triangle as an example [16]. Naidu and Prasad showed that every convergent sequence in a 2-metric space need not be a Cauchy sequence. Different authors proved that there is no relation between 2-metric and ordinary metric and that the contraction mappings in ordinary metric spaces and 2-metric spaces are unrelated. For fixed point theorems in 2-metric spaces [1], [14], [15], [16]. In 2007, Huang and Zang introduced the concept of cone metric space by generalizing ordinary metric space [7]. Several authors proved many fixed and common fixed point theorems in cone metric spaces. Liu and Xu proved fixed point theorems in cone metric spaces over banach algebra. For fixed point theorems in cone metric spaces, cone b-metric spaces. See [3], [5], [8], [9], [11], [18], [19]. In 2012, B.Singh et.al, introduced cone 2-metric space by generalizing 2-metric and cone metric and proved some fixed point theorems. For some results on cone 2-metric space, see [2], [6], [17]. Recently, Zead Mustafa et. al, generalized the above 2-metric and b-metric spaces and introduced ππππ-metric space [10]. They proved some fixed point theorems in partially ordered ππππ-metric space.
In this work, we generalized ππππ-metric space and cone metric space as cone ππππ-metric space giving example for the existance of such spaces. We then proved a fixed point theorem using Hardy and Rogerβs [4] type contractive condition over banach algebra. An example is given supporting our main theorem.
2. PRELIMINARY NOTES
Let π΄π΄ always represents a real Banach Algebra. Then we have the following properties: For all π₯π₯,π¦π¦,π§π§ β ππ andππ β β: 1. (π₯π₯π¦π¦)π§π§=π₯π₯(π¦π¦π§π§)
2. π₯π₯(π¦π¦+π§π§) =π₯π₯π¦π¦+π₯π₯π§π§and (π₯π₯+π¦π¦)π§π§=π₯π₯π¦π¦+π₯π₯π§π§ 3. ππ(π₯π₯π¦π¦) = (πππ₯π₯)π¦π¦=π₯π₯(πππ¦π¦)
4. Ηπ₯π₯π¦π¦Ηβ€Ηπ₯π₯ΗΗπ¦π¦Η
Throughout this paper, we shall assume that the Banach Algebra has a unit (multiplicative identity) ππ such that
πππ₯π₯=π₯π₯ππ=π₯π₯ for all π₯π₯ β π΄π΄. An element π₯π₯ β π΄π΄ is said to be invertible if there is an element π¦π¦ β π΄π΄ such that
π₯π₯π¦π¦=π¦π¦π₯π₯=ππ. The inverse of π₯π₯ is denoted by π₯π₯β1.
Preposition 2.1 [8]: Let π΄π΄ be a real Banach algebra with a unit ππ and π₯π₯ β π΄π΄. If the spectral radius ππ(π₯π₯) of π₯π₯ is less than
1. ππ.ππ. ,ππ(π₯π₯) = limπ₯π₯βππβπ₯π₯ππβ1ππ < 1, then ππ β π₯π₯ is invertible and (ππ β π₯π₯)β1=ββππ=0π₯π₯ππ.
Corresponding Author: P. Rama Bhadra Murthy*
Remark 2.2 [18] For all π₯π₯ β π΄π΄, we have ππ(π₯π₯)β€ βπ₯π₯β.
Definition 2.3 [18] Let π΄π΄ be a Banach algebra and ππ be a subset of πΈπΈ. ππ is called a cone if: 1. ππ is closed, nonempty and ππ β {ππ}.
2. ππ,ππ β π π ,ππ,ππ β₯0,π₯π₯,π¦π¦ β ππ β πππ₯π₯+πππ¦π¦ β ππ. 3. π₯π₯ β ππ and βπ₯π₯ β ππ β π₯π₯=ππ
For a cone ππ β π΄π΄, define a partial ordering βΌ w.r.t ππ as follows: 1. π₯π₯ βΌ π¦π¦ iff π¦π¦ β π₯π₯ β ππ.
2. π₯π₯ βΊ π¦π¦ will stand for π₯π₯ βΌ π¦π¦ and π₯π₯ β π¦π¦.
3. π₯π₯ βͺ π¦π¦ means π¦π¦ β π₯π₯ β ππππππππ, where ππππππππ denotes interior of ππ.
If ππππππππ β β , then ππ is called a solid cone.
The cone ππ is called normal if there is a number ππ> 0 such that for all π₯π₯,π¦π¦ β π΄π΄, we have
ππ βΌ π₯π₯ βΌ π¦π¦ βΉ βπ₯π₯β β€ ππβπ¦π¦β.
The least positive number satisfying the above is called the normal constant of ππ. Throughout this paper, we shall assume that ππ is a solid cone.
Definition 2.4: Let ππ be a non-empty set and π π β₯1. Suppose the mapping ππ:ππΓππΓππ β π΄π΄satisfies: 1. for every pair of distinct points π₯π₯,π¦π¦ β ππ, there exists a point π§π§ β ππ such that ππ(π₯π₯,π¦π¦,π§π§)β ππ. 2. ππ βΌ ππ(π₯π₯,π¦π¦,π§π§), for all π₯π₯,π¦π¦,π§π§ β ππ and ππ(π₯π₯,π¦π¦,π§π§) =ππ if and only if at least two of π₯π₯,π¦π¦,π§π§ are equal. 3. ππ(π₯π₯,π¦π¦,π§π§) =ππ(ππ(π₯π₯,π¦π¦,π§π§)) for all π₯π₯,π¦π¦,π§π§ β ππ and for all permutations ππ(π₯π₯,π¦π¦,π§π§) of π₯π₯,π¦π¦,π§π§.
4. ππ(π₯π₯,π¦π¦,π§π§)βΌ π π [ππ(π₯π₯,π¦π¦,π€π€) +ππ(π₯π₯,π€π€,π§π§) +ππ(π€π€,π¦π¦,π§π§)] for all permutations π₯π₯,π¦π¦,π§π§,π€π€ β ππ.
Then ππ is called a cone ππππ-metric on ππ, and (ππ,ππ) is called a cone ππππ-metric space.
Example 2.5: Let ππ=β,π΄π΄=β2 and β(π₯π₯,π¦π¦)β= |π₯π₯| + |π¦π¦| for all (π₯π₯,π¦π¦)β π΄π΄. Let ππ= {(π₯π₯,π¦π¦):π₯π₯,π¦π¦ β₯0} be a normal cone. Define ππ1:ππ2β π΄π΄ as ππ1(πΌπΌ,π½π½) = (βπΌπΌ β π½π½βππ,ππβπΌπΌ β π½π½βππ) for all πΌπΌ,π½π½ β π΄π΄ where ππ β₯0 and ππ> 1 are constants.
Now define ππ:ππ3β π΄π΄ as ππ(π₯π₯,π¦π¦,π§π§) =ππ1(πΌπΌ,π½π½) where πΌπΌ,π½π½ β ππ are such that βπΌπΌ β π½π½β= min{βπ₯π₯ β π¦π¦β,βπ¦π¦ β π§π§β,βπ§π§ β π₯π₯β}.
Then it follows from [12] that (ππ,ππ) is a cone ππππ-metric space with π π = 3ππβ1 but not a cone 2-metric space.
Definition 2.6 Let {π₯π₯ππ} be a sequence in cone ππππ-metric space over banach algebra (ππ,ππ). Then 1. {π₯π₯ππ} is said to be ππππ-convergent to π₯π₯ β ππ if limππββππ(π₯π₯ππ,π₯π₯,ππ) =ππ for all ππ β ππ. 2. {π₯π₯ππ} is said to be ππππ-Cauchy sequence in ππ if limππββππ(π₯π₯ππ,π₯π₯ππ,ππ) =ππ for all π₯π₯ β ππ. 3. (ππ,ππ) is said to be ππππ-complete if every ππππ-Cauchy sequence is a ππππ-convergent in ππ.
Definition 2.7 [18] Let ππ be a solid cone in a Banach algebra π΄π΄. A sequence {π’π’ππ}β ππ is a ππ βsequence if for each ππ with ππ βͺ ππ there exist ππ β β such that π’π’ππβͺ ππ for all ππ β₯ ππ.
Let (ππ,ππ) be a cone metric space over banach algebra. The following properties are often used (particularly when dealing with non-normal cone).
(ππ1) If π₯π₯ βΌ π¦π¦ and π¦π¦ βΌ π§π§, then π₯π₯ βͺ π§π§.
(ππ2) If ππ βΌ π₯π₯ βͺ ππ for each ππ β ππππππππ, then π₯π₯=ππ. (ππ3) If π₯π₯ βΌ π¦π¦+ππ for each ππ β ππππππππ, then π₯π₯ βΌ π¦π¦.
(ππ4) If ππ βΌ π₯π₯ βΌ π¦π¦ and ππ βΌ π’π’ then ππ βΌ π’π’π₯π₯ βΌ π’π’π¦π¦.
(ππ5) If ππ βΌ π₯π₯ππ βΌ π¦π¦ππ for each ππ β β and limππββπ₯π₯ππ =π₯π₯ and limππββπ¦π¦ππ =π¦π¦, then ππ βΌ π₯π₯ βΌ π¦π¦. (ππ6) If ππ βΌ ππ(π₯π₯ππ,π₯π₯)βΌ ππππ and ππππβ ππ, then π₯π₯ππ β π₯π₯.
(ππ7) If π’π’ βΌ πππ’π’, where π’π’ β ππ and 0 <ππ< 1, then π’π’=ππ.
(ππ8) If ππ β ππππππππ, ππ βΌ π’π’ππ and βπ’π’ππβ β0 then there exist an ππ β β such that π’π’ππ βͺ ππ for all ππ β₯ ππ. (ππ9) If {π₯π₯ππ} and {π¦π¦ππ} are ππ βsequences and ππ,ππ> 0, then {πππ₯π₯ππ+πππ¦π¦ππ} is also a ππ βsequence.
(ππ10) If {π’π’ππ} is a ππ βsequence in ππ and ππ β ππ is an arbitrarily given vector, then {πππ’π’ππ} is also a ππ βsequence. (ππ11) If {π₯π₯ππ} and {π¦π¦ππ} are sequences in π΄π΄ with π₯π₯ππβ π₯π₯ and π¦π¦ππ β π¦π¦ as ππ β β,
where π₯π₯,π¦π¦ β π΄π΄, then we have π₯π₯πππ¦π¦ππ β π₯π₯π¦π¦ as ππ β β.
(ππ12) If {π₯π₯ππ} is a ππ βsequence in ππ converging to π₯π₯ β ππ, then {ππ(π₯π₯ππ,π₯π₯)} and {ππ(π₯π₯ππ,π₯π₯ππ+ππ)} for any ππ β β are also ππ βsequences.
(ππ15) If π₯π₯,π¦π¦ β π΄π΄ and π₯π₯,π¦π¦ commute, then the following holds: (ππ) ππ(π₯π₯π¦π¦)β€ ππ(π₯π₯)ππ(π¦π¦)
(ππ) ππ(π₯π₯+π¦π¦)β€ ππ(π₯π₯) +ππ(π¦π¦) and (ππ) |ππ(π₯π₯)β ππ(π¦π¦)|β€ ππ(π₯π₯ β π¦π¦).
(ππ16) If ππ β π΄π΄ such that 0β€ ππ(ππ) < 1, then we have
ππ((ππ β π₯π₯)β1)β€(1β ππ(ππ))β1.
3. MAIN RESULTS
Lemma 3.1 Let (ππ,ππ) be a complete cone ππ2-metric space over banach algebra π΄π΄with coefficient π π β₯1 and ππ be a solid cone. If {π₯π₯ππ} is any sequence in ππ, then we have
ππ(π₯π₯ππ+1,π₯π₯ππ,ππ)βΌ ππππ(π₯π₯ππ,π₯π₯ππβ1,ππ)βΉ ππ(π₯π₯ππ+1,π₯π₯ππ,π₯π₯ππ) =ππ (3.1) for all ππ β ππ and ππ β ββ{0} where ππ β ππ with ππ(ππ)β[0,1
π π ).
Proof: Since ππ(π₯π₯ππ+1,π₯π₯ππ,ππ)βΌ ππππ(π₯π₯ππ,π₯π₯ππβ1,ππ), repeated application of this to ππ(π₯π₯ππ+1,π₯π₯ππ,ππ) gives
ππ(π₯π₯ππ+1,π₯π₯ππ,ππ)βΌ ππππππ(π₯π₯1,π₯π₯0,ππ). (3.2)
If ππ<ππ, then ππ(π₯π₯ππ+1,π₯π₯ππ,π₯π₯ππ)βΌ ππππ(π₯π₯ππ,π₯π₯ππβ1,π₯π₯ππ)
βΌ ππ2ππ(π₯π₯
ππβ1,π₯π₯ππβ2,π₯π₯ππ) β¦ β¦ β¦ β¦
βΌ ππππβππππ(π₯π₯
ππ+1,π₯π₯ππ,π₯π₯ππ) =ππ.
β΄ ππ(π₯π₯ππ+1,π₯π₯ππ,π₯π₯ππ) =ππ for all ππ with ππ<ππ. (3.3)
If ππ>ππ, then
ππ(π₯π₯ππ+1,π₯π₯ππ,π₯π₯ππ)βΌ π π [ππ(π₯π₯ππ+1,π₯π₯ππ,π₯π₯ππβ1) +ππ(π₯π₯ππ+1,π₯π₯ππβ1,π₯π₯ππ) +ππ(π₯π₯ππβ1,π₯π₯ππ,π₯π₯ππ)] =π π ππ(π₯π₯ππ+1,π₯π₯ππ,π₯π₯ππβ1) Using (3.3) βΌ π π 2[ππ(π₯π₯ππ+1,π₯π₯ππ,π₯π₯ππβ2) +ππ(π₯π₯ππ+1,π₯π₯ππβ2,π₯π₯ππβ1) +ππ(π₯π₯ππβ2,π₯π₯ππ,π₯π₯ππβ1)] =π π 2ππ(π₯π₯ππ+1,π₯π₯ππ,π₯π₯ππβ2) Using (3.3) βΌ π π 3ππ(π₯π₯ππ+1,π₯π₯ππ,π₯π₯ππβ3)
β¦. β¦ β¦ β¦ βΌ π π ππβππππ(π₯π₯ππ+1,π₯π₯ππ,π₯π₯ππ) =ππ.
β΄ ππ(π₯π₯ππ+1,π₯π₯ππ,π₯π₯ππ) =ππ for all ππ with ππ>ππ. (3.4)
Since the case is obvious with ππ=ππ, from (3.3) and (3.4) we have,
ππ(π₯π₯ππ+1,π₯π₯ππ,π₯π₯ππ) =ππ for all ππ β ββ{0}.
Lemma 3.2 If the hypothesis of Lemma 3.1 hold, then for any ππ>ππ, we have
ππ(π₯π₯ππ,π₯π₯ππ,ππ)βΌ π π ππππ(ππ β π π ππ)β1ππ(π₯π₯1,π₯π₯0,ππ). (3.5)
Proof For any ππ>ππ,
ππ(π₯π₯ππ,π₯π₯ππ,ππ)βΌ π π [ππ(π₯π₯ππ,π₯π₯ππ,π₯π₯ππ+1) +ππ(π₯π₯ππ,π₯π₯ππ+1,ππ) +ππ(π₯π₯ππ+1,π₯π₯ππ,ππ)] =π π ππππππ(π₯π₯1,π₯π₯0,ππ) +π π ππ(π₯π₯
ππ+1,π₯π₯ππ,ππ) Using (3.1) and (3.2)
βΌ π π ππππππ(π₯π₯1,π₯π₯0,ππ) +π π 2[ππ(π₯π₯
ππ+1,π₯π₯ππ,π₯π₯ππ+2) +ππ(π₯π₯ππ+1,π₯π₯ππ+2,ππ) +ππ(π₯π₯ππ+2,π₯π₯ππ,ππ)]
βΌ π π ππππππ(π₯π₯1,π₯π₯0,ππ) +π π 2ππππ+1ππ(π₯π₯1,π₯π₯0,ππ) +π π 2ππ(π₯π₯
ππ+2,π₯π₯ππ,ππ) β¦ β¦ β¦ β¦
βΌ π π ππππππ(π₯π₯1,π₯π₯0,ππ) +π π 2ππππ+1ππ(π₯π₯1,π₯π₯0,ππ) +π π 3ππππ+2ππ(π₯π₯1,π₯π₯0,ππ) β¦ β¦
+π π ππβππβ2ππππβ3ππ(π₯π₯1,π₯π₯0,ππ) +π π ππβππβ2ππ(π₯π₯
ππβ2,π₯π₯ππ,ππ)
βΌ(π π ππππ+π π 2ππππ+1+π π 3ππππ+2+β―+π π ππβππβ2ππππβ3)ππ(π₯π₯1,π₯π₯0,ππ)
+π π ππβππβ1ππππβ2ππ(π₯π₯
1,π₯π₯0,ππ) +π π ππβππβ1ππ(π₯π₯ππβ1,π₯π₯ππ,ππ)
βΌ(π π ππππ+π π 2ππππ+1+π π 3ππππ+2+β―+π π ππβππβ2ππππβ3
+π π ππβππβ1ππππβ2)ππ(π₯π₯1,π₯π₯0,ππ) +π π ππβππβ1ππππβ1ππ(π₯π₯1,π₯π₯0,ππ) βΌ(π π ππππ+π π 2ππππ+1+β―+π π ππβππβ1ππππβ2+π π ππβππππππβ1)ππ(π₯π₯
1,π₯π₯0,ππ)
=π π ππππ(1 +π π ππ+π π 2ππ2+β―+π π ππβππβ1ππππβππβ1)ππ(π₯π₯ 1,π₯π₯0,ππ)
βΌ π π πππποΏ½οΏ½(π π ππ)ππ β
ππ=1
οΏ½ ππ(π₯π₯1,π₯π₯0,ππ)
βΌ π π ππππ(ππ β π π ππ)β1ππ(π₯π₯
Theorem 3.3 Let (ππ,ππ) be a compelte cone ππ2-metric space over banach algebra π΄π΄ with coefficient π π β₯1 and ππ be a solid cone. Let ππ:ππ β ππ be a mapping satisfying
ππ(πππ₯π₯,πππ¦π¦,ππ)βΌ ππ1ππ(π₯π₯,π¦π¦,ππ) +ππ2ππ(π₯π₯,πππ₯π₯,ππ) +ππ3ππ(π¦π¦,πππ¦π¦,ππ) +ππ4ππ(π₯π₯,πππ¦π¦,ππ) +ππ5ππ(π¦π¦,πππ₯π₯,ππ) (3.6) for all π₯π₯,π¦π¦,ππ β ππ where
2π π ππ(ππ1) + (π π + 1)ππ(ππ2+ππ3)(π π 2+π π )ππ(ππ4+ππ5) < 2. (3.7)
Then ππ has a unique fixed point in ππ.
Proof Let π₯π₯0β ππ be arbitrary. Define a sequence {π₯π₯ππ} by π₯π₯ππ+1=πππ₯π₯ππ for all ππ β ββ{0}.
If π₯π₯ππ =π₯π₯ππ+1 for some ππ, then it can be proved that π₯π₯ππ is a fixed point of ππ. Assume π₯π₯ππ β π₯π₯ππ+1 for all ππ β₯0.
ππ(π₯π₯ππ+1,π₯π₯ππ,ππ) =ππ(πππ₯π₯ππ,πππ₯π₯ππβ1,ππ)
βΌ ππ1ππ(π₯π₯ππ,π₯π₯ππβ1,ππ) +ππ2ππ(π₯π₯ππ,π₯π₯ππ+1,ππ) +ππ3ππ(π₯π₯ππβ1,π₯π₯ππ,ππ) +ππ4ππ(π₯π₯ππ,π₯π₯ππ,ππ) +ππ5ππ(π₯π₯ππβ1,π₯π₯ππ+1,ππ)
βΌ ππ1ππ(π₯π₯ππ,π₯π₯ππβ1,ππ) +ππ2ππ(π₯π₯ππ,π₯π₯ππ+1,ππ) +ππ3ππ(π₯π₯ππβ1,π₯π₯ππ,ππ)
+ππ5π π [ππ(π₯π₯ππβ1,π₯π₯ππ+1,π₯π₯ππ) +ππ(π₯π₯ππβ1,π₯π₯ππ,ππ) +ππ(π₯π₯ππ,π₯π₯ππ+1,ππ)]
ππ.ππ.,(ππ β ππ2β π π ππ5)ππ(π₯π₯ππ+1,π₯π₯ππ,ππ)βΌ(ππ1+ππ3+π π ππ5)ππ(π₯π₯ππ,π₯π₯ππβ1,ππ)
ππ.ππ., ππ(π₯π₯ππ+1,π₯π₯ππ,ππ)βΌ πΌπΌππ(π₯π₯ππ,π₯π₯ππβ1,ππ) (3.8)
where πΌπΌ= (ππ β ππ2β π π ππ5)β1(ππ1+ππ3+π π ππ5)β ππ.
Because of symmetry of the cone ππ2-metric ππ, we have
ππ(π₯π₯ππ+1,π₯π₯ππ,ππ) =ππ(π₯π₯ππ,π₯π₯ππ+1,ππ)
=ππ(πππ₯π₯ππβ1,πππ₯π₯ππ,ππ)
βΌ ππ1ππ(π₯π₯ππβ1,π₯π₯ππ,ππ) +ππ2ππ(π₯π₯ππβ1,π₯π₯ππ,ππ) +ππ3ππ(π₯π₯ππ,π₯π₯ππ+1,ππ) +ππ4ππ(π₯π₯ππβ1,π₯π₯ππ+1,ππ) +ππ5ππ(π₯π₯ππ,π₯π₯ππ,ππ)
βΌ ππ1ππ(π₯π₯ππβ1,π₯π₯ππ,ππ) +ππ2ππ(π₯π₯ππβ1,π₯π₯ππ,ππ) +ππ3ππ(π₯π₯ππ,π₯π₯ππ+1,ππ) +ππ4[ππ(π₯π₯ππβ1,π₯π₯ππ+1,π₯π₯ππ) +ππ(π₯π₯ππβ1,π₯π₯ππ,ππ) +ππ(π₯π₯ππ,π₯π₯ππ+1,ππ)]
ππ.ππ.,(ππ β ππ3β π π ππ4)ππ(π₯π₯ππ+1,π₯π₯ππ,ππ)βΌ(ππ1+ππ2+π π ππ4)ππ(π₯π₯ππ,π₯π₯ππβ1,ππ)
Or ππ(π₯π₯ππ+1,π₯π₯ππ,ππ)βΌ π½π½ππ(π₯π₯ππ,π₯π₯ππβ1,ππ) (3.9)
where π½π½= (ππ β ππ3β π π ππ4)β1(ππ1+ππ2+π π ππ4)β ππ.
Using (ππ14) and (ππ15) we have
ππ(πΌπΌ)β€ππ(ππ1)+ππ(ππ3)+π π ππ(ππ5)
1βππ(ππ2)βπ π ππ(ππ5) and ππ(π½π½)β€
ππ(ππ1)+ππ(ππ3)+π π ππ(ππ4)
1βππ(ππ3)βπ π ππ(ππ4)
If ππ(πΌπΌ) >1
π π and ππ(π½π½) >
1
π π then we get
π π ππ(ππ1) +ππ(ππ2) +π π ππ(ππ3) + (π π 2+π π )ππ(ππ5) > 1 and π π ππ(ππ1) +ππ(ππ2) +π π ππ(ππ3) + (π π 2+π π )ππ(ππ4) > 1.
Upon adding the above two inequalities we get
2π π ππ(ππ1) + (π π + 1)ππ(ππ2+ππ3)(π π 2+π π )ππ(ππ4+ππ5) > 2
which is a contradiction to the hypothesis (3.7). Thus we have either ππ(πΌπΌ) <1
π π or ππ(π½π½) <
1
π π or both. Take
ππ=οΏ½
πΌπΌ ππππ ππ(πΌπΌ) <1π π
π½π½ ππππ ππ(π½π½) <1π π
πΌπΌπππππ½π½ ππππ ππππππβπππππππππππ’π’ππ.
Then ππ(π₯π₯ππ+1,π₯π₯ππ,ππ)βΌ ππππ(π₯π₯ππ,π₯π₯ππβ1,ππ) for all ππ β₯0 where ππ β ππ with ππ(ππ)β οΏ½0,1 π π οΏ½.
Keeping in view Lemma (3.1) and (3.2), for ππ>ππ we have,
ππ(π₯π₯ππ,π₯π₯ππ,ππ)βΌ π π ππππ(ππ β π π ππ)β1ππ(π₯π₯1,π₯π₯0,ππ).
Now we see that this π₯π₯ is a fixed point of ππ. For that considering ππ(πππ₯π₯,π₯π₯,ππ) with triangle inequality and (3.6), we have
ππ(πππ₯π₯,π₯π₯,ππ)βΌ π π [ππ(πππ₯π₯,π₯π₯,πππ₯π₯ππ) +ππ(πππ₯π₯,πππ₯π₯ππ,ππ) +ππ(πππ₯π₯ππ,π₯π₯,ππ)]
βΌ π π ππ(π₯π₯ππ+1,π₯π₯,πππ₯π₯) +π π [ππ1ππ(π₯π₯,π₯π₯ππ,ππ) +ππ2ππ(π₯π₯,πππ₯π₯,ππ) +ππ3ππ(π₯π₯ππ,π₯π₯ππ+1,ππ) +ππ4ππ(π₯π₯,π₯π₯ππ+1,ππ) +ππ5ππ(π₯π₯ππ,πππ₯π₯,ππ)] +π π ππ(π₯π₯ππ+1,π₯π₯,ππ)
βΌ π π ππ(π₯π₯ππ+1,π₯π₯,πππ₯π₯) +π π ππ1ππ(π₯π₯,π₯π₯ππ,ππ) +π π ππ2ππ(π₯π₯,πππ₯π₯,ππ) +π π ππ3ππ(π₯π₯ππ,π₯π₯ππ+1,ππ)
+π π ππ4ππ(π₯π₯,π₯π₯ππ+1,ππ) +π π 2ππ5[ππ(π₯π₯ππ,πππ₯π₯,ππ) +ππ(π₯π₯ππ,π₯π₯,ππ) + +ππ(π₯π₯,πππ₯π₯,ππ)+π π ππ(π₯π₯ππ+1,π₯π₯,ππ)
Letting ππ β β, we get ππ(πππ₯π₯,π₯π₯,ππ)βΌ(π π ππ2+π π 2ππ5)ππ(πππ₯π₯,π₯π₯,ππ).
Because of Symmetry, we also get ππ(πππ₯π₯,π₯π₯,ππ)βΌ(π π ππ3+π π 2ππ4)ππ(πππ₯π₯,π₯π₯,ππ).
Upon adding the above two inequalities, we get
ππ(πππ₯π₯,π₯π₯,ππ)βΌ[2π π (ππ2+ππ3) +π π
2
2(ππ4+ππ5)]ππ(πππ₯π₯,π₯π₯,ππ) (3.10)
Using (3.7) we have ππ οΏ½π π
2(ππ2+ππ3) +
π π 2
2 (ππ4+ππ5)οΏ½< 1
ππ.ππ.,ππ(πππ₯π₯,π₯π₯,ππ) =ππ for all ππ β ππ.
ππ.ππ.,πππ₯π₯=π₯π₯ for all ππ β ππ.
β΄ π₯π₯is a fixed point of ππ.
The Uniqueness can be proved easily by taking πππ₯π₯=π₯π₯ and πππ¦π¦=π¦π¦ in (3.6).
Example 3.4: Let π΄π΄=β2,ππ= {(π₯π₯,π¦π¦):π₯π₯,π¦π¦ β₯0} with β(π₯π₯,π¦π¦)β= |π₯π₯| + |π¦π¦| for all (π₯π₯,π¦π¦)β π΄π΄ and ππ= {(πΌπΌ, 0):πΌπΌ β₯ 0}β{(0,2)}. Define the 2-metric ππ:πππ₯π₯πππ₯π₯ππ β π΄π΄ as follows:
ππ(π΄π΄,π΅π΅,πΆπΆ) =
β© β¨
β§πποΏ½ππ(π΄π΄,π΅π΅,πΆπΆ)οΏ½ πππππππππππππππ π πππππππππ’π’ππππππππππππ
(1,1) ππππ πππ€π€πππππππ΄π΄,π΅π΅,πΆπΆππππππ (0,0), (0,2) ππππππ π΄π΄,π΅π΅,πΆπΆπππππππππππ π ππππππππππ
(β,β) ππππβπππππ€π€πππ π ππ. where β= square of the area of triangle π΄π΄,π΅π΅,πΆπΆ.
Consider the following:
πποΏ½(πΌπΌ, 0), (π½π½, 0), (0,2)οΏ½ βΌ πποΏ½(πΌπΌ, 0), (π½π½, 0), (πΎπΎ, 0)οΏ½+πποΏ½(πΌπΌ, 0), (πΎπΎ, 0), (0,2)οΏ½+πποΏ½(πΎπΎ, 0), (π½π½, 0), (0,2)οΏ½
ππ.ππ., (πΌπΌ β π½π½)2βΌ(πΌπΌ β πΎπΎ)2+ (πΎπΎ β π½π½)2
which shows that ππ is not a cone 2-metric, but a cone ππ2-metric with π π = 2. Define ππ:ππ β ππ as
ππ(π₯π₯, 0) =οΏ½οΏ½π₯π₯2, 0οΏ½ πππππ₯π₯ β₯10
(0,0) πππππ₯π₯< 10 and ππ(0,2) = (0,0).
Take ππ1=οΏ½1
4, 0οΏ½,ππ2=ππ3=οΏ½
1
18, 0οΏ½ and ππ4= 5 =οΏ½ 1 24, 0οΏ½.
Clearly, each ππππ(ππ= 1,2,3,4,5) and π π = 2 satisfies (3.7) and ππ satisfies the condition (3.6). Hence, by our main result,
ππ has unique fixed point which is (0,0).
ACKNOWLEDGEMENT
The research was supported by Council of Scientific and Industrial Research (CSIR), New Delhi.
REFERENCES
1. Ashok K.Sharma., A note on Fixed points in 2-metric spaces. Indian J. Pure Appl. Math. Vol.11, No.12 (1980).1580-1583.
2. B.Singh et al., Cone 2-metric space and fixed point theorem of contractive mappings. Commentationes Mathematicae. Vol.52, No.2 (2012), 143-151.
4. G.E.Hardy and T.D.Rogers., A generalization of fixed point theorem of Reich. Cana. Math. Bull. Vol.16. No.2 (1973), 201-206.
5. Hao Liu And Shaoyuan Xu., Fixed point theorems of Quasicontractions on cone metric spaces with Banach algebras. Abstract & Appl. Anal Vol.2013, Article ID 187348, 5pages.
6. Huang and Xu., Fixed point theorems of contractive mappings in cone b-metric spaces and applications. Fixed Point Theory Appl., 2013:112.
7. Huang and Zhang., Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. Vol.332 (2007), 1468-1476.
8. Liu and Xu., Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings. Fixed Point Theory Appl., 2013:320.
9. Ljiljana Gajic and Vladimir Rakocevic., Quasi-Contractions on a Non-normal cone metrics space. Funct. Anal. Appl. Vol.46. No.1 (2012), 62-65.
10. Mustafa et.al., ππ2-metric spaces and some fixed point theorems. Fixed Point Theory Appl., 2014:144.
11. N. Hussain, M.H. Shah., KKM mappings in cone b-metric spaces. Comput. Math. Appl., Vol.62 (2011), 1677-1684.
12. S.Czerwik., contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav, Vol.1 (1993), 5-11. 13. Shatanawi et al., Contraction conditions using comparision functions on b-metric spaces. Fixed Point Theory
appl. 2014:135.
14. S.N.Lal and A.K.Singh., An alalogue of Banachβs contraction principle for 2-metric spaces. Bull. Austral. Math. Soc.Vol.17. No.8 (1986), 974-993.
15. S.V.R. Naidu and J.Rajendra Prasad., Fixed point theorems in 2-metric spaces. Indian J. Pure Appl. Math. Vol.17.No.8 (1986), 974-993.
16. V.S.Gahler., 2-metrische Raume und ihre topologisehe Struktur. Math.Nachr. Vol.26 (1962), 115-118.
17. Wang et al., Fixed point theorems on cone 2-metric spaces over Banach algebras and an application. Fixed Point Theory Appl., 2015:204.
18. Xu and Radenovic., Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality. Fixed Point Theory Appl., 2014:102.
19. Z.Kadelburg et al., Remarks on βQuasi-contration on a cone metric space.β Appl. Math. Lett., Vol.22(2009), 1674-1679.
Source of support: Council of Scientific and Industrial Research (CSIR), New Delhi, India. Conflict of interest: None Declared
