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N. Appa Rao et al Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 3( Version 1), March 2014, pp.300-308
www.ijera.com 300 | P a g e
A Coupled Fixed Point Theorem for Geraghty Contractions in
Partially Ordered Metric Spaces
K.P.R. Sastry1, Ch. Srinivasa Rao
2, N. Appa Rao
3 and S.S.A. Sastri
4
1 8-28-8/1, Tamil street, Chinna Waltair, Visakhapatnam-530 017, India.
2 Department of Mathematics Mrs. A.V.N. College, Visakhapatnam-530 001, India.
3 Department of Basic Engineering, Chalapathi Institute of Engineering and Technology, Lam, Guntur-530 026,
India. 4 Department of Mathematics, GVP College of Engineering, Madhurawada, Visakhapatnam -530048, India.
Abstract In this paper we establish results on the existence and uniqueness of coupled fixed points of Geraghty
contraction on a partially ordered set with a metric, with the continuity of the altering distance function dropped.
Our results are improvements over the results of GVR Babu and P.Subhashini [3].
Key words: Geraghty contraction, altering distance function, fixed point, mixed monotone property, partially
ordered set.
Mathematical subject classification (2010): 47H10, 54H25
I. Introduction Delbosco [8] and Skof [23] introduced the concept of an altering distance function, which alters the
distance between two points in a metric space. This technique is made famous by Khan, Swaleh and Sessa [15].
Afterwards many researchers [2, 6, 9, 10, 15, 22] applied this concept to obtain the existence of fixed points.
Also, G.V.R. Babu and P. Subhashini [3] applied this concept to obtain coupled fixed point via Geraghty
contraction.
Throughout this paper β+ denotes the interval 0,β .
II. Preliminaries Notation: Let Π€ = π:β+ β β+/Ο is non decreasing and Ο π‘ = 0 β π‘ = 0
Definition 2.1: [23] π β Π€ is called an altering distance function if π is continuous.
Definition 2.2: [14] Let (π,β€) be partially ordered set and πΉ:π Γ π β π be a map. We say that π has the
mixed monotone property if πΉ(π₯, π¦) is non-decreasing in π₯ and is non-increasing inπ¦.
i.e. π₯1 , π₯2 β π, π₯1 β€ π₯2 β πΉ(π₯1 , π¦) β€ πΉ(π₯2 , π¦) and π¦1 , π¦2 β π, π¦1 β€ π¦2 β πΉ(π₯, π¦1) β₯ πΉ(π₯, π¦2).
Definition 2.3: [12] Let (π,β€) be a partially ordered set and πΉ:π Γ π β π be a map. For anyπ₯, π¦ β π. A point
(π₯, π¦) β π Γ π is called a coupled fixed point of πΉ if π₯ = πΉ(π₯, π¦) and π¦ = πΉ(π¦, π₯).
Define the partial order β€1 on π Γ π as follows:
π₯, π¦ β€1 π’, π£ ππ π₯ β€ π’ πππ π¦ β₯ π£ β π₯, π¦,π’, π£ β π.
We say that (π₯, π¦) and (π’, π£) are comparable, if either π₯, π¦ β€1 π’, π£ or π’, π£ β€1 π₯, π¦ We denote the class of all altering distance functions π:β+ β β+ by Π€. We use the following notation as
mentioned in [11] to define Geraghty contraction.
π = π½: β+ β 0,1 / Ξ² π‘π β 1 β π‘π β 0
Definition 2.4: [11] Let (π,π) be a metric space. A self map π:π β π is said to be a Geraghty contraction if
β π½ β π such that π π π₯ , π π¦ β€ π½ π π₯, π¦ π π₯, π¦ β π₯, π¦ β π
Remark 2.5: It is trivial to see that every contraction map is Geraghty contraction. The following example
(GVR Babu and P Subhashini [3]) shows that a Geraghty contraction need not be a contraction.
RESEARCH ARTICLE OPEN ACCESS
N. Appa Rao et al Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 3( Version 1), March 2014, pp.300-308
www.ijera.com 301 | P a g e
Example 2.6: [3] Let π = β+ with usual metric. We define π:π β π by π π₯ =π₯
1+π₯ β π₯ β π and π½: β+ β
0,1 by π½ π‘ = 2
2+π‘ ππ π‘ > 0
0 ππ π‘ = 0
Then clearly π½ β π. We observe that π is a Geraghty contraction. For,
π π π₯ , π π¦ = π₯
1 + π₯β
π¦
1 + π¦ =
π₯ β π¦
1 + π₯ 1 + π¦ β€
2 π₯ β π¦
2 + π₯ β π¦ = π½ π π₯, π¦ π π₯, π¦ βπ₯, π¦ β π
But π is not a contraction.
The following theorem is proved in [11].
Theorem 2.7 (Geraghty, [11]): Let π:π β π be a self map of a complete metric space π. If β π½ β π such
that π π π₯ , π π¦ β€ π½ π π₯, π¦ π π₯, π¦ β π₯,π¦ β π, then for any choice of initial point π₯0, the iteration
π₯π = π(π₯πβ1) for π = 1,2,3,β¦ converges to the unique fixed point π§ of π in π.
In 2010 Amini-Harandi and Emami [1] extended Theorem 2.7 to complete metric spaces with partial
order.
Theorem 2.8 [1]: Let π,β€ be a poset and suppose that there exists a metric π on π such that π,π is a
complete metric space. Let π:π β π be an increasing mapping and β π½ β π
such that π π π₯ , π π¦ β€ π½ π π₯, π¦ π π₯, π¦ β π₯, π¦ β π with π₯ β₯ π¦.
Assume that either (i) π is continuous or (ii) π is such that if π₯π an increasing sequence π₯π β π₯ in π, then
π₯π β€ π₯ β π.
Besides suppose thatif for each π₯, π¦ β π,β π§ β π which is comparable to π₯ and π¦. Then π has a unique fixed
point.
Guo and Lakshmikantham [13] introduced the mixed monotone operators. In 2006, Gnana Bhaskar and
Lakshmikantham [12] established the existence of coupled fixed points for mixed monotone operators in metric
spaces with partial order. For more literature on the existence of coupled fixed points of different contraction
conditions in partially ordered metric spaces, we refer [4, 7, 14, 16, 18, 19, 20, 21].
Definition 2.9: [12] Let π be a non-empty set and πΉ:π Γ π β π be a mapping. An element π₯, π¦ β π Γ π is
said to be coupled fixed point of πΉ if πΉ π₯, π¦ = π₯ and πΉ π¦, π₯ = π¦.
The following theorem is due to Gnana Bhaskar and Lakshmikantham [12].
Theorem 2.10: [12] Let (π,β€) be a poset and suppose that there exists a metric π on π such that (π,π) is a
complete metric space. Let πΉ:π Γ π β π be a mapping having the mixed monotone property on π. Assume that
there exists π β [0,1) such that
π πΉ π₯, π¦ ,πΉ(π’, π£) β€π
2 π π₯,π’ + π(π¦, π£) β¦β¦ (1)
for all π₯, π¦,π’, π£ β π with π₯ β₯ π’, π¦ β€ π£.
Also suppose that either (i) πΉ is continuous or (ii) π has the following properties:
(a) If π₯π is a non-decreasing sequence in π with π₯π β π₯, then π₯π β€ π₯ β π β π«+
(b) If π¦π is a non- increasing sequence in π with π¦π β π¦, then π¦π β₯ π¦ β π β π«+
If β π₯0 , π¦0 β π such that π₯0 β€ πΉ(π₯0 , π¦0) and π¦0 β₯ πΉ(π₯0 , π¦0), then β π₯, π¦ β π such that π₯ = πΉ π₯, π¦ and
π¦ = πΉ π¦, π₯ Recently Choudhury and Kundu [5] extended Theorem 2.7 to Geraghty contractions in the context of
coupled fixed points in metric spaces with partial order.
Theorem 2.11: [5] Let (π,β€) be a poset and suppose that there exists a metric π on π such that (π,π) is a
complete metric space. Let πΉ:π Γ π β π be a mapping having the mixed monotone property on π. Assume that
there exists π½ β π such that
π πΉ π₯, π¦ ,πΉ(π’, π£) β€ π½ π π₯ ,π’ +π(π¦ ,π£)
2
π π₯ ,π’ +π(π¦ ,π£)
2 β¦β¦ (2)
whenever π₯, π¦,π’, π£ β π and π₯, π¦ and (π’, π£) are comparable..
Also suppose that either (i) πΉ is continuous or (ii) π has the following properties:
(a) If π₯π is a non-decreasing sequence in π with π₯π β π₯, then π₯π β€ π₯ β π β π«+
(b) If π¦π is a non- increasing sequence in π with π¦π β π¦, then π¦π β₯ π¦ β π β π«+
If β π₯0, π¦0 β π such that π₯0 β€ πΉ(π₯0 , π¦0) and π¦0 β₯ πΉ(π₯0 , π¦0), then πΉ has a fixed point. That is β π₯, π¦ β π such
that π₯ = πΉ π₯, π¦ and π¦ = πΉ π¦, π₯ .
N. Appa Rao et al Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 3( Version 1), March 2014, pp.300-308
www.ijera.com 302 | P a g e
G.V.R. Babu and P. Subhashini [3] proved two theorems which extend the coupled fixed point results
established by Gnana Bhaskar and Lakshmikantham [12] and Choudhury and Kundu [5], to the case of
Geraghty contraction maps by using an altering distance function.
Theorem 2.12 (GVR Babu and P Subhashini, Theorem 2.1, [3]): Let (π,β€) be a poset and suppose that there
exists a metric π on π such that (π,π) is a complete metric space. Let πΉ:π Γ π β π be a continuous map
having the mixed monotone property on π. Suppose there exists an altering distance function π and π½ β π such
that
π π πΉ π₯, π¦ ,πΉ(π’, π£) β€ π½ π π₯ ,π’ +π(π¦ ,π£)
2 π max π π₯,π’ ,π(π¦, π£) β¦β¦ (3)
β π₯, π¦,π’, π£ β π whenever π₯, π¦ and (π’, π£) are comparable.
If β π₯0, π¦0 β π such that π₯0 β€ πΉ(π₯0 , π¦0) and π¦0 β₯ πΉ(π₯0 , π¦0), then πΉ has a coupled fixed point. That is β π₯, π¦ βπ such that π₯ = πΉ π₯, π¦ and π¦ = πΉ π¦, π₯ .
Theorem 2.13 (GVR Babu and P Subhashini, Theorem 2.2, [3]): Let (π,β€) be a poset and suppose that there
exists a metric π on π such that (π,π) is a complete metric space. Let πΉ:π Γ π β π be a continuous map
having the mixed monotone property on π. Suppose there exists an altering distance function π and π½ β π such
that
π π πΉ π₯, π¦ ,πΉ(π’, π£) β€ π½ π π₯ ,π’ +π(π¦ ,π£)
2 π max π π₯,π’ ,π(π¦, π£) β¦β¦ (4)
β π₯, π¦,π’, π£ β π whenever π₯, π¦ and (π’, π£) are comparable..
Further assume that π has the following properties:
(a) If π₯π is a non-decreasing sequence in π with π₯π β π₯, then π₯π β€ π₯ β π β π«+
(b) If π¦π is a non- increasing sequence in π with π¦π β π¦, then π¦π β₯ π¦ β π β π«+
If β π₯0, π¦0 β π such that π₯0 β€ πΉ(π₯0 , π¦0) and π¦0 β₯ πΉ(π¦0 , π₯0), then πΉ has a coupled fixed point. That is, β π₯, π¦ βπ such that π₯ = πΉ π₯, π¦ and π¦ = πΉ π¦, π₯ . In this paper, we establish Theorem 2.12 and Theorem 2.13 without using the continuity of alter
distance function π.
III. Main results
Theorem 3.1: Let (π,β€) be a poset and suppose that there exists a metric π on π such that (π,π) is a metric
space. Let πΉ:π Γ π β π be a mapping having the mixed monotone property on π and there exist π β Π€ and
π½ β π such that
π π πΉ π₯, π¦ ,πΉ(π’, π£) β€ π½ π π₯ ,π’ +π(π¦ ,π£)
2 π max π π₯,π’ ,π(π¦, π£) β¦ (5)
For all π₯, π¦,π’, π£ β π whenever π₯, π¦ and (π’, π£) are comparable..
Suppose β π₯0, π¦0 β π such that π₯0 β€ πΉ(π₯0 , π¦0) and π¦0 β₯ πΉ(π¦0 , π₯0). Define the sequences π₯π and π¦π in π by
π₯π+1 = πΉ(π₯π , π¦π) and π¦π+1 = πΉ(π¦π , π₯π) for all π = 0, 1, 2,β¦ (6)
Then π₯π is an increasing sequence, π¦π is a decreasing sequence and π₯π and π¦π are Cauchy sequences.
Proof: First we prove that π₯π β€ π₯π+1 and π¦π β₯ π¦π+1 for all π = 0, 1, 2,β¦ (7)
and then we show that π₯π and π¦π are Cauchy sequences.
We have π₯0 β€ πΉ(π₯0 , π¦0) and π¦0 β₯ πΉ(π¦0 , π₯0) .
Hence π₯0 β€ π₯1 and π¦0 β₯ π¦1
β΄ (7) is true for π = 0.
Assume that (7) is true for some positive integer π.
By using the mixed monotone property of πΉ, we have
π₯π+2 = πΉ π₯π+1, π¦π+1 β₯ πΉ π₯π , π¦π = π₯π+1 and π¦π+2 = πΉ π¦π+1, π₯π+1 β€ πΉ π¦π , π₯π = π¦π+1
β΄ (7) is true for π + 1.
Therefore by mathematical induction (7) follows.
We now show that limπββ max π π₯π , π₯π+1 ,π π¦π , π¦π+1 = 0.
We have π₯π β€ π₯π+1 and π¦π β₯ π¦π+1 for all π = 0, 1, 2,β¦
Now π π π₯π+1, π₯π = π π πΉ π₯π , π¦π ,πΉ(π₯πβ1, π¦πβ1)
β€ π½ π‘π π max π π₯π , π₯πβ1 ,π π¦π , π¦πβ1 β¦. (8)
β€ π max π π₯π , π₯πβ1 ,π π¦π , π¦πβ1
π π π¦π , π¦π+1 = π π πΉ π¦πβ1 , π₯πβ1 ,πΉ(π¦π , π₯π)
β€ π½ π‘π π max π π¦πβ1 , π¦π ,π π₯πβ1, π₯π β¦. (9)
N. Appa Rao et al Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 3( Version 1), March 2014, pp.300-308
www.ijera.com 303 | P a g e
where π‘π =π π¦πβ1 ,π¦π +π π₯πβ1 ,π₯π
2
β€ π max π π¦πβ1, π¦π ,π π₯πβ1, π₯π From (8) and (9), we have
max Ο π π₯π , π₯π+1 ,π π¦π , π¦π+1 β€ π½ π‘π π max π π₯π , π₯πβ1 ,π π¦π , π¦πβ1 Since π is increasing, we get
Ο max π π₯π , π₯π+1 ,π π¦π , π¦π+1 β€ π½ π‘π π max π π₯π ,π₯πβ1 ,π π¦π , π¦πβ1 β¦ (10)
If π½ π‘π = 0, for some π, then Ο max π( π₯π , π₯π+1 ,π( π¦π , π¦π+1 = 0 and hence from (10) we have
Ο max π π₯π , π₯π+1 ,π π¦π , π¦π+1 = 0 πππ π β₯ π.
Consequently max π π₯π , π₯π+1 ,π π¦π , π¦π+1 = 0 for π β₯ π
So that limπββ max π π₯π , π₯π+1 ,π π¦π , π¦π+1 = 0.
Hence, we may suppose that π½ π‘π > 0 β π β¦.β¦ (11)
Again from non-decreasing property of π and (10), we have
max π π₯π , π₯π+1 ,π π¦π , π¦π+1 β€ max π π₯π , π₯πβ1 ,π π¦π , π¦πβ1 β΄ max π π₯π , π₯π+1 ,π π¦π , π¦π+1 is a non-negative and decreasing sequence of real and hence it converges to
a real number π (say) π β₯ 0.
Now, we prove that π = 0. If possible suppose that π > 0.
Again from (10), we have
π ππ+1 β€ π½ π‘π π ππ where ππ = max π π₯π , π₯πβ1 ,π( π¦π , π¦πβ1 β₯ π > 0
i.e π ππ+1
π ππ β€ π½ π‘π < 1 β΅ π ππ > π π > 0
Let π ππ decreases to π , where limπββ π ππ = π .
Now π β€ ππ+1 β π π β€ π ππ+1 β π β π(π) β€ π
Now π(π) β€ π β€ π(ππ+1) β€ π½ π‘π π(ππ) β¦.. (12)
Case (i): π½ π‘π β 1. Then π‘π β 0 β΅ π½ β π
β π π₯π ,π₯πβ1 +π π¦π ,π¦πβ1
2 β 0
β π π₯π , π₯πβ1 + π π¦π , π¦πβ1 β 0 β ππ β€ π π₯π , π₯πβ1 + π π¦π , π¦πβ1 β 0 β ππ β 0 β π = 0
Case (ii): limπββ π½ π‘π β 1.
Then β ν > 0 such that π½ π‘π < 1 β ν for infinitely many π.
Then from (10), we have π π β€ π β€ π ππ+1 β€ π½ π‘π π ππ β€ 1 β ν π(ππ) for infinitely many π.
On letting π β β, we get π π β€ π β€ 1 β ν π β π = 0 and π π = 0 β π = 0 and π = 0.
β΄ 0 = π = limπββ max π( π₯π , π₯π+1 ,π( π¦π , π¦π+1 β¦. (13)
Next, we have to prove that π₯π and π¦π are Cauchy sequences.
If possible, assume that either π₯π or π¦π fails to be Cauchy.
Then either limπ ,πββ π π₯π , π₯π β 0 or limπ ,πββ π π¦π , π¦π β 0
Hence max limπ ,πββ π π₯π , π₯π , limπ ,πββ π π¦π , π¦π β 0
i.e. limπ ,πββ max limπ ,πββ π π₯π ,π₯π , limπ ,πββ π π¦π , π¦π β 0
i.e β ν > 0, for which we can find sub sequences π(π) and π(π) of positive integers with π(π) > π(π) >
π such that max π π₯π (π), π₯π(π) ,π π¦π (π), π¦π(π) β₯ ν β¦.. (14)
Further, we choose π(π) to be the smallest +ve integer such that π(π) > π(π) satisfying (14).
Hence, we have max π π₯π (π), π₯π(π) ,π π¦π (π), π¦π(π) β₯ ν and
max π π₯π (π), π₯π π β1 ,π π¦π (π), π¦π π β1 < ν ..β¦.. (15)
Now, we prove that
I: limπββ max π π₯π(π), π₯π (π) ,π π¦π(π), π¦π (π) = ν
II: limπββ max π π₯π π β1 ,π₯π π β1 ,π π¦π π β1, π¦π π β1 = ν
III: limπββ max π π₯π (π), π₯π π β1 ,π π¦π (π), π¦π π β1 = ν
First we prove I:-
From the triangular inequality and (11), we have
π π₯π π , π₯π π β€ π π₯π π , π₯π π β1 + π π₯π π β1 ,π₯π (π) < π π₯π π , π₯π π β1 + ν β¦ (16)
π π¦π π , π¦π π β€ π π¦π π , π¦π π β1 + π π¦π π β1, π¦π π < π π¦π π , π¦π π β1 + ν β¦ (17)
From (14), (16) and (17)
ν β€ max π π₯π(π), π₯π (π) ,π π¦π(π), π¦π (π) < max π π₯π(π), π₯π π β1 ,π π¦π(π), π¦π π β1 β¦ (18)
On letting π β β, we get
ν β€ limπββ
max π π₯π(π), π₯π (π) ,π( π¦π(π), π¦π (π) β€ ν
N. Appa Rao et al Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 3( Version 1), March 2014, pp.300-308
www.ijera.com 304 | P a g e
β΄ limπββ
max π π₯π(π), π₯π (π) ,π π¦π(π), π¦π (π) = ν
β΄ (I) holds.
Now, we prove (II):-
π π₯π π β1, π₯π π β1 β€ π π₯π π β1 , π₯π π + π π₯π π , π₯π π β1
< π π₯π π β1, π₯π π + ν ππ¦ (15)
π π¦π π β1, π¦π π β1 β€ π π¦π π β1, π¦π π + π π¦π π , π¦π π β1
< π π¦π π β1, π¦π π + ν ππ¦ (15)
β΄ max π π₯π π β1, π₯π π β1 ,π π¦π π β1, π¦π π β1 β€ max π π₯π π β1 , π₯π π ,π π¦π π β1, π¦π π + ν
β΄ lim sup max π π₯π π β1, π₯π π β1 ,π π¦π π β1, π¦π π β1 β€ ν β¦ (19)
π π₯π π , π₯π π β€ π π₯π π , π₯π π β1 + π π₯π π β1 ,π₯π π β1 + π π₯π π β1, π₯π π
π π¦π π , π¦π π β€ π π¦π π , π¦π π β1 + π π¦π π β1, π¦π π β1 + π π¦π π β1, π¦π π
β΄ max π π₯π π , π₯π π ,π π¦π π , π¦π π β€ max π π₯π π , π₯π π β1 ,π π¦π π β1, π¦π π +
max π π₯π π β1, π₯π π β1 ,π π¦π π β1, π¦π π β1 + max π π₯π π β1, π₯π π ,π π¦π π β1, π¦π π
Letting π β β, from (12), we get
0 β€ lim inf max π π₯π π β1, π₯π π β1 ,π π¦π π β1, π¦π π β1
β€ lim sup max π π₯π π β1, π₯π π β1 ,π π¦π π β1, π¦π π β1
β€ ν
β΄ lim max π π₯π π β1, π₯π π β1 ,π π¦π π β1 , π¦π π β1 = ν
β΄ (II) holds.
π π₯π π , π₯π π β€ π π₯π π , π₯π π β1 + π π₯π π β1 ,π₯π π
β€ π π₯π π , π₯π π β1 + π π₯π π β1 , π₯π π β1 + π π₯π π β1, π₯π π
π π¦π π , π¦π π β€ π π¦π π , π¦π π β1 + π π¦π π β1, π¦π π
β€ π π¦π π , π¦π π β1 + π π¦π π β1, π¦π π β1 + π π¦π π β1, π¦π π
β΄ max π π₯π π , π₯π π ,π π¦π π , π¦π π β€ max π π₯π π , π₯π π β1 ,π π¦π π , π¦π π +
max π π₯π π β1, π₯π π ,π π¦π π β1 , π¦π π
β€ max π π₯π π , π₯π π β1 ,π π¦π π , π¦π π β1 + max π π₯π π β1, π₯π π β1 ,π π¦π π β1, π¦π π β1 +
max π π₯π π β1, π₯π π ,π π¦π π β1 , π¦π π
On letting π β β, from (I), (II), we get (III). β΅ ν β€ 0 + lim max π π₯π π β1, π₯π π ,π π¦π π β1, π¦π π β€ ν
Now, we have ν β€ lim inf max π π₯π π β1, π₯π π β1 ,π π¦π π β1, π¦π π β1
β€ lim sup max π π₯π π β1, π₯π π β1 ,π π¦π π β1, π¦π π β1 = ν
β΄ lim inf max π π₯π π β1, π₯π π β1 ,π π¦π π β1, π¦π π β1 = ν
Since π₯π π β1 β₯ π₯π π β1 and π¦π π β1 β€ π¦π π β1, from (5), we get
π π π₯π π , π₯π π = π π πΉ π₯π π β1, π¦π π β1 ,πΉ π₯π π β1 , π¦π π β1
β€ π½ π π₯π π β1, π₯π π β1 + π π¦π π β1, π¦π π β1
2 π max π π₯π π β1, π₯π π β1 ,π π¦π π β1, π¦π π β1
β¦β¦. (20)
Similarly π π π¦π π , π¦π π = π π πΉ π¦π π β1 , π₯π π β1 ,πΉ π¦π π β1, π₯π π β1
β€ π½ π π₯π π β1, π₯π π β1 + π π¦π π β1, π¦π π β1
2 π max π π₯π π β1, π₯π π β1 ,π π¦π π β1, π¦π π β1
β¦β¦.. (21)
From (20) and (21), we get π π π β€ π½ ππ .π(ππ) where
π π = max π π₯π π , π₯π π ,π π¦π π , π¦π π and ππ =π π₯π π β1 ,π₯π π β1 +π π¦π π β1 ,π¦π π β1
2
ππ = max π π₯π π β1, π₯π π β1 ,π π¦π π β1, π¦π π β1
Now, π½ ππ β 1 β ππ β 0 (by hypothesis)
β ππ β 0 ππ π β β β ν = 0 by (16) , a contradiction.
β΄ limπββ
π½(ππ) β 1
β΄ β a positive integer π and πΏ β (0,1) such that π½ ππ < πΏ for π β₯ π β¦. (22)
β΄ π ν < π π π β€ π½ ππ π(ππ) β¦.. (23)
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From (22), we have π ν < π π π < πΏπ(ππ) < πΏπ(ν + π) for a given π > 0 and large π.
π ν β€ π ν + 0 < π(π π) < πΏπ(ν + π)
This being true for every π > 0 follows that 0 β€ π ν < π ν + 0 < πΏπ(ν + 0)
β΄ 0 β€ π ν < π ν + 0 = 0 (β΅ 0 < πΏ < 1)
β΄ π ν = 0 β ν = 0, a contradiction.
Hence both π₯π πππ π¦π are Cauchy sequences.
Theorem 3.2: In addition to the hypothesis of Theorem 3.1, suppose that (π,π) is complete and either (a) πΉ is
continuous or (ii) π has the following properties:
(a) If π₯π is a non-decreasing sequence in π with π₯π β π₯, then π₯π β€ π₯ β π β π«+
(b) If π¦π is a non- increasing sequence in π with π¦π β π¦, then π¦π β₯ π¦ β π β π«+
Then πΉ has a coupled fixed point in π. i.e. β π₯, π¦ β π such that π₯ = πΉ π₯, π¦ and π¦ = πΉ π¦, π₯ .
Proof: Let the sequences π₯π and π¦π be as defined in (6) of Theorem 3.1. Then from Theorem 3.1, both the
sequences π₯π and π¦π are Cauchy. Since (π,π) is complete, β π₯, π¦ β π such that
limπββ π₯π = π₯ πππ limπββ π¦π = π¦.
(a) Suppose πΉ is continuous. Then it follows that
π₯ = limπββ
π₯π = limπββ
πΉ π₯πβ1, π¦πβ1 = πΉ limπββ
π₯πβ1 , limπββ
π¦πβ1 = πΉ(π₯, π¦)
π¦ = limπββ π¦π = limπββ πΉ π¦πβ1, π₯πβ1 = πΉ limπββ π¦πβ1 , limπββ π₯πβ1 = πΉ(π¦, π₯)
Thus (π₯, π¦) is a coupled fixed point of πΉ in π.
(b) Suppose that π has the properties (i) and (ii). Let ν > 0. Since limπββ π₯π = π₯ and limπββ π¦π = π¦,β a
positive integer π such that π(π₯, π₯π) < ν and π π¦, π¦π < ν β π β₯ π.
Since π₯π β€ π₯ and π¦π β₯ π¦, β π, from (5), we have
π π πΉ π₯, π¦ ,πΉ(π₯π , π¦π) β€ π½ π π₯, π₯π + π(π¦, π¦π)
2 π max π π₯, π₯π ,π(π¦, π¦π)
= 0 ππ π½ π π₯ ,π₯π +π(π¦ ,π¦π )
2 = 0 ππ π max π π₯, π₯π ,π(π¦, π¦π) = 0
< π max π π₯, π₯π ,π(π¦, π¦π) ππ‘βπππ€ππ π
β€ π ν πππ πππ π β₯ π
β΄ π πΉ π₯, π¦ ,πΉ(π₯π , π¦π) < ν for all π β₯ π (β΅ π is increasing)
β΄ πΉ(π₯π , π¦π) β πΉ(π₯, π¦). But πΉ π₯π , π¦π = π₯π+1 β π₯ ππ π β β
β΄ π₯ = πΉ(π₯, π¦) β¦.. (24)
Again from (5), we have
π π πΉ π¦, π₯ ,πΉ π¦π , π₯π = π π πΉ π¦π , π₯π ,πΉ(π¦, π₯)
β€ π½ π π¦π , π¦ + π(π₯π , π₯)
2 π max π π¦π , π¦ ,π(π₯π , π₯)
= 0 ππ π½ π π¦π ,π¦ +π(π₯π ,π₯)
2 = 0 ππ π max π π¦π , π¦ ,π(π₯π , π₯) = 0
< π max π π¦π , π¦ ,π(π₯π , π₯) ππ‘βπππ€ππ π
< π ν πππ πππ π β₯ π
β΄ π πΉ π¦, π₯ ,πΉ π¦π , π₯π < ν β π β₯ π (β΅ π is increasing)
β΄ πΉ π¦π , π₯π β πΉ(π₯, π¦). But πΉ π¦π , π₯π = π¦π+1 β π¦ ππ π β β
β΄ π¦ = πΉ π¦, π₯ β¦.. (25)
β΄ (24) and (25) shows that (π₯, π¦) is a coupled fixed point for πΉ in π.
Lemma 3.3: Let (π,β€) be a poset and suppose that there exists a metric π on π such that (π,π) is a complete
metric space. Let πΉ:π Γ π β π be a mapping having the mixed monotone property on π. Suppose there exists
an altering distance function π β Π€ and π½ β π such that
π π πΉ π₯, π¦ ,πΉ(π’, π£) β€ π½ π π₯ ,π’ +π(π¦ ,π£)
2 π max π π₯,π’ ,π(π¦, π£) β¦ (5)
for π₯, π¦,π’, π£ β π whenever π₯, π¦ and (π’, π£) are comparable.
Suppose (π₯, π¦) is a coupled fixed point of πΉ. i.e. π₯ = πΉ(π₯, π¦) or π¦ = πΉ(π¦, π₯). Let (π’, π£) β π Γ π such that
π’,π£ β€1 (π₯, π¦) β¦ (26)
Construct the sequences π’π and π£π by π’0 = π’, π£0 = π£,π’π+1 = πΉ π’π , π£π , π£π+1 = πΉ(π£π ,π’π).
Then π’π β π₯ and π£π β π¦ as π β β.
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Proof: First we prove that π’π , π£π β€1 (π₯, π¦) for all π = 0, 1, 2,β¦. i.e π₯ β₯ π’π and π¦ β₯ π£π for all π = 1, 2,β¦ (27)
From (26), π₯ β₯ π’ = π’0 and π¦ β€ π£ = π£π
β΄ (27) is true for π = 0.
Assume that (27) is true for some positive integer π.
Hence π’π+1 = πΉ π’π , π£π β€ πΉ π₯, π¦ = π₯ and π£π+1 = πΉ π£π ,π’π β₯ πΉ π¦,π₯ = π¦
Therefore by mathematical induction (27) is true for all π = 1, 2,β¦
Since π₯ β₯ π’πβ1 and π¦ β€ π£πβ1 and from (5), we have,
π π π₯,π’π = π π(πΉ π₯, π¦ ,πΉ(π’πβ1, π£πβ1 β€ π½ π π₯ ,π’πβ1 +π(π¦ ,π£πβ1)
2 π max π π₯,π’πβ1 ,π(π¦, π£πβ1)
and π π π¦, π£π = π π(πΉ π¦, π₯ ,πΉ(π£πβ1,π’πβ1 )
β€ π½ π π¦ ,π£πβ1 +π π₯ ,π’πβ1
2 π max π π¦, π£πβ1 + π π₯,π’πβ1
β΄ max π π π₯,π’π ,π π π¦, π£π β€
π½ π π¦ ,π£πβ1 +π π₯ ,π’πβ1
2 π max π π¦, π£πβ1 ,π π₯,π’πβ1 β¦. (28)
Case (i): π max π π₯,π’πβ1 ,π π¦, π£πβ1 = 0 for some π
Then π₯ = π’πβ1 and π¦ = π£πβ1 and from (28), we get π₯ = π’π and π¦ = π£π . Similarly π₯ = π’π or π¦ = π£π for π β₯ π.
β΄ π’π β π₯ and π£π β π¦ as π β β.
Case (ii): π max π π₯,π’πβ1 ,π π¦, π£πβ1 = 0 for all π β₯ 1.
Then from (28), we have max Ο π π₯,π’π π π π¦, π£π β€ Ο πππ₯ π π₯,π’πβ1 ,π π¦, π£πβ1
Since π is increasing, we get max π π₯,π’π ,π π¦, π£π β€ max π π₯,π’πβ1 ,π π¦, π£πβ1
i.e. max π π₯,π’π ,π π¦, π£π is a deceasing sequence of reals and hence deceases to π (say), π β₯ 0.
i.e. limπββ max π π₯,π’π ,π π¦, π£π = π.
Again from (28), we have
π π β€ π max π π₯,π’π ,π π¦, π£π β€
π½ π π₯ ,π’πβ1 +π π¦ ,π£πβ1
2 π max π π₯,π’πβ1 ,π π¦, π£πβ1 β¦ (29)
Let π π = max π π₯,π’π ,π π¦, π£π , ππ =π π₯ ,π’πβ1 +π π¦ ,π£πβ1
2
Then max π π₯,π’πβ1 ,π π¦,π£πβ1 = π πβ1
β΄ π π β€ π π π β€ π½ ππ π(π πβ1)
Suppose π > 0. Now, π½ ππ β 1 β ππ β 0 (by hypothesis)
β π π β 0 ππ π β β β π = 0, β΅ π πβ1 β π , a contradiction.
β΄ limπββ
π½(ππ) β 1
β΄ β a positive integer π and πΏ β (0,1) such that π½ ππ < πΏ for π β₯ π
β΄ 0 < π π β€ π π π < πΏπ π πβ1 β€ πΏπ(π + π) for a given π > 0 and large π.
β΄ π π β€ π π + 0 β€ π π π < πΏπ(π + π)
β΄ 0 < π π β€ π π + 0 β€ πΏπ π + 0 < π π + 0 β΅ 0 < πΏ < 1 , a contradiction.
β΄ π = 0. Therefore π(π₯,π’π) β 0 and π(π¦, π£π) β 0 as π β β.
β΄ π’π β π₯ and π£π β π¦ as π β β.
Lemma 3.4: Let (π,β€) be a poset and suppose that there exists a metric π on π such that (π,π) is a complete
metric space. Let πΉ:π Γ π β π be a mapping having the mixed monotone property on π. Suppose there exists
an altering distance function π β Π€ and π½ β π such that
π π πΉ π₯, π¦ ,πΉ(π’, π£) β€ π½ π π₯ ,π’ +π(π¦ ,π£)
2 π max π π₯,π’ ,π(π¦, π£) β¦ (5) for π₯, π¦,π’, π£ β π whenever π₯, π¦
and (π’, π£) are comparable.
Suppose (π₯, π¦) is a coupled fixed point of πΉ. i.e. π₯ = πΉ(π₯, π¦) or π¦ = πΉ(π¦, π₯). Let (π’, π£) β π Γ π such that
π₯, π¦ β€1 (π’, π£) β¦ (30)
Construct the sequences π’π and π£π by π’0 = π’, π£0 = π£,π’π+1 = πΉ π’π , π£π , π£π+1 = πΉ(π£π ,π’π).
Then π’π β π₯ and π£π β π¦ as π β β.
The proof is similar to that of Lemma 3.3.
Definition 3.5: Let (π’, π£) β π Γ π. Define the sequences π₯π πππ π¦π as follows:
π’0 = π’,π£0 = π£,π’1 = πΉ π’0, π£0 and π’π+1 = πΉ(π’π , π£π) for all π β₯ 1
π£1 = πΉ π£0 ,π’0 and π£π+1 = πΉ(π£π ,π’π) for all π β₯ 1
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Then the sequence π’π , π£π is called the coupled iterative sequence of π’, π£ .
Theorem 3.6: Let (π,β€) be a poset and suppose that there exists a metric π on π such that (π,π) is a metric
space. Let πΉ:π Γ π β π be a continuous map having the mixed monotone property on π. Suppose there exists
an altering distance function π β Π€ and π½ β π such that
π π πΉ π₯, π¦ ,πΉ(π’, π£) β€ π½ π π₯ ,π’ +π(π¦ ,π£)
2 π max π π₯,π’ ,π(π¦, π£) β¦ (5)
for π₯, π¦,π’, π£ β π whenever π₯, π¦ and (π’, π£) are comparable.
Suppose (π₯, π¦) is a coupled fixed point of πΉ. i.e. π₯ = πΉ(π₯, π¦) or π¦ = πΉ(π¦, π₯). Let (π’, π£) β π Γ π such that
π’,π£ πππ (π₯, π¦) are comparable. Let π’π , π£π be the coupled iterative sequence of π’, π£ . Then π’π β π₯ πππ π£π β π¦.
Proof: Case(i): If π’, π£ β€ (π₯, π¦), then result follows from Lemma 3.3.
Case(ii): If π₯, π¦) β€ (π’, π£), then the result follows from Lemma 3.4.
Theorem 3.7: Suppose the hypothesis of Theorem 3.2 holds. Then there does not exists a pair (π’, π£) β π Γ π
such that (π’, π£) is comparable to two distinct coupled fixed points of πΉ.
Proof: Suppose (π₯, π¦) and (π₯ β² , π¦β²) are two coupled fixed points of πΉ. Let (π’, π£) be comparable with (π₯, π¦) and
(π₯ β² , π¦β²). Let π’π , π£π be the coupled iterative sequence of (π’, π£).
Case(i): Suppose π’, π£ β€1 (π₯, π¦) and π’, π£ β€1 (π₯ β² , π¦β²).
Then by Lemma 3.3, π’π β π₯ πππ π£π β π¦ and also π’π β π₯β² πππ π£π β π¦β². β΄ π₯ = π₯β² and π¦ = π¦β². Consequently π₯, π¦ = (π₯ β² , π¦β²).
Case(ii): Suppose π₯, π¦ β€1 (π’, π£) and π₯ β² , π¦β² β€1 (π’, π£).
Then by Lemma 3.4, π’π β π₯ πππ π£π β π¦ and also π’π β π₯β² πππ π£π β π¦β². β΄ π₯ = π₯β² and π¦ = π¦β². Consequently π₯, π¦ = (π₯ β² , π¦β²).
Case(iii): Suppose (π₯, π¦) β€1 π’, π£ β€1 (π₯ β² , π¦β²).
Then by Lemma 3.4, π’π β π₯ πππ π£π β π¦ and also π’π β π₯β² πππ π£π β π¦β². β΄ π₯ = π₯β² and π¦ = π¦β². Consequently π₯, π¦ = (π₯ β² , π¦β²).
Case(iv): Suppose (π₯ β² , π¦β²) β€1 π’, π£ β€1 (π₯, π¦).
In this case also π₯, π¦ = (π₯ β² , π¦β²) as in Case (iii).
Hence π₯, π¦ and (π₯ β² , π¦β²) cannot be distinct, a contradiction.
Theorem 3.8: Suppose the hypothesis of Theorem 3.2 holds. Further assume that
(H): For π₯, π¦ , (π§, π‘) β π Γ π there exists (π’, π£) β π Γ π which is comparable to (π₯, π¦) and (π§, π‘). Then πΉ has
unique coupled fixed point.
Proof: By Theorem 3.2, πΉ has a coupled fixed point (π₯, π¦).
Suppose (π₯ β² , π¦β²) is also a coupled fixed point of πΉ.
Then by (H) there exists (π’, π£) β π Γ π which is comparable to (π₯, π¦) and (π₯ β² , π¦β²).
Then by Theorem 3.7, π₯, π¦ = (π₯ β² , π¦β²).
Thus πΉ has unique coupled fixed point.
Corollary 3.9: Suppose the hypothesis of Theorem 3.2 holds. Further assume that (π,β€) is a lattice. Then πΉ has
a unique coupled fixed point.
Proof: Since (π,β€) is a lattice, condition (H) holds.
Consequently by Theorem 3.8, πΉ has a unique coupled fixed point.
Note: If we replace the argument π π₯ ,π’ +π(π¦ ,π£)
2 in π½ by max π π₯,π’ ,π(π¦, π£) still the results hold good.
Conclusion: Under the hypothesis of Theorem 3.2, the fixed point set π of πΉ decomposes the set π into
pointwise disjoint sets ππ /π β π in the following way:
For π β π, write ππ = π β π: π ππ ππππππππππ π€ππ‘β π
Then (i) ππ β β , since π β ππ
(ii) ππ and ππ are disjoint whenever π, π β π and π β π
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(iii) π = πππβπ may be a proper sub set of π
in which case π β π does not contain any fixed point of πΉ.
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