Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 2213-2234
© Research India Publications
http://www.ripublication.com
On Generalization of Banach Contraction Principle
in Partially Ordered Metric Spaces
R.K.Sharma
Department of Mathematics
Govt. Holkar Science College (DAVV), Indore-452001, Madhya Pradesh, India
V. Raich
Department of Mathematics
Govt. Holkar Science College (DAVV), Indore-452001, Madhya Pradesh,
India.
C.S.Chauhan
Department of Applied Mathematics
Institute of Engineering & Technology(DAVV), Indore-452001, Madhya Pradesh,
India
Abstract
Some common fixed point theorems of four self-mappings satisfying
contraction type condition in partially ordered complete metric spaces have
been proved. The purpose of this paper is to unify and generalize the earlier
results of Al-Muhiameed et al. and Aydi. Some relevant examples are also
given to justify the results.
Keywords: Partially ordered metric spaces, common fixed point,
compatibility, weakly compatibility of maps, weakly increasing maps, weakly
annihilator maps, dominating maps.
AMS Subject classification: Primary 54H25, Secondary 47H10.
2214
R.K.Sharma, V. Raich and C.S.Chauhan
1. INTRODUCTION
After the classical result of Jungck [9] on common fixed point theorem for two
commuting mappings, various classes of commutativity of maps namely, weakly
commutativity of maps introduced by Sessa [17], compatibility of maps introduced by
Jungck [10], compatibility of type (A) of maps introduced by Jungck et al. [12],
compatibility of type (B) and of type (P) of maps are introduced by Pathak et al.
[13,14], weakly compatibility of maps introduced by Jungck & Rhoades [11] etc. and
established results regarding common fixed points in metric spaces.
Replacing the Cauchy condition for convergence of a contractive iteration by an
equivalent functional condition, Geraghty [8] generalized the Banach contraction
principle in a complete metric space. After that this result is extended for generalized
contraction type by Harandi et al. [5] in the setting of partially ordered metric space.
Recently many results regarding fixed point and common fixed points of maps have
been established in partially ordered metric spaces by many researchers along with
some applications to matrix equations, ordinary differential equations and integral
equations, [2, 5, 7, 15, 16].
Altun & Simsek [4] introduced the notion of weakly increasing mappings and gave
results on common fixed point in ordered cone metric spaces. Further, Aydi [6] has
presented some existence and uniqueness of common fixed point theorems for three
weakly increasing self-mappings.
Recently Al-Muhiameed et al. [3] extended the result of Aydi [6] for four maps as
opposed to three by using the notions of weakly increasing, partially weakly
increasing, weak annihilator, dominating of maps along with the compatibility, weak
compatibility of maps and well orderedness of two elements in partially ordered
metric space.
In this paper, we generalize the result of Al-Muhiameedet al. [3] and Aydi [6] in the
setting of partially ordered complete metric spaces.
2. PRELIMINARIES:
Throughout the paper (𝑋, 𝑑) stands for partially ordered metric space. We start this
section by some basic notations, definitions and results which are used in sequel.
Definition 2.1: Let (𝑋, ≤) be a partially ordered set and 𝑓, 𝑔: 𝑋 → 𝑋 are said to be
On Generalization of Banach Contraction Principle in Partially Ordered Metric Spaces 2215
(2.1.1) weakly increasing if 𝑓𝑥 ≤ 𝑔𝑓𝑥 and 𝑔𝑥 ≤ 𝑓𝑔𝑥 for all 𝑥 ∈ 𝑋. (cf. [4])
(2.1.2) partially weakly increasing if 𝑓𝑥 ≤ 𝑔𝑓𝑥 for all 𝑥 ∈ 𝑋. (cf. [1])
(2.1.3) weakly increasing with respect to 𝐻: 𝑋 → 𝑋 if 𝑓𝑋 ⊆ 𝐻𝑋, 𝑔𝑋 ⊆ 𝐻𝑋, if and
only if for all 𝑥 ∈ 𝑋, 𝑓𝑥 ≤ 𝑔𝑦, ∀ 𝑦 ∈ 𝐻 −1 (𝑓𝑥), 𝑔𝑥 ≤ 𝑓𝑦, ∀ 𝑦 ∈ 𝐻 −1 (𝑔𝑥). (cf. [6])
Remark 2.2:[1, 6] (2.2.1) A pair (𝑓, 𝑔) of self-maps of 𝑋 is weakly increasing if and
only if the pairs (𝑓, 𝑔) and (𝑔, 𝑓) are partially weakly increasing.
(2.2.2) If 𝐻: 𝑋 → 𝑋 is the identity map then the pair (𝑓, 𝑔) is weakly increasing with
respect to 𝐻 implies that the pair (𝑓, 𝑔) is weakly increasing.
(2.2.3) A pair (𝑓, 𝑔) of self-maps of 𝑋 is weakly increasing ⇒ the pair (𝑓, 𝑔) is
partially weakly increasing but the converse is not true. Following example shows
that:
Example 2.3:[1] Let 𝑋 = [0, 1] be endowed with usual ordering and 𝑓, 𝑔: 𝑋 → 𝑋 be
define by 𝑓𝑥 = 𝑥 2 and 𝑔𝑥 = √𝑥 . Since 𝑓𝑥 = 𝑥 2 ≤ 𝑔𝑓𝑥 = 𝑔𝑥 2 = 𝑥, clearly (𝑓, 𝑔) is
partially weakly increasing. But 𝑔𝑥 = √𝑥 ≰ 𝑥 = 𝑓𝑔𝑥 for all 𝑥 ∈ 𝑋 ⇒ (𝑔, 𝑓) is not
partially weakly increasing.
Definition 2.4: Let (𝑋, ≤) be a partially ordered set and 𝑓, 𝑔: 𝑋 → 𝑋 then
(2.4.1) 𝑓 is called weak annihilator of 𝑔 if 𝑓𝑔𝑥 ≤ 𝑥 for all 𝑥 ∈ 𝑋. (cf. [1])
(2.4.2) 𝑓 is called dominating if 𝑥 ≤ 𝑓𝑥 for all 𝑥 ∈ 𝑋. (cf. [1])
Example 2.5:[1] Let 𝑋 = [0, 1] be endowed with usual ordering and 𝑓, 𝑔: 𝑋 → 𝑋 be
define by 𝑓𝑥 = 𝑥 2 and 𝑔𝑥 = 𝑥 3 . Since 𝑓𝑔𝑥 = 𝑥 6 ≤ 𝑥, ∀𝑥 ∈ 𝑋 thus 𝑓 is a weak
annihilator of 𝑔.
Example 2.6:[1] Let 𝑋 = [0, 1] be endowed with usual ordering and 𝑓: 𝑋 → 𝑋 be
1
1
define by 𝑓𝑥 = 𝑥 3 since 𝑥 ≤ 𝑥 3 = 𝑓𝑥 for all 𝑥 ∈ 𝑋 so 𝑓 is a dominating map.
Definition 2.7:[3] A subset 𝑊 of a partially ordered set 𝑋 is called well-ordered if
every pair of elements of 𝑊 are comparable.
Definition 2.8: Let (𝑋, 𝑑) be a metric spaces, then maps 𝑓, 𝑔: 𝑋 → 𝑋 are said to be
(2.8.1) compatible if lim 𝑑(𝑓𝑔𝑥𝑛 , 𝑔𝑓𝑥𝑛 ) = 0, whenever {𝑥𝑛 } is a sequence in 𝑋 such
𝑛→∞
that lim 𝑓𝑥𝑛 = lim 𝑔𝑥𝑛 = u for some 𝑢 ∈ 𝑋. (cf.[10])
𝑛→∞
𝑛→∞
R.K.Sharma, V. Raich and C.S.Chauhan
2216
(2.8.2) weakly compatible if they commute at their coincidence points, that is, if
𝑓𝑥 = 𝑔𝑥 for some 𝑥 ∈ 𝑋 then 𝑓𝑔𝑥 = 𝑔𝑓𝑥.(cf. [11])
Geraghty [8] generalized the Banach contraction principle in metric spaces and
proved that:
If ℱ is the family of functions 𝛼: 𝑅 + → [0, 1) such that 𝛼(𝑡𝑛 ) → 1 implies 𝑡𝑛 → 0;
Theorem 2.9:[8] Let 𝑓: 𝑋 → 𝑋 be a contraction of a complete metric space 𝑋
satisfying
𝑑(𝑓(𝑥), 𝑓(𝑦)) ≤ 𝛼(𝑑(𝑥, 𝑦))𝑑(𝑥, 𝑦), ∀ 𝑥, 𝑦 ∈ 𝑋, where 𝛼 ∈ ℱ which
(2.9.1)
need not be continuous. Then for any arbitrary point 𝑥0 the iteration 𝑥𝑛 = 𝑓(𝑥𝑛−1 ),
𝑛 ≥ 1 converges to a unique fixed point of 𝑓 in 𝑋.
Harandi et al. [5] extended the Theorem 2.9 in the context of partially ordered metric
spaces, they proved.
Theorem 2.10:[5] Let (𝑋, ≤) be a partially ordered set and there exists a metric 𝑑 in
𝑋 such that (𝑋, 𝑑) is a complete metric space. Let 𝑓: 𝑋 → 𝑋 be a non-decreasing
mapping such that there exists 𝑥0 ∈ 𝑋 with 𝑥0 ≤ 𝑓𝑥0 satisfying (2.9.1) for all 𝑥, 𝑦 ∈
𝑋 with 𝑥 ≤ 𝑦,
(2.10.1) either 𝑓 is continuous or there exists a non-decreasing sequence {𝑥𝑛 } in 𝑋
such that 𝑥𝑛 → 𝑥 then 𝑥𝑛 ≤ 𝑥, ∀ 𝑛.
(2.10.2) for any 𝑥, 𝑦 ∈ 𝑋, there exists 𝑢 ∈ 𝑋 which is comparable to 𝑥 and 𝑦. Then 𝑓
has a unique fixed point.
By increasing number of mappings up to three Aydi [6] generalized the result of
Harandi et al. [5] in partially ordered metric space:
Theorem 2.11:[6] Let (𝑋, ≤) be a partially ordered set and there exists a metric 𝑑 in
𝑋 such that (𝑋, 𝑑) is a complete metric space. Let 𝑓, 𝑔, 𝐻: 𝑋 → 𝑋 are continuous
mappings such that:
(2.11.1) 𝑓𝑋 ⊆ 𝐻𝑋, 𝑔𝑋 ⊆ 𝐻𝑋.
(2.11.2) ∀𝑥, 𝑦 ∈ 𝑋, 𝐻𝑥 and 𝐻𝑦 are comparable such that
𝑑(𝑓𝑥, 𝑔𝑦) ≤ 𝛼(𝑑(𝐻𝑥, 𝐻𝑦))𝑑(𝐻𝑥, 𝐻𝑦), where 𝛼 ∈ ℱ.
(2.11.3) the pairs (𝑓, 𝐻 ) and (𝑔, 𝐻) are compatible.
On Generalization of Banach Contraction Principle in Partially Ordered Metric Spaces 2217
(2.11.4) 𝑓 and 𝑔 are weakly increasing with respect to 𝐻 then 𝑓, 𝑔 and 𝐻 have a
coincidence point. Moreover, if
(2.11.5) for any 𝑥, 𝑦 ∈ 𝑋, there exists 𝑢 ∈ 𝑋 such that 𝑓𝑥 ≤ 𝑓𝑢, 𝑓𝑦 ≤ 𝑓𝑢 then 𝑓, 𝑔
and 𝐻 have a unique common fixed point.
Aydi [6] shows that the Theorem 2.11 is also valid if the conditions of continuity of
𝑓, 𝑔, 𝐻 and the compatibility of the pairs (𝑓, 𝐻 ) and (𝑔, 𝐻) are replaced by
(2.11.6) 𝑋 is regular (i.e. if {𝑥𝑛 } a non-decreasing sequence in 𝑋 with respect to ≤
such that 𝑥𝑛 → 𝑥 as 𝑛 → ∞ then 𝑥𝑛 ≤ 𝑥, ∀ 𝑛.)
(2.11.7) 𝐻𝑋 is a closed subspace of (𝑋, 𝑑).
Recently, Al-Muhiameed et al. [3] generalized the Theorem 2.11 for four maps as
opposed to three maps by using the notions of weakly increasing, partially weakly
increasing, weak annihilator and dominating of maps along with compatibility, weak
compatibility and well orderedness of two elements in complete partially ordered
metric space:
Theorem 2.12: [3] Let (𝑋, ≤) be a partially ordered set and there exists a metric 𝑑 in
𝑋 such that (𝑋, 𝑑) is a complete metric space. Let 𝑓, 𝑔, 𝑆, 𝑇: 𝑋 → 𝑋 such that:
(2.12.1) 𝑓𝑋 ⊆ 𝑇𝑋 and 𝑔𝑋 ⊆ 𝑆𝑋.
(2.12.2) for every comparable elements 𝑥, 𝑦 ∈ 𝑋, 𝑑(𝑓𝑥, 𝑔𝑦) ≤
𝛼(𝑑(𝑆𝑥, 𝑇𝑦))𝑑(𝑆𝑥, 𝑇𝑦)
where 𝛼 ∈ ℱ.
(2.12.3) the pairs (𝑇, 𝑓 ) and (𝑆, 𝑔 ) are partially weakly increasing.
(2.12.4) 𝑓 and 𝑔 are dominating and weak annihilator maps of 𝑇 and 𝑆 respectively.
(2.12.5) there exist a non-decreasing sequence {𝑥𝑛 } with 𝑥𝑛 ≤ 𝑦𝑛 for all 𝑛 and 𝑦𝑛 →
𝑢 implies that 𝑥𝑛 ≤ 𝑢 .
(2.12.6) either pair (𝑓, 𝑆 ) is compatible, pair ( 𝑔, 𝑇) is weakly compatible and 𝑓 or 𝑆
is continuous map.
OR
pair ( 𝑔, 𝑇) is compatible, pair (𝑓, 𝑆 ) is weakly compatible and 𝑔 or 𝑇 is
continuous
map.
R.K.Sharma, V. Raich and C.S.Chauhan
2218
Then 𝑓, 𝑔, 𝑆 and 𝑇 have a common fixed point. Moreover, the common fixed point of
𝑓, 𝑔, 𝑆 and 𝑇 is unique if and only if the set of common fixed point of 𝑓, 𝑔, 𝑆 and 𝑇
is well ordered.
In this paper we prove some common fixed point theorems in the setting of partially
ordered metric spaces. Our results generalize the results of Al-Muhiameed et al. [3],
Aydi [6] and others. Also the main result is discussed with relevant example.
3. MAIN RESULT:
We prove the following.
Theorem 3.1: Let (𝑋, ≤) be a partially ordered set and there exists a metric 𝑑 in 𝑋
such that (𝑋, 𝑑) is a complete metric space. Let 𝑓, 𝑔, 𝑆, 𝑇: 𝑋 → 𝑋 satisfying the
conditions (2.12.1), (2.12.3)-(2.12.6) and
(3.1.1) 𝑑(𝑓𝑥, 𝑔𝑦) ≤ 𝛼(𝑀(𝑥, 𝑦))𝑀(𝑥, 𝑦), where
1
𝑀(𝑥, 𝑦) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑥, 𝑇𝑦), 𝑑(𝑓𝑥, 𝑆𝑥), 𝑑(𝑔𝑦, 𝑇𝑦), (𝑑(𝑆𝑥, 𝑔𝑦) + 𝑑(𝑓𝑥, 𝑇𝑦))}
2
for all 𝑥, 𝑦 ∈ 𝑋 with 𝑥 ≤ 𝑦 and 𝛼 ∈ ℱ.
Then 𝑓, 𝑔, 𝑆 and 𝑇 have a common fixed point. Moreover, the common fixed point of
𝑓, 𝑔, 𝑆 and 𝑇 is unique if and only if the set of common fixed point of 𝑓, 𝑔, 𝑆 and 𝑇 is
well ordered.
Proof: Let 𝑥0 be an arbitrary point in 𝑋, since 𝑓𝑋 ⊆ 𝑇𝑋 then there exists 𝑥1 ∈ 𝑋 such
that 𝑓𝑥0 = 𝑇𝑥1 , since 𝑔𝑋 ⊆ 𝑆𝑋 then there exists 𝑥2 ∈ 𝑋 such that 𝑔𝑥1 = 𝑆𝑥2 . On
continuing this process, we can construct sequences {𝑥𝑛 } and {𝑦𝑛 } in 𝑋 such that
𝑦2𝑛−1 = 𝑓𝑥2𝑛−2 = 𝑇𝑥2𝑛−1 and 𝑦2𝑛 = 𝑔𝑥2𝑛−1 = 𝑆𝑥2𝑛 for all 𝑛 = 1,2,3 … .
From conditions (2.12.3) and (2.12.4), we have
𝑥2𝑛−2 ≤ 𝑓𝑥2𝑛−2 = 𝑇𝑥2𝑛−1 ≤ 𝑓𝑇𝑥2𝑛−1 ≤ 𝑥2𝑛−1 and
𝑥2𝑛−1 ≤ 𝑔𝑥2𝑛−1 = 𝑆𝑥2𝑛 ≤ 𝑆𝑔𝑥2𝑛 ≤ 𝑥2𝑛 .
Thus ∀ 𝑛 ≥ 1 we obtain 𝑥1 ≤ 𝑥2 ≤ 𝑥3 ≤ ⋯ ≤ 𝑥𝑛 ≤ 𝑥𝑛+1 . i.e. a non-decreasing
sequence.
Now we prove that { 𝑦𝑛 } is a Cauchy sequence in 𝑋. For this let us consider that
𝑑(𝑦2𝑛 , 𝑦2𝑛+1 ) > 0 for every 𝑛. If not then 𝑦2𝑛 = 𝑦2𝑛+1 , for some 𝑛, therefore using
(3.1.1), we have
On Generalization of Banach Contraction Principle in Partially Ordered Metric Spaces 2219
𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ) = 𝑑(𝑓𝑥2𝑛 , 𝑔𝑥2𝑛+1 ) ≤ 𝛼(𝑀(𝑥2𝑛 , 𝑥2𝑛+1 ))𝑀(𝑥2𝑛 , 𝑥2𝑛+1 )
where
𝑀(𝑥2𝑛 , 𝑥2𝑛+1 ) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑥2𝑛 , 𝑇𝑥2𝑛+1 ), 𝑑(𝑓𝑥2𝑛 , 𝑆𝑥2𝑛 ), 𝑑(𝑔𝑥2𝑛+1 , 𝑇𝑥2𝑛+1 ),
1
(𝑑(𝑆𝑥2𝑛 , 𝑔𝑥2𝑛+1 ) + 𝑑(𝑓𝑥2𝑛 , 𝑇𝑥2𝑛+1 ))}
2
= 𝑚𝑎𝑥{ 𝑑(𝑦2𝑛 , 𝑦2𝑛+1 ), 𝑑(𝑦2𝑛+1 , 𝑦2𝑛 ), 𝑑(𝑦2𝑛+2 , 𝑦2𝑛+1 ),
1
(𝑑(𝑦2𝑛 , 𝑦2𝑛+2 ) + 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+1 ))}
2
1
= 𝑚𝑎𝑥 {0, 0, 𝑑(𝑦2𝑛+2 , 𝑦2𝑛+1 ), (𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ))}
2
= 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ).
Hence 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ) = 𝑑(𝑓𝑥2𝑛 , 𝑔𝑥2𝑛+1 ) ≤ 𝛼(𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ))𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ).
Using 0 ≤ 𝛼 < 1, we deduce that 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ) < 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ) which is a
contradiction. Hence we must have 𝑦2𝑛+1 = 𝑦2𝑛+2 . By the similar arguments, we
obtain 𝑦2𝑛+2 = 𝑦2𝑛+3 and so on. Thus {𝑦𝑛 } turns out to be a constant sequence and
𝑦2𝑛 is the common fixed point of 𝑓, 𝑔, 𝑆 and 𝑇.
Now we suppose 𝑑(𝑦2𝑛 , 𝑦2𝑛+1 ) > 0 for every 𝑛, since 𝑥 = 𝑥2𝑛 and 𝑦 = 𝑥2𝑛+1 are
comparable elements so using (3.1.1), we have
𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ) = 𝑑(𝑓𝑥2𝑛 , 𝑔𝑥2𝑛+1 ) ≤ 𝛼(𝑀(𝑥2𝑛 , 𝑥2𝑛+1 ))𝑀(𝑥2𝑛 , 𝑥2𝑛+1 )
where
𝑀(𝑥2𝑛 , 𝑥2𝑛+1 ) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑥2𝑛 , 𝑇𝑥2𝑛+1 ), 𝑑(𝑓𝑥2𝑛 , 𝑆𝑥2𝑛 ), 𝑑(𝑔𝑥2𝑛+1 , 𝑇𝑥2𝑛+1 ),
1
(𝑑(𝑆𝑥2𝑛 , 𝑔𝑥2𝑛+1 ) + 𝑑(𝑓𝑥2𝑛 , 𝑇𝑥2𝑛+1 ))}
2
= 𝑚𝑎𝑥{ 𝑑(𝑦2𝑛 , 𝑦2𝑛+1 ), 𝑑(𝑦2𝑛+1 , 𝑦2𝑛 ), 𝑑(𝑦2𝑛+2 , 𝑦2𝑛+1 ),
1
(𝑑(𝑦2𝑛 , 𝑦2𝑛+2 ) + 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+1 ))}
2
= 𝑚𝑎𝑥{ 𝑑(𝑦2𝑛 , 𝑦2𝑛+1 ), 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ),
1
(𝑑(𝑦2𝑛 , 𝑦2𝑛+1 ) + 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ))}
2
(3.1)
R.K.Sharma, V. Raich and C.S.Chauhan
2220
= 𝑚𝑎𝑥{𝑑(𝑦2𝑛 , 𝑦2𝑛+1 ), 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 )}
Now
𝑀(𝑥2𝑛 , 𝑥2𝑛+1 ) = either 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ) or 𝑑(𝑦2𝑛 , 𝑦2𝑛+1 ).
If
𝑀(𝑥2𝑛 , 𝑥2𝑛+1 ) = 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ) then from (3.1), we have
𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ) = 𝑑(𝑓𝑥2𝑛 , 𝑔𝑥2𝑛+1 ) ≤ 𝛼(𝑑(𝑦2𝑛+2 , 𝑦2𝑛+1 ))𝑑(𝑦2𝑛+2 , 𝑦2𝑛+1 ).
Using 0 ≤ 𝛼 < 1, we deduce that 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ) < 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ) which is a
contradiction. Hence 𝑀(𝑥2𝑛 , 𝑥2𝑛+1 ) = 𝑑 (𝑦2𝑛 , 𝑦2𝑛+1 ) and from (3.1), we have
𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ) = 𝑑(𝑓𝑥2𝑛 , 𝑔𝑥2𝑛+1 ) ≤ 𝛼(𝑑(𝑦2𝑛 , 𝑦2𝑛+1 ))𝑑(𝑦2𝑛 , 𝑦2𝑛+1 )
(3.2)
Using 0 ≤ 𝛼 < 1, we deduce that 𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 ) ≤ 𝑑(𝑦2𝑛 , 𝑦2𝑛+1 ).
By using similar arguments for 𝑥 = 𝑥2𝑛−1 and 𝑦 = 𝑥2𝑛 in (3.1.1), we have
𝑑(𝑦2𝑛 , 𝑦2𝑛+1 ) ≤ 𝑑(𝑦2𝑛−1 , 𝑦2𝑛 )
Hence for any
𝑛, 𝑑(𝑦𝑛+2 , 𝑦𝑛+1 ) ≤ 𝑑(𝑦𝑛+1 , 𝑦𝑛 ) ≤ 𝑑(𝑦𝑛 , 𝑦𝑛−1 ) ≤ ⋯ ≤ 𝑑(𝑦1 , 𝑦0 )
implies that the sequence {𝑑(𝑦𝑛+1 , 𝑦𝑛 )} is monotonically non-increasing sequence.
Hence there exists 𝑟 ≥ 0 such that lim 𝑑(𝑦𝑛+1 , 𝑦𝑛 ) = 𝑟
𝑛→∞
(3.3)
Using (3.2), we have
𝑑(𝑦2𝑛+1 , 𝑦2𝑛+2 )
≤ 𝛼(𝑑(𝑦2𝑛 , 𝑦2𝑛+1 )) < 1
𝑑(𝑦2𝑛 , 𝑦2𝑛+1 )
Letting 𝑛 → ∞ and using (3.3), we have lim 𝛼(𝑑(𝑦2𝑛 , 𝑦2𝑛+1 )) = 1. Since 𝛼 ∈
𝑛→∞
ℱ yields that 𝑟 = 0. Consequently lim 𝑑(𝑦𝑛+1 , 𝑦𝑛 ) = 0.
𝑛→∞
(3.4)
Now we claim that {𝑦2𝑛 } is a Cauchy sequence. Suppose on the contrary that {𝑦2𝑛 } is
not a Cauchy sequence then there is 𝜀 > 0 and there exist even integers 2𝑚𝑘 ,
2𝑛𝑘 with 2𝑚𝑘 > 2𝑛𝑘 > 𝑘 for all 𝑘 > 0 such that
𝑑(𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 ) ≥ 𝜖 and 𝑑(𝑦2𝑚𝑘 −2 , 𝑦2𝑛𝑘 ) < 𝜖
From triangle inequality, we have
𝜖 ≤ 𝑑(𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 )
(3.5)
On Generalization of Banach Contraction Principle in Partially Ordered Metric Spaces 2221
≤ 𝑑(𝑦2𝑛𝑘 , 𝑦2𝑚𝑘 −2 ) + 𝑑(𝑦2𝑚𝑘−1 , 𝑦2𝑚𝑘 −2 ) + 𝑑(𝑦2𝑚𝑘−1 , 𝑦2𝑚𝑘 )
< 𝜖 + 𝑑(𝑦2𝑚𝑘−1 , 𝑦2𝑚𝑘 −2 ) + 𝑑(𝑦2𝑚𝑘 −1 , 𝑦2𝑚𝑘 )
Letting 𝑘 → ∞ and using (3.4), we have lim 𝑑(𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 ) = 𝜖.
𝑘→∞
(3.6)
Now for all 𝑘 > 0 from (3.4) and (3.5), we have
𝜖 ≤ 𝑑(𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 ) ≤ 𝑑(𝑦2𝑚𝑘 , 𝑦2𝑚𝑘 −1 ) + 𝑑(𝑦2𝑚𝑘 −1 , 𝑦2𝑛𝑘 ) implies that
𝜖 ≤ lim 𝑑(𝑦2𝑚𝑘 −1 , 𝑦2𝑛𝑘 ). On the other hand from (3.4) and (3.6), we have
𝑘→∞
𝑑(𝑦2𝑚𝑘 −1 , 𝑦2𝑛𝑘 ) ≤ 𝑑(𝑦2𝑚𝑘 −1 , 𝑦2𝑚𝑘 ) + 𝑑(𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 ) implies that
lim 𝑑(𝑦2𝑚𝑘−1 , 𝑦2𝑛𝑘 ) ≤ 𝜖. Hence lim 𝑑(𝑦2𝑚𝑘−1 , 𝑦2𝑛𝑘 ) = 𝜖.
𝑘→∞
𝑘→∞
(3.7)
Similarly for all 𝑘, from (3.4) and (3.5), we have
𝑑(𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 ) ≤ 𝑑(𝑦2𝑚𝑘 , 𝑦2𝑛𝑘+1 ) + 𝑑(𝑦2𝑛𝑘+1 , 𝑦2𝑛𝑘 ) implies that
𝜖 ≤ lim 𝑑(𝑦2𝑚𝑘 , 𝑦2𝑛𝑘+1 ). On the other hand from (3.4) and (3.6), we have
𝑘→∞
𝑑(𝑦2𝑚𝑘 , 𝑦2𝑛𝑘+1 ) ≤ 𝑑(𝑦2𝑛𝑘 , 𝑦2𝑛𝑘+1 ) + 𝑑(𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 ) implies that
lim 𝑑(𝑦2𝑚𝑘−1 , 𝑦2𝑛𝑘 ) ≤ 𝜖. Hence lim 𝑑(𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 +1 ) = 𝜖.
𝑘→∞
𝑘→∞
(3.8)
Now taking 𝑥 = 𝑥2𝑛𝑘 and 𝑦 = 𝑥2𝑚𝑘 −1 and using (3.1.1),∀ 𝑘 > 0, we have
𝑑(𝑦2𝑛𝑘+1 , 𝑦2𝑚𝑘 ) = 𝑑(𝑓𝑥2𝑛𝑘 , 𝑔𝑥2𝑚𝑘 −1 ) ≤ 𝛼 (𝑀(𝑥2𝑛𝑘 , 𝑥2𝑚𝑘 −1 )) 𝑀(𝑥2𝑛𝑘 , 𝑥2𝑚𝑘 −1 )
(3.9)
where
𝑀(𝑥2𝑛𝑘 , 𝑥2𝑚𝑘−1 )
= 𝑚𝑎𝑥{ 𝑑(𝑆𝑥2𝑛𝑘 , 𝑇𝑥2𝑚𝑘 −1 ), 𝑑(𝑓𝑥2𝑛𝑘 , 𝑆𝑥2𝑛𝑘 ), 𝑑(𝑔𝑥2𝑚𝑘 −1 , 𝑇𝑥2𝑚𝑘−1 ),
1
(𝑑(𝑆𝑥2𝑛𝑘 , 𝑔𝑥2𝑚𝑘 −1 ) + 𝑑(𝑓𝑥2𝑛𝑘 , 𝑇𝑥2𝑚𝑘−1 ))}
2
= 𝑚𝑎𝑥{ 𝑑(𝑦2𝑛𝑘 , 𝑦2𝑚𝑘 −1 ), 𝑑(𝑦2𝑛𝑘+1 , 𝑦2𝑛𝑘 ), 𝑑(𝑦2𝑚𝑘 , 𝑦2𝑚𝑘−1 ),
1
(𝑑(𝑦2𝑛𝑘 , 𝑦2𝑚𝑘 ) + 𝑑(𝑦2𝑛𝑘 +1 , 𝑦2𝑚𝑘 −1 ))}
2
R.K.Sharma, V. Raich and C.S.Chauhan
2222
= 𝑚𝑎𝑥{ 𝑑(𝑦2𝑛𝑘 , 𝑦2𝑚𝑘 −1 ), 𝑑(𝑦2𝑛𝑘+1 , 𝑦2𝑛𝑘 ), 𝑑(𝑦2𝑚𝑘 , 𝑦2𝑚𝑘−1 ),
1
(𝑑(𝑦2𝑛𝑘 , 𝑦2𝑚𝑘 −1 ) + 𝑑(𝑦2𝑚𝑘 , 𝑦2𝑛𝑘+1 ))}
2
Letting 𝑘 → ∞ and using (3.4), (3.7) and (3.8), we have
lim 𝑀(𝑥2𝑛𝑘 , 𝑥2𝑚𝑘 −1 ) = max {𝜖, 0, 0,
𝑘→∞
Therefore,
𝑑(𝑓𝑥2𝑛𝑘 ,𝑔𝑥2𝑚𝑘 −1 )
𝑀(𝑥2𝑛𝑘 ,𝑥2𝑚𝑘−1 )
(𝜖 + 𝜖)
} = 𝜖.
2
< 𝛼 (𝑀(𝑥2𝑛𝑘 , 𝑥2𝑚𝑘−1 )) < 1. (since𝑦2𝑛𝑘 +1 ≠ 𝑦2𝑚𝑘 )
Using the fact that 𝜖 = lim 𝑑(𝑓𝑥2𝑛𝑘 , 𝑔𝑥2𝑚𝑘 −1 ) = lim 𝑀(𝑥2𝑛𝑘 , 𝑥2𝑚𝑘 −1 ), we get
𝑘→∞
𝑘→∞
lim 𝛼 (𝑀(𝑥2𝑛𝑘 , 𝑥2𝑚𝑘 −1 )) = 1, Since 𝛼 ∈ ℱ, hence lim 𝑀(𝑥2𝑛𝑘 , 𝑥2𝑚𝑘 −1 ) = 0 which
𝑘→∞
𝑘→∞
is a contradiction. Hence {𝑦2𝑛 } is a Cauchy sequence. By completeness of 𝑋 there
exist a point 𝑢 in 𝑋 such that {𝑦𝑛 } and its sub-sequences {𝑦2𝑛+1 } and {𝑦2𝑛 } are also
converges to 𝑢. i.e.
lim 𝑦2𝑛+1 = lim 𝑇 𝑥2𝑛+1 = lim 𝑓𝑥2𝑛 = 𝑢
𝑛→∞
𝑛→∞
𝑛→∞
(3.10)
and
lim 𝑦2𝑛+2 = lim 𝑆 𝑥2𝑛+2 = lim 𝑔𝑥2𝑛+1 = 𝑢.
𝑛→∞
𝑛→∞
𝑛→∞
(3.11)
Suppose that 𝑆 is continuous and by compatibility of (𝑓, 𝑆), we have
(3.12)
lim 𝑓𝑆 𝑥2𝑛+2 = lim 𝑆𝑓𝑥2𝑛+2 = 𝑆𝑢
𝑛→∞
𝑛→∞
Again since 𝑥2𝑛+1 ≤ 𝑔𝑥2𝑛+1 = 𝑆𝑥2𝑛+2 , using (3.1.1), we have
𝑑(𝑓𝑆𝑥2𝑛+2 , 𝑔𝑥2𝑛+1 ) ≤ 𝛼(𝑀(𝑆𝑥2𝑛+2 , 𝑥2𝑛+1 ))𝑀(𝑆𝑥2𝑛+2 , 𝑥2𝑛+1 )
(3.13)
where
𝑀(𝑆𝑥2𝑛+2 , 𝑥2𝑛+1 )
= 𝑚𝑎𝑥{ 𝑑(𝑆𝑆𝑥2𝑛+2 , 𝑇𝑥2𝑛+1 ), 𝑑(𝑓𝑆𝑥2𝑛+2 , 𝑆𝑆𝑥2𝑛+2 ), 𝑑(𝑔𝑥2𝑛+1 , 𝑇𝑥2𝑛+1 ),
1
(𝑑(𝑆𝑆𝑥2𝑛+2 , 𝑔𝑥2𝑛+1 ) + 𝑑(𝑓𝑆𝑥2𝑛+2 , 𝑇𝑥2𝑛+1 ))}
2
On Generalization of Banach Contraction Principle in Partially Ordered Metric Spaces 2223
Letting 𝑛 → ∞ and using (3.10), (3.11) and (3.12), we have
lim 𝑀(𝑆𝑥2𝑛+2 , 𝑥2𝑛+1 )
𝑛→∞
1
= 𝑚𝑎𝑥{ 𝑑(𝑆𝑢, 𝑢), 𝑑(𝑆𝑢, 𝑆𝑢), 𝑑(𝑢, 𝑢), (𝑑(𝑆𝑢, 𝑢) + 𝑑(𝑆𝑢, 𝑢))}
2
= 𝑚𝑎𝑥{ 𝑑(𝑆𝑢, 𝑢), 0,0, 𝑑(𝑆𝑢, 𝑢)} = 𝑑(𝑆𝑢, 𝑢)
Therefore from (3.13) as 𝑛 → ∞, we have
𝑑(𝑆𝑢, 𝑢) ≤ 𝛼(𝑑(𝑆𝑢, 𝑢))𝑑(𝑆𝑢, 𝑢) < 𝑑(𝑆𝑢, 𝑢), yields that 𝑆𝑢 = 𝑢.
Since
(3.14)
𝑥2𝑛+1 ≤ 𝑔𝑥2𝑛+1 and 𝑔𝑥2𝑛+1 → 𝑢 as 𝑛 → ∞, 𝑥2𝑛+1 ≤ 𝑢 .
From (3.1.1), we have
𝑑(𝑓𝑢, 𝑔𝑥2𝑛+1 ) ≤ 𝛼(𝑀(𝑢, 𝑥2𝑛+1 ))𝑀(𝑢, 𝑥2𝑛+1 )
(3.15)
where
𝑀(𝑢, 𝑥2𝑛+1 ) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑢, 𝑇𝑥2𝑛+1 ), 𝑑(𝑓𝑢, 𝑆𝑢), 𝑑(𝑔𝑥2𝑛+1 , 𝑇𝑥2𝑛+1 ),
1
(𝑑(𝑆𝑢, 𝑔𝑥2𝑛+1 ) + 𝑑(𝑓𝑢, 𝑇𝑥2𝑛+1 ))}
2
Letting 𝑛 → ∞ and using (3.10), (3.11) and (3.14), we have
1
lim 𝑀(𝑢, 𝑥2𝑛+1 ) = 𝑚𝑎𝑥{ 𝑑(𝑢, 𝑢), 𝑑(𝑓𝑢, 𝑢), 𝑑(𝑢, 𝑢), (𝑑(𝑢, 𝑢) + 𝑑(𝑓𝑢, 𝑢))}
𝑛→∞
2
1
= 𝑚𝑎𝑥{ 0, 𝑑(𝑓𝑢, 𝑢), 0, (0 + 𝑑(𝑓𝑢, 𝑢))} = 𝑑(𝑓𝑢, 𝑢)
2
Therefore from (3.15) as 𝑛 → ∞, we have
𝑑(𝑓𝑢, 𝑢) ≤ 𝛼(𝑑(𝑓𝑢, 𝑢))𝑑(𝑓𝑢, 𝑢) < 𝑑(𝑓𝑢, 𝑢), yields that 𝑓𝑢 = 𝑢.
Since 𝑓𝑋 ⊆ 𝑇𝑋 then there exists a point 𝑤 ∈ 𝑋 such that 𝑢 = 𝑓𝑢 = 𝑇𝑤. Since 𝑢 ≤
𝑓𝑢 = 𝑇𝑤 ≤ 𝑓𝑇𝑤 ≤ 𝑤 implies that 𝑢 ≤ 𝑤. Using (3.1.1), we have
𝑑(𝑇𝑤, 𝑔𝑤) = 𝑑(𝑓𝑢, 𝑔𝑤) ≤ 𝛼(𝑀(𝑢, 𝑤))𝑀(𝑢, 𝑤)
(3.16)
where
1
𝑀(𝑢, 𝑤) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑢, 𝑇𝑤), 𝑑(𝑓𝑢, 𝑆𝑢), 𝑑(𝑔𝑤, 𝑇𝑤), (𝑑(𝑆𝑢, 𝑔𝑤) + 𝑑(𝑓𝑢, 𝑇𝑤))}
2
R.K.Sharma, V. Raich and C.S.Chauhan
2224
1
= 𝑚𝑎𝑥{ 𝑑(𝑆𝑢, 𝑓𝑢), 𝑑(𝑓𝑢, 𝑆𝑢), 𝑑(𝑔𝑤, 𝑇𝑤), (𝑑(𝑆𝑢, 𝑔𝑤) + 𝑑(𝑓𝑢, 𝑇𝑤))}
2
1
= 𝑚𝑎𝑥{ 𝑑(𝑢, 𝑢), 𝑑(𝑢, 𝑢), 𝑑(𝑔𝑤, 𝑇𝑤), (𝑑(𝑇𝑤, 𝑔𝑤) + 𝑑(𝑇𝑤, 𝑇𝑤))}
2
1
= 𝑚𝑎𝑥{ 0, 0, 𝑑(𝑔𝑤, 𝑇𝑤), (𝑑(𝑇𝑤, 𝑔𝑤) + 0)} = 𝑑(𝑔𝑤, 𝑇𝑤)
2
Therefore from (3.16), we have
𝑑(𝑇𝑤, 𝑔𝑤) = 𝑑(𝑓𝑢, 𝑔𝑤) ≤ 𝛼(𝑑(𝑔𝑤, 𝑇𝑤))𝑑(𝑔𝑤, 𝑇𝑤) < 𝑑(𝑇𝑤, 𝑔𝑤), yields that
𝑇𝑤 = 𝑔𝑤.
Now by weakly compatibility of the pair (𝑔, 𝑇), 𝑔𝑢 = 𝑔𝑓𝑢 = 𝑔𝑇𝑤 = 𝑇𝑔𝑤 = 𝑇𝑓𝑢 =
𝑇𝑢, i.e. 𝑢 is a coincidence point of 𝑔 and 𝑇.
Next since 𝑥2𝑛 ≤ 𝑓𝑥2𝑛 and 𝑓𝑥2𝑛 → 𝑢 as 𝑛 → ∞ implies that 𝑥2𝑛 ≤ 𝑢, from (3.1.1),
we have
𝑑(𝑓𝑥2𝑛 , 𝑔𝑢) ≤ 𝛼(𝑀(𝑥2𝑛 , 𝑢))𝑀(𝑥2𝑛 , 𝑢)
(3.17)
where
1
𝑀(𝑥2𝑛 , 𝑢) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑥2𝑛 , 𝑇𝑢), 𝑑(𝑓𝑥2𝑛 , 𝑆𝑥2𝑛 ), 𝑑(𝑔𝑢, 𝑇𝑢), (𝑑(𝑆𝑥2𝑛 , 𝑔𝑢)
2
+ 𝑑(𝑓𝑥2𝑛 , 𝑇𝑢))}
Letting 𝑛 → ∞ and using (3.10) and (3.11), we have
1
lim 𝑀(𝑥2𝑛 , 𝑢) = 𝑚𝑎𝑥{ 𝑑(𝑢, 𝑔𝑢), 𝑑(𝑢, 𝑢), 𝑑(𝑔𝑢, 𝑔𝑢), (𝑑(𝑢, 𝑔𝑢) + 𝑑(𝑢, 𝑔𝑢))}
𝑛→∞
2
= 𝑚𝑎𝑥{ 𝑑(𝑢, 𝑔𝑢), 0,0, 𝑑(𝑢, 𝑔𝑢))} = 𝑑(𝑢, 𝑔𝑢)
Therefore from (3.17) as 𝑛 → ∞, we have
𝑑(𝑢, 𝑔𝑢) ≤ 𝛼(𝑑(𝑢, 𝑔𝑢))𝑑(𝑢, 𝑔𝑢) < 𝑑(𝑢, 𝑔𝑢),yields that 𝑔𝑢 = 𝑢.
Hence 𝑓𝑢 = 𝑔𝑢 = 𝑆𝑢 = 𝑇𝑢 = 𝑢 i.e. 𝑢 is the common fixed point of 𝑓, 𝑔, 𝑆 and 𝑇.
The proof is similar if 𝑓 is continuous instead of 𝑆. Similar result follows if the pair
( 𝑔, 𝑇) is compatible, pair (𝑓, 𝑆 ) is weakly compatible and 𝑔 or 𝑇 is continuous map.
On Generalization of Banach Contraction Principle in Partially Ordered Metric Spaces 2225
For the uniqueness of common fixed point suppose that the set of common fixed
points of 𝑓, 𝑔, 𝑆 and 𝑇 is well ordered and let 𝑢 and 𝑣 be any two common fixed
points of 𝑓, 𝑔, 𝑆 and 𝑇 then from (3.1.1), we have
(3.18)
𝑑(𝑓𝑢, 𝑔𝑣) ≤ 𝛼(𝑀(𝑢, 𝑣))𝑀(𝑢, 𝑣)
where
1
𝑀(𝑢, 𝑣) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑢, 𝑇𝑣), 𝑑(𝑓𝑢, 𝑆𝑢), 𝑑(𝑔𝑣, 𝑇𝑣), (𝑑(𝑆𝑢, 𝑔𝑣) + 𝑑(𝑓𝑢, 𝑇𝑣))}
2
1
= 𝑚𝑎𝑥 {𝑑(𝑢, 𝑣), 0,0, (𝑑(𝑢, 𝑣) + 𝑑(𝑢, 𝑣))} = 𝑑(𝑢, 𝑣)
2
From (3.18), we have
𝑑(𝑢, 𝑣) ≤ 𝛼(𝑑(𝑢, 𝑣))𝑑(𝑢, 𝑣) < 𝑑(𝑢, 𝑣), yields that 𝑢 = 𝑣 i.e. common fixed points
of 𝑓, 𝑔, 𝑆 and 𝑇 is unique.
Conversely, if 𝑓, 𝑔, 𝑆and 𝑇 have only one common fixed point then the set of common
fixed point of 𝑓, 𝑔, 𝑆 and 𝑇 being singleton is well ordered.
If we take 𝑆 = 𝑇 in Theorem 3.1, we have the following corollary:
Corollary 3.2: Let (𝑋, ≤) be a partially ordered set and there exists a metric 𝑑 in 𝑋
such that (𝑋, 𝑑) is a complete metric space. Let 𝑓, 𝑔, 𝑇: 𝑋 → 𝑋 such that 𝑓𝑋 ⊆
𝑇𝑋, 𝑔𝑋 ⊆ 𝑇𝑋 satisfying (2.12.5) and
(3.2.1) 𝑑(𝑓𝑥, 𝑔𝑦) ≤ 𝛼(𝑀(𝑥, 𝑦))𝑀(𝑥, 𝑦) where
1
𝑀(𝑥, 𝑦) = 𝑚𝑎𝑥{ 𝑑(𝑇𝑥, 𝑇𝑦), 𝑑(𝑓𝑥, 𝑇𝑥), 𝑑(𝑔𝑦, 𝑇𝑦), (𝑑(𝑇𝑥, 𝑔𝑦) + 𝑑(𝑓𝑥, 𝑇𝑦))}
2
for all 𝑥, 𝑦 ∈ 𝑋 with 𝑥 ≤ 𝑦 and 𝛼 ∈ ℱ.
(3.2.2) pairs ( 𝑇, 𝑓 ) and ( 𝑇, 𝑔) are partially weakly increasing.
(3.2.3) dominating maps 𝑓 and 𝑔 are weak annihilators of 𝑇.
(3.2.4) either pair (𝑓, 𝑇) is compatible, pair (𝑔, 𝑇 ) is weakly compatible and 𝑓 or 𝑇 is
continuous maps.
OR
R.K.Sharma, V. Raich and C.S.Chauhan
2226
pair ( 𝑔, 𝑇) is compatible, pair (𝑓, 𝑇) is weakly compatible and 𝑔 or 𝑇 is
continuous maps.
Then 𝑓, 𝑔 and 𝑇 have a common fixed point. Moreover, the common fixed point of
𝑓, 𝑔 and 𝑇 is unique if and only if the set of common fixed point of 𝑓, 𝑔 and 𝑇 is
well ordered.
Again on taking 𝑓 = 𝑔 in Theorem 3.1, we have the following corollary:
Corollary 3.3: Let (𝑋, ≤) be a partially ordered set and there exists a metric 𝑑 in 𝑋
such that (𝑋, 𝑑) is a complete metric space. Let 𝑓, 𝑆, 𝑇: 𝑋 → 𝑋 such that 𝑓𝑋 ⊆
𝑇𝑋, 𝑓𝑋 ⊆ 𝑆𝑋 satisfying (2.12.5) and
(3.3.1) 𝑑(𝑓𝑥, 𝑓𝑦) ≤ 𝛼(𝑀(𝑥, 𝑦))𝑀(𝑥, 𝑦) where
1
𝑀(𝑥, 𝑦) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑥, 𝑇𝑦), 𝑑(𝑓𝑥, 𝑆𝑥), 𝑑(𝑓𝑦, 𝑇𝑦), 2 (𝑑(𝑆𝑥, 𝑓𝑦) +
𝑑(𝑓𝑥, 𝑇𝑦))}
for all 𝑥, 𝑦 ∈ 𝑋 with 𝑥 ≤ 𝑦 and 𝛼 ∈ ℱ.
(3.3.2) pairs (𝑇, 𝑓) and (𝑆, 𝑓) are partially weakly increasing.
(3.3.3) dominating map 𝑓 is weak annihilators of 𝑇 and 𝑆.
(3.3.4) either pair (𝑓, 𝑆) is compatible, pair (𝑓, 𝑇) is weakly compatible and 𝑓 or 𝑆 is
continuous map.
OR
pair ( 𝑓, 𝑇) is compatible, pair (𝑓, 𝑆) is weakly compatible and 𝑓 or 𝑇 is
continuous map.
Then 𝑓, 𝑆 and 𝑇 have a common fixed point. Moreover, the common fixed point of 𝑓,
𝑆 and 𝑇 is unique if and only if the set of common fixed point of 𝑓, 𝑆 and 𝑇 is well
ordered.
Further taking 𝑓 = 𝑔 and 𝑆 = 𝑇 in Theorem 3.1, we have the following corollary:
Corollary 3.4: Let (𝑋, ≤) be a partially ordered set and there exists a metric 𝑑 on 𝑋
such that (𝑋, 𝑑) is a complete metric space. Let 𝑓, 𝑇: 𝑋 → 𝑋 such that 𝑓𝑋 ⊆ 𝑇𝑋
satisfying (2.12.5) and
(3.4.1) 𝑑(𝑓𝑥, 𝑓𝑦) ≤ 𝛼(𝑀(𝑥, 𝑦))𝑀(𝑥, 𝑦) where
On Generalization of Banach Contraction Principle in Partially Ordered Metric Spaces 2227
1
𝑀(𝑥, 𝑦) = 𝑚𝑎𝑥{ 𝑑(𝑇𝑥, 𝑇𝑦), 𝑑(𝑓𝑥, 𝑇𝑥), 𝑑(𝑓𝑦, 𝑇𝑦), 2 (𝑑(𝑇𝑥, 𝑓𝑦) +
𝑑(𝑓𝑥, 𝑇𝑦))}
for all 𝑥, 𝑦 ∈ 𝑋 with 𝑥 ≤ 𝑦 and 𝛼 ∈ ℱ.
(3.4.2) pair (𝑇, 𝑓) is partially weak increasing.
(3.4.3) dominating map 𝑓 is weak annihilator of 𝑇.
(3.4.4) pair ( 𝑓, 𝑇) is compatible and 𝑓 or 𝑇 is continuous map.
Then 𝑓 and 𝑇 have a common fixed point. Moreover, the common fixed point of 𝑓
and 𝑇 is unique if and only if the set of common fixed point of 𝑓 and 𝑇 is well
ordered.
Remark 3.5: If we take 𝑓 = 𝑔 and 𝑆 = 𝑇 = Identity map in Theorem 3.1, 𝑓 = 𝑔 and
𝑇 = Identity map in Cor. 3.2 and 𝑆 = 𝑇 = Identity map in Cor. 3.3 respectively then
we have Theorem 2.10 [5].
Now by omitting the conditions of continuity of 𝑓, 𝑔, 𝑆 and 𝑇, compatibility of pairs
( 𝑓, 𝑆) and ( 𝑔, 𝑇) by other conditions in the Theorem 3.1, we have the following
result:
Theorem 3.6: Let (𝑋, ≤) be a partially ordered set and there exists a metric 𝑑 in 𝑋
such that (𝑋, 𝑑) is a complete metric space. Let 𝑓, 𝑔, 𝑆, 𝑇: 𝑋 → 𝑋 satisfying the
conditions (2.11.6), (2.12.1), (2.12.3), (2.12.4), (3.1.1) and
(3.6.1) 𝑆𝑋 ∪ 𝑇𝑋 is closed subspace of 𝑋.
(3.6.2) pairs ( 𝑓, 𝑆) and ( 𝑔, 𝑇) are weakly compatible.Then 𝑓, 𝑔, 𝑆 and 𝑇 have a
unique common fixed point.
Proof: We take the sequences {𝑥𝑛 } and {𝑦𝑛 } defined in the proof of theorem 3.1.
Then as the proof of theorem 3.1,{𝑦𝑛 } is a Cauchy sequence in closed subspace 𝑆𝑋 ∪
𝑇𝑋, then there exists 𝑢 ∈ 𝑆𝑋 ∪ 𝑇𝑋 such that 𝑦𝑛 → 𝑢 as 𝑛 → ∞. Further, the
subsequences {𝑇𝑥2𝑛+1 } = {𝑓𝑥2𝑛 } = {𝑦2𝑛+1 } and {𝑆𝑥2𝑛+2 } = {𝑔𝑥2𝑛+1 } = {𝑦2𝑛+2 } of
{𝑦𝑛 } also converge to the point 𝑢.
Now, since 𝑢 ∈ 𝑆𝑋 ∪ 𝑇𝑋, we have 𝑢 ∈ 𝑆𝑋 or 𝑢 ∈ 𝑇𝑋.
Case I: If 𝑢 ∈ 𝑇𝑋, then we can find 𝑤 ∈ 𝑋 such that 𝑢 = 𝑇𝑤, since 𝑥2𝑛 ≤ 𝑓𝑥2𝑛 and
𝑓𝑥2𝑛 → 𝑢 as 𝑛 → ∞, we have 𝑥2𝑛 ≤ 𝑢 also since dominating map 𝑓 is weak
R.K.Sharma, V. Raich and C.S.Chauhan
2228
annihilator with respect to 𝑇 therefore 𝑥2𝑛 ≤ 𝑢 = 𝑇𝑤 ≤ 𝑓𝑇𝑤 ≤ 𝑤. Now we claim
that 𝑢 = 𝑔𝑤, if not then from (3.1.1), we have
(3.19)
𝑑(𝑓𝑥2𝑛 , 𝑔𝑤) ≤ 𝛼(𝑀(𝑥2𝑛 , 𝑤))𝑀(𝑥2𝑛 , 𝑤)
where
𝑀(𝑥2𝑛 , 𝑤) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑥2𝑛 , 𝑇𝑤), 𝑑(𝑓𝑥2𝑛 , 𝑆𝑥2𝑛 ), 𝑑(𝑔𝑤, 𝑇𝑤),
1
(𝑑(𝑆𝑥2𝑛 , 𝑔𝑤) + 𝑑(𝑓𝑥2𝑛 , 𝑇𝑤))}
2
1
= 𝑚𝑎𝑥{ 𝑑(𝑆𝑥2𝑛 , 𝑢), 𝑑(𝑓𝑥2𝑛 , 𝑆𝑥2𝑛 ), 𝑑(𝑔𝑤, 𝑢), (𝑑(𝑆𝑥2𝑛 , 𝑔𝑤) + 𝑑(𝑓𝑥2𝑛 , 𝑢))}.
2
Letting 𝑛 → ∞ and using (3.10) and (3.11), we have
1
lim 𝑀(𝑥2𝑛 , 𝑤) = 𝑚𝑎𝑥{ 𝑑(𝑢, 𝑢), 𝑑(𝑢, 𝑢), 𝑑(𝑔𝑤, 𝑢), (𝑑(𝑢, 𝑔𝑤) + 𝑑(𝑢, 𝑢))}
𝑛→∞
2
1
= 𝑚𝑎𝑥{ 0, 0, 𝑑(𝑔𝑤, 𝑢), (𝑑(𝑢, 𝑔𝑤) + 0)} = 𝑑(𝑔𝑤, 𝑢)
2
Therefore from (3.19), as 𝑛 → ∞, we have
𝑑(𝑢, 𝑔𝑤) ≤ 𝛼(𝑑(𝑔𝑤, 𝑢))𝑑(𝑔𝑤, 𝑢) < 𝑑(𝑢, 𝑔𝑤), yields that 𝑢 = 𝑔𝑤.
Therefore 𝑢 = 𝑔𝑤 = 𝑇𝑤, i.e. 𝑤 is the coincidence point of 𝑔 and 𝑇. Now by weakly
compatibility of (𝑔, 𝑇), we have 𝑔𝑢 = 𝑔𝑇𝑤 = 𝑇𝑔𝑤 = 𝑇𝑢 i.e.𝑢 is a coincidence point
of 𝑔 and 𝑇.
Next we claim that 𝑔𝑢 = 𝑢, if not then from (3.1.1), we have
𝑑(𝑓𝑥2𝑛 , 𝑔𝑢) ≤ 𝛼(𝑀(𝑥2𝑛 , 𝑢))𝑀(𝑥2𝑛 , 𝑢)
(3.20)
where
1
𝑀(𝑥2𝑛 , 𝑢) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑥2𝑛 , 𝑇𝑢), 𝑑(𝑓𝑥2𝑛 , 𝑆𝑥2𝑛 ), 𝑑(𝑔𝑢, 𝑇𝑢), (𝑑(𝑆𝑥2𝑛 , 𝑔𝑢)
2
+ 𝑑(𝑓𝑥2𝑛 , 𝑇𝑢))}
1
= 𝑚𝑎𝑥{𝑑(𝑆𝑥2𝑛 , 𝑔𝑢), 𝑑(𝑓𝑥2𝑛 , 𝑆𝑥2𝑛 ), 𝑑(𝑔𝑢, 𝑔𝑢), (𝑑(𝑆𝑥2𝑛 , 𝑔𝑢)
2
+ 𝑑(𝑓𝑥2𝑛 , 𝑔𝑢))}
On Generalization of Banach Contraction Principle in Partially Ordered Metric Spaces 2229
Letting 𝑛 → ∞ and using (3.10) and (3.11), we have
1
lim 𝑀(𝑥2𝑛 , 𝑢) = 𝑚𝑎𝑥{ 𝑑(𝑢, 𝑔𝑢), 𝑑(𝑢, 𝑢), 𝑑(𝑔𝑢, 𝑔𝑢), (𝑑(𝑢, 𝑔𝑢) + 𝑑(𝑢, 𝑔𝑢))}
𝑛→∞
2
= max{𝑑(𝑢, 𝑔𝑢), 0, 0, 𝑑(𝑢, 𝑔𝑢)} = 𝑑(𝑔𝑢, 𝑢)
Therefore from (3.20) as 𝑛 → ∞, we have
𝑑(𝑢, 𝑔𝑢) ≤ 𝛼(𝑑(𝑔𝑢, 𝑢))𝑑(𝑔𝑢, 𝑢) < 𝑑(𝑢, 𝑔𝑢), yields that 𝑢 = 𝑔𝑢.
i.e. 𝑢 is the common fixed point of𝑔 and 𝑇.
Since 𝑔𝑋 ⊆ 𝑆𝑋 then there exists a point 𝑦 ∈ 𝑋 such that 𝑢 = 𝑔𝑢 = 𝑆𝑦, since 𝑢 ≤
𝑔𝑢 = 𝑆𝑦 ≤ 𝑔𝑆𝑦 ≤ 𝑦 implies that 𝑢 ≤ 𝑦. Using (3.1.1), we have
𝑑(𝑓𝑦, 𝑆𝑦) = 𝑑(𝑓𝑦, 𝑔𝑢) ≤ 𝛼(𝑀(𝑦, 𝑢))𝑀(𝑦, 𝑢)
(3.21)
where
1
𝑀(𝑦, 𝑢) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑦, 𝑇𝑢), 𝑑(𝑓𝑦, 𝑆𝑦), 𝑑(𝑔𝑢, 𝑇𝑢), (𝑑(𝑆𝑦, 𝑔𝑢) + 𝑑(𝑓𝑦, 𝑇𝑢))}
2
1
= 𝑚𝑎𝑥{ 𝑑(𝑆𝑦, 𝑔𝑢), 𝑑(𝑓𝑦, 𝑆𝑦), 𝑑(𝑔𝑢, 𝑔𝑢), (𝑑(𝑆𝑦, 𝑔𝑢) + 𝑑(𝑓𝑦, 𝑔𝑢))}
2
1
= 𝑚𝑎𝑥{ 0, 𝑑(𝑓𝑦, 𝑆𝑦), 0, (0 + 𝑑(𝑓𝑦, 𝑆𝑦))} = 𝑑(𝑓𝑦, 𝑆𝑦)
2
Therefore from (3.21), we have
𝑑(𝑓𝑦, 𝑆𝑦) ≤ 𝛼(𝑑(𝑓𝑦, 𝑆𝑦))𝑑(𝑓𝑦, 𝑆𝑦) < 𝑑(𝑓𝑦, 𝑆𝑦), yields that 𝑓𝑦 = 𝑆𝑦.
i.e. y is the coincidence point of (𝑓, 𝑆).
Since 𝑥2𝑛+1 ≤ 𝑔𝑥2𝑛+1 and 𝑔𝑥2𝑛+1 → 𝑢 as 𝑛 → ∞, we have 𝑥2𝑛+1 ≤ 𝑢 also
dominating map 𝑔 is weak annihilator with respect to 𝑆 therefore 𝑥2𝑛+1 ≤ 𝑢 = 𝑆𝑦 ≤
𝑔𝑆𝑦 ≤ 𝑦.
Now we claim that 𝑢 = 𝑓𝑦, if not then from (3.1.1), we have
𝑑(𝑓𝑦, 𝑔𝑥2𝑛+1 ) ≤ 𝛼(𝑀(𝑦, 𝑥2𝑛+1 ))𝑀(𝑦, 𝑥2𝑛+1 )
where
𝑀(𝑦, 𝑥2𝑛+1 ) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑦, 𝑇𝑥2𝑛+1 ), 𝑑(𝑓𝑦, 𝑆𝑦), 𝑑(𝑔𝑥2𝑛+1 , 𝑇𝑥2𝑛+1 ),
(3.22)
R.K.Sharma, V. Raich and C.S.Chauhan
2230
1
(𝑑(𝑆𝑦, 𝑔𝑥2𝑛+1 ) + 𝑑(𝑓𝑦, 𝑇𝑥2𝑛+1 ))}
2
= 𝑚𝑎𝑥{ 𝑑(𝑢, 𝑇𝑥2𝑛+1 ), 𝑑(𝑓𝑦, 𝑢), 𝑑(𝑔𝑥2𝑛+1 , 𝑇𝑥2𝑛+1 ),
1
(𝑑(𝑢, 𝑔𝑥2𝑛+1 ) + 𝑑(𝑓𝑦, 𝑇𝑥2𝑛+1 ))}
2
Letting 𝑛 → ∞ and using (3.10) and (3.11), we have
1
lim 𝑀(𝑦, 𝑥2𝑛+1 ) = 𝑚𝑎𝑥{ 𝑑(𝑢, 𝑢), 𝑑(𝑓𝑦, 𝑢), 𝑑(𝑢, 𝑢), (𝑑(𝑢, 𝑢) + 𝑑(𝑓𝑦, 𝑢))}
𝑛→∞
2
1
= 𝑚𝑎𝑥{ 0, 𝑑(𝑓𝑦, 𝑢), 0, (0 + 𝑑(𝑓𝑦, 𝑢))} = 𝑑(𝑓𝑦, 𝑢)
2
Therefore from (3.22) as 𝑛 → ∞, we have
𝑑(𝑓𝑦, 𝑢) ≤ 𝛼(𝑑(𝑓𝑦, 𝑢))𝑑(𝑓𝑦, 𝑢) < 𝑑(𝑓𝑦, 𝑢), yields that 𝑢 = 𝑓𝑦.
Hence 𝑢 = 𝑓𝑦 = 𝑆𝑦, i.e. 𝑦 is the coincidence point of 𝑓 and 𝑆. Now by weakly
compatibility of pair (𝑓, 𝑆), 𝑓𝑢 = 𝑓𝑆𝑦 = 𝑆𝑓𝑦 = 𝑆𝑢, i.e.𝑢 is a coincidence point of 𝑓
and 𝑆.
Next we claim that 𝑓𝑢 = 𝑢, if not then from (3.1.1), we have
𝑑(𝑓𝑢, 𝑢) = 𝑑(𝑓𝑢 𝑔𝑢) ≤ 𝛼(𝑀(𝑢, 𝑢))𝑀(𝑢, 𝑢)
where
1
𝑀(𝑢, 𝑢) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑢, 𝑇𝑢), 𝑑(𝑓𝑢, 𝑆𝑢), 𝑑(𝑔𝑢, 𝑇𝑢), (𝑑(𝑆𝑢, 𝑔𝑢) + 𝑑(𝑓𝑢, 𝑇𝑢))}
2
1
= max { 𝑑(𝑆𝑢, 𝑔𝑢), 𝑑(𝑓𝑢, 𝑆𝑢), 𝑑(𝑔𝑢, 𝑔𝑢), (𝑑(𝑆𝑢, 𝑔𝑢) + 𝑑(𝑓𝑢, 𝑔𝑢))}
2
1
= max { 𝑑(𝑓𝑢, 𝑢), 𝑑(𝑓𝑢, 𝑆𝑢), 𝑑(𝑔𝑢, 𝑔𝑢), (𝑑(𝑓𝑢, 𝑢) + 𝑑(𝑓𝑢, 𝑢))}
2
= max{ 𝑑(𝑓𝑢, 𝑢), 0,0, 𝑑(𝑓𝑢, 𝑢)} = 𝑑(𝑓𝑢, 𝑢)
Therefore from (3.23), we have
𝑑(𝑓𝑢, 𝑢) ≤ 𝛼(𝑑(𝑓𝑢, 𝑢))𝑑(𝑓𝑢, 𝑢) < 𝑑(𝑓𝑢, 𝑢), yields that 𝑓𝑢 = 𝑢.
(3.23)
On Generalization of Banach Contraction Principle in Partially Ordered Metric Spaces 2231
Consequently 𝑓𝑢 = 𝑔𝑢 = 𝑆𝑢 = 𝑇𝑢 = 𝑢 i.e. 𝑢 is a common fixed point of 𝑓, 𝑔, 𝑆 and
𝑇.
Case II: If 𝑢 ∈ 𝑆𝑋, then similarly as case I, it can be proved that 𝑢 is the common
fixed point of 𝑓, 𝑔, 𝑆 and 𝑇.
For the uniqueness of common fixed point suppose that 𝑢∗ is another common fixed
point of 𝑓, 𝑔, 𝑆 and 𝑇, using (3.1.1), we have
𝑑(𝑢, 𝑢∗ ) = 𝑑(𝑓𝑢 𝑔𝑢∗ ) ≤ 𝛼(𝑀(𝑢, 𝑢∗ ))𝑀(𝑢, 𝑢∗ )
(3.24)
where
1
𝑀(𝑢, 𝑢∗ ) = 𝑚𝑎𝑥{ 𝑑(𝑆𝑢, 𝑇𝑢∗ ), 𝑑(𝑓𝑢, 𝑆𝑢), 𝑑(𝑔𝑢∗ , 𝑇𝑢∗ ), (𝑑(𝑆𝑢, 𝑔𝑢∗ ) + 𝑑(𝑓𝑢, 𝑇𝑢∗ ))}
2
1
= max { 𝑑(𝑢, 𝑢∗ ), 0,0, (𝑑(𝑢, 𝑢∗ ) + 𝑑(𝑢, 𝑢∗ ))} = 𝑑(𝑢, 𝑢∗ )
2
Therefore from (3.24), we have
𝑑(𝑢, 𝑢∗ ) ≤ 𝛼(𝑑(𝑢, 𝑢∗ ))𝑑(𝑢, 𝑢∗ ) < 𝑑(𝑢, 𝑢∗ ), yields that 𝑢 = 𝑢∗ i.e. common fixed
points of 𝑓, 𝑔, 𝑆 and 𝑇 is unique.
Corollary 3.7: Here note that the Theorem 3.6 remains true if the condition (3.6.1) is
replaced by 𝑆𝑋 ∩ 𝑇𝑋 is closed subspace of 𝑋.
Proof: If 𝑆𝑋 ∩ 𝑇𝑋 is closed subspace of 𝑋 then as in the proof of theorem 3.6 we
have
𝑢 ∈ 𝑆𝑋 ∩ 𝑇𝑋 , implies that 𝑢 ∈ 𝑆𝑋 and 𝑢 ∈ 𝑇𝑋. Hence the result remains true.
Corollary 3.8: If we take 𝑇 = 𝑆 in Theorem 3.6 then we have Theorem 2.11 [6].
Now we provide the following example in support of Theorem 3.1.
Example 3.9: Let 𝑋 = [0, ∞) be endowed with the usual metric 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦|,let
“≤” be the usual ordering on 𝑅, we define a new ordering “≼” on 𝑋 such that 𝑥 ≼
𝑦 ⇔ 𝑦 ≤ 𝑥, ∀ 𝑥, 𝑦 ∈ 𝑋.
1, 𝑆𝑥 = 𝑒
8𝑥
Define
𝑥
𝑥
4
8
𝑓𝑥 = ln (1 + ) , 𝑔𝑥 = ln (1 + ) , 𝑇𝑥 = 𝑒 4𝑥 −
− 1.
Then 𝑓𝑋 = [0, ∞), 𝑔𝑋 = [0, ∞), 𝑆𝑋 = [0, ∞), 𝑇𝑋 = [0, ∞), we have 𝑓𝑋 ⊆ 𝑇𝑋 and
𝑔𝑋 ⊆ 𝑆𝑋.
R.K.Sharma, V. Raich and C.S.Chauhan
2232
𝑥
𝑥
𝑥
Now for each 𝑥 ∈ 𝑋 we have 1 + 4 ≤ 𝑒 𝑥 and 1 + 8 ≤ 𝑒 𝑥 so 𝑓𝑥 = ln (1 + 4) ≤ 𝑥 and
𝑥
𝑔𝑥 = ln (1 + ) ≤ 𝑥 which implies that 𝑥 ≼ 𝑓𝑥 and 𝑥 ≼ 𝑔𝑥 so 𝑓, 𝑔 are dominating
8
maps.
Also for each 𝑥 ∈ 𝑋, we have
𝑓𝑇𝑥 = ln (1 +
𝑇𝑥
𝑒 4𝑥 − 1
3 + 𝑒 4𝑥
3𝑒 −2𝑥 + 𝑒 2𝑥
) = ln (1 +
) = ln (
) = ln (𝑒 2𝑥 .
)
4
4
4
4
3𝑒 −2𝑥 +𝑒 2𝑥
= 2𝑥 + ln (
4
) ≥ 𝑥 which implies that 𝑓𝑇𝑥 ≼ 𝑥.
And
𝑔𝑆𝑥 = ln (1 +
𝑆𝑥
𝑒 8𝑥 − 1
7 + 𝑒 8𝑥
7𝑒 −4𝑥 + 𝑒 4𝑥
) = ln (1 +
) = ln (
) = ln (𝑒 4𝑥 .
)
8
8
8
8
7𝑒 −4𝑥 +𝑒 4𝑥
= 4𝑥 + ln (
8
) ≥ 𝑥 which implies that 𝑔𝑆𝑥 ≼ 𝑥. Thus 𝑓 and 𝑔 are weak
annihilators of 𝑇 and 𝑆, respectively.
Since 𝑓𝑇𝑥 ≼ 𝑥 and 𝑥 ≼ 𝑓𝑥 so 𝑓𝑇𝑥 ≼ 𝑓𝑥 which implies that (𝑇, 𝑓) is partially weakly
increasing. Similarly 𝑔𝑆𝑥 ≼ 𝑥 and 𝑥 ≼ 𝑔𝑥 so 𝑔𝑆𝑥 ≼ 𝑔𝑥 which implies that (𝑆, 𝑔) is
partially weakly increasing.
1
1
Now there exists a non-decreasing sequence {𝑥𝑛 } = {𝑛} in 𝑋 such that 𝑥𝑛 = 𝑛 → 0,
𝑓𝑥𝑛 = ln (1 +
𝑥𝑛
Also 𝑓𝑆𝑥𝑛 = ln (1 +
Therefore,
maps.
1
8
) = ln (1 + 4𝑛) → 0, 𝑆𝑥𝑛 = 𝑒 8𝑥𝑛 − 1 = 𝑒 𝑛 − 1 → 0, as 𝑛 → ∞.
4
𝑆𝑥𝑛
4
) → 0 and 𝑆𝑓𝑥𝑛 = 𝑒 8𝑓𝑥𝑛 − 1 → 0.
lim 𝑑(𝑓𝑆𝑥𝑛 , 𝑆𝑓𝑥𝑛 ) = 0, i.e., the pair (𝑓, 𝑆) is compatible and continuous
𝑛→∞
Also here 0 is the coincidence points of the pair (𝑔, 𝑇) and we have
𝑔𝑇(0) = 𝑔(0) = 0 = 𝑇(0) = 𝑇𝑔(0), i.e., the pair (𝑔, 𝑇) is weakly compatible
mappings.
1
1
Now, we define 𝛼(𝑡) = 1+𝑡 if 𝑡 ∈ (0, ∞) and 𝛼(𝑡) = 0 if 𝑡 = 0 then for 𝑡𝑛 = 𝑛 ,
On Generalization of Banach Contraction Principle in Partially Ordered Metric Spaces 2233
lim 𝛼(𝑡𝑛 ) = lim
𝑛→∞
1
1
𝑛→∞ 1+
1
𝑛
→ 1 ⇒ lim 𝑡𝑛 = 𝑛 → 0 thus 𝛼 ∈ ℱ.
𝑛→∞
Now from (3.1.1) and using mean value theorem we have for 𝑥, 𝑦 ∈ 𝑋,
𝑥
𝑦
𝑥
𝑦
𝑑(𝑓𝑥, 𝑔𝑦) = |𝑓𝑥 − 𝑔𝑦| = |ln (1 + ) − ln (1 + )| ≤ |1 + − 1 − |
4
8
4
8
≤
1
1
1
1
|2𝑥 − 𝑦| = |8𝑥 − 4𝑦| ≤ |𝑒 8𝑥 − 𝑒 4𝑦 | ≤ |𝑆𝑥 − 𝑇𝑦|
8
2
2
2
1
= 𝑑(𝑆𝑥, 𝑇𝑦)
2
1
≤ 2 𝑀(𝑥, 𝑦) ≤ 𝛼(𝑀(𝑥, 𝑦))𝑀(𝑥, 𝑦) holds if
1
2
≤ 𝛼(𝑀(𝑥, 𝑦)) < 1 ∀ 𝑥, 𝑦 ∈ 𝑋.
Thus all the conditions of Theorem 3.1 are satisfied and 0 is the unique common fixed
point of 𝑓, 𝑔, 𝑆 and 𝑇.
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