Mathematics 2019 , 7, x doi: FOR PEER REVIEW www.mdpi.comjournal mathematics [620039]

Mathematics 2019 , 7, x; doi: FOR PEER REVIEW www.mdpi.com/journal/ mathematics
Article 1
Some common fixed point theorems in ordered partial metric spaces via 𝓕-generalized 2
contractive type mappings 3
Pooja Dhawan 1* and Jatinderdeep Kaur2 4
1 Thapar Institute of Engineering and Technology, Patiala, Punjab (147004), India; 5
pdhawan12@g mail.com 6
* Correspondence: pdhawan12@g mail.com 7
Received: date; Accepted: date; Published: date 8
9
Abstract: In the present work, The concept of ℱ-generalized contractive t ype mappings by using 10
𝐶-class functions is introduced and some common fixed point results for weakly isotone increasing 11
set-valued mappings in the setting of ordered partial metric spaces are studied. These results 12
improve and generalize various results existing in literature. The effectiveness of obtained results 13
have been verified with the help of some compara tive examples. 14
15
Keywords: Fixed point; Ordered partial metric space; ℱ-generalized contractive type mappings. 16
17
1. Introduction 18
The study of common fi xed points was initiated by Gerald Jungck [10] in 1986 and this conc ept 19
attracted many researchers to prove the existence of fixed p oints by using various metrical 20
contractions. On the other hand, the notion of partial metric spaces was presented by S.G. Matthews 21
[14] which has been considered as one of the most interesting, robust and outstanding 22
generalizatio ns of metric spaces. Many authors generalized this notion in different ways (See [5], 23
[12, 13], [11], [14], [15], [18]). In 2010, Hong [9] de fined the concept of approximative values to prove 24
the existence of common fixed points for multivalued operators i n the framework of ordered metric 25
spaces. After that, Erduran [8] extended this concept and studied some fixed point results for 26
multivalued mappings in partial metric spaces. In 2014, Arslan Hojat Ansari [3] introduced 𝐶-class 27
functions defi ned on R. 28
In this paper, The notion of ℱ-generalized contractive type mappings is in troduced and some 29
common fi xed point theorems for multivalued mappings in ordered partial metric spaces by using 30
𝐶-class functions and 𝐹-generalized contractive type mappings are o btained. 31
32
Definition 1.1. [14] Let 𝑈 be a non empty set. A function 𝑝∶𝑈× 𝑈→𝑅+ is said to be a partial 33
metric on 𝑈 if the following postulates hold true for all 𝑢,𝑣,𝑤∈𝑈: 34
(p1)𝑢=𝑣 if and only if 𝑝(𝑢,𝑢)=𝑝(𝑣,𝑣)=𝑝(𝑢,𝑣); 35
(p2) 𝑝(𝑢,𝑢)≤𝑝(𝑢,𝑣) (small self -distance axiom ); 36
(p3) 𝑝(𝑢,𝑣) = 𝑝(𝑣,𝑢) (symmetry ); 37
(p4) 𝑝(𝑢,𝑤)≤𝑝(𝑢,𝑣)+ 𝑝(𝑣,𝑤)− 𝑝(𝑣,𝑣) (modified triangle inequality ). 38
The pair (𝑈,𝑝) is then called a partial metric space (in short PMS). 39
Each partial metric p on 𝑈 generates a 𝑇0 topology 𝜏𝑝 on 𝑈 which has a base, the family of open 40
𝑝-balls {𝐵𝑝(𝑢,𝜖),𝑢∈𝑈,𝜖>0}, where 41
𝐵𝑝(𝑢,𝜖)= {v∈𝑈: 𝑝(𝑢,𝑣)< 𝑝(𝑢,𝑢)+ 𝜖} for all 𝑢∈𝑈 and 𝜖>0. 42

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If 𝑝 is a partial metric defi ned on 𝑈, then the mapping 𝑑𝑝: 𝑈× 𝑈→𝑅+ given by 43
𝑑𝑝(𝑢,𝑣)=2𝑝(𝑢,𝑣)−𝑝(𝑢,𝑢)−𝑝(𝑣,𝑣) is a metric on 𝑈. 44
45
Defi nition 1.2 . [16] For a partial metric space (𝑈,𝑝), a sequence {𝑢𝑛} in 𝑈 is said to be 46
(i) convergent if there exists a point 𝑢∈𝑈 such that 𝑝(𝑢,𝑢)= lim
𝑛→∞𝑝(𝑢𝑛,𝑢); 47
(ii) a cauchy sequence if the limit lim
𝑛,𝑚→∞𝑝(𝑢𝑛,𝑢𝑚) exists (and is fi nite). 48
Defi nition 1.3. [16] A partial metric space (𝑈,𝑝) is said to be complete if every cauchy sequence {𝑢𝑛} 49
in 𝑈 converges w. r. t. 𝜏𝑝 to a point 𝑢∈𝑈 such that 𝑝(𝑢,𝑢)=lim
𝑛,𝑚→∞𝑝(𝑢𝑛,𝑢𝑚). 50
Lemma 1.4. [16] Let (𝑈,𝑝) be a partial metric space. Then 51
(i) {𝑢𝑛} is said to be a cauchy sequence in (𝑈,𝑝) if it is a cauchy sequence in the metric space 52
(𝑈,𝑑𝑝); 53
(ii) (𝑈,𝑝) is complete if the metric space (𝑈,𝑑𝑝) is complete. Also, 54
lim
𝑛→∞𝑑𝑝(𝑢𝑛,𝑢)= 0 if 𝑝(𝑢,𝑢)= lim
𝑛→∞𝑝(𝑢𝑛,𝑢)= lim
𝑛,𝑚→∞𝑝(𝑢𝑛,𝑢𝑚). 55
Lemma 1.5. [6] Let (𝑈,𝑝) be a partial metric space and let {𝑢𝑛} be a sequence in 𝑈 such 56
that lim
𝑛→∞𝑝(𝑢𝑛,𝑢𝑛+1)=0. 57
If the sequence {𝑢2𝑛} is not a cauchy sequence in (𝑈,𝑝), then there exist 𝜖>0 and two sequences 58
{𝑢𝑚(𝑘)} and {𝑢𝑛(𝑘)} of positive integers with 𝑛(𝑘)> 𝑚(𝑘)> 𝑘 such that the following four 59
sequences 60
𝑝(𝑢2𝑚(𝑘),𝑢2𝑛(𝑘)+1), 𝑝(𝑢2𝑚(𝑘),𝑢2𝑛(𝑘)), 𝑝(𝑢2𝑚(𝑘)−1,𝑢2𝑛(𝑘)+1), 𝑝(𝑢2𝑚(𝑘)−1,𝑢2𝑛(𝑘)) 61
tend to 𝜖>0 when k→∞. 62
63
Lemma 1.6 . [2]. If the sequence {𝑢𝑛} with lim
𝑛→∞𝑑𝑝(𝑢𝑛+1, 𝑢𝑛) = 0 is not a cauchy sequence in (𝑈,𝑝), 64
then for each 𝜖>0, there exist two sequences {𝑚(𝑘)} and {𝑛(𝑘)} of positive integers with 𝑛(𝑘)> 65
𝑚(𝑘)> 𝑘 such that the following four sequence s 66
𝑝(𝑢𝑚(𝑘),𝑢𝑛(𝑘)+1), 𝑝(𝑢𝑚(𝑘),𝑢𝑛(𝑘)), 𝑝(𝑢𝑚(𝑘)−1,𝑢𝑛(𝑘)+1), 𝑝(𝑢𝑚(𝑘)−1,𝑢𝑛(𝑘)) 67
tend to 𝜖>0 when k→∞. 68
69
Let 𝐶𝐵𝑝(𝑈) be a family of all nonempty, closed and bounded subsets of the partial metric space 70
(𝑈,𝑝). Note that closedness is obvious as 𝜏𝑝 is the topology induced by 𝑝 and boundedness is 71
given as follows: 𝐴1 is a bounded subset in (𝑈,𝑝) if there exists 𝑀≥0 and 𝑢0∈ 𝑈 such that for 72
each 𝑎1∈𝐴1, we have , 𝑎1∈ 𝐵𝑝(𝑢0,𝑀) 𝑖.𝑒. 𝑝(𝑢0,𝑎1)< 𝑝(𝑎1,𝑎1)+𝑀. 73
74
For all 𝐴1, 𝐴2∈ 𝐶𝐵𝑝(𝑈) and u∈ U, 75
𝑝(𝑢,𝐴1) = inf {𝑝(𝑢, 𝑣∶ 𝑣 ∈𝐴1}, 76
𝛿𝑝(𝐴1, 𝐴2)=sup{𝑝(𝑎1, 𝐴2): 𝑎1 ∈ 𝐴1}, 77
𝛿𝑝(𝐴2, 𝐴1)=sup{𝑝(𝐴1, 𝑎2): 𝑎2 ∈ 𝐴2}, 78
and 79

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𝑃ℎ(𝐴1, 𝐴2) = max { 𝛿𝑝(𝐴1, 𝐴2),𝛿𝑝(𝐴2, 𝐴1)}. 80
Note that 𝑝(𝑢,𝐴1)= 0 implies 𝑑𝑝(𝑢,𝐴1)= 0 where 𝑑𝑝(𝑢,𝐴1)= inf {𝑑𝑝(𝑢,𝑎1) : 𝑎1∈𝐴1} 81
Corollary 1.7. [1] Let (𝑈,𝑝) be a partial metric space and let 𝐴1 be any nonempty set in (𝑈,𝑝), 82
then 𝑎1∈𝐴1̅̅̅ iff 𝑝(𝑎1, 𝐴1) = 𝑝(𝑎1,𝑎1) where 𝐴1̅̅̅ denotes the closure of 𝐴1 w. r. t. the partial 83
metric 𝑝. we say that 𝐴1 is closed in (𝑈,𝑝) iff 𝐴1̅̅̅= 𝐴1. 84
85
Proposition 1.8. [4] Let (𝑈,𝑝) be a partial metric space. For all 𝐴1, 𝐴2,𝐴3∈𝐶𝐵𝑝(𝑈) we have 86
(h1) 𝑃ℎ(𝐴1, 𝐴1)≤𝑃ℎ(𝐴1, 𝐴2), 87
(h2) 𝑃ℎ(𝐴1, 𝐴2) = 𝑃ℎ(𝐴2, 𝐴1), 88
(h3) 𝑃ℎ(𝐴1, 𝐴2)≤ 𝑃ℎ(𝐴1, 𝐴3) + 𝑃ℎ(𝐴3, 𝐴2) – 𝑖𝑛𝑓 𝑎3∈𝐴3𝑝(𝑎3,𝑎3), 89
(h4) 𝑃ℎ(𝐴1, 𝐴2) = 0 ⟹ 𝐴1= 𝐴2. 90
The mapping 𝑃ℎ: 𝐶𝐵𝑝(𝑈)×𝐶𝐵𝑝(𝑈)→ [0, +∞) is called the Partial Haus dorff metric induced by p. 91
Every Hausdorff metric is a Partial Hausdor ff metric but the converse need not be true ( Example 2.6 , 92
[4]). 93
94
Defi nition 1.9. [19] For a nonempty set 𝑈, The space (𝑈,𝑝) is called an ordered partial metric 95
space if (𝑈,𝑝) is a partial metric space and (𝑈,≼) is a partially ordered set. 96
Let (𝑈,≼) be a partially ordered set. Then 𝑢,𝑣∈ 𝑈 are called comparable if 𝑢≼ 𝑣 or 𝑣≼ 𝑢. 97
98
Defi nition 1.10. [9] Let 𝐴1 and 𝐴2 be any two nonempty subsets of an ordered set (𝑈,≼). The 99
relation ≼2 between 𝐴1 and 𝐴2 is defi ned as follows: 100
𝐴1≼2 𝐴2 if 𝑎1≼ 𝑎2 for each 𝑎1∈𝐴1 and 𝑎2∈𝐴2. 101
102
Defi nition 1.11. [7] Let (𝑈,≼) be a partially ordered set. Two maps 𝑆,𝑇: 𝑈→ 2𝑈 are said to be 103
weakly isotone increasing if for any 𝑢∈ 𝑈, we have 𝑆𝑢≼2 𝑇𝑣 for all 𝑣∈ 𝑆𝑢 and 𝑇𝑢≼2 𝑆𝑣 for 104
all 𝑣∈ 𝑇𝑢. 105
In particular, the mappings 𝑆,𝑇: 𝑈→ 𝑈 are called weakly isotone increasing if 𝑆𝑢≼𝑇𝑆𝑢 and 106
𝑇𝑢≼ 𝑆𝑇𝑢 hold for each 𝑢∈ 𝑈. 107
108
Defi nition 1.12. [8] An ordered partial metric space is said to have a sequential limit comparison 109
property if for every nonincreasing sequence (or nondecreasing sequence) {𝑢𝑛} in U, we have 𝑢𝑛→ 110
𝑢 implies 𝑢≤ 𝑢𝑛 (or 𝑢𝑛≤𝑢) respectively. 111
112
Defi nition 1.13. [8] A subset 𝐴 of set 𝑈 is said to be approximative if the set 𝑃𝐴(𝑢)={𝑣∈ 𝐴∶ 113
𝑝(𝑢,𝑣)= 𝑝(𝐴,𝑢)} ∀ 𝑢∈ 𝑈 is nonempty. 114
A set -valued mapping 𝑇 is said to have approximate values in 𝑈 if 𝑇𝑢 is an approximation for 115
each 𝑢∈ 𝑈. 116
117
Defi nition 1.14. [17] Denot e by Υ the set of all functions 𝜉: [0,+∞)4→[0,+∞) with the following 118
properties: 119
1) 𝜉 is nondecreasing in third and fourth components. 120
2) 𝜉(𝑠1, 𝑠2,𝑠3,𝑠4) = 0 iff 𝑠1𝑠2𝑠3𝑠4= 0. 121
3) 𝜉 is continuous. 122

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123
The following functions belong to Υ: 124
(1) 𝜉(𝑠1, 𝑠2,𝑠3,𝑠4) = 𝐿 min {𝑠1, 𝑠2,𝑠3,𝑠4} where 𝐿> 0, 125
(2) 𝜉(𝑠1, 𝑠2,𝑠3,𝑠4) = 𝑠1𝑠2𝑠3𝑠4, 126
(3) 𝜉(𝑠1, 𝑠2,𝑠3,𝑠4) = 𝑙𝑛(1+ 𝑠1𝑠2𝑠3𝑠4), 127
(4) 𝜉(𝑠1, 𝑠2,𝑠3,𝑠4) = 𝑒𝑥𝑝(𝑠1𝑠2𝑠3𝑠4) – 1. 128
129
For two mappings 𝑆,𝑇: 𝑈→ 2𝑈, we defi ne 130
𝑀(𝑢,𝑣)=max {𝑝(𝑢,𝑣),𝑝(𝑢,𝑇𝑢),𝑝(𝑣,𝑆𝑣),1
2[𝑝(𝑣,𝑇𝑢)+ 𝑝(𝑢,𝑆𝑣)]}. 131
In 2014, the concept of 𝐶-class functions was introduced by A. H. Ansari [3]. By using this concept, 132
Many fi xed point theorems in the literature can be generalized. 133
134
Defi nition 1.15. [3] A mapping ℱ: [0,∞)2→𝑅 is called a 𝐶-class function if it is continuous and 135
satisfi es the following axioms: 136
(1) ℱ(𝑡1,𝑡2)≤ 𝑡1, 137
(2) ℱ(𝑡1,𝑡2)= 𝑡1 implies that either 𝑡1= 0 or 𝑡2= 0 for all 𝑡1, 𝑡2∈[0,∞). 138
We denote 𝐶-class functions by 𝒞. 139
140
Example 1.16. [3] For all 𝑡1, 𝑡2∈[0,∞), following are some elements of class 𝒞: 141
(1) ℱ(𝑡1,𝑡2) = 𝑡1−𝑡2, ℱ(𝑡1,𝑡2) = 𝑡1⟹ 𝑡2= 0; 142
(2) ℱ(𝑡1,𝑡2) = m 𝑡1, 0 < m < 1, ℱ(𝑡1,𝑡2) = 𝑡1⟹ 𝑡1= 0; 143
(3) ℱ(𝑡1,𝑡2) = 𝑡1
(1+𝑡2)r , r ∈ (0, ∞), ℱ(𝑡1,𝑡2) = 𝑡1⟹ 𝑡1= 0 or 𝑡2= 0; 144
(4) ℱ(𝑡1,𝑡2) = 𝑙𝑜𝑔(𝑡2+𝑎𝑡1)
(1 + 𝑡2) , 𝑎 > 1, ℱ(𝑡1,𝑡2) = 𝑡1⟹ 𝑡1= 0 or 𝑡2= 0; 145
(5) ℱ(𝑡1,𝑡2) = 𝑙𝑛(1 + 𝑎𝑡1)
2,𝑎 > 𝑒, ℱ(𝑡1,𝑡2) = 𝑡1⟹ 𝑡1= 0; 146
(6) ℱ(𝑡1,𝑡2) = (𝑡1+ 𝑙)(1/(1+𝑡2)𝑟)− 𝑙,𝑙> 1,r ∈ (0,∞),ℱ(𝑡1,𝑡2) = 𝑡1⟹ 𝑡2 = 0; 147
(7) ℱ(𝑡1,𝑡2) = 𝑡1 𝑙𝑜𝑔 𝑡2+𝑎 𝑎,𝑎 > 1, ℱ(𝑡1,𝑡2) = 𝑡1⟹ 𝑡1= 0 or 𝑡2= 0; 148
(8) ℱ(𝑡1,𝑡2) = 𝑡1- (1+𝑡1
2+𝑡1 )(𝑡2
1+𝑡2), ℱ(𝑡1,𝑡2) = 𝑡1⟹ 𝑡2= 0; 149
(9) ℱ(𝑡1,𝑡2) = 𝑡1 𝛽(𝑡1) where 𝛽: [0,∞)→ [0, 1) is continuous, ℱ(𝑡1,𝑡2) = 𝑡1⟹ 𝑡1 = 0; 150
(10) ℱ(𝑡1,𝑡2) = 𝑡1- 𝑡2
k+𝑡2 ,ℱ(𝑡1,𝑡2) = 𝑡1⟹ 𝑡2 = 0; 151
(11) ℱ(𝑡1,𝑡2) = 𝑡1- 𝜑(𝑡1), ℱ(𝑡1,𝑡2) = 𝑡1⟹ 𝑡1= 0 where 𝜑: [0,∞)→[0,∞) is a continuous function 152
such that 𝜑(𝑡2) = 0 iff 𝑡2 = 0; 153
(12) ℱ(𝑡1,𝑡2) = 𝑡1 h(𝑡1,𝑡2), ℱ(𝑡1,𝑡2) = 𝑡1⟹ 𝑡1 = 0 where h : [[0,∞)×[0,∞) →[0,∞) is a 154
continuous function such that h(𝑡2,𝑡1) < 1 for all 𝑡2,𝑡1 > 0; 155
(13) ℱ(𝑡1,𝑡2) = 𝑡1- (2+𝑡2
1+𝑡2 )𝑡2, ℱ(𝑡1,𝑡2) = 𝑡1⟹ 𝑡2 = 0; 156
(14) ℱ(𝑡1,𝑡2) = √ln(1+𝑡1𝑛),𝑛 ℱ(𝑡1,𝑡2) = 𝑡1 ⟹ 𝑡1= 0; 157

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(15) ℱ(𝑡1,𝑡2) = 𝜙( 𝑡1), ℱ(𝑡1,𝑡2) = 𝑡1 ⟹ 𝑡1= 0 where 𝜙: [0,∞)→[0,∞) is a upper semicontinuous 158
function such that 𝜙(0) = 0 and 𝜙(𝑡2) < 𝑡2 for 𝑡2 > 0; 159
(16) ℱ(𝑡1,𝑡2) = 𝜗(𝑡1) where 𝜗: 𝑅+×𝑅+→𝑅+ is a generalized Mizoguchi -Takahashi type function, 160
ℱ(𝑡1,𝑡2) = 𝑡1 ⟹ 𝑡1= 0; 161
(17) ℱ(𝑡1,𝑡2) = 𝑡1
Γ(1
2) ∫𝑒−𝑢
√𝑢+𝑡2∞
0 where Γ is the Euler Gamma function. 162
Let Ψ be the family of continuous and monotone nondecreasing functions 𝜓: [0,∞)→[0,∞) such 163
that 𝜓(t) = 0 iff t = 0 and Φ1 the family of continuous functions 𝜑: [0,∞)→[0,∞) such that 𝜑(t) = 164
0 iff t = 0 and Φu be the family of continuous functions 𝜑: [0,∞)→[0,∞) such that 𝜑(0) ≥ 0. 165
166
Note that Φ1⊂Φu. 167
168
2. Main results 169
In this section, ℱ-generalized ( 𝜓,𝜑,𝜉)-contractive type mappings are de fined and some common 170
fixed point theorems have been proved. 171
172
Defi nition 2.1. Let 𝑈 be an ordered partial metric space. Two mappings 𝑆,𝑇: 𝑈→ 2𝑈are said to be 173
ℱ-generalized ( 𝜓,𝜑,𝜉)-contractive type mappings if, 174
𝜓(𝐻𝑝(𝑇𝑢,𝑆𝑣)) ≤ℱ(𝜓(𝑀(𝑢,𝑣)),𝜑(𝜓(𝑀(𝑢,𝑣))) 175
+𝜉(𝑝(𝑢,𝑇𝑢),𝑝(𝑣,𝑆𝑣),𝑝(𝑣,𝑇𝑢)−𝑝(𝑣,𝑣),𝑝(𝑢,𝑆𝑣)− 𝑝(𝑢,𝑢)), 176
for all 𝑢,𝑣∈ 𝑈 with 𝑢 and 𝑣 comparable and 𝜓∈Ψ,𝜑∈Φu,𝜉∈Υ. 177
178
Defi nition 2.2. Limit comparison property : A nonempty set U is said to hold limit comparison 179
property if for a sequence {𝑢𝑛} ∈ U, 𝑢𝑛 is comparable to 𝑢 for all 𝑛 ∈𝑁. 180
181
Theorem 1. Let ( 𝑈,≼) be a complete ordered partial metric space with the limit comparison property. 182
Assume that 𝑆,𝑇: 𝑈→ 2𝑈are weakly isotone increasing ℱ-generalized ( 𝜓,𝜑,𝜉)-contractive type mappings 183
and satisfy approximative property. Suppose that there exists 𝑢0∈𝑈 such that {𝑢0}≼2 T𝑢0. 184
Then T, S have a common fi xed point 𝑢∈𝑈 such that p(u, u) = 0. 185
186
Proof. Firstly, it is proved that if u is a fi xed point of T such that p(u, u) = 0, then it is a common 187
fixed point of 𝑇 and 𝑆. 188
By using given contractive condition and property 2) of 𝜉, 189
𝜓(𝑝(𝑢,𝑆𝑢))≤ 𝜓(𝐻𝑝(𝑇𝑢,𝑆𝑢)) 190
≤ℱ(𝜓(𝑀(𝑢,𝑢),𝜑(𝜓(𝑀(𝑢,𝑢)))) 191
+ 𝜉(𝑝(𝑢,𝑇𝑢),𝑝(𝑢,𝑆𝑢),𝑝(𝑢,𝑇𝑢)− 𝑝(𝑢,𝑢),𝑝(𝑢,𝑆𝑢)− 𝑝(𝑢,𝑢)) 192
= ℱ(𝜓(𝑀(𝑢,𝑢),𝜑(𝜓(𝑀(𝑢,𝑢)))) + 𝜉(0,𝑝(𝑢,𝑆𝑢),0,𝑝(𝑢,𝑆𝑢)−0) 193
= ℱ(𝜓(𝑀(𝑢,𝑢),𝜑(𝜓(𝑀(𝑢,𝑢)))) (2.1) 194
where 195
𝑀(𝑢,𝑢)= 𝑀𝑎𝑥 {𝑝(𝑢,𝑢),𝑝(𝑢,𝑇𝑢),𝑝(𝑢,𝑆𝑢),𝑝(𝑢,𝑆𝑢)+ 𝑝(𝑢,𝑇𝑢)
2} 196

Mathematics 2019 , 7, x FOR PEER REVIEW 6 of 11
≤𝑀𝑎𝑥 {𝑝(𝑢,𝑢),𝑝(𝑢,𝑢),𝑝(𝑢,𝑆𝑢),𝑝(𝑢,𝑆𝑢)+ 𝑝(𝑢,𝑢)
2} 197
= 𝑝(𝑢,𝑆𝑢). 198
thus by (2.1), 199
𝜓(𝑝(𝑢,𝑆𝑢))≤ ℱ(𝜓(𝑝(𝑢,𝑆𝑢),𝜑(𝜓(𝑝(𝑆𝑢,𝑢)))) 200
= ℱ(𝜓(𝑝(𝑢,𝑆𝑢)) 𝜑(𝜓(𝑝(𝑢,𝑆𝑢)))) 201
This implies that, 𝜓(𝑝(𝑢,𝑆𝑢)) = 0 or 𝜑(𝜓(𝑝(𝑢,𝑆𝑢)))= 0, therefore 𝑝(𝑆𝑢,𝑢)=0. Since 𝑆𝑢 satisfy 202
approximative pro perty , therefore there exist 𝑣∈𝑃𝑆𝑢(𝑢) such that 𝑝(𝑣,𝑢)= 𝑝(𝑢,𝑆𝑢)= 0 𝑖.𝑒,𝑣= 203
𝑢. Thus 𝑢∈𝑆𝑢. 204
Let 𝑢0∈ 𝑈, if 𝑢0∈ 𝑇𝑢0, the proof is complete. Otherwise, from the fact that 𝑇𝑢0 has 205
approximative property, it follows that there exists 𝑢1∈ 𝑇𝑢0, with 𝑢1≠𝑢0 such that 206
𝑝(𝑢0,𝑢1)= 𝑖𝑛𝑓 𝑢∈ 𝑇𝑢0𝑝(𝑢,𝑢0)= 𝑝(𝑇𝑢0,𝑢0). 207
Again if 𝑢1∈ 𝑆𝑢1, the proof is complete. Otherwise, since 𝑆𝑢1 has approximative property, it 208
follows that there exist 𝑢2∈ 𝑆𝑢1 with 𝑢2≠𝑢1 such that 209
𝑝(𝑢1,𝑢2)= 𝑖𝑛𝑓 𝑢∈ 𝑆𝑢1𝑝(𝑢,𝑢1)= 𝑝(𝑆𝑢1,𝑢1). 210
By repeating this process, we can fi nd a sequence {𝑢𝑛} in U, such that 𝑢2𝑛+1∈𝑇𝑢2𝑛 and 211
𝑝(𝑢2𝑛+1,𝑢2𝑛)= 𝑝(𝑇𝑢 2𝑛,𝑢2𝑛) and 𝑢2𝑛+2∈ 𝑆𝑢2𝑛+1 with 𝑝(𝑢2𝑛+2,𝑢2𝑛+1)= 𝑝(𝑆𝑢2𝑛+1,𝑢2𝑛+1), 212
213
On the other hand 214
𝑝(𝑇𝑢2𝑛,𝑢2𝑛) ≤𝑠𝑢𝑝 𝑢∈𝑆𝑢2𝑛−1𝑝(𝑇𝑢2𝑛,𝑢) 215
≤𝐻𝑝(𝑇𝑢2𝑛,𝑆𝑢2𝑛−1). 216
Therefore 217
𝑝(𝑢2𝑛+1,𝑢2𝑛) ≤𝐻𝑝(𝑇𝑢2𝑛,𝑆𝑢2𝑛−1). (2.2) 218
and similarly 219
𝑝(𝑢2𝑛+2,𝑢2𝑛+1)≤𝐻𝑝(𝑆𝑢2𝑛+1,𝑇𝑢2𝑛). (2.3) 220
Since 𝑢0 ≼2 𝑇𝑢0 and 𝑢1 ∈ 𝑇𝑢0⇒𝑢0 ≼2 𝑢1. Also, since 𝑇 and 𝑆 are isotone increasing, 221
therefore, 𝑇𝑢0≼2 𝑆𝑣 for all 𝑣∈ 𝑇𝑢0 and thus 𝑇𝑢0 ≼2 𝑆𝑢1. In particular, 𝑢1 ≼2 𝑢2. Continuing 222
this process, we obtain 223
𝑢1 ≼ 𝑢2≼⋯≼𝑢𝑛≼ 𝑢𝑛+1 ≼⋯ 224
Now it is required to show that lim n→∞ 𝑝(𝑢𝑛+1,𝑢𝑛) = 0. 225
Using (2.2) and the fact that 𝑇 and 𝑆 are ℱ-generalized ( 𝜓,𝜑,𝜉)-contractive mappings, we get 226
𝜓(𝑝(𝑢2𝑛+1,𝑢2𝑛))≤𝜓(𝐻𝑝(𝑇𝑢2𝑛,𝑆𝑢2𝑛−1)) 227
≤ℱ(𝜓(𝑀(𝑢2𝑛,𝑢2𝑛−1),𝜑(𝜓(𝑀(𝑢2𝑛,𝑢2𝑛−1)))) 228
+ 𝜉(𝑝(𝑢2𝑛,𝑇𝑢2𝑛),𝑝(𝑢2𝑛−1,𝑆𝑢2𝑛−1),𝑝(𝑢2𝑛,𝑆𝑢2𝑛−1)−𝑝(𝑢2𝑛,𝑢2𝑛) 229
, 𝑝(𝑢2𝑛+1,𝑇𝑢2𝑛)− 𝑝(𝑢2𝑛,𝑢2𝑛)) (2.4) 230
= ℱ(𝜓(𝑀((𝑢2𝑛,𝑢2𝑛−1),𝜑(𝜓(𝑀(𝑢2𝑛,𝑢2𝑛−1)))) 231
+ 𝜉(𝑝(𝑢2𝑛,𝑢2𝑛+1),𝑝(𝑢2𝑛−1,𝑆𝑢2𝑛−1),𝑝(𝑢2𝑛,𝑢2𝑛)−𝑝(𝑢2𝑛,𝑢2𝑛) 232
, 𝑝(𝑢2𝑛+1,𝑢2𝑛+1)− 𝑝(𝑢2𝑛−1,𝑢2𝑛−1)) 233
≤ℱ(𝜓(𝑀((𝑢2𝑛,𝑢2𝑛−1),𝜑(𝜓(𝑀(𝑢2𝑛,𝑢2𝑛−1)))) 234
where 235
𝑀(𝑢2𝑛,𝑢2𝑛−1)= 𝑀𝑎𝑥 {𝑝(𝑢2𝑛,𝑢2𝑛−1),𝑝(𝑢2𝑛,𝑇𝑢 2𝑛),𝑝(𝑢2𝑛−1,𝑆𝑢2𝑛−1),𝑝(𝑢2𝑛−1,𝑇𝑢 2𝑛)+ 𝑝(𝑢2𝑛,𝑆𝑢2𝑛−1)
2} 236
≤ 𝑀𝑎𝑥 {𝑝(𝑢2𝑛,𝑢2𝑛−1),𝑝(𝑢2𝑛,𝑢2𝑛+1),𝑝(𝑢2𝑛−1,𝑢2𝑛),𝑝(𝑢2𝑛−1,𝑢2𝑛+1)+ 𝑝(𝑢2𝑛,𝑢2𝑛)
2} 237

Mathematics 2019 , 7, x FOR PEER REVIEW 7 of 11
≤ 𝑀𝑎𝑥 {𝑝(𝑢2𝑛,𝑢2𝑛−1),𝑝(𝑢2𝑛,𝑢2𝑛+1),𝑝(𝑢2𝑛−1,𝑢2𝑛),𝑝(𝑢2𝑛−1,𝑢2𝑛−1)+ 𝑝(𝑢2𝑛,𝑢2𝑛+1)
2} 238
239
= 𝑀𝑎𝑥 {𝑝(𝑢2𝑛,𝑢2𝑛−1),𝑝(𝑢2𝑛,𝑢2𝑛+1)} 240
If 𝑀𝑎𝑥 {𝑝(𝑢2𝑛,𝑢2𝑛−1),𝑝(𝑢2𝑛,𝑢2𝑛+1)}= 𝑝(𝑢2𝑛,𝑢2𝑛+1), then by (2.4), 241
𝜓(𝑝(𝑢2𝑛,𝑢2𝑛+1))≤ℱ(𝜓(𝑝(𝑢2𝑛,𝑢2𝑛−1),𝜑(𝜓(𝑝(𝑢2𝑛+1,𝑢2𝑛)))) 242
which implies that 𝜓(𝑝(𝑢2𝑛,𝑢2𝑛+1)) = 0 or 𝜑(𝜓(𝑝(𝑢2𝑛+1,𝑢2𝑛))) = 0. Therefore 𝑝(𝑢2𝑛,𝑢2𝑛+1) = 0 243
which is a contradiction. 244
Thus, 𝑝(𝑢2𝑛,𝑢2𝑛−1)≤ 𝑀(𝑢2𝑛,𝑢2𝑛−1)≤𝑝(𝑢2𝑛,𝑢2𝑛−1) and so 𝑀(𝑢2𝑛,𝑢2𝑛−1)=𝑝(𝑢2𝑛,𝑢2𝑛−1). 245
246
Also, by using (2.4), we get 247
𝜓(𝑝(𝑢2𝑛,𝑢2𝑛+1)) )≤ℱ(𝜓(𝑝(𝑢2𝑛,𝑢2𝑛−1)),𝜑(𝜓(𝑝(𝑢2𝑛,𝑢2𝑛−1))))≤𝜓(𝑝(𝑢2𝑛,𝑢2𝑛−1)). (2.5) 248
Proceeding as above, 249
𝜓(𝑝(𝑢2𝑛+1,𝑢2𝑛+2)) )≤ℱ(𝜓(𝑝(𝑢2𝑛,𝑢2𝑛+1)),𝜑(𝜓(𝑝(𝑢2𝑛,𝑢2𝑛+1))))≤𝜓(𝑝(𝑢2𝑛,𝑢2𝑛+1)). (2.5) 250
By (2.5) and (2.6), 251
𝑝(𝑢𝑛+1,𝑢𝑛)≤ 𝑝(𝑢𝑛,𝑢𝑛−1) for each 𝑛∈ 𝑁. 252
Therefore, the sequence {𝑝(𝑢𝑛,𝑢𝑛+1)} is a nonnegative and non -increasing sequence and thus there 253
exists r > 0 such that 254
𝑙𝑖𝑚 𝑛→∞ 𝑝(𝑢𝑛,𝑢𝑛+1)= 𝑟. 255
Now since 𝜑 is lower semicontinuous, 256
𝜑(𝜓(𝑟)) ≤ 𝑙𝑖𝑚 𝑛→∞ inf𝜑(𝜓(𝑝(𝑢𝑛,𝑢𝑛−1))) 257
Therefore, by (2.5), we obtain 258
𝜓(𝑟) ≤ ℱ(𝜓(𝑟),𝜑(𝜓(𝑟))) 259
This implies 𝜓(𝑟) = 0 or 𝜑(𝜓(𝑟)) = 0. Hence 𝑟= 0. 260
Next it remains to show that {𝑢𝑛} is a Cauchy sequence in 𝑈, i.e. to prove that 261
𝑙𝑖𝑚 𝑛,𝑚→∞𝑝(𝑢𝑛,𝑢𝑚)= 0. 262
Assume that the sequence {𝑢2𝑛} is not a Cauchy sequence in (𝑈,𝑝), then by Lemma 1.5 , there exist 263
𝜖>0 and two sequences {𝑢𝑚(𝑘)} and {𝑢𝑛(𝑘)} of {𝑢𝑛} with 𝑛(𝑘)> 𝑚(𝑘)> 𝑘 such that the 264
sequences 265
𝑝(𝑢2𝑚(𝑘),𝑢2𝑛(𝑘)+1), 𝑝(𝑢2𝑚(𝑘),𝑢2𝑛(𝑘)), 𝑝(𝑢2𝑚(𝑘)−1,𝑢2𝑛(𝑘)+1), 𝑝(𝑢2𝑚(𝑘)−1,𝑢2𝑛(𝑘)) 266
tend to 𝜖>0 when k→∞. 267
Using the given contractive condition, 268
𝜓(𝑝(𝑢2𝑚(𝑘),𝑢2𝑛(𝑘)+1))≤𝜓(𝑃ℎ(𝑇𝑢2𝑚(𝑘)−1,𝑆𝑢2𝑛(𝑘))) 269
≤ ℱ(𝜓(𝑀(𝑢2𝑚(𝑘)−1,𝑢2𝑛(𝑘))),𝜑(𝜓(𝑀(𝑢2𝑚(𝑘)−1,𝑆𝑢2𝑛(𝑘))))) (2.7) 270
where 271

Mathematics 2019 , 7, x FOR PEER REVIEW 8 of 11
𝑀(𝑢2𝑚(𝑘)−1,𝑢2𝑛(𝑘))= 𝑀𝑎𝑥 {𝑝(𝑢2𝑚(𝑘)−1,𝑢2𝑛(𝑘)),𝑝(𝑢2𝑚(𝑘)−1,𝑇𝑢2𝑚(𝑘)−1),𝑝(𝑢2𝑛(𝑘),𝑆𝑢2𝑛(𝑘))
,𝑝(𝑢2𝑛(𝑘),𝑇𝑢 2𝑚(𝑘)−1)+ 𝑝(𝑢2𝑚(𝑘)−1,𝑆𝑢2𝑛(𝑘))
2} 272
≤ 𝑀𝑎𝑥 {𝑝(𝑢2𝑚(𝑘)−1,𝑢2𝑛(𝑘)),𝑝(𝑢2𝑚(𝑘)−1,𝑢2𝑚(𝑘)),𝑝(𝑢2𝑛(𝑘),𝑢2𝑛(𝑘)+1)
,𝑝(𝑢2𝑛(𝑘),𝑢2𝑚(𝑘))+ 𝑝(𝑢2𝑚(𝑘)−1,𝑢2𝑛(𝑘)+1)
2} 273
→ 𝑀𝑎𝑥 {𝜖,0,0,𝜖} 274
= 𝜖 as 𝑘→∞. 275
Thus by (2.7) and for any 𝑘→∞, 276
𝜓(𝜖)≤ ℱ(𝜓(𝜖),𝜑(𝜓(𝜖))) 277
This implies that 𝜓(𝜖)= 0 or 𝜑(𝜓(𝜖))= 0 and thus 𝜖 = 0 which is a contradiction. Therefore the 278
sequence {𝑢𝑛} is a cauchy sequence. As (𝑈,𝑝) is complete, the space (𝑈,𝑑𝑝) is complete. Therefore, 279
𝑙𝑖𝑚 𝑛→∞𝑑𝑝(𝑢𝑛,𝑢) = 0 for some 𝑢∈𝑈. Now by Lemma 1.4 , 280
𝑝(𝑢,𝑢)= 𝑙𝑖𝑚 𝑛→∞𝑝(𝑢𝑛,𝑢)=𝑙𝑖𝑚 𝑚,𝑛→∞𝑝(𝑢𝑛,𝑢𝑚) = 0. 281
Since U has limit comparison property, therefore for 𝑛∈𝑁, 𝑢𝑛 is comparable to 𝑢, therefore, 282
𝑝(𝑢2𝑛+2,𝑇𝑢) ≤𝑠𝑢𝑝 𝑢∈𝑆𝑢2𝑛+1𝑝(𝑢,𝑇𝑢)≤ 𝐻𝑝(𝑆𝑢2𝑛+1,𝑇𝑢) 283
Thus, 284
𝜓(𝑝(𝑢2𝑛+2,𝑇𝑢))≤𝜓(𝐻𝑝(𝑆𝑢2𝑛+1,𝑇𝑢)) 285
≤ℱ(𝜓(𝑀(𝑢2𝑛+1,𝑢)) 𝜑(𝜓(𝑀(𝑢2𝑛+1,𝑢)))) 286
+ 𝜉(𝑝(𝑢2𝑛+1,𝑆𝑢2𝑛+1),𝑝(𝑢,𝑇𝑢),𝑝(𝑢2𝑛+1,𝑇𝑢)−𝑝(𝑢2𝑛+1,𝑢2𝑛+1),𝑝(𝑢,𝑆𝑢2𝑛+1)
−𝑝(𝑢,𝑢)) (2.8) 287
≤ℱ(𝜓(𝑀(𝑢2𝑛+1,𝑢)),𝜑(𝜓(𝑀(𝑢2𝑛+1,𝑢)))) 288
+ 𝜉(𝑝(𝑢2𝑛+1,𝑢2𝑛+2),𝑝(𝑢,𝑇𝑢),𝑝(𝑢2𝑛+1,𝑇𝑢)−𝑝(𝑢2𝑛+1,𝑢2𝑛+1),𝑝(𝑢,𝑢2𝑛+2)
−𝑝(𝑢,𝑢)) 289
where 290
𝑝(𝑢,𝑇𝑢)≤𝑀(𝑢2𝑛+1,𝑢)= 𝑀𝑎𝑥 {𝑝(𝑢2𝑛+1,𝑢),𝑝(𝑢2𝑛+1,𝑆𝑢2𝑛+1),𝑝(𝑢,𝑇𝑢),𝑝(𝑢2𝑛+1,𝑇𝑢)+ 𝑝(𝑢,𝑆𝑢2𝑛+1)
2} 291
≤ 𝑀𝑎𝑥 {𝑝(𝑢2𝑛+1,𝑢),𝑝(𝑢2𝑛+1,𝑢2𝑛+2),𝑝(𝑢,𝑇𝑢),𝑝(𝑢2𝑛+1,𝑇𝑢)+ 𝑝(𝑢,𝑢2𝑛+2)
2} 292
Taking limit 𝑛→∞, we get 𝑙𝑖𝑚 𝑛→∞ 𝑀(𝑢2𝑛+1,𝑢)= 𝑑(𝑢,𝑇𝑢). Since 𝜑 is lower semicontinuous, 293
taking limit 𝑛→∞, in (2.8) implies 294
𝜓(𝑝(𝑢,𝑇𝑢))≤ℱ(𝜓(𝑝(𝑢,𝑇𝑢)),𝜑(𝜓(𝑝(𝑢,𝑇𝑢)))) 295
which further implies that 𝜓(𝑝(𝑢,𝑇𝑢))= 0 or 𝜑(𝜓(𝑝(𝑢,𝑇𝑢)))= 0. 296
Thus, 𝑝(𝑢,𝑇𝑢) = 0. Since 𝑇𝑢 has approximative property, there exist 𝑣∈ 𝑃𝑇𝑢 such that 𝑝(𝑣,𝑢)= 297
0 𝑖.𝑒 𝑣= 𝑢, therefore 𝑢∈𝑇𝑢. Thus 𝑢 is a fi xed point of 𝑇. This completes the proof. 298
299
By putting ℱ(𝑡1,𝑡2)= 𝑡1− 𝑡2, the following result holds: 300
Corollary 2. 1. Let 𝑈 be a complete ordered partial metric space satisfying limit comparison property. Let 𝑆, 301
𝑇: 𝑈→ 2𝑈 are two weakly isotone increasing mappings holding approximative property such that 302
𝜓(𝐻𝑝(𝑇𝑢,𝑆𝑣)) ≤𝜓(𝑀(𝑢,𝑣))−𝜑(𝜓(𝑀(𝑢,𝑣))) 303

Mathematics 2019 , 7, x FOR PEER REVIEW 9 of 11
+𝜉(𝑝(𝑢,𝑇𝑢),𝑝(𝑣,𝑆𝑣),𝑝(𝑣,𝑇𝑢)−𝑝(𝑣,𝑣),𝑝(𝑢,𝑆𝑣)− 𝑝(𝑢,𝑢)), 304
Suppose that there exists 𝑢0∈𝑈 such that { 𝑢0}≼2 T𝑢0. Then T, S have a common fixed point 𝑢∈𝑈 such 305
that p(u, u) = 0. 306
On putting ℱ(𝑡1,𝑡2)= 𝑚𝑡1 and 𝜓(t) = t , the following result is obtained: 307
Corollary 2.2 . Let 𝑈 be a complete ordered partial metric space satisfying limit comparison property. Let 𝑆, 308
𝑇: 𝑈→ 2𝑈 are two weakly isotone increasing mappings holding approximative property and there exists 309
𝑘∈ [0,1) such that 310
𝐻𝑝(𝑇𝑢,𝑆𝑣) ≤𝑚 𝑀(𝑢,𝑣)+𝜉(𝑝(𝑢,𝑇𝑢),𝑝(𝑣,𝑆𝑣),𝑝(𝑣,𝑇𝑢)−𝑝(𝑣,𝑣),𝑝(𝑢,𝑆𝑣)− 𝑝(𝑢,𝑢)), 311
for all 𝑢,𝑣∈𝑈 with 𝑢 𝑎𝑛𝑑 𝑣 comparable and 𝜉∈Υ. Suppose that there exists 𝑢0∈𝑈 such that { 𝑢0}≼2 312
T𝑢0. Then T, S have a common fixed point 𝑢∈𝑈 such that p(u, u) = 0. 313
314
By putting 𝑆 = 𝑇 in Theorem 1 , the following Corollary holds: 315
Corollary 2. 3. Let ( 𝑈,≼) be a complete ordered partial metric space with the limit comparison property. 316
Assume that 𝑇: 𝑈→ 2𝑈is a weakly isotone increasing ℱ-generalized ( 𝜓,𝜑,𝜉)-contractive type mappings 317
and satisfy approximative property. Suppose that there exists 𝑢0∈𝑈 such that { 𝑢0}≼2 T𝑢0. 318
Then T has a fixed point 𝑢∈𝑈 such that p(u, u) = 0. 319
320
Example 2.4 . Let 𝑈={0,1,1
2} equipped with partial metric 𝑝 defined by 𝑝(𝑢,𝑣)= max {𝑢,𝑣}, for 321
each 𝑢,𝑣∈𝑈. Defi ne the partial order on 𝑈 by 322
𝑢 ≼𝑣 ⟺𝑝(𝑢,𝑢)= 𝑝(𝑢,𝑣)⟺𝑢=max {𝑢,𝑣}⟺𝑣 ≤𝑢. 323
It is easy to check that (U,≼) is a totally ordered set and (𝑈,𝑝) is a complete partial metric space. 324
Also, the mappings 𝑇 and 𝑆 are defi ned as 325
𝑇𝑢 ={{0} 𝑖𝑓 𝑢∈{0,1
2},
{0,1
2} 𝑖𝑓 𝑢=1.} and 𝑆𝑢 ={{0} 𝑖𝑓 𝑢∈{0,1
2},
{1
2} 𝑖𝑓 𝑢=1.} 326
Note that T and S are weakly isotone increasing as for ,𝑧∈ 𝑆𝑢; 𝑤∈𝑇𝑣 ⟹𝑤= 0. Thus, 𝑤 ≤𝑧⟹ 327
𝑧≼𝑤. Hence, for each 𝑢∈𝑈; Su≼2 Tv for each 𝑣∈ 𝑆𝑢. Similarly, for each 𝑢∈𝑈, it can be easily 328
shown that Tu≼2 Sv for all 𝑣∈ 𝑇𝑢 329
Let 𝜓(𝑡) = 2𝑡 and 𝜑(𝑡)= 𝑡
2,ℱ(𝑡1,𝑡2)= 1
2𝑡1 and 𝜉(𝑠1, 𝑠2,𝑠3,𝑠4) = 𝑠1𝑠2𝑠3𝑠4. 330
Next it is proved that the mappings T and S are ℱ-generalized ( ψ,φ,ξ)-contractive type mappings 331
Following cases arises: 332
Case I : If 𝑢,𝑣∈{0,1
2}. Then, 333
𝜓(𝐻𝑝(𝑇𝑢,𝑆𝑣))= 𝜓(𝐻𝑝({0},{0})) 334
= 𝜓(0) 335
= 0≤ 𝐹(𝜓(𝑀(𝑢,𝑣)),𝜑(𝜓(𝑀(𝑢,𝑣))) 336
+ 𝜉(𝑝(𝑢,𝑇𝑢),𝑝(𝑣,𝑆𝑣),𝑝(𝑣,𝑇𝑢)−𝑝(𝑣,𝑣),𝑝(𝑢,𝑆𝑣)−𝑝(𝑢,𝑢)) 337
Case II: If 𝑢= 𝑣= 1. Then, 338
𝜓(𝐻𝑝(𝑇𝑢,𝑆𝑣))= 𝜓(𝐻𝑝({0,1
2},{1
2})) 339
= 𝜓(1
2)= 1. 340

Mathematics 2019 , 7, x FOR PEER REVIEW 10 of 11
Now , 341
𝑀(𝑢,𝑣)= 𝑀𝑎𝑥 {𝑝(𝑢,𝑣),𝑝(𝑢,𝑇𝑢),𝑝(𝑣,𝑆𝑣),𝑝(𝑢,𝑆𝑣)+ 𝑝(𝑣,𝑇𝑢)
2} 342
≤𝑀𝑎𝑥 {𝑝(1,1),𝑝(1,{0,1
2}),𝑝(1,1
2),𝑝(1,{0,1
2})+ 𝑝(1,1
2)
2} 343
=𝑀𝑎𝑥 {1,1,1,1
2(1+1)} 344
=1. 345
So, 346
𝐹(𝜓(𝑀(𝑢,𝑣)),𝜑(𝜓(𝑀(𝑢,𝑣))) 347
+ 𝜉(𝑝(𝑢,𝑇𝑢),𝑝(𝑣,𝑆𝑣),𝑝(𝑣,𝑇𝑢)−𝑝(𝑣,𝑣),𝑝(𝑢,𝑆𝑣)−𝑝(𝑢,𝑢)) 348
=1
2 𝜓(1)+𝜉(1,1,1−1,1−1) 349
=1. 350
Thus, the contractive condition is proved. Similarly, the remaining cases can be discussed and 351
proved. Hence, all the hypotheses of Theorem 1 are fulfi lled. Therefore, T, S have a common fi xed 352
point 𝑢 = 0. 353
354
References 355
1. Altun, I., Sola, F., Simsek, H.; Generalized contractions on partial metric spaces, Topology and its Applications . 356
157, 2778 -2785( 2010 ). 357
2. Ansari, A.H.; Note on 𝜑−𝜓−contrac tive type mappings and related fi xed points, The 358
2nd Regional Conference on Mathematics and Applications, Payame Noor University . 377-380(2014 ). 359
3. Aydi, H., Abbas, M., Vetro, C.; Partial Hausdorff metric and Nadlers fi xed point theorem on partial metric 360
spaces , Topology and its Applications . 159, 3234 -3242( 2012 ). 361
4. Bari, C. Di , Milojevic, M., Radenovic, S., Vetro, P.; Common fi xed points for self -mappings on partial metric 362
spaces, Fixed Point Theory and Applications . 2012 , 2012:140. 363
5. Chen, C. , Zhu, C.; Fixed point theorems for weakly 𝐶-contractive mappings in partial metric spaces, Fixed 364
Point Theory and Applications . 2013 , 2013:107. 365
6. Dhage, B.C., O'Regan, D., Agarwal, R.P.; Common fi xed point theorems for a pair of countably condensing 366
mappings in ordered Banach spaces, Journal of Applied Mathematics and Stochastic Analysis . 16, 243 -248(2003 ). 367
7. Erduran, A.; Common fi xed point of g-approximative multivalued mapping in ordered partial metric space, 368
Fixed Point Theory and Applications . 2013 , 2013:36. 369
8. Hong, S.; Fixed points of multivalued operators in ordered metric spaces with application, Nonlinear Analysis . 370
72, 3929 -3942( 2010 ). 371
9. Jungck, G.; Compatible mappings and common fi xed point , International Journal of Mathematics and 372
Mathematical Sciences . 9, 771 -779(1986 ). 373
10. Kadelburh, Z., Nashine, H.K., Radenovvic, S.; Fixed point results under various contractive conditions in 374
partial metric spaces, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas . 375
September 2013, 107(2), 241 – 256(2013 ). 376
11. Ili𝑐́, D., Pavlovi 𝑐́, V. and Rakocevi 𝑐́, V.; Some new extensions of Banach's contraction principle to partial 377
metric space, Applied Mathematics Letters . 24, 1326 -1330( 2011 ). 378
12. Matthews, S.G.; Partial metric topology, Proceedings of the 8th Summer Conference 379

Mathematics 2019 , 7, x FOR PEER REVIEW 11 of 11
on General Topology and Applications, Annals of the New York Academy of Sciences . 728, 183 -197(1994 ). 380
13. Nashine, H.K., Kadelburg , Z., Radenovi 𝑐́, S., Kim, J.K..; Fixed point theorems un der Hardy -Rogers weak 381
contractive conditions on 0 -complete ordered partial metric spaces, Fixed Point Theory and Applications . 2012 , 382
2012:180. 383
14. Nashine, H.K., Kadelburg , Z., and Radenovi 𝑐́, S.; Common fi xed point theorems for weakly isotone 384
increasing mappings in ordered partial metric spaces, Mathematical and Computer Modelling . 57, 385
2355 -2365( 2013 ). 386
15. Nazari, E.,. Mohitazar, N.; Common fi xed points for multivalued mappings in ordered partial metric spaces, 387
International Journal of Applied Mathematical Research . 4(2), 259-26(2015 ) . 388
16. Oltra, S., Valero, O.; Banach's fi xed point theorem for partial metric spaces, Rend Istit Math Univ Trieste . 36, 389
17-26(2004 ). 390
17. Ran, A.C.M., . Reurings, M.C.B .; A fi xed point theorem in partially ordered sets and some application to 391
matrix equations, Proceedings of the American Mathematical Society . 132, 1435 -1443( 2004 ). 392