# Existence of Fixed Points of Four Maps for a New GeneralizedF-Contraction and an Application

Authors: Muhammad Nazam; Hassen Aydi; Mohd Salmi Noorani; Haitham Qawaqneh
Journal: Journal of Function Spaces
 (2019)

---

## Abstract

We initiate the concept of a new generalizedF-contraction satisfying some contractive conditions involving four maps on a partial metric space. We set up an example to elucidate our main result. An application is derived where a system of elliptic boundary value equations has a common solution.


---

## Body

## 1. Introduction

The Banach Contraction Principle (BCP) has many useful applications in various directions like differential, integral, functional, and matrix equations (both linear and nonlinear). This contraction principle is being generalized extensively in different distance spaces. One of its best generalizations is theF-contraction presented by Wardowski [1]. Aydi el al. [2] (see also [3]) modified the concept of F-contractions via α-admissibility. Many authors have formed different generalizations of Wardowski [1] result (see [4–6]). Recently, in [7] authors provided a fixed point result for a C
´iric
´ type F-contraction. Berinde [8] and Berinde and Vetro [9] have established some common fixed point results for mappings satisfying compatible conditions (resp., implicit contractions) in a complete metric space.Here, we discuss a new generalizedF-contraction based on four self-mappings related to elliptic boundary value problem; in particular, we collect a common fixed point result for four self-mappings to show the existence of a common solution for operators satisfying an elliptic boundary value problem. It is remarked that the notion of the F-contraction in partial metric spaces is more general with respect to the metric space.Definition 1 (see [10]).
LetI be a nonempty set. If the function p
:
I
×
I
→
[
0
,
∞
) satisfies the following properties:(p
1)
η
=
ϱ
⇔
p
η
,
η
=
p
η
,
ϱ
=
p
ϱ
,
ϱ
(p
2)
p
η
,
η
≤
p
η
,
ϱ
(p
3)
p
η
,
ϱ
=
p
ϱ
,
η
(p
4)
p
η
,
ς
≤
p
η
,
ϱ
+
p
ϱ
,
ς
-
p
ϱ
,
ϱ,
for allη
,
ϱ
,
ς
∈
I, then p is called a partial metric on IExample 2.
LetI
=
[
0
,
∞
). Define the classical partial metric p
:
I
×
I
→
I as p
(
η
,
ϱ
)
=
max
⁡
{
η
,
ϱ
}. Note that p is not a metric on I. Indeed p
(
η
,
η
)
=
η
>
0 for each η
>
0.Matthews [10] explored the following aspects of a partial metric p on I.(1)
The functiond
:
I
×
I
→
[
0
,
∞
) defined by(1)
d
η
,
ϱ
=
2
p
η
,
ϱ
-
p
η
,
η
-
p
ϱ
,
ϱ
for allη
,
ϱ
∈
I, defines a metric on I (called the induced metric by p)
(2)
Open balls:B
p
ρ
,
ϵ
=
x
∈
I
:
p
ρ
,
x
<
p
ρ
,
ρ
+
ϵ; ϵ
>
0
(3)
Each partial metricp on I generates a T
0 topology T
[
p
] on I. The base of topology T
p consists of the set B
p
ρ
,
ϵ
;
ρ
∈
I
,
ϵ
>
0
(4)
A sequence{
σ
n
}
n
∈
N in (
I
,
p
) converges to σ
∈
I iff (2)
p
σ
,
σ
=
l
i
m
n
→
∞
p
σ
,
σ
n
(5)
A sequence{
σ
n
}
n
∈
N in (
I
,
p
) is said Cauchy if l
i
m
n
,
m
→
∞
p
(
σ
n
,
σ
m
) exists and is finite
(6)
(
I
,
p
) is said to be complete if every Cauchy sequence {
σ
n
}
n
∈
N in I converges, with respect to T
[
p
], to σ
∈
I so that(3)
p
σ
,
σ
=
l
i
m
n
,
m
→
∞
p
σ
n
,
σ
mAfter the result of Matthews [10], numerous authors obtained various applicable fixed point results in partial metric spaces. Kumamet al. [11] showed the existence of fixed points for weak α
-
ψ-contraction mappings in partial metric spaces. Radenovic
´
et al. [12] investigated the existence of fixed points via simulation functions. Similarly various authors have established (common) fixed point theorems under the notion of a partial metric; see, for example, [13–19].Lemma 3 (see [10]).
Given a sequence{
σ
n
}
n
∈
N in a partial metric space (
I
,
p
), the following are provided.(1)
{
σ
n
}
n
∈
N is Cauchy in (
I
,
p
) iff it is Cauchy in (
I
,
d
)
(2)
(
I
,
p
) is complete iff the metric space (
I
,
d
) is complete
(3)
{
σ
n
}
n
∈
N converges to σ in (
I
,
p
) iff(4)
l
i
m
n
→
∞
p
σ
,
σ
n
=
p
σ
,
σ
=
l
i
m
n
,
m
→
∞
p
σ
n
,
σ
mDefinition 4 (see [20]).
Givenf
,
g
:
I
→
I, the following is provided. If f
(
σ
)
=
g
(
σ
) for some σ
∈
I, then σ is said to be a coincidence point of f and g.
Such mapsf and g are called weakly compatible if they commute at their coincidence points; i.e., if f
(
σ
)
=
g
(
σ
), then f
g
(
σ
)
=
g
f
(
σ
).Example 5.
LetI
=
R. Consider g
,
f
:
I
→
I as(5)
g
σ
=
6
σ
-
5
a
n
d
f
σ
=
5
σ
-
4
.Here, f
,
g are weakly compatible and the coincidence point is σ
=
1.Definition 6.
The pair of mappings(
f
,
g
) is compatible in a metric space (
I
,
d
) iff(6)
l
i
m
n
→
∞
d
f
g
σ
n
,
g
f
σ
n
=
0
,
w
h
e
n
e
v
e
r
σ
n
i
s
a
s
e
q
u
e
n
c
e
i
n
I
s
o
t
h
a
t
l
i
m
n
→
∞
f
σ
n
=
l
i
m
n
→
∞
g
σ
n
=
ς
for
s
o
m
e
ς
∈
I
.Wardowski [1] considered functions F
:
(
0
,
∞
)
→
R verifying the following:(F
1)
F is strictly increasing
(F
2)
For each positive sequence{
σ
n
}, we have (7)
l
i
m
n
→
∞
σ
n
=
0
i
f
f
l
i
m
n
→
∞
F
σ
n
=
-
∞
(F
3)
There existsθ
∈
(
0,1
) such that l
i
m
α
→
0
+
(
α
)
θ
F
(
α
)
=
0
The family of the above functions satisfying (F
1) – (F
3) is denoted by F.Example 7.
The following functions are inF:(a)
F
(
ς
)
=
ln
⁡
(
ς
)
(b)
F
(
ς
)
=
ς
+
ln
⁡
(
ς
)
(c)
F
(
ς
)
=
ln
⁡
(
ς
2
+
ς
)
(d)
F
(
ς
)
=
-
1
/
ςDefinition 8 (see [1]).
A self-mapq on the metric space (
I
,
d
) is an F
d-contraction if there are F
∈
F and τ
>
0 so that(8)
d
q
σ
,
q
ς
>
0
⇒
τ
+
F
d
q
σ
,
q
ς
≤
F
d
σ
,
ς
,
f
o
r
a
l
l
σ
,
ς
∈
I
.Definition 9.
Let(
I
,
p
) be a partial metric space. The mapping q
:
I
→
I is an F
p-contraction if there exist F
∈
F and τ
>
0 such that(9)
p
q
σ
,
q
ς
>
0
⇒
τ
+
F
p
q
σ
,
q
ς
≤
F
p
σ
,
ς
,
f
o
r
a
l
l
σ
,
ς
∈
I
.Following example indicates that an F
p-contraction is more general than an F
d-contraction.Example 10.
Considerp
(
σ
,
ς
)
=
max
⁡
σ
,
ς for all σ
,
ς
∈
I
=
[
0,1
]. Here, d
(
σ
,
ς
)
=
σ
-
ς for all σ
,
ς
∈
I. Define F
:
(
0
,
∞
)
→
R by F
(
ς
)
=
ln
⁡
(
ς
) and q by(10)
q
ς
=
ς
5
i
f
ς
∈
0,1
;
0
i
f
ς
=
1Note that, for all σ
,
ς
∈
I with σ
≤
ς or ς
≤
σ, we have(11)
τ
+
F
p
q
σ
,
q
ς
≤
F
p
σ
,
ς
i
m
p
l
i
e
s
τ
+
F
σ
5
≤
F
σ
o
r
τ
+
F
ς
5
≤
F
ς
.So q is an F
p-contraction, but q is neither continuous, nor an F
d-contraction. Indeed, for σ
=
1 and ς
=
5
/
6, d
(
q
(
σ
)
,
q
(
ς
)
)
>
0 and we have(12)
τ
+
F
d
q
σ
,
q
ς
≤
F
d
σ
,
ς
,
τ
+
F
d
q
1
,
q
5
6
≤
F
d
1
,
5
6
,
τ
+
F
d
0
,
1
6
≤
F
1
6
,
1
6
<
1
6
.It is a contradiction for each τ
>
0.Lemma 11.
Let(
I
,
d
) be a metric space. If there are two sequences {
σ
n
} and {
s
n
} so that(13)
l
i
m
n
→
∞
d
σ
n
,
s
n
=
0
a
n
d
l
i
m
n
→
∞
σ
n
=
ξ
f
o
r
s
o
m
e
ξ
∈
I
,then l
i
m
n
→
∞
s
n
=
ξ.Proof.
By triangle inequality, we have(14)
d
s
n
,
ξ
≤
d
s
n
,
σ
n
+
d
σ
n
,
ξ
.Taking n
→
∞, the result follows.


## 2. Fixed Point Theorems

First, we introduce a new generalizedF-contraction involving four self-mappings f
,
g
,
h
,
q. By imposing the conditions f
(
I
)
⊆
q
(
I
) and g
(
I
)
⊆
h
(
I
), we generate a Cauchy sequence {
ξ
n
} whose terms satisfy new generalized F-contractions. We then (under some assumptions) prove that this sequence converges to a point ξ such that ξ
=
f
(
ξ
)
=
g
(
ξ
)
=
h
(
ξ
)
=
q
(
ξ
).Note that, in view of Example10, the results in this article are independent of remarks given in [21]. Consider(15)
M
η
,
ϱ
=
max
⁡
p
h
η
,
q
ϱ
,
p
f
η
,
h
η
,
p
g
ϱ
,
q
ϱ
,
p
h
η
,
g
ϱ
+
p
f
η
,
q
ϱ
2
.Definition 12.
Letf
,
g
,
h
,
q be self-mappings defined on a partial metric space (
I
,
p
). These maps form a new generalized F-contraction if there are F
∈
F and τ
>
0 such that, for all η
,
ϱ
∈
I,(16)
p
f
η
,
g
ϱ
>
0
i
m
p
l
i
e
s
τ
+
F
p
f
η
,
g
ϱ
≤
F
M
η
,
ϱ
.The following theorem is our main result.Theorem 13.
Letf
,
g
,
h
,
q be four self-mappings defined on a complete partial metric space (
I
,
p
). Assume that the mappings f
,
g
,
h
,
q form a new generalized F-contraction with f
(
I
)
⊆
q
(
I
) and g
(
I
)
⊆
h
(
I
). Assume that either (
i
) or (
i
i
) holds:(i)
(
f
,
h
) is a compatible pair, f or h is continuous on (
I
,
d
), and (
g
,
q
) is a weakly compatible pair
(ii)
(
g
,
q
) is a compatible pair, g or q is continuous on (
I
,
d
), and (
f
,
h
) is a weakly compatible pair
Thenf
,
g
,
h
,
q have a unique common fixed point.Proof.
(i) Letσ
0
∈
I. As f
(
I
)
⊆
q
(
I
), there exists σ
1
∈
I such that f
(
σ
0
)
=
q
(
σ
1
). Since g
(
σ
1
)
∈
h
(
I
), we can choose σ
2
∈
I such that g
(
σ
1
)
=
h
(
σ
2
). In general σ
2
n
+
1 and σ
2
n
+
2 are chosen in I such that f
(
σ
2
n
)
=
q
(
σ
2
n
+
1
) and g
(
σ
2
n
+
1
)
=
h
(
σ
2
n
+
2
). Define a sequence {
ξ
n
} in I as ξ
2
n
=
f
(
σ
2
n
)
=
q
(
σ
2
n
+
1
) and ξ
2
n
+
1
=
g
(
σ
2
n
+
1
)
=
h
(
σ
2
n
+
2
) for each n
≥
0. We prove that {
ξ
n
} is Cauchy. Assume that p
(
f
(
σ
2
n
)
,
g
(
σ
2
n
+
1
)
)
>
0; then by (16), we get(17)
F
p
ξ
2
n
,
ξ
2
n
+
1
=
F
p
f
σ
2
n
,
g
σ
2
n
+
1
≤
F
M
σ
2
n
,
σ
2
n
+
1
-
τ
,for each n
≥
1, where(18)
M
σ
2
n
,
σ
2
n
+
1
=
max
⁡
p
h
σ
2
n
,
q
σ
2
n
+
1
,
p
f
σ
2
n
,
h
σ
2
n
,
p
g
σ
2
n
+
1
,
q
σ
2
n
+
1
,
p
h
σ
2
n
,
g
σ
2
n
+
1
+
p
f
σ
2
n
,
q
σ
2
n
+
1
2
=
max
⁡
p
ξ
2
n
-
1
,
ξ
2
n
,
p
ξ
2
n
,
ξ
2
n
-
1
,
p
ξ
2
n
+
1
,
ξ
2
n
,
p
ξ
2
n
-
1
,
ξ
2
n
+
1
+
p
ξ
2
n
,
ξ
2
n
2
=
max
⁡
p
ξ
2
n
-
1
,
ξ
2
n
,
p
ξ
2
n
,
ξ
2
n
+
1
.For if M
(
σ
2
n
,
σ
2
n
+
1
)
=
p
(
ξ
2
n
,
ξ
2
n
+
1
), then(19)
F
p
ξ
2
n
,
ξ
2
n
+
1
≤
F
p
ξ
2
n
,
ξ
2
n
+
1
-
τ
,which is a contradiction to (
F
1
). Therefore,(20)
F
p
ξ
2
n
,
ξ
2
n
+
1
≤
F
p
ξ
2
n
-
1
,
ξ
2
n
-
τ
,for each n
≥
1. Similarly,(21)
F
p
ξ
2
n
-
1
,
ξ
2
n
≤
F
p
ξ
2
n
-
2
,
ξ
2
n
-
1
-
τ
,for each n
≥
1
. Hence, by (20) and (21), we have(22)
F
p
ξ
n
,
ξ
n
+
1
≤
F
p
ξ
n
-
1
,
ξ
n
-
τ
,for all n
∈
N
∗ and(23)
F
p
ξ
n
-
1
,
ξ
n
≤
F
p
ξ
n
-
2
,
ξ
n
-
1
-
τ
.Again, we get(24)
F
p
ξ
n
,
ξ
n
+
1
≤
F
p
ξ
0
,
ξ
1
-
n
τ
,which implies l
i
m
n
→
∞
F
p
(
ξ
n
,
ξ
n
+
1
)
=
-
∞. By (
F
2
), we have(25)
l
i
m
n
→
∞
p
ξ
n
,
ξ
n
+
1
=
0
.By the property (
F
3
) of an F-contraction, there exists κ
∈
0,1 such that(26)
l
i
m
n
→
∞
p
ξ
n
,
ξ
n
+
1
κ
F
p
ξ
n
,
ξ
n
+
1
=
0
.The inequality (24) implies(27)
p
ξ
n
,
ξ
n
+
1
κ
F
p
ξ
n
,
ξ
n
+
1
-
F
p
ξ
0
,
ξ
1
≤
-
p
ξ
n
,
ξ
n
+
1
κ
n
τ
≤
0
.Using (25) and (26) and taking n
→
∞ in (27), we get(28)
l
i
m
n
→
∞
n
p
ξ
n
,
ξ
n
+
1
κ
=
0
.By (28), there is n
1
∈
N, such that n
p
(
ξ
n
,
ξ
n
+
1
)
κ
≤
1 for each n
≥
n
1; that is,(29)
p
ξ
n
,
ξ
n
+
1
≤
1
n
1
/
κ
f
o
r
e
a
c
h
n
≥
n
1
.Using (29), we get, for m
>
n
≥
n
1,(30)
p
ξ
n
,
ξ
m
≤
p
ξ
n
,
ξ
n
+
1
+
p
ξ
n
+
1
,
ξ
n
+
2
+
p
ξ
n
+
2
,
ξ
n
+
3
+
⋯
+
p
ξ
m
-
1
,
ξ
m
-
∑
ρ
=
n
+
1
m
-
1
p
ξ
ρ
,
ξ
ρ
≤
p
ξ
n
,
ξ
n
+
1
+
p
ξ
n
+
1
,
ξ
n
+
2
+
p
ξ
n
+
2
,
ξ
n
+
3
+
⋯
+
p
ξ
m
-
1
,
ξ
m
=
∑
i
=
n
m
-
1
p
ξ
i
,
ξ
i
+
1
≤
∑
i
=
n
∞
p
ξ
i
,
ξ
i
+
1
≤
∑
i
=
n
∞
1
i
1
/
k
.Since the series ∑
i
=
n
∞
(
1
/
i
1
/
κ
) converges, we get l
i
m
n
,
m
→
∞
p
(
ξ
n
,
ξ
m
)
=
0; i.e., {
ξ
n
} is Cauchy in I
,
p. By Lemma 3(1), it is also Cauchy in I
,
d, which is a complete metric space. Hence, there is ξ
∈
I so that l
i
m
n
→
∞
d
(
ξ
n
,
ξ
)
=
0. Moreover, Lemma 3(3) implies(31)
l
i
m
n
→
∞
p
ξ
,
ξ
n
=
p
ξ
,
ξ
=
l
i
m
n
,
m
→
∞
p
ξ
n
,
ξ
m
.Since l
i
m
n
,
m
→
∞
p
(
ξ
n
,
ξ
m
)
=
0, we deduce from (31) that(32)
p
ξ
,
ξ
=
0
=
l
i
m
n
→
∞
p
ξ
,
ξ
n
.We also have the following:(33)
l
i
m
n
→
∞
f
σ
2
n
=
l
i
m
n
→
∞
q
σ
2
n
+
1
=
l
i
m
n
→
∞
g
σ
2
n
+
1
=
l
i
m
n
→
∞
h
σ
2
n
+
2
=
ξ
.The map h is continuous, so(34)
l
i
m
n
→
∞
h
f
σ
2
n
=
h
ξ
=
l
i
m
n
→
∞
h
2
σ
2
n
+
2
.The pair (
f
,
h
) is compatible on (
I
,
d
); then(35)
l
i
m
n
→
∞
d
f
h
σ
2
n
,
h
f
σ
2
n
=
0
a
n
d
s
o
b
y
L
e
m
m
a
11
,
l
i
m
n
→
∞
f
h
σ
2
n
=
h
ξ
.Now put x
=
h
(
σ
2
n
) and y
=
σ
2
n
+
1 in (16) and suppose on contrary that p
(
h
(
ξ
)
,
ξ
)
>
0 to obtain(36)
F
p
f
h
σ
2
n
,
g
σ
2
n
+
1
≤
F
M
h
σ
2
n
,
σ
2
n
+
1
-
τ
,where(37)
M
h
σ
2
n
,
σ
2
n
+
1
=
max
⁡
p
h
2
σ
2
n
,
q
σ
2
n
+
1
,
p
f
h
σ
2
n
,
h
2
σ
2
n
,
p
g
σ
2
n
+
1
,
q
σ
2
n
+
1
,
p
h
2
σ
2
n
,
g
σ
2
n
+
1
+
p
f
h
σ
2
n
,
q
σ
2
n
+
1
2
.We claim that l
i
m
n
→
∞
p
(
f
h
(
σ
2
n
)
,
g
(
σ
2
n
+
1
)
=
p
(
h
(
ξ
)
,
ξ
). Since(38)
d
f
h
σ
2
n
,
g
σ
2
n
+
1
=
2
p
f
h
σ
2
n
,
g
σ
2
n
+
1
-
p
f
h
σ
2
n
,
f
h
σ
2
n
-
p
g
σ
2
n
+
1
,
g
σ
2
n
+
1
,we have(39)
d
f
h
σ
2
n
,
g
σ
2
n
+
1
+
p
f
h
σ
2
n
,
f
h
σ
2
n
+
p
g
σ
2
n
+
1
,
g
σ
2
n
+
1
=
2
p
f
h
σ
2
n
,
g
σ
2
n
+
1
.Letting n
→
∞, we obtain from (39) that(40)
l
i
m
n
→
∞
2
p
f
h
σ
2
n
,
g
σ
2
n
+
1
=
d
h
ξ
,
ξ
+
p
h
ξ
,
h
ξ
+
p
ξ
,
ξ
=
2
p
h
ξ
,
ξ
-
p
h
ξ
,
h
ξ
-
p
ξ
,
ξ
+
p
h
ξ
,
h
ξ
+
p
ξ
,
ξ
;
t
h
u
s
,
l
i
m
n
→
∞
p
f
h
σ
2
n
,
g
σ
2
n
+
1
=
p
h
ξ
,
ξ
.The upper limit in (36) yields that(41)
F
p
h
ξ
,
ξ
≤
F
p
h
ξ
,
ξ
-
τ
<
F
p
h
ξ
,
ξ
,a contradiction. Hence, p
(
h
(
ξ
)
,
ξ
)
=
0, so h
(
ξ
)
=
ξ. Now, assuming, on the contrary, p
(
f
(
ξ
)
,
ξ
)
>
0, we have by (16)(42)
F
p
f
ξ
,
g
σ
2
n
+
1
≤
F
M
ξ
,
σ
2
n
+
1
-
τ
,where(43)
M
ξ
,
σ
2
n
+
1
=
max
⁡
p
h
ξ
,
q
σ
2
n
+
1
,
p
f
ξ
,
h
ξ
,
p
g
σ
2
n
+
1
,
q
σ
2
n
+
1
,
p
h
ξ
,
g
σ
2
n
+
1
+
p
f
ξ
,
q
σ
2
n
+
1
2
.Taking the upper limit in (42), we have(44)
F
p
f
ξ
,
ξ
≤
F
p
f
ξ
,
ξ
-
τ
<
F
p
f
ξ
,
ξ
,a contradiction. Hence, p
(
f
(
ξ
)
,
ξ
)
=
0; i.e., f
(
ξ
)
=
ξ. Since f
(
I
)
⊆
q
(
I
), there is l
∈
I such that f
(
ξ
)
=
q
(
l
). Assume on contrary that p
(
g
(
l
)
,
q
(
l
)
)
>
0 and so, by (16), we obtain(45)
F
p
q
l
,
g
l
=
F
p
f
ξ
,
g
l
≤
F
M
ξ
,
l
-
τ
,where(46)
M
ξ
,
l
=
max
⁡
p
h
ξ
,
q
l
,
p
f
ξ
,
h
ξ
,
p
g
l
,
q
l
,
p
h
ξ
,
g
l
+
p
f
ξ
,
q
l
2
=
p
g
l
,
q
l
.Thus, by (45), we have(47)
F
p
q
l
,
g
l
≤
F
p
g
l
,
q
l
-
τ
,a contradiction. Hence, p
(
g
(
l
)
,
q
(
l
)
)
=
0. By (
p
1
) and (
p
2
) we have g
(
l
)
=
q
(
l
). As g and q are weakly compatible so g
(
ξ
)
=
g
f
(
ξ
)
=
g
q
(
l
)
=
q
g
(
l
)
=
q
f
(
ξ
)
=
q
(
ξ
). Thus ξ is a coincidence point of g and q. If p
(
ξ
,
g
(
ξ
)
)
>
0, from (16) we have(48)
F
p
f
σ
2
n
,
g
ξ
≤
F
M
σ
2
n
,
ξ
-
τ
,where(49)
M
σ
2
n
,
ξ
=
max
⁡
p
h
σ
2
n
,
q
ξ
,
p
f
σ
2
n
,
h
σ
2
n
,
p
g
ξ
,
q
ξ
,
p
h
σ
2
n
,
g
ξ
+
p
f
σ
2
n
,
q
ξ
2
.Taking the upper limit in (48), we have(50)
F
p
ξ
,
g
ξ
≤
F
p
ξ
,
g
ξ
-
τ
<
F
p
ξ
,
g
ξ
,a contradiction, so p
(
g
(
ξ
)
,
ξ
)
=
0. Hence g
(
ξ
)
=
ξ, so ξ is a common fixed point of the four mappings f
,
g
,
h
,
q. Similarly, ξ is a common fixed point of f
,
g
,
h
,
q in the case that (ii) holds. If ω
≠
ξ is another common fixed point of f
,
g
,
h
,
q, then, by (16), we have(51)
F
p
ξ
,
ω
=
F
p
f
ξ
,
g
ω
≤
F
M
ξ
,
ω
-
τ
,where(52)
M
ξ
,
ω
=
max
⁡
p
h
ξ
,
q
ω
,
p
f
ξ
,
h
ξ
,
p
g
ω
,
q
ω
,
p
h
ξ
,
g
ω
+
p
f
ξ
,
q
ω
2
.Thus, from (51),(53)
F
p
ξ
,
ω
<
F
p
ξ
,
ω
,which is a contradiction. Thus ξ
=
ω.Theorem13 is supported by the following.Example 14.
Take the partial metricp
(
η
,
ϱ
)
=
max
⁡
η
,
ϱ on I
=
[
0,1
]. Consider f
,
g
,
h
,
q
:
I
→
I as h
(
η
)
=
(
η
/
3
)
8,(54)
f
η
=
η
3
16
i
f
η
∈
0,1
;
0
i
f
η
=
1
,
g
η
=
η
3
8
i
f
η
∈
0,1
;
0
i
f
η
=
1
a
n
d
q
η
=
η
3
4
i
f
η
∈
0,1
;
0
i
f
η
=
1Note that f
(
I
)
⊆
q
(
I
), g
(
I
)
⊆
h
(
I
), h is continuous on (
I
,
d
), (
g
,
q
) is a weakly compatible pair, and (
f
,
h
) is a pair of compatible mappings. Choose F
:
(
0
.
∞
)
→
R as F
(
u
)
=
l
n
(
u
), for each u
>
0. Let η
,
ϱ
∈
I be such that p
(
f
(
η
)
,
g
(
ϱ
)
)
>
0 and η
≤
ϱ. Take τ
≤
l
n
(
81
)
. Then(55)
τ
+
F
p
f
η
,
g
ϱ
=
τ
+
l
n
max
⁡
η
3
16
,
ϱ
3
8
≤
l
n
81
+
l
n
1
81
×
ϱ
3
4
=
ln
⁡
max
⁡
η
3
8
,
ϱ
3
4
=
ln
⁡
p
h
η
,
q
ϱ
≤
F
M
η
,
ϱ
.Thus, the contractive condition (16) holds for all η
,
ϱ
∈
I. Hence, all the hypotheses of Theorem 13 are satisfied (for case (
i
)) and f
,
g
,
h
,
q have a unique common fixed point η
=
0.Following results can be derived as consequences of Theorem13.Corollary 15.
Letf
,
g
,
h
,
q be self-mappings defined on a complete partial metric space (
I
,
p
) such that f
(
I
)
⊆
q
(
I
) and g
(
I
)
⊆
h
(
I
). If there are F
∈
F and τ
>
0 such that, for all η
,
ϱ
∈
I,(56)
p
f
η
,
g
ϱ
>
0
i
m
p
l
i
e
s
τ
+
F
p
f
η
,
g
ϱ
≤
F
R
η
,
ϱwhere, for a
i
>
0
i
=
1,2
,
3,4,(57)
R
η
,
ϱ
=
a
1
p
h
η
,
q
ϱ
+
a
2
p
f
η
,
h
η
+
a
3
p
g
ϱ
,
q
ϱ
+
a
4
p
h
η
,
g
ϱ
+
p
f
η
,
q
ϱ
,where a
1
+
a
2
+
a
3
+
2
a
4
<
1. If either condition (i) or (ii) in the statement of Theorem 13 holds, then f
,
g
,
h
,
q have a unique common fixed point.Proof.
Recall that(58)
a
1
p
h
η
,
q
ϱ
+
a
2
p
f
η
,
h
η
+
a
3
p
g
ϱ
,
q
ϱ
+
a
4
p
h
η
,
g
ϱ
+
p
f
η
,
q
ϱ
≤
a
1
+
a
2
+
a
3
+
2
a
4
M
η
,
ϱ
<
M
η
,
ϱ
.Applying (
F
1
), we see that the inequality (56) reduces to the inequality (16).Corollary 16.
Letf
,
g
,
h
,
q be self-mappings defined on a complete partial metric space (
I
,
p
) such that f
(
I
)
⊆
q
(
I
), g
(
I
)
⊆
h
(
I
). Assume there exist F
∈
F and τ
>
0 such that, for all η
,
ϱ
∈
I and a
i
>
0
(
i
=
1,2
,
3
),(59)
p
f
η
,
g
ϱ
>
0
i
m
p
l
i
e
s
τ
+
F
p
f
η
,
g
ϱ
≤
F
a
1
p
h
η
,
q
ϱ
+
a
2
p
f
η
,
h
η
+
a
3
p
g
ϱ
,
q
ϱwhere a
1
+
a
2
+
a
3
<
1. If either condition (i) or (ii) in the statement of Theorem 13 holds, then f
,
g
,
h
,
q have a unique common fixed point.Proof.
Using(60)
a
1
p
h
η
,
q
ϱ
+
a
2
p
f
η
,
h
η
+
a
3
p
g
ϱ
,
q
ϱ
≤
a
1
+
a
2
+
a
3
M
η
,
ϱ
<
M
η
,
ϱand (
F
1
), the inequality (59) reduces to the inequality (16). Thus, this proof follows the pattern of the proof of Theorem 13.Corollary 17.
Letf be a self-mapping on a complete partial metric space (
I
,
p
). Suppose there are F
∈
F and τ
>
0 such that, for all η
,
ϱ
∈
I,(61)
p
f
η
,
f
ϱ
>
0
i
m
p
l
i
e
s
τ
+
F
p
f
η
,
f
ϱ
≤
F
C
p
η
,
ϱwhere(62)
C
p
η
,
ϱ
=
max
⁡
p
η
,
ϱ
,
p
f
η
,
η
,
p
ϱ
,
f
ϱ
,
p
η
,
f
ϱ
+
p
f
η
,
ϱ
2
.If either f or F is continuous, then f has a unique fixed point.Proof.
Set, in Theorem13, g
=
f and h
=
q
=
I (identity mapping).Corollary 18 (see [7, Theorem 2.2]).
Letf be a self-mapping on a complete metric space (
I
,
d
). Suppose there exist F
∈
F and τ
>
0 such that, for all η
,
ϱ
∈
I,(63)
d
f
η
,
f
ϱ
>
0
i
m
p
l
i
e
s
τ
+
F
d
f
η
,
f
ϱ
≤
F
C
d
η
,
ϱwhere(64)
C
d
η
,
ϱ
=
max
⁡
d
η
,
ϱ
,
d
f
η
,
η
,
d
ϱ
,
f
ϱ
,
d
η
,
f
ϱ
+
d
f
η
,
ϱ
2
.If either f or F is continuous, then f has a unique fixed point.Proof.
Setp
(
η
,
η
)
=
0, for all η
∈
I, in Corollary 17.Remark 19.
In Example10, f is not an F
d-contraction. Hence we cannot apply Corollary 18, so condition (61) is more general than [7] and hence that of Wardowski (see [1]) is not applicable.Corollary 20.
Letf be a self-mapping on a complete partial metric space (
I
,
p
). If there are F
∈
F and τ
>
0 such that, for all η
,
ϱ
∈
I,(65)
p
f
η
,
f
ϱ
>
0
i
m
p
l
i
e
s
τ
+
F
p
f
η
,
f
ϱ
≤
F
H
p
η
,
ϱ
,where(66)
H
p
η
,
ϱ
=
a
1
p
η
,
ϱ
+
a
2
p
f
η
,
η
+
a
3
p
ϱ
,
f
ϱ
+
a
4
2
p
η
,
f
ϱ
+
p
f
η
,
ϱprovided a
i
>
0
(
i
=
1,2
,
3,4
) such that a
1
+
a
2
+
a
3
+
2
a
4
<
1. If either f or F is continuous, then f has a unique fixed point.Proof.
Setg
=
f and h
=
q
=
I in Corollary 15.Corollary 21 (see [22, Theorem 3.1]).
Letf be a self-mapping on a complete metric space (
I
,
d
). Assume there are F
∈
F and τ
>
0 such that(67)
d
f
η
,
f
ϱ
>
0
i
m
p
l
i
e
s
τ
+
F
d
f
η
,
f
ϱ
≤
F
H
d
η
,
ϱ
,for all η
,
ϱ
∈
I, where(68)
H
d
η
,
ϱ
=
a
1
d
η
,
ϱ
+
a
2
d
f
η
,
η
+
a
3
d
ϱ
,
f
ϱ
+
a
4
2
d
η
,
f
ϱ
+
d
f
η
,
ϱprovided a
i
>
0
(
i
=
1,2
,
3,4
) and a
1
+
a
2
+
a
3
+
2
a
4
<
1. If f or F is continuous, then f has a unique fixed point.Proof.
Set in Corollary20
p
(
η
,
η
)
=
0 for all η
∈
I.Remark 22.
We have seen in Example10 that f is not an F
d-contraction, so we cannot apply Corollary 21. Hence condition (65) is more general than the contractive condition provided by Cosentino et al. [22].Remark 23.
The reader interested in fixed points of multivalued contraction mappings in a partial metric space and their applications may be referred to Khan et al. [23].


## 3. Applications

Denote byI
=
C
(
[
0,1
]
,
R
) the set of all continuous functions defined on [
0,1
]. Define the partial metric p
:
I
×
I
→
[
0
,
∞
) by(69)
p
ω
,
υ
=
s
u
p
s
∈
a
,
b
ω
s
-
υ
s
+
η
,
f
o
r
a
l
l
η
≥
0
a
n
d
ω
,
υ
∈
C
a
,
b
,
R
.Note that (
I
,
p
) is complete.We consider a system of elliptic boundary value equations given by(70)
-
d
2
u
d
t
2
=
H
t
,
h
u
t
,
t
∈
0,1
u
0
=
u
1
=
0
a
n
d
-
d
2
v
d
t
2
=
K
t
,
q
v
t
,
t
∈
0,1
v
0
=
v
1
=
0
,where H
,
K
:
0,1
×
I
→
R are continuous mappings. The Green function associated to (70) is defined by(71)
G
t
,
ζ
=
t
1
-
ζ
,
0
≤
t
≤
ζ
≤
1
ζ
1
-
t
,
0
≤
ζ
≤
t
≤
1
.Note that the system (70) has a solution if and only if the operators f
,
g
:
I
→
I, defined by(72)
f
u
t
=
∫
0
1
G
t
,
ζ
H
ζ
,
h
u
ζ
d
ζ
;
(73)
g
v
t
=
∫
0
1
G
t
,
ζ
K
ζ
,
q
v
ζ
d
ζ
,have a common fixed point.Theorem 24.
LetI
=
C
(
[
0,1
]
,
R
) and let p be as in (69). If(A1)
there areτ
>
0, k
∈
(
0,1
), and continuous mappings H
,
K
:
0,1
×
I
→
R such that (74)
H
t
,
h
u
t
-
K
t
,
q
v
t
≤
8
ln
⁡
k
p
h
u
t
,
q
v
t
e
-
τ
,
for eacht
∈
[
0,1
] and h
(
u
(
t
)
)
,
q
(
v
(
t
)
)
∈
I such that(75)
k
p
h
u
t
,
q
v
t
>
e
τ
;
(A2)
ln
⁡
(
p
(
f
(
u
(
t
)
)
,
g
(
v
(
t
)
)
)
)
≤
|
f
(
u
(
t
)
)
-
g
(
v
(
t
)
)
|, for all t
∈
[
0,1
];
(A3)
there is a sequence{
σ
n
} in I such that l
i
m
n
→
∞
p
(
f
h
(
σ
n
)
,
S
f
(
σ
n
)
)
=
0
andl
i
m
n
→
∞
p
(
g
q
(
σ
n
)
,
T
g
(
σ
n
)
)
=
0, whenever(76)
l
i
m
n
→
∞
f
σ
n
=
l
i
m
n
→
∞
h
σ
n
=
t
a
n
d
l
i
m
n
→
∞
g
σ
n
=
l
i
m
n
→
∞
q
σ
n
=
t
f
o
r
s
o
m
e
t
∈
I
,then the system (70) has at least one solution (say) u
∗
(
·
)
∈
C
2
[
0,1
].Proof.
We note thatu
∗
(
·
)
∈
C
2
[
0,1
] is a solution of (70) iff u
∗
(
·
) is a solution of the integral equations given by (72) and (73).
Letu
(
·
)
,
v
(
·
)
∈
C
(
[
0,1
]
). By assumption (
A
1
), we have(77)
f
u
t
-
g
v
t
=
∫
0
1
G
t
,
ζ
H
ζ
,
h
u
ζ
-
K
ζ
,
q
v
ζ
d
ζ
≤
∫
0
1
G
t
,
ζ
H
ζ
,
h
u
ζ
-
K
ζ
,
q
v
ζ
d
ζ
≤
∫
0
1
G
t
,
ζ
8
ln
⁡
k
p
h
u
t
,
q
v
t
e
-
τ
d
ζ
=
8
ln
⁡
k
p
h
u
t
,
q
v
t
e
-
τ
s
u
p
t
∈
0,1
∫
0
1
G
t
,
ζ
d
ζ
.Since ∫
0
1
G
(
t
,
ζ
)
d
ζ
=
-
t
2
/
2
+
t
/
2 for all t
∈
[
0,1
], we have s
u
p
t
∈
[
0,1
]
∫
0
1
‍
G
(
t
,
ζ
)
d
ζ
=
1
/
8. Taking the supremum over t in the resulting inequality and using assumption (A2), we get (78)
ln
⁡
p
f
u
t
,
g
v
t
≤
ln
⁡
k
p
h
u
t
,
q
v
t
e
-
τ
≤
ln
⁡
a
1
p
h
u
t
,
q
v
t
+
a
2
p
f
u
t
,
h
u
t
+
a
3
p
g
v
t
,
q
v
t
e
-
τ
,for all u
(
·
)
,
v
(
·
)
∈
C
(
[
0,1
]
). F
(
x
)
=
l
n
(
x
) verifies all the hypotheses of Corollary 16(a) and so its conclusion ensures that the problem (70) has a solution.



---

*Source: 5980312-2019-04-07.xml*