Some Fixed Point Results of Ciric-Type Contraction Mappings on Ordered G-Partial Metric Spaces ()
1. Introduction and Preliminary Definitions
The Banach contraction principle has been generalized and extended in many directions for some decades. Of all the generalizations, Ciric [1] [2] generalizations seem outstanding. Cho Song Wong [3] dealt with a pair of operators in which the control functions in the generalized contraction maps are upper semi-continuous, while Ciric considered a single operator and took the control function to be a constant. If the control function is an upper semi-continuous, then the result of Ciric [1] is invalid. In Kiany and Amini-Harandi [4] , a condition is imposed on the control function and the mapping is termed a Ciric generalized quasi-contraction mapping. In this work, we introduce the concept of generalized quasi-contraction mappings in the new framework of G-partial metric spaces.
Rodriguez-Lopez and Nieto [5] , Ran and Reuring [6] presented some new results for the existence of the fixed point for some mappings in partially ordered metric spaces. The main idea in [5] [6] involves combing the ideas of an iterative technique in the contraction mapping principle with those in the monotone technique. In this work, the existence of a unique fixed point for generalized contraction mappings in ordered G-partial metric spaces is proved.
Matthew [7] generalized the notion of metric spaces by introducing the concept of nonzero self-distance and thus, defined a generalized metric space known as partial metric space, as follows:
Definition 1.1. [7] . A partial metric space is a pair (X, p), where X is a nonempty set and 
such that:
(p1) 
(p2) if 
then 
(p3) 
(p4) 
He was able to establish a relationship between partial metric spaces and the usual metric spaces when
Mustafa and Sims [8] also extended the concepts of metric to G-metric by assigning a positive real number to every triplet of an arbitrary set as follows:
Definition 1.2. [8] . Let X be a nonempty set, and let
be a function satisfying:
(G1) 
(G2) 
for all 
with 
(G3) 
(G4) 
(symmetry in all three variables)(G5) 
for all 
(rectangle inequality).
Then, the function G is called a generalized metric, or more specifically, a G-metric on X, and the pair 
is a G-metric space.
Mustafa [8] gave an example to show the relationship between G-metric spaces and ordinary metric spaces as: For any G-metric G on X, if 
then 
is a metric space.
In this work, the idea of the nonzero self-distance of partial metric spaces and the rectangle inequality of G-metric spaces are combined to develop a new generalized metric space which is defined as the following:
Definition 1.3. Let X be a nonempty set, and let 
be a function satisfying the following:
(Gp1) 
(small self-distance)(Gp2) 
iff 
(equality)(Gp3) 
(symmetry in all three variables)(Gp4) 
(Rectangle inequality).
The function 
is called a G-partial metric and the pair 
is called a G-partial metric space.
Definition 1.4. A G-partial metric space is said to be symmetric if 
for all
.
In this work, we will assume that 
is symmetric. The following proposition establishes the relation between G-partial metric spaces and (partial) metric spaces.
Definition 1.5. Let 
be a G-partial metric space. Define the functions 
and 
by 
and 
Then 1) (X, p) is a partial metric space.
2) (X, d) is a metric space.
Proof 1) From (Gp1), we have that for all 
hence (p1) is satisfied.
If 
then
By (Gp1), it must follow that 
From the symmetry of 
and by (Gp2), 
hence (p2) is satisfied.
(p3) follows from (Gp3) and the triangle inequality (p4) is easily verifiable using (Gp4).
2) Since (X, p) is a partial metric space, then
defines a metric on X and so 
also defines a metric on X.
Example 1.6. Let 
and define the function 
as 
Then 
is a G-partial metric space.
We state the following definitions and motivations.
Definition 1.7. A sequence 
of points in a G-partial metric space 
converges to some 
if 
Definition 1.8. A sequence 
of points in a G-partial metric spaces 
is Cauchy if the numbers 
converges to some 
as n, m, l approach infinity.
The proof of the following result follows from the above definitions:
Proposition 1.9. Let 
be a sequence in G-partial metric space X and
. If 
converges to 
then 
is a Cauchy sequence.
Definition 1.10. A G-partial metric space 
is said to be complete if every Cauchy sequence in 
converges to an element in
.
Definition 1.11. [6] . If 
is a partially ordered set and T: X → X, then T is monotone non-decreasing if for every
, 
implies
.
Definition 1.12. Let 
be a partially ordered set. Then two elements 
are said to be totally ordered or ordered if they are comparable, i.e. 
or
.
Gordji et al. [9] proved the existence of a unique fixed point for contraction type maps in partially ordered metric spaces using a control function. Kiany and Amini-Harandi [4] proved the existence of a unique fixed point for a generalized Ciric quasi-contraction mapping in what they tagged a generalized metric space. The map they considered extend that of Gordji et al., albeit the space they considered was not endowed with an order. Saadati et al. [10] considered the concept of Omega-distances on a complete partially ordered G-metric space and proved some fixed point theorems. Turkoglu et al. [11] and Sastry et al. [12] proved some fixed point theorems for generalized contraction mappings in cone metric spaces and metric spaces respectively.
In this work, the existence of unique fixed points of the two generalized contraction mappings below is proved in ordered G-partial metric spaces, extending thus the results in [2] [4] [9] [11] .
Definition 1.13. Let 
be a G-partial metric space. The self-map T: X→ X is said to be a generalized Ciric quasi-contraction if
(1)
for any 
where 
is a mapping.
Definition 1.14. Let 
be a G-partial metric space. The self-map T: X→ X is said to be a generalized G-contraction if for all 
(2)
where 
are functions such that
2. Main Results
Theorem 2.1. Let 
be a partially ordered set and suppose there exists a G-partial metric 
in X such that 
is a complete G-partial metric space. Let 
be a self-mapping in X such that for each 
satisfying 
(3)
where 
are functions such that
(4)
Suppose T is a non-decreasing map such that there exists an 
with
. Also suppose that X is such that for any non-decreasing sequence 
converging to x, 
for all 
Then T has a fixed point. Moreover, if for each
, there exists 
which is comparable to u and v, then T has a unique fixed point.
Proof. Fix 
Let 
be defined by 
, 
, ···,
. Since 
and T is non-decreasing, then 
This implies that 
for each
.
Since 
for each 
then by (3) we have
Thus, with 
evaluated at
, we have
(5)
Since 
then (5) becomes 
Consequently, 
For 
we get,
(6)
Take the limit as 
in (6) yields 
which implies that 
is a Cauchy sequence. Since X is a complete space then there exists 
such that 
converges to 
and
Next we prove that 
is the fixed point of T. From (3) and (4), since
, for all
,
where 
are evaluated at 
Take limit as 
yields
Since 
then 
Hence 
For uniqueness, suppose 
and 
are two fixed points of T, and there exists 
which is comparable to 
and 
Monotonicity of T implies that 
is comparable to 
and 
for
.
Moreover
where 
are evaluated at 
Taking the limit as 
and by symmetry we get,
(7)
Consequently, 
Similarly, 
Finally for all 
with 
where 
we have,
Letting 
yields 
Hence 
Theorem 2.1 can be viewed as an extension of results of Turkoglu et al. ([11] , Theorem 2.1) to the setting of G-partial metric spaces endowed with an order. The following corollary can be obtained:
Corollary 2.2. Let 
be a partially ordered set and let there exist a G-partial metric 
in X such that 
is a complete G-partial metric space. Let 
be a self-mapping in X such that for each 
satisfying 
where 
Suppose T is a non-decreasing map such that there exists an 
with
. Also suppose that X is such that for any non-decreasing sequence 
converging to
, 
for all 
Then T has a fixed point. Moreover, if for each 
there exists 
which is comparable to 
and 
then T has a unique fixed point.
Proof: Observe that
where 
and 
are chosen such that for any 
one and only one of 
is non-null. In such case,
Thus, the proof of the corollary follows from Theorem 2.1.
Theorem 2.3. Let 
be a partially ordered set and suppose there exists a G-partial metric 
in X such that 
is a complete G-partial metric space. Let 
be a generalized Ciric quasi-contraction map such that 
satisfies 
for each 
for any 
with 
Assume that there exists an 
with the bounded orbit, that is the sequence 
defined by 
for all n, is bounded. Furthermore, if T is an increasing map such that there exists an 
with 
and if any non-decreasing sequence 
satisfies 
for all n, then T has a fixed point. Moreover, if for each 
there exists 
which is comparable to 
and 
then T has a unique fixed point.
Proof. Starting with 
such that 
and with T non-decreasing, we have
We prove that there exists 0 < c < 1 such that
(8)
On the contrary, assume that
for some subsequence 
of 
Since by our assumption the sequence 
is bounded, then the subsequence 
is bounded too. Since the sequence is monotonic and bounded then it converges. Let 
From our assumption, 
a contradiction. Thus (8) holds.
Now, we show that 
is a Cauchy sequence. To prove the claim, we show by induction that for each 
(9)
where K is a bound for the bounded sequence 
When 
From the axiom (Gp1), 
Thus
Thus (9) holds for 
Suppose that (9) holds for each k < n; let us show that it holds for k = n. Since T is a generalized Ciric quasicontraction map,
(10)
From axiom (Gp1), 
Hence (10) becomes
From the induction hypothesis, 
Thus,
(11)
We also have from the definition of T and the induction hypothesis,
The inequality (11) becomes
(12)
Repeating the same process,
Thus (9) holds for each 
From (9) we deduce that 
is a Cauchy sequence.
Since X is complete then there exists 
such that 
and
Now we prove that q is the fixed point of T. To show that, we claim that there exists 0 < b < 1 such that 
On the contrary, we assume 
for some subsequences 
Since 
then 
a contradiction.
Since T is a generalized quasi-contraction mapping we have
Letting 
we have, 
Also
. Hence 
Since b < 1, q = Tq.
The uniqueness of the fixed point follows from the quasicontractive condition.
Theorem 2.3 is an extension of Theorem 2.3 of Gordji et al. [4] to G-partial metric space in the sense that, if
in (1), then we get
which is the G-partial metric version of the map of Gordji [9] .
The proof of Corollary 2.4 follows from Theorem 2.3.
Corollary 2.4. Let 
be a partially ordered set such that there exists a G-partial metric on X such that 
is a complete G-partial metric space. Let 
be an increasing mapping such that there exists 
with 
Suppose that there exists 
such that
for all comparable 
If T is continuous and if for each 
there exists 
which is comparable to x and y. Then T has a unique fixed point.
Example 2.5. Let 
and a G-partial metric defined by 
for all 
On the set X, we consider the usual ordering 
Clearly, 
is a complete G-partial metric space and
is a partially ordered set. Define a function 
as follows: 
for all 
Define 
by 
for each 
Then we have,
for each 
Thus, all of the hypotheses of Theorem 2.3 are satisfied and so T has a unique fixed point (0 is the unique fixed point of T).