ABSTRACT

We introduce the concept of generalized quasi-contraction mappings in G-partial metric spaces and prove some fixed point results in ordered G-partial metric spaces. The results generalize and extend some recent results in literature.

Keywords:Fixed Points, Generalized Quasi-Contraction Maps, Bounded Orbit, Partially Ordered Set, 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).

References