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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.

The Banach contraction principle has been generalized and extended in many directions for some decades. Of all the generalizations, Ciric [

Rodriguez-Lopez and Nieto [

Matthew [

Definition 1.1. [

(p2) if

He was able to establish a relationship between partial metric spaces and the usual metric spaces when

Mustafa and Sims [

Definition 1.2. [

(G2)

(G4)

Then, the function G is called a generalized metric, or more specifically, a G-metric on X, and the pair

Mustafa [

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

(Gp1)

The function

Definition 1.4. A G-partial metric space is said to be symmetric if

In this work, we will assume that

Definition 1.5. Let

2) (X, d) is a metric space.

Proof 1) From (Gp1), we have that for all

hence (p1) is satisfied.

If

By (Gp1), it must follow that

From the symmetry of

(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

Example 1.6. Let

We state the following definitions and motivations.

Definition 1.7. A sequence

Definition 1.8. A sequence

The proof of the following result follows from the above definitions:

Proposition 1.9. Let

Definition 1.10. A G-partial metric space

Definition 1.11. [

Definition 1.12. Let

Gordji et al. [

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 [

Definition 1.13. Let

for any

Definition 1.14. Let

where

Theorem 2.1. Let

where

Suppose T is a non-decreasing map such that there exists an

Then T has a fixed point. Moreover, if for each

Proof. Fix

This implies that

Since

Thus, with

Since

Consequently,

For

Take the limit as

Next we prove that

where

Take limit as

Since

For uniqueness, suppose

Moreover

where

Taking the limit as

Consequently,

Similarly,

Finally for all

Letting

Theorem 2.1 can be viewed as an extension of results of Turkoglu et al. ([

Corollary 2.2. Let

where

Suppose T is a non-decreasing map such that there exists an

Proof: Observe that

where

Thus, the proof of the corollary follows from Theorem 2.1.

Theorem 2.3. Let

Assume that there exists an

Proof. Starting with

We prove that there exists 0 < c < 1 such that

On the contrary, assume that

for some subsequence

Now, we show that

where K is a bound for the bounded sequence

From the axiom (Gp1),

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,

From axiom (Gp1),

Hence (10) becomes

From the induction hypothesis,

We also have from the definition of T and the induction hypothesis,

The inequality (11) becomes

Repeating the same process,

Thus (9) holds for each

Since X is complete then there exists

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

Since T is a generalized quasi-contraction mapping we have

Letting

Also

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. [

in (1), then we get

which is the G-partial metric version of the map of Gordji [

The proof of Corollary 2.4 follows from Theorem 2.3.

Corollary 2.4. Let

for all comparable

Example 2.5. Let

for each