#### Abstract

Using the concepts of -metric, partial metric, and -metric spaces, we define a new concept of generalized partial -metric space. Topological and structural properties of the new space are investigated and certain fixed point theorems for contractive mappings in such spaces are obtained. Some examples are provided here to illustrate the usability of the obtained results.

#### 1. Introduction and Mathematical Preliminaries

The concept of a -metric space was introduced by Czerwik in [1, 2]. After that, several interesting results about the existence of fixed point for single-valued and multivalued operators in (ordered) -metric spaces have been obtained (see, e.g., [3–13]).

*Definition 1 (see [1]). *Let be a (nonempty) set and a given real number. A function is a -metric on if, for all , the following conditions hold:) if and only if ,(),().

In this case, the pair is called a -metric space.

The concept of a generalized metric space, or a -metric space, was introduced by Mustafa and Sims [14].

*Definition 2 (see [14]). *Let be a nonempty set and a function satisfying the following properties:() if ;(), for all with ;(), for all with ;(), where is any permutation of (symmetry in all the three variables);(), for all (rectangle inequality).

Then, the function is called a -metric on and the pair is called a -metric space.

Aghajani et al. in [15] introduced the class of generalized -metric spaces (-metric spaces) and then they presented some basic properties of -metric spaces.

The following is their definition of -metric spaces.

*Definition 3 (see [15]). *Let be a nonempty set and a given real number. Suppose that a mapping satisfies() if ,() for all with ,() for all with ,(), where is a permutation of (symmetry),() for all (rectangle inequality).

Then is called a generalized -metric and the pair is called a generalized -metric space or a -metric space.

On the other hand, Matthews [16] has introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. In partial metric spaces, self-distance of an arbitrary point need not to be equal to zero.

*Definition 4 (see [16]). *A partial metric on a nonempty set is a mapping such that, for all :() if and only if ,(),
(),
().

In this case, is called a partial metric space.

For a survey of fixed point theory, its applications, and comparison of different contractive conditions and related results in both partial metric spaces and -metric spaces we refer the reader to, for example, [17–27] and references mentioned therein.

Recently, Zand and Nezhad [28] have introduced a new generalized metric space (-metric spaces) as a generalization of both partial metric spaces and -metric spaces.

We will use the following definition of a -metric space.

*Definition 5 (see [29]). *Let be a nonempty set. Suppose that a mapping satisfies() if ;() for all with ;(), where is any permutation of , and (symmetry in all the three variables);() for all (rectangle inequality).

Then is called a -metric and is called a -metric space.

As a generalization and unification of partial metric and -metric spaces, Shukla [30] presented the concept of a partial -metric space as follows.

*Definition 6 (see [30]). *A partial -metric on a nonempty set is a mapping such that, for all :() if and only if ,(),(),().

A partial -metric space is a pair such that is a nonempty set and is a partial -metric on . The number is called the coefficient of .

In a partial -metric space , if and , then , but the converse may not be true. It is clear that every partial metric space is a partial -metric space with coefficient and every -metric space is a partial -metric space with the same coefficient and zero self-distance. However, the converse of these facts needs not to be hold.

*Example 7 (see [30]). *Let , a constant, and defined by
Then is a partial -metric space with the coefficient , but it is neither a -metric nor a partial metric space.

Note that in a partial -metric space the limit of a convergent sequence may not be unique (see [30, Example 2]).

In [31], Mustafa et al. introduced a modified version of ordered partial -metric spaces in order to obtain that each partial -metric generates a -metric .

*Definition 8 (see [31]). *Let be a (nonempty) set and a given real number. A function is a partial -metric if, for all , the following conditions are satisfied:(),(),(),().

The pair is called a partial -metric space.

Since , from , we have Hence, a partial -metric in the sense of Definition 8 is also a partial -metric in the sense of Definition 6.

The following example shows that a partial -metric on (in the sense of Definition 8) is neither a partial metric nor a -metric on .

*Example 9 (see [31]). *Let be a metric space and , where and are real numbers. Then is a partial -metric with .

Proposition 10 (see [31]). *Every partial -metric defines a -metric , where
**
for all .*

Hence, the importance of our definition of partial -metric is that by using it we can define a dependent -metric which we call the -metric associated with .

Now, we present some definitions and propositions in a partial -metric space.

*Definition 11 (see [31]). *Let be a partial -metric space. Then, for an and an , the -ball with center and radius is

Lemma 12 (see [31]). *Let be a partial -metric space. Then,*(A)*if , then ;*(B)*if , then .*

Proposition 13 (see [31]). *Let be a partial -metric space, , and . If then there exists a such that .*

Thus, from the above proposition the family of all open -balls is a base of a -topology on which we call the -metric topology.

The topological space is but need not be .

The following lemma shows the relationship between the concepts of -convergence, -Cauchyness, and -completeness in two spaces and .

Lemma 14 (see [31]). *(1) A sequence is a -Cauchy sequence in a partial -metric space if and only if it is a -Cauchy sequence in the -metric space .**(2) A partial -metric space is -complete if and only if the -metric space is -complete. Moreover, if and only if
*

Now, we introduce the concept of generalized partial -metric space, a -metric space, as a generalization of both partial -metric space and -metric space.

*Definition 15. *Let be a nonempty set. Suppose that the mapping satisfies the following conditions:() if ;() for all with ;(), where is any permutation of , or (symmetry in all three variables);() + + + + for all (rectangle inequality).

Then is called a -metric and is called a -metric space.

Since , so from we have the following inequality: The -metric space is called symmetric if holds for all . Otherwise, is an asymmetric -metric.

Now we present some examples of -metric space.

*Example 16. *Let and let be given by , where . Obviously, is a symmetric -metric space which is not a -metric space. Indeed, if , then . It is easy to see that are satisfied. Now we show that holds. For each , we have
so
Thus,
which implies the required inequality

*Example 17. *Let . Let
Define by
It is easy to see that is an asymmetric -metric space with coefficient .

Proposition 18. *Every -metric space defines a -metric space , where
**
for all .*

*Proof. *Let . Then we have

With straightforward calculations, we have the following proposition.

Proposition 19. *Let be a -metric space. Then for each it follows that*(1);(2) + − + ;(3) + + ;(4) + − + , .

Lemma 20. *Let be a -metric space. Then,*(A)*if , then ;*(B)*if , then .*

*Proof. *If , then from we have , so from we obtain (A). To prove (B), on the contrary, if , then from (A) we have , a contradiction.

*Definition 21. *Let be a -metric space. Then for an and an , the -ball with center and radius is

By motivation of Proposition 4 in [31], we provide the following proposition.

Proposition 22. *Let be a -metric space, , and . If , then there exists a such that .*

*Proof. *Let ; if , then we choose . Suppose that ; then, by Lemma 20, we have . Now, we consider two cases.

Case 1. If , then for we choose . If , then we consider the set

By Archmedean property, is a nonempty set; then by the well ordering principle, has a least element . Since , we have and we choose . Let ; by property we have
Hence, .

Case 2. If , then, from property , we have , and. for , we consider the set

Similarly, by the well ordering principle there exists an element such that , and we choose . One can easily obtain that .

For , we consider the set
and by the well ordering principle there exists an element such that and we choose . Let . By property we have
Hence, .

Thus, from the above proposition the family of all open -balls is a base of a -topology on which we call the -metric topology.

The topological space is , but need not be .

*Definition 23. *Let be a -metric space. Let be a sequence in .(1)A point is said to be a limit of the sequence , denoted by , if .(2) is said to be a -Cauchy sequence, if exists (and is finite).(3) is said to be -complete if every -Cauchy sequence in is -convergent to an .

Using the above definitions, one can easily prove the following proposition.

Proposition 24. *Let be a -metric space. Then, for any sequence in X and a point , the following statements are equivalent:*(1) *is* *-convergent to* .(2)*, as* .(3)*, as* .

Based on Lemma 2.2 of [27], we prove the following essential lemma.

Lemma 25. *(1) A sequence is a -Cauchy sequence in a -metric space if and only if it is a -Cauchy sequence in the -metric space .**(2) A -metric space is -complete if and only if the -metric space is -complete. Moreover, if and only if
*

*Proof. *First, we show that every -Cauchy sequence in is a -Cauchy sequence in . Let be a -Cauchy sequence in . Then, there exists such that, for arbitrary , there is with
for all . Hence,
for all . Hence, we conclude that is a -Cauchy sequence in .

Next, we prove that -completeness of implies -completeness of . Indeed, if is a -Cauchy sequence in , then it is also a -Cauchy sequence in . Since the -metric space is -complete we deduce that there exists such that . Hence,
therefore; .

On the other hand,
Also, from ,
Hence, we obtain that is a -convergent sequence in .

Now, we prove that every -Cauchy sequence in is a -Cauchy sequence in . Let . Then, there exists such that for all . Since
then
Consequently, the sequence is bounded in and so there exists such that a subsequence is convergent to ; that is,
Now, we prove that is a Cauchy sequence in . Since is a -Cauchy sequence in , for given , there exists such that , for all . Thus, for all ,
Therefore, .

On the other hand,
for all . Hence, , and consequently is a -Cauchy sequence in .

Conversely, let be a -Cauchy sequence in . Then, is a -Cauchy sequence in and so it is -convergent to a point with
Then, for given , there exists such that
Therefore,
whenever . Therefore, is -complete.

Finally, let . So
On the other hand,

*Definition 26. *Let and be two generalized partial -metric spaces and let be a mapping. Then is said to be -continuous at a point if, for a given , there exists such that and imply that . The mapping is -continuous on if it is -continuous at all . For simplicity, we say that is continuous.

From the above definition, with straightforward calculations, we have the following proposition.

Proposition 27. *Let and be two generalized partial -metric spaces. Then a mapping is -continuous at a point if and only if it is sequentially continuous at ; that is, whenever is -convergent to , is -convergent to .*

*Definition 28. *A triple is called an ordered generalized partial -metric space if is a partially ordered set and is a generalized partial -metric on .

We will need the following simple lemma of the -convergent sequences in the proof of our main results.

Lemma 29. *Let be a -metric space and suppose that , and are -convergent to , , and , respectively. Then we have
**
In particular, if are constant, then
*

*Proof. *Using the rectangle inequality, we obtain
Taking the lower limit as in the first inequality and the upper limit as in the second inequality we obtain the desired result.

If , then

Let denote the class of all real functions satisfying the condition

In order to generalize the Banach contraction principle, Geraghty proved the following result.

Theorem 30 (see [32]). *Let be a complete metric space and let be a self-map. Suppose that there exists such that
**
holds for all . Then f has a unique fixed point and for each the Picard sequence converges to z.*

In [33], some fixed point theorems for mappings satisfying Geraghty-type contractive conditions are proved in various generalized metric spaces.

As in [33], we will consider the class of functions , where if and has the property

Theorem 31 (see [33]). *Let and let be a complete metric type space. Suppose that a mapping satisfies the condition
**
for all and some . Then has a unique fixed point and for each converges to in .*

The aim of this paper is to present certain new fixed point theorems for hybrid rational Geraghty-type and -contractive mappings in partially ordered -metric spaces. Our results improve and generalize many comparable results in literature. Some examples are established to prove the generality of our results.

#### 2. Main Results

Recall that denotes the class of all functions satisfying the following condition:

Theorem 32. *Let be a partially ordered set and suppose that there exists a generalized partial -metric on such that is a -complete -metric space and let be an increasing mapping with respect to with for some . Suppose that
**
for all comparable elements , where
**
If is continuous, then has a fixed point.*

*Proof. *Put . Since and is an increasing function we obtain by induction that
*Step **1.* We will show that . Since for each , then by (49) we have
because
Therefore, is decreasing. Then there exists such that . Letting in (52) we have
Since , we deduce that ; that is,
*Step **2.* Now, we prove that the sequence is a -Cauchy sequence. By rectangular inequality and (49), we have

Letting in the above inequality and applying (55) we have