#### Abstract

We discuss topological structure of -metric-like spaces and demonstrate a fundamental lemma for the convergence of sequences. As an application we prove certain fixed point results in the setup of such spaces for different types of contractive mappings. Finally, some periodic point results in -metric-like spaces are obtained. Two examples are presented in order to verify the effectiveness and applicability of our main results.

#### 1. Introduction

There are a lot of generalizations of the concept of metric space. The concepts of -metric space and partial metric space were introduced by Czerwik [1] and Matthews [2], respectively. Combining these two notions, Shukla [3] introduced another generalization which is called a partial -metric space.

On the other hand, Amini-Harandi [4] introduced a new extension of the concept of partial metric space, called a metric-like space. The concept of -metric-like space which generalizes the notions of partial metric space, metric-like space, and -metric space was introduced by Alghamdi et al. in [5]. They established the existence and uniqueness of fixed points in a -metric-like space as well as in a partially ordered -metric-like space. In addition, as an application, they derived some new fixed point and coupled fixed point results in partial metric spaces, metric-like spaces, and -metric spaces (see also [6–14]).

The aim of this paper is to examine more closely the topological structure of these spaces. In this context, we demonstrate a fundamental lemma for the convergence of sequences in -metric-like spaces and by using it we prove some fixed point results in the setup of such spaces. Finally, some periodic point results in -metric-like spaces are obtained. Two examples are presented in order to verify the effectiveness and applicability of our main results.

#### 2. Preliminaries

##### 2.1. Definitions and Basic Properties of Certain Types of Spaces

To clarify the issue, we first recall definitions of -metric, partial metric, partial -metric, and metric-like spaces.

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

The pair is called a -metric space with coefficient .

*Definition 2 (see [2]). *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 .

*Definition 3 (see [3]). *A partial -metric on a nonempty set is a mapping such that for some real number and 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 .

*Definition 4 (see [4]). *A metric-like on a nonempty set is a mapping such that for all implies ,,.

The pair is called a metric-like space.

Every partial metric space is a metric-like space. Below we give some other examples of metric-like spaces.

*Example 5 (see [15]). *Let . Then the mapping defined by is a metric-like on .

*Example 6 (see [15]). *Let ; then the mappings (), defined by
are metric-like on , where and .

*Definition 7 (see [5]). *Let be a nonempty set and a given real number. A function is -metric-like if, for all , the following conditions are satisfied: implies ,,.

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

In a -metric-like space if and , then , but the converse may not be true and may be positive for . It is clear that every partial -metric space is a -metric-like space with the same coefficient and every -metric space is also a -metric-like space with the same coefficient . However, the converses of these facts need not hold.

*Example 8. *Let , a constant, and be defined by
Then is a -metric-like space with coefficient , but it is not a partial -metric space. Indeed, for any we have , so of Definition 3 is not satisfied.

The following propositions help us to construct some more examples of -metric-like spaces.

Proposition 9. *Let be a metric-like space and , where is a real number. Then is -metric-like with coefficient .*

*Proof. *The proof follows from the fact that , where .

From the above proposition and Examples 5 and 6 we have the following examples of -metric-like spaces.

*Example 10. *Let . Then the mapping defined by , where is a real number, is -metric-like on with coefficient .

*Example 11. *Let . Then the mappings (), defined by
are -metric-like on , where , and .

Proposition 12. *Let be a nonempty set such that and are -metric and partial -metric, respectively, , and is a metric-like on . Then the mappings (), defined by
**
for all are -metric-like on .*

*Proof. *Let be a partial -metric space and a -metric space with . Then conditions , , and are obvious for the function . For instance, if are arbitrary then, as is partial -metric and is -metric on , we have
Therefore, is satisfied and so is a -metric-like space. Similarly, one can show that and are -metric-like spaces.

From the above proposition and Examples 5 and 6 we have the following examples.

*Example 13. *Let . Then the mapping defined by , where is a real number, is -metric-like on with coefficient .

*Example 14. *Let . Then the mappings (), defined by
are -metric-like on with coefficient , where , , and .

Each -metric-like on generates a topology on whose base is the family of all open -balls , where for all and .

Now, we define the concepts of Cauchy sequence and convergent sequence in a -metric-like space.

*Definition 15 (see [5]). *Let be a -metric-like space with coefficient , and let be any sequence in and . Then

(i)the sequence is said to be convergent to with respect to , if ;(ii)the sequence is said to be a Cauchy sequence in if exists and is finite;(iii) is said to be a complete -metric-like space if for every Cauchy sequence in there exists such that

It is clear that the limit of a sequence in a -metric-like space is usually not unique (since already partial metric spaces share this property).

We start our work by proving the following crucial lemma.

Lemma 16. *Let be a -metric-like space with coefficient , and suppose that and are convergent to and , respectively. Then one has
**
In particular, if , then one has .**Moreover, for each one has
**
In particular, if , then
*

*Proof. *Using the triangle inequality in a -metric-like space it is easy to see that
Taking the lower limit as in the first inequality and the upper limit as in the second inequality we obtain the first desired result. If , then by the triangle inequality we get and . Therefore, we have . Similarly, using again the triangle inequality the other assertions follow.

##### 2.2. Contraction Conditions and Fixed Point Results

It is well known that a self-map on a metric space is said to be a Banach contraction mapping, if there exists a number such that for all . A mapping is called a quasicontraction if for some constant and for every This concept was introduced and studied by Ćirić in 1974 [16]. A result of Ćirić shows that every quasicontraction in a complete metric space has a unique fixed point.

The existence of fixed points in partially ordered metric spaces was first investigated in 2004 by Ran and Reurings [17] and then by Nieto and Rodríguez-López [18].

In this paper, we establish some fixed point theorems for quasicontractive type mappings in a partially ordered complete -metric-like space. We investigate also the so-called -property for mappings in such spaces.

#### 3. Main Results

##### 3.1. Fixed Points of Quasicontraction-Type Mappings

Throughout this paper, let be a partially ordered set, and let be a -metric-like space (we will say, for short, that is a partially ordered -metric-like space). Further, let be the fixed point set of , the lower fixed point set of , and In this section, we obtain some fixed point results for quasicontractions defined on a partially ordered complete -metric-like space.

We will also make use of the following notion.

*Definition 17. *An ordered -metric-like space is said to have the sequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) in , implies that , for all .

Theorem 18. *Let be a complete partially ordered -metric-like space. If is a nondecreasing map such that, for all elements with ,
**
where , then provided that there exists an and one of the following two conditions is satisfied:*(a)* is a continuous self-map on ,*(b)* has the sequential limit comparison property.**Moreover, has a unique fixed point if and only if each pair of fixed points of is comparable.*

*Proof. *Since and is nondecreasing, therefore for each . Define a sequence in with and so for all . If there exists a positive integer such that , then which implies that is a fixed point of . Assume that for every positive integer . Since , therefore by replacing by and by in (15), we have
If, for some , , then according to the above inequality which is a contradiction.

Hence, for all , and therefore
where . Obviously, . Repeating the above process, we get
for all , and so, for , we have
Since, by assumption, , it follows that . Since is complete, there exists an element such that
If is a continuous self-map on , then . Indeed, by the triangle inequality we have
Taking the limit as in the above inequality, the desired result is obtained.

If condition (b) is fulfilled then for all , and it follows that
On taking the upper limit as , (using Lemma 16, as ) it follows that
and hence , or equivalently, .

Suppose that the fixed points of are comparable. Let be another fixed point of such that . Assume, for example, that . Using (15), we obtain that
which further implies that . The converse is trivial.

*Example 19. *Let be endowed with the usual order. Define by
for all . Then is a complete -metric-like space with coefficient (see Example 14).

Let be defined by . It is easy to see that is a nondecreasing and continuous self-map on , and . Using the Mean Value Theorem for any with and that , we have
Thus (15) is satisfied with . Thus all conditions of Theorem 18 are satisfied. Moreover, is the unique fixed point of .

Theorem 20. *Let be a partially ordered complete -metric-like space with coefficient . If a nondecreasing map satisfies
**
for all with , where
**
and is a continuous and nondecreasing function such that for all and , then provided that there exists an and one of the following two conditions is satisfied:*(a)* is a continuous self-map on .*(b)* has the sequential limit comparison property.**Moreover, has a unique fixed point if and only if the fixed points of are comparable.*

*Proof. *Define the sequence as given in the proof of Theorem 18. If there exists a positive integer such that , then is a fixed point of . Assume that for every positive integer . Since , therefore by replacing by and by in (27), we obtain
Therefore, for all , and is nonincreasing and bounded from below. Hence, there exists such that .

From the above argument we have
If , we get
which is, because of , possible only if . So we have

Next, we show that is a Cauchy sequence. If not, then there exists for which we can find subsequences and of the sequence where is the smallest index for which with
Then,
From (33) and (34), we obtain
Taking the upper and lower limits as , from (32) we conclude that
Note that
Using (34) and (32), we get
On the other hand,
Using (36) and (32), we get
From (38) and (40), we have
Also,
Taking the upper limit as , we conclude that
or equivalently,
As is nondecreasing and , from (27) we have
where
wherefrom on taking the upper limit as , from (36) and (41), we have
Thus, from (44) and (45), we have
which gives , a contradiction. Hence is a Cauchy sequence in . Since is complete, there exists an element such that

If is a continuous self-map on , then . If assumption (b) is satisfied then for all ; it follows that
where
On taking the upper limit as in (50), from Lemma 16 we obtain
and hence , implying that .

Now, suppose that the fixed points of are comparable. Let be another fixed point of . We will show that . If not, then without loss of generality, we assume that . Using (27), we obtain
where
Thus,
which implies that , which yields that . The converse is trivial.

*Example 21. *Let be endowed with the usual order . Define by
for all . Then is a complete ordered -metric-like space with coefficient (see Example 11).

Let be defined by . It is easy to see that is a nondecreasing and continuous self-map on , and . Using the Mean Value Theorem for the function for any with , we have
which implies that
Hence
Now we define by , and, for any with , we have
since is an increasing function and .

Thus, all conditions of Theorem 20 are satisfied. Moreover, is the unique fixed point of .

##### 3.2. Coincidence Points of Four Mappings under Generalized Weakly Contractive Conditions

*Definition 22. *Let and be two self-maps on partially ordered set . A pair is said to be(i)weakly increasing if and , for all [19, 20],(ii)partially weakly increasing if , for all [21].

Let be a nonempty set and a given mapping. For every , let .

*Definition 23. *Let be a partially ordered set and mappings such that and . The ordered pair is said to be(a)weakly increasing with respect to if and only if, for all , for all , and for all [22],(b)partially weakly increasing with respect to if , for all [23].

*Remark 24. *In the above definition (i) if , we say that is weakly increasing (partially weakly increasing) with respect to and (ii) if (the identity mapping on , then the above definition reduces to the weakly increasing (partially weakly increasing) mapping (see [22, 24]).

The study of unique common fixed points of mappings satisfying weakly contractive conditions has been at the center of vigorous research activity. Motivated by the work in [21, 23–30], we prove some coincidence point results for nonlinear generalized -weakly contractive mappings in partially ordered -metric-like spaces.

Recall [31] that an altering distance function is a mapping which satisfies that is increasing and continuous, if and only if .

Let be an ordered -metric-like space and four self-mappings. Throughout this subsection, unless otherwise stated, for all , let Suppose that and ; let be an arbitrary point of . Choose such that and such that . Continuing in this way, construct a sequence defined by and , for all . The sequence in is said to be a Jungck-type iterative sequence with initial guess .

Theorem 25. *Let be a partially ordered -metric-like space with sequential limit comparison property, four mappings such that and , and and -complete subsets of . Suppose that, for comparable elements , one has
**
where are altering distance functions. Then, the pairs and have a coincidence point in provided that the pairs and are weakly compatible, and the pairs and are partially weakly increasing with respect to and , respectively.*

*Proof. *Let be a Jungck-type iterative sequence with initial guess in ; that is, , for all .

As and , and the pairs and are partially weakly increasing with respect to and , so we have
Continuing this process, we obtain , for . We will complete the proof in three steps.*Step I.* We will prove that .

Define . Suppose , for some . Then, . If , then gives . Indeed,
where
Thus,
which implies that ; that is, . Similarly, if , then gives . Consequently, the sequence becomes constant for and hence .

Suppose that
for each . We now claim that the following inequality holds:
for each .

Let , and for an , . Then, as , using (62) we obtain that
where
If, for some , , (69) implies that
which is possible only if ; that is, , a contradiction to (67). Hence, and
Therefore, (68) is proved for .

Similarly, it can be shown that
Hence, is a nonincreasing sequence of nonnegative real numbers. Therefore, there is an such that
Taking the limit as in (68), we obtain