#### Abstract

The aim of this paper is to define modified weak --contractive mappings and to establish fixed point results for such mappings defined on partial metric spaces using the notion of triangular -admissibility. As an application, we prove new fixed point results for graphic weak -contractive mappings. Moreover, some examples and an application to integral equation are given here to illustrate the usability of the obtained results.

#### 1. Introduction and Preliminaries

The concept of partial metric space was introduced by Matthews [1] in 1994. Partial metric space is a generalized metric space in which each object does not necessarily have to have a zero distance from itself. A motivation behind introducing the concept of a partial metric was to obtain appropriate mathematical models in the theory of computation and, in particular, to give a modified version of the Banach contraction principle [2, 3]. Subsequently, several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions on partial metric spaces (e.g., [4–10]). For a recent survey on the existence of fixed points in different spaces with generalized distance functions, the reader may check [11–18]. We start by recalling some definitions and properties of partial metric spaces.

*Definition 1. *A *partial metric *on a nonempty set is a function such that, for all ,(p1);
(p2);
(p3);
(p4).

A partial metric space is a pair such that is nonempty set and is a partial metric on .

From the above definition, if , then . But if , may not be in general. A trivial example of a partial metric space is the pair , where is defined as . For more examples of partial metric spaces, we refer to [6, 7].

Each partial metric on generates a topology on which has as a base the family of open -balls , where for all and . A sequence in converges to a point , with respect to , if and only if . A sequence in is called Cauchy sequence if exists and is finite. A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .

A sequence in is called -Cauchy sequence if . A partial metric space is said to be -complete if every -Cauchy sequence in converges, with respect to , to a point such that .

The partial metric space , where denotes the set of rational numbers and the partial metric is given by , provides an example of a 0-complete partial metric space which is not complete.

If is a partial metric on , then the function given by is a metric on . A partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if . Some more properties of partial metric spaces can be found in [3, 7, 19].

Samet et al. [20] defined the notion of -admissible mappings and proved the following result.

*Definition 2. *Let be a self-mapping on and a function. One says that is an -admissible mapping if

Denote with the family of nondecreasing functions such that for all , where is the th iterate of .

Theorem 3. *Let be a complete metric space and be -admissible mapping. Assume that
**
where . Also, suppose that the following assertions hold:*(i)*there exists such that ;*(ii)*either is continuous or for any sequence in with for all and as , one has for all . ** Then, has a fixed point.*

Very recently, Salimi et al. [21] modified the notions of --contractive mappings and -admissible mappings as follows.

*Definition 4 (see [21]). *Let be a self-mapping on and two functions. One says that is an -admissible mapping with respect to if
Note that if we take , then this definition reduces to Definition 2. Also, if we take, , then we say that is an -subadmissible mapping.

The following result properly contains Theorem 3 and Theorems 2.3 and 2.4 of [22].

Theorem 5 (see [21]). *Let be a complete metric space and be -admissible mapping with respect to . Assume that
**
where and
**
Also, suppose that the following assertions hold:*(i)*there exists such that ;*(ii)*either is continuous or for any sequence in with for all and as , one has for all . ** Then, has a fixed point.*

In fact, Banach contraction principle and Theorem 5 hold for the following example, but Theorem 3 does not hold.

*Example 6 (see [21]). *Let be endowed with the usual metric for all , and let be defined by . Also, define, by and by .

In this paper, we define modified weak --contractive mappings and establish fixed point results for such mappings defined on ordinary as well as ordered partial metric spaces using the notion of triangular -admissibility. As an application, we prove new fixed point results for graphic weak -contractive mappings. Moreover, some examples and an application to integral equation are given here to illustrate the usability of the obtained results.

#### 2. Modified Weak --Contractions

Recently, Karapınar et al. [23] introduced the notion of triangular -admissible mapping as follows.

*Definition 7 (see [23]). *Let and . One says that is a triangular -admissible mapping if(T1) implies ;(T2) imply .

Lemma 8 (see [23]). *Let be a triangular -admissible mapping. Assume that there exists such that . Define sequence by . Then,
*

Motivated by Samet et al. [20] and Salimi et al. [21], we introduce the following mapping.

*Definition 9. *Let be a partial metric space and , two mappings. If there exists an upper semicontinuous from the right nondecreasing function with for all such that
where
Then, we say that is a modified weak --contractive mapping.

*Definition 10. *Let be a partial metric space. Let and . One says that is an -continuous function on if for given and in ,

*Example 11. *Let and . Assume that and defined by
Clearly, is not continuous, but is an -continuous on . Indeed, if as and , then and so .

Theorem 12. *Let be a 0-complete partial metric space and a modified weak --contractive and triangular -admissible mapping. Suppose that the following assertions hold:*(i)*there exists such that ;*(ii)* is an -continuous function on .** Then, has a fixed point.*

*Proof. *Let such that . Define a sequence in by for all . Then, by Lemma 8, we have
If for some , then is a fixed point for , and the result is proved. Hence, we suppose that for all . Now, by (8) and (12) with , , we get
On the other hand,
and then, . So, from (13), we have
Now, if for some , then
which is a contradiction. Hence, for all , we have

This implies that the sequence is decreasing, and so, by (17), there is such that
Now, we show that must be equal to .

In fact, if , then we get
which is a contradiction. Hence,
We prove that is a 0-Cauchy sequence. Suppose, to the contrary, that is not a 0-Cauchy sequence. Then, there is and sequences and such that, for all positive integers ,
Now, for all , we have
Taking the limit as in the above inequality and using (20), we get
Again, from
taking the limit as and by (20) and (23), we deduce
Also, since
then by taking limit as in the last inequality and applying (20) and (23), we deduce
Similarly,
Then, by (8) and (12), we obtain
where
Taking limit supremum as in the above inequality and applying (20), (23), (25), (27), and (28), we get
which is a contradiction. Hence, is a 0-Cauchy sequence.

Since is orbitally -continuous on , as and , then we have
So, is a fixed point of .

Theorem 13. *Let be a 0-complete partial metric space and a modified weak --contractive and -admissible mapping. Suppose that the following assertions hold:*(i)*there exists such that ;*(ii)* for all and if is a sequence in such that for all and as , then for all . ** Then, has a fixed point.*

*Proof. *Condition (ii) implies property (T2) in definition of triangular -admissible map. Indeed, if and , then applying (ii) to defined by
we get for , and hence, . Thus, as in Theorem 12, we obtain a 0-Cauchy sequence such that as . Since,
for all and as , then from (ii), we have
for all . Then, from (8), we get
where
By taking limit supremum as in the above inequality, we have
which is a contradiction. Hence, . That is, .

*Example 14. *Let be endowed with the partial metric for all , and let be defined by
Define and by

Clearly, is a 0-complete partial metric space. We show that is a triangular -admissible mapping. Let , if , then or . On the other hand, for all , we have . It follows that . Also, if , and then . That is, . Hence, the assertion holds. In reason of the above arguments, .

Now, if is a sequence in such that for all and as , then , and hence, . This implies that for all .

Let . Then, . We get
That is,
Hence, all conditions of Theorem 13 hold and is a fixed point of .

Corollary 15. *Let be a 0-complete partial metric space and be such that for some . Assume that
**
hold for all . Also, suppose that one of the following assertions holds:*(i)* is triangular -admissible and -continuous on ;*(ii)* is -admissible, for all and if is a sequence in such that for all and as , then for all . ** Then, has a fixed point.*

Corollary 16. *Let be a 0-complete partial metric space and be such that for some . Assume that
**
hold for all where . Also, suppose that one of the following assertions holds:*(i)* is triangular -admissible and -continuous on ;*(ii)* is -admissible, for all and if is a sequence in such that for all and as , then for all . ** Then, has a fixed point.*

*Proof. *Let . Then, . Hence, by (44), we have
Thus, . Hence, conditions of Corollary 15 hold, and has a fixed point.

Similarly, we have the following corollary.

Corollary 17. *Let be a 0-complete partial metric space and be such that for some . Assume that
**
hold for all where . Also, suppose that one of the following assertions holds:*(i)* is triangular -admissible and -continuous on ;*(ii)* is -admissible, for all and if is a sequence in such that for all and as , then for all . **Then, has a fixed point.*

Corollary 18 (Matthews [1]). *Let be a 0-complete partial metric space and a given mapping satisfying
**
for all , where . Then, has a unique fixed point.*

#### 3. Fixed Point Results in Partially Ordered Partial Metric Spaces

Fixed point theorems for monotone operators in partially ordered metric spaces are widely investigated and have found various applications in differential and integral equations (see [24, 25] and references therein).

Theorem 19 (see [24]). *Let be a partially ordered complete metric space and a continuous increasing self-mapping such that for some . Assume that
**
hold for all with , where . Then, has a fixed point.*

Theorem 20. *Let be a partially ordered 0-complete partial metric space and an increasing self-mapping such that for some . Assume that
**
hold for all with . Now, if is a continuous mapping on , then has a fixed point.*

*Proof. *Define, by
At first we prove that is a triangular -admissible mapping. Let , then . As is an increasing mapping, we have . That is, . Also, let and , then and . So, from transitivity, we have . That is, . Thus, is a triangular -admissible mapping. Also, there exists such that which implies . Let , then . Now, from (49), we have . That is,
Hence, all conditions of Theorem 12 are satisfied, and has a fixed point.

Theorem 21 (see [25]). *Let be a partially ordered complete metric space and an increasing mapping such that
**
for all with , where . Suppose that the following assertions hold:*(i)*there exists such that ;*(ii)*if is a sequence in such that for all and as , then for all . **Then, has a fixed point.*

Theorem 22. *Let be a partially ordered 0-complete partial metric space and an increasing mapping such that
**
for all with . Suppose that the following assertions hold:*(i)*there exists such that ;*(ii)*if is a sequence in such that for all and as , then for all . **Then, has a fixed point.*

*Proof. *Define as in proof of Theorem 20. Assume for all such that as . Then, for all . Hence, by (ii), we get for all and so for all . Proceeding as in proof of Theorem 20, we can deduce that is a modified weak --contractive and -admissible mapping and there exists such that . Hence, all conditions of Theorem 13 hold, and has a fixed point.

*Remark 23. *Similarly, we may obtain more fixed point results on ordered partial metric spaces as immediate consequences of Corollaries 15–17.

#### 4. Fixed Point Results for Graphic Contractions

Consistent with Jachymski [26], let be a partial metric space, and let denote the diagonal of the Cartesian product . Consider a directed graph such that the set of its vertices coincides with and the set of its edges contains all loops, that is, . We assume that has no parallel edges, so we can identify with the pair . Moreover, we may treat as a weighted graph (see [26]) by assigning to each edge the distance between its vertices. If and are vertices in a graph , then a path in from to of length is a sequence of vertices such that , and for . A graph is connected if there is a path between any two vertices. is weakly connected if is connected (see for details [11, 26–28]).

Recently, some results have appeared providing sufficient conditions for a mapping to be a Picard operator if is endowed with a graph. The first result in this direction was given by Jachymski [26].

*Definition 24 (see [26]). *One says that a mapping is a Banach -contraction or simply -contraction if preserves edges of , that is,
and decreases weights of edges of in the following way:

*Definition 25 (see [26]). *A mapping is called -continuous if given and sequence

*Definition 26. *Let be a partial metric space endowed with a graph and a self-mapping. If there exists an upper semicontinuous from the right function with for all such that
where
Then, is called a weak -graphic contractive mapping.

Theorem 27. *Let be a 0-complete partial metric space endowed with a graph and a weak -graphic contractive mapping. Suppose that the following assertions hold:*(i)*there exists such that ;*(ii)* is -continuous on ;*(iii)* and imply for all , that is, is a quasi-order [26]. ** Then, has a fixed point.*

*Proof. *Define by
At first we prove that is a triangular -admissible mapping. Let , then . As is a weak -graphic contractive mapping, we have . That is, . Also, let and , then and . So, from (iii), we have . That is, . Thus, is a triangular -admissible mapping. Let be -continuous on . Then,
That is,
which implies is -continuous on . From (i), there exists such that . That is, .

If , then . Now, since is a weak -graphic contractive mapping, so . That is,
Hence, all conditions of Theorem 12 are satisfied, and has a fixed point.

If is a connected graph, then condition (iii) of Theorem 27 is automatically satisfied. Thus, we have the following result.

Corollary 28. *Let be a 0-complete partial metric space endowed with a graph and a weak -graphic contractive mapping. Suppose that the following assertions hold:*(i)*there exists such that ;*(ii)* is -continuous on ; *(iii)* is a connected graph. ** Then, has a fixed point.*

Theorem 29. *Let be a 0-complete partial metric space endowed with a graph and a weak -graphic contractive mapping. Suppose that the following assertions hold:*(i)*there exists such that ;*(ii)*if is a sequence in such that for all and as , then for all . ** Then, has a fixed point.*

*Proof. *Define, as in proof of Theorem 27. Condition (ii) implies that is a quasi-order, that is, and imply for all (see Remark 3.1 [26]). Let, as and for all . Then, is a sequence in such that for all and as . So, by (ii), we have for all . That is, . All other conditions of Theorem 13, follow similarly as in proof of Theorem 27 and consequently has a fixed point.

Theorem 3.2(2°) in [26] and Theorem 2.3(2) in [29] are extended to weak -graphic contractive maps defined on a 0-complete partial metric space as follows.

Corollary 30. * there exists such that ;*(iis)* is a quasi-order and if is a sequence in such that for all and as , then there is a subsequence with for all . **Then, has a fixed point.*

* Proof. *Condition (iis) implies that of in Theorem 29 (see Remark 3.1 [26]). Now, the conclusion follows from Theorem 29.

Corollary 31. *Let be a 0-complete partial metric space and -chainable for some , that is, given , there is and a sequence such that , and for . Suppose that is a mapping satisfying
**
Then, has a fixed point.*

*Proof. *Consider the graph with and . Then, -chainability of means is connected, and hence, is quasi-order. If , then
so , hence is a -contraction. Let be in with , then for sufficiently large , so there is such that . Thus, by Corollary 30, has a fixed point.

*Definition 32. *Let be a partially ordered partial metric space endowed with a graph and a self-mapping. If there exists an upper semicontinuous from the right function with for all such that, for all , with , where
where
Then, we say is an ordered weak -graphic contractive mapping.

Theorem 33. *Let be a partially ordered 0-complete partial metric space endowed with a graph and an ordered weak -graphic contractive mapping. Suppose that the following assertions hold:*(i)*there exists such that with ,*(ii)*either is -continuous in and and imply or; *(iii)*if is a sequence in such that with for all and as , then with for all . ** Then, has a fixed point.*

*Proof. *Define by
At first, we prove that is a triangular -admissible mapping. Let , then with . As is an ordered weak -graphic contractive mapping, we have where . That is, . Also, let and , then with and with . So from (ii), we have . Also, and implies . Hence, . Thus, is a triangular -admissible mapping. Let be -continuous on . Then,
That is,
which implies is -continuous on . From (i), there exists such that . That is, .

Let , then with . Now, since is an ordered weak -graphic contractive mapping, then . That is,
Hence, all conditions of Theorem 12 (or 13) are satisfied, and has a fixed point.

*Remark 34. *All our results established above are new even in the setting of complete metric spaces.

#### 5. Application to Existence of Solutions of Integral Equations

Fixed point theorems for monotone operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations (see [30, 31] and references therein). In this section, we apply our result to the existence of a solution of an integral equation. Let be the set of real continuous functions defined on , and let be defined by for all . Then, is a complete metric space. Also, assume this metric space endowed with a graph .

Consider the integral equation and let be defined by We assume that (A) is continuous; (B) is continuous; (C) is continuous; (D)there exists an upper semicontinuous from the right nondecreasing function with for all such that for all , (F) there exist such that ;(G) if is a sequence in such that for all and as , then for all ;(H) assume

Theorem 35. *Under assumptions , the integral equation (72) has a solution in .*

*Proof. *Consider the mapping defined by (73). Let . Then, from , we deduce
Then,