Abstract
In the current paper, we first introduce a new class of contractions via a new notion called p-cyclic contraction mapping by combining the ideas of cyclic contraction mapping and p-contraction mapping. Then, we give a new definition of a cyclically 0-complete pair to weaken the completeness condition on the partial metric spaces. Following that, we prove some best proximity point results for p-cyclic contraction mappings on where is a cyclically 0-complete pair in the setting of partial metric spaces. Hence, we generalize and unify famous and well-known results in the literature of metric fixed point theory. Additionally, we present some nontrivial examples to compare our results with earlier. Finally, we investigate the sufficient conditions for the existence of a solution to nonlinear Fredholm integral equations by the results in the paper.
MSC:
54H25; 47H10; 45B05
1. Introduction
Banach [1] proved a fixed point result, which is known as Banach contraction principle, in 1922. In this result, it has been shown that every self mapping on a complete metric space such that there is such that
for all has a unique fixed point in .
Banach contraction principle has been considered the beginning of metric fixed point theory. Then, many authors have generalized and improved it in different ways [2,3,4]. In this sense, Popescu [5] introduced the concept of p-contraction mapping and obtained a fixed point result for these mappings. According to this result, every p-contraction mapping on a complete metric space , that is, there exists q in such that
for all has a unique fixed point.
Recently, Kirk et al. [6] has obtained a new generalization of Banach contraction principle via a new concept of cyclic mapping. In their result, the cyclic mapping may not be continuous, unlike the Banach’s result. This is the important feature of their result. Then, many researchers have studied to obtain some fixed point results for cyclic mappings [7,8].
On the other hand, one of the interesting generalizations of Banach contraction principle has been obtained by taking into account nonself mappings. Let be subsets of a metric space and be a nonself mapping. If , cannot have a fixed point. Then, since for all , it is reasonable to search a point satisfying . This point is said to be a best proximity point of . Note that a best proximity point of is an optimal solution for the problem . Additionally, a best proximity point turns into a fixed point in the case of . Therefore, many authors have studied on this topic [9,10,11,12,13,14,15].
Taking into account the ideas of both best proximity point and cyclic mapping the famous concept of cyclic contraction mapping was introduced by Eldred and Veeremani [16]. In this way, these ideas were unified.
Until now, some properties such as bounded compactness have been used to guarantee the existence of best proximity points for cyclic contraction mappings. Recently, introducing a nice notion called cyclically completeness, Karpagam and Agrawal [17] show the existence of best proximity point of a cyclic contraction mapping without using the property of bounded compactness. Then, many authors have obtained some best proximity point results with the help of this concept [18,19].
In 1994, motivated by the experience of computer science, Matthews, Ref. [20], relaxed the condition implies in a metric space by introducing the partial metric spaces. Following that, many authors obtained both various fixed point results and best proximity point results in the settings of partial metric spaces [21,22,23]. Very recently, Romaguera [24] introduced the concept of 0-complete partial metric space. Hence, a weaker form of completeness on partial metric spaces has been obtained.
In this paper, we aim to extend and unify some famous results in the literature of metric fixed point theory, such as the main results of Eldred-Veeremani [16] and Popescu [5]. Hence, we first introduce a new class of contractions via a new notion called p-cyclic contraction mapping by combining the ideas of cyclic contraction mapping and p-contraction mapping. Then, we give a new definition of a cyclically 0-complete pair to weaken the completeness condition on the partial metric spaces. Following, we prove some best proximity point results for p-cyclic contraction mappings on where is a cyclically 0-complete pair in a partial metric space. Additionally, we present some nontrivial examples to show the effectiveness of our work. Finally, we investigate the sufficient conditions for the existence of a solution to nonlinear Fredholm integral equations by the results in the paper.
2. Preliminaries
In this section, we give some definitons, lemmas and theorems which are important in our main result. We begin this section with the following result for cyclic mappings, which was obtained by Kirk et al. [6].
Theorem 1
([6]). Let be closed subsets of a complete metric space and be a cyclic mapping, that is, and . Then, Υ has a fixed point in if there is such that
for all and
Taking into account in Theorem 1 the famous concept of cyclic contraction mapping was introduced by Eldred and Veeremani [16]. Then, they obtain a best proximity point result as follows:
Definition 1
([16]). Let be subsets of a metric space and be a cyclic mapping. Then, is called cyclic contraction mapping if there exists q in such that
for all and .
Theorem 2
([16]). Let be a metric space, where are closed and be a cyclic contraction mapping. If either D or E is boundedly compact, then Υ has a best proximity point in
Next, we recall the concept of cyclically completeness.
Definition 2
([17]). Let be a metric space and . A sequence in with and is called a cyclically Cauchy sequence if for each there is satisfying
for all with m is odd, n is even.
Definition 3
([17]). A pair of subsets of a metric space is said to be cyclically complete if for every cyclically Cauchy sequence in , either or are convergent.
Now, we give the definition of the partial metric space and its topological properties.
Definition 4
([20]). Let and be a mapping satisfying the following conditions:
- ()
- if and only if
- ()
- ()
- ()
for all . Then, θ is called a partial metric. Additionally, is called a partial metric space.
It can be easily seen that every metric space is a partial metric space, but the converse may not be true (see for more details [25,26,27,28]). Now, assume that is a partial metric on . Then, there is an topology on . Additionally, the family open -balls
is a base for the topology where
for all and . If we take a sequence and , then it is clear that converges to w.r.t. if and only if
The sequence is said to be Cauchy sequence if exists and is finite. Additionally, is said to be a complete partial metric space if every Cauchy sequence converges to a point in w.r.t. such that
Definition 5
([24]). Let be a partial metric space and be a sequence in Λ.
- (i)
- is called 0-Cauchy sequence if
- (ii)
- is called 0-complete partial metric space if every 0-Cauchy sequence converges to a point ϰ in w.r.t. such that
If is a partial metric on , then the mapping defined by
for all is an ordinary metric on .
The following lemma shows the relation between a partial metric and ordinary metric
Lemma 1.
Let be a partial metric space.
- (i)
- is a Cauchy sequence in if and only if is a Cauchy sequence in
- (ii)
- is a complete partial metric space if and only if is a complete metric space,
- (iii)
- Given a sequence in Λ and Then, we have
3. Main Results
We first give the definition of p-cyclic contraction mapping on partial metric spaces.
Definition 6.
Let be subsets of a partial metric space and be a cyclic mapping. Then, Υ is said to be p-cyclic contraction mapping if there is q in such that
for all and where
The following example shows that the classes of cyclic contractions and p-contractions are proper subsets of the class of p-cyclic contractions.
Example 1.
Let and be a function defined by
for . It is clear that is a partial metric space. Consider the following subsets
and
Then, we have . Let define a mapping by
Now, we shall show that Υ is a p-cyclic contraction mapping. Then, we have the following four conditions:
Case 1: Let and . In this case, we have
Case 2: Let and . In this case, we have
Case 3: Let and . In this case, we have
Case 4: Let and . In this case, we have
Hence, Υ is a p-cyclic contraction mapping for . However, Υ is not neither a cyclic contraction mapping nor a p-contraction mapping. If we take and , then we have
for all which implies that Υ is not a cyclic contraction mapping. Additionally, if we take and , then we have
for all , which implies that Υ is not a p-contraction mapping.
Then, we restate the definition of cyclically Cauchy sequence in the settings of partial metric spaces.
Definition 7.
Let be subsets of a partial metric space . A sequence in with and is called a cyclically Cauchy sequence if for each there is such that
for all with m is odd, n is even.
Note that, if in Definition 7, then the definition of cyclically Cauchy sequence turns into the definition of 0-Cauchy sequence.
Now, we introduce the definition of cyclically 0-complete pair in a partial metric spaces.
Definition 8.
Let be subsets of a partial metric space . A pair is said to be cyclically 0-complete pair if for every cyclically Cauchy sequence in , either the sequence has a convergent subsequence to a point w.r.t. such that
or has a convergent subsequence to a point w.r.t. such that
Remark 1.
If D or E is a closed subset of 0-complete partial metric space and , then is a cyclically 0-complete pair. However, if is a cyclically 0-complete pair, then D and E are not necessarily 0-complete. The following example shows this fact.
Example 2.
Let and be a function defined as
for all . Then, is a partial metric space. Let’s take the subsets and of In this case, we have . Now, we claim that is a cyclically 0-complete pair. For this, let’s take a cyclically Cauchy sequence in with and . Then, for each there is such that
for all with m is odd, n is even. Hence, we have that converges to w.r.t. and
that is, the sequence has a subsequence satisfying (2). However, neither D nor E is 0-complete. Indeed, if we take a sequence in D, then we have . Hence, the sequence is a 0-Cauchy sequence in D, but it is not convergent in D. Hence, D is not 0-complete. Similarly, we can show that E is not 0-complete by considering the sequence in E.
Now, we give a new definition in partial metric spaces.
Definition 9.
Let be a subset of a partial metric space . Then, D is called 0-boundedly compact if every bounded sequence has a convergent subsequence to a point w.r.t. such that
Remark 2.
Note that if either D or E is a 0-boundedly compact, then the pair is a cyclically 0-complete pair. However, the converse may not be true. Example 2 can be given to show this fact.
Proposition 1.
Let be a partial metric space and . Suppose that is a p-cyclic contraction mapping. If for any sequence defined by with the initial point , there is such that
then Υ has a best proximity point in .
Proof.
Assume that is an arbitrary sequence defined by with the initial point . Since is a p-cyclic contraction mapping, there is q in such that
for all . Now, if there is such that
then from (4) we get
Hence, we have
Additionally, since we get
Then, we obtain
Hence, and are best proximity points of . □
Remark 3.
If for the sequence mentioned in Proposition 1 there is such that , then Υ has a best proximity point in . Therefore, we will investigate the condition for all in the rest of paper.
Proposition 2.
Let be a partial metric space and . Assume that is a p-cyclic contraction mapping. Then, for every sequence created as in Proposition 1, we have as .
Proof.
Let be a sequence constructed as in Proposition 1. Since is a p-cyclic contraction mapping, considering Remark 3 we have
and so we get
for all where and . By using the last inequality, we have
for all . Hence, we get
□
The following proposition is crucial for our main result.
Proposition 3.
Let be subsets of a partial metric space . Assume that is a p-cyclic contraction mapping. Then, every sequence created as in Proposition 1 is bounded.
Proof.
Let be a sequence constructed as in Proposition 1. Hence, from Proposition 2 the sequence converges to as , and so the sequence is bounded. Then, there exists such that
for all . Since is a p-cyclic contraction mapping, considering Remark 3, we have
for each which implies that
Let
Hence, is bounded. Additionally, we get
Hence, is bounded. Therefore, is bounded. □
Theorem 3.
Let be subsets of a partial metric space where is a cyclically 0-complete pair. If is a p-cyclic contraction mapping, Υ has a best proximity point.
Proof.
Let be a sequence constructed as in Proposition 1 with the initial point . If there is such that
then, from Proposition 1 has a best proximity point. Now assume
for all . In this case, using Proposition 2, we have
Now, let us show that is a cyclically Cauchy sequence. Assume with . Since is a p-cyclic contraction mapping, we get
for all with . Additionally, from Proposition 2 we obtain
for all where and . Therefore, we have
Since is a bounded sequence, considering the last inequality we obtain
for all with and for some . Hence, we have
Now, since is a cyclically 0-complete pair, without loss of the generality we can assume that has a subsequence such that
for some . Moreover, we have
Taking limit in last inequality and using the equality (6) we have
Additionally, we get
Taking limit in last inequality, from (7) we get
Therefore, we have and so . Hence, is a best proximity point of in If has a subsequence such that
for some . Then, by the similar way, it can be shown that is a best proximity point of in E. □
Example 3.
Let and be a function defined by
for . It is clear that is a partial metric space. Let
and
then . Now, we show that the pair is a cyclically 0-complete pair. Let be a cyclically Cauchy sequence in with and . Then, we have
Hence, we get
which implies that and . Then, we have
that is, the sequence has a subsequence satisfying (2). If we define a mapping by
then, it is clear that Υ is a p-cyclic contraction mapping for . Hence, all conditions of Theorem 3 are satisfied, and so Υ has a best proximity point in .
Corollary 1.
Let be subsets of a partial metric space and is a p-cyclic contraction mapping. If D or E is 0-boundedly compact, then Υ has a best proximity point.
Proof.
From Remark 2, we know that if D or E is 0-boundedly compact, then the pair is a cyclically 0-complete pair. Considering Theorem 3, we obtain that has a best proximity point. □
Using Theorem 3 we obtain the following corollary which is a generalization of the main result of Popescu [5].
Corollary 2.
Let be a 0-complete partial metric space and be a mapping. If there exists q in such that
then Υ has a fixed point.
Proof.
Let be a 0-complete partial metric space. If we take and , taking into Remark 1 we can say that is a cyclically 0-complete pair. Additionally, from inequality (8) is p-cyclic contraction mapping. Since all hypotheses of Theorem 3 are satisfied, we conclude that there exists a point such that
which implies that . □
4. Application
In this section, we will consider the following nonlinear Fredholm integral equation
where the functions and are continuous. In mathematics and other sciences such as physics, chemistry, biology, etc., some problems can be modeled by this kind of integral equations. In general, to find an exact solution to these integral equations may not be possible. Hence, it can be used the iterative methods as an alternative way to approach the solution [29,30,31]. We investigate the existence a solution of nonlinear Fredholm integral equations by taking into account Corollary 2. Now, we consider the space as the positive cone of , that is,
Define a partial metric on as
Then, is a 0-complete partial metric space.
Theorem 4.
Assume the following conditions hold:
- (i)
- the mapping defined by
- (ii)
- there is q in such that
Then, the integral Equation (9) has a positive solution.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Acknowledgments
The author is thankful to the referees and editor for making valuable suggestions leading to the better presentations of the paper.
Conflicts of Interest
The author declares no conflict of interest.
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