Abstract
Partial -metric spaces are characterised by a modified triangular inequality and that the self-distance of any point of space may not be zero and the symmetry is preserved. The spaces with a symmetric property have interesting topological properties. This manuscript paper deals with the existence and uniqueness of fixed point points for contraction mappings using triangular weak -admissibility with regard to and -class functions in the class of partial -metric spaces. We also introduce an example to demonstrate the obtained results.
1. Introduction and Preliminaries
The contribution of fixed point theorems has been appeared as a really effective and successful strategy for understanding many mathematical issues, which rise from actual applications. Fixed point theory is an essential tool to solve several equations in diverse fields, such as integral and differential (fractional) equations appeared in economics, game theory, engineering, physics, and chemistry. For instance, see [1–8]. As the significance and accessibility of Banach contraction mapping (BCM), various authors have amplified and generalized this contraction and set up various interesting supplements and alterations. In this continuation, Czerwik [9] and Bakhtin [10] originated the idea of a -metric space as a suitable generalization of metric spaces. They demonstrated the truth of new weak contraction mapping standards in -metric spaces that generalize the celebrated Banach contraction. Subsequently, several authors have dealt with related fixed point results in this setting (see [11–17]). Furthermore, Matthews [18] presented the idea of a partial metric space by considering the distance between any point to itself that requires not be equal to zero. For more related works in this setting, see [19–21]. Recently, Shukla [22] studied the connection, as topological properties, between a partial metric and a -metric space to introduce an unification of the concept of a partial -metric space.
Definition 1 ([22]). Given a nonempty set . A function is denominated as a -partial metric if for all
If , then . The converse does not hold in general. Now, consider and . Take as
for all . Then, is a partial -metric space (Here, ). It is neither a partial metric, nor a -metric space. Let be a partial metric and be a metric (with ) on a nonempty set . Then, defined by for all , is a partial -metric on .
Definition 2 ([22, 23]). Let be a partial -metric space with . Let be a sequence in and . Then, (i) converges to if (ii) is Cauchy in if exists and is finite(iii) is complete if for every Cauchy sequence , there is so that
Given a nonempty set . Given and . Then, is called -admissible [24] if for all with , we have . The conception of triangular -admissibility was initiated in [25]. This concept has been weakened as follows.
Definition 3 ([26]). Given a nonempty set . Given . is triangular weak -admissible with regard to if (1), implies that , for all (2) and , imply that .
Let . Given as . Take as and . Here, is triangular weak -admissible with regard to
Lemma 4 ([26]). Given . Let be triangular weak -admissible with regard to . Assume that there is such that . Define in by . Then, for all integers
Let be a metric space. Given . Following [27], the mapping is said to be --continuous if every sequence in with for all and , we have . This concept could be extended to -partial metric spaces as follows.
Definition 5. Let be a -partial metric space. Given . The mapping is said to be --continuous if every sequence in with for all and , we have .
The conception of -class functions is posed as follows.
Definition 6 ([28]). We say that is a -class function if it is continuous and for , we have (i)(ii) yields that either , or Denote by the set of -class functions. The following functions are elements in (for all ): (1)(2) (with (3) (with is continuous)Denote by the set of functions [29] verifying that: (i) is monotone and nondecreasing(ii)Let be the set of functions so that (i) is continuous(ii)In this paper, we initiate the concept of generalized -contraction self-mappings via -class functions and the connotation of triangular weak -admissibility with regard to . The goal is to prove some related fixed point results in the context of partial -metric spaces.
2. Main Results
We start with the following.
Definition 7. Let be a complete -partial metric space with coefficient and . If there are, , , and , so that for all with , where either or then is called a generalized -contraction.
Theorem 8. Let be a complete -partial metric space (with ) and . Assume that (a) is a generalized -contraction(b) is triangular weak -admissible(c)There is such that (d) is continuous
Then, possesses a unique fixed point.
Proof. Let be in order that . Realize a sequence in in order that for all . If for some , , the is a fixed point. Now, suppose that for all . By using assumption (b), the triangular weak -admissibility and Lemma 4, we get that for all integers ,
In particular, for all .
Case 1. Here, we consider . We need three steps.
Step 1. In view of assumption (a), one writes for all , where
If for some , then we get
This implicates that either or , so ; hence, . It is a contradiction. Thus. for all . By (5), we have
Since is nondecreasing, we get for all . Thus, is nonincreasing. Thus, there is in order that . We claim that . We have
At the limit, we acquire that
This infers that either , or . That is, , i.e.,
To prove that is a Cauchy sequence in the -partial metric space , we argue by contradiction. For this, suppose that there exist and subsequences and of positive integers with such that
Using and applying (15) and (16), one easily gets
In view of assumption (a), we have where
Letting and having in mind properties of , and , we deduce that
That is, so either or . Thus, , which is, a contradiction. Hence, is a Cauchy sequence.
Step 2. We claim that has a fixed point. The sequence is Cauchy in the complete -partial metric space, so there is in order that
For each , we have
By assumption (d) and Definition 5, as . Using (25), we deduce that . We will show that . Suppose that , then we obtain that . This implies that where
So, Hence, . We can write
Then,
Thus, or . Hence, . It is a contradiction. Consequently, has a fixed point.
Step 3. We demonstrate that the fixed point of is unique.
Given so that and . From (5),
This infers that either or . This implies that , so .
Case 1. We consider
Again, in view of assumption (a), one writes
for all , where
Then,
We deduce that
That is,
Let Then,
This emphasizes that is a bounded decreasing sequence and bounded below, so it must be a convergent sequence.
For each , we must have,
Again, for all , we get
Therefore, is a bounded sequence in . By Bolzano Weierstrass theorem that every bounded sequence of real numbers has a convergent subsequence, must have a convergent subsequence, like which converges to some So,
This confirms that the subsequence itself converges to . So, the sequence must converge to . Hence, for all ,
as
This concludes that is a Cauchy sequence. As the subsequence of converges to , the limit of must be . Also, we have
This infers that . Hence, is a fixed point of .
Next, we assay the uniqueness of . Debating by contradiction, let be another fixed point of , then . We have
Thus, , which is a contradiction. Then, .
Now, the continuity of in Theorem 8 has been dropped.
Theorem 9. Let be a complete -partial metric space (with ) and . Assume that: (1) is a generalized -contraction(2) is triangular weak -admissible(3)There is in order that (4)If is a sequence in in order that and as , then there is in order that
Then, possesses a unique fixed point.
Proof. As in the proof of Theorem 8, we build the sequence by for all , which converges to in order that for all . By condition (4), there is a subsequence of in order that for all . Consequently,
for all , where
Case 1.
By letting , then
Case 2.
By letting , then
By using (44), we get
which infers that Hence, .
Following Theorem 8 and Theorem 9, we have
Corollary 10. Let be a complete -partial metric space (with ). Let be a self-mapping on . Given . Assume that (1)for all with and , we have (2) is triangular weak -admissible(3)There is in order that (4) is continuous
Then, admits a unique fixed point.
Proof. We consider as .
Corollary 11. Let be a complete -partial metric space (with ). Let be a self-mapping on . Given . Assume that (1)for all with and , we have (2) is triangular -admissible(3)There is in order that (4)If is a sequence in in order that and as , then there is in order that
Then, possesses a unique fixed point.
Proof. Take in Theorem 9.
Now, by taking where is continuous, we have
Corollary 12. Let be a complete -partial metric space (with ). Assume is a self-mapping on . Given . Assume that for all with and where and is continuous. If (1) is triangular weak -admissible(2)There is in order that (3) is continuousthen possesses a unique fixed point.
Corollary 13. Let be a complete -partial metric space (with ) and be a self-mapping on . Given . Assume that where and is continuous, for all with and . If (1) is triangular weak -admissible(2)There is in order that (3) is a sequence in so that and as , then there is of in order that then possesses a unique fixed point.
3. Consequences
Let be the family of nonnegative functions defined on so that: (1) is Lebesgue-integrable on every compact of (2)for every
Theorem 14. Let be a complete -partial metric space (with ) and be a self-mapping on . Suppose there are , , and such for all with and , we get
Also, suppose that (1) is triangular weak -admissible(2)There is in order that (3) is continuous
Then, possesses a unique fixed point.
Proof. Take and . By using Theorem 8, possesses a fixed point.
Theorem 15. Let be a complete -partial metric space (with ) and . Suppose there are , , and such for all with and , we have Assume that: (1) is triangular weak -admissible(2)There is in order that (3)If is a sequence in in order that and as , then there is in order that
Then, possesses a unique fixed point.
Proof. Consider and in Theorem 9.
Example 1. Given and as for all . Choose and . Consider . Given as
Put for all . We are going to demonstrate the following: (1) is a complete partial -metric space(2) is triangular weak -admissible(3)There is in order that (4)If is a sequence in in order that and as
Proof. Note that is a complete partial -metric space (Here, ).
If , then , So , then . So and so . Hence, .
Recall that for all . In this case,
All hypotheses of Corollary 11 hold, and so, admits a unique fixed point.
4. Conclusion
We ensured the existence of a unique fixed point for generalized contraction type mappings involving triangular weak -admissibility with regard to a function and -class functions in the class of partial -metric spaces. Some illustrated examples have been also provided.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no competing interests regarding the publication of this paper.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgments
The authors would relish to acknowledge the grant: UKM grant DIP-2014-034; FRGS/1/2014/ST06/UKM/01/1 for financial bolster.