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Unique Fixed Point Theorems for Generalized Contractive Mappings in Partial Metric Spaces
We establish some unique fixed point theorems in complete partial metric spaces for generalized weakly -contractive mappings, containing two altering distance functions under certain assumptions. Also, we discuss some examples in support of our main results.
1. Introduction and Preliminaries
An abstract metric space was first introduced and studied by the French mathematician Frechet  in 1906. Many researchers have generalized the concept of metric space as cone metric space, semimetric space, quasimetric space, and so forth, along with the generalization of contraction mappings with applications (see [2–7]). The best approximations of functions in locally convex spaces were discussed by Mishra et al.  and Mishra . The degree of approximation of signals in Lp-space is established in .
Matthews [11, 12] initiated the concept of partial metric space as another generalization of metric space to study the denotational semantics of dataflow networks. Also, Matthews  generalized the Banach contraction principle to the class of partial metric spaces as follows: let be a complete partial metric space, and then a self-mapping on , satisfying where , has a unique fixed point.
After the Matthews  historical contribution, several researchers have established some more fixed point theorems in partial metric spaces and also discussed its topological properties (see [13–15] and references therein).
First, we recall some useful definitions and results, which is useful throughout the paper.
Definition 1 (see [11, 12]). Let be a nonempty set, and a mapping satisfying the following conditions is called a partial metric space on :, ,,,for all , and the pair is called a partial metric space. In the rest of the paper, represents a partial metric space equipped with a partial metric , unless otherwise stated. Let be a partial metric space and then let a function be defined as which is a metric on . Consider the function such that then is a metric on , and both of the above metrics and are equivalent .
Remark 2 (see ). In a partial metric space , (1) but if , then may not be zero,(2) for all , for all .
Example 3 (see ). Consider a mapping such that for all . Then will satisfy all the property of partial metric, and hence is a partial metric space but fails to be the condition of for all nonzero . Therefore is not a metric space.
Example 4 (see [16, 18]). Let be three mappings and for any arbitrary mapping such that , , ,for all , where and are a metric space and a partial metric space, respectively. Then each is the partial metric on .
Definition 5 (see ). In a partial metric space , (1) a sequence is said to be convergent to a point if and only if .
(2) A sequence is called Cauchy sequence if and only if is finite.
(3) If every Cauchy sequence converges to a point such that then is known as complete partial metric space.
The generalization of contractive mappings into -contractive mappings has been introduced by Chatterjea .
Shukla and Tiwari  have introduced the concept of weakly -contractive mappings.
Definition 8 (see ). A self-mapping on a complete metric space is said to be weakly -contractive mapping or a weak -contraction, if the following inequality holds: for all and is a continuous function with if and only if .
Lemma 10 (see ). If is not a Cauchy sequence in and two sequences and of positive integers such that , then the four sequences tend to , when .
If is a Cauchy sequence in the metric space , we haveand therefore, by definition of , we have
2. Main Results
Theorem 12. Let be a complete partial metric space and and be two altering distance functions such that . Then the self-continuous nondecreasing mapping on , satisfying the conditionfor all and is a continuous function such that if and only if , has a unique fixed point in .
Proof. First we prove that if fixed point of exists, then it will be unique. On the contrary, we consider two fixed points of such that . Then by (12), we have By the property of , we obtain Using Remark 2, we obtain , which is a contradiction with respect to . Thus, we conclude that has a unique fixed point in .
Next, we show that the mappings , satisfying (12), have a fixed point. We choose an arbitrary point in . If , then the theorem follows trivially. Now, we suppose that and we choose such that . Since is a nondecreasing function, then we have . Again, let . Then we get Proceeding with this work, we obtained a sequence in such that and Supposing that for some , then by Remark 2 we have Again, we suppose that . Firstly, we prove that the sequence is nonincreasing. Suppose this is not true, and thenPutting and in (12) and using , we have Using above, we get Using (18) above, we have which contradicts our assumption that for all . Thus, we deduce that is a nonincreasing sequence. ThereforeSince is a monotonically decreasing and bounded below sequence in , then there exists such thatUsing (23) and letting in (20), we get Then (23) reduces toNow, we have required proving that the sequence is a Cauchy sequence in the metric space and so in by Lemma 11. On the contrary, that is, the sequence not being a Cauchy sequence in , sequences in Lemma 10 tend to , when . Now, we put and in (12). We have Taking and applying Lemma 10 in the above inequality, we have which is a contradiction with respect to . Thus is a Cauchy sequence in and so in . Since is complete, is also complete (by Lemma 11). Therefore, the Cauchy sequence converges in ; that is, ; then by Lemma 11, we haveBy Lemma 11, we get . So, by definition of , we get Using (24) and taking in the above inequality, we obtainFrom (28) and (30), we getBy , we obtain Taking and using (31), (24), and Lemma 9 in the above inequality, we haveFrom , we haveBy (33) and (34), we getFrom (35) and (12), we obtain Using (31) and property of in the above inequality, we obtain Thus, is a unique fixed point of in .
Example 13. Let be a complete partial metric space defined by ∀. Consider a self-map on such that . Also, we define such that , , respectively, and such that .
Corollary 14. Let be a complete partial metric space. Then the self-continuous nondecreasing mapping on , satisfying the conditionfor all and and which are the same as in Theorem 12, has a unique fixed point in .
Theorem 18. Let be a complete partial metric space and and be two altering distance functions such that . Then the two self-continuous nondecreasing mappings and on , satisfying the conditionfor all and is a continuous function such that if and only if , having a unique common fixed point in .
Proof. First, we show that the common fixed point of and is unique, if it exists. On the contrary, we assume two common fixed points of and such that . Then by (41), we get Property of implies that which contradicts our assumption that . Therefore, we conclude that and have a unique common fixed point in .
Now, we prove that the mappings and , satisfying (41), have a common fixed point in . We choose an arbitrary point in . If and , then theorem follows trivially. So, we suppose that and . Then we construct a sequence in , in such a way that and .
Let us assume that and . Then, we can prove that and have a common fixed point in . Firstly, we show that is nonincreasing sequence. Suppose this is not true, and thenPutting and in (41) and using , we get Using above, we get By (44) and (46), we obtain which is a contradiction with respect to and . Therefore is a nonincreasing sequence in . Thus, we haveSince is a monotonically decreasing sequence in , then there exists such thatLetting in (46) and using (49), consequently we get Then (49) will get reduced toNow, we have to show that is a Cauchy sequence in the partial metric space . By similar arguments as used in case of proving Theorem 12 we find that the sequence is a Cauchy sequence. Putting and in (41), we have Taking and using Lemma 10 in the above inequality, we obtain which contradicts our assumption that . Thus is a Cauchy sequence in and so in . Further, by similar arguments of Theorem 12, we obtainBy substituting , in (41), we obtainLetting and using (54) with property of nondecreasing function in the above inequality, we obtain Hence is a fixed point of . Similarly, if we take and in (41) and use (54), we obtain . By uniqueness of the fixed point, is a unique common fixed point of and .
Again, if or , then we will show that the mappings and have a common fixed point in .
Here, we suppose that ∀. Then by Remark 2, , for all . Let , and thenFrom (41), we get Using , , and (57) above, we obtain Similarly, we can show that Thus becomes a constant sequence. So for all . Hence is a common fixed point of and .
Finally, we assume that . Then by Remark 2, we have . Let , and thenUsing (58), (61), and with property of nondecreasing function , we have Using similar property of , as used in first case, we have Thus, becomes a constant sequence. So . Hence is a common fixed point of and .
Example 19. Let , , , , and all be the same as in Example 13 and a self-mapping on defined as . Then is a unique common fixed point of and . One can compute the solution similarly as done in Example 13.
Corollary 20. Two self-continuous nondecreasing mappings and on a complete partial metric space , satisfying the conditionfor all and and , are the same as in Theorem 18, having a unique common fixed point in .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to express their deep gratitude to the anonymous learned referee(s) and the editor for their valuable suggestions and constructive comments, which resulted in the subsequent improvement of this research article. Special thanks are due to our great Master and friend academician Professor M. Mursaleen, Editor of the Journal of Function Spaces, for his efforts to send the reports of the paper timely. The authors are also grateful to all the editorial board members and reviewers of esteemed journal, that is, Journal of Function Spaces. The first author Lakshmi Narayan Mishra acknowledges the Ministry of Human Resource Development, New Delhi, India, for supporting this research article. All the authors carried out the proof of theorems. Each author contributed equally in the development of the paper. Vishnu Narayan Mishra conceived of the study and participated in its design and coordination. The second author Shiv Kant Tiwari is grateful to Ms. Jagrati Bilthare for her valuable suggestions during the preparation of this paper. All the authors read and approved the final version of paper.
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