Abstract
We study the sufficient conditions for the existence of a unique common fixed point of generalized --Geraghty contractions in an -complete partial -metric space. We give an example in support of our findings. Our work generalizes many existing results in the literature. As an application of our findings we demonstrate the existence of the solution of the system of elliptic boundary value problems.
1. Introduction and Preliminaries
The Banach contraction mapping principle is very important in modern mathematics. For decades, several authors have studied existence of fixed points by contraction mappings, such as fuzzy mappings and others, and also get some important results, for details we can see [1–5, 5–12]. Now the theory of fixed point has been applied in many applied mathematics [13, 14] besides integral equations and differential equations [15]. For decades, people have done a lot of research on this issue and got a lot of important results [16–20].
As is well known, the existence of the solution of boundary value problems is an important of differential equations. In this paper we study the sufficient conditions for the existence of a unique common fixed point of generalized --Geraghty contractions in an -complete partial -metric space. As an application of our findings we demonstrate the existence of the solution of the system of elliptic boundary value problems.
We first give some conceptions of this paper. In 1973, Geraghty studied a generalization of Banach contraction principle. In 2013, Cho introduced the notion of -Geraghty contractive type mappings and established some unique fixed point theorems for such mappings in complete metric spaces. Popescu defined the concept of triangular -orbital admissible mappings and proved the unique fixed point theorems for the mentioned mappings which are generalized -Geraghty contraction type mappings. On the other hand, Karapinar proved the existence of a unique fixed point theorem for a triangular -admissible mapping which is a generalized --Geraghty contraction type mapping. Shukla [21] introduced the concept of partial -metric space and established some fixed point theorems. We have Figure 1 where arrows stand for inclusions. The inverse inclusions do not hold.
In this paper, we introduce the notion of generalized --Geraghty contraction type mappings and develop some new common fixed point theorems for such mappings in an -complete partial -metric space. An example and an application are given to support the theory.
We denote the set of natural numbers, rational numbers, , , and by , , , , and , respectively.
First we recall some definitions and properties of a partial -metric space.
Shukla generalized the notion of -metric, as follows.
Definition 1 (see [21]). Let be a nonempty set and be a real number. A mapping is said to be a partial -metric if it satisfies following axioms, for all , if and only if ; The triplet is called a partial -metric space.
Remark 1. The self-distance , referred to the size or weight of , is a feature used to describe the amount of information contained in .
Remark 2. Obviously, every partial metric space is a partial -metric space with coefficient and every -metric space is a partial -metric space with zero self-distance. However, the converse of this fact needs not to hold.
Example 3. Let and ; the mapping defined byis a partial -metric on with . For , , so is not a -metric on .
Let such that . Then following inequality always holds:Since and ,This shows that is not a partial metric on .
Example 4 (see [21]). Let be a nonempty set and be a partial metric defined on . The mapping defined bydefines a partial -metric.
Definition 5. Let be a partial -metric space. The mapping defined bydefines a metric on , called induced metric.
In partial -metric space , we immediately have a natural definition for the open balls:
Remark 6. The open balls in a partial -metric space may not be open set.
The following example justifies Remark 6.
Example 7. Let and define as follows: , , , , . Then is a partial -metric, but, for any , does not lie in . This implies that is not an open set in .
Definition 8. Let be a partial -metric space. (1)A sequence in is called a Cauchy sequence if exists and is finite.(2)A partial -metric space is said to be complete if every Cauchy sequence in converges with respect to topology induced by its convergence, to a point such that
Lemma 9 (see [21]). Let be a partial -metric space. Then (1)every Cauchy sequence in is also a Cauchy sequence in and vice versa;(2)a partial -metric is complete if and only if the metric space is complete;(3)a sequence in converges to a point if and only if
Remark 10. We know that in a metric space limit of a convergent sequence is always unique but in a partial -metric space the limit of a convergent sequence may not be unique. Indeed, if , let be any constant. Define by for all , then is a partial -metric space with arbitrary coefficient . Define the sequence in by for all . One can note that if then ; thus for all . Hence, the limit of a convergent sequence is not unique.
Definition 11 (see [22]). Let and be two mappings. We say that is -admissible if implies
Definition 12 (see [22]). Let and be two mappings. Then is said to be triangular -admissible if satisfies the following conditions: is -admissible. and imply
Definition 13 (see [23]). Let and be two mappings. Then is said to be -orbital admissible if implies
Definition 14 (see [23]). Let and be two mappings. Then is said to be triangular -orbital admissible if is -orbital admissible and and imply
Let denote the class of all mappings such that, for any bounded sequence of positive reals, implies
We let denote the class of the functions satisfying the following conditions:(1) is nondecreasing.(2) is continuous.(3) if and only if
2. Main Results
Throughout this paper we let be a partial -metric space, be a mapping, and
Definition 15. The space is said to be -complete if every Cauchy sequence in satisfying for all converges in .
Remark 16. If is a complete partial -metric space, then is also an -complete partial -metric space but the converse is not true. The following example explains this fact.
Example 17. Let and the partial -metric be defined by , for all . Define by
It is easy to see that is not a complete partial -metric space, but is an -complete partial -metric space. Indeed, if is a Cauchy sequence in such that , for all , then , for all . Since is a closed subset of , we see that is a complete partial -metric space and then there exists such that as .
Definition 18. Let be a partially ordered set. Two mappings are said to be weakly increasing mappings, if and hold for all .
Example 19. Let . Define by Then, are weakly increasing mappings.
Definition 20. The self-mappings are said to be -orbital admissible if the following condition holds.
and imply and
We note that Definitions 13 and 18 are particular cases of Definition 20 (set and define whenever or , respectively, in Definition 20).
Definition 21. Let be two mappings. The pair is said to be triangular -orbital admissible, if(i)the self-mappings are -orbital admissible,(ii), and imply and
Example 22. Let and for all be a partial -metric with :Define byThen it is easy to show that the mappings satisfy conditions (i) and (ii) in Definition 21.
Lemma 23. Let be two mappings such that the pair is triangular -orbital admissible. Assume that there exists such that Define a sequence in by and , where . Then for with , we have .
Proof. Since and are -orbital admissible self-mappings,Applying the above argument repeatedly, we obtain with Since are triangular -orbital admissible mappings,
Definition 24. We say the self-mapping is an --continuous mapping if whenever is a sequence in with for all and such that , then .
Now, we introduce the concept of generalized --Geraghty contractions as follows.
Definition 25. The self-mappings defined on are called generalized --Geraghty contractions with respect to , if there exist , , and such thatfor satisfying .
The main result of this section is given by the following:
Theorem 26. Let be an -complete partial -metric space. Let be generalized --Geraghty contractions satisfying the following conditions: (i)there exists such that ;(ii)the mappings are triangular -orbital admissible;(iii)(a)the mappings are --continuous(b) is a sequence in such that for all and as , then there exists a subsequence of such that for all . Then and have a common fixed point in In addition, if is also a common fixed point of the pair such that , then .
Proof. Firstly we prove that the self-mappings have at most one common fixed point. Suppose that and are two different common fixed points of and . Then . It follows that , and . Since , contractive condition (16) implies which is a contradiction. Hence, the pair has at most one common fixed point.
(a). By assumption (i) and Lemma 23, we have For , we have where If , then by (29) we havewhich is a contradiction. Thus we conclude that By (29), we get that . Since is nondecreasing, we have This implies that Hence, we deduce that the sequence is nonincreasing. Therefore, there exists such that Now, we shall prove that . Suppose that . By (16), we have which implies Hence This implies that Since , we havewhich yieldsa contradiction. Now, we claim that is a Cauchy sequence in . Suppose, on thw contrary, that is not a Cauchy sequence; that is, . Then there exists for which we can find two subsequences and of such that is the smallest index for which , This means that By the triangle inequality, we have Thus,for all . In the view of (33) and (29), we haveAgain by triangle inequality, we have and By (29) and (34), we deduce thatAlso by application of triangle inequality, it follows thatBy Lemma 23, since , we haveSimilarly, we can show thatThus, concluding above arguments we have which is a contradiction. Thus is a Cauchy sequence in . Since is an -complete partial -metric space, by Lemma 9(2), is an -complete -metric space. Therefore, the sequence converges to some . By Lemma 9(3), there exists such that if and only ifSince , thus, considering (29) and axiom with , we conclude thatCombining (42) and (43), we haveNow implies that and . As and are --continuous mappings, we Thus and so , and, similarly, . Therefore and have a common fixed point .
(b). From (a) we know that the sequence in defined by and , where with , for all converges to There exists a subsequence of such that for all . Therefore, which implieswhereSince then by letting we have Suppose that . By (47), we haveLetting in above inequality, we obtain that So Hence , and due to and we obtain so . Similarly we can show that Thus and have a common fixed point .
Remark 27. We note that Theorem 26 is more general than the results established in [24–26].
Example 28. Let . Define a function by . Clearly, is a complete partial -metric space with the constant Let be a function on defined by for all Then Also, be a function on defined by Then Define the mappings by Also, we define the function by If is a Cauchy sequence such that for all , then . Since is a complete partial -metric space, then the sequence converges in Thus is an -complete partial -metric space. Let and , and thus and , and so and Thus, is -orbital admissible. Let be such that , and . Then we have , which implies that and . Therefore is triangular -orbital admissible. Let be a Cauchy sequence such that as and for all Then for all So Hence is -continuous. Similarly, we can show that is -continuous. Let . Then Let such that . Then and hence with . Thus all conditions of Theorem 26 are satisfied. Hence and have a common fixed point
3. Consequences
Corollary 29. Let be an -complete partial -metric space. Assume that (i)there exist and such that, for all with the self-mappings satisfy the following inequality: (ii) are triangular -orbital admissible mappings;(iii)there exists such that (iv)(a) and are -continuous mappings;(b) is a sequence in with for all such that as , then there exists a subsequence of such that for all . Then and have a common fixed point In addition, if is also a common fixed point of the pair such that , then .
Proof. Define by for all .
Corollary 30. Let be an -complete -metric space. Let be a generalized --Geraghty contractions with respect to satisfying the following conditions: (i)There exists such that .(ii)The mappings are triangular -orbital admissible.(iii)(a)The mappings are --continuous.(b) is a sequence in such that for all and as , then there exists a subsequence of such that for all . Then and have a common fixed point in In addition, if is also a common fixed point of the pair such that , then .
Proof. Set for all in Theorem 26.
Corollary 31. Let be a partially ordered set and be an ordered complete partial -metric space. Assume that the weakly increasing mappings satisfy the following conditions: (i)there exist and such that for all comparable ( or );(ii)there exists such that (iii)(a)either or is continuous;(b) is a nondecreasing sequence such that as , then there exists a subsequence of such that for all . Then and have a common fixed point . In addition, if is also a common fixed point of the pair such that , then .
Proof. Define the relation on by Proof follows from the proof of Theorem 26.
Definition 32. The self-mappings defined on is called a generalized --Geraghty contraction if there exist , , and such that for satisfying , where
Corollary 33. Let be an -complete partial -metric space. Let be a generalized --Geraghty contraction satisfying the following conditions: (i)there exists such that ;(ii)the mapping is triangular -orbital admissible;(iii)(a)the mapping is --continuous;(b) is a sequence in such that for all and as , then there exists a subsequence of such that for all . Then has a fixed point in In addition, if is also a common fixed point of the pair such that , then .
Proof. Set in Theorem 26.
We extend Definition 25 for all as follows
Definition 34. The self-mappings defined on are called generalized -Geraghty contractions, if there exist , , and such that for .
Theorem 35. Let be two -continuous generalized -Geraghty contractions defined on a complete partial -metric space ; then and have a common fixed point.
Proof. The arguments follow as the same lines in proof of Theorem 26.
4. Application
In this section, we present an application on existence of a solution of a pair of elliptic boundary value problems. Let be the space of all continuous function defined on . Consider the following pair of differential equations:where are continuous functions. The Green function associated with (62) is defined by Define the function by It is known that is a complete partial -metric space with constant . Now, define the operators defined byfor all Remark that (62) has a solution if and only if operators and have a common fixed point.
Theorem 36. Assume that there exist continuous functions such that, for all and , we have where is defined by (9) such that
Proof. It is well known that is a solution of (62) if and only if is a common solution of the integral equations given by (65). Define the mappings by (65). Hence the solution of (62) is equivalent to find a common fixed point of and . Let . By (i), we get Since for all , then we have , which implies thatIt can easily be proved that . Thus, Define the functions and byNote that is continuous, nondecreasing, positive in , , and
Hence andfor all . Therefore all assumptions of Theorem 35 are satisfied with Hence and have a common fixed point ; that is, which is a solution of (62).
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The research is supported by the Scientific Research Foundation of Education Bureau of Hebei Province (Grant no. QN2016191) and by Doctoral Fund of Hebei University of Architecture (Grant no. B201801).