International Journal of Analysis
Volume 2017 (2017), Article ID 8247925, 14 pages
https://doi.org/10.1155/2017/8247925

Research Article

Geraghty Type Generalized -Contractions and Related Applications in Partial -Metric Spaces

Deepak Singh,¹ Varsha Chauhan,² and R. Wangkeeree³

¹Department of Applied Sciences, NITTTR, Ministry of HRD, Government of India, Bhopal 462002, India
²Department of Mathematics, NRI Institute of Research & Technology, Bhopal, India
³Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Correspondence should be addressed to R. Wangkeeree; rabianw@nu.ac.th

Received 31 October 2016; Accepted 6 December 2016; Published 15 March 2017

Academic Editor: Ying Hu

Copyright © 2017 Deepak Singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The purpose of this paper is to introduce new concepts of -admissible Geraghty type generalized -contraction and to prove that some fixed point results for such mappings are in the perspective of partial -metric space. As an application, we inaugurate new fixed point results for Geraghty type generalized graphic -contraction defined on partial metric space endowed with a directed graph. On the other hand, one more application to the existence and uniqueness of a solution for the first-order periodic boundary value problem is also provided. Our findings encompass various generalizations of the Banach contraction principle on metric space, partial metric space, and partial -metric space. Moreover, some examples are presented to illustrate the usability of the new theory.

1. Introduction and Preliminaries

There are a lots of extensions and generalizations of Banach contraction principle [1]. In 1973, Geraghty [2] introduced an interesting contraction and investigated the existence and uniqueness of such mappings in the setting of complete metric spaces by considering an auxiliary function. Later, the -metric concept was launched by Bakhtin [3] as a generalization of a metric to extend the celebrated Banach contraction principle, which was extensively extended by Czerwik [4, 5]. On the other hand, in 1994, Matthews [6] introduced the concept of partial metric spaces wherein the distance of a point from itself may not be zero. Later on, in [7], Karapinar et al. implanted fixed point theorems involving rational expressions in the frame of partial metric space. In 2012, Samet et al. [8] introduced the concept of -contractive and -admissible mappings and established various fixed point results for such type of mappings in the context of complete metric spaces. In 2015, Chandok [9] generalized the concept of -admissible mappings by introducing -admissible mappings.

In recent times, Shukla [10] generalized the concept of both -metric and partial metric space by presenting the partial -metric space. After that, in [11], Mustafa et al. introduced a modified version of partial -metric space. On the other hand, in 2012, Wardowski [12] introduced a new contraction called -contraction and proved a fixed point result as a generalization of the Banach contraction principle. After this, Secelean [13] described a large class of functions by replacing condition instead of condition in the definition of -contraction presented by Wardowski [12]. Very recently, Piri and Kumam [14] improved the result of Secelean [13] by replacing condition instead of condition .

In this work, we introduce the notion of -admissible Geraghty type generalized -contraction and set up some fixed point results concerning such contractions. Moreover, some examples and applications are presented in support of our results. Our proposed definitions and related applications are different as given in [15]. In the sequel, , , and will represent the set of all real numbers, natural numbers, and positive integers, respectively. Some useful definitions and auxiliary results, which will be required in the sequel, are recollected here.

Definition 1 (see [4]). Let be a nonempty set and be a given real number. A function is called a -metric if for all the following conditions are satisfied: iff ...The pair is called a -metric space. The number is called the coefficient of .

Definition 2 (see [6]). A partial metric on a nonempty set is a function such that for all iff ;;;.A partial metric space is a pair such that is a nonempty set and is a partial metric on .

Definition 3 (see [10]). Let be a nonempty set and be a given real number. A function is called a partial -metric if for all the following conditions are satisfied: iff ....The pair is called a partial -metric space. The number is called the coefficient of .

In the following definition, Mustafa et al. [11] modified Definition 3 in order to find whether each partial -metric generates a -metric .

Definition 4 (see [11]). Let be a nonempty set and be a given real number. A function is called a partial -metric if for all the following conditions are satisfied: iff ....The pair is called a partial -metric space. The number is called the coefficient of .

Example 5 (see [10]). Let , be a constant and be defined byfor all Then, is a partial -metric space with the coefficient , but it is neither a -metric nor a partial metric space.

Remark 6. The class of partial -metric space is effectively larger than the class of partial metric space, since a partial metric space is a special case of a partial -metric space when . Also, the class of partial -metric space is effectively larger than the class of -metric space, since a -metric space is a special case of a partial -metric space when the self-distance .

Proposition 7 (see [10]). Let be a nonempty set and let be a partial metric and be a -metric with the coefficient on . Then, the function , defined by for all , is a partial -metric on with the coefficient .

Proposition 8 (see [10]). Let be a partial metric space and . Then, is a partial -metric space with the coefficient , where is defined by .

Proposition 9 (see [11]). Every partial -metric defines a -metric , where

Definition 10 (see [11]). A sequence in a partial -metric space is said to be (1)-convergent to a point if ;(2)a -Cauchy sequence if exists (and is finite).A partial -metric space is said to be -complete if every -Cauchy sequence in   -converges to a point , such that

Lemma 11 (see [11]). Let be a partial -metric space. Then, (1)a sequence is a -Cauchy sequence in if and only if it is a -Cauchy sequence in the -metric space ;(2) is -complete if and only if the -metric space is complete. Moreover, if and only if

Definition 12 (see [2]). Let denote the class of the functions which satisfy the condition

Definition 13 (see [16]). Let be a self-mapping on and be a function. One says that is an -admissible mapping if

Definition 14 (see [9]). Let be a nonempty set,  , and . One says that is -admissible if and imply and , for all .

Let be the set of functions such that(1) is nondecreasing;(2) is continuous;(3)

On the other hand, Wardowski [12] introduced the -contraction as follows.

Definition 15. Let be a mapping satisfying the following: is strictly increasing, that is, for such that implies .For each sequence of positive numbers, if and only if .There exists such that .

We denote the set of all functions satisfying ()–() by . In [13], Secelean replaced condition by an equivalent but a more simple condition .,  or also by the following.There exists a sequence of positive real numbers such that . Most recently, Piri and Kumam [14] used the following condition instead of . is continuous on . We denote the set of all functions satisfying , , and by .

We introduce the following function.

Definition 16. Let be the set of all continuous functions satisfying the following: for all , if for , where , then there exists such that .

2. Main Results

2.1. Fixed Point Results for -Admissible Geraghty Type Generalized -Contraction

We start this section by introducing the following definition.

Definition 17. Let be a partial -metric space and be a self-mapping. Also, suppose that , where the pair is defined as in Definition 14. One says that is an -admissible Geraghty type generalized -contraction on a partial -metric space , if there exist ,  ,  , and such that, for all and with , whereand is a constant.

Our main result of this paper is the following one.

Theorem 18. Let be a complete partial -metric space. Let be a self-mapping on satisfying the following conditions: (1) is -admissible.(2)There exists such that and .(3) is an -admissible Geraghty type generalized -contraction on .(4) is continuous.Then, has a unique fixed point ; moreover, .

Proof. Let such that and . Define a sequence in by for all . If for any , then is a fixed point of . Consequently, assume that for all .
Since is an -admissible mapping, it follows from (2) that ,  . By induction, we getSimilarly, for all
By taking and in (5) and due to (), property of and , we arrive atwhereIf , for all , from (8) and by the definition of functions and , we deduce that a contradiction, since . Thus, it follows that . Again, from (8) and by the definition of functions and , we havewhich givesHence, is a decreasing sequence of positive real numbers. Repeated use of (11) givesSince , letting the limit as in (13), we getMoreover, from (), we have the following:Now, we will prove that is a -Cauchy sequence in . From Lemma 11, we need to prove that is a -Cauchy sequence in the -metric space . Suppose on the contrary that there exists such that for an integer there exists integer such thatFor every integer , let be the least positive integer exceeding satisfying (16) such thatDue to triangle inequality and from (16), we getwhich on making and using (17) give rise toAlso, from (17) and (19), we havewhich givesFurthermore,which yieldsUtilizing Proposition 9, we haveThen, from the above inequality along with (19), we acquireAnalogously, we deduce thatSince , therefore, due to inequality (5), we haveUtilizing the definition of and along with inequalities (25)-(26) gives rise toIndeed, so thatBy repeating the above technique, one can easily arrive at From (27) together with (28), we have which implies thatwhich yieldsUtilizing the definition of and , we obtaina contradiction. Thus, we have proved that is a -Cauchy sequence in the -metric space ; then, from Lemma 11,   is a -Cauchy sequence in the partial -metric space As is complete, by Lemma 11,  -metric space is -complete. Therefore, the sequence converges to some point ; that is, Again, from Lemma 11,Next, we will show that is the fixed point of reasoning by contradiction. Suppose that ; inequality (5) implies thatin whichBy repeating the same process as mentioned above, we haveMaking limit in (37) and due to inequalities (38) and (39), property of function together with continuity of , this gives rise to Using the definitions of and , the above inequality turns intowhich implieswhich is impossible. This contradiction proves that ; that is, . Hence, we assert that is a fixed point of .
To prove the uniqueness of the fixed point , let be another fixed point of ; that is, , such that From (5), we obtain that whereTherefore, using the definition of and along with the value of and , the above inequality turns into the following: which gives a contradiction. Hence, ; that is, . Thus, we conclude that the fixed point of is unique. Next, we will prove that . If , then, from (5) and by applying the routine calculation as mentioned above, we can get a contradiction. Thus, .
This completes the proof of the theorem.

Now, we present an example which shows the superiority of our assertion.

Example 19. Let be equipped with the partial order relation defined byand the function is defined byfor all , where . It is obvious that is a complete partial -metric space. Let the mapping be defined by And we define the mapping by By the definition of , it is clear that and . Also, there exists in such that and .
Define by . And let be given by . Let for all .
Without loss of generality, we may take such that . In order to check the contractive condition (5) of Theorem 18, we have to consider the following cases.
Case 1. If , thenFor RHS, utilizing the definitions of and , one can easily verify that and Hence,for all and using the above hypotheses that . Figures 1 and 2 demonstrate that the RHS expression (with green curve) dominates the LHS expression (with pink curve) for , which validates our inequality.
From Figures 1 and 2, it is easy to find that condition (5) holds for all with and .
Case 2. If , then . From (5), we arrive at Figures 3 and 4 show that the RHS expression (with green curve) overshadows the LHS expression (with pink curve), which authenticates our inequality.
From Figures 3 and 4, it is clear that condition (5) holds for all .
Case 3. If and , then Case is similar to Case , and therefore we skip the details.
Thus, all the conditions of Theorem 18 are fulfilled and is a unique fixed point of the involved mapping (see Figure 5).

Figure 1: Plot of inequality for Case with and .

Figure 2: Plot of inequality for Case with and .

Figure 3: Plot of inequality for Case with .

Figure 4: Plot of inequality for Case with .

Figure 5: Fixed point of the mapping .

Remark 20. If in Definition 17, then we say that is -admissible Geraghty type generalized -contraction on .

The following example highlights the above remark.

Example 21. Let and mapping is defined byWe define the mappings by It is easy to observe that is an -admissible but not a -admissible mapping.

Remark 22. If in Definition 17, then we say that is -admissible Geraghty type generalized -contraction on .

The following example validates the aforesaid remark.

Example 23. Let and mapping is given byWe define the mappings by One can verify that is a -admissible but not an -admissible mapping.

Remark 24. If in Definition 17, then we say that is Geraghty type generalized -contraction on .

Remark 25. If in Definition 17, then we say that is -admissible Geraghty type -contraction on .

2.2. Fixed Point Results for Geraghty Type Generalized -Contraction with a Partial Order

Definition 26. Let be a partially ordered set and be a given mapping. One says that is nondecreasing with respect to if

Theorem 27. Let be a complete ordered partial -metric space. Let be a nondecreasing mapping with respect to . Suppose that the following conditions hold: (1)There exists such that .(2) is a Geraghty type generalized -contraction on .(3) is continuous.Then, has a unique fixed point ; moreover, .

Proof. Let be defined by Due to condition (1), we have and . Using the nondecreasing property of , we arrive atSimilarly, we obtain that .
Thus, is -admissible. Hence, it is easy to conclude that is -admissible Geraghty type generalized -contraction. It follows from Theorem 18 that has a unique fixed point if it is continuous.

3. Some Consequences

From Theorem 18, if , we deduce the following theorem.

Theorem 28. Let be a complete partial metric space. Let be a self-mapping on satisfying the following conditions: (1) is -admissible.(2)There exists such that and .(3)There exist , , , and such that, for all with ,  where  and is defined as in Definition 16.(4) is continuous.Then, has a unique fixed point ; moreover, .

The following theorem is an another version of our main result.

Theorem 29. Let be a complete partial -metric space. Let be a self-mapping on satisfying the following conditions: (1) is -admissible.(2)There exists such that and .(3)There exist , , , and such that, for all and with , one has where and are defined as in Definition 17.(4) is continuous.Then, has a unique fixed point ; moreover, .

Proof. The proof of this theorem can be completed on the lines of the proof of Theorem 18, and hence we skip the details.

If and , in Theorem 29, then we have the following corollary.

Corollary 30. Let be a complete partial -metric space. Let be a self-mapping on satisfying the following conditions: (1) is -admissible.(2)There exists such that and .(3)There exist , , and such that, for all and with , one has (4) is continuous.Then, has a unique fixed point ; moreover, .

By using ,   for all , in Theorem 18, we get the following one.

Corollary 31. Let be a complete partial -metric space with . Let be a self-mapping on . There exist , , , and such that, for all with ,where and are defined as in Theorem 18 and . Then, has a unique fixed point in ; moreover, .

Remark 32. In view of Remark 6, Corollary 31 generalizes the result of Wardowski [12] for partial metric space and partial -metric space along with Geraghty type contraction.

In [12], the author guaranteed that the -contraction is the updated version of Banach contraction principle. Wardowski concluded that the Banach contractions are a particular case of -contractions and the author supported his finding by presenting some -contractions which are not Banach contractions.

In view of the aforementioned, we generalized and extended the following results existing in the literature.

Remark 33. Take and , where in Theorem 28 is akin to Theorem of Matthews [6] in the sense of -contraction.

Remark 34. By introducing Theorem 18, we generalized the results of Chandok [9] and obtained the -contraction version of [9] in partial -metric spaces.

Remark 35. Taking and in Theorem 28 is akin to Theorem of Altun and Sadarangani [17] for -contraction.

Remark 36. By taking ,  , where along with zero self-distance in Theorem 28, we extend the Banach contraction principle of Banach [1] in the sense of -contraction.

4. Applications

4.1. Fixed Point Results for Graphic Contractions

Following Jachymski [18], let be a partial metric space and let denote the diagonal of the cartesian product of . Consider a directed graph , where denotes the set of its vertices coinciding with and denotes the set of its edges containing all loops, which gives that is the superset of . Suppose that has no parallel edges. 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 called connected if there is a path between any two vertices (for more details, see [16, 19]).

Definition 37 (see [18]). A mapping is called -continuous if given and sequence such that as , , for all , imply

Definition 38. 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:

Next, we introduce the following definition.

Definition 39. Let be a partial metric space endowed with a graph and is a self-mapping. One says that is Geraghty type generalized graphic -contraction on . There exist , , , and such that, for all with , one haswhere and is defined as in Definition 16.

Theorem 40. Let be a complete partial metric space endowed with a graph . is self-mapping on satisfying the following conditions: (1)For all ,  .(2)There exists such that .(3) is -continuous on .(4) is Geraghty type generalized graphic -contraction on .Then, has a fixed point .

Proof. Define by At first, we will prove that is an -admissible mapping. Let such that ; then, by the definition of , we obtain that and from (1) we get . Again, from the definition of , we have .
Similarly, .
Thus, is an -admissible mapping. From (2), there exists such that , which yields and .
From (4), is Geraghty type generalized graphic -contraction on and ,  , implies Hence, all the conditions of Theorem 28 are satisfied and has a fixed point. This concludes the proof.

The following example illustrates Theorem 40 involving a directed graph.

Example 41. Let be endowed with the function defined by Then, is a complete partial metric space. Consider as follows: Define by . And let be given by . Let for all .
Let be a directed graph such that and
It is easy to deduce that preserves edges in and is -continuous. Also, there exists such that Without loss of generality, we may take such that . To prove that the contractive condition (68) of Theorem 40 holds, consider the following cases.
Case 1. If such that , then we have to consider the following possibilities.
Case 1(i). If and , we get in which and
Case 1(ii). If and , we have in which and
Case 1(iii). If and , we obtain in which and
Case 2. If such that , then inequality (68) turns intoin which and .
Case 3. If such that and , then we have to consider the following six subcases.
Case 3(i). If and , we get Indeed, and
Case 3(ii). If and , we getIndeed, and
Case 3(iii). If and , we get Indeed, and
Case 3(iv). If and , we get Indeed, and
Case 3(v). If and , we get Case 3(vi). If and , we get Indeed, and
Figure 6 represents the graph with all the possible cases.
Hence, all the conditions of Theorem 40 are satisfied and Theorem 40 ensures the existence of fixed point of , which is .

Figure 6: The graph defined in Example 41.

It needs to be mentioned that the following definition will be useful in the following subsection.

Definition 42. Let denote the class of those functions which satisfies the following conditions: (i) is continuous, subadditive, and nondecreasing.(ii)For each ,  .(iii).(iv) is positive in with For example, and are in .

4.2. Application to Solutions of Ordinary Differential Equations

Consider the following first-order periodic boundary value problem:where and is a continuous function.

Consider the space of all real continuous functions on Let be given by for all , where and . Obviously, the space is a complete partial -metric space with parameter . The space can also be equipped with the following order relation:

Definition 43. An element is called a lower solution for problem (84) if

Theorem 44. Consider problem (84) and assume the following: (1)There exists and functions such that, for any , with and , for all with and (2)there exists such that and for all , where the function is defined by(3)For each and , one hasThen, the existence of a lower solution for the periodic boundary value problem provides a unique solution for (84).

Proof. Problem (84) is equivalent to the integral equationin whichIt is easy to note that if is a fixed point of , then is a solution of problem (84).
For any such that and , where , we have which yieldswhereConsequently, by passing to logarithms, the above inequality turns intoThis givesfor ,  . Define by for all . Hence,Now, using condition (2) and in account of the definition of and , we haveFrom condition (3), we getThus, is an -admissible mapping. Finally, let be a lower solution for (84). Then, from [20], we have . Hence, all the hypotheses of Corollary 30 are satisfied and thus the mapping has a fixed point that is a solution in of problem (84).

5. Open Problems for Future Reading

(i)In Theorem 18, can -admissible Geraghty type generalized -contraction be improved by cyclic contraction?(ii)Can Theorem 18 be extended and generalized by replacing Geraghty contraction by hybrid rational Geraghty contraction?

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

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