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Jamal, Noor; Abdeljawad, Thabet; Sarwar, Muhammad; Mlaiki, Nabil; Sumati Kumari, Panda

doi:https://doi.org/10.1155/2020/9307302

International Journal of Mathematics and Mathematical Sciences

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 9307302 | https://doi.org/10.1155/2020/9307302

Some Valid Generalizations of Boyd and Wong Inequality and -Weak Contraction in Partially Ordered Metric Spaces

Noor Jamal,¹Thabet Abdeljawad,^2,3,4Muhammad Sarwar,¹Nabil Mlaiki,²and Panda Sumati Kumari⁵

Academic Editor: Aloys Krieg

Received21 Mar 2020

Revised29 May 2020

Accepted15 Jun 2020

Published13 Jul 2020

Abstract

In this manuscript, we use -weak contraction to generalize coincidence point results which are established in the context of partially ordered b-metric spaces. The presented work explicitly generalized some recent results from the existing literature. Examples are also provided to show the authenticity of the established work.

1. Introduction

Banach contraction principle [1] is one of the most important result in analysis. It is widely considered as the source of fixed point theory. Banach principle has been modified by many researchers by changing either the contractive condition or the underlying spaces. Boyd and Wong [2] modified the contraction condition in metric spaces with control function

The -weak contraction condition was initiated by Alber and Guerre-Delabriere in the Hilbert spaces, see [3]. Rhoudes, in [4], showed that every -weak contractive mapping in metric spaces has a unique fixed point. The generalized -weak contractive condition in complete metric spaces was introduced by Zhang and Song in [5]. In [6], Dutta and Choudhury studied unique fixed point results with the help of -weak contraction in metric spaces. Doric, in [7], studied the result of [6] with generalized -weak contraction in metric spaces. The concept of fixed point results for two pairs of mappings was introduced by Jungck in [8], for commuted mappings. Jungck in [8, 9] states this new assumption, with compatibility and weak compatibility of mappings. Some further work for generalized -weak contraction in metric spaces can be found in [6, 10, 11]. Ran and Reurring, in [12], introduced partially ordered metric spaces. A b-metric space with partial ordering is called partially ordered b-metric space. Recently, -weak contraction in partially ordered b-metric spaces gain the attention of many researcher. In this direction, fixed point and coincidence point results are discussed by many authors. For more details of fixed point and coincidence point results and their applications, comparison of different contraction conditions, and related results in b-metric spaces, we refer the reader to [13–40] along with the references mentioned therein. To show that a sequence is Cauchy in metric-type spaces, Jovanovic et al. [41] proved Lemma 1. As b-metric spaces have discontinuous structure, therefore, in this manuscript, we give a generalization of Boyd and Wong [2] inequality and -weak contraction, to establish coincidence and fixed point results.

2. Preliminaries

Definition 1. (see [42]). Let be a partially ordered set, and assume that the map satisfies the following conditions, For all and : if and only if Then, is called partially ordered -metric space.

Lemma 1. (see [41]). “Let be b-metric space and . When a sequence satisfies the following condition,for some, and .
Then, is -Cauchy sequence in .

Definition 2. (see [9]). The pair of mappings is compatible in the metric space if and only ifwhenever in sequence such that

Definition 3. (see [8]). In the metric space , the pair of mappings , is weakly compatible if is commutative at the point of coincidence (i.e., whenever ).

Definition 4. (see [43]). The pair of mappings , defined on a partially ordered set is called(a)Weakly increasing if and , for all (b)Partially weakly increasing if ,

Definition 5. (see [44, 45]). Let , , be three mappings on partially ordered set such that and . Then, pair is said to be(i)Weakly increasing with respect to if and only if for all , , for all and , and for all (ii)Partially weakly increasing w.r.t if and only if for all .

Theorem 1. (see [45]). Let self-mappings be continuous mappings in partially ordered complete metric space with . Assume that the pair is weakly compatible, whereas is partially weakly increasing with respect to such that and are comparable elements and satisfying the following condition:where are altering distance functions. Then, mappings , have coincidence point.

Theorem 2. (see [46]). Let , be mappings on a partially ordered complete -metric space . Assume that , is partially weakly increasing and satisfying the following condition:where are altering distance and lower semicontinuous functions, respectively, with andwhere . Furthermore, assume that either or is continuous mapping or space is regular. Then, have a common fixed point.

Theorem 3. (see [19]). “Let be partially ordered complete -metric space and , , , and be continuous mappings such that and . Suppose for comparable elements and , the following condition holdwith altering distance functions, and . Let the pairs and be compatible and the pairs and be partially weakly increasing w.r.t and , respectively. Then, , and have a coincidence point.”

Theorem 4. (see [18]). “Let , , , be continuous mappings on a partially ordered complete -metric space with and . Assume that compatible pairs and and comparable elements and satisfy the following condition:where , while and the pairs of mappings and are partially weakly increasing w.r.t and , respectively. When and are comparable, then .”

Theorem 5. (see [20]). Let the mappings , be defined on a partially ordered complete -metric space and be -weakly isotone increasing. Let be a function such that for all and satisfy the condition:where . If the mappings or are continuous in . Then, has a common fixed point.

3. Main Results

We begin with the following result.

Theorem 6. Let , , , be continuous mappings on a partially ordered complete -metric space, with and . Assume that compatible pairs and and comparable elements and satisfy the following contractive condition:where and satisfy for all , , andwhere and the pairs and are partially weakly increasing w.r.t , respectively. Then, and have a coincidence point. Furthermore, if is a coincidence point for comparable elements and , then .

Proof. Let . Since and , then there exist such that and . The sequence is constructed as follows:The pairs of mappings and are partially weakly increasing w.r.t and , respectively. Therefore,By repeating this process, we deduce thatAssume thatWe discuss three steps: Step I: at this step, we have to prove that . Since and are comparable, therefore, from (11), we have with To show that , suppose that Here, we discuss three possible cases of . Case (1): if , from (17), we have Since for , therefore which contradict assumption (19). Case (2): if , since for , therefore, from (17), one can write which is held unless , which implies that , but it contradicts (16). Therefore, Case (3): if . Using triangle inequality and assumption (19), we have Since for all , thus, from (17), one has This is again contradiction. Hence, for all cases, we conclude that Step II: in this step, we have to prove that the sequence is -Cauchy by using Lemma 1, for all three cases. Case (1): if . From (17), one can write As for , so Taking , we have Case (2): if , from (26), we have . From (17), one has Since , for , therefore Thus, Let . Then, Case (3): if . Using triangle inequality and (26), it is reduced to case (2). Therefore, (2) holds for three cases. Similarly, if and , one can easily show that Hence, Define . Suppose , for some , then . In case , then . We have If Then, from (17), Since for all and , therefore, This is only possible if . Hence, . Thus, . Similarly, one can show for the remaining two cases. By the same process, if , then , which gives . Therefore, is a constant sequence with , and (2) is also valid for constant sequence. Hence, is -Cauchy sequence. The completeness of space implies that there exist such that . Consequently, the subsequences will also converge to : Step III: now, we show that is a coincidence point of , sinceAs from the compatibility of , we haveMoreover, and .
As and are continuous mappings, thereforeUsing triangle inequality twice, one can writeTaking limit and using (41) and (42) in the above inequality, we obtainThus, . Thus, . Therefore, is coincidence at point of .
Similarly, .
From comparability of element and (11), one can writewhereIf , thenThis is possible if . Hence, .
On the same process, from the other two cases, we can show that .
In the following, the result condition of continuity and compatibility is relaxed for mappings.

Theorem 7. Let , , , be four mappings on regular partially ordered complete -metric spaces such that , , and , are -closed subsets of . Suppose the pairs and are partially weakly increasing with respect to , , respectively, whereas pairs are weak compatible and satisfy the following condition:where and with for , where , andwith . Then, , and have a coincidence point. Furthermore, if are comparable, then .

Proof. Following the lines of the proof of Theorem 6, one can easily show that there exists a sequence which converges to some . Therefore,If and but and are -closed subsets of , then there must be some , from which and . Hence, from the construction of the sequence given in Theorem 6, we haveNow, we prove that .
From regularity of , we have and .
Thus, from (49), one can writewhereSince for all and and using (54) in (53), we obtainThis is only possible if . It implies that . Hence, . Now, from weak compatibility of and , we can write . Thus, . Similarly, we can show that . Furthermore, following the last lines of the proof of Theorem 6, we have .
By putting (identity mapping) in Theorems (6 and 7), then the following corollary is obtained with

Corollary 1. Let , be two mappings on . Let be partially weakly increasing, whereas with for and , satisfy the following condition with comparable elements

Also,(i) or is continuous Or(ii) is regular

Then, b is a common fixed point of , .

Example 1. Let . Define , for all , then clearly is a partially ordered -metric on with and partially ordering is defined byAssume that are continuous mappings defined as follows:Since . Therefore, . Therefore, . Thus, the pair is partially weakly increasing. Now, we show thatDefine , where and take . Here, we discuss the following cases. Case (1): if , then Using mean value theorem for , where , we obtain Case (2): if , then . Therefore, . Hence, condition of Corollary 1 holds, and 0 is coincidence point.

Remark 1. If we put in Theorems 6 and 7, we can obtain Theorems (2.1, 2.2) of [18], respectively.

Remark 2. As are partially weakly increasing, therefore is -weakly isotone increasing. Thus, by substitution , in Corollary 1, we get Theorem 2.1 of [20]. Moreover, if and in Corollary 1, then Corollary 2.4 of [20] is obtained.

Remark 3. Clearly, the conditions of our Corollary 1 holds for Example 2.8 of [20] and the corresponding conclusion holds. By substituting in Example 2, then condition of Theorem 2.1 of [20] does not hold but our condition (57) of Corollary 1 holds.
In the remaining part of this manuscript, we discuss coincidence point of two compatible pairs of mappings with generalized -weak contractive condition.
Throughout the rest of this paper, we consider the following, , and for all , we defineand are altering distance functions and is partially ordered complete -metric space.

Theorem 8. Let , , , be continuous self-mappings on , where and . Suppose that the pairs and are compatible, whereas the pairs and are partially weakly increasing with respect to and , respectively, and satisfy the following condition:

Then , and have a coincidence point. Moreover, if are comparable then .

Proof. Let . Since and , so there exist such that and . Construct the sequence as follows:Since the pairs and are partially weakly increasing with respect to and , therefore,Repeating the above process, we can writeAssume thatWe discuss the proof in three steps. Step I: first, we prove that . Since and are comparable, therefore (64) implies that where To show that , suppose that Here, we discuss three possible cases of . Case (1): if , then As is nondecreasing, therefore which contradicts assumption (69). Thus, Case (2): if , then This is only possible if . However, if . Hence, , which implies that which contradicts (68). Hence, Case (3): if . Using triangle inequality and (69), we have Then, which implies that , Therefore, . So, contradicts (68). Therefore, in all three cases, we concluded that Step II: in this step, we will show that the sequence is -Cauchy sequence by using Lemma 1 for all three cases. Case (1): if . From (69), one can write Since for , therefore Since is nondecreasing, therefore where . Thus, Case(2): when , then from (79), one can write From (69), one has Since for , therefore From nondecreasing property of , we have where with . Case (3): when . Then, by using triangle inequality and (79) it will be converted to case (2). Therefore, (2) holds for all cases. Similarly, if and , again we have Hence, Let . Assume that for , then . When , it implies that ; therefore, If Then, from (64), From the above, we can obtain Since is nondecreasing, therefore It implies that . Therefore, . Hence, . Therefore, the sequence is constant sequence for . Therefore, (2) also holds for constant sequence . Thus, by Lemma 1, is a -Cauchy sequence. From the completeness of one can say that -Cauchy sequence converges to some . Consequently, . Step III: now, we show that is a coincidence point of and :Since the pair is compatible, so we haveMoreover, and . Also, and are continuous mapping; therefore,Using triangle inequality, one can writeAgain, applying triangle inequality on the second term, we haveBy taking limit and using (96) and (97), we can writeTherefore, . Thus, .
Similarly, we can show that .
Since and are comparable, therefore from (64), we can obtainwhereThus,which implies that . Hence, .
In the forthcoming, result condition of continuity and compatibility for mapping is relaxed.

Theorem 9. Let , , , , with , and , are -closed subsets of regular partially ordered complete -metric space . Assume that the pairs and are partially weakly increasing with respect to and , respectively, whereas and are weakly compatible and satisfy the following condition:

Then, these four mappings have a coincidence point. If and are comparable then .

Proof. As Theorem 8, one can easily construct a sequence which converges to some . Thus,If , . Since and are -closed subsets , therefore, there exist some , such that and . Hence,Now, we prove that .
Since is regular, thereforeHence, (104) implies thatwhereFrom (108) and (109), we havewhich implies that . Hence, . Thus, . Now, from weak compatibility of and , one can write . Thus, . Similarly, we can show . By following the last lines of the proof of Theorem 8 we have .
By taking , in Theorem 8, the following corollary is obtained with

Corollary 2. Let , , be three continuous mappings on where . Let pairs and be compatible, whereas be partially weakly increasing with respect to and satisfying the following condition:then , and have a coincidence point.

If we put in Theorem 8, the following corollary is obtained with

Corollary 3. Let , , be three continuous mappings on such that . Assume that pairs and are compatible, whereas is partially weakly increasing with respect to , and satisfies the following condition:then , and have a coincidence point.

When we put and in Theorem 8, the following corollary is obtained with

Corollary 4. Let , be mappings on such that . Assume that pair is compatible, whereas is partially weakly increasing with respect to and satisfies the following condition:

either and are continuous or space is regular.

Then, have a coincidence point.

If we put (identity mapping) in Theorem 8, then the following corollary is obtained with

Corollary 5. Let , be mappings on such that the pair is partially weakly increasing and satisfies the following condition:

either and are continuous or space is regular. Then, have a common fixed point.

Example 2. Let if and only if , is partially ordering on . Define -metric on by , with . Suppose , and be continuous mappings defined byFirst, we check the compatibility of . Let be sequence which converges to some . Then,The continuity of , implies thatSince limit of the sequence is unique, therefore , which implies that . Also, and are continuous, soHence, the pair is compatible. Similarly, one can check the compatibility of the pair .
Now, we have to prove that is partially weakly increases with respect to . Assume that , for ,Therefore, . We know that. Hence,Therefore,With the same process one can show that is partially weakly increases with respect to .
Next, we check the validity of the following inequity:Let and , where and .
Apply mean value theorem to in intervals and in intervals . TakeAll conditions of Theorem 8 are satisfied, and consequently 0 is a coincidence point.

Remark 4. Note that the conditions of our Theorem 8 holds for example of [19] and the corresponding conclusion holds. However, when , then inequality of Theorem 3 does not hold for our Example 2. Hence, Theorem 8 is a genuine generalization of Theorem 3.

Remark 5. By substitution, in Theorem 8 and 9 and corollaries 3.7, 3.8, and 3.9, and we get the results of [19].

Remark 6. Theorems 2.4 and 2.6 of [45] are special case of Corollary 4. By putting in Corollary 2, one can get Corollary 2.7 of [45].

Remark 7. Theorems 3.1 and 4.3 and Corollary 3.3 of [46] are special cases of Corollary 5.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

All the authors have contributed equally in all parts.

Acknowledgments

The authors would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group no. RG-DES-2017-01-17.

References

S. Banach, “Sur les operations dans les ensembles abstraits et leur application aux equations integrals,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.
View at: Publisher Site | Google Scholar
D. W. Boyd and T. S. W. Wong, “On nonlinear contractions,” Proceedings of the American Mathematical Society, vol. 20, no. 2, pp. 458–464, 1969.
View at: Publisher Site | Google Scholar
Y. I. Alber and S. Guerre-Delabriere, “Principle of weak contractive mapes in Hilbert space,” New Results in Operator Theory and Its Applications, Operator Theory: Advances and Applications, vol. 98, Birkhuser, Basel, Switzerland, 1997.
View at: Google Scholar
B. E. Rhoudes, “Some theorem on weakly contractive maps,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 4, pp. 2683–2693, 2001.
View at: Publisher Site | Google Scholar
Q. Zhang and Y. Song, “Fixed point theory for generalized -weak contractions,” Applied Mathematics Letters, vol. 22, pp. 75–78, 2009.
View at: Google Scholar
W. Dutta and B. S. Choudhury, “A generalization of contraction principle in metric spaces,” Fixed Point Theory and Applications, vol. 18, 2008.
View at: Google Scholar
D. Doric, “Common fixed point for generalized -weak contractions,” Applied Mathematics Letters, vol. 22, pp. 101174–101900, 2009.
View at: Google Scholar
G. Jungck, “Common fixed point for noncontinuious nonself maps,” Far East Journal of Mathematical Sciences, vol. 4, pp. 199–215, 1996.
View at: Google Scholar
G. Jungck, “Compatible mappings and common fixed points (2),” International Journal of Mathematics and Mathematical Sciences, vol. 11, no. 2, pp. 771–779, 1986.
View at: Publisher Site | Google Scholar
M. Abbas and D. Doric, “Common fixed point for generalized -weak contractions,” Math. Un. Nis. Serbia., vol. 10, pp. 1–10, 2010.
View at: Publisher Site | Google Scholar
P. Parvateesam, Murthy, K. Tas, and U. D. Patel, “Common fixed point theorems for generalized -weak contraction,” Journal of Inequalities and Applications, vol. 139, 2015.
View at: Google Scholar
A. C. M. Ran and M. C. B. Reurring, “Fixed point theorems,” Proceedings of the American Mathematical Society, vol. 132, pp. 1435–1443, 2004.
View at: Google Scholar
A. Aghajani, M. Abbas, and J. R. Roshan, “Common fixed point results for four mapps satisfy generalized contractive condition in partially ordered - metric space,” Mathematica Slovaca, vol. 64, no. 4, pp. 941–960, 2014.
View at: Google Scholar
E. Karapinar, H. Piri, and H. H. AlSulami, “Fixed points of generalized F-Suzuki type contraction in complete b-metric spaces,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 969726, 8 pages, 2015.
View at: Publisher Site | Google Scholar
H. Afshari, H. Aydi, and E. Karapinar, “Existence of fixed points of set-valued mappings in b-metric spaces,” East Asian Mathematical Journal, vol. 32, no. 3, pp. 319–332, 2016.
View at: Publisher Site | Google Scholar
H. Alsulami, S. Gulyaz, E. Karapinar, and I. Erhan, “An ulam stability result on quasi--metric-like spaces,” Open Mathematics, vol. 14, no. 1, pp. 1087–1103, 2016.
View at: Publisher Site | Google Scholar
H. Aydi, M.-F. Bota, E. Karapinar, and S. Moradi, “A common fixed point for weak -contractions on b-metric spaces,” Fixed Point Theory, vol. 13, no. 2, pp. 337–346, 2012.
View at: Google Scholar
H. Huang, S. Randenovic, and J. Vujakovic, “Coincidence point results in partially ordered b-metric spaces,” Fixed Point Theory and Applications, vol. 63, 2015.
View at: Google Scholar
J. R. Roshan, V. Parvaneh, and I. Altun, “Coincidence point results in partially ordered b-metric spaces,” Applied Mathematics and Computation, vol. 226, pp. 725–737, 2014.
View at: Google Scholar
J. R. Roshan, V. Parvaneh, and Z. Kadelburg, “Fixed point theorem for weakly isotone increasing mapping in ordered b-metric spaces,” Journal of Nonlinear Sciences and Applications, vol. 7, pp. 229–245, 2014.
View at: Google Scholar
J. Roshan, V. Parvaneh, S. Radenovic, and M. Rajovic, “Coincidence point results for generalized -weak contractions in ordered b-metric space,” Fixed Point Theory and Applications, vol. 68, 2015.
View at: Google Scholar
M. Kutbi, E. Karapinar, J. Ahmed, and A. Azam, “Some fixed point results for multi-valued mappings in b-metric spaces,” Journal of Inequalities and Applications, vol. 2014, no. 1, p. 126, 2014.
View at: Publisher Site | Google Scholar
M.-F. Bota, E. Karapinar, and O. Mlesnite, “Ulam-Hyers stability results for fixed point problems via -contractive mapping in b-metric space,” Journal of Industrial & Management Optimization, vol. 2013, Article ID 825293, 6 pages, 2013.
View at: Publisher Site | Google Scholar
M.-F. Bota and E. Karapinar, “A note on “some results on multi-valued weakly Jungck mappings in b-metric space”,” Journal of Industrial & Management Optimization, vol. 11, no. 9, pp. 1711-1712, 2013.
View at: Publisher Site | Google Scholar
M.-F. Bota, C. Chifu, and E. Karapinar, “Fixed point theorems for generalized -ciric-type contractive multivalued operators in -metric spaces,” Journal of Nonlinear Sciences and Applications, vol. 9, no. 3, pp. 1165–1177, 2016.
View at: Google Scholar
M. Sarwar, N. Jamal, and Y. Li, “Coincidence point results via generalized -weak contractions in partial ordered -metric spaces with application,” The Journal of Nonlinear Sciences and Applications, vol. 10, no. 7, pp. 3719–3731, 2017.
View at: Publisher Site | Google Scholar
N. Jamal, M. Sarwar, and M. Imdad, “A modified Perry’s conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations,” Razmadze Mathematical Institute, vol. 171, pp. 182–194, 2017.
View at: Publisher Site | Google Scholar
R. Allahyari, R. Arab, and A. S. Haghighi, “Coincidence point result of almost generalized contractive condition to partially ordered complete b-metric space and application to quadratic integral equation,” Journal of Inequalities and Applications, vol. 2014, p. 355, 2014.
View at: Google Scholar
S. Gülyaz-Özyurt, “On some alpha-admissible contraction mappings on Branciari b-metric spaces,” Advances in the Theory of Nonlinear Analysis and Its Application, vol. 1, no. 1, pp. 1–13, 2017.
View at: Publisher Site | Google Scholar
J. R. Roshan, V. Parvaneh, N. Shobkolaei, S. Sedghi, and W. Shatanawi, “Common fixed points of almost generalized -contractive mappings in ordered b-metric spaces,” Fixed Point Theory and Applications, vol. 159, 2013.
View at: Google Scholar
U. Aksoy, E. Karapinar, and I. M. Erhan, “Fixed points of generalized alpha-admissible contractions on b-metric spaces with an application to boundary value problems,” Journal of nonlinear and convex analysis, vol. 17, no. 6, pp. 1095–1100, 2016.
View at: Google Scholar
Z. D. Mitrović, H. Aydi, N. Mlaiki et al., “Some new Observations and Results for convex Contractions of istratescu’s type,” Symmetry, vol. 11, no. 12, p. 1457, 2019.
View at: Publisher Site | Google Scholar
A. Mukheimer, N. Mlaiki, K. Abodayeh, and W. Shatanawi, “New theorems on extended b-metric spaces under new contractions,” Nonlinear Analysis: Modelling and Control, vol. 24, no. 6, 2019.
View at: Publisher Site | Google Scholar
V. Parvaneh, N. Hussain, M. Khorshidi, N Mlaiki, and H. Aydi, “Fixed non-smooth equations based method for -norm problems with applications to compressed sensing,” Nonlinear Analysis: Theory, Methods & Applications, vol. 7, no. 10, p. 887, 2019.
View at: Publisher Site | Google Scholar
Q. Kiran, N. Alamgir, N. Mlaiki, and H. Aydi, “On some new fixed-point results in complete extended b-metric spaces,” Mathematics, vol. 7, p. 476, 2019.
View at: Google Scholar
N. Mlaiki and Y. Rohen, “Some coupled fixed point theorems in partially ordered metric space,” Journal of Nonlinear Sciences and Applications (JNSA), vol. 10, no. 4, pp. 1731–1743, 2017.
View at: Google Scholar
N. Souayah and N. Mlaiki, “A coincident point principle for two weakly compatible mappings in partial S-metric spaces,” Journal of Nonlinear Sciences and Applications (JNSA), vol. 9, no. 5, pp. p2217–2223, 2016.
View at: Publisher Site | Google Scholar
K. Abodayeh, N. Mlaiki, T. Abdeljawad, and W. Shatanawi, “Relations between partial metric spaces and M-metric spaces, Caristi Kirk’s theorem in M-metric type spaces,” Journal of Mathematical Analysis, vol. 7, no. 3, pp. p1–12, 2016.
View at: Google Scholar
H. Afshari, H. Aydi, and E. Karapinar, “On generalized Geraghty contractions on b-metric spaces,” Georgian Mathematical Journal, vol. 27, no. 1, pp. 9–21, 2020.
View at: Publisher Site | Google Scholar
H. Afshari, S. Rezapour, and N. Shahzad, “Absolute retractivity of the common fixed points set of two multifunctions,” Topological Methods in Nonlinear Analysis, vol. 40, pp. 429–436, 2012.
View at: Google Scholar
M. Jovanovic, Z. Kadelburg, and S. Radenovic, “Common fixed point results in metric-type spaces,” Fixed Point Theory and Applications, Article ID 978121, p. 15, 2010.
View at: Google Scholar
S. Czerwik, “Nonlinear set-valued contraction mappings in b-metric spaces,” Atti Sem. Mat. Fis. Un. Modena., vol. 46, pp. 263–276, 1998.
View at: Google Scholar
M. Abbas, T. Nazir, and S. Radenovic, “Common fixed point of four mapps in partially ordered metric space,” Applied Mathematics Letters, vol. 24, pp. 1520–1526, 2011.
View at: Google Scholar
J. Esmaily, S. M. Vaezpour, and B. E. Rhoades, “Coincidence point theorem of generalized weak contractive condition of ordered metric,” Applied Mathematics and Computation, vol. 219, pp. 1536–1548, 2012.
View at: Google Scholar
H. K. Nashine and B. Samet, “Fixed point results for mappings satisfying -weakly contractive condition in partially ordered metrics spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 6, pp. 2201–2209, 2011.
View at: Publisher Site | Google Scholar
S. Radenovic and Z. Kadelburg, “Fixed point results in partially ordered metric spaces,” Applied Mathematics and Computation, vol. 60, pp. 1776–1783, 2010.
View at: Google Scholar

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Copyright © 2020 Noor Jamal 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.

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