Fixed Point Results for Single and Multivalued Maps on Partial Extended <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.20973 12.533" width="8.20973pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g></svg>-</span>Metric Spaces

Aydi, Hassen; Batul, Samina; Aslam, Muhammad; Sagheer, Dur-e-Shehwar; Ameer, Eskandar

doi:https://doi.org/10.1155/2022/2617972

Journal of Function Spaces

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

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Fixed-Point Theory and Applications to Probabilistic Functional Analysis

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Research Article | Open Access

Volume 2022 | Article ID 2617972 | https://doi.org/10.1155/2022/2617972

Fixed Point Results for Single and Multivalued Maps on Partial Extended -Metric Spaces

Hassen Aydi,^1,2,3Samina Batul,⁴Muhammad Aslam,⁵Dur-e-Shehwar Sagheer,⁴and Eskandar Ameer⁶

Academic Editor: Pasquale Vetro

Received27 Aug 2021

Accepted08 May 2022

Published23 May 2022

Abstract

This article is based on the concept of partial extended -metric spaces, which is inspired by the notions of new extended -metric spaces and partial metric spaces. Fixed point results for single and multivalued mappings on such spaces are also presented. Few examples are also provided to elaborate the concepts.

1. Introduction and Preliminaries

Many problems in science and engineering defined by nonlinear equations can be solved by reducing them to an equivalent fixed point problem. Banach contraction principal provided a platform for researchers to prove the existence of fixed point of mappings on different abstract spaces. Fixed point theorems are developed for single-valued or set-valued mappings of abstract metric spaces. One can see a lot of literature on metric fixed point theory [1–9]. Recently, Shukla and Panicker [10] investigated some fixed point results on generalized enriched nonexpansive mappings defined on Banach spaces. These results are generalization of many existing results.

A partial metric space was introduced by Matthews [11, 12]. It is the generalization of a metric space, in which may be nonzero and for every . Matthews proved a fixed point theorem for contractive mappings in that space and also discussed its topological properties. After this development, many researchers worked in this direction (see [4, 9, 13–15]). Aydi et al. [16] initiated the idea of a partial Hausdorff metric space, and they proved fixed point theorems in partial metric spaces for multivalued mappings. A beautiful piece of work in this aspect can be seen in [17]. Authors used partial metric spaces by incorporating a strictly increasing -mapping to introduce multivalued -contractions. This paper extends some results in partial metric spaces. Shukla [18] presented the notion of a partial -metric space and proved some related fixed point results. A lot of research can be seen in literature on -metric spaces (see [1, 2, 7, 8, 19–23]), which is a stimulation towards the concept of extended -metric spaces introduced by Kamran et al. [6] who proved the Banach contraction theorem in the setting of extended -metric spaces. Afterwards, many authors have focused on the subject and generalized different results of metric spaces in extended -metric spaces (see [3, 24–28]). Very recently, Aydi et al. [5] have introduced the concept of new extended -metric spaces by replacing the triangle inequality of extended -metric spaces with a modified functional inequality. In this setting, they proved few fixed point theorems for nonlinear contractive mappings. Motivated by the above-mentioned research, we initiate the idea of partial extended -metric spaces and we extend the result of Kamran et al. [6] in setting of the partial extended -metric spaces. We have also presented few examples to elaborate the idea.

To reach the goal of proving some fixed point results in the new notion, first we give some definitions.

Definition 1 [21]. Consider a nonempty set with a real number . The function is called a -metric if it satisfies the following properties for each :
()
()
()
Here, the pair is called a -metric space.
In the following, we give the definition of a new extended -metric space (in the sense of Aydi et al. [5]).

Definition 2. Let be a nonempty set and given . The function is called a new extended -metric if it satisfies the following conditions for all : (1)(2)(3)The pair is called a new extended -metric space.

Definition 3 [11]. Let be a nonempty set. The function is said to be a partial metric on if satisfies the conditions listed below:
()
()
()
()
The pair is called a partial metric space.
Shukla [18] has presented a partial -metric space as follows.

Definition 4 [18]. Let be a nonempty set and be a given real number. The function is called a partial -metric with coefficient if for all , the following conditions are satisfied:
(PB1): if and only if
(PB2):
(PB3):
(PB4):
The is called a partial -metric space.

2. Partial Extended -Metric Space

In this section, we first elaborate the idea of a partial extended -metric space (in the sense of Aydi et al. [5]), and then with the help of an example, we explain our new definition.

Definition 5. Let be nonempty set and be a function. The function is called a partial extended -metric if for all , it satisfies the following:
(PE1):
(PE2):
(PE3):
(PE4):

The partial extended -metric space on generates a topology with a base of the family of open -balls where for all and A sequence in a partial new extended -metric space converges with respect to to a point if and only if

Remark 6. (1)If , then the above definition coincides with a partial -metric space(2)If , then the partial new extended -metric becomes a partial metricThe following is an example of a partial extended -metric space, which shows that the notion of a partial extended -metric space is a generalization of a partial -metric space.

Example 7. Consider and define by Then, is a partial extended -metric on , where is defined by

Example 8. Consider the set , and define the function by Define in the following manner: Then, is a partial extended -metric on . In fact, the first three axioms of Definition 5 hold, so there is only need to check the triangular inequality. To do this, proceed as follows: Similarly, the remaining pairs can be verified as above; hence, for all , holds. Hence, is a partial extended -metric space.

Remark 9. Note that in the above examples, the function depends on all the three variables , , and . So from this fact, it is clear that a partial extended -metric needs not to be a partial extended -metric.
In partial extended -metric spaces, the notion of Cauchyness and convergence are defined as follows.

Definition 10. Let be a partial extended -metric space. (1)A sequence is said to be a Cauchy sequence if exists and is finite(2)A sequence converges to if and only ifIt is worth to mention that if is a partial extended -metric on , the defined by is an extended -metric on .

Definition 11. A partial extended -metric space is complete if for each Cauchy sequence in , there is such that

Theorem 12. Let be a complete partial extended -metric space and be a continuous functional. Consider a self map such that where Also assume that there exists such that where . Then, has a unique fixed point.

Proof. For an arbitrary , take the iterative sequence defined by , which satisfies (12). Inequality (11) implies Thus, by applying (11) successively, we obtain

By use of triangular inequality, we have (for )

Continuing this process, we get

By using (14), one writes

Therefore,

Set

For , we conclude that

Consider the series

Take

Thus, we have

By using the condition by ratio test, the series is convergent. So we conclude that Hence, by (20), is a Cauchy sequence in . Since is complete, we have such that . Now, we prove that is a fixed point of . Consider

Taking limit as , we get due to the fact that is continuous. We also conclude and . Hence, is a fixed point of . To verify its uniqueness, assume that there exists another fixed point of , that is, . By (11), we have

Since , . Also, we have and . Thus, ; that is, the fixed point of is unique.

Definition 13. Let be a self map and The set is called the orbit of . The function is said to be -orbitally lower semicontinuous at if and , which implies

Theorem 14. Let be a complete partial extended -metric space and represent a continuous functional. Consider a self map and such that where . Also for such , where , . Then, . Further, is a fixed point of if is -orbitally lower semi continuous at .

Proof. For a given , define an orbit with , , By the use of successive iteration of the inequality (28), we get Proceeding as in Theorem 12, we can prove is a Cauchy sequence. As is complete, then . Since is -orbitally lower semi continuous at , Thus, Hence, is a fixed point of .

Example 15. Take . Define by and by Then, represents a complete partial extended -metric on .
Let be defined by We have For each and , we get Since all the conditions of Theorem 14 are satisfied, the mapping has a fixed point.

3. Partial Hausdorff Distance via the Extended -Metric Space

Shukla and Panicker [10] introduced the notion of a partial Hausdorff metric. Baig and Pathak [29] has proved well-known Nadler’s fixed point theorem for multivalued mappings on weak partial metric spaces. This is further used by Kanwal et al. [30] for establishing a fixed point result on weak partial -metric spaces. In this section, we establish a fixed point result for set-valued mappings on partial new extended -metric spaces. We first give some requisite definitions. Let be a partial extended -metric space. Denote by the collection of all nonempty bounded and closed subsets of with respect to the partial extended -metric . For and , define the following:

Remark 16. Let be a partial extended -metric space and be any nonempty set in , then where is the closure of . Further, is closed in if and only if .

Definition 17. Let be a partial extended -metric space. For , define The mapping is called a partial Hausdorff distance function induced by

Corollary 18. If is partial extended -metric space and , then

Lemma 19. Let be a partial extended -metric space, , and . Then, for any , there exists such that

Theorem 20. Let be a complete partial extended -metric space so that is a continuous functional. If is a multivalued mapping such that for all , we have where . Also, there exists such that for every with , we have . Then, has a fixed point.

Proof. Given and , recall that , so by use of Lemma 19, we have with By using (42), we get For , by Lemma 19, we have with Again by (42), we get Thus, Continuing this process, we have a sequence in such that and Also, By using (PE4) of the partial extended -metric space, we have (for ) Continuing this process, we get By using (49), one writes This further gives Consider For , we conclude that Consider the series Substitute Then, we have Therefore, by ratio test, the series is convergent. So we conclude that By using (55), is a Cauchy sequence in . Since is complete, with respect to , that is, By (42), we get . Therefore, . Now, gives that By using continuity of , we get By (59), Thus we have which implies
The following corollary follows immediately from Theorem 20 by setting for all .

Corollary 21. Let be a complete partial -metric space and be a continuous functional. Suppose that is a multivalued mapping such that for all , we have where and Then, has a fixed point.

Remark 22. By setting for all , the main result of [16] becomes a special case of Theorem 20.

4. Conclusion

In this article, the notion of a partial extended -metric space was introduced with suitable examples. Fixed point results for single-valued mappings endowed on partial extended -metric spaces are established. These results are extensions of many existing results in literature.

The idea of a partial Hausdorff distance in partial extended -metric spaces was presented. Furthermore, some fixed point theorems involving multivalued mappings have been proved. In the future, one can prove the above results under the platform of dislocated partial extended -metric spaces.

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.

Authors’ Contributions

All authors contributed equally to the writing of this paper.

Acknowledgments

M. Aslam extends his appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia, for funding this work through research group program under grant number R.G. P-2/98/43.

References

M. U. Ali, T. Kamran, and M. Postolache, “Solution of Volterra integral inclusion in b-metric spaces via new fixed point theorem,” Nonlinear Analysis: Modelling and Control, vol. 2017, no. 1, pp. 17–30, 2017.
View at: Publisher Site | Google Scholar
M. U. Ali and A. Pitea, “Existence theorem for integral inclusions by a fixed point theorem for multivalued implicit-type contractive mappings,” Nonlinear Analysis: Modelling and Control, vol. 26, no. 2, pp. 334–348, 2021.
View at: Publisher Site | Google Scholar
T. Abdeljawad, R. P. Agarwal, E. Karapınar, and P. S. Kumari, “Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space,” Symmetry, vol. 11, no. 5, p. 686, 2019.
View at: Publisher Site | Google Scholar
T. Abdeljawad, E. Karapinar, and K. Tas, “Fixed point theorems for operators on partial metric spaces,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1894–1899, 2011.
View at: Publisher Site | Google Scholar
H. Aydi, A. Felhi, T. Kamran, E. Karapınar, and M. U. Ali, “On nonlinear contractions in new extended b-metric spaces,” Applications and Applied Mathematics, vol. 14, no. 1, 2019.
View at: Google Scholar
T. Kamran, M. Samreen, and Q. U. L. Ain, “A generalization of b-metric space and some fixed point theorems,” Mathematics, vol. 5, no. 2, p. 19, 2017.
View at: Publisher Site | Google Scholar
E. Karapinar, “A short survey on the recent fixed point results on b-metric spaces,” Constructive Mathematical Analysis, vol. 1, no. 1, pp. 15–44, 2018.
View at: Publisher Site | Google Scholar
E. Karapınar and C. Chifu, “Results in wt-distance over b-metric spaces,” Mathematics, vol. 8, no. 2, p. 220, 2020.
View at: Publisher Site | Google Scholar
E. Karapinar, “A note on common fixed point theorems in partial metric spaces,” Mathematical Notes, vol. 12, no. 2, pp. 185–191, 2011.
View at: Publisher Site | Google Scholar
R. Shukla and R. Panicker, “Some fixed point theorems for generalized enriched nonexpansive mappings in Banach spaces,” Rendiconti del Circolo Matematico di Palermo Series, vol. 2, pp. 1–15, 2022.
View at: Publisher Site | Google Scholar
S. G. Matthews, Partial Metric Topology, Research Report 212, Department of Computer Science, University of Warwick, 1992.
S. G. Matthews, “Partial metric topology, general topology and its applications,” in Proceedings of the 8th Summer Conference, Queen’s College, 1992.
View at: Google Scholar
E. Karapinar and I. M. Erhan, “Existence and uniqueness of a common fixed point on partial metric spaces,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1900–1904, 2011.
View at: Publisher Site | Google Scholar
E. Karapinar, “Generalizations of Caristi Kirk’s theorem on partial metric spaces,” Fixed Point Theory and Applications, vol. 2011, Article ID 4, 2011.
View at: Publisher Site | Google Scholar
S. Oltra, S. Romaguera, and E. A. Sanchez Perez, “Bicompleting weightable quasi-metric spaces and partial metric spaces,” Rendiconti del Circolo Matematico di Palermo, vol. 51, no. 1, pp. 151–162, 2002.
View at: Publisher Site | Google Scholar
H. Aydi, M. Abbas, and C. Vetro, “Partial Hausdorff metric and Nadlerʼs fixed point theorem on partial metric spaces,” Topology and its Applications, vol. 159, no. 14, pp. 3234–3242, 2012.
View at: Publisher Site | Google Scholar
S. Kumar and S. Luambano, “On some fixed point theorems for multivaluedF-contractions in partial metric spaces,” Demonstratio Mathematica, vol. 54, no. 1, pp. 151–161, 2021.
View at: Publisher Site | Google Scholar
S. Shukla, “Partial b-metric spaces and fixed point theorems,” Mediterranean Journal of Mathematics, vol. 11, no. 2, pp. 703–711, 2014.
View at: Publisher Site | Google Scholar
B. Alqahtani, A. Fulga, and E. Karapınar, “Non-unique fixed point results in extended b-metric space,” Mathematics, vol. 6, no. 5, p. 68, 2018.
View at: Publisher Site | Google Scholar
B. Alqahtani, E. Karapınar, and A. Ozturk, “On (α,ψ)-k-contractions in the extended b-metric space,” Univerzitet u Nišu, vol. 32, no. 15, pp. 5337–5345, 2018.
View at: Publisher Site | Google Scholar
I. A. Bakhtin, “The contraction principle in quasimetric spaces,” Functional Analysis, vol. 30, pp. 26–37, 1989.
View at: Google Scholar
N. Bourbaki, Topologie générale, Herman, Paris, France, 1974.
T. Kamran, M. Postolache, A. Ghiura, S. Batul, and R. Ali, “The Banach contraction principle in -algebra-valued b-metric spaces with application,” Fixed Point Theory and Applications, vol. 1, 7 pages, 2016.
View at: Google Scholar
H. Aydi, M. Aslam, D. E. S. Sagheer, S. Batul, R. Ali, and E. Ameer, “Kannan-type contractions on new extended -metric spaces,” Journal of Function Spaces, vol. 2021, Article ID 7613684, 12 pages, 2021.
View at: Publisher Site | Google Scholar
M. Samreen, W. Ullah, and E. Karapinar, “Multivalued -contractions on extended -metric spaces,” Journal of Function Spaces, vol. 2020, Article ID 5989652, 6 pages, 2020.
View at: Publisher Site | Google Scholar
L. Subashi, “Some topological properties of extended -metric space,” in Proceeding of the 5th international virtual conference on advanced scientific results, pp. 164–167, 2017.
View at: Publisher Site | Google Scholar
L. Subashi, “Some results on extended -metric spaces and Pompeiu-Hausdorff metric,” Journal of Progressive research in Mathematics, vol. 12, p. 2395, 2017.
View at: Google Scholar
N. Zikria, M. Samreen, T. Kamran, and S. S. Yesilkaya, “Periodic and fixed points for Caristi G-type-contractions in extended -gauge spaces,” Journal of Function Spaces, vol. 2021, Article ID 1865172, 9 pages, 2021.
View at: Publisher Site | Google Scholar
I. Beg and H. K. Pathak, “A variant of Nadler’s theorem on weak partial metric spaces with application to a homotopy result,” Vietnam Journal of Mathematics, vol. 46, no. 3, pp. 693–706, 2018.
View at: Publisher Site | Google Scholar
T. Kanwal, A. Hussain, P. Kumam, and E. Savas, “Weak partial b-metric spaces and Nadler’s theorem,” Mathematics, vol. 7, no. 4, p. 332, 2019.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Hassen Aydi 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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