Admissible Hybrid <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3240995pt" id="M1" height="12.223pt" version="1.1" viewBox="-0.0657574 -11.8989 16.1846 12.223" width="16.1846pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g198-27" d="M407 519C374 464 344 443 287 443C240 443 200 472 200 528C200 573 239 648 335 648C442 648 532 615 613 583C557 535 504 459 456 342C449 343 444 343 439 343C392 343 348 315 313 278L330 262C369 291 396 294 425 294C430 294 434 294 437 293C400 197 365 125 317 78C266 94 218 109 175 109C107 109 59 82 59 51C59 16 88 -9 171 -9C229 -9 283 5 333 34C432 -1 479 -15 530 -15C655 -15 733 49 752 162L732 163C707 75 636 24 528 24C481 24 427 41 371 60C440 112 494 189 529 281C533 280 543 280 547 280C595 280 634 306 665 339L651 356C626 337 590 330 558 329C555 329 552 329 547 330C587 445 623 519 663 566C708 551 751 540 796 540C864 540 912 567 912 598S881 658 800 658C753 658 702 645 650 612C566 646 464 687 356 687C223 687 159 596 159 528C159 457 223 413 287 413C358 413 396 452 425 511L407 519ZM871 598C871 583 843 570 806 570C771 570 735 580 694 595C725 618 759 628 800 628C855 628 871 614 871 598ZM283 51C252 31 214 21 171 21C116 21 100 35 100 51C100 66 128 79 165 79C207 79 235 70 283 51Z"/></g></svg>-</span>Contractions in 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="M2" 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

Kiran, Quanita; Azhar, Mamuna; Aydi, Hassen; Gaba, Yaé Ulrich

doi:https://doi.org/10.1155/2021/5579577

Advances in Mathematical Physics

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

Research Article | Open Access

Volume 2021 | Article ID 5579577 | https://doi.org/10.1155/2021/5579577

Admissible Hybrid -Contractions in Extended -Metric Spaces

Quanita Kiran,¹Mamuna Azhar,²Hassen Aydi,^3,4,5and Yaé Ulrich Gaba^6,7,8,9

Academic Editor: Ricardo Weder

Received13 Jan 2021

Revised25 Jun 2021

Accepted18 Jul 2021

Published06 Aug 2021

Abstract

In this paper, we define the notation of admissible hybrid -contractions in the setting of extended -metric spaces, which unifies and generalizes previously existing results in literature. Furthermore, as an application, we discuss Ulam-Hyers stability and well-posedness of a fixed point problem.

1. Introduction

The source of metric fixed point theory is considered to be the Banach Contraction Principle which is a very important mechanism for finding the existence and solutions for many problems including differential and integral equations. Afterwards, numerous papers on generalizations and extensions of the Banach’s theorem for both single-valued and multivalued mappings have been published, either by changing the contraction conditions or by changing the structure of metric space to more generalized form, e.g., see [1] and all the references therein.

The concept of a -metric space was accomplished by the works of Bourbaki [2], Bakhtin [3], and Czerwik [4]. Subsequently, several articles have appeared in literature which dealt with the fixed point theorems by taking into account more general forms, of a metric spaces, i.e., -metric space, see [5], and the applications of relaxed triangular inequalities, like NEM (nonlinear elastic matching), ice floes, etc., were also utilized in various directions [6, 7]. Following the idea of -metric spaces, a number of authors have presented several results in this direction, see [8, 9]. To have some insight about miscellaneous generalizations of a metric, we refer the readers to [10–15] for some works on -metric spaces.

In 2017, Kamran et al.[16] generalized the structure of a -metric space and referred it as an extended -metric space. He weakened the triangle inequality of a metric and established fixed point results for a class of contractions. Thereafter, many researchers have studied and generalized fixed point results for single and multivalued mappings. Proving extensions of the Banach contraction principle from metric spaces to -metric spaces and hence to extended -metric spaces is useful to prove existence and uniqueness theorems for different types of integral and differential equations. Keeping the length of paper concise, we refer to [17–35] and to references mentioned therein. For more topological properties of extended -metric spaces, see [20].

The main purpose of this paper is to merge different linear and nonlinear results existing in literature in setup of an extended -metric space, which is a real generalization of a -metric and a standard metric space. We express our results in a more refined form by combining the notations, like admissible mappings, simulation functions, and hybrid contractions. We will prove fixed point results involving a certain type of mappings. The obtained results generalize [36]. Moreover, we prove Ulam-Hyers stability [37–39] and well-posedness of fixed point problems as well.

2. Preliminaries

In this section, we recollect some definitions and results from literature along with some examples.

Definition 1 [40]. Let be a nonempty set and . A function is called an extended -metric, if it satisfies the following properties for all:
()
()
()
The pair is called an extended -metric space.

Example 1. Let and defined by . Define as
for
for
for
Note that is an extended -metric space.

Example 2. Let , , and as
such that
=
=
=
=
Here, is an extended -metric on .

Note that the extended -metric space becomes a -metric space, whenever , where and a standard metric space for.

Definition 2 [16]. Let be an extended -metric space. The sequence in is termed as follows:(i)Cauchy if and only if as (ii)Convergent if and only if there exists such that as and we write Note that the extended -metric space is complete if every Cauchy sequence is convergent.

The -metric is not continuous in general and so the same for an extended -metric. We define the concept of -orbital continuity (in case of an extended -metric space) as used in [41].

Definition 3 [42]. Given a mapping . Suppose that there exists some such that . The set is called the orbit of . A self-mapping is called orbitally continuous if for some implies thatMoreover, if every Cauchy sequence of the form as , converges in , then an extended -metric space is called -orbitally complete.

Definition 4 [40]. Let be an extended -metric space. A function is called an extended -comparison function if it is increasing and also there exists a mapping such that for some , , converges for each and for every . Here, for . We say that is an extended -comparison function for at and denotes the collection of all extended -comparison functions by.

Example 3 [40]. Let be an extended -metric space and be a self mapping on . Assume that exists. Define such that , with . Then, the series, converges by ratio test.
Here, and for.

The notation of -admissible mappings also played a vital role in fixed point theory, see [43, 44].

Definition 5 [36]. Let be a mapping. A function is -orbital admissible if.
An -orbital admissible, mapping is called triangular -orbital admissible, if and for all.

Example 4. Let and such that and . Consider given as and otherwise. Clearly, is -orbital admissible.

Definition 6 [45]. A mapping satisfying the following conditions
() for all
() If , are the sequences in such that
. Then, is termed as a simulation function
Let represents the set of all simulation functions defined above.

The main idea of the simulation function is very useful and effective. For a self-mapping on a metric space, the contraction can be represented as , where and . By letting and , the corresponding simulation function for Banach’s fixed point theorem is . It is clear that for many other well-known results (Geraghty, Boyd-Wong, etc.), one can find a corresponding simulation function, see [36, 46–51]. In other words, a simulation function can be considered as a generator of different contraction type inequalities.

Definition 7. A self-mapping , defined on a metric space , is called a -contraction with respect to , if it satisfies

Every -contraction defined on a complete metric space has a 0 fixed point, as described in [46]. A -contraction generalizes the Banach contraction principle by assuming and for all . It also unifies several known type of contractions. Many authors have extended their work on -contractions in order to prove a more generalized version (see [47]).

The notion of admissible hybrid contractions is introduced [36, 46, 48–50] in order to generalize and unify the several existing fixed point results in the literature. The main goal of this paper is to investigate the existence and uniqueness of a fixed point of admissible hybrid -contractions in the context of an extended -metric space. We shall also list some existing results in the literature as corollaries and consequences of our main results. Consequently, the results in the class of -metric spaces and standard metric spaces become a special case of our obtained results.

3. Main Results

Definition 8. Let be an extended -metric space. A self-mapping is called an admissible hybrid contraction if there exist an extended -comparison function and such thatwhere and , with , andwhereand

Definition 9. Let be an extended -metric space. A self-mapping is said to be an admissible hybrid -contraction, if there exist an extended -comparison function , , and such that

Further, we discuss the existence and uniqueness of a fixed point of an admissible hybrid -contraction mapping.

Note that we assume that is continuous and , throughout Section 3 and Section 4.

Theorem 10. Let be an extended -metric space. Let be an admissible hybrid -contraction, which satisfies the following axioms:(i)The function is triangular -orbital admissible(ii)There exists such that(iii)Either is continuous or(iv) is continuous and for any Then, possesses a fixed point.

Proof. Let be any arbitrary point. We start from and iteratively, we construct a sequence such that for all . Suppose that there exists some such that , we find that is a fixed point of , and in this way, the proof is completed. Thus, we can assume that for any . By condition (i), as is an admissible hybrid -contraction, so by assuming and in equation (3), we havewhich gives,By condition (ii), as is triangular -orbital admissible and , so☐

Case 1. Consider so thatFrom equation (10),Suppose . As is an increasing function, so above inequality can be written asAs, soSince , we getwhich is a contradiction. Thus, for every , , and thus equation (10) becomesLet , by triangular inequality, we haveSince is an extended -comparison function, the seriesis convergent for every .
Denote and .
Thus, for , the above inequality becomesLetting , we getwhich implies that is a Cauchy sequence in a complete extended -metric space. Therefore, it is convergent, so there exists such thatNow, we prove that is a fixed point of . By condition (iii), if is continuous, we haveTherefore, is a fixed point of . Now, consider that is continuous. It follows that . We shall now prove that . Contrarily, suppose that, , from equation (3)It implies thatAs, sowhich leads to a contradiction. Thus, .

Case 2. Consider . Let and . One writesUsing triangular inequality, we obtainBy using the following inequality, one getsSo, we haveand by equation (7),which gives,Suppose that . As is an increasing function, soso, we have two cases here that or which is a contradiction. Thus, we obtainUtilizing the same arguments as of Case 1, we obtain that is a Cauchy sequence in an extended -metric space. Therefore, it is convergent, so there exists such thatNow, further we claim that is a fixed point of . By condition (iii), if is continuous, thenThus,Hence, is a fixed point of . Now, consider is continuous. It follows that . Now, we shall validate that . Contrarily, suppose that , thenIt implies thatwhereTherefore, we havewhich is a contradiction. So, .

Theorem 11. Assume that all the postulates of Theorem 10 hold. Additionally, suppose that , for any . Then, has a unique fixed point.

Proof. Assume that the fixed point of the mapping is not unique. Let be another fixed point of , where . For case , by equation (7), we getThis yields thatwhich is a contradiction. Similarly, for case , we obtain , which also gives a contradiction. Thus, ; that is, has a unique fixed point.☐

Example 1. Let and be such that and for all . Consider the mapping defined byandNote that(i) is an extended complete -metric space with (ii) is triangular -orbital admissible mapping(iii)For , and therefore, (iv) is continuous(v) is continuous, where Furthermore, for , we get (vi)Consider , then , and hence, . For all , we haveHence,Now, consider , , and , we haveHence,In all other cases, we have , soHence, we acquire that is an admissible hybrid -contraction. It follows all the hypothesis of Theorem 11, and so, is the fixed point of .

Let be the collection of all auxiliary functions , which are continuous and if and only if .

Corollary 12. Let be an extended -metric space, and . Suppose that there exist two functions with , for all , such thatwhere is defined as . Additionally, suppose that(i) is triangular -orbital admissible(ii)There exists such that(iii)Either is continuous or(iv) is continuous and for any(v)If , thenThen, has a unique fixed point.

Corollary 13. Let be an extended -metric space. Suppose that there exists a function , where such that all is continuous and if and only if , such thatwhere is defined as . Furthermore, we assume that(i) is triangular -orbital admissible(ii)There exists such that (iii)Either is continuous or(iv) is continuous and for any (v)If , thenThen, has a unique fixed point.

Corollary 14. Let be an extended -metric space. Suppose that there exist a function such that exists and , for every , with property thatwhere is defined as . Moreover, we suppose that(i) is triangular -orbital admissible(ii)There exists such that (iii)Either is continuous or(iv) is continuous mapping and for any(v)If , thenThen, has a unique fixed point.

4. Application

In this section, we explore Ulam-Hyers stability and well posedness of the fixed point problem in the setup of an extended -metric space (see [36] and references therein).

Definition 15. Let be a self-mapping defined on an extended -metric space. Consider the fixed point problemThe fixed point problem (53) is well-posed if(i)(ii)If is a sequence such that , as , then , as

Theorem 16. Let be an extended -metric space. Suppose that all the assumptions of Theorem 11 hold, and . Additionally, we assume that for any sequence , , as , we have , for all , where . If , where , then the fixed point problem (53) is well-posed.

Proof. As , by equation (7),ConsiderIn this way, we obtainor we can writeFrom here, we getAs , we have that . So, .
Thus, the fixed point problem (53) is well-posed.☐

Definition 17. The fixed point problem is called generalized Ulam-Hyers stable if and only if there exists which is increasing, continuous at 0 with , such that for every and for each ,there exists a solution of the fixed point problem such that .
If there exists such that , for each , then the fixed point problem is referred to be Ulam-Hyers stable.

Theorem 18. Let be an extended -metric space. Suppose that all the assumptions of Theorem 11 hold and .
Furthermore, we assume that , for all satisfying (60), where . If , where , then the fixed point problem is Ulam-Hyers stable.

Proof. By (7),ConsiderThus, we getor we can writeTherefore, we obtainThus, we getHence,where for all and .☐

5. Conclusion

In this research paper, we consolidated and refined several existing results in literature by bringing up the notation of admissible hybrid -contractions in the setup of an extended -metric space. Accordingly, all the exhibited results are authentic in context of complete -metric spaces by letting , where , and in context of complete metric spaces by letting . Furthermore, the paper generalizes the results of [36, 45, 46, 51]. Numerous fixed point results can be concluded in standard -metric spaces via a partial order or a cyclic contraction. Moreover, one can derive results in extended -metric spaces using [52–55].

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Disclosure

The statements made and views expressed are solely the responsibility of the author.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this manuscript.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The fourth author would like to acknowledge that this publication was made possible by a grant from Carnegie Corporation of New York.

References

T. van An, N. van Dung, Z. Kadelburg, and S. Radenović, “Various generalizations of metric spaces and fixed point theorems,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, vol. 109, no. 1, pp. 175–198, 2015.
View at: Publisher Site | Google Scholar
N. Bourbaki, “Topologie Generale Herman: Paris, France,” 1974.
View at: Google Scholar
I. A. Bakhtin, “The contraction mapping principle in almost metric spaces,” Funct. Ana, vol. 30, pp. 26–37, 1989.
View at: Google Scholar
S. Czerwik, “Nonlinear set-valued contraction mappings in -metric spaces,” Atti del Seminario Matematico e Fisico dell'Università di Modena, vol. 46, pp. 263–276, 1998.
View at: Google Scholar
A. Hussain, T. Kanwal, and A. Al-Rawashdeh, “Global best approximate solutions for set valued contraction in -metric spaces with applications,” Communications in mathematics and applications., vol. 9, no. 3, pp. 293–313, 2018.
View at: Google Scholar
R. Fagin and L. Stockmeyer, “Relaxing the triangle inequality in pattern matching,” International Journal of Computer Vision, vol. 30, no. 3, article 187208, pp. 219–231, 1998.
View at: Publisher Site | Google Scholar
R. McConnell, R. Kwok, J. Curlander, W. Kober, and S. Pang, “ correlation and dynamic time warping: two methods for tracking ice floes in SAR images,” IEEE Transactions on Geoscience and Remote sensing, vol. 29, no. 6, pp. 1004–1012, 1991.
View at: Publisher Site | Google Scholar
T. Abdeljawad, N. Mlaiki, H. Aydi, and N. Souayah, “Double controlled metric type spaces and some fixed point results,” Mathematics, vol. 6, no. 12, p. 320, 2018.
View at: Publisher Site | Google Scholar
N. Mlaiki, H. Aydi, N. Souayah, and T. Abdeljawad, “Controlled metric type spaces and the related contraction principle,” Mathematics, vol. 6, no. 10, p. 194, 2018.
View at: Publisher Site | Google Scholar
Q. Mahmood, A. Shoaib, T. Rasham, and M. Arshad, “Fixed point results for the family of multivalued F-contractive mappings on closed ball in complete dislocated -Metric spaces,” Mathematics, vol. 7, no. 1, p. 56, 2019.
View at: Publisher Site | Google Scholar
M. Samreen, K. Waheed, and Q. Kiran, Multivalued -contractions and fixed points, vol. 32, no. 4, University of Niš, 2018.
H. A. Hammad, H. Aydi, and N. Mlaiki, “Contributions of the fixed point technique to solve the 2D Volterra integral equations, Riemann–Liouville fractional integrals, and Atangana–Baleanu integral operators,” Advances in Difference Equations, vol. 2021, no. 1, Article ID 97, 2021.
View at: Publisher Site | Google Scholar
M. Shoaib, T. Abdeljawad, M. Sarwar, and F. Jarad, “Fixed point theorems for multi-valued contractions in -metric spaces with applications to fractional differential and integral equations,” IEEE Access, vol. 7, pp. 127373–127383, 2019.
View at: Publisher Site | Google Scholar
H. A. Hammad, H. Aydi, and Y. U. Gaba, “Exciting fixed point results on a novel space with supportive applications,” Journal of Function Spaces, vol. 2021, Article ID 6613774, 12 pages, 2021.
View at: Publisher Site | Google Scholar
M. U. Ali, H. Aydi, and M. Alansari, “New generalizations of set valued interpolative Hardy-Rogers type contractions in b-metric spaces,” Journal of Function Spaces, vol. 2021, Article ID 6641342, 8 pages, 2021.
View at: Publisher Site | Google Scholar
T. Kamran, M. Samreen, and Q. U. L. Ain, “Generalization of -metric space and some fixed point theorems,” Mathematics, vol. 5, no. 19, pp. 1–7, 2017.
View at: Publisher Site | Google Scholar
H. R. Marasi and H. Aydi, “Existence and uniqueness results for two-term nonlinear fractional differential equations via a fixed point technique,” Journal of Mathematics, vol. 2021, Article ID 6670176, 7 pages, 2021.
View at: Publisher Site | Google Scholar
L. Subashi and N. Gjini, “Fractals in extended -metric space,” Journal of Progressive Research in Mathematics, vol. 12, pp. 2057–2065, 2017.
View at: Google Scholar
L. Subashi and N. Gjini, “Some results on extended -metric spaces and Pompeiu-Hausdorff metric,” Journal of Progressive Research in Mathematics, vol. 12, pp. 2021–2029, 2017.
View at: Google Scholar
L. Subashi, “Some topological properties of extended -metric space,” in Proceedings of the 5th International Virtual Conference on Advanced Scientific Results, vol. 5, pp. 164–167, 2017.
View at: Publisher Site | Google Scholar
N. Mlaiki, N. Souayah, T. Abdeljawad, and H. Aydi, “A new extension to the controlled metric type spaces endowed with a graph,” Advances in Difference Equations, vol. 2021, no. 1, Article ID 94, 2021.
View at: Publisher Site | Google Scholar
W. Shatanawi, A. Mukheimer, and K. Abodayeh, “Some fixed point theorems in extended -metric spaces,” Applied Mathematics and Physics, vol. 80, no. 4, pp. 71–78, 2018.
View at: Google Scholar
H. Huang, Y. M. Singh, M. S. Khan, and S. Radenović, “Rational type contractions in extended -metric spaces,” Symmetry, vol. 13, no. 4, p. 614, 2021.
View at: Publisher Site | Google Scholar
H. Aydi, E. Karapinar, and W. Shatanawi, “Coupled fixed point results for -weakly contractive condition in ordered partial metric spaces,” Computers and Mathematics with Applications, vol. 62, no. 12, pp. 4449–4460, 2011.
View at: Publisher Site | Google Scholar
H. Aydi, M. Jleli, and B. Samet, “On positive solutions for a fractional thermostat model with a convex–concave source term via -Caputo fractional derivative,” Mediterranean Journal of Mathematics, vol. 17, no. 1, article 16, 2020.
View at: Publisher Site | Google Scholar
H. Aydi, A. Felhi, T. Kamran, E. Karapinar, and M. U. Ali, “On nonlinear contractions in new extended -metric spaces,” Applications and Applied Mathematics: An International Journal (AAM), vol. 14, pp. 537–547, 2019.
View at: Google Scholar
I. C. Chifu and E. Karapinar, “On contractions via simulation functions on extended -metric spaces,” Miskolc Mathematical Notes, vol. 21, no. 1, pp. 127–141, 2020.
View at: Publisher Site | Google Scholar
T. Abdeljawad, E. Karapinar, S. K. Panda, and N. Mlaiki, “Solutions of boundary value problems on extended-Branciari -distance,” Journal of Inequalities and Applications, vol. 2020, no. 1, Article ID 2373, 2020.
View at: Publisher Site | Google Scholar
M. Alghamdi, S. Gulyaz-Ozyurt, and E. Karapınar, “A note on extended Z-contraction,” Mathematics, vol. 8, no. 2, p. 195, 2020.
View at: Publisher Site | Google Scholar
V. Parvaneh, M. R. Haddadi, and H. Aydi, “On best proximity point results for some type of mappings,” Journal of Function Spaces, vol. 2020, Article ID 6298138, 6 pages, 2020.
View at: Publisher Site | Google Scholar
B. Alqahtani, A. Fulga, E. Karapinar, and V. Rakocevic, “Contractions with rational inequalities in the extended -metric space,” Journal of Inequalities and Applications, vol. 2019, no. 1, Article ID 2176, 2019.
View at: Publisher Site | Google Scholar
E. Karapınar, P. Kumari, and D. Lateef, “A new approach to the solution of the Fredholm integral equation via a fixed point on extended -metric spaces,” Symmetry, vol. 10, no. 10, p. 512, 2018.
View at: Publisher Site | Google Scholar
K. Jain, J. Kaur, and F. Khojasteh, “Some fixed point results in -metric spaces and b-metric-like spaces with new contractive mappings,” Bulletin of Mathematical Analysis And Applications, vol. 10, no. 2, pp. 55–135, 2021.
View at: Publisher Site | Google Scholar
B. Alqahtani, A. Fulga, and E. Karapinar, “Common fixed point results on an extended -metric space,” Journal of Inequalities and Applications, vol. 2018, no. 1, Article ID 1745, 2018.
View at: Publisher Site | Google Scholar
H. Qawaqneh, M. Md Noorani, W. Shatanawi, H. Aydi, and H. Alsamir, “Fixed point results for multi-valued contractions in b-metric spaces and an application,” Mathematics, vol. 7, no. 2, p. 132, 2019.
View at: Publisher Site | Google Scholar
I. C. Chifu and E. Karapinar, “Admissible hybrid Z-contractions in -metric spaces,” Axioms, vol. 9, no. 1, p. 2, 2020.
View at: Publisher Site | Google Scholar
S. M. Ulam, Problems in Modern Mathematics, vol. 1, Science Editions John Wiley and Sons, Inc, New York, 1964.
D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America., vol. 27, no. 4, pp. 222–224, 1941.
View at: Publisher Site | Google Scholar
A. Boutiara, S. Etemad, A. Hussain, and S. Rezapour, “The generalized U-H and U-H stability and existence analysis of a coupled hybrid system of integro-differential IVPs involving -Caputo fractional operators,” Advances in Difference Equations, vol. 2021, no. 1, Article ID 3253, 2021.
View at: Publisher Site | Google Scholar
M. Samreen, T. Kamran, and M. Postolache, “Extended -metric space, extended -comparison function and nonlinear contractions,” UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, vol. 80, pp. 21–28, 2018.
View at: Google Scholar
T. L. Hicks and B. E. Rhoades, “A Banach type fixed point theorem,” Japanese Journal of Mathematics, vol. 24, pp. 327–330, 1979.
View at: Google Scholar
L. B. Ćirić, “A generalization of Banach’s contraction principle,” Proceedings of the American Mathematical Society., vol. 45, no. 2, pp. 267–273, 1974.
View at: Publisher Site | Google Scholar
O. Popescu, “Some new fixed point theorems for -Geraghty contraction type maps in metric spaces,” Wm Stukely MD FRS, vol. 2014, no. 1, article 776, 2014.
View at: Publisher Site | Google Scholar
E. Karapinar, P. Kumam, and P. Salimi, “On -Meir-Keeler contractive mappings,” Fixed Point Theory and Applications, vol. 2013, no. 1, 2013.
View at: Publisher Site | Google Scholar
F. Zarinfar, F. Khojasteh, and S. M. Vaezpour, “A new approach to the study of fixed point theorems via simulation functions,” Filomat, vol. 29, pp. 1189–1194, 2015.
View at: Google Scholar
E. Karapinar and A. Fulga, “New hybrid contractions on -metric Spaces,” Mathematics, vol. 7, no. 7, p. 578, 2019.
View at: Publisher Site | Google Scholar
X. Li, A. Hussain, M. Adeel, and E. Savas, “Fixed point theorems for -contraction and applications to nonlinear integral equations,” IEEE Access, vol. 7, pp. 120023–120029, 2019.
View at: Publisher Site | Google Scholar
E. Karapinar and A. Fulga, “An admissible hybrid contraction with an Ulam type stability,” Demonstratio Mathematica, vol. 52, no. 1, pp. 428–436, 2019.
View at: Publisher Site | Google Scholar
K. Abodayeh, E. Karapinar, A. Pitea, and W. Shatanawi, “Hybrid contractions on Branciari type distance spaces,” Mathematics, vol. 7, no. 10, p. 994, 2019.
View at: Publisher Site | Google Scholar
E. Karapinar, A. Petrusel, and G. Petrusel, “On admissible hybrid Geraghty contractions,” Carpathian Journal of Mathematics, pp. 435–444, 2020.
View at: Google Scholar
E. Karapinar and B. Samet, “Generalized α–ψ contractive type mappings and related fixed point theorems with applications,” Abstract and Applied Analysis, vol. 2012, Article ID 793486, 17 pages, 2012.
View at: Publisher Site | Google Scholar
M. Aslantas, H. Sahin, and D. Turkoglu, “Some Caristi type fixed point theorems,” The Journal of Analysis, vol. 29, no. 1, pp. 89–103, 2021.
View at: Publisher Site | Google Scholar
H. Sahin, M. Aslantas, and I. Altun, “Feng Liu type approach to best proximity point results for multivalued mappings,” Journal of Fixed Point Theory and Applications., vol. 22, no. 1, p. 11, 2020.
View at: Publisher Site | Google Scholar
I. Altun, M. Aslantas, and H. Sahin, “Best proximity point results for p-proximal contractions,” Acta Mathematica Hungarica, vol. 162, no. 2, pp. 393–402, 2020.
View at: Publisher Site | Google Scholar
M. Aslantas, H. Sahin, and I. Altun, “Best proximity point theorems for cyclic p-contractions with some consequences and applications,” Nonlinear Analysis: Modelling and Control, vol. 26, no. 1, pp. 113–129, 2021.
View at: Publisher Site | Google Scholar

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