Fixed Point Results for <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.199599pt" id="M1" height="16.4602pt" version="1.1" viewBox="-0.0657574 -12.2606 51.9332 16.4602" width="51.9332pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,5.937,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g><g transform="matrix(.017,0,0,-0.017,15.686,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,22.474,0)"><path id="g117-183" d="M619 0V50H365V548H307V50H53V0H619Z"/></g><g transform="matrix(.012,0,0,-0.012,34.032,4.134)"><path id="g198-7" d="M896 662C863 643 826 624 783 624C682 624 539 680 429 680C328 680 253 659 202 616C163 583 138 537 138 484C138 399 208 349 283 349C396 349 466 417 492 554L472 555C436 442 385 379 286 379C234 379 179 410 179 491C179 578 257 634 382 634C475 634 622 574 731 574C796 574 862 610 907 647L896 662ZM657 568C577 499 520 440 477 341L463 310C368 277 286 233 247 186L260 171C320 211 380 237 437 256C349 77 281 31 184 31C124 31 83 59 78 94H80C87 83 101 81 109 81C137 81 153 104 153 131S130 180 103 180C68 180 43 154 43 111C43 34 123 0 192 0C335 0 469 75 546 282C599 291 640 293 661 294C653 267 650 244 650 224V206L709 226C706 235 706 245 706 252C706 271 706 289 714 312C758 333 786 368 786 387C786 403 775 409 763 409C744 409 712 391 685 352C644 351 604 345 564 337C595 433 622 506 671 554L657 568Z"/></g><g transform="matrix(.017,0,0,-0.017,45.77,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-</span>Contractions in Orthogonal <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06579971pt" id="M2" height="11.8441pt" version="1.1" viewBox="-0.0657574 -11.7783 15.9607 11.8441" width="15.9607pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><use xlink:href="#g198-7"/></g></svg>-</span>Metric Spaces with Applications

Ahmad, Jamshaid; Al-Rawashdeh, Ahmed Saleh; Al-Mazrooei, Abdullah Eqal

doi:https://doi.org/10.1155/2022/8532797

Journal of Function Spaces

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Abstract Introduction and Preliminaries Applications Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Advances in Nonlinear Analysis and Applications

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Research Article | Open Access

Volume 2022 | Article ID 8532797 | https://doi.org/10.1155/2022/8532797

Fixed Point Results for -Contractions in Orthogonal -Metric Spaces with Applications

Jamshaid Ahmad,¹Ahmed Saleh Al-Rawashdeh,²and Abdullah Eqal Al-Mazrooei³

Academic Editor: Hemant Kumar Nashine

Received05 Apr 2022

Revised24 May 2022

Accepted06 Jul 2022

Published14 Sept 2022

Abstract

The main objective of this article is to introduce the notion of -contraction in the setting of orthogonal -metric space and present some new fixed point results in such newly introduced space. We also furnish an example to manifest the originality of the obtained results. As application of our foremost result, we look into the solution of a nonlinear fractional differential equation.

1. Introduction and Preliminaries

One of the most important part in the theory of fixed points is the underlying space as well as the contractive mapping and contractive condition. In 1905, a French mathematician Maurice Fréchet gave the study of metric space which plays a virtual role in the pioneer result in this theory. In the last few decadess, many researchers have presented interesting generalizations of metric spaces. Most of the generalizations are made by making some changes in the triangle inequality of the original definition. Some of these well-known generalizations of metric space are -metric space due to Czerwik [1], rectangular metric space due to Branciari [2], and JS-metric space due to Jleli and Samet [3]. After all such generalizations, Jleli and Samet [4] gave a compulsive extension of a metric space which is known as -metric space in this way.

Suppose be the set of functions satisfying

for all , if and only if

Definition 1 (see [4]). Let be nonempty set, and let . Suppose that there exists such that
(D₁) , if and lnly if
(D₂) , for all
(D₃) for all , and with , we have for all Then, is said to be an -metric space (-MS).

Example 1 (see [4]). Let Then, defined by with and is an -metric.

Definition 2 (see [4]). Let be an -MS.
Let be a sequence in . Then, is said to be an -convergent to if is convergent to with regard to an -metric . (i)A sequence is -Cauchy, ifFor more particulars in this way, we mention the researchers to [5–9].
On the other hand, Gordji et al. [10] initiated the conception of the orthogonal set (in short, -set).

Definition 3. Let be a non empty and . If with the binary relation satisfies the condition: then, it is said to be an orthogonal set. Moreover, is said to be an orthogonal point. We represent this -set by .

Example 2 (see [10]). Let . Define on by if there exists such that . It is very simple to note that for all . Thus, (,) is an -set.

Definition 4 (see [10]). Let be an -set. A sequence {} is called orthogonal sequence (in essence -sequence) if In this direction, these researchers [11–14] utilized -sets in different ways to obtain their results.
Very recently, Kanwal et al. [15] combined both concepts and introduced the notion of orthogonal -metric space in this way.

Definition 5 (see. [15]). Let (,) be an -set and be an -metric on Then, triplet is claimed to be an orthogonal -metric space (-MS).

Example 3 (see [15]). Let . Define -metric by for all and Define if or Then, for all 0 so () is an -set. Then, is an -MS.
From now onwards, we represent () as an -set and an -MS.

Definition 6 (see [15]). A mapping : is professed to be orthogonality continuous (in short -continuous) at if for each -sequence {} in if then Also is -continuous on if is -continuous in each .

Definition 7 (see [15]). A set of is professed to be orthogonally -complete (in short --complete) if every Cauchy O-sequence is -convergent.
In 2012, Samet et al. [16] gave the conception of -admissible in this way.

Definition 8. A mapping is called -admissible whenever implies

Later on, Ramezani [17] extended the above concept to orthogonal set as follows.

Definition 9. A mapping is called orthogonally -admissible whenever and implies
We give the following property: (OH) for any
On the other side, Wardowski [18] introduced the following new notion of -contraction in 2012.

Definition 10. Let be a metric space. A mapping is said to be a -contraction if there exists such that for : where is a mapping satisfying (), (), and (), that is,
() there exists such that

We represent the set of the functions satisfying ()-(). Later on, Secelean [19] and Piri and Kumam [20] replaced the conditions () and () with some weaker symmetrical conditions repectively. Popescu and Stan [21] obtained two new fixed point results for -contractions with these weaker symmetrical conditions. Many authors [22–29] used the concept of -contraction introduced by Wardowski in order to define and prove new results on fixed points in complete metric spaces.

In this article, we introduce the notion of -contraction in the background of orthogonal -metric space (-MS) and obtain some new results in such newly introduced space. We also furnish an example to manifest the originality of the obtained results. As application of our foremost result, we look into the solution of a nonlinear fractional differential equation.

2. Main Results

Definition 11. Let be an -MS. A mapping : is said to be -contraction if there exists and such that

Theorem 12. Let be an -complete -MS and is an -contraction such that following assertions hold: (i) is -preserving, that is, implies (ii) is orthogonally -admissible(iii)there exists such that and (iv) is -continuousThen, there exists such that . Furthermore, if has the property (OH), then this fixed point is unique.

Proof. By assertion (iii), there exists such that and Let the sequence {} is defined as for all As is -preserving, so {} is an -sequence in . As is orthogonally -admissible, so it follows that for all . If for any it is then evident that is a fixed point of . So we suppose that for all Hence we have for all As is -preserving, we get for all This means that {} is an -sequence. Hence, we presume that for all From (8) and (11), we get hence, for all Consequently, we have for all Now by applying and by (), we have From the condition (), there exists such that From (14) and (16), we have Taking , we have Hence, there exists such that for all This yields for Since is convergent, so there exists such that for Hence, by (21) and (), we get Using () and (22), we get which, from (), gives that Hence, is a Cauchy -sequence in . The -completeness of guarantees that there exists such that Now, we prove that is fixed point of . By -continuity of gives as n Thus, Thus, is a fixed point of . Lastly, we assume that is once more fixed point of such that From (OH), we have or and Thus, from (8), we have which implies that that is a contradiction. Thus, Hence, the fixed point is unique.

Corollary 13. Let be an -complete -MS. If is -preserving and -continuous satisfying then, there exists such that which is unique.

Example 4. Define the sequence as follows: for all Let equipped with the metric defined by with and For all , define if and only if . Hence, is an -complete O-MS. Define by and by Clearly, then is not a contraction in the sense of [18].
It is easy to check that is -continuous and is -preserving. Let the mapping defined by It is easy to show that Now, to prove is an -contraction, that is for The above condition is equivalent to So, we have to check that For every we have Thus, the inequality (8) is satisfied with . Hence, is an (,)-contraction. Thus, Theorem 12 implies that is a unique fixed point of .

Example 5. Define the sequence as follows: for all Let equipped with the metric defined by with and For all , define if and only if . Hence, is an -complete -MS. Define by Clearly, then is not a contraction in the sense of [18].

It is easy to check that is -continuous and is -preserving. Let the mapping defined by

It is easy to show that Now, to prove is an -contraction, that is, for The above condition is equivalent to

So, we have to check that

For every we have

For every , we have

Thus, the inequality (8) is satisfied with . Thus, Corollary 13 implies that is a unique fixed point of .

Theorem 14. Let be an -complete -metric space, be an -admissible mapping and there exist and such that for all and Assume that there exists such that then, there exists such that . Furthermore, if has the property (OH), then this fixed point is unique.

Proof. Suppose that Fix . Since satisfies the inequality (49), for all This implies that is -set. Now, it is obviously that is -complete and is -continuous and -preserving. Thu, by applying Theorem 12, has a unique fixed point in .

Corollary 15 (see [30]). Let be an -complete -metric space, be an -admissible mapping and there exist such that for all and then, there exists unique such that .
If we take for and in Definition 5, then -MS reduces to -metric space and we get the following result.

Theorem 16. Let be an -complete metric space and is an -contraction such that following assertions hold: (i) is -preserving, that is, implies (ii) is orthogonally -admissible(iii)there exists such that and (iv) is -continuousThen, there exists such that . Furthermore, if has the property (OH), then this fixed point is unique.
If we take for all in Theorem 16, then we get the following result of Mani et al. [31].

Corollary 17. Let be an -complete metric space, and there exist such that Assume that the following conditions hold: (i) is -preserving, that is, implies (ii) is orthogonally -admissible(iii)there exists such that and (iv) is -continuousThen, there exists such that . Furthermore, if has the property (OH), then this fixed point is unique.

2.1. Periodic Point Result

Let be an -MS and : such that then , for every Although, the converse of this reality is not correct generally. The mapping fulfilling for all is called to have property

Definition 18. Let be an -MS and : The set is said to be an orbit of A mapping is said to be an orbitally -continuous at if for each -sequence in A mapping : is orbitally continuous on if is orbitally -continuous at all

Theorem 19. Let be an -complete -MS and . Suppose that there exist and such that and is -preserving. Then, has the property supplied that is orbitally continuous on .

Proof. Let be fixed and there exists such that () is satisfied for . Then, by (), there exists such that Now, we manifest that Define {} in such that If there exists such that then is a fixed point of . Now, we assume that , for all Using inequality (14), we get This implies that Thus, we have for all Now, by applying the limit as and using (), we have From the condition (), there exists such that From (59) and (61), we have Taking , we have Hence, there exists such that for all This yields for Since is convergent, so there exists such that for Hence, by (64) and (), we get Using () and (67), we get which, from (), gives that Hence, is a Cauchy -sequence in . As and is -complete, there exists such that Now, by the orbital -continuity of , we have Thus, has a fixed point and is true for Suppose We assume on the contrary that but , then Now, Thus, we have for all Taking we have which, from () we get which is contradiction. So,

3. Applications

Consider a nonlinear differential equation of fractional order

via the integral boundary conditions where represents the Caputo fractional derivative of order given by

where represents the integer part of the real number and is continuous. We take with supremum norm Thus, is a Banach space. Recall, the Riemann–Liouville fractional integral of order is given by

Lemma 20 (see [15]). The Banach space equipped with the -metric given by and orthogonal relation where is an orthogonal -metric space.

Theorem 21. Suppose that (i)there exists such thatfor all (ii)there exists is defined byfor all such that where (). Then, (76) has at least one solution

Proof. It is well known that is a solution of (76) iff is a solution of the integral equation Then, problem (76) is equivalent to find which is a fixed point of . Suppose that a relation on , by for all . With this relation, is orthogonal because for all there exists for all such that We examine for all So, the triplet is a complete -MS. It is clear from the definition that is -continuous. We first prove that is -preserving. Let for all . Now, we get which yields that , i.e., is -preserving. Next, for all , we have which implies that From the above inequality, we obtain that Taking natural log function on both sides, we have that is that is Thus, where Now, consider given by for each , then . Thus, for all and
Now, let () be a Cauchy -sequence converging in . Thus, we get , for all and for .

We will have two possible cases to discuss:

Case 1. If , for all and . Then, for all , there exists a sequence of positives that converges to . Thus, we obtain for all , i.e., for all and .

Case 2. If for all , has to be thrown out. Hence, by Corollary 13, Equation (76) has a unique solution.

4. Conclusion

In this article, we introduced the notion of -contraction in the context of orthogonal -metric space and established some new fixed point theorems in this newly introduced space. We have given soeme examples to manifest the authenticity of the obtained results. As application of our foremost result, we looked into the solution of a nonlinear fractional differential equation.

In this direction, our future work will focus on studying the fixed points for -contraction in the context of orthogonal -metric space endowed with a graph.

Data Availability

No such data were used for this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The second author gratefully acknowledges with thanks the Department of Research Affairs at UAE University, as this article is financially supported by the grant: UPAR (2019), Fund No. 31S397 (COS).

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Copyright © 2022 Jamshaid Ahmad 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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