New Fixed Point Theorems for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4618998pt" id="M1" height="12.7225pt" version="1.1" viewBox="-0.0657574 -12.2606 47.5902 12.7225" width="47.5902pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-230" d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"/></g><g transform="matrix(.017,0,0,-0.017,12.383,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.017,0,0,-0.017,26.29,0)"><path id="g113-246" d="M615 290C615 392 563 448 529 448C511 448 491 441 484 434L483 425C518 400 546 349 546 273C546 158 504 46 417 46C381 46 356 77 356 142C356 184 380 293 392 334L387 340L307 323C305 288 296 228 287 188C265 94 221 52 180 52C144 52 112 89 112 171C112 278 164 372 241 430L226 454C106 394 23 288 23 155C23 40 79 -12 139 -12C184 -12 245 32 285 85C292 31 322 -12 376 -12C423 -12 470 18 501 45C570 104 615 184 615 290Z"/></g><g transform="matrix(.017,0,0,-0.017,37.24,0)"><use xlink:href="#g117-33"/></g></svg>Contraction on <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5269pt" id="M2" height="16.7875pt" version="1.1" viewBox="-0.0657574 -12.2606 37.9217 16.7875" width="37.9217pt"><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="g113-233" d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"/></g><g transform="matrix(.017,0,0,-0.017,15.407,0)"><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,-0.017,22.196,0)"><path id="g113-234" d="M538 96L524 119C492 88 454 63 446 63C439 63 434 70 440 101C463 223 491 341 518 448H508L433 422L401 276C355 192 240 56 188 56C163 56 154 89 163 133C184 233 207 338 235 443L230 448L152 424L58 17C40 -60 23 -143 23 -185C23 -241 42 -261 63 -261C92 -261 117 -241 125 -227L124 -221C109 -209 89 -165 89 -103C89 -55 96 -13 105 26H107C121 -3 134 -12 151 -12C172 -12 194 -5 221 16C279 61 330 124 384 194H386C381 159 377 141 368 105C343 3 363 -12 383 -12C416 -12 486 35 538 96Z"/></g><g transform="matrix(.017,0,0,-0.017,31.721,0)"><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>Generalized Metric Spaces

Kari, Abdelkarim; Al-Rawashdeh, Ahmed

doi:https://doi.org/10.1155/2023/8069112

Journal of Function Spaces

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

Research Article | Open Access

Volume 2023 | Article ID 8069112 | https://doi.org/10.1155/2023/8069112

New Fixed Point Theorems for Contraction on -Generalized Metric Spaces

Abdelkarim Kari¹and Ahmed Al-Rawashdeh²

Academic Editor: Ozgur Ege

Received21 Mar 2023

Revised28 Sept 2023

Accepted30 Sept 2023

Published21 Oct 2023

Abstract

In this paper, we consider a new extension of the Banach contraction principle, which is called the contraction inspired by the concept of contraction in -generalized metric spaces and to study the existence and uniqueness of fixed point for the mappings in metric space. Moreover, we discuss some illustrative examples to highlight the improvements that were made, and we also give an iterated application of linear integral equations.

1. Introduction

Fixed point theory is an important and fascinating subject, and it provides essential tools for solving problems arising in various branches of mathematical analysis, see [1–5]. Fixed point theory guarantees the uniqueness and existence of the solution of integral and differential equations.

In 1922, a Polish mathematician Banach introduced a contraction principle [6], which was one of the most applicable results in mathematics. In recent times, many generalizations and improvement of the Banach contraction principle have appeared in the literature (see [7–13]).

The metric function has been generalized many times by modifying the associated axioms. Specifically, Bakhtin [14] and Czerwik [15] presented -metric spaces in a way that the triangle inequality was replaced by the -triangle inequality: for all pairwise distinct points and . One of these generalizations was given by Branciari [16]. Any metric space is a generalized metric space, but in general, generalized metric space might not be a metric space for details [17, 18]. Various fixed point results have been established on such spaces; for example, Nazam et al. [19–21] have proved some fixed point and common fixed point results in partial metric and S-metric spaces. For more details, see [12, 22, 23]).

Huang and Zhang [24] redefined the concept of -metric spaces and convergence in an ordered Banach space E with a normal solid cone. Shatanawi et al. in [23] defined E-metric spaces and characterized the cone metric spaces in a more general way by defining ordered normed spaces. Also, Mehmood et al. in [25] deduced more results about -metric spaces.

Jleli et al. [11, 22] introduced the notion of -contraction and proved a fixed point theorem which generalizes the Banach contraction principle in a different way than in the known results from the literature. Later, Kari et al. [26] proved new type of fixed-point theorems in a rectangular metric space and generalized asymmetric metric space by using a modified generalized -contraction maps.

In 2014, Hussain and Salimi [27] introduced the notion of an -contraction and stated fixed point theorems for -contractions. On the other hand, Hussain et al. [28] establish some new fixed-point theorems for generalized -contractions mappings in complete -metric spaces.

In this paper, we introduce the notion of a generalized -contraction to generalize an -contraction in generalized metric space. Also, examples are given to illustrate the obtained results we derive some useful corollaries of these results.

2. Preliminaries

Definition 1 (see [16]). Let be a nonempty set and be a mapping such that for all and for all distinct points , each of them different from and (i)(ii) (iii). Then, is called a generalized metric space

Definition 2 (see [16]). Let be a generalized metric space and be a sequence in , and . Then, (i)We say that the sequence converges to if and only if (ii)We say that is the Cauchy if

Lemma 3 (see [16]). Let be a generalized metric space and be a Cauchy sequence with pairwise disjoint elements in . If converges to both and , then .

Definition 4 (see [16]). The generalized metric space is said to be complete if every Cauchy sequence in converges to an .
In [11, 22], authors defined the following collections functions. Let be the family of all functions : such that
is increasing
For each sequence is continuous.
Let be the family of all functions such that
is increasing
for each sequence there exist and such that

Definition 5. Let be a generalized metric space and be a mapping. is said to be a -contraction if there exist and such that for any where

Theorem 6 (see [11]). Let be a complete generalized metric space and let be a contraction. Then, has a unique fixed point.

Remark 7. The sets and are different.

Example 8. Define by Then, , but for any , Since , so, does not satisfy the condition , then,

Example 9 (see [29]). Define by Then, , but for any , So, does not satisfy the condition , then,

Definition 10 (see [28]). Let and :. We say that is a triangular -admissible mapping if
for all
for all
for all
for all .

Definition 11 (see [28]). Let be a generalized metric space and let : be two mappings. Then,
is a -continuous mapping on if for a given point and sequence in and for all then
is a subcontinuous mapping on , if for given point and sequence in and for all then
is a -continuous mapping on , if for given point and sequence in such that with and for all , we have

Definition 12 (see [30]). Let be a rectangular -metric space and let : be two mappings. Then, the space is said to be
-complete, if every Cauchy sequence in with for all converges in
subcomplete, if every Cauchy sequence in with for all converges in
-complete, if every Cauchy sequence in with and for all converges in

Definition 13 (see [30]). Let be a generalized metric space and let be two mappings. Then, the space is said to be
-regular, if , for all implies for all
subregular, if , for all implies for all
-regular, if , and for all imply that and for all

3. Main Results

In this section, we introduce a new notion of generalized -contraction in the context of -generalized metric spaces as follows.

Definition 14. Let denote the set of all functions satisfying the following: for all with there exists such that .

Example 15. If where , then,

Example 16. If where , then,

Definition 17. Let be a -complete generalized metric space, and let be a self-mapping on , where are two functions. We say that is an -contraction, if for all with and , we have where , and

Example 18. Let and for all . So,
Define by Then, is a complete generalized metric space. Define by Then, is an -continuous triangular admissible mapping.

Case 1. Since , we get Thus, which implies that Thus, Thus,

Case 2. Similarly, we conclude that Hence, is an -contraction.

Theorem 19. Let be a complete generalized metric space. Let , satisfying the following conditions: (I) is a triangular admissible mapping(II) is an contraction(III)There exists such that and (IV) is an continuousThen, has a fixed point. Moreover, has a unique fixed point when and for all fixed points

Proof. Let such that and
Define a sequence by Since is a triangular -admissible mapping, then and .
Continuing this process, we have and for all By and one has Suppose that there exists such that Then, is a fixed point of , and we have nothing to prove. Hence, we assume that , i.e., for all We have Indeed, suppose that for some so we have Denote Then, (11) implies that where Then, and there exists such that Thus, Let Then, we have which is a contradiction, so Continuing this process, we get which is a contradiction. Thus, as follows, we can assume that (22) and (23) hold.
Substituting and in (11), for all , we have where Then, and there exists such that Let Then, It is a contradiction. Therefore, Using , we get Therefore, is a nonnegative strictly decreasing sequence of real numbers. Consequently, there exists such that Now, we claim that . Arguing by contradiction, we assume that Since is a nonnegative strictly decreasing sequence of real numbers, then we have By property of , we get By letting in inequality (43), we obtain It is a contradiction. Therefore, Substituting and in (11), for all , we have where Since we have and there exists such that Then, Take and . Thus, one can write By , we get By (40), we have which implies that Therefore, the sequence is a nonnegative strictly decreasing sequence of real numbers. Thus, there exists such that We assume that . By (45) and then Taking the in (51), and using the properties of , we obtain Therefore, which is a contradiction. Therefore, Next, we shall prove that is a Cauchy sequence, i.e, for all . Suppose to the contrary, we assume that there exist and a sequence and of natural numbers such that ,
and Now, using (40), (51), (61), and the quadrilateral inequality, we find Then, By quadrilateral inequality, we have Letting in the above inequalities, we obtain Now, by quadrilateral inequality, we have Letting in the above inequalities, we obtain By quadrilateral inequality, we have Letting in the above inequalities, we obtain By the quadrilateral inequality, we find Letting in the above inequalities, we obtain From (11) and by setting and , we have Taking the limit as , we have Applying (11) with and , we obtain
As is a continuous function So, there exist such that . Then, Letting in the above inequality, applying the continuity of , we have Therefore, which is a contradiction. Then, Hence, is a Cauchy sequence in . By completeness of , there exists such that Now, we show that . Arguing by contradiction, we assume that Now, by quadrilateral inequality we get, By letting in inequality (82) and (83), we obtain Therefore, Since as for all and since is an -continuous, we conclude that . Then, So
For uniqueness, now, suppose that are two fixed points of such that . Therefore, we have Applying (11) with and , we have where Therefore, we have which implies that which is a contradiction. Therefore, , and hence, the proof is complete.

Consequently, we have the following:

Corollary 20. Let be a complete generalized metric space, and let be two functions. Let be a self-mapping satisfying the following conditions: (i)(ii) is continuous. Then, has a unique fixed point

Proof. Define a function by Clearly,
Taking Thus, is an -contraction, and is a triangular admissible mapping. As in the proof of Theorem 19, has a unique fixed point .
It is clear that, if is a fixed point of , then is also a fixed point of for every . The notion of property introduced first by Jeong and Rhoades [31] that if a mapping satisfies for each , then it is said that has property or that has no periodic points.

Theorem 21. Let : be two functions, and let be an -generalized complete metric space. Let be a mapping satisfying the following conditions: (i) is a triangular -admissible mapping(ii) is an -contraction(iii) and for all . Then, has the property

Proof. Let for some fixed . As and and is a triangular -admissible mapping, then Continuing this process, we have for all . By and , we get Assume that Fix , i.e.,
Applying (11) with and , we get which implies that Thus, there exists such that Then, where As , taking the limit as Since is an increasing and contentious function, therefore, which is a contradiction. So, Then, . Therefore, has the property ().

Assuming the following conditions, we prove that Theorem 19 still holds for not necessarily continuous.

Theorem 22. Let be two functions, and let be an complete generalized metric space.
Let be a mapping, satisfying the following assertions: (i) is triangular admissible(ii) is contraction(iii)There exists such that and (iv) is -regularThen, has a fixed point. Moreover, has a unique fixed point whenever and for all

Proof. Let such that and . Similar to the proof of Theorem 19, we can conclude that where From and hold for all
Suppose that for some From Theorem 19, we know that the members of the sequence are distinct. Hence, we have , i.e., for all Thus, we can apply (11), to and for all to get Therefore, where Thus, Since Thus, and there exist such that If , then by (110) and the fact that and are continuous and by taking the limit as in (106), we obtain Using (85), we get It is a contradiction. Therefore, , that is, is a fixed point of , and so Thus, is a fixed point of The proof of the uniqueness is similar to that of Theorem 19.

Definition 23. Let be a -generalized metric space, and let be a self-mapping on . Suppose that are two functions. We say that is an -contraction, if for all with and , we have where and

Theorem 24. Let be a -complete generalized metric space, and let be two functions. Let be a self-mapping satisfying the following conditions: (i) is a triangular admissible mapping(ii) is an -contraction(iii)There exists such that and (iv)is a-continuousThen, has a fixed point. Moreover, has a unique fixed point when and for all

Proof. Let such that and . Similar to the proof of Theorem 19, we can conclude that By , there exist and such that Suppose that . So, there exists such that Taking , we have Suppose now that . Let be an arbitrary positive number. So, there exists such that Taking , we have Thus, in all cases, there exist and such that By induction, we obtain Letting in the above inequality, we obtain So, there exists such that By , there exist and such that
Suppose that . So, there exists such that Taking , we have Suppose now that . Let be an arbitrary positive number. So, there exists such that So by taking , we have Thus, in all cases, there exist and such that By induction, we obtain Letting in the above inequality, we obtain So, there exists such that If and with , then If and with , then Therefore, As , the series converges. Therefore, by taking the limit as in (136), we get Hence, is a Cauchy sequence. Since is complete, there exists such that Since is -continuous, Then, This proves that is a fixed point of .

Corollary 25. Let be a -complete generalized metric space. Let be two functions. Let be a self-mapping satisfying the following conditions: (i)(ii) is a triangular -admissible mapping(iii)There exists such that and (iv) is a continuousThen, has a fixed point. Moreover, has a unique fixed point when and for all

Proof. Define a function by So, , and is an -contraction. As in the proof of Theorem 24, has a unique fixed point .

Example 26. Let and . Define by Then, is a complete generalized metric space. Define by Then, is an -continuous triangular admissible mapping.

Case 1. Since and , we get Thus, On the other hand, which implies that Thus, As Thus,

Case 2. As and , Thus, On the other hand, which implies that Thus, As Thus, where Hence, conditions (11) and (115) are satisfied. Therefore, has a unique fixed point .

4. Application to Nonlinear Integral Equations

In this section, we endeavour to apply Theorems 19 and 24 to prove the existence and uniqueness of the integral equation of the Fredholm type.

where , , and are continuous functions and for some constant depending on the parameters and .

Theorem 27. Suppose the function is such that and . Then, the equation (160) has a unique solution and .

Proof. Let and defined by . Define given by Then, is a complete generalized metric space. Assume that and . Then, we get Thus, As and for any , then we can take natural exponential sides and get Since , which implies that Hence, for all with and . Then, satisfies conditions (11) and (115) which are hold.

5. Concluding Remarks

The paper deals with contraction in -generalized metric spaces, which is an extension of the Banach contraction principle. We prove fixed point theorems of some generalized contractions which are defined on generalized metric spaces that satisfy a -complete generalized metric space condition. Our generalized results are based on -contraction. Finally, we present an application dealing with the existence of solutions for integral equation of the Fredholm type. Further, we also need to illustrate some generalizations of the introduced contraction mappings for generalized metric spaces with a graph. Some open problems for the future, for example, fixed circle problem or fixed figure problem of the contraction mappings for generalized metric spaces.

Data Availability

No underlying data was collected.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Acknowledgments

The authors acknowledge with thanks the Department of Research Affairs at UAEU. This project is financially supported by UPAR-2019 (Fund No. 31S397).

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Copyright © 2023 Abdelkarim Kari and Ahmed Al-Rawashdeh. 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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