Fixed Point Results for Generalized <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="M1" height="16.7875pt" version="1.1" viewBox="-0.0657574 -12.2606 39.9025 16.7875" width="39.9025pt"><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="g113-245" d="M642 419L635 448C586 435 552 411 532 392C518 379 509 351 496 304C484 260 462 196 445 150C424 93 388 40 314 30L413 508L406 510L344 491L257 36C205 46 175 84 175 169C175 195 180 241 185 277C189 304 190 331 190 358C190 413 175 448 141 448C111 448 69 429 23 373L30 347C51 365 68 375 81 375C93 375 108 368 108 327C108 307 104 268 101 239C98 211 95 180 95 155C95 33 159 -8 248 -14L213 -186C206 -220 221 -254 230 -261L261 -230L306 -11C339 -4 366 6 389 20C431 46 473 87 503 155C533 224 557 286 571 325C586 367 606 393 642 419Z"/></g><g transform="matrix(.017,0,0,-0.017,33.7,0)"><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>Contraction Mapping in Rectangular <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

Mamud, Mustefa Abduletif; Tola, Kidane Koyas

doi:https://doi.org/10.1155/2022/9370083

Abstract and Applied Analysis

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

Research Article | Open Access

Volume 2022 | Article ID 9370083 | https://doi.org/10.1155/2022/9370083

Fixed Point Results for Generalized -Contraction Mapping in Rectangular -Metric Spaces

Mustefa Abduletif Mamud¹and Kidane Koyas Tola¹

Academic Editor: Douglas R. Anderson

Received18 Dec 2021

Revised11 Mar 2022

Accepted15 Mar 2022

Published05 Apr 2022

Abstract

In this paper, we introduce generalized -contraction mappings in the setting of rectangular -metric spaces and established existence and uniqueness of fixed points for the mappings introduced. Our results extend and generalize related fixed point results in the existing literature. We derive some consequences and corollaries from our obtained results. Also, we provide examples in support of our main findings. Furthermore, we determined a solution to an integral equation by applying our obtained results.

1. Introduction

Fixed point theory is one of the most important topics in mathematics, especially in analysis. Due to its application in various disciplines like engineering, computer science, biological sciences, and economics, many researchers took interest in fixed point theory and its application. The Banach contraction principle is a fundamental result in fixed point theory [1]. For one century, due to its importance and simplicity, several authors have obtained many interesting extensions and generalizations of the Banach contraction principle in several direction. One possible direction is the notion of best proximity point results. In this line of research, [2–4] obtained interesting best proximity point results and derived fixed results as consequences of their works in which Banach fixed point is one of them. Also, taking the key role of the notion of the metric in mathematics and hence in quantitative sciences, it has been extended and generalized in several distinct directions by many authors. In particular, -metric spaces introduced by Bakhtin [5] and Czerwik [6] are one generalization of metric spaces. In 2000, Branciari [7] introduced the concept of rectangular metric space by replacing the sum on the right hand side of the triangular inequality in the definition of a metric space by a three-term expression proved an analogue of the Banach contraction principle in such space. Also, various fixed point results were established on such spaces (see [8–13] and references therein). Recently, George et al. [14] announced the notion of rectangular -metric space as a generalization of metric, -metric space, and rectangular metric space; many authors initiated and studied many existing fixed point theorems in such spaces (see [15–20]). Recently, in 2019, Baiya and Kaewcharoen [21] established fixed point theorems for generalized contractions in complete rectangular metric spaces and proved the existence of fixed points, and in 2020, Kari et al. [22] introduced the notion of -contraction in -rectangular metric spaces and studied the existence and uniqueness of fixed point for the mappings introduced. Inspired and motivated by the works of Baiya and Kaewcharoen [21] and Kari et al. [22] in this paper, we introduce a new notion of generalized -contraction mappings and establish some fixed point results for such mappings in complete rectangular -metric spaces. Our results extend and generalize related fixed point results in the existing literature.

2. Preliminaries

In this section, we recall some basic definitions and results that will be used to prove our main results.

Notation: We need the following symbols and class of functions to prove certain results of this section: (1);(2) is the set of all natural numbers;(3), such that, is nondecreasing, for each sequence , if and only if and there exist and such that ;(4), such that, is continuous and nondecreasing}.

Definition 1 (see [23]). Let be a nonempty set and be a function satisfying the following conditions: (i) if and only if (ii)(iii) for all

Then, is called a metric, and the pair is called a metric space.

The following is the definition of the notion of -metric space.

Definition 2 (see [6]). Let be a nonempty set and be a given real number. A function is said to be a -metric if and only if for all , the following conditions are satisfied: (i) if and only if (ii)(iii)The pair is called a -metric space.

Definition 3 (see [7]). Let be a nonempty set, and let be a mapping such that for all and all distinct points each distinct from and : (i) if and only if (ii)(iii) (rectangular inequality).Then, is called a rectangular metric, and the pair is called a rectangular metric space.

Definition 4 (see [14]). Let be a nonempty set, be a given real number, and be a mapping such that for all and all distinct points each distinct from and : (i) if and only if (ii)(iii) (-rectangular inequality).Then, is called a -rectangular metric, and the pair is called a -rectangular metric space.

Definition 5 (see [14]). Let be a -rectangular metric space and be a sequence in ; we say that: (i) -rectangular converges to if as (ii) is a -rectangular Cauchy sequence if as (iii) is a -rectangular complete if every -rectangular Cauchy sequence in is -rectangular convergent.

Lemma 6 (see [22]). Let be a rectangular -metric space, and let be a sequence in . If is not a Cauchy sequence, then, there exists and two subsequences and of , such that is the smallest index with for which and . Then, the following hold: (1)(2)(3)(4)

Definition 7 (see [24]). Let be a nonempty set and be a function. A mapping is said to be -admissible; if for all , implies .

Definition 8 (see [25]). Let be a nonempty set and be a function. A mapping is said to be -orbital admissible; if for all , implies .

Definition 9 (see [25]). Let be a nonempty set and and be a function. We say that is triangular -orbital admissible if (i) is -orbital admissible;(ii)For all and .

Lemma 10 (see [25]). Let be a nonempty set, , and . Suppose that is a triangular -orbital admissible mapping and assume that there exists such that . Define a sequence in by for all . Then, for all with .

Theorem 11 (see [26]). Let be a complete rectangular metric space and . Suppose that there exist and such that for all where Then, has a fixed point.

Theorem 12 (see [21]). Let be a complete rectangular metric space, , and . Suppose that the following conditions hold: (i)There exist and such that for all where (ii)There exists such that (iii) is a triangular -orbital admissible mapping;(iv) is continuous.Then, has a fixed point.

Theorem 13 (see [21]). Let be a complete rectangular metric space and and be a function. Suppose that the following conditions hold: (i)There exist and such that for all where (ii)There exists such that (iii) is a triangular -orbital admissible mapping;(iv)If is a sequence in such that for all and as , then there exists a subsequence of such that for all Then, has a fixed point.

Theorem 14 (see [27]). Let be a complete rectangular metric space and and be a function. Suppose that the following conditions hold: (i)There exist and such that for all where (ii)There exists such that and (iii) is a triangular -orbital admissible mapping;(iv)If is a sequence in such that for all and as , then there exists a subsequence of such that for all . Then has a fixed point in and converges to .

3. Main Results

In this section, we introduce generalized -contraction mapping in the setting of rectangular -metric spaces and prove fixed point results for the mappings introduced.

We start this section by giving the definition of generalized -contraction mapping in the setting of rectangular -metric space as follows:

Definition 15. Let be a rectangular -metric space with parameter . Suppose that , be a function and . A self-mapping is called generalized -contraction mapping if it satisfies for all in the following condition: where Now, we state and prove the following fixed point theorem.

Theorem 16. Let be a complete rectangular -metric space, , and .
Suppose that the following conditions hold: (i) is a triangular -orbital admissible mapping;(ii) is generalized -contraction mapping;(iii)There exists such that ;(iv) is continuous.Then, has a fixed point in .

Proof. By (iii), there exists such that . We define an iterative sequence in by , for . Suppose that for some . Since the point forms a fixed point of that completes the proof.
From now on, we suppose that for all . By condition (iii), we have . Using Lemma 10, we obtain that From (9) to (11), for all , we have the following: where If , then by (12), we get , which is a contradiction. Hence, . Using (12), we have . Since is nondecreasing, we have for all .
Hence, is decreasing sequence of nonnegative real numbers. Hence, converges to a nonnegative real number, say That is We will prove that . Suppose that . Since is nondecreasing, by using (12) and (14), we obtain that By taking limit as in the above inequality, we get , which contradicts to the assumption that for all . Hence, we have and therefore Now, we shall prove that for all or for all . Assume on the contrary that there exist such that . Since , for each . Without loss of generality, we may assume that . Using (9) and (11), we obtain that where If , then from (17), we obtain that which is a contradiction. Hence, .
By (17), we obtain that Since is nondecreasing from the above inequality, we get By using (9), we have where If , then by (22), we obtain that which is a contradiction. Hence, By (22), we have Since is nondecreasing, we get By continuing this process, we obtain the following inequality: which is a contradiction. Hence for all .
We now prove that is bounded. Since is bounded, there exists such that If for all , then from and Lemma 10, we obtain that From above the inequality, we get for all This implies that is decreasing. Therefore, is bounded.
If for some , then from and Lemma 10, we obtain that Therefore, . This implies that is bounded. We next prove that . Suppose that . So there exists a subsequence of such that for some . Using (9) and Lemma 10, we have where Letting in (35), we obtain that which is a contradiction. Therefore, We now prove that is a -rectangular -Cauchy sequence in . Suppose that is not a Cauchy sequence. Then, there exists and two subsequences and of , such that is the smallest index with for which This implies that By applying -rectangular inequality, using (39) and (40), we get that Letting in above inequality, using (16) and (38), we get that Also, Letting in above inequality, using (16) and Lemma 6, we obtain For each , we have By using (16) and (42), we obtain that By (42) and (46), there exists a positive integer such that and , for all .
By Lemma 10 and using (9), for all we get that Letting in the above inequality, by (42), (44), and (46) and the continuity of , we obtain that which is a contradiction. Therefore, is a rectangular -Cauchy sequence in . Since is complete rectangular -metric space, it follows that converges to some . Since is continuous, we have Therefore, is a fixed point of .

By removing the continuity assumption in Theorem 16, we get the following fixed point result.

Theorem 17. Let be a complete rectangular -metric spaces, and .
Suppose that the following conditions hold: (i) is generalized -contraction mapping;(ii) is a triangular -orbital admissible mapping;(iii)There exists such that ;(iv)If is a sequence in such that for all and as , then there exists a subsequence of such that for all .Then, has a fixed point.

Proof. As in the proof of Theorem 16, we can construct a sequence in such that , for all and for all . The sequence is rectangular -Cauchy, and since is complete, there exists such that as . By (iv), there exists a subsequence of such that for all . We can suppose that for all . By (9), we obtain that where Taking limit as Now, we prove that . Suppose that . Therefore, It follows that By using (50), (52), and (54) and , we obtain that which is a contradiction. Thus, , and hence, is a fixed point of .

Now, we prove the uniqueness of fixed point. For this, we need the following additional condition.

(U): For all , we have where denotes the set of all fixed points of .

Theorem 18. Adding condition (U) to the hypothesis of Theorem 16 (or Theorem 17), we obtain the uniqueness of fixed point of .

Proof. From the proof of Theorem 16 (or Theorem 17), . We argue by contradiction that there exist be in such that and with . By condition (U), we have . So, by (9), we get where From (56), we have which is a contradiction. Therefore,

The following are corollaries to our main results.

Corollary 19. Let be a complete rectangular -metric space, , , , and . Suppose that the following conditions hold: (i)where (ii) is a triangular -orbital admissible mapping;(iii)There exists such that ;(iv)For every pair and of fixed points of , ;(v)Either is continuous or is a sequence in such that for all and as , then there exists a subsequence of such that for all Then, has a unique fixed point.

proof The result follows by taking in Theorem 16 (or Theorem 17).

Remark 20. By taking in Corollary 19, we get Thoerem 2.1 (or Theorem 2.2) of Baiya and Kaewcharoen [21]. Thus, this work generalizes the work of Baiya and Kaewcharoen [21].

Corollary 21. Letbe a complete rectangularmetric space, , and. Suppose that the following condition hold:where Then, has a unique fixed point.

proof. The result follows by taking , for all and in Theorem 16 (or Theorem 17).

Corollary 22. Let be a complete rectangular -metric space, , and . Suppose that the following condition hold: where Then, has a unique fixed point.
proof The result follows by taking for all in Theorem 16 (or Theorem 17).

Remark 23. By taking and in Corollary 22, we get Theorem 11 of Kari et al. [22].

Corollary 24. Let be a complete rectangular-metric space, , , and . Suppose that the following conditions hold: (i);(ii) is a triangular -orbital admissible mapping;(iii)There exists such that ;(iv)For every pair and of fixed points of , ;(v)Either is continuous or is a sequence in such that for all and as , then there exists a subsequence of such that for all .Then, has a unique fixed point.

proof. The result follows by taking and in Theorem 16 (or Theorem 17).

We now present examples for supporting our main results.

Example 1. Let be a finite set defined as .
Defined as

Clearly, is complete rectangular -metric spaces with . But is neither a metric nor a rectangular metric space because Let be the mapping defined by Define , by and .

We next illustrate that all conditions in Theorem 16 and Theorem 17 hold. Taking , we have . We next prove that is an -orbital admissible.

Let such that . Therefore, and then . By the definition of , we obtain that

It follows that is an -orbital admissible. Let such that and . By the definition of , we have . This yields

and implies ,

and implies .

This implies that is triangular -orbital admissible. Let be a sequence such that for all and as . By the definition of , for each , we get that . We obtain that . Thus, we have for each . We next prove that is generalized -contraction mapping. Let be such that . So we consider the following cases: and or and . Since = , we divide the proof into three cases as follows:

Case 1 For and , we have

This implies that

Case 2 For and , we have

This implies that

Case 3: For and , we have

This implies that

Hence, all assumptions in Theorem 16 and Theorem 17 are satisfied, and thus, has a fixed point which is . Finally let . Clearly , therefore, by the definition of , we have . So, all assumptions in Theorem 18 are satisfied, and thus, has a unique fixed point which is

Example 2. Let where .
Define by.

Then, is a complete rectangular -metric space with parameter . But, is neither metric nor rectangular metric because

Define mapping by

Define , by and with .

We next illustrate that all conditions in Theorem 17 hold.

Clearly, is a triangular -orbital admissible mapping. Taking , we have . Let be a sequence such that for all and as . By the definition of , for each , we get that . We obtain that . Thus, we have for each . We next prove is generalized -contraction mapping. Let be such that . So, we consider only the following case: and or and . Since = , we have

Hence, all assumptions in Theorem 17 are satisfied, and thus, has a fixed point which is . Finally let . Clearly ; therefore, by the definition of , we have . So, all assumptions in Theorem 18 are satisfied, and thus, has a unique fixed point which is .

3.1. Application to Integral Equation

In this section, we use Corollary 22 to show that there is a solution to the following integral equation: where and are continuous function. Let be the set of all continuous functions defined on . Define by which for all , then is a complete rectangular -metric space with parameter . Now, we prove the following result.

Theorem 25. Suppose the following hypothesis hold: (i)there is continuous function such that (ii)There is such that

Then, the integral Equation (78) has a unique solution in .

Proof. For and , define the mapping by .
For , from condition (i) and (ii), for all , we have Therefore, all conditions of Corollary 22 are satisfied and as a result the mapping has a unique fixed point in , which is a solution of the integral equation in (78).

4. Conclusion

In 2020, Kari et al. [22] introduced the notion of -contraction in -rectangular metric spaces and study the existence and uniqueness of fixed point for the mappings introduced. In 2019, Baiya and Kaewcharoen [21] established fixed point theorems for generalized contractions in complete rectangular metric spaces and proved the existence of fixed points. Inspired and motivated by the works of Kari et al. [22] and Baiya and Kaewcharoen [21] in this paper, we introduce generalized -contraction mappings in rectangular -metric spaces and prove the existence and uniqueness of fixed point for the mappings introduced. Our results extend and generalize related fixed point results in the existing literature. We have also supported the main result of this paper by applicable examples. Furthermore, we determined a solution to an integral equation by applying our obtained results.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally. All authors read and approved the final manuscript.

Acknowledgments

The authors would like to thank the College of Natural Sciences, Jimma University, for funding this research work.

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Copyright

Copyright © 2022 Mustefa Abduletif Mahmud and Kidane Koyas Tola. 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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