Generalization Contractive Mappings on 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="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 Space

Patil, Jayshree; Hardan, Basel; Hamoud, Ahmed A.; Bachhav, Amol; Emadifar, Homan; Günerhan, Hatira

doi:https://doi.org/10.1155/2022/7291001

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Research Article | Open Access

Volume 2022 | Article ID 7291001 | https://doi.org/10.1155/2022/7291001

Generalization Contractive Mappings on Rectangular -Metric Space

Jayshree Patil,¹Basel Hardan,²Ahmed A. Hamoud,³Amol Bachhav,⁴Homan Emadifar,⁵and Hatira Günerhan⁶

Academic Editor: Remi Léandre

Received06 Feb 2022

Revised21 Apr 2022

Accepted26 Apr 2022

Published14 May 2022

Abstract

In this paper, we introduce new coincidence fixed point theorems for generalized -contractive mappings fulfilling kind of an admissibility provision in a Hausdorff -rectangular metric space with the support of C-functions. We applied our results to establish the existence of a solution for some integralitions. Finally, an example is presented to clarify our theorem.

1. Introduction

One of the main results in the development area of fixed point theorems is the Banach contraction principle [1]. It has been evolutionized and generalized in several directions. From those directions, we faced some new kinds of metric spaces in literature as the one established by Branciari [2] and gave the principle of a rectangular metric space in which the replacement of the triangle inequality with a weaker hypothesis called quadrilateral inequality and an analogue of the Banach contraction principle is shown. Then next, fixed point results in these spaces were studied by many authors. For more informations on fixed point theorems, in rectangular metric space, see [3–10]. In fact, a contraction principle in rectangulat -metric space and its properties appeared by George [11]. Many definitions of various mathematical concepts and terms in rectangular -metric space can be found in [12–19]. Separation of the Hausdorff space from rectangular -metric space is not useful for our theorem, as Hausdorff space plays an important role in Theorem 16 and its corollaries. On another hand, Samet in [20] introduced the principle of ()-contractive mapping. Newly, two separate evaluations of -admissible mapping were introduced in which the researcher Ansari in [21] used the notion of -functions, whereas Budhia et al. in [22] used a rectangular metric space. By ideas from [21, 22], we prove several coincide fixed point results in rectangular b-metric space. That should be considered as development of [23], which are applied to find the existence and uniqueness of a solution for many problems in different mathematical branches. Moreover, one of the most attractive research subjects in fixed point theorem is the investigation of the existence and uniqueness of coincidence points of various operators in the setting of metric spaces (see [24–29]).

2. Mathematical Preliminaries

We recall some basic notions and needful results on the work in the literature.

Definition 1 (see [2]). Suppose is a nonempty set. A function is a rectangular metric (RM) on if, for all and all distinct points such that , the following terms hold (i), if and only if (ii)(iii) (rectangular inequality)

Then, is called a rectangular-metric space.

Definition 2 (see [11, 30]). Let be a nonempty set and the mapping satisfies (i), if and only if for all (ii) for all (iii)there exists a real number such that For all and all .
Then, is called a RM on and is called a rectangular -metric space with coefficient .

Remark 3 (see [31]). The type of rectangular -metric space is greater than the type of metric space, where a -metric space is a metric space when .

Example 1 (see [31]). Let and , such that . Then, is a MS, with .

Remark 4 (see [11]). Every metric space is a rectangular metric space, and every rectangular metric space is a rectangular -metric space, with . However, the opposite of implying above is not valid.

Example 2 (see [11]). Suppose, and as Consider . Hence, is a rectangular -metric space, with , but is not rectangular metric space, take

In 1914, German mathematician Felix Hausdorff [32] defined a new distance idea called Hausdorff metric, denoted by , as where

The Hausdorff distance is the largest one of all the distances measured from one set to another.

Definition 5 (see [33]). A Hausdorff space is a topological space where for any two distinct points , there exist neighbourhoods of each are disjoint from each other.

Definition 6 (see [11]). Let be a rectangular -metric space, be a sequence in , and . Then, (i)a sequence is said to be convergent in and converges to , if for all there exists such that , for all and this truth is acted via (ii)a sequence is said to be Cauchy sequence in if for all there exists such that , for all or equivalently, if , for all (iii) is said to be a complete rectangular -metric space if every Cauchy sequence in converges to some

The next lemmas are helpful in providing principle outcomes.

Lemma 7 (see [34]). Let be a rectangular -metric space with and let be a Cauchy sequence in such that when it was . Then, be able convergence at most one point.

Lemma 8 (see [34]). Let be a rectangular -metric space with . (i)Suppose that the sequences where , such that , for all . Thus, we have(ii)Suppose and are a Cauchy sequence in where , for all . where . Thus, we have

Definition 9 (see [35, 36]). Let be a self-mapping on a metric space and let be a function. is called a -admissible function if

Definition 10 (see [35, 36]). Let be a self-mapping on a metric space . A map is called a ()-contractive mapping if there exist two functions and such that where is a nondecreasing functions such that where is the iteration of .
For more informations for -admissible and ()-contractive mappings, see [35–37].

Definition 11 (see [38]). Let be a self-mapping on a metric space and let are two mappings. A function is called -admissible with regard to if

Observe that, if . Thus, this definition led to Definition 9. Likewise, if we pick , then we state that is a -subadmissible functions.

Definition 12 ([21]). A -function is a continuous functions such that (i)(ii)For all .

Definition 13 (see [39]). A nondecreasing continuous map is called an altering distance mapping whenever .

Definition 14 (see [29]). Let be a rectangular metric space and let in Definition 10. is said to be -orderly with respect to if for with , for all where ; therefore, , for all .

Remark 15. In altering the type of distance mapping, we denoted it by symbol .

In the next section, we present a new coincide FPT by the generalization ()-contractive mappings on rectangular -metric space fulfilling -admissibility by the concept of -functions.

A lot of authors used altering functions to prove the existence and uniqueness of the fixed point; see [3, 22, 34]. We will use generalizing some of them to prove our results.

3. Main Results

We will introduce new results of coincide fixed point in RMS. Let us start with the following.

Theorem 16. Let be a Hausdorff rectangular -metric space, with . Assume that be an -admissible function with respect to such that . Assume that is a complete rectangular metric space. Let C-functions and such that, for all , where Let that (a), for all (b)either is continuous or is -orderly with respect to Then, there exist such that , for some ; i.e., is a periodic point. If all periodic point satisfies we can decide that and have a fixed point. The fixed point is unique if for all , such that

Proof. We shall prove the existence of coincide point of and . Assume that be an arbitrary point such that

Consider the iteration

such that , for all . Since satisfied Definition 11 and by (16) using (15), we have

In the beginning, we will show that ; i.e., is nonincreasing.

By (11), we get where

We have two cases.

Case 1. If for some . Thus, inequality (18) will become Since is nondecreasing function, was obtained; is a nonincreasing sequence which and satisfies Also, since are continuous functions, Thus, By Definition 13, we get ; hence,

Case 2. If for some . Thus, inequality (18) becomes By Definition 13 we get either or and then , but this is a contradiction with .

In the next step, we will show that .

By (11), we have

Well, it could be

since is altering distance. We find that

We obtain , when . Therefore, by (27)

Consequently, by Definition 13, we get

The next lemma is useful for the rest and its proof is classical. We omit it.

Lemma 17. Let be a rectangular -metric space with and let be a sequence in such that where for all . If is not a -Cauchy sequence, then there exist and two subsequences , where . Also, such that for the next sequences it satisfies

Now, replace by and by in (11). Then, where

Taking in (36), we obtain

This implies that or ; thus, , but this is a contradiction with the fact . Hence, is a Cauchy sequence in a rectangular -metric space. Since is complete in a rectangular -metric space, then there exist such that .

For the case that is continuous and by relation (16) we have

is Hausdorff; then . Hence, and have a common fixed point. On the other case, assume that is -orderly with respect to , then from , for all we have where

Since .

Then, and . Hence, and have a periodic coincide fixed point.

Now, we will prove that and have a coincide fixed point. Assume that is the coincide of and such that . When , then , is a coincide of and . We shall show is the coincide of and in case . Assume that and , for all , such that for a periodic point . Therefore, from (11) and (12), where

Take . Then,

Either , i.e., ; this leads to which is a contradiction. Hence, our assumption that is a coincide fixed point of and is not true. Accordingly, and have a coincide fixed point.

To make sure the uniqueness of the coincide fixed point, let us assume that such that are two coincide fixed points of and . By the inequality and (11) and (12), we get where

Then, . Applying it in (46), we get

Thus, either or , which implies that . Hence, . This proves the uniqueness coincide fixed point of and on .

Corollary 18. Let be a complete Hausdorff rectangular -metric space, with . Let be a self-mapping such that satisfies For all , such that . Then, and have a unique coincide fixed point in .

Corollary 19. Let be a complete Hausdorff rectangular -metric space, with . Let be a self-mappings such that satisfies For all such that . Then, and have a unique coincide fixed point in .

Corollary 20. Let be a complete RMS Hausdorff, with . Let be the self-mappings such that satisfies where For all such that . Then, and have a unique coincide fixed point in .

Theorem 21. Let be a complete Haudorff rectangular -metric space, with . Let be the self-mappings such that satisfies For all , such that Then, and have a unique coincide fixed point.

Proof. Assume that be an arbitrary point. Since , consider the sequence such that for all . Let that for all . We have from (11) where The remainder of the proof is identical as the proof of Theorem 16.

4. Applications

Definition 22. Let be the class of functions such the following are satisfying (i) is Lebesgue integral function for all compact subset of (ii) for each

Theorem 23. Let be a complete Hausdorff rectangular -metric space, with . Let be a self-mappings. Assume that is a complete rectangular -metric space and that the next condition holds , such that T and F satisfy inequality (11), where Then, and have a unique coincide fixed point.

Proof. Let that and . Then, . Hence, by Theorem 16, and F have a unique coincide fixed point.

Theorem 24. Let be a complete RMS Hausdorff with . Let be a self-mappings, such that For all and , such that and satisfy inequality (11), where Then, and have a unique coincide fixed point.

Proof. Suppose . Hence, by Theorem 23, and have a unique fixed point.

Theorem 25. Let be a complete Hausdorff rectangular -metric space, with . Assume that be a self-mappings. Let For all , such that and satisfy inequality (11), where Then and have a unique coincide fixed point.

Proof. Suppose and . Then, . Hence, by Theorem 21, and have a unique coincide fixed point.

Example 3. Suppose and define such that Consider since and for all . Assume be rectangular -metric space, with , where

It is very well maybe the next step is to: (i) and if (ii) and if (iii) and if

Consider the function and defined as and . Then, Theorem 16 has been fulfilled. Hence, is a unique coincide fixed point of and .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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