Common Fixed-Point Theorems of Generalized <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 50.2543 16.7875" width="50.2543pt"><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="g113-245" 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,-0.017,17.163,0)"><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,-0.017,23.951,0)"><path id="g113-253" d="M559 271C559 395 493 448 433 448C370 448 304 394 275 297C259 245 235 121 219 37C157 54 113 110 113 201C113 310 180 386 276 429L265 457C131 410 23 326 23 186C23 83 86 5 211 -10C180 -159 167 -220 158 -252L165 -261C198 -255 236 -246 251 -218L287 -5C421 9 559 119 559 271ZM484 273C484 144 400 40 293 33L352 331C364 389 382 406 405 406C431 406 484 370 484 273Z"/></g><g transform="matrix(.017,0,0,-0.017,33.94,0)"><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><g transform="matrix(.017,0,0,-0.017,39.877,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g></svg>Weakly Contractive Mappings in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4618998pt" id="M2" height="12.7225pt" version="1.1" viewBox="-0.0657574 -12.2606 18.3204 12.7225" width="18.3204pt"><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><g transform="matrix(.017,0,0,-0.017,8.053,0)"><use xlink:href="#g117-33"/></g></svg>Metric-Like Spaces and Application

Guan, Hongyan; Li, Jianju

doi:https://doi.org/10.1155/2021/6680381

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

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Recent Advances in Fixed Point Theory in Abstract Spaces

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Research Article | Open Access

Volume 2021 | Article ID 6680381 | https://doi.org/10.1155/2021/6680381

Common Fixed-Point Theorems of Generalized Weakly Contractive Mappings in Metric-Like Spaces and Application

Hongyan Guan¹and Jianju Li¹

Academic Editor: Naeem Saleem

Received06 Dec 2020

Revised26 Dec 2020

Accepted30 Dec 2020

Published20 Jan 2021

Abstract

In this paper, we prove some common fixed-point theorems of generalized weakly contractive mappings in metric-like spaces. We also give two examples to support our results. Meanwhile, we present an application to the existence of solutions for a system of integral equations by means of our results.

1. Introduction

The Banach contraction mapping theorem [1] popularly known as Banach contraction mapping principle is a rewarding result in fixed-point theory. It has widespread applications in both pure and applied mathematics. This celebrated principle has been generalized by several authors. Recently, Saleem et al. [2] obtained fixed-point results of Suzuki-type generalized multivalued -almost contractions and coincidence and common fixed-point results of Suzuki-type generalized multivalued -almost contraction mapping in the setting of metric spaces. In 2019, Li et al. [3] defined a new contractive-type mapping called -contraction and proved some fixed-point and Suzuki-type fixed-point results in the context of complete metric spaces. Another extension idea is the promotion of spaces. In 1993, Czerwik [4] introduced the concept of the metric space, and he also obtained some fixed-point theorems of contractive mappings in metric space. Since then, some papers have generalized the fixed-point of different contractive conditions in metric space. For example, Aydi et al. [5] proved common fixed-point results for weak contractions on metric space, and Abbas et al. [6] also proved common fixed-point theorems of four mappings in metric space. In [7], Hussain et al. established some best proximity point and coupled best proximity point results in the context of complete -metric spaces based on the concepts of -proximal admissible mappings and simulation function. Lately, Lael et al. [8] replied to an open problem related to a -metric version of Banach’s fixed-point theorem and obtained some new fixed-point theorems for multivalued mappings in -metric spaces.

Inspired by Czerwik’s work, some researchers had studied the fixed-point theorems of various new types of contractive conditions in the generalized metric space and metric space. In 2011, Hussain and Shah in [9] introduced the notion of cone metric spaces, which means that it is a generalization of metric spaces and cone metric spaces. They also considered topological properties of cone metric spaces and results on KKM mappings in the setting of cone metric spaces. Fernandez et al. [10] introduced the concept of -cone metric spaces over a Banach algebra as a generalization of -cone metric spaces over a Banach algebra and -metric spaces and studied some coupled common fixed-point theorems for generalized Lipschitz mappings in this framework. In 2019, Kanwal et al. [11] generalized Nadler’s theorem in weak partial -metric space by using weak partial Hausdorff -metric spaces. In 2020, Abbas et al. [12] introduced the concepts of -contraction and monotone -contraction correspondence in fuzzy -metric spaces and obtained fixed-point results for these contractive mappings. Ansari et al. [13] introduced the concept of inverse -class function in -metric setting and established some fixed-point theorems. Recently, Saleem et al. [14] proved some new fixed-point theorems, coincidence point theorems, and common fixed-point theorems for multivalued -contractions involving a binary relation that is not necessarily a partial order, in the context of generalized metric spaces (in the sense of Jleli and Samet).

The concept of contractive mappings was introduced by Rhoades [15]. Afterwards, some researchers introduced a few and weakly contractive conditions and discussed the existence of fixed and common fixed-point for these mappings [16–18]. In particular, Aghajani et al. [19] presented several common fixed-point results of generalized weak contractive mappings in partially ordered –metric spaces.

Our main concern is to study common fixed-point results involving generalized –weakly contractive conditions in metric-like space. Furthermore, we apply the given results to obtain existence of solutions of integral equations.

2. Preliminaries

Throughout this paper, let denote the set of all positive integers, and . Put

In order to get our main results, we introduce some definitions and lemmas as follows.

Definition 1. (see [4]). Let be a nonempty set and be a given real number. A mapping is said to be a metric if and only if, for all , the following conditions are satisfied:(1) if and only if (2)(3)

The pair is called a metric space with parameter .

In general, the class of metric space is effectively larger than that of metric space, since a metric is a metric with . We can find several examples of metric spaces which are not metric spaces (see [20]).

Definition 2. (see [21]). Let be a nonempty set and be a given real number. A mapping is said to be a metric-like if and only if, for all , the following conditions are satisfied:(1) implies (2)(3)

The pair is called a metric-like space with parameter .

Remark 1. We should note that in a metric-like space , if and , then . But the converse need not be true, and may be positive for .

Example 1. Let , and let the mapping be defined by for all . Then, is a metric-like space with parameter .

Proof. We can infer from the convexity of the function () that holds. Then, for , we haveIn this case, is a metric-like space with parameter .

Definition 3. (see [21]). Let be a metric-like space with parameter and be a sequence in .(1)The sequence is said to be convergent to if .(2)The sequence is said to be a Cauchy sequence if and only if exists and is finite.(3) is said to be complete if for each Cauchy sequence in , there exists an such that .

Definition 4. (see [22]). Let and be two self-mappings on a nonempty set . If , for some , then is said to be the coincidence point of and , where is called the point of coincidence of and . Let denote the set of all coincidence points of and .

Definition 5. (see [22]). Let and be two self-mappings defined on a nonempty set . Then, and are said to be weakly compatible if they commute at every coincidence point, that is, for every .

Lemma 1. (see [21]). Let be a metric-like space with . We assume that and are convergent to and , respectively. Then, we haveIn particular, if , then we have . Moreover, for each , we haveIn particular, if , then

Lemma 2. (see [23]). Let be a metric-like space with . Then,(1)If , then (2)If is a sequence such that , then we have (3)If , then

3. Main Results

In this section, we will show the existence and uniqueness of common fixed-point for generalized weakly contractive mappings in complete metric-like space. Meanwhile, we give two examples to support our results.

Theorem 1. Let be a complete metric-like space with parameter and let be given self-mappings satisfying where is a closed subset of . If there are functions and such thatwherethen and have a unique coincidence point in . Moreover, and have a unique common fixed-point provided that and are weakly compatible.

Proof. Let . As , there exists such that . Now we define the sequences and in by for all . If for some , then we have and and have a coincidence point. Without loss of generality, we assume that (by Lemma 2, we know that ) for all . Applying (6) with and , we obtainwhereIf , for some , in view of (9) and (10), we haveIt follows from inequality (8) and the above inequalities thatwhich implies , that is, , a contradiction. Hence, and is a nonincreasing sequence, and so there exists such that
By virtue of (9) and (10), we haveIt follows thatNow suppose that . By taking the limit as in (12), we have , a contradiction. This yields thatNow we shall prove that . Suppose on the contrary that . It follows that there exists for which one can find sequences and of where is the smallest index for which , and
In view of the triangle inequality in metric-like space, we getUsing equality (15) and taking the upper limit as in the above inequality, we obtainAs the same arguments, we deduce the following results:In view of (18), we haveUsing (19) and (20), we obtainSimilarly, we deduce thatIt follows thatThrough the definition of , we havewhich yields thatAlso,It is easy to show thatApplying (6) with and , we getIn light of (26), one can obtainwhich implies thata contradiction to (28). It follows that is a Cauchy sequence in and . Since is complete metric-like space, there exists such thatFurthermore, we have since is closed. It follows that one can choose a such that , and one can write (32) asIf , taking and in contractive condition (6), we getwhereand then we obtainTaking the upper limit as in (34),which implies that
It follows that . That is, . Therefore, is a point of coincidence for and . We also conclude that the point of coincidence is unique. Assume on the contrary that there exist and ; applying (6) with and , we obtain thatHence, . That is, the point of coincidence is unique. Considering the weak compatibility of and , it can be shown that is a unique common fixed-point. This completes the proof.

Theorem 2. Let be a complete metric-like space with parameter and let be given self-mappings satisfying where is a closed subset of . If there are functions and such thatwherethen and have a unique coincidence point in . Moreover, and have a unique common fixed-point provided that and are weakly compatible.

Proof. It is the same as the proof of Theorem 1, and we also define the sequences and in by for all . We also suppose that for each , and it follows from (39) thatwhereIf we assume that for some ,then from inequalities (42) and (43), we get thatIn view of (41), we have the following inequality:which gives that , a contradiction to . It follows that . Hence, is a nonincreasing sequence. Consequently, the limit of the sequence is a nonnegative number, say . That is, .
According to (42) and (43), we haveSo,If , then letting in above inequality, we obtain that which implies that , i.e.,
Now we prove that . If not, as the proof of Theorem 1, there exists for which one can find sequences and of so that is the smallest index for which , and the following inequalities hold:We deduce the following equations according to the definitions of and :using (49), one can obtain thatTaking and in (39), we getTherefore, we haveand we conclude that which gives a contradiction to (52). Hence,
The completeness of ensures that there exists such thatIn view of the hypothesis is closed, we obtain that . It follows that one can choose such that , and we write the above equality asIf , putting and into contractive condition (39), we havewhereConsequently, we getTaking the upper limit as in (57), we havewhich implies that . That is, is a point of coincidence for and . Using the same technique in the proof of Theorem 1, it can be proved that is a unique common fixed-point. This completes the proof.

The following examples support Theorems 1 and 2.

Example 2. Let be endowed with the metric-like for all and . Define mappings by and
The control functions are defined as for all . It is clear that and is closed. For all , we have It follows thatIt is easy to see thatTherefore, the conditions of Theorem 1 are satisfied. It is obvious that 0 is the unique common fixed-point of and .

Example 3. Let and for all , so is a metric-like space with parameter . Let be defined by the formulasThe control functions are defined as and , for all , where .
Now we consider four cases: Case 1. . It is clear that so we have Case 2. . It follows that and we obtain that Case 3. . By concise calculation, we get It follows that Case 4. . It is easy to see that Obviously, for all , we obtain It follows from Theorem 2 that and have a unique common fixed-point in . It is easy to show that 0 is the unique common fixed-point of and .

According to Theorems 1 and 2, one can get the following results.

Corollary 1. Let be a complete metric-like space with constant and let be given self-mappings satisfying where is a closed subset of . If the following condition is satisfied:where L ∈ (0,1) is a constant and then and have a unique coincidence point in . Moreover, and have a unique common fixed-point provided that and are weakly compatible.

Corollary 2. Let be a complete metric-like space with constant and let be given self-mappings satisfying where is a closed subset of . If the following condition is satisfied:then and have a unique coincidence point in . Moreover, if and are weakly compatible, then and have a unique common fixed-point.

4. Application

In this section, we will use Corollary 1 to show that there is a solution to the following system of integral equations:

Let be the set of real continuous functions defined on for . Define the metric-like mapping : byfor all , where . It is evident that with is a complete metric-like space. Consider the mappings by

Theorem 3. Consider the system of integral equations (75) and suppose that the following conditions hold:(i) is continuous(ii)If for all , then we have (iii)There exists a continuous function such that for all (iv)There exists constant such that for all ,

Then, the system of integral equations (75) has a unique solution in .

Proof. Obviously, by condition (ii), are weakly compatible. Let ; from conditions (i), (iii), and (iv), for all , we havewhich implies thatConsequently, letting and , all of the conditions of Corollary 1 are satisfied. As a result, the mappings have a unique common fixed-point in , which is the solution of the system of integral equations (75).

5. Conclusions

In this paper, we introduced new generalized weakly contractive mappings and obtained common fixed-point theorems in the framework of metric-like space. Further we provided examples that elaborated the useability of our results. As an application of our result, we obtained a solution of the system of integral equations.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

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

Acknowledgments

This study was financially supported by the Science and Research Project Foundation of Liaoning Province Education Department (nos. LQN201902 and LJC202003).

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Copyright © 2021 Hongyan Guan and Jianju Li. 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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