Common Fixed Point Results for Generalized <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.3443pt" id="M1" height="12.2127pt" version="1.1" viewBox="-0.0657574 -7.8684 20.5604 12.2127" width="20.5604pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><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(.012,0,0,-0.012,9.318,4.134)"><path id="g50-116" d="M357 391C357 416 325 451 270 451C241 451 179 431 146 400C109 365 97 334 97 307C97 248 143 213 196 180C250 146 261 123 261 101C261 75 237 40 190 40C153 40 110 72 86 109C81 116 68 119 55 112C37 102 24 87 24 66C24 30 85 -12 137 -12C229 -12 331 62 331 140C331 189 304 218 236 260C197 284 164 309 164 346C164 381 194 400 220 400C259 400 282 380 304 350C310 342 321 339 330 344C347 353 357 372 357 391Z"/></g><g transform="matrix(.009,0,0,-0.009,13.894,-.086)"><path id="g176-113" d="M575 299C575 406 528 454 412 454C385 454 342 447 313 440L330 514L318 525C293 510 258 487 242 466L227 419C194 403 158 384 123 361L126 327C155 342 183 355 220 364L107 -148C96 -198 76 -210 19 -215L14 -253L259 -241V-206L240 -205C184 -202 183 -188 192 -141L220 -1C242 -10 269 -12 284 -12C352 5 434 48 484 102C542 165 575 238 575 299ZM479 290C479 170 380 41 305 41C281 41 249 60 237 72L300 395C329 399 353 401 374 401C426 401 479 377 479 290Z"/></g></svg> Contractive Mappings and Applications

Zoto, Kastriot; Vardhami, Ilir

doi:https://doi.org/10.1155/2018/1282414

Journal of Function Spaces

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

Research Article | Open Access

Volume 2018 | Article ID 1282414 | https://doi.org/10.1155/2018/1282414

Common Fixed Point Results for Generalized Contractive Mappings and Applications

Kastriot Zoto¹and Ilir Vardhami²

Academic Editor: Tomonari Suzuki

Received21 Feb 2018

Accepted17 Apr 2018

Published03 Jun 2018

Abstract

A class of -admissible mappings and general type contraction mappings on -metric-like space are defined. Some fixed point results dealing with such a class of contractions are obtained. The generalized contractions considered in this work cover and unify many particular types of contractions. Finally, we present application to the existence of solutions for a system of integral equations by means of -metric-like spaces.

1. Introduction

The study of new classes of spaces and their basic properties are always favorite topics of interest. Recently some authors have introduced some generalizations of metric spaces in several ways and have studied fixed point problems in these spaces, as well as their applications. In this context, Matthews [1] introduced the notion of partial metric space where self-distance of an arbitrary point needs not be equal to zero. Hitzler and Seda [2] and Amini-Harandi [3] made further generalizations under the name of dislocated, respectively, metric-like space. Further, Shukla et al. introduced in [4] the notion of complete metric space. This concept was further extended by Alghamdi et al. [5] under the name of -metric-like spaces. This class of spaces has received significant attention lately.

Also these generalizations have been associated with new and generalized classes of contractive mappings. In this direction Samet et al. [6] introduced the concept of -admissible and -contractive mappings, further extended to the -contractive mappings, -contractive mappings, and -admissible mappings. Many papers dealing with the above notions have been considered to prove fixed point results (e.g., see [7–25]).

In this paper, following the above discussion, we introduce the notion of -admissible mapping and also introduce the concepts of quasi-contractive mappings and rational contractive mappings in the larger framework of -metric-like space. The considered contractive conditions not only generalize the known ones but also include and unify a huge number of existing results on the topic in the corresponding literature. Finally, we apply the given results to obtain existence of solutions of integral equations.

2. Preliminaries

Definition 1 (see [8]). Let be a nonempty set and be a given real number. A mapping is called a -metric if, for all , the following conditions are satisfied: (i) if and only if ;(ii);(iii).

The pair is called a -metric space with parameter .

Definition 2 (see [26]). Let be a nonempty set. A mapping is called metric-like if, for all , the following conditions are satisfied: (i) implies ;(ii);(iii).

The pair is called a metric-like space.

Definition 3 (see [5]). Let be a nonempty set and be a given real number. A mapping is called -metric-like if, for all , the following conditions are satisfied: (i) implies ;(ii);(iii).

The pair is called a -metric-like space.

In a -metric-like space , if and , then , but the converse needs not be true, and may be positive for .

Example 4 (see [18]). Let and be a constant. Define a function by or . Then is a -metric-like space with parameter . Clearly, is neither a -metric, nor metric-like, nor partial -metric space.

Definition 5 (see [5]). Let be a -metric-like space with parameter , and let be any sequence in and . Then (1)The sequence is said to be convergent to if ;(2)The sequence is said to be a Cauchy sequence in if exists and is finite;(3) is said to be a complete -metric-like space if, for every Cauchy sequence in , there exists an such that .

The limit of a sequence in a -metric-like space need not be unique.

Proposition 6 (see [5]). Let be a -metric-like space with parameter , and let be any sequence in with such that . Then (i) is unique,(ii) for all .

In 2012, Samet et al. [6] introduced the class of -admissible mappings.

Definition 7. Let be a nonempty set, , and be mappings. We say that is an -admissible mapping if implies that , for all .

In 2014 a new notion of -admissible mapping was introduced by Rosa and Vetro [27].

Definition 8. For a nonempty set , let and be mappings. The mapping is called -admissible if, for all such that , we have .

Definition 9 (see [28]). Let be a -metric-like space with parameter , and let be a function and arbitrary constants , such that and . A self-mapping is -admissible if implies , for all .

Examples 3.3 and 3.4 in [28] illustrate Definition 9.

Lemma 10 (see [9]). Let be a -metric-like space with parameter and be a given mapping. Suppose that is continuous at . Then for all sequence in such that , we have ; that is,

Lemma 11. Let be a sequence in a -metric-like space with parameter , such that for some , where . Then (1),(2) is a Cauchy sequence in and .

Proof. For the proof of the lemma, one can use the following clear inequalities:where and .

Lemma 12 (see [5]). Let be a -metric-like space with parameter and suppose that and are -convergent to and , respectively. Then we have In particular, if , then we have .
Moreover, for each , we have In particular, if , then

The following result is useful.

Lemma 13 (see [28]). Let be a -metric-like space with parameter . Then (1)If , then ;(2)If is a sequence such that , then we have(3)If , then .

Lemma 14 (see [29]). Let be a complete -metric-like space with parameter and let be a sequence such that If for the sequence , , then there exist and sequences and of positive integers with , such that

3. Main Results

We begin this section with the following definition.

Definition 15. Let be a -metric-like space with parameter , and let and be given mappings and arbitrary constant such that . The mapping is -admissible if, for all implies .

Remark 16. Taking in definition we obtain an -admissible mapping defined in [27]. Taking as the identity mapping on , we deduce the definition of -admissible mapping as in [28]. For and the definition reduces to the definition of an -admissible mapping in a metric space [6].
In the sequel, according to [28] we shall consider the following properties in case of

Let be a complete -metric-like space with parameter and be a function. ThenIf is a sequence in such that as and , then there exists a subsequence of such that for all .For all , we have or , where denotes the set of all coincidence points of and .

Example 17. Let . We define the mappings by for all , and by Then is -admissible.

Extending the well known definition of quasi-contraction from Cirić, we introduce the notion of a generalized -quasi-contraction in the setting of a -metric-like space.

Definition 18. Let be a complete -metric-like space with parameter and be given mappings. We say that is a generalized -quasi-contraction if is a -admissible mapping satisfyingfor all and .

Remark 19. If we take (), the definition reduces to the definition of an quasi-contraction. If we take , the definition reduces to the -quasi-contraction in the setting of metric spaces.

Theorem 20. Let be a complete -metric-like space with parameter and be self-mappings on such that , or is a closed subset of , and is a given mapping. Suppose that the following conditions are satisfied: (i) is -admissible mapping;(ii) is a generalized contractive mapping;(iii)there exists such that ;(iv)properties and are satisfied. Then and have a unique point of coincidence in . Moreover, if and are weakly compatible, then and have a unique common fixed point.

Proof. By hypothesis (iii), there exists an such that . We define a sequence and in by for all . If for some , then and and have a point of coincidence. Without loss of generality, one can suppose that (by Lemma 13; i.e., ) for each .
Since is -admissible mapping, we have Hence, by induction, we get for all .
By condition (11), we haveIf for some , then, from inequality (13), we have a contradiction since .
Hence, for all , , and also by inequality (13), we getwhere .
Then, by Lemma 11, we haveand the sequence is a Cauchy sequence, and . By completeness of the sequence converges to a point . By hypothesis, since is closed, then . Therefore, there exists such that . That isSince property is satisfied, there exists a subsequence of such that (that is ) for all . If , applying contractive condition (ii) of theorem, with and , we obtainTaking the upper limit as in the above inequality, and using (16), (17), and Lemmas 12 and 13, we have Hence, from above we get ; that is, and is a point of coincidence of and . By using condition (11) and property one can be convinced that the point of coincidence is unique and by weak compatibility of and it follows that is a unique common fixed point.

Example 21. Let and ; for all is a -metric-like space with parameter Define the mappings and on by It is clear that . For such that ; then and this implies that . By definitions we have and ; that is, is -admissible mapping;
For , we have where .

Obviously the other assumptions of theorem can be verified and is the unique common fixed point of and .

Theorem 22. Let be a complete -metric-like space with parameter and be self-mappings on such that or is a closed subset of , and a given mapping. Suppose that the following conditions are satisfied: (i) is -admissible mapping;(ii)for all , and constants , , where ;(iii)there exists such that ;(iv)properties and are satisfied.Then and have a unique point of coincidence in . Moreover, if and are weakly compatible, then and have a unique common fixed point.

Proof. This theorem can be considered as a corollary of Theorem 20, since, for all , inequality (22) is a special case of (11).

If we consider (where ) in Theorem 20, then we deduce the following corollary.

Corollary 23. Let be a complete -metric-like space with parameter , and , be self-mappings on such that or is a closed subset of and satisfy the condition for all and .
Then and have a unique point of coincidence in . Moreover, if and are weakly compatible, then and have a unique common fixed point.

Remark 24. Theorem 20 generalizes Theorem 18 in [30]. For and, for each , Theorems 20 and 22 reduce to Theorems 3.2 and 3.13 of [31]. In Theorem 22, by choosing the constants in certain manner, we obtain, as particular cases, certain classes of types of classical contractions (such as Kannan, Chatterjea, Reich, and Zamfirescu contractions).

Contraction-type mappings have been generalized in several directions. A series of generalizations start with Samet et al. [6] with the concept of -admissible mappings and -contractive mapping. Later many authors used these classes of mappings under weakly and generalized weakly contractivity conditions and discussed fixed point results in various spaces.

The contractivity conditions considered in the second part of this section are constructed via auxiliary functions defined with the families , and , respectively: is an increasing and continuous function; is continuous function and for all ; satisfying the condition implies that as

Let be a -metric-like space with parameter . For two self-mappings , we define the set with the involvement of rational terms.for all .

We introduce now the notion of rational contraction in the setting of -metric-like spaces.

Definition 25. Let be a -metric-like space with parameter and two self-mappings. Also, let and . We say that is called a generalized contractive mapping, if there exist such thatfor all with , where is defined as in (24).

Remark 26. If we take as the identity mapping on , then we obtain the definition of contractive mapping as in [28].
Taking in the Definition 25, we obtain -contractive mappings.
For and the definition reduces to the definition of an -contractive mapping.
The definition reduces to a -contractive mapping if we take .
The definition reduces to an contractive mapping if we take .

Theorem 27. Let be a complete -metric-like space with parameter and , be self-mappings on such that or is a closed subset of , and a given mapping. Suppose that the following conditions are satisfied: (i) is admissible mapping;(ii) is a generalized contractive mapping;(iii)there exists such that ;(iv)properties and are satisfied.Then and have a unique point of coincidence in . Moreover, if and are weakly compatible, then and have a unique common fixed point.

Proof. From the similar arguments as in proof of Theorem 20, we construct the sequences and in by for all . Supposing that (which by Lemma 13 implies ) for each , we getBy (26) and condition (25) we havewhereIf we assume that, for some , then, from inequality (28), we getAgain, by (30) and using condition (27) and property of , we obtain Hence as a result we haveThen by (27) using (32) we obtain which gives a contradiction, since we have assumed that and property for all . So , for all . Hence, the sequence of nonnegative numbers is nonincreasing. Thus it converges to a nonnegative number, say . That is and also . If , then letting in (27) we get which implies ; that is,Now for the sequence we shall show that . Suppose, on the contrary, that . Then, using Lemma 14, we get that there exists for which we can find subsequences and of , with , such that the following hold: From the definition of , we haveTaking the upper limit as in (36), and using (34), (35), we getApplying (25) with and , we obtainTaking the upper limit in (38), using (35) and (37), we obtain which implies that , a contradiction with . Hence, ; that is, is a Cauchy sequence in . From the completeness of , there exists such thatBy hypothesis, since is closed, by (40), . Therefore, there exists such that . And (40) can be written as Since property is satisfied there exists a subsequence of such that (that is ) for all . If , applying contractive condition (26), with and , we obtainwhereTaking the upper limit in (43) and using Lemma 13 and (40), we obtainTaking the upper limit as in (42), and using (44) and Lemma 13, we obtainIn view of property of from (45), we get which implies that . Hence, is a point of coincidence for and . Similarly as in Theorem 20 by using condition (26) and property and weak compatibility it can be shown that is a unique common fixed point.

By taking , where , in Theorem 27, we obtain the following result.

Corollary 28. Let be a complete -metric-like space with parameter and , be self-mappings on such that or is a closed subset of , and a given mapping. Suppose that the following conditions are satisfied: (i) is admissible mapping;(ii)there exist functions such that(iii)there exists such that ;(iv)properties and are satisfied. Then and have a unique point of coincidence in . Moreover, if and are weakly compatible, then and have a unique common fixed point.

Theorem 29. Let be a complete -metric-like space with parameter and , be self-mappings on such that or is a closed subset of , and a given mapping. Suppose that the following conditions are satisfied: (i) is admissible mapping;(ii)there exist functions such that(iii)there exists such that ;(iv)properties and are satisfied. Then and have a unique point of coincidence in . Moreover, if and are weakly compatible, then and have a unique common fixed point.

Proof. This corollary is a special case of Theorem 27 since inequality (47) implies inequality (25).

By taking and , where in Theorem 27, we obtain the following result.

Corollary 30. Let be a complete -metric-like space with parameter and , be self-mappings on such that or is a closed subset of , and . Suppose that the following conditions are satisfied: (i) is a admissible mapping;(ii)there exists function such that(iii)there exists an such that ;(iv)properties and are satisfied.Then and have a unique point of coincidence in . Moreover, if and are weakly compatible, then and have a unique common fixed point.

If we take in Theorem 27, then we get the following result.

Corollary 31. Let be a complete -metric-like space with parameter and , be self-mappings on such that or is a closed subset of , and . Suppose that the following conditions are satisfied: (i) is a admissible mapping;(ii)there exist function such that(iii)there exists an such that ;(iv)properties and are satisfied. Then and have a unique point of coincidence in . Moreover, if and are weakly compatible, then and have a unique common fixed point.

Corollary 32. Let be a complete -metric-like space with parameter and , be self-mappings on such that or is a closed subset of , and . Suppose that the following conditions are satisfied: (i) is a admissible mapping;(ii)there exist functions such that(iii)there exists an such that ;(iv)properties and are satisfied. Then and have a unique point of coincidence in . Moreover, if and are weakly compatible, then and have a unique common fixed point.

Proof. It follows from Corollary 28 by taking .

Corollary 33. Let be a complete -metric-like space with parameter and , be self-mappings on such that or is a closed subset of , and . Suppose that the following conditions are satisfied: (i) is a admissible mapping;(ii)there exists function such that for all , where is defined as in (32) and ;(iii)there exists an such that ;(iv)properties and are satisfied. Then and have a unique point of coincidence in . Moreover, if and are weakly compatible, then and have a unique common fixed point.

Proof. In Theorem 27 take , where .

Corollary 34. Let be a complete -metric-like space with parameter and be two self-maps of and such that they satisfy the conditionfor all , where is defined as in (32) and . Then and have a unique common fixed point in .

Proof. In Theorem 27 take , where and .

Remark 35. Our results generalize, extend, and improve the results appearing in the literature [3, 7, 10, 12, 16, 21, 26, 28, 29, 31–34]. It is clear that several more corresponding results can be derived by our main theorems by choosing constant and the mappings , in a suitable way.

4. Application

In this section we will use Theorem 27 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 . We define a -metric-like by for all , where .

It is evident that is a complete -metric-like space.

Consider the mappings by and let be a given function.

Theorem 36. Consider the system of integral equations (53) and suppose that the following assertions hold: (i) (that is , ) is continuous;(ii)if for all , then we have (iii)there exists such that for all ;(iv)for all and , (v)properties and are satisfied;(vi)there is a continuous function such that for all , ;(vii)there exist constant , such that for all Then the system of integral equations (53) has a unique solution in .

Proof. We define a function by It is easy to see that the function is admissible. By condition (ii) and are weakly compatible.
Let be such that , that is, for all , then from conditions (iv), (vi), and (vii), for all , we have which implies thatTherefore, taking and , where , and in view of assertion (v) all of the conditions of Theorem 27 are satisfied, and, as a result, the mappings and have a unique common fixed point in , which is a solution of the system of integral equations in (53).

5. Conclusions

In this manuscript, we defined new rational contraction using a larger class of -admissible mappings and auxiliaries functions in the framework of -metric-like spaces. The presented main theorems of the paper cover and unify a huge number of published results and also complement the previous work on the topic in the related literature.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

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

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Copyright © 2018 Kastriot Zoto and Ilir Vardhami. 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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