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Journal of Function Spaces
Volume 2018, Article ID 1282414, 11 pages
https://doi.org/10.1155/2018/1282414
Research Article

Common Fixed Point Results for Generalized Contractive Mappings and Applications

1Department of Mathematics and Computer Sciences, Faculty of Natural Sciences, University of Gjirokastra, Gjirokastra, Albania
2Department of Mathematics, Faculty of Natural Sciences, University of Tirana, Tirana, Albania

Correspondence should be addressed to Kastriot Zoto; moc.oohay@toirtsakotoz

Received 21 February 2018; Accepted 17 April 2018; Published 3 June 2018

Academic Editor: Tomonari Suzuki

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.

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 [725]).

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 obtainwhere