A Fixed Point Theorem for Three Pairs of Mappings Satisfying Contractive Condition of Integral Type in Dislocated Metric Space

Abstract

The aim of this paper is to study a common fixed point theorem for three pairs of self-mappings satisfying a contractive condition of integral type in the setting of dislocated metric space. We notice that our established theorem generalizes the main result of Branciari (2002) in the context of dislocated metric space.

1. Introduction

In 1922, Banach proved a fixed point theorem for contraction mapping in complete metric space. Banach contraction theorem is one of the pivotal results of functional analysis. It has many applications in various fields of mathematics such as differential equations and integral equations. There are many generalizations of Banach contraction theorem in the literature. One of the most interesting generalizations of it is that of Branciari [1]. Branciari [1] proved a fixed point theorem for a single map satisfying an analogue of Banach contraction principle of integral type. Furthermore, authors in [2, 3] proved fixed point theorems satisfying more general contractive conditions of integral type in metric space.

The notion of dislocated metric (-metric) space was introduced by Hitzler and Seda in [4, 5]. This notion plays a vital role in logic programming semantics, electronic engineering, and computer science [5]. Compatible mappings were introduced by Jungck in [6]. The same author in [7, 8] generalized the concept of compatible mappings and introduced the concept of weakly compatible mappings. Since then several papers have been published containing fixed point results for weakly compatible maps in dislocated metric space (see [9–11]). Moreover, Al-Thagafi and Shahzad [12] initiated the concept of occasionally weakly compatible (owc) mappings. In the present days occasionally weakly compatible mappings become an interesting research topic in the field of metric fixed point theory.

In this paper, we have proved a common fixed point theorem for six self-mappings satisfying a new type of contractive condition of integral type using the idea of weakly compatible and commuting maps in the frame work of dislocated metric space.

2. Preliminaries

Throughout this paper will represent the set of nonnegative real numbers.

Definition 1 (see [5]). Let be a nonempty set and a function satisfying the following conditions:; implies ;; for all .If satisfies the conditions from to then it is called metric on ; if satisfies conditions to then it is called dislocated metric (-metric) on . The pair is called dislocated metric space.

Clearly every metric space is a dislocated metric space but the converse is not necessarily true as clear from the following example.

Example 2. Let define the distance function by Clearly is a dislocated metric space but not a metric space.

The following definitions are required in the sequel which can be found in [5].

Definition 3. A sequence in -metric space is called Cauchy sequence if for there exists a positive integer such that, for , one has

Definition 4. A sequence in dislocated metric space is called dislocated convergent (-convergent) if In such a case is called the dislocated limit (d-limit) of the sequence .

Definition 5. A -metric space is said to be complete if every Cauchy sequence in converges to a point in .

Definition 6. Let be a -metric space. A mapping is called contraction if there exist such that

Lemma 7 (see [5]). Limit in -metric space is unique.

Theorem 8 (see [5]). Let be a complete -metric space a contraction. Then has a unique fixed point.

Definition 9 (see [9]). Let and be two self-mappings on a nonempty set ; then(1)any point is said to be fixed point of if ;(2)any point is called coincidence point of and if and one calls a point of coincidence of and ;(3)a point is called common fixed point of and if .

Definition 10 (see [13]). Let and be two mappings on a nonempty set . Then and are said to be a commuting pair if

In [6] Jungck introduced the concept of compatible mappings which generalize the concept of commuting maps.

Definition 11. Let and be self-mappings on a nonempty set . Then and are said to be compatible mappings if whenever there exists a sequence in such that

Clearly compatible mappings commute at their coincidence sequence.

Jungck in [7] further generalized the concept of compatible maps as follows.

Definition 12 (see [7]). Let and be two self-mappings on a nonempty set . Then and are said to be weakly compatible if they commute at all of their coincidence points; that is, for some and then .

Obviously compatible mappings are weakly compatible but the converse is not true.

Example 13. Let , with the usual metric . Define by where denotes the integral part of . In the above if , then and are not compatible but they are weakly compatible as they commute at their coincidence points; that is, .

The following concept, introduced by Al-Thagafi and Shahzad in [12], is a proper generalization of nontrivial weakly compatible maps which do have a coincidence point.

Definition 14. Let and be two self-mappings on a nonempty set . Then and are said to be occasionally weakly compatible (owc) if there exists at least one coincidence point of and at which they commute; that is, implies that for any coincidence point .

The following example shows that the weakly compatible maps form a proper subclass of occasionally weakly compatible maps.

Example 15. Let with the usual metric. Define by and for all . Then and but . Therefore are an owc pair but not weakly compatible.

Lemma 16 (see [14]). Let , , and be self-mappings on a nonempty set with , , and having a unique point of coincidence in . If and are weakly compatible. Then , , and have a unique common fixed point.

In 2002, Branciari [1] proved the following theorem which is one of the interesting generalizations of Banach contraction principle.

Theorem 17. Let be a complete metric space . Let be a mapping such that for all the following condition holds: where is a Lebesgue integrable mapping which is summable on each compact subset of , nonnegative and such that for any . Then has a unique fixed point.

Definition 18 (see [15]). A map is called comparison function if it satisfies the following: is monotonic increasing.The sequence converges to zero for all where stands for th iterate of . If satisfies which converge for all , then is called -comparison function.

Thus every comparison function is -comparison function. A prototype example for comparison function is

Some more examples and properties of comparison and -comparison function can be found in [15].

3. Main Result

Theorem 19. Let be a complete -metric space. Let be self-mappings satisfying the following conditions: (1), ;(2)the pairs and are weakly compatible mappings;(3);where for all and is a -comparison function with being a Lebesgue integrable mapping which is summable on each compact subset of , nonnegative and such that for any . Then , , , and have a unique common fixed point. Moreover, if and are commuting pairs, then , , , , , and have a unique common fixed point.

Proof. Using condition we construct the Jungck sequence by the rule Now using we have Using the defined construction of the sequence we have
Finally we have If , then we have which is a contradiction. Therefore ; hence we have Also Similarly proceeding we have Since is -comparison function so by taking limit implies . Therefore Hence is a Cauchy sequence in complete -metric space. So there must exist such that Also the subsequences and converge to . So , , , and converge to . Using since so there must exist such that . Now using we have Taking limit we get the following inequality: Since , therefore ; which is a contradiction; therefore . Hence .
Also since so there must exist such that . Again using we have Taking limit we have Since , so which is again a contradiction. Therefore . Hence . Thus is the point of coincidence of , , , and . Now we have to show that the point of coincidence of , , , and is unique.
Let be two distinct points of coincidence of , and . Then . Consider Using we have Since implies , therefore : which is a contradiction; therefore . Hence point of coincidence of , , , and is unique. Also since and are weakly compatible so by Lemma 16, , , and have a unique common fixed point in . That is, .
Now if and are commuting pairs then which implies that is the fixed point of but as proved above fixed point of is unique. Therefore which implies that is the fixed point of . Also Using the similar argument as above we can get that . Thus is the fixed point of . Similarly one can easily show that and . Hence , and have a common fixed point in .
Uniqueness. Let be two distinct common fixed points of , , , , , and ; then consider Since then Hence which is again a contradiction. Therefore . Thus common fixed point of , , , , , and is unique.

We deduce the following corollary from Theorem 19.

Corollary 20. In Theorem 19 if and all other conditions of the above theorem hold, then again , , , , , and have a unique common fixed point.

Remarks. Corollary 20 is the result of Chauhan and Utereja [16] for weakly compatible mappings.

Theorem 19 is a generalization of the main result of Branciari [1] in dislocated metric space.

Example 21. Let with dislocated metric on which is defined by for all . Define , , , , , and as Satisfy all the conditions of Theorem 19 for having as the unique common fixed point of , , , , , and .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the editor and anonymous reviewer for their careful reviews, valuable comments, and remarks to improve this paper.