Unique Coincidence and Fixed Point Theorem for 𝑔-Weakly C-Contractive Mappings in Partial Metric Spaces

Abstract

We prove that every map satisfying the -weakly C-contractive inequality in partial metric space has a unique coincidence point. Our results generalize several well-known existing results in the literature.

1. Introduction and Preliminaries

The Banach contraction principle is the source of metric fixed point theory. This principle had been extended by many authors in different directions (see [1]).

Chatterjea [2] introduced the following contraction which has been named later as C-contraction.

Definition 1 (see [2]). Let be a metric space and a mapping. Then is called a C-contraction if there exists such that holds for all , .

Under this kind of contractive inequality, Chatterjea [2] established the following fixed point result.

Theorem 2 (see [2]). Every C-contraction in a complete metric space has a unique fixed point.

As a generalization of C-contractive mapping, Choudhury [3] introduced the concept of weakly C-contractive mapping and proved that every weakly C-contractive mapping in a complete metric space has a unique fixed point.

Definition 3 (see [3]). Let be a metric space and a mapping. Then is called a weakly C-contractive if satisfies for all , , where is a continuous mapping such that if and only if .

Under this kind of contraction, Choudhury [3] established the following fixed point result.

Theorem 4 (see [3, Theorem 2.1]). Every weakly C-contraction in a complete metric space has a unique fixed point.

Recently, Harjani et al. [4] studied some fixed point results for weakly C-contractive mappings in a complete metric space endowed with a partial order. Moreover, Shatanawi [5] proved some fixed point and coupled fixed point theorems for a nonlinear weakly C-contraction type mapping in metric and ordered metric spaces.

In another aspect, the notion of a partial metric space has been introduced by Matthews [6] in 1994 as a generalization of the usual metric in such a way that each object does not necessarily have to have a zero distance from itself. A motivation behind introducing the concept of a partial metric was to obtain appropriate mathematical models in the theory of computation and, in particular, to give a modified version of the Banach contraction principle (see, e.g., [7, 8]). Subsequently, several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions on partial metric spaces (e.g., [9–13]).

We recall some definitions and properties of partial metric spaces.

Definition 5. A partial metric on a nonempty set is a function such that for all , , , (p1); (p2); (p3); (p4).
A partial metric space is a pair such that is nonempty set and is a partial metric on .

From the above definition, if , then . But if , may not be in general. A famous example of a partial metric space is the pair , where is defined as . For some more examples of partial metric spaces, we refer to [8, 12].

Each partial metric on generates a topology on which has as a base the family of open -balls: , where for all and . A sequence in converges to a point , with respect to if and only if . A sequence in is called Cauchy sequence if exists and is finite.

Definition 6 (see [6, 13]). Let be a partial metric space. Then,(i)a sequence in a partial metric space converges to a point if and only if ;(ii)a sequence in a partial metric space is called a Cauchy sequence if there exists (and is finite) ;(iii)a partial metric space is said to be complete if every Cauchy sequence in converges to a point ; that is, .

If is a partial metric on , then the function given by is a metric on .

Lemma 7 (see [6, 13]). Let be a partial metric space. Then,(a) is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space ;(b) is complete if and only if the metric space is complete. Furthermore, if and only if

Moreover, Bhaskar and Lakshmikantham [14] presented coupled fixed point theorems for contractions in partially ordered metric spaces. This concept attracted many mathematician and for more related work on coupled fixed and coincidence points results we refer the readers to recent works in [4, 9, 10, 15–19].

Definition 8. Let , be two mappings. One says that is a coincidence point of and if .

In this paper, we extend the concept of a weakly C-contractive mapping to the context of partial metric space and define -weakly C-contractive map. Moreover, we prove that every -weakly C-contractive mapping in a complete partial metric space has a unique coincidence point. Our result generalizes several well-known results in the literature.

2. Unique Coincidence and Fixed Point Theorem

Definition 9. Let be a partial metric space and a map. Then, the mapping is said to be -weakly C-contractive if for all , , , , where is a continuous mapping such that if and only if .

Now we state and prove our main result.

Theorem 10. Let be a complete partial metric space and a -weakly C-contraction mapping. Suppose that . Then, and have a unique coincidence point in .

Proof. Let be arbitrary point in . Since , we can construct sequence in as
Set .
If there exists such that , then by (p1) and (p2) we have . Hence, and have a coincidence point in . Now assume that for all . Thus by (5), we have
By property (p4), we have
Thus from (7), we have
From (9), we have either or since for all .
If then since and by (10), we have which is a contradiction. Thus, we have and therefore By the above inequalities, we have that is a non increasing sequence of positive real numbers. Therefore, there is some such that Then taking the limit as in (10), we have Then, and therefore By continuity of , we conclude that Letting in (9) and (15), (19), and the continuity of , we conclude that . Thus,
Next, we will prove that Suppose the contrary; that is,
Then there exists an for which we can find subsequences , of such that is the smallest integer for which This means that From the above two inequalities and (p4), we have Letting and using (20), we get By (p3) and (p4), we have
Letting in the above inequalities and using (20) and (26), we have
Therefore, from (5), we have
Letting in the above and using (28) and the continuity of , we conclude that which is a contradiction. Thus, we have
Therefore, is a Cauchy sequence in the complete partial metric space .
By Lemma 7, we have that
Thus, is a Cauchy sequence in the complete metric space . Hence, by Lemma 7, is a Cauchy sequence in the complete metric space . Again, by Lemma 7, there exists such that which implies that
Next, we prove that .
Letting in we have
Also, letting in we have
From (36) and (38), we have
Now, we prove that . By (5), we have
Letting in the above and using (39) and the continuity of , we conclude that
Hence, . By (p1) and (p2), we have .
Thus, is a coincidence point of and .
To prove the uniqueness of the coincidence point of and , suppose that is another coincidence point of and . From (5), we have
Therefore, we have . Hence, . By (p1) and (p2), we have .
Thus, and have a unique coincidence point.

As an immediate consequence of the above theorem, we have the following fixed point result.

Corollary 11. Let be a complete partial metric space and a weakly C-contraction mapping. That is, satisfies for all , , where is a continuous mapping such that if and only if .
Then, there exists a unique such that .

Corollary 12. Let be a complete partial metric space. Suppose that the mapping satisfies the following contractive condition: for all , , , , where , are nonnegative constants with . Then, has a unique fixed point.

Proof. Take , where , are nonnegative constants with .

Corollary 13 (see [10, Corollary 2.7]). Let be a complete partial metric space. Suppose that the mapping satisfies the following contractive condition: for all , , where . Then, has a unique fixed point.

Proof. Take , where .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.