Abstract

This paper is aimed at presenting some coincidence point results using admissible mapping in the framework of the partial -metric spaces. Observed results of the article cover a number of existing works on the topic of “investigation of nonunique fixed points.” We express an example to indicate the validity of the observed outcomes.

1. Introduction and Preliminaries

In 1974, Ćirić [1] published the first paper on nonunique fixed point theory. Despite Banach’s theorem, Ćirić [1] focused only on the existence of a fixed point, but not the uniqueness. The motivation of Ćirić [1] was inspired by Banach’s motivation. As it is known, Banach’s fixed point theorem is abstracted from Picard’s paper, in which Picard [2] analyzed both the existence and uniqueness of the solution of the certain differential equation (see [3–5]). On the other hand, not all differential or integral equations have a unique solution. In the differential/integral equations, nonunique solutions are also crucial, for example, periodic solutions. Consequently, Ćirić [1] investigated the corresponding fixed point theorems that would be a tool in finding periodic solutions of the differential/integral equations. In the last five decades, a number of nonunique fixed point results have been reported in two ways: either proposing a new contraction type or changing the structure. The first example for the changing the contraction inequality, in the standard set-up, was given by Achari [6] in 1976 and Pachpatte [7] in 1973. Fifteen years later, Ćirić and Jotić [8] proposed a new type of contraction inequalities in the context of complete metric space. This trend was followed by the attractive results [9–13]. On the other hand side, in [14–17], the authors observed several characterizations of the unique fixed point results in the setting of complete -metric spaces. Indeed, among the several extensions of metric structure, the true extension is the -metric space. For this reason, observed nonunique fixed theorems in the context of -metric space is very interesting and important, see also [18–20]. In addition, in [21–23], the characterization of fixed point theorems in partial metric spaces is crucial due to the potential application in the domain theory of computer science. Regarding the applied mathematics, nonunique fixed point results in cone metric spaces have taken attention [24].

In this paper, we consider a nonunique fixed point theorem in the context of the very general frame, partial -metric spaces. An illustrative example is a set-up to indicate the validity of the main theorem.

Let be a nonempty set, a real number , and .In this case, the triplet forms a partial -metric space, on short -ms.Undoubtedly, -metric spaces (and ordinary metric spaces) are closely related to partial -metric spaces. Definitely, a -metric space () is a partial -metric space with zero self-distance and a partial metric space is a partial -metric space with . Moreover, a partial -metric can define a -metric space. Indeed, for example, let be a partial -metric on . Then, the functions , where are -metrics on

Definition 1. A function is a partial -metric on if for all , it satisfies the following conditions:

Example 1. (see [25]). Let be a partial metric on the set . Then, the functions are given for all by (1) is a partial -metric on (where is a -metric () on )(2) for , define a partial -metrics on with coefficient

Remark 2. From and , it follows that if are such that , then

Definition 3. (see [26, 27]). Let be a sequence on the (1) is -convergent to if (2) is -Cauchy if exists and is finite(3) is --Cauchy if (4) is -complete if every -Cauchy sequence in is -convergent(5) is --complete if every --Cauchy sequence we can find such thatMoreover, in [26], the following interesting results were proved.

Lemma 4. (see [26]). Every -complete -ms is --complete.

Lemma 5. (see [26]). The -ms is --complete if and only if the -metric space is complete, where the -metric was defined in (3).
They also showed that the converse affirmation does not hold.
Let to self-mappings on the set We say that (i) commutes with on if for all (ii)a point is a point of coincidence of and if we can find such that (iii)a point is a common fixed point of and if We will use the following notations: In [28], the notion of --admissible mapping was introduced as follows: (i)Let the function and . The mapping is said to be --admissible iffor all
In case that , the mapping is said to be -admissible.
Let be a -ms and . The space is -regular if for every sequence in such that and , there exists a subsequence of such that for all .

Lemma 6. Let such that is a --admissible. If there exists such that , then where the sequence in is defined by , for each .

Proof. By the assumption , since the mapping is --admissible, we get and by induction, it follows that for .

2. Main Results

Following the idea in [29], we state the following results useful in the sequel.

Lemma 7. Let be a -ms. If is a sequence in such that there exists in , satisfying the inequality for any , then the sequence is and is --Cauchy.

Proof. First of all, by (12), we get for all . On the other hand, by using , we can derive that (1)If , by (13) and (14), we getTherefore, is a --Cauchy sequence. (2)If , thus (as ). Moreover, there exits such that . This means . Again, by (13) together with (14), we haveThereby, letting by Case (i), we get that the sequence is --Cauchy sequence, which means that On the other hand, and using (13), we have Finally, combining relations (19) and (17) and keeping in mind , we have Thereupon, the sequence is --Cauchy.

Theorem 8. Let be a complete -ms and two mappings . Suppose that there exists such that for all , such that when . Suppose also that (a) and is a --complete -ms(b) is --admissible, and there exists such that (c) is -regularThen, the mappings and have a point of coincidence.

Proof. Let be an arbitrary point in , such that . Thus, since , there exists such that Thereupon, and we can find such that . In this way, we can build a sequence as follows: for all . Letting and in (ref1T1) and taking into account Lemma 6, we have Keeping in mind (22), we get which is equivalent with Therefore, we get for any .Let now be a sequence in , with , . First of all, we mention that for every . Indeed, if we suppose that there exists such that , thus by (22), we have so that is a point of coincidence. Thus, for every and (28) can be rewritten as Therefore, according to Lemma 7, the sequence is --Cauchy. Since the space is --complete, it follows that there is such that But, on the other hand, since and the space is --complete, we can find , with . Thus, Supposing that for and and taking into account the -regularity of the space , we have If , the above inequality becomes Letting and taking into account (28) and (30), we get and by , , we have .If , we find that On the other hand, by , and then, , as . This proves that , that is, is a point of coincidence for and .