Generalized Contractive Mappings and Weakly -Admissible Pairs in -Metric Spaces

Abstract

The aim of this paper is to present some coincidence and common fixed point results for generalized (, )-contractive mappings using partially weakly --admissibility in the setup of -metric space. As an application of our results, periodic points of weakly contractive mappings are obtained. We also derive certain new coincidence point and common fixed point theorems in partially ordered -metric spaces. Moreover, some examples are provided here to illustrate the usability of the obtained results.

1. Introduction and Mathematical Preliminaries

The concept of a generalized metric space, or a -metric space, was introduced by Mustafa and Sims [1]. In recent years, many authors have obtained different fixed point theorems for mappings satisfying various contractive conditions on -metric spaces. For a survey of fixed point theory, its applications, different contractive conditions, and related topics in -metric spaces we refer the reader to [1–33] and the references mentioned therein.

Recall that very recently Samet et al. [33] and Jleli and Samet [22] proved that several results in G-metric spaces can be deduced from the usual one. Later on, Agarwal and Karapnar [23] and Asadi et al. [25] suggested some new contraction mapping type to fail the approaches in [22, 33].

Definition 1 (-metric space [1]). Let be a nonempty set and let be a function satisfying the following properties:(G1) if ;(G2), for all with ;(G3), for all with ;(G4) (symmetry in all three variables);(G5), for all (rectangle inequality).
Then, the function is called a -metric on and the pair is called a -metric space.

Definition 2 (see [1]). Let be a -metric space and let be a sequence of points of . A point is said to be the limit of the sequence if  . In this case, one says that the sequence is -convergent to . Thus, if in a -metric space , then, for any , there exists a positive integer such that , for all , .

Definition 3 (see [1]). Let be a -metric space. A sequence is called -Cauchy if for every , there is a positive integer such that , for all , , ; that is, if , as , , .

Lemma 4 (see [1]). Let be a -metric space. Then, the following are equivalent:(1) is -convergent to ;(2), as ;(3), as ;(4), as , .

Lemma 5 (see [34]). If is a -metric space, then is a -Cauchy sequence if and only if for every , there exists a positive integer such that , for all .

Definition 6 (see [1]). A -metric space is said to be -complete (or complete -metric space) if every -Cauchy sequence in is -convergent in .

Proposition 7 (see [1]). Let be a -metric space. Then for each it follows that(1)if then ,(2),(3),(4).

Definition 8 (see [1]). Let and be two -metric spaces. Then a function is -continuous at a point if and only if it is -sequentially continuous at ; that is, whenever is -convergent to , is -convergent to .

The concept of an altering distance function was introduced by Khan et al. [35] as follows.

Definition 9. The function is called an altering distance function, if the following properties are satisfied.(1) is continuous and nondecreasing.(2) if and only if .

Samet et al. [36] defined the notion of -admissible mappings in the framework of metric spaces as follows.

Definition 10. Let be a self-mapping on and let be a function. We say that is an -admissible mapping if For more details on -admissible mappings we refer the reader to [37–39].

Definition 11 (see [40]). Let be a -metric space and let be a self-mapping on and let be a function. We say that is a --admissible mapping if

Definition 12. Let be an arbitrary set, , and . A mapping is called an -dominating map on if or for each in .

Example 13. Let . Let be defined by and let be defined by . Then, for all . That is, . Thus, is an -dominating map.

Definition 14. Let be a -metric space. We say that is -regular if and only if the following hypothesis holds.
For any sequence in with such that as , it follows that or or , for all .

Definition 15. Let be a set and let be given mappings. We say that the pair is partially weakly --admissible if and only if for all .

Let be a nonempty set and a given mapping. For every , let .

Definition 16. Let be a set and let be given mappings. We say that the pair is partially weakly --admissible with respect to if and only if for all , , where .

If , we say that is partially weakly --admissible with respect to .

If (the identity mapping on , then the previous definition reduces to the partially weakly --admissible pair.

Following is an example of mappings , , , , , and for which ordered pairs , , and are partially weakly --admissible with respect to , , and , respectively.

Example 17. Let . We define functions by Also, let .

Jungck in [41] introduced the following definition.

Definition 18 (see [42]). Let be a -metric space and let . The pair is said to be compatible if and only if , whenever is a sequence in such that for some .

The aim of this paper is to prove some coincidence and common fixed point theorems for nonlinear weakly -contractive mappings , , and which are partially weakly -admissible with respect to , , and , respectively, in a -metric space.

2. Main Results

Let be a metric space and let be six self-mappings. In the rest of this paper, unless otherwise stated, for all , let From now on, let be a function having the following property: For example, one can take by .

Our first result is the following.

Theorem 19. Let be a -complete -metric space. Let be six mappings such that , , and . Suppose that, for every three elements , , and with , one has where are altering distance functions. Let , , , , , and be continuous, the pairs , , and compatible, and the pairs , , and partially weakly -admissible with respect to , , and , respectively. Then, the pairs , , and have a coincidence point in . Moreover, if , then is a coincidence point of , , , , , and .

Proof. Let be an arbitrary point. Since , we can choose such that . Since , we can choose such that . Also, as , we can choose such that .
Continuing this process, we can construct a sequence defined by for all .
Now, since , , and and , , and are partially weakly -admissible with respect to , , and , respectively, we obtain that
Continuing this process, from (5), we get for all .
Define . Suppose , for some . Then, . In the case that , then gives . Indeed, If then
Thus, which implies that .
Analogously, for other values of , we can get this result.
Similarly, if , then gives . Also, if , then implies that . Consequently, is a coincidence point of the pairs , , and . Indeed, let . Then, we know that .
So,
This means that , , and .
On the other hand, the pairs , , and are compatible. So, they are weakly compatible. Hence, , , and or, equivalently, , , and .
Now, since , we have , , and .
In the other cases, when (), similarly, one can show that () is a coincidence point of the pairs , , and .
So, suppose that for each ; that is, for each .
We complete the proof in three steps as follows.
Step  1. We will prove that
Since , using (6), we obtain that
Since is a nondecreasing function, we get that
If , then (17) becomes
If then, from (G3) and (G4) in Definition 1, and then (17) will be
If then, again from (G3) and (G4), and then (17) becomes
If then, again from (G3) and (G4), and then (17) becomes
Finally, if then and then (17) becomes
Similarly it can be shown that
Hence, we conclude that is a nondecreasing sequence of nonnegative real numbers. Thus, there is an such that
Reviewing the above argument, from (17), we have
In general, we can show that
Letting in (34), we get that
Letting and using (6), (35), and the continuity of and , we get , and hence . This gives us that from our assumptions about . Also, from Definition 1, part (G3), we have
Step  2. We will show that is a -Cauchy sequence in . So, we will show that, for every , there exists such that, for all , .
Suppose the above statement is false. Then, there exists for which we can find subsequences and of such that satisfying that and is the smallest number such that (38) holds; that is, From rectangle inequality, Hence, in (40), if , using (36) and (39), we have
Also, Hence, in (42), if , using (36) and (38), we have
On the other hand, Hence, in (44), if , from (43), we have
Also, Hence, in (46), if , using (36) and (38), we have
In a similar way, we have Therefore, from (48) by taking limit when , using (36) and (45), we get that Further, we can obtain that
Also, or, equivalently,
Also, Hence, in (53), if , using (36) and (38), we have Since , putting , , and in (6), for all , we have where
If = , from (54) and (41), if in (55), we have which is a contradiction to (43).
If from (37), (45), (49), and (54), if in (55), we have which is a contradiction to (47).
If from (37), (50), and (54), if in (55), we have which is a contradiction to (52).
If from (37) and (54), if in (55), we have
If from (37) and (54), if in (55), we have
Hence, (63) and (65) yield that which is a contradiction. Consequently, is a -Cauchy sequence.
Step  3. We will show that , , , , , and have a coincidence point.
Since is a -Cauchy sequence in the complete -metric space , there exists such that Hence,
As is compatible, so
Moreover, from , , and the continuity of and , we obtain
By the rectangle inequality, we have
Taking limit as in (70), using (68) and (69), we obtain which implies that ; that is, is a coincidence point of and .
Similarly, we can obtain that and .
Now, let . By (6), we have where
Let ; that is,