Fixed Points of Generalized -Suzuki Type Contraction in Complete -Metric Spaces

Abstract

We introduce the notion of generalized -Suzuki type contraction in -metric spaces and investigate the existence of fixed points of such mappings. The presented results generalize and improve several results of the topics in the literature.

1. Introduction and Preliminaries

The concept of -metric was introduced by Czerwik [1] as a generalization of metric (see also Bakhtin [2, 3]) to extend celebrated Banach contraction mapping principle. Following this initial paper of Czerwik [1], a number of researchers in nonlinear analysis investigate the topology of the paper and proved several fixed point theorems in the context of complete -metric spaces (see, e.g., [4–8] and the related references therein).

Definition 1 (see [1]). Let be a nonempty set and let be a given real number. A mapping is said to be a -metric if for all the following conditions are satisfied: if and only if ;;.In this case, the pair is called a -metric space (with constant ).

Remark 2. It is clear that the notion of -metric is a real generalization of usual metric since a -metric space is a metric space when . For more details and examples on -metric, see, for example, [1, 3, 9, 10].

Example 3. Let and let a mapping be defined as follows: Then, is a -metric space with coefficient . But it is not a metric space since the triangle inequality is not satisfied. Indeed,

Definition 4 (see [11]). Let be a -metric space. Then, a sequence in is called(A)convergent if and only if there exists such that as and in this case we write ;(B)Cauchy if and only if as .

Remark 5 (see [11]). Notice that in a -metric space the following assertions hold:(A)a convergent sequence has a unique limit;(B)each convergent sequence is Cauchy;(C)in general, a -metric is not continuous;(D)in general, a -metric does not induce a topology on .

Definition 6 (see [11]). The -metric space is complete if every Cauchy sequence in converges in .

Definition 7. Let and be -metric spaces; a mapping is called (A)continuous at a point , if for every sequence in such that , then ;(B)continuous on , if it is continuous at each point .

2. Main Result

In this section we state and prove our main results. Throughout the paper, we assume that -metric is continuous.

Inspired by the notion of -contraction, defined by Wardowski [12], we introduce the notion of generalized -Suzuki type contraction as follows. , , denote the set of natural numbers, set of real numbers, and the set of non-negative real numbers.

Definition 8. Let be a -metric space with constant . A mapping is called a generalized -Suzuki type contraction if there exists such that for all with where and are real numbers with and is a mapping satisfying the following conditions: is strictly increasing; that is, for all such that ,  ;for every sequence of positive real numbers if and only if .

The following is the main result of this paper.

Theorem 9. Let be a complete -metric space with constant . If is a generalized -Suzuki type contraction, then has a fixed point .

Proof. Fix . We establish an iterative sequence in the following way: Throughout the proof, we assume that Indeed, if there exists such that , then the proof is completed trivially.
Due to assumption (5), we have Thus, by hypotheses of theorem, we have which is equivalent to Since , inequality (8) turns into From , we conclude that Therefore, is a decreasing sequence of real numbers which is bounded from below. Therefore, converges and We will show that . Suppose, on the contrary, that . In other words, for every there exists , such that From , we find that On the other hand, we have due to (6). Since is generalized -Suzuki type contraction, we derive which yields that Taking the fact that into account, we find
Analogously, again by (6), we have . Owing to the fact that is generalized -Suzuki type contraction, we conclude that It implies that since . Furthermore, by combining (17) and (20), we get
Iteratively, we obtain that
By letting , we find that Consequently, from (), we derive that . Thus, there exists such that and from (4) we get This is a contradiction with the definition of . Hence, we have
In what follows, we will prove that Suppose, on the contrary, that there exist and sequences and of natural numbers such that From the triangle inequality, we have
Owing to (25), there exists such that
Taking (29) into account, (28) yields that So from , we obtain
On the other hand, we can easily get that from (27) and (29). Since is generalized -Suzuki type contraction, for all we find that Taking (31) into account, (33) turns into Regarding (25) and , we obtain that From , we get that This is a contradiction with relations in (27). Hence, ; that is, is a Cauchy sequence in . On account of the completeness of , there exists such that We claim that, for every , We will prove the claim above by the method of reductio ad absurdum. Suppose, on the contrary, that there exists such that From (10) and , we have It follows from (39) and (40) that This is a contradiction. Hence, (38) holds. Since is generalized -Suzuki type contraction, (38) yields that, for every , either or holds. On account of , the limits in (25) and (37) imply that Thus, letting in (42), we conclude that Again by using , we observe that By regarding the triangle inequality with (4), we derive that By letting in the inequality above together with the limits in (37) and (46), we conclude that . Thus, is a fixed point of ; that is, .
Let us analyze the second case (42). Regarding (4), we have As it was discussed above, from (25), (37), and , we conclude that From equivalently, we get Again by the triangle inequality together with (4), we find that By letting in the inequality above together with the limits in (37) and (50), we obtain . Thus, is a fixed point of and that completes the proof.

Definition 10. Let be a -metric space with constant . A mapping is said to be a -Suzuki type contraction if there exists such that for all with where is a mapping satisfying the following conditions.() is strictly increasing; that is, for all such that ,  .()For every sequence of positive real numbers if and only if .

Theorem 11. Let be complete -metric space with constant and let be a -Suzuki type contraction mapping. Then, has a fixed point.

Proof. By taking and in Theorem 9, the proof is complete.

Corollary 12. Let be a self-mapping on a complete metric space . Let and be real numbers such that . Assume that there exists such that, for all with , where is satisfied in conditions and . Then, has a fixed point ; that is, .

Proof. Since any metric space is a -metric space with constant , so from Theorem 9 the proof is complete.

Corollary 13. Let be a self-mapping on a complete metric space . Assume that there exists such that, for all with , where is satisfied in conditions and . Then, has a fixed point ; that is, .

Proof. Since any metric space is a -metric space with constant , so by taking and in Theorem 9 the proof is complete.

Theorem 14. Let be complete -metric space with constant and let be a continuous self-mapping on . Let and be real numbers such that . Assume that there exists such that for all where is satisfied in conditions and . Then, has a fixed point ; that is, .

Proof. Choose . Set If there exists such that , the proof is complete. So we assume that So from the assumption of theorem, we have and hence Since , we get
So from , we conclude that Therefore, is a decreasing sequence of real numbers which is bounded from below. Therefore, converges and We will show that . Arguing by contradiction, we assume . For every , there exists , such that Hence, from , we get
On the other hand (57), we have So from assumption of theorem, we obtain and hence
Since , we get
Also from (57), we have , and thus, by assumption of theorem, we have and therefore, Since , we get
Now by using (64) and continuing similar method as used in (68) and (71), we obtain
This implies that So from (), we have , so that there exists such that and so from (56) we get
This is a contradiction with definition of . So, and from (62) we have
Now, we claim that Arguing by contradiction, we assume that there exist and sequences and of natural numbers such that So
From (76), there exists such that
It follows from (79) and (80) that So, from , we obtain
On the other hand from (78), we have Hence, from (82) and assumption of theorem, we have
Using (79) and , we obtain
Thus, from this and , we get
This is contradiction with relation (78). Hence, . By completeness of , there exists such that Since is continuous, we get Since , therefore and hence .

Theorem 15. Let be complete -metric space with constant and let be a continuous self-mapping on . Assume that there exists such that for all where is satisfied in conditions and . Then, has a fixed point ; that is, .

Proof. By taking and in Theorem 14, the proof is complete.

Theorem 16. Let be complete metric space and let be a continuous self-mapping on . Let and be real numbers such that . Assume that there exists such that for all where is satisfied in conditions and . Then, has a fixed point ; that is, .

Proof. It is sufficient to take in Theorem 14.

Theorem 17. Let be complete metric space and let be a continuous self-mapping on . Assume that there exists such that for all where is satisfied in conditions and . Then, has a unique fixed point ; that is, .

Proof. Since every metric space is a -metric space with constant , from Theorem 15   has a fixed point . Indeed, if there is another fixed point of , such that , therefore . Since , from assumption of theorem we obtain This is contradiction. So has a unique fixed point.

Remark 18. Theorem 17 gives all consequence of Theorem 2.1 of [12] without assumption used by [12]. Notice also that the results in [12] can be also concluded from the main theorem in [13] that is also a proper extension of the results in [12] in different aspect, more precisely, in complete metric-like spaces.

Conflict of Interests

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

Acknowledgment

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant no. 5513035HiCi. The authors, therefore, acknowledge the technical and financial support of KAU.