#### Abstract

We consider some Nemytzki-Edelstein-Meir-Keeler type results in the context of b-metric spaces. In some cases, we assume that the b-metric is continuous. Our results generalize several known ones in existing literature. We also present some examples to illustrate the usability of our results.

#### 1. Definitions, Notations, and Preliminaries

Let be a metric space and be a self-mapping. The following Meir-Keeler conditions are well known: For each , there exists such thatoror is contractive andIt is clear that (1) implies (2) and (2) implies (3), while the converse is not true. One says that the mapping defined on the metric space is contractive if holds, whenever . For more details, see [1] (pages 30-33 and 56-58) and [2]. In 1969, Meir-Keeler [2] proved the following.

Theorem 1 ([2], Theorem). Let be a complete metric space and let be a self-mapping on satisfying (1). Then has a unique fixed point, say , and, for each , .

For other fixed point results via generalized Meir-Keeler contractions, see [35]. Inspired from Meir-Keeler theorem, Ćirić proved the next slightly more general result.

Theorem 2 ([1], Theorem 2.5). Let be a complete metric space and let be a self-mapping on satisfying (2). Then has a unique fixed point, say , and, for each ,

The following example shows that Ćirić result is a proper generalization of Meir-Keeler theorem.

Example 3. Let be a subset of reals with the Euclidean metric and let be a self-mapping on defined by Then one can verify that satisfies (2), while it does not satisfy the Meir-Keeler condition (1). For all details, see [1].

Remark 4. Theorems 1 and 2 are true if the self-mapping satisfies condition (3). For more details, see [1], pages 30-33.

On the other hand, Bakhtin [6] and Czerwik [7] introduced the concept of b-metric spaces (a generalization of metric spaces) and proved the Banach contraction principle. The definition of a b-metric space is the following.

Definition 5 (Bakhtin [6] and Czerwik [7]). Let be a nonempty set and let be a given real number. A function is said to be a b-metric if and only if, for all , the following conditions are satisfied:(b1) if and only if (b2)(b3)

The triplet is called a b-metric space with coefficient

In the last period, many authors obtained several fixed point results for single-valued or set-valued mappings in the context of b-metric spaces. For more details, see [5, 835]. It should be noted that the class of b-metric spaces is effectively larger than that of standard metric spaces, since a b-metric is a metric when The following example shows that, in general, a b-metric does not necessarily need to be a metric.

Example 6. Let be a metric space and where is a real number. Then is a b-metric with , but is not a metric on

The concepts of b-convergence, b-completeness, b-Cauchy, and b-closed set in b-metric spaces have been initiated in [6, 7].

The following two lemmas are very significant in the class of b-metric spaces.

Lemma 7 ([21], Lemma 3.1). Let be a sequence in a b-metric space such thatfor some , and each . Then is a b-Cauchy sequence in a b-metric space

Lemma 8 ([30], Lemma 2.2). Let be a sequence in a b-metric space such thatfor some , and each . Then is a b-Cauchy sequence in a b-metric space

Since in general a b-metric is not continuous, we need the following two lemmas.

Lemma 9 ([36], Lemma 2.1). Let be a b-metric space with . Suppose that and are b-convergent to and , respectively. Then In particular, if , then we have Moreover, for each , we have

Lemma 10 (see [37]). Let be a b-metric space and be a sequence in such that If is not b-Cauchy, then there exist and two sequences and of positive integers such that, for the following four sequences we have

Essential to the proofs of fixed point theorems for the most contractive conditions in the context of b-metric spaces are the above two lemmas (see, for example, [3, 5, 9, 13, 17, 23, 25, 28]). However, it is not hard to show that the proofs of the most fixed point theorems in the context of b-metric spaces become simpler and shorter if they are based on Lemma 8.

#### 2. Main Result

To our knowledge, it is not known whether Meir-Keeler and Ćirić theorems hold in the context of b-metric spaces. Also, it is not known that if there are examples such that condition (1) or (2) or (3) holds in the context of b-metric spaces, but has no fixed point.

Our first result generalizes Lemma 1 of [2]. For some results also see recent paper [38].

Lemma 11. Let be a b-complete b-metric space and such that condition (1) holds. If is a b-Cauchy sequence for each , then has a unique fixed point, say and

Proof. Since is b-complete, each has a limit point, say Since condition (1) implies the continuity of , we haveThus, is a fixed point, and therefore all are equal.

Remark 12. If condition (1) holds on -metric spaces , we do not know whether every sequence is b-Cauchy.
However, with a stronger condition than (1), we have a positive response. It will be the subject of Theorem 13.

Now, we announce a Meir-Keeler type result in the context of b-metric spaces.

Theorem 13. Let be a complete b-metric space and let be a self-mapping on satisfying the following condition.
Given there exists such that where is given.
Then has a unique fixed point, say , and, for each

Proof. It is clear that, for all with , we obtainwhere
Let be an arbitrary point. Define the sequence by for all If, for some , , then is a fixed point of . From now on, suppose that for all From condition (15), we obtainFurther, according to ([30], Lemma 2.2.), the sequence is b-Cauchy in the b-metric space By b-completeness of , there exists such thatFinally, (15) and (17) imply that ; that is, is the unique fixed point of in .

Example 14. Let and define as follows: , for all , , , and Then is a b-complete b-metric space, but it is not a metric space. Let be defined by We shall check that, for all , the contractive condition (15) holds. For this, we distinguish three cases.
(a) Obviously, condition (15) holds.
(b) Since , i.e., , which is true, hence, again (15) holds.
(c) Now, we have , i.e., , which is also true because
Therefore, condition (15) holds for each However, condition (14) is not true for Indeed, for and , it becomes or equivalently Take . Then there exists such that (for example, any ). But is false.

Now, we give an example supporting Theorem 13.

Example 15. Let Then is a b-complete b-metric space. Let be defined as . Taking , we get, for and satisfying , Hence, all the conditions of Theorem 13 are satisfied. The mapping has a unique fixed point, which is

Let be the class of all mappings which satisfy the condition: , whenever Note that As an example, consider the mapping given by for and

The following is Geraghty type result in the context of b-metric spaces (see, for instance, [17], where authors use Lemma 1.4.).

Theorem 16. Let be a complete b-metric space. Suppose that the mapping satisfies the condition for all and some Then has a unique fixed point and for each the Picard sequence converges to in

Proof. Since , we getIn view of , the result follows according to Lemma 8. and condition (15).

It is well known that, in compact metric spaces, fixed point results can be obtained under the strict contractive condition ( whenever ). In the case of b-metric spaces with a continuous b-metric, the following results of Nemytzki and Edelstein can be obtained in the same way as in the metric case (see [1], pages 56-58).

Theorem 17. Let be a compact b-metric space with continuous b-metric and let be a self-mapping. Suppose that the following condition holds: Then has a unique fixed point, say , and, for each

Proof. Define a function bySince is continuous, is also continuous. So, as is a compact b-metric space, there exists a point such thatIf we assume that , then as is contractive ( whether ), one writeswhich is a contradiction. Therefore, is a fixed point of The uniqueness is obvious.

Theorem 18. Let be a b-metric space with a continuous b-metric and let be a self-map. Suppose that the following condition holds: If there exists a point such that the sequence contains a convergent subsequence to , then is the unique fixed point of

Proof. Consider the real sequence If for some , then for is a stationary sequence and so Thus, implies Assume now that for all Then as is contractive, is a strictly decreasing sequence of positive reals. Therefore, it converges. Since as and is continuous, we haveThus Since and are subsequences of the convergent sequence , they have the same limit. Therefore,Hence If not, as is contractive, we havewhich is a contradiction.

Remark 19. The two previous theorems are known in literature as Nemytzki and Edelstein theorems, respectively. It is clear that Edelstein theorem extends the result of Nemytzki.

In the sequel, we consider contractive mappings in the context of b-metric spaces. Namely, we first introduce the following.

Definition 20. A mapping of a b-metric space into itself is said to be contractive, if and only if there exists such thatThe following results extend ones from standard metric spaces to b-metric spaces, with a continuous b-metric

Theorem 21. Let be a b-metric space with continuous b-metric and an contractive self-mapping on If, for some , the sequence has a convergent subsequence to , then is a periodic point; that is, there exists a positive integer such that

Proof. Since has a cluster point, there exist positive integers and with such that ; that is, , where Then the sequence is nonincreasing due to the fact that is contractive. Thus, this sequence converges and soSince and the b-metric are both continuous, we haveandThus,Hence, . Otherwise, as is contractive, we would have .

Now, consider a class of mappings of a b-metric space into itself which satisfy the following condition.

For every there exists a positive integer such thatA mapping satisfying (38) is called an eventually contractive mapping.

It is obvious that any contractive mapping is eventually contractive (it satisfies (38) with ), but the implication is not reverse.

Contractive and contractive mappings are continuous. However, eventually contractive mappings need not be continuous nor orbitally continuous. Recall that a mapping is said to be orbitally continuous if, for each , implies

Now, we announce the next result.

Theorem 22. Let be a b-metric space with continuous b-metric and be an eventually contractive and orbitally continuous mapping. If, for some , the sequence of iterates has a subsequence converging to , then is the unique fixed point of and

Proof. If for some , then for all Thus, and so implies that
Assume now that for all Consider Since is orbitally continuous, for any fixed positive integer , we haveAssume that As is eventually contractive, there exists such thatHence Sincefor arbitrary , there exists a sufficiently large such thatFor , we haveSince for all and is eventually contractive, there exists some positive integer such thatthat is, for all So, we obtain thatHenceThus we getHencewhich contradicts the choice of Therefore, ; that is,
Now, we show that Let be arbitrary. Since as , there exists a sufficiently large such that . If , then and hence for all Assume that As is eventually contractive and , there is such thatSince , we haveTherefore,

Corollary 23. If is a compact b-metric space with a continuous b-metric and is a continuous and eventually contractive self-mapping on , then has a unique fixed point, say . Also, for every

A mapping of a b-metric space into itself is said to be an eventually contractive mapping if there exists such that, for every , there is a positive integer such that For eventually contractive mappings in the context of b-metric spaces, we announce the next result.

Theorem 24. Let be a b-metric space with a continuous b-metric and be an eventually contractive and orbitally continuous self-map on If, for some , the sequence has a convergent subsequence to , then is a periodic point of

Proof. The proof is very similar to the ones in the previous theorem. Therefore, it is omitted.

Corollary 25. If is a compact b-metric space with a continuous b-metric and is an eventually contractive and continuous self-map on , then the set of periodic points of is not empty.

The following two results extend ones from standard metric spaces to b-metric spaces (see [15]).

Theorem 26. Let be a compact b-metric space with continuous b-metric and a continuous self-map on such that for every there exists a positive integer such that Then has a unique fixed point.

Proof. Define by Since and are continuous, is also continuous. Therefore, attains its minimum on in some point, say Assume that Then, by hypothesis, there exists a positive integer such thatthat is, , which contradicts the choice of Therefore,

Theorem 27. Let be a compact b-metric space with a continuous b-metric and be a continuous self-map on such that, for every , Then the set of periodic points of is not empty.

Proof. Let Since is compact, has a cluster point. Therefore, there exist positive integers and such thatDefine Since is continuous, is continuous and so is continuous. Therefore, there is some such that ThenAssume that Then, as , by hypothesis there exists such thatThus, , which contradicts the choice of Therefore,

#### Data Availability

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#### Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.