Fundamental Questions and New Counterexamples for b -Metric Spaces and Fatou Property

: In this paper, we give new examples to show that the continuity actually strictly stronger than the Fatou property in b -metric spaces. We establish a new ﬁxed point theorem for new essential and fundamental sufﬁcient conditions such that a ´Ciri´c type contraction with contraction constant λ ∈ [ 1 s , 1 ) in a complete b -metric space with s > 1 have a unique ﬁxed point. Many new examples illustrating our results are also given. Our new results extend and improve many recent results and they are completely original and quite different from the well known results on the topic in the literature.


Introduction
The notion of b-metric space, introduced by Bakhtin [1] (see also Czerwik [2]), is one of interesting generalizations of standard metric spaces. Later, in 1998, Czerwik improved and generalized this notion in [3] from the constant s = 2 to a constant s ≥ 1. In the last years, a lot of fixed point results in the framework of b-metric space were studied by many authors, see e.g., [4][5][6][7][8][9][10][11][12] and references therein.
What follows we recall the notion of b-metric space.
In this case, the pair (W, ρ) is called a b-metric space.
Clearly, every metric space is a b-metric space, but the converse is not true, see [3]. The basic topological properties (convergence, completeness, continuity, etc.) in b-metric spaces have been observed by the mimic of the standard metric versions as follows.
Definition 2 (see [3]). Let (W, ρ) be a b-metric space and {z n } be a sequence in W. Then (i) {z n } is said to converge to z ∈ W if lim n→∞ ρ(z n , z) = 0; (ii) {z n } is called Cauchy if lim m,n→∞ ρ(z m , z n ) = 0; (iii) (W, ρ) is said to be complete if every Cauchy sequence converges. T is called continuous on W if T is continuous at every point of W.
It is known that the continuity of a metric plays a crucial role in metric fixed point theory. However, it is worth mentioning that a b-metric fail to be continuous in general. So the continuity can be deemed as one of the main differences between a metric and a b-metric. In the past, some examples of b-metric spaces with discontinuous b-metrics were given, but these examples are similar to each other; see e.g., [4][5][6]10,13]. In 2014, Amini-Harandi [4] introduced the following notion of Fatou property for studying new fixed point results in b-metric spaces.
Definition 4 (see [4]). Let (W, ρ) be a b-metric space. We say that ρ has the Fatou property if whenever {z n } ⊂ W with lim n→∞ ρ(z n , z) = 0 and any y ∈ W.
It is obvious that the continuity implies the Fatou property. Some examples of b-metric spaces with Fatou property were provided, see [4] (Examples 2.5 and 2.6). However these given b-metrics are still continuous. In fact, [4] (Example 2.3) does not enjoy Fatou property, so it certainly fails to have continuity. Thus the following problem arises from the relationship between Fatou property and continuity. Question 1. Is the continuity actually strictly stronger than the Fatou property? In other words, does there exist an example that a b-metric is discontinuous as well as satisfying Fatou property?
The set of fixed points of T is denoted by F (T). Throughout this paper, we denote by N and R, the sets of positive integers and real numbers, respectively. Recall that a selfmap T : W → W is called (i) a Banach type contraction, if there exists a nonnegative number λ < 1 such that ρ(Tx, Ty) ≤ λρ(x, y) for all x, y ∈ W.
In this case, λ is called the contraction constant of T.
It is worth mentioning that Banach type contraction, Kannan type contraction and Chatterjea type contraction are independent and different from each other in general and they are allĆirić type contractions. In 1974,Ćirić established the following famous fixed point theorem (so-calledĆirić fixed point theorem [14]) in the setting of metric spaces (i.e., b-metric space with s = 1).
Clearly,Ćirić fixed point theorem is an actually generalization of the Banach contraction principle [15], Kannan's fixed point theorem [16] and Chatterjea's fixed point theorem [17]. Due to the importance and application potential to quantitative sciences, the generalizations ofĆirić fixed point theorem have been investigated heavily by many authors in various distinct directions over the past 20 years; see, e.g., [4,5,8,12,[18][19][20][21] and the related references therein. Recently, Amini-Harandi [4] proved a generalization ofĆirić fixed point theorem in the setting of b-metric spaces with Fatou property. Later, He et al. [8] and Zhao et al. [12] respectively improved the results of Amini-Harandi without Fatou property assumption.
In fact, the ranges of the contraction constants are almost limited to [0, 1 s ) used in all known fixed point results forĆirić type contractions in the setting of b-metric spaces with s ≥ 1; see, e.g., [4,8,9,12,13]. In [6], Dung and Hang successfully generalized the Banach contraction principle from metric spaces to b-metric spaces with contraction constant λ ∈ [0, 1). Unfortunately, Theorem 2 is not always true if s > 1 and the contraction constant λ ∈ 1 s , 1 , see [6] (Theorem 2.6) and [22] (Remark 3.7). Motivated by that reason, the following question arises naturally.

Question 2.
Can we give some new essential and fundamental sufficient conditions such that aĆirić type contraction with contraction constant λ ∈ [ 1 s , 1) in a complete b-metric space with s > 1 have a unique fixed point?
In this work, our questions will be answered affirmatively. In Section 2, we successfully establish one new example to show that there exists a b-metric such that it has the Fatou property as well as is discontinuous. So we prove that the continuity is strictly stronger than the Fatou property, that is a positive answer to Question 1. In Section 3, we first construct a new simple counterexample to show that Theorem 2 is not always true for λ ∈ [ 1 s , 1). Furthermore, we give three sufficient conditions to demonstrate that aĆirić type contraction with contraction constant λ ∈ [ 1 s , 1) in a complete b-metric space with s > 1 have a unique fixed point. From this, we successfully give a complete answer to Question 2. Finally, we give three examples to show that three sufficient conditions are independent of each other. Our new results extend and improve many recent results and they are completely original and quite different from the well known results on the topic in the literature.

Some New Counterexamples to Answer Question 1
Now we construct two new examples that every b-metric is discontinuous. The first example tell us that there exists a b-metric such that it is discontinuous but fails to have the Fatou property. The second example shows that there exists a b-metric such that it has the Fatou property as well as is discontinuous.
for any x, y ∈ W.
Then the following hold: Proof.
(a) It is obvious that (b1) and (b2) of Definition 1 are satisfied. Now we prove that (b3) holds. For any x, y, z ∈ W, let us consider the following possible cases: Case 2. Suppose that xy = 0. Without loss of generality, we may assume that x = 0.
Hence, by Cases 1 and 2, we prove that (W, ρ) is a b-metric space with s = α. Next, we verify the completeness of W. Let d(x, y) = |x − y| for all x, y ∈ W. Then (W, d) is a complete metric space. Due to the fact that d(x, y) ≤ ρ(x, y) ≤ αd(x, y) for all x, y ∈ W, we can easily prove that (W, ρ) is complete.
(b) Let a n = 1 n , n ∈ N and b = 1. So, we have a n = 0 for any n ∈ N and a n → 0 as n → ∞. Since we show that ρ does not satisfy the Fatou property. (c) The conclusion is an immediate consequence of (b).
The following example gives a positive answer to Question 1.
for any x, y ∈ W.
Then the following hold: (a) (W, ρ) is a complete b-metric space with s = β; (b) ρ has the Fatou property; (c) ρ is discontinuous on W.
Case A2. Suppose that xy = 0. Without loss of generality, we may assume that x = 0.
Hence, by Cases A1 and A2, we prove that (W, ρ) is a b-metric space with s = β. Following a similar argument as in the proof of Example 1, we can show that (W, ρ) is complete.
Let z, y ∈ W be given. Let {z n } be a sequence in W such that z n converges to z. If there exists n ∈ N such that z n = z for all n ≥ n, then ρ(z, y) = lim inf n→∞ ρ(z n , y) and (1) holds. Otherwise, we may assume that z n = z for all n ∈ N. We consider the following three cases.
Passing to the limit as n → ∞ in the above inequality, we obtain (1). Case B2. Assume that z = 0 and y = 0. Thus there exists n 0 ∈ N such that z n = 0 for all n ≥ n 0 . Hence we obtain β ρ(z, z n ) + ρ(z n , y) for all n ≥ n 0 .
Passing to the limit as n → ∞ in the above inequality, we get (1). Case B3. Assume that z = 0. In this case, if y = 0, then z = y and (1) always holds. So we suppose y = 0. Since z n = z for all n ∈ N, we have Passing to the limit as n → ∞ in the above inequality, we obtain Therefore, by the above three cases, we prove that ρ has the Fatou property.
(c) Let x = 0, y = 1, and x n = 1 n , n ∈ N. Thus we have lim n→∞ ρ(x n , y) = lim This shows that ρ is not continuous.

Remark 1.
In Example 1, the b-metric is discontinuous and the b-metric space does not enjoy Fatou property. In Example 2, the b-metric space enjoys Fatou property but the b-metric ρ is not continuous. It shows that continuity is strictly stronger than Fatou property.

Some New Results forĆirić Type Contraction and Answers to Question 2
In this section, we first give a new simply counterexample to show that Theorem 2 is not always true if s > 1 and the contraction constant λ ∈ 1 s , 1 .
for any x, y ∈ W.
Let T : W → W be a map defined by Then the following hold: (a) (W, ρ) is a complete b-metric space with s = 2; (b) T is aĆirić type contraction with contraction constant λ = 2 3 (note: λ ∈ 1 s , 1 ); (c) T has no fixed point in W.
Proof. Clearly, the conclusion (c) is true and the conclusion (a) immediately follows from Example 1 with α = 2. To see (b), we first note that Tx = 0 for any x ∈ W. So we always have ρ(Tx, Ty) = |Tx − Ty| for all x, y ∈ W.

Case 2.
Suppose that xy = 0. Then, without loss of generality, we may assume x = 0.
Hence, by Cases 1 and 2, we prove that T is aĆirić's type contraction with contraction constant λ = 2 3 .
The following lemmas are crucial in this paper.
Hence, we obtain L(0, n) ≤ M for all n ∈ N. The proof is completed. Lemma 3. Let (W, ρ) be a b-metric space with s ≥ 1 and T : W → W be aĆirić type contraction with contraction constant λ ∈ [0, 1). Then for each x ∈ W, {T n x} n∈N∪{0} is a Cauchy sequence in W (here, T 0 = I is the identity map).
Proof. Let z 0 ∈ W be given. Let {z n } n∈N∪{0} be a sequence defined by z n = Tz n−1 = T n z 0 for all n ∈ N. We claim that {z n } n∈N∪{0} is Cauchy in W.
For any m, n ∈ N and m < n, by applying Lemma 2, we have Since λ < 1, the last inequalities imply that lim m,n→∞ ρ(z m , z n ) = 0, which show that {z n } n∈N∪{0} is Cauchy in W.
By Lemmas 1 and 3, we establish the following new fixed point theorem forĆirić type contractions in a complete b-metric space. This new fixed point theorem gives a positive answer to Question 2. Notice that the conclusion (a) in Theorem 3 is actually the originalĆirić fixed point theorem (i.e., Theorem 1), but we give a new proof by using Lemma 3 for the sake of completeness and the readers convenience. Proof. Given z 0 ∈ W and let {z n } n∈N∪{0} be a sequence defined by z n = Tz n−1 = T n z for all n ∈ N. Applying Lemma 3, {z n } n∈N∪{0} is a Cauchy sequence in W. By the completeness of W, there exists v ∈ W such that z n → v as n → ∞.
Since z n → v as n → ∞ and ρ is continuous, by taking the limit as n → ∞ in the last inequality, we get which implies v ∈ F (T) = ∅. Next, we verify that F (T) is a singleton set. If u ∈ F (T), then we have ρ(Tu, v) = ρ(u, Tv) = ρ(u, v) and ρ(u, Tv) = ρ(v, Tv) = 0. Since which implies ρ(u, v) = 0 and hence u = v. So F (T) = {v} is a singleton set which means that T has the unique fixed point v in W. Since z ∈ W is arbitrary given, the sequence {T n z} n∈N∪{0} must converge to v. If (D2) holds, since T is aĆirić type contraction with contraction constant λ ∈ [0, 1), we have for all n ∈ N. Since z n → v as n → ∞and ρ has the Fatou property, we get On the other hand, since By applying Lemma 1 and taking into account (2)-(4), we obtain Hence, we know from the last inequalities that Therefore v ∈ F (T). Finally, suppose that (D3) holds. We first claim the inequality (5) holds, where for all n ∈ N. Since T is aĆirić type contraction with contraction constant λ ∈ [0, 1), we have ρ(z n+1 , Tv) = ρ(z n , Tv) ≤ λ max {ρ(z n , v), ρ(z n , z n+1 ), ρ(v, Tv), ρ(z n , Tv), ρ(z n+1 , v)} for all n ∈ N. Note that for any n ∈ N, if ρ(z n+1 , Tv) ≤ λρ(v, Tv), since If ρ(z n+1 , Tv) ≤ λρ(z n , Tv), then which deduces Therefore, by above, we prove (5) holds. Since z n → v as n → ∞, we have So, by passing to the limit as n → ∞ in (5), we obtain lim n→∞ ρ(z n+1 , Tv) = 0. Due to the uniqueness of the limit, we get Tv = v.
Following the same argument as the proof of (a), we can show that T has the unique fixed point v in W and the sequence {T n z} n∈N∪{0} converges to v. The proof is completed.
In [23] (Definition 12.7), the notion of strong b-metric space was introduced. Now we recall this notion as follows.
In this case, the pair (W, ρ) is called a strong b-metric space.
It is obvious that every strong b-metric space is a b-metric space and the strong b-metric is continuous (see [6] (Remark 1.7)). From Theorem 3 we obtain the following result immediately.
Then T has a unique fixed point.
We now construct three examples to illustrate Theorem 3 and show the complete independence of these three conditions in Theorem 3. First, we give a example which satisfies (D2) in Theorem 3, but neither of (D1) nor (D3) is satisfied. for any x, y ∈ W.
for any x ∈ W.
We give new examples to show that the continuity actually strictly stronger than the Fatou property in b-metric spaces. We establish a new fixed point theorem for new essential and fundamental sufficient conditions such that aĆirić type contraction with contraction constant λ ∈ [ 1 s , 1) in a complete b-metric space with s > 1 have a unique fixed point. Many new examples illustrating our results are also given. Our new results extend and improve many recent results and they are completely original and quite different from the well known results on the topic in the literature.
Author Contributions: All authors contributed equally to this work. All authors read and approved the final manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.