ANSWERS TO KIRK-SHAHZAD’S QUESTIONS ON STRONG b-METRIC SPACES

ANSWERS TO KIRK-SHAHZAD’S QUESTIONS ON STRONG b-METRIC
SPACES

TRAN VAN AN AND NGUYEN VAN DUNG



Abstract. In this paper, two open questions on strong b-metric spaces posed by Kirk and
Shahzad [13, Chapter 12] are investigated. A counter-example is constructed to give a negative
answer to the first question, and a theorem on the completion of a strong b-metric space is
arXiv:1503.08126v2 [math.GN] 2 Apr 2015




proved to give a positive answer to the second question.




1. Introduction and preliminaries

In 1993, Czerwik [4] introduced the notion of a b-metric which is a generalization of a metric
with a view of generalizing the Banach contraction map theorem.
Definition 1.1 ([4]). Let X be a non-empty set and d : X × X −→ [0, +∞) be a function such
that for all x, y, z ∈ X,

(1) d(x, y) = 0 if and only if x = y.
(2) d(x, y) = d(y, x).
(3) d(x, z) ≤ 2[d(x, y) + d(y, z)].

Then d is called a b-metric on X and (X, d) is called a b-metric space.

After that, in 1998, Czerwik [5] generalized this notion where the constant 2 was replaced by
a constant s ≥ 1, also with the name b-metric. In 2010, Khamsi and Hussain [12] reintroduced
the notion of a b-metric under the name metric-type.
Definition 1.2 ([12], Definition 6). Let X be a non-empty set, K > 0 and D : X × X −→
[0, +∞) be a function such that for all x, y, z ∈ X,

(1) D(x, y) = 0 if and only if x = y.
(2) D(x, y) = D(y, x).
(3) D(x, z) ≤ K[D(x, y) + D(y, z)].

Then D is called a metric-type on X and (X, D, K) is called a metric-type space.
Definition 1.3 ([12], Definition 7). Let (X, D, K) be a b-metric space.

(1) A sequence {xn } is called convergent to x, written as lim xn = x, if lim D(xn , x) = 0.
n→∞ n→∞
(2) A sequence {xn } is called Cauchy if lim D(xn , xm ) = 0.
n,m→∞
(3) (X, D, K) is called complete if every Cauchy sequence is a convergent sequence.

From Definition 1.2.(3), it is easy to see that K ≥ 1. Also in 2010, Khamsi [11] introduced
another definition of a metric-type where the condition (3) in Definition 1.2 was replaced by
D(x, z) ≤ K[D(x, y1 ) + . . . + D(yn , z)]
2000 Mathematics Subject Classification. Primary 47H10, 54H25; Secondary 54D99, 54E99.
1

, 2 TRAN VAN AN AND NGUYEN VAN DUNG


for all x, y1 , . . . , yn , z ∈ X, see [11, Definition 2.7]. In the sequel, the metric-type in the sense of
Khamsi and Hussain [12] will be called a b-metric to avoid the confusion about the metric-type
in the sense of Khamsi [11]. Note that every metric-type is a b-metric.
The same relaxation of the triangle inequality in Definition 1.2 was also discussed in 2003 by
Fagin et al. [8], who called this new distance measure nonlinear elastic matching. The authors of
that paper remarked that this measure had been used, for example, in [9] for trademark shapes
and in [3] to measure ice floes. In 2009, Xia [15] used this semimetric distance to study the
optimal transport path between probability measures.
In recent times, b-metric spaces were studied by many authors, especially fixed point theory on
b-metric spaces [1], [7], [10], [13, Chapter 12], [14]. Some authors were also studied topological
properties of b-metric spaces. In [2], An et al. showed that every b-metric space with the
topology induced by its convergence is a semi-metrizable space and thus many properties of
b-metric spaces used in the literature are obvious. Then, the authors proved the Stone-type
theorem on b-metric spaces and get a sufficient condition for a b-metric space to be metrizable.
Notice that a b-metric space is always understood to be a topological space with respect to the
topology induced by its convergence and a b-metric need not be continuous [2, Examples 3.9 &
3.10]. This fact suggests a strengthening of the notion of b-metric spaces which remedies this
defect.
Definition 1.4 ([13], Definition 12.7). Let X be a non-empty set, K ≥ 1 and D : X × X −→
[0, +∞) be a function such that for all x, y, z ∈ X,

(1) D(x, y) = 0 if and only if x = y.
(2) D(x, y) = D(y, x).
(3) D(x, z) ≤ D(x, y) + KD(y, z).

Then D is called a strong b-metric on X and (X, D, K) is called a strong b-metric space.
Remark 1.5 ([13], page 122). (1) Every strong b-metric is continuous.
(2) Every open ball B(a, r) = {x ∈ X : D(a, x) < r} of a strong b-metric space (X, D, K) is
open.

In [13, Chapter 12], Kirk and Shahzad surveyed b-metric spaces, strong b-metric spaces, and
related problems. An interesting work was attracted many authors is to transform results of
metric spaces to the setting of b-metric spaces. It is only fair to point out that some results seem
to require the full use of the triangle inequality of a metric space. In this connection, Kirk and
Shahzad [13, page 127] mentioned an interesting extension of Nadler’s theorem due to Dontchev
and Hager [6]. Recall that for a metric space (X, d) and A, B ⊂ X, x ∈ X,
dist(x, A) = inf{d(x, a) : a ∈ A}
δ(A, B) = sup{dist(x, A) : x ∈ B}
and these notation are understood similarly on b-metric spaces.
Theorem 1.6 ([13], Theorem 12.7). Let (X, d) be a complete metric space, T : X −→ X be a
map from X into a non-empty closed subset of X, and x0 ∈ X such that

(1) dist(x0 , T x0 ) < r(1 − k) for some r > 0 and some k ∈ [0, 1).
(2) δ(T x ∩ B(x0 , r), T y) ≤ kd(x, y) for all x, y ∈ B(x0 , r).

Then T has a fixed point in B(x0 , r).

Based on the definition of δ(A, B) and the proof of Theorem [13, Theorem 12.7], the assump-
tion (2) in the above theorem is implicitly understood as