The notion of \(w\)-distance on a metric space was introduced and studied bu Kada et al. in [2].

Definition 1. Let \(\left(X,d\right)\) be a metric space. A \(w\)-distance on \(X\) is a function \(p:X\times X\to\left[0,\infty\right)\) satisfying the following conditions:

(P1) \(p\left(x,y\right)\le p\left(x,z\right)+p\left(z,y\right)\) ;

(P2) \(p\left(x,\cdot\right):X\to\left[0,\infty\right)\) is a lower semicontinuous function for all \(x\in X\) ;

(P3)  for all \(\varepsilon>0\) there exists a \(\delta>0\) such that \(p\left(z,x\right)\le\delta\) and \(p\left(z,y\right)\le\delta\) imply \(d\left(x,y\right)\le\varepsilon\) ;

for all \(x,y,z\in X\).  

We recall that a real-valued function \(f\) defined on a metric space \(\left(X,d\right)\) is lower semicontinuous at a point \(x\in X\)  if for any sequence \(\ \left\{x_n\right\}\subseteq X\) converging to \(x\) we have that either \(\liminf_{x_n\to x}f\left(x_n\right)=+\infty\) or \(f\left(x\right)\le\liminf_{x_n\to x}f\left(x_n\right)\).

The following lemma shall be used to prove the main result.

Lemma 1. [4] Let \(\left(X,d\right)\) be a metric space with a \(w\)-distance \(p\). If \(\left\{x_n\right\}\) is a sequence  in \(X\) such that  \[\lim_{n\ \to\infty}\sup_{m>n}p\left(x_n,x_m\right)=0\]  then \(\ \left\{x_n\right\}\) is Cauchy.

We prove the following statement, which is the main result of this note.

Theorem 1. A metric space \(\left(X,d\right)\) with a \(w\)-distance \(p\)  is complete if and only if :

(1) every sequence \(\ \left\{x_i\right\}\ \subseteq X\) such that \[\sum_{_{i=1}}^{\infty}p\left(x_i,x_{i+1}\right)<\infty\] converges to some \(x\in X\).

Proof. \(\left(\Rightarrow:\right)\) Let \(\left(X,d\right)\) be complete,  and let \(\ \left\{x_i\right\}\subseteq X\) be a sequence such that   \(\sum_{_{i=1}}^{\infty}p\left(x_i,x_{i+1}\right)<\infty\).  Then for all \(\varepsilon>0\) there exists \(N_{\varepsilon}\in\mathbb{N}\) such that \(\sum_{_{i=n}}^{\infty}p\left(x_i,x_{i+1}\right)<\varepsilon\)  for all \(n\ge N_{\varepsilon}\).  Hence,  for all \(m,n\in\mathbb{N}\) such that \(m>n\ge N_{\varepsilon}\) we have \(\)\[p\left(x_n,x_m\right)\le\sum_{_{i=n}}^{m-1}p\left(x_i,x_{i+1}\right)\le\sum_{_{i=n}}^{\infty}p\left(x_i,x_{i+1}\right)<\varepsilon\]which implies that   \(\lim_{n\ \to\infty}\sup_{m>n}p\left(x_n,x_m\right)=0\), so by Lemma 1,\(\ \left\{x_i\right\}\) is Cauchy. Since \(X\) is complete,\(\ \left\{x_i\right\}\) converges to some \(x\in X\).

\(\left(\Leftarrow:\right)\) Now suppose that (1) holds, but \(X\) is not complete, so there exists a Cauchy sequence \(\ \left\{x_i\right\}\subseteq X\) which is not convergent.  Let \(F=\ \left\{x_i:i\in\mathbb{N}\right\}\), and let \(p:X\times X\to\left[0,\infty\right)\) be defined as \[p\left(x,y\right)=\begin{cases} d(x,y),\; \mathrm{if}\, x,y\in F,\\ 2\,diam\, F,\; \mathrm{otherwise}.\end{cases}\]

    Since \(\ \left\{x_i\right\}\) is Cauchy sequence which is not convergent, the set \(F\) is closed and bounded, so   \(p\) is a \(w\)-distance  on \(X\) (see [2, Example 7]).  Let \(i_j\) be the least natural number such that \(p(x_n,x_m)=d\left(x_n,x_m\right)\le\frac{1}{2^j}\) for all \(m,n\in\N\) such that \(m>n\ge i_j\).  Then we have \[\sum_{j=1}^{\infty}\ p\left(x_{i_j},x_{i_{j+1}}\right)\le\sum_{j=1}^{\infty}\frac{1}{2^j}<\infty\]

which by (1) means that \(\ \left\{x_i\right\}\) has a convergent subsequence \(\ \left\{x_{i_j}\right\}\) which is impossible (since its limit would be the limit of the whole sequence). \(\square\)

Remark. In [5] the authors characterized completeness of metric spaces with a \(w\)-distance via generalized Banach's contraction, i.e. the weak contraction. In [3] the author of the present paper introduced the  functions   \(\delta_p\) and \(\alpha_p\) (\(p\)-diameter and Kuratowski \(p\)-measure of noncompactness)  on such spaces and studied the metric completenes via those functions. Hence, we can now formulate our second main result, which represents an analogue of [1, Theorem I.5.1] for metric spaces with a \(w\)-distance, and summarizes all known characterizations of completeness for such spaces.  For the definition of weak contraction we refer the reader to [5], and for the definitions of   \(\delta_p,\alpha_p\)  to [3].

Theorem 2. Let \(\left(X,d\right)\)  be a metric space with a \(w\)-distance \(p\). The following conditions are equivalent.

(i) \(X\) is complete;

(ii) Every weak contraction on \(X\) has a uniqe fixed point;

(iii) Every sequence \(F_n\) of nonempty closed subsets in \(X\) such that \(F_{n+1}\subseteq F_n\) for all \(n\in\mathbb{N}\) and \(\lim_{n\to\infty}\delta_p\left(F_n\right)=0\) has a singleton intersection;

(iv) Every sequence \(F_n\) of nonempty closed subsets in \(X\) such that \(F_{n+1}\subseteq F_n\) for all \(n\in\mathbb{N}\) and \(\lim_{n\to\infty}\alpha_p\left(F_n\right)=0\) has a nonempty compact intersection;

(v)  Every sequence \(\ \left\{x_i\right\}\ \) such that \(\sum_{_{i=1}}^{\infty}p\left(x_i,x_{i+1}\right)<\infty\) converges.  

Proof.  (i)\(\Leftrightarrow\)(ii) is proven in [5, Theorem 4], (i)  \(\Leftrightarrow\) (iii)  \(\Leftrightarrow\) (iv) is proven in [3, Theorem 3.1] and (i)  \(\Leftrightarrow\) (v) is Theorem 1 of the present paper.

[3] Aleksandar Kostić : Measures of noncompactness on w-distance spaces, author's preprint, DOI: 10.13140/RG.2.2.14302.46402