**2. Results**

We start this section by recalling some known concepts.

As usual, we denote by Ψ the family of nondecreasing functions *ψ* : [0, ∞) → [0, ∞) such that ∑∞*<sup>n</sup>*=<sup>1</sup>*ψ<sup>n</sup>*(*t*) < ∞ for all *t* ≥ 0.

Let X be a set, T : X→X and *α* : X ×X → [0, <sup>∞</sup>). Following [1] (Definition 2.2), we say that T is *α*-admissible if *<sup>α</sup>*(*ζ*, *η*) ≥ 1 implies *α*(T *ζ*, T *η*) ≥ 1; *ζ*, *η* ∈ X .

As in the metric case [1] (Definition 2.1), given a quasi-metric space (X , *ρ*) we say that a mapping T : X→X is an *<sup>α</sup>*–*ψ*-contractive mapping if there exist two functions *α* : X ×X → [0, ∞) and *ψ* ∈ Ψ such that *<sup>α</sup>*(*ζ*, *η*)*ρ*(T *ζ*, T *η*) ≤ *ψ*(*ρ*(*ζ*, *η*)) for all *ζ*, *η* ∈ X .

The following slight modification of condition (iii) in Theorem 2.2 of [1] constitutes a crucial ingredient in obtaining our main result:

Let (X , *ρ*) be a quasi-metric space and *α* : X ×X → [0, <sup>∞</sup>). We say that (X , *ρ*) has property (A) (with respect to *α*) if for any sequence (*ζn*)*n*∈<sup>N</sup> in X satisfying *<sup>α</sup>*(*ζ<sup>n</sup>*, *ζn*+<sup>1</sup>) ≥ 1 for all *n* ∈ N and such that *ρ*(*ζ*, *ζn*) → 0 as *n* → ∞ for some *ζ* ∈ X , it follows that *<sup>α</sup>*(*ζ*, *ζn*) ≥ 1 for all *n* ∈ N.

**Definition 1.** *Given a quasi-metric space* (X , *ρ*), *an <sup>α</sup>–ψ-contractive mapping* T : X→X *will be called an <sup>α</sup>–ψ-SVV contractive mapping if: (i)* T *is α-admissible; (ii) there exists ζ*0 ∈ X *such that <sup>α</sup>*(*ζ*0, T *ζ*0) ≥ 1; *(iii)* (X , *ρ*) *has property (A) (with respect to <sup>α</sup>*).

By using the preceding definition, Theorem 2.2 of [1] can be reformulated as follows: *Every <sup>α</sup>*–*ψ*-*SVV contractive mapping on a complete metric space has a fixed point.*

Our first result provides a quasi-metric extension of Theorem 2.2 of [1] (its proof is only an adaptation of the original proof of Samet, Vetro, and Vetro).

**Theorem 1.** *Every <sup>α</sup>–ψ-SVV contractive mapping on a left K-complete quasi-metric space has a fixed point.*

**Proof of Theorem 1.** Let T be an *<sup>α</sup>*–*ψ*-*SVV* contractive mapping on a Hausdorff left K-complete quasi-metric space (X , *ρ*). Then, there exists an *α*-admissible function such that T is *<sup>α</sup>*–*ψ*-contractive, (X , *ρ*) has property (A), and *<sup>α</sup>*(*ζ*0, T *ζ*0) ≥ 1 for some *ζ*0 ∈ X .

For each *n* ∈ N let *ζn* := T *<sup>n</sup>ζ*0. If there exists *m* ∈ N such that *ζ<sup>m</sup>*−<sup>1</sup> = *ζ<sup>m</sup>*, then *ζm* is a fixed point of T . Assume then that *ζn* = *ζm* for all *n*, *m* ∈ N∪{0}. Since *<sup>α</sup>*(*ζ*0, *ζ*1) ≥ 1 and T is *α*-admissible we deduce that *<sup>α</sup>*(*ζ<sup>n</sup>*, *ζn*+<sup>1</sup>) ≥ 1 for all *n* ∈ N∪{0}. As in the proof of Theorem 2.1 of [1] we obtain *ρ*(*ζ<sup>n</sup>*, *ζn*+<sup>1</sup>) ≤ *ψ<sup>n</sup>*(*ρ*(*ζ*0, *ζ*1)) and deduce that (*ζn*)*n*∈<sup>N</sup> is a left K-Cauchy sequence in (X , *ρ*) (see [1] (p. 2156)). Since (X , *ρ*) is left K-complete there exists *θ* ∈ X such that *ρ*(*<sup>θ</sup>*, *ζn*) → 0 as *n* → ∞. From property (A) it follows that *<sup>α</sup>*(*<sup>θ</sup>*, *ζn*) ≥ 1 for all *n* ∈ N∪{0}. We shall show that *θ* is a fixed point of T . Indeed,foreach*n*∈ N∪{0}wehave:*ρ*(T*θ*,*ζn*+<sup>1</sup>)= *ρ*(T*θ*,T*ζn*)≤ *<sup>α</sup>*(*<sup>θ</sup>*,*ζn*)*ρ*(T*θ*,T*ζn*)≤ *ψ*(*ρ*(*<sup>θ</sup>*,*ζn*)).

 Since *ρ*(*<sup>θ</sup>*, *ζn*) > 0, we deduce that *ψ*(*ρ*(*<sup>θ</sup>*, *ζn*)) < *ρ*(*<sup>θ</sup>*, *ζn*) (see e.g., [1] (Lemma 2.1)), and, hence, *ρ*(T *θ*, *ζn*) → 0 as *n* → ∞. Since (X , *ρ*) is Hausdorff we conclude that *θ* = T *θ*.

As for metric spaces [1] (Theorem 2.1), a slight modification of the proof of Theorem 1 shows the following result where the property (A) is replaced by continuity of T . More precisely we have

**Theorem 2.** *Let* (X , *ρ*) *be a Hausdorff left K-complete quasi-metric space and* T : X→X *be an <sup>α</sup>–ψ-contractive mapping such that*

*(i)* T *is α-admissible;*

*(ii) there exists ζ*0 ∈ X *such that <sup>α</sup>*(*ζ*0, T *ζ*0) ≥ 1;

*(iii)* T *is continuous.*

#### *Then* T *has a fixed point.*

Theorems 1 and 2 can not be generalized to T1 left K-complete quasi-metric spaces (see e.g., [23] (Example 5)).

Let us recall that if *ρ* is a quasi-metric on a set X , then the function *ρs* defined on X ×X by *ρ<sup>s</sup>*(*ζ*, *η*) = max{*ρ*(*ζ*, *η*), *ρ*(*η*, *ζ*)} is a metric on X . We give an example for a quasi-metric space (X , *ρ*) where we can apply both Theorem 1 and Theorem 2 but not [1] (Theorem 2.2) because the metric space (X , *ρ<sup>s</sup>*) is not complete.

**Example 1.** *Let* X := {0}∪{1/*n* : *n* ∈ N}∪{*n* : *n* ∈ N\{1}}. *It is routine to check that* (X , *ρ*) *is a Hausdorff quasi-metric space where (the quasi-metric) ρ is defined as follows:*

*ρ*(*ζ*, *ζ*) = 0 *for all ζ* ∈ X . *ρ*(0, 1/*n*) = 1/*n for all n* ∈ N. *ρ*(1/*<sup>n</sup>*, 1/*m*) = 1/*n whenever n* < *m*. *ρ*(0, *n*) = 2−*<sup>n</sup> for all n* ∈ N\{1}. *ρ*(*<sup>n</sup>*, *m*) = |<sup>2</sup>−*<sup>n</sup>* − <sup>2</sup>−*<sup>m</sup>*| *for all n*, *m* ∈ N\{1}, *and ρ*(*ζ*, *η*) = 1 *otherwise.*

:

 =

*n* ∈

*Observe that* (X , *ρ*) *is left K-complete: The sequence* (1/*n*)*n*∈<sup>N</sup> *is left K-Cauchy and converges to* 0, *whereas the sequence* (*n*)*n*∈<sup>N</sup> *is Cauchy in the metric space* (X , *ρ<sup>s</sup>*), *and hence left K-Cauchy in* (X , *ρ*), *and also converges to* 0. *However*, *we have ρ*(*<sup>n</sup>*, 0) = 1 *for all n* ∈ N, *and thus the metric space* (X , *ρ<sup>s</sup>*) *is not complete. Now*TX→XT0T1*all*N,*and*T*all*

 *define* 0, + *for*  ∈ (1/*n*)  *for*  ∈ N\{1}. *We show that* T *is an <sup>α</sup>–ψ-SVV contractive mapping for α given by α*(0, *n*) = *<sup>α</sup>*(*<sup>n</sup>*, *n* + 1) = 1 *for all* N,*and<sup>α</sup>*(*ζ*,*η*)0*otherwise;andψ*Ψ*givenby ψ*(*t*)*t*/2*forallt*≥0.

*n*

 = =

*n*

*n*

*Indeed, since α*(1, *T*1) = *α*(1, 2) = 1, *we deduce by the definition of* T *and the construction of α that* T *is α-contractive. Also, the property (A) is clearly satisfied since ρ*(0, *n*) → 0 *as n* → <sup>∞</sup>, *and α*(0, *n*) = 1 *for all n* ∈ N. *It remains to check that* T *is an <sup>α</sup>–ψ-contractive mapping. To this end, it suffices to consider the followingtwocases:*

*Case 1. ζ* = 0, *η* = *n*, *n* ∈ N*. Thus, we obtain*

*as*

=

 ∈

*n*

=

*n*

$$
\mu(\zeta,\eta)\rho(T\zeta,T\eta) = \mathfrak{a}(0,n)\rho(0,n+1) = 2^{-(n+1)} \le \frac{1}{2}\rho(0,n) = \psi(\rho(\zeta,\eta)).
$$

*Case 2. ζ* = *n*, *η* = *n* + 1, *n* ∈ N. *Thus, we obtain*

$$\begin{aligned} \alpha(\zeta,\eta)\rho(T\zeta,T\eta) &= \alpha(n,n+1)\rho(Tn,T(n+1)) = \rho(n+1,n+2) \\ &= 2^{-(n+2)} = \frac{1}{2}\rho(n,n+1) = \psi(\rho(\zeta,\eta)). \end{aligned}$$

*Therefore, all conditions of Theorem 1 are satisfied. Clearly, we can also apply Theorem 2 because* T *is continuous (with respect to τρ*).

Now, we present an easy example where we can apply Theorem 1 but not Theorem 2.

**Example 2.** *Let* X := {0, ∞} ∪ N*. Clearly* (X , *ρ*) *is a Hausdorff left K-complete quasi-metric space where (the quasi-metric) ρ is defined as follows:*

*ρ*(*ζ*, *ζ*) = 0 *for all ζ* ∈ X . *ρ*(0, 1/*n*) = 1/*n for all n* ∈ N, *and ρ*(*ζ*, *η*) = 1 *otherwise.*

*Now define* T : X→X *as* T 0 = 0, T ∞ = <sup>∞</sup>, *and* T *n* = ∞ *for all n* ∈ N. *Since ρ*(0, *n*) → 0 *as n* → <sup>∞</sup>, *but ρ*(T 0, T *n*) = *ρ*(0, ∞) = 1, *we conclude that* T *is not continuous. However, it is obvious that* T *is an <sup>α</sup>–ψ-SVV contractive mapping for α given by <sup>α</sup>*(<sup>∞</sup>, ∞) = 1, *and <sup>α</sup>*(*ζ*, *y*) = 0 *otherwise, and any ψ* ∈ Ψ.

In our main result (Theorem 3 below), we prove that Theorem 1 characterizes left K-completeness of Hausdorff quasi-metric spaces. However, Theorem 2 does not provide such characterization even in the case of metric spaces, as Suzuki and Takahashi constructed in [24] an example of a non-complete metric space for which every continuous self map has fixed points.

**Theorem 3.** *A Hausdorff quasi-metric space is left K-complete if and only if every <sup>α</sup>–ψ-SVV contractive mapping has a fixed point.*

**Proof of Theorem 3.** Let (X , *ρ*) be a Hausdorff left K-complete quasi-metric space. By Theorem 1, every *<sup>α</sup>*–*ψ*-*SVV* contractive mapping on (X , *ρ*) has a fixed point.

Conversely, suppose that (X , *ρ*) is a Hausdorff quasi-metric space which is not left K-complete. Then there exists a left K-Cauchy sequence (*ζn*)*n*∈<sup>N</sup> (of distinct points) in (*ζ*, *ρ*) which is not convergen<sup>t</sup> for *τρ*. Put A = {*ζn* : *n* ∈ N}. Since *ρ*(*ζ*1, A\{*ζ*1}) > 0, there exists *h*1 ∈ N, with *h*1 > 1, such that *<sup>ρ</sup>*(*ζj*, *ζk*) < *ρ*(*ζ*1, A\{*ζ*1})/2 whenever *h*1 ≤ *j* ≤ *k*. Similarly, there exists *h*2 ∈ N, with *h*2 > max{2, *h*1}, such that *<sup>ρ</sup>*(*ζj*, *ζk*) < *ρ*(*ζ*2, A\{*ζ*2})/2 whenever *h*2 ≤ *j* ≤ *k*. In this way we obtain a subsequence (*hn*)*n*∈<sup>N</sup> of (*n*)*n*∈<sup>N</sup> such that *hn* > max{*<sup>n</sup>*, *hn*−<sup>1</sup>} and *<sup>ρ</sup>*(*ζj*, *ζk*) < *ρ*(*ζ<sup>n</sup>*, A\{*ζn*})/2 whenever *hn* ≤ *j* ≤ *k*.

Define T : X→X and *α* : X ×X → [0, ∞) as follows: T *ζn* = *ζhn* for *n* ∈ N, and T *ζ* = *ζ*1 for *ζ* ∈ X \A, and *<sup>α</sup>*(*ζ*, *η*) = 1 if *ζ* = *ζn* and *η* = *ζm* for *n*, *m* ∈ N with *n* < *m*, and *<sup>α</sup>*(*ζ*, *η*) = 0 otherwise.Wefirstnotethat*<sup>α</sup>*(*ζ*1,T*ζ*1)=1because1<*h*1.

Moreover T is *α*-admissible. Indeed, if *<sup>α</sup>*(*ζ*, *η*) ≥ 1, then *ζ* = *ζn* and *η* = *ζm* with *n* < *m*. So *α*(T *ζ*, T *η*) = *<sup>α</sup>*(*ζhn* , *ζhm* ) = 1 because *hn* < *hm*.

Next, we show that T is *<sup>α</sup>*–*ψ*-contractive for *ψ* ∈ Ψ given by *ψ*(*t*) = *t*/2. Indeed, by the construction of *α* it suffices to check the case that *ζ* = *ζn* and *η* = *ζm* with *n* < *m*. Thus, we obtain

$$\begin{aligned} \alpha(\zeta,\eta)\rho(\mathcal{T}\zeta,\mathcal{T}\eta) &= \alpha(\zeta\_{n},\zeta\_{m})\rho(\zeta\_{h\_{n}},\zeta\_{h\_{m}}) < \frac{1}{2}\rho(\zeta\_{n},\mathcal{A}\backslash\{\zeta\_{n}\}) \\ &\leq \frac{1}{2}\rho(\zeta\_{h},\zeta\_{m}) = \frac{1}{2}\rho(\zeta\_{\prime}\eta) = \psi(\rho(\zeta,\eta)). \end{aligned}$$

Finally, note that (X , *ρ*) trivially satisfies the property (A) because the only convergen<sup>t</sup> sequences in A are those that are eventually constant.

We have shown that T is an *<sup>α</sup>*–*ψ*-*SVV* contractive mapping without fixed point. This contradiction concludes the proof.

**Corollary 1.** *A metric space is complete if and only every <sup>α</sup>–ψ-SVV contractive mapping has a fixed point.*

**Author Contributions:** Investigation, S.R. and P.T.; Writing–original draft, S.R. and P.T. All authors contributed equally in writing this article. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially funded by Ministerio de Ciencia, Innovación y Universidades, under gran<sup>t</sup> PGC2018-095709-B-C21 and AEI/FEDER, UE funds.

**Acknowledgments:** The authors thank the reviewers for their useful suggestions and comments.

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