On weak partial-quasi k-metric spaces

Abstract

In this paper, we introduce the concept of weak partial-quasi k-metrics, which generalizes both k-metric and weak metric. Also, we present some examples to support our results. Furthermore, we obtain some fixed point theorems in weak partial-quasi k-metric spaces.

Keywords

weak metric k-metric partial-quasi k-metric weak partial-quasi k-metric fixed point theorem

1 Introduction

Since Ribeiro introduced the notion of weak metric spaces in 1943 [1], Bakhtin introduced the notion of b-metric spaces and presented the contraction mapping in b-metric spaces in 1989 [2]. In particular, Matthews gave the concepts of the partial metric spaces and quasi-metric spaces in 1994 [3]. It is interesting that the distance of a point to itself may not be zero in partial metric spaces, also he generalized Banach contraction theorem. After that, it is widely recognized that partial metric spaces played an important role in the theory of computation. A number of generalized metric spaces were given by topological researchers in the past few decades [4 –14]. For example, weak partial metric spaces [5], partial semi-metric spaces [4], partial-quasi metric spaces [6], quasi-partial metric spaces [7], quasi-partial b-metric spaces [8], etc.

In addition, the term “b-metric” has no justification for the letter “b” and says nothing about the constant “k” laid in the basis of the definition of such “metrics”. Following Heckmann [4] and Shukla [11], and dropping the symmetry condition, in this paper, we generalize the concept of k-metrics, weak metrics and partial-quasi metrics by introducing the weak partial-quasi k-metrics. Also we give some fixed point theorems in these spaces.

2 Preliminaries

Throughout this paper, X always denotes a nonempty set and the letters $ℝ$ , $ℝ^{+}$ , $ℕ^{+}$ always denote the set of real numbers, of all positive real numbers and of positive integers, respectively.

First, we recall some basic notions and results that will be used in the following sections (see more details in [4 –13]).

Definition 2.1. [1] A weak metric is a function δ : X × X → [0, + ∞) satisfying the following conditions: ∀x, y, z ∈ X,

(W1) δ (x, x) =0;

(W2) δ (x, z) ≤ δ (x, y) + δ (y, z).

A weak metric space is a pair (X, δ) such that δ is a weak metric on X.

Definition 2.2. [2] A k-metric is a function b : X × X → [0, + ∞) satisfying the following conditions for some number k ≥ 1: ∀x, y, z ∈ X,

(B1) x = y ⇔ b (x, y) =0;

(B2) b (x, y) = b (y, x);

(B3) b (x, z) ≤ k [b (x, y) + b (y, z)].

Definition 2.3. [4] A weak partial metric is a function p^w : X × X → [0, + ∞) satisfying the following conditions: ∀x, y, z ∈ X,

(WP1) p^w (x, x) = p^w (x, y) = p^w (y, y) ⇔x = y;

(WP2) p^w (x, y) = p^w (y, x);

(WP3) p^w (x, z) ≤ p^w (x, y) + p^w (y, z) - p^w (y, y).

A weak partial metric is called a partial metric if it also satisfies

(WP4) ∀x, y ∈ X, p^w (x, x) ≤ p^w (x, y).

Definition 2.4. A partial-quasi k-metric is a function p_q : X × X → [0, + ∞) satisfying the following conditions for some number k ≥ 1: ∀x, y, z ∈ X,

(PQ1) x = y ⇔ p_q (x, x) = p_q (x, y) and p_q (y, y) = p_q (y, x);

(PQ2) p_q (x, x) ≤ p_q (x, y) ∧ p_q (y, x);

(PQ3) p_q (x, z) ≤ k [p_q (x, y) + p_q (y, z)] - p_q (y, y).

Apparently, if p^w is a weak partial metric on X, then it is easy to see that $P^{w} (x, y) \geq \frac{P^{w} (x, x) + P^{w} (y, y)}{2}$ for all x, y ∈ X, namely weak partial spaces are not far from small self-distance axiom. Moreover, each partial metric space is a weak partial metric space, but the converse may not be true. Moreover, we claim that ${\hat{p}}^{w}$ is an ordinary metric on X, where p^w is a weak partial metric and $\begin{matrix} {\hat{p}}^{w} (x, y) & = {p^{w} (x, y) - p^{w} (x, x)} \lor {p^{w} (x, y) - p^{w} (y, y)} \\ = p^{w} (x, y) - p^{w} (x, x) \land p^{w} (y, y) . \end{matrix}$

Definition 2.5. A weak partial-quasi k-metric is a function $p_{k}^{w} : X \times X \to [0, + \infty)$ satisfying the following conditions for some number k ≥ 1: ∀x, y, z ∈ X,

(WPK1) x = y $\Leftrightarrow p_{k}^{w} (x, x) = p_{k}^{w} (x, y)$ ;

(WPK2) $p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) \leq p_{k}^{w} (x, y)$ ;

(WPK3) $p_{k}^{w} (x, z) \leq k [p_{k}^{w} (x, y) + p_{k}^{w} (y, z) - p_{k}^{w} (y, y)] - (k - 1) p_{k}^{w} (x, x) \land p_{k}^{w} (z, z)$ .

A weak partial-quasi k-metric space is a pair $(X, p_{k}^{w})$ such that $p_{k}^{w}$ is a weak partial-quasi k-metric on X, the number k is called the coefficient of $(X, p_{k}^{w})$ .

Example 2.6. Let X = [0, + ∞) and define $p_{k}^{w}$ : X × X → [0, + ∞) by $p_{k}^{w} (x, y) = max {x, y} + \frac{y - x}{2}$ for all x, y ∈ X. Then $(X, p_{k}^{w})$ is a weak partial-quasi k-metric space.

We verify the conditions (WPK1)-(WPK3) one by one.

(WPK1): (⇒) Suppose x = y. It is clear that $p_{k}^{w} (x, x) = p_{k}^{w} (x, y) = x$ .

(←) For any x, y ∈ X, we claim that if x ≤ y, then $max {x, y} + \frac{y - x}{2} = \frac{3 y - x}{2}$ ; if x > y, then $max {x, y} + \frac{y - x}{2} = \frac{y + x}{2}$ .

Suppose $p_{k}^{w} (x, x) = p_{k}^{w} (x, y)$ . Then $x = max {x, y} + \frac{y - x}{2}$ .

If x ≤ y, then we have $x = \frac{3 y - x}{2}$ , which implies that x = y. If x > y, then we have $x = \frac{y + x}{2}$ , which also implies that x = y.

Therefore, $p_{k}^{w}$ satisfies the rule (WPK1).

(WPK2): By assumption, we have $p_{k}^{w} (x, x) = x$ , $p_{k}^{w} (y, y) = y$ . Thus, $p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) = min {x, y}$ .

If x ≤ y, then by min {x, y} = x and $max {x, y} + \frac{y - x}{2} = \frac{3 y - x}{2}$ , we have $x \leq \frac{3 y - x}{2}$ . Namely, $p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) \leq p_{k}^{w} (x, y)$ .

If x > y, then by min {x, y} = y and $max {x, y} + \frac{y - x}{2} = \frac{y + x}{2}$ , we have $y < \frac{y + x}{2}$ . Namely, $p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) < p_{k}^{w} (x, y)$ .

Therefore, $p_{k}^{w}$ satisfies the rule (WPK2).

(WPK3): First, we claim that max {a, c} ≤ max {a, b} + max {b, c} - b for all a, b, c ∈ [0, + ∞). Then we have $\begin{matrix} p_{k}^{w} (x, z) & = max {x, z} + \frac{z - x}{2} \\ \leq max {x, y} + max {y, z} - y + \frac{z - y}{2} + \frac{y - x}{2} \\ = max {x, y} + \frac{y - x}{2} + max {y, z} + \frac{z - y}{2} - max {y, y}, \end{matrix}$ for all x, y, z ∈ X. Thus we have

$p_{k}^{w} (x, z) \leq k [p_{k}^{w} (x, y) + p_{k}^{w} (y, z) - p_{k}^{w} (y, y)] - (k - 1) p_{k}^{w} (x, x) \land p_{k}^{w} (z, z)$ .

Hence, ( $X, p_{k}^{w})$ is a weak partial-quasi k-metric space with k = 1.

Example 2.7. Let $X = [0, \frac{π}{8}]$ and define $p_{k}^{w}$ : X × X → [0, + ∞) by $p_{k}^{w} (x, y) = sin 2 | x - y | + x$ for all x, y ∈ X. Then $(X, p_{k}^{w})$ is a weak partial-quasi k-metric space.

We verify the conditions (WPK1)-(WPK3) one by one.

(WPK1): (⇒) Suppose x = y. It is clear that $p_{k}^{w} (x, x) = p_{k}^{w} (x, y) = x$ .

(←) Suppose $p_{k}^{w} (x, x) = p_{k}^{w} (x, y)$ . Then x = sin 2|x - y| + x, which implies that sin |x - y|=0, namely x = y.

(WPK2): First, we claim that min {x, y} ≤ x and min {x, y} ≤ y. Since $p_{k}^{w} (x, x) = x$ , $p_{k}^{w} (y, y) = y$ , we have $p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) = min {x, y} \leq x \leq sin 2 | x - y | + x = p_{k}^{w} (x, y) .$ Therefore, $p_{k}^{w}$ satisfies rule (WPK2).

(WPK3): First, we claim that sin 2x ≥ x for all $x \in [0, \frac{π}{8}]$ and | sin x| ≤ |x| for all $x \in ℝ$ , respectively. Then we have $\begin{matrix} p_{k}^{w} (x, z) & = sin 2 | x - z | + x \\ \leq sin 2 (| x - y | + | y - z |) + x \\ \leq 2 (| x - y | + | y - z |) + x \\ \leq 2 sin 2 | x - y | + 2 sin 2 | y - z | + x \\ = 2 [(sin 2 | x - y | + x) + (sin 2 | y - z | + y) - y] - x \\ \leq 2 [(sin 2 | x - y | + x) + (sin 2 | y - z | + y) - y] - min {x, z} \\ = 2 [p_{k}^{w} (x, y) + p_{k}^{w} (y, z) - p_{k}^{w} (y, y)] - p_{k}^{w} (x, x) \land p_{k}^{w} (z, z), \end{matrix}$ for all x, y, z ∈ X. Hence, ( $X, p_{k}^{w})$ is a weak partial-quasi k-metric space with k = 2.

Apparently, if (X, p^w) is a weak partial-quasi k-metric space, then it is neither a metric space nor a weak metric space, also neither a k-metric space nor a partial metric space. The above conclusions can be illustrated by the following example:

Example 2.8. Let X = [0, + ∞) and define $p_{k}^{w}$ : X × X → [0, + ∞) by $p_{k}^{w} (x, y) = \frac{x + y}{2}$ for all x, y ∈ X. Then $(X, p_{k}^{w})$ is a weak partial-quasi k-metric space with k = 1.

In fact, it is trivial that $p_{k}^{w}$ satisfies (WPK1)-(WPK3).

First, we have $p_{k}^{w} (1, 1) = 1 \neq 0$ and $p_{k}^{w} (1, 0) = \frac{1}{2} < p_{k}^{w} (1, 1)$ , which shows that a weak partial-quasi k-metric space is neither a metric space nor a weak metric space. Furthermore, by Example 2.7, we have $p_{k}^{w} (x, y) = sin 2 | x - y | + x$ and $p_{k}^{w} (y, x) = sin 2 | y - x | + y$ , which shows that a weak partial-quasi k-metric space may be neither a weak partial metric space nor a partial-quasi metric space.

Remark 2.9.

$p_{k}^{w} (x, x) + p_{k}^{w} (y, y) \leq p_{k}^{w} (x, y) + p_{k}^{w} (y, x)$ for all x, y ∈ X, where p^w is a weak partial-quasi k-metric on X;

If $p_{k}^{w} (x, y) = 0$ for all x, y ∈ X, then x = y, where p^w is a weak partial-quasi k-metric on X.

Indeed, by (WPK3), set z = x. We have $p_{k}^{w} (x, x) \leq k [p_{k}^{w} (x, y) + p_{k}^{w} (y, x) - p_{k}^{w} (y, y)] - (k - 1) p_{k}^{w} (x, x)$ . Therefore, $p_{k}^{w} (x, x) + p_{k}^{w} (y, y) \leq p_{k}^{w} (x, y) + p_{k}^{w} (y, x)$ .

Furthermore, by (WPK2), we have $p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) \leq p_{k}^{w} (x, y)$ . If $p_{k}^{w} (x, y) = 0$ , then $p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) = 0$ . Thus $p_{k}^{w} (x, x) = 0$ or $p_{k}^{w} (y, y) = 0$ , then we have $p_{k}^{w} (x, x) = p_{k}^{w} (x, y)$ or $p_{k}^{w} (y, y) = p_{k}^{w} (x, y)$ , which implies that x = y by (WPK1).

Proposition 2.10. Let X be a nonempty set, $(X, p_{k}^{w})$ a weak partial-quasi k-metric space with coefficient k ≥ 1. Define the function ${\hat{p}}_{k}^{w} : X \times X \to [0, + \infty)$ as follows: ${\hat{p}}_{k}^{w} (x, y) = [p_{k}^{w} (x, y) + p_{k}^{w} (y, x)] - 2 p_{k}^{w} (x, x) \land p_{k}^{w} (y, y)$ for all x, y ∈ X. Then ${\hat{p}}_{k}^{w}$ is a k-metric on X.

Proof. We verify the conditions (B1)-(B3) one by one.

(B1): (⇒): Suppose x, y ∈ X. It is clear that ${\hat{p}}_{k}^{w} (x, x) = 0$ .

(←): Suppose ${\hat{p}}_{k}^{w} (x, y) = 0$ . Then $p_{k}^{w} (x, y) + p_{k}^{w} (y, x) = 2 p_{k}^{w} (x, x) \land p_{k}^{w} (y, y)$ . By (WPK2), we have $p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) \leq p_{k}^{w} (x, y)$ . Then $p_{k}^{w} (x, y) + p_{k}^{w} (y, x) \leq 2 p_{k}^{w} (x, y),$ which implies that $p_{k}^{w} (y, x) \leq p_{k}^{w} (x, y)$ . Similarly, we have $p_{k}^{w} (x, y) \leq p_{k}^{w} (y, x)$ . Thus $p_{k}^{w} (x, y) = p_{k}^{w} (y, x) = p_{k}^{w} (x, x) \land p_{k}^{w} (y, y)$ . Without loss of generality, if $p_{k}^{w} (x, x) \leq p_{k}^{w} (y, y)$ , then we have $p_{k}^{w} (x, y) = p_{k}^{w} (x, x)$ , so x = y by (WPK1).

(B2): It is clear that ${\hat{p}}_{k}^{w} (x, y) = {\hat{p}}_{k}^{w} (y, x)$ for all x, y ∈ X.

(B3): First, we claim that min {a, c} ≥ min {a, b} + min {b, c} - b for all $a, b, c \in ℝ$ . By (WPK3), we have $\begin{matrix} k [{\hat{p}}_{k}^{w} (x, y) + {\hat{p}}_{k}^{w} (y, z)] \\ = k [p_{k}^{w} (x, y) + p_{k}^{w} (y, x) - 2 p_{k}^{w} (x, x) \land p_{k}^{w} (y, y)] \\ + k [p_{k}^{w} (y, z) + p_{k}^{w} (z, y) - 2 p_{k}^{w} (y, y) \land p_{k}^{w} (z, z)] \\ = k [p_{k}^{w} (x, y) + p_{k}^{w} (y, z)] + k [p_{k}^{w} (z, y) + p_{k}^{w} (y, x)] - 2 {kp}_{k}^{w} (x, x) \land p_{k}^{w} (y, y) \\ - 2 {kp}_{k}^{w} (y, y) \land p_{k}^{w} (z, z) \\ \geq [p_{k}^{w} (x, z) + k p_{k}^{w} (y, y) + (k - 1) p_{k}^{w} (x, x) \land p_{k}^{w} (z, z)] + [p_{k}^{w} (z, x) + k p_{k}^{w} (y, y) \\ + (k - 1) p_{k}^{w} (x, x) \land p_{k}^{w} (z, z)] - 2 k [p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) + p_{k}^{w} (y, y) \land p_{k}^{w} (z, z)] \\ = [p_{k}^{w} (x, z) + p_{k}^{w} (z, x)] + 2 k p_{k}^{w} (y, y) + 2 (k - 1) p_{k}^{w} (x, x) \land p_{k}^{w} (z, z) \\ - 2 k [p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) + p_{k}^{w} (y, y) \land p_{k}^{w} (z, z)] \\ = [p_{k}^{w} (x, z) + p_{k}^{w} (z, x) - 2 p_{k}^{w} (x, x) \land p_{k}^{w} (z, z)] + 2 k [p_{k}^{w} (y, y) + p_{k}^{w} (x, x) \land p_{k}^{w} (z, z) \\ - p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) - p_{k}^{w} (y, y) \land p_{k}^{w} (z, z)] \\ \geq {\hat{p}}_{k}^{w} (x, z), \end{matrix}$ for all x, y, z ∈ X. Hence, ${\hat{p}}_{k}^{w}$ is a k-metric. □

Lemma 2.11. Let $(X, p_{k}^{w})$ be a weak partial-quasi k-metric space and $(X, {\hat{p}}_{k}^{w})$ the corresponding k-metric space. Then $| p_{k}^{w} (x, x) - p_{k}^{w} (y, y) | \leq {\hat{p}}_{k}^{w} (x, y)$ , where ${\hat{p}}_{k}^{w} (x, y) = [p_{k}^{w} (x, y) + p_{k}^{w} (y, x)] - 2 p_{k}^{w} (x, x) \land p_{k}^{w} (y, y)$ for all x, y ∈ X.

Proof. By Remark 2.9 (1), we have $\begin{matrix} | p_{k}^{w} (x, x) - p_{k}^{w} (y, y) | \\ = p_{k}^{w} (x, x) \lor p_{k}^{w} (y, y) - p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) \\ = p_{k}^{w} (x, x) \lor p_{k}^{w} (y, y) + p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) - 2 p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) \\ = [p_{k}^{w} (x, x) + p_{k}^{w} (y, y)] - 2 p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) \\ \leq [p_{k}^{w} (x, y) + p_{k}^{w} (y, x)] - 2 p_{k}^{w} (x, x) \land p_{k}^{w} (y, y) \\ = {\hat{p}}_{k}^{w} (x, y), \end{matrix}$ completing the proof. □

3 Fixed point theorem on weak partial-quasi k-metric spaces

Definition 3.1. Let $(X, p_{k}^{w})$ be a weak partial-quasi b-metric space, and {x_n} a sequence in X.

A sequence {x_n} converges to a point x ∈ X if $p_{k}^{w} (x, x) = lim_{n \to + \infty} p_{k}^{w} (x, x_{n})$ ;

A sequence {x_n} is called a Cauchy sequence if $lim_{n, m \to + \infty} p_{k}^{w} (x_{n}, x_{m})$ exists and is finite;

$(X, p_{k}^{w})$ is said to be complete if every Cauchy sequence {x_n} in X converges to a point x ∈ X such that

$lim_{n, m \to + \infty} p_{k}^{w} (x_{n}, x_{m}) = lim_{n \to + \infty} p_{k}^{w} (x_{n}, x) = p_{k}^{w} (x, x)$ .

Lemma 3.2. Let $(X, p_{k}^{w})$ be a weak partial-quasi k-metric space and $(X, {\hat{p}}_{k}^{w})$ the corresponding k-metric space defined in Proposition 2.10, i.e., ${\hat{p}}_{k}^{w} (x, y) = [p_{k}^{w} (x, y) + p_{k}^{w} (y, x)] - 2 p_{k}^{w} (x, x) \land p_{k}^{w} (y, y)$ for all x, y ∈ X. Then the following statements hold.

A sequence is a Cauchy sequence in $(X, p_{k}^{w})$ if and only if it is a Cauchy sequence in $(X, {\hat{p}}_{k}^{w})$ .

$(X, p_{k}^{w})$ is complete if and only if $(X, {\hat{p}}_{k}^{w})$ is complete. Furthermore,

$lim_{n \to + \infty} {\hat{p}}^{w} (x_{n}, x) = 0$ if and only if $p_{k}^{w} (x, x) = lim_{n \to + \infty} p_{k}^{w} (x_{n}, x) = lim_{n, m \to + \infty} p_{k}^{w} (x_{n}, x_{m})$ .

Proof. (1) (⇒) Let {x_n} be a Cauchy sequence in $(X, p_{k}^{w})$ . There exists a ∈ X such that $lim_{n, m \to + \infty} p_{k}^{w} (x_{n}, x_{m}) = a$ . Then for any ɛ > 0, there exists $n_{0} \in ℕ^{+}$ such that $| p_{k}^{w} (x_{n}, x_{m}) - a | < \frac{ɛ}{4}, \forall n, m \geq n_{0} .$ Then we have $\begin{matrix} | {\hat{p}}_{k}^{w} (x_{n}, x_{m}) | \\ = | p_{k}^{w} (x_{n}, x_{m}) + p_{k}^{w} (x_{m}, x_{n}) - 2 p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x_{m}, x_{m}) | \\ = | (p_{k}^{w} (x_{n}, x_{m}) - a) + (p_{k}^{w} (x_{m}, x_{n}) - a) - 2 (p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x_{m}, x_{m}) - a) | \\ \leq | p_{k}^{w} (x_{n}, x_{m}) - a | + | p_{k}^{w} (x_{m}, x_{n}) - a | + 2 | p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x_{m}, x_{m}) - a | \\ < \frac{ɛ}{4} + \frac{ɛ}{4} + \frac{2 ɛ}{4} = ɛ, \end{matrix}$ which implies that {x_n} is a Cauchy sequence in $(X, {\hat{p}}_{k}^{w})$ .

(←) Suppose {x_n} is a Cauchy sequence in $(X, {\hat{p}}_{k}^{w})$ and let ɛ > 0. Then there exists $n_{ɛ} \in ℕ^{+}$ such that ${\hat{p}}_{k}^{w} (x_{n}, x_{m}) < \frac{ɛ}{2}, \forall n, m \geq n_{ɛ} .$ Set ɛ = 1. Then there exists $n_{0} \in ℕ^{+}$ such that ${\hat{p}}_{k}^{w} (x_{n}, x_{m}) < \frac{1}{2}, \forall n, m \geq n_{0} .$ We prove that {x_n} is a Cauchy sequence in $(X, p_{k}^{w})$ in the following steps.

Step 1: By Lemma 2.11, we have $\begin{matrix} p_{k}^{w} (x_{n}, x_{n}) \\ = p_{k}^{w} (x_{n}, x_{n}) - p_{k}^{w} (x_{n_{0}}, x_{n_{0}}) + p_{k}^{w} (x_{n_{0}}, x_{n_{0}}) \\ \leq | p_{k}^{w} (x_{n}, x_{n}) - p_{k}^{w} (x_{n_{0}}, x_{n_{0}}) | + p_{k}^{w} (x_{n_{0}}, x_{n_{0}}) \\ \leq {\hat{p}}_{k}^{w} (x_{n}, x_{n_{0}}) + p_{k}^{w} (x_{n_{0}}, x_{n_{0}}) \\ < \frac{1}{2} + p_{k}^{w} (x_{n_{0}}, x_{n_{0}}) \end{matrix}$ for all n ≥ n₀, which implies that the sequence ${p_{k}^{w} (x_{n}, x_{n})}$ is bounded in $ℝ$ . Therefore, the sequence ${p_{k}^{w} (x_{n}, x_{n})}$ has a subsequence ${p_{k}^{w} (x_{n_{k}}, x_{n_{k}})}$ that is convergent, and we denote $lim_{n_{k} \to + \infty} p_{k}^{w} (x_{n_{k}}, x_{n_{k}}) = a$ .

On the other hand, since $lim_{n, m \to + \infty} {\hat{p}}_{k}^{w} (x_{n}, x_{m}) = 0$ for all n, m ≥ n_ɛ, we have that $lim_{n, m \to + \infty} [p_{k}^{w} (x_{n}, x_{m}) + p_{k}^{w} (x_{m}, x_{n}) - 2 p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x_{m}, x_{m})] = 0 .$

In addition, by (WPK2), we deduced that $lim_{n, m \to + \infty} [p_{k}^{w} (x_{n}, x_{m}) - p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x_{m}, x_{m})] = 0$ and $lim_{n, m \to + \infty} [p_{k}^{w} (x_{m}, x_{n}) - p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x_{m}, x_{m})] = 0 .$ Therefore, given ɛ > 0, there exists $n_{1} \in ℕ^{+}$ , such that $| p_{k}^{w} (x_{n}, x_{m}) - p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x_{m}, x_{m}) | < \frac{ɛ}{2}, \forall n, m \geq n_{1} .$ Moreover, $| p_{k}^{w} (x_{n}, x_{n}) - p_{k}^{w} (x_{m}, x_{m}) | \leq {\hat{p}}_{k}^{w} (x_{n}, x_{m}) < \frac{ɛ}{2}$ for all n, m ≥ n_ɛ, which implies that the sequence ${p_{k}^{w} (x_{n}, x_{n})}$ is a Cauchy subsequence. Thus $lim_{n \to + \infty} p_{k}^{w} (x_{n}, x_{n}) = a$ , namely, for any ɛ > 0, there exists $n_{2} \in ℕ^{+}$ , such that $| p_{k}^{w} (x_{n}, x_{n}) - a | < \frac{ɛ}{2}, \forall n \geq max {n_{ɛ}, n_{2}} .$

Step 2: By Step 1, we have $\begin{matrix} | p_{k}^{w} (x_{n}, x_{m}) - a | \\ = | p_{k}^{w} (x_{n}, x_{m}) - p_{k}^{w} (x_{n} x_{n}) \land p_{k}^{w} (x_{m}, x_{m}) + p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x_{m}, x_{m}) - a | \\ \leq | p_{k}^{w} (x_{n}, x_{m}) - p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x_{m}, x_{m}) | + | p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x_{m}, x_{m}) - a | \\ < \frac{ɛ}{2} + \frac{ɛ}{2} = ɛ, \end{matrix}$ for all n, m ≥ n₃, where n₃ = max {n₀, n₁, n₂, n_ɛ}.

From Step 1 and Step 2, we obtain that $lim_{n, m \to + \infty} p_{k}^{w} (x_{n}, x_{m}) = a$ , which implies that {x_n} is a Cauchy sequence in $(X, p_{k}^{w})$ .

(2)(←) Let {x_n} be a Cauchy sequence in $(X, p_{k}^{w})$ . It is clear that {x_n} is a Cauchy sequence in $(X, {\hat{p}}_{k}^{w})$ by Lemma 3.2 (1). Since $(X, {\hat{p}}_{k}^{w})$ is complete, there exists x ∈ X such that $lim_{n \to + \infty} {\hat{p}}_{k}^{w} (x_{n}, x) = 0$ . This shows that {x_n} is a convergent sequence in $(X, {\hat{p}}_{k}^{w})$ , and we have $lim_{n \to + \infty} [p_{k}^{w} (x_{n}, x) + p_{k}^{w} (x, x_{n}) - 2 p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x, x)] = 0 .$ Also, $| p_{k}^{w} (x_{n}, x_{n}) - p_{k}^{w} (x, x) | \leq {\hat{p}}_{k}^{w} (x_{n}, x),$ which implies that $lim_{n \to + \infty} p_{k}^{w} (x_{n}, x_{n}) = p_{k}^{w} (x, x)$ .

In addition, we have $lim_{n \to + \infty} [p_{k}^{w} (x_{n}, x) - p_{k}^{w} (x_{n} x_{n}) \land p_{k}^{w} (x, x)] + lim_{n \to + \infty} [p_{k}^{w} (x, x_{n}) - p_{k}^{w} (x, x) \land p_{k}^{w} (x_{n}, x_{n})] = 0 .$ Thus $lim_{n \to + \infty} p_{k}^{w} (x_{n}, x) = lim_{n \to + \infty} p_{k}^{w} (x, x_{n}) = p_{k}^{w} (x, x) .$ Moreover, by (WPK3) we have $\begin{matrix} p_{k}^{w} (x_{n}, x_{m}) \\ \leq lim_{n, m \to + \infty} k [(p_{k}^{w} (x_{n}, x) + p_{k}^{w} (x, x_{m}) - p_{k}^{w} (x, x)) - (k - 1) p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x_{m}, x_{m})] \\ = p_{k}^{w} (x, x) . \end{matrix}$ On the other hand, since $p_{k}^{w} (x, y) \geq p_{k}^{w} (x, x) \land p_{k}^{w} (y, y)$ , we have $lim_{n, m \to + \infty} p_{k}^{w} (x_{n}, x_{m}) \geq lim_{n, m \to + \infty} p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x_{m}, x_{m}) .$ Hence, $lim_{n, m \to + \infty} p_{k}^{w} (x_{n}, x_{m}) = p_{k}^{w} (x, x)$ . This implies that $(X, p_{k}^{w})$ is complete.

(⇒) Let {x_n} be a Cauchy sequence in $(X, {\hat{p}}_{k}^{w})$ . Then {x_n} is a Cauchy sequence in $(X, p_{k}^{w})$ by Lemma 3.2 (1). Since $(X, p_{k}^{w})$ is complete, there exists x ∈ X such that $lim_{n, m \to + \infty} p_{k}^{w} (x_{n}, x_{m}) = lim_{n \to + \infty} p_{k}^{w} (x_{n}, x) = lim_{n \to + \infty} p_{k}^{w} (x, x_{n}) = p_{k}^{w} (x, x) .$ Then, for a given ɛ > 0, there exists $n_{ɛ} \in ℕ^{+}$ such that $| p_{k}^{w} (x_{n}, x) - p_{k}^{w} (x_{n}, x_{n}) | \lor | p_{k}^{w} (x, x_{n}) - p_{k}^{w} (x, x) | < \frac{ɛ}{2}, \forall n \geq n_{ɛ} .$ Thus, we have $\begin{matrix} | {\hat{p}}_{k}^{w} (x_{n}, x) | \\ = | p_{k}^{w} (x_{n}, x) + p_{k}^{w} (x, x_{n}) - 2 p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x, x) | \\ \leq | p_{k}^{w} (x_{n}, x) - p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x, x) | + | p_{k}^{w} (x, x_{n}) - p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x, x) | \\ < \frac{ɛ}{2} + \frac{ɛ}{2} = ɛ \end{matrix}$ for all n ≥ n_ɛ. This implies $lim_{n \to + \infty} {\hat{p}}_{k}^{w} (x_{n}, x) = 0$ . Similarly, we also have $lim_{n \to + \infty} {\hat{p}}_{k}^{w} (x, x_{n}) = 0$ . Then we can deduce $lim_{n, m \to + \infty} {\hat{p}}_{k}^{w} (x_{n}, x_{m}) = 0$ . Hence, $(X, {\hat{p}}_{k}^{w})$ is complete. □

Theorem 3.3. Let $(X, p_{k}^{w})$ be a complete weak partial-quasi k-metric space with coefficient k ≥ 1, and T : X → X a function satisfying $p_{k}^{w} (Tx, Ty) \leq λ p_{k}^{w} (x, y)$ for all x, y ∈ X, where $λ \in [0, \frac{1}{k})$ . Then T has a unique fixed point x^* ∈ X, and $p_{k}^{w} (x^{*}, x^{*}) = 0 .$

Proof. First we define the sequence in the following way:

x₀ = x, and x_n+1 = Tx_n = Tⁿ⁺¹x₀ for all $n \in ℕ^{+}$ , x₀ ∈ X.

Case 1: If x_n+1 = x_n for some $n \in ℕ^{+}$ , then we have x_n = x_n+1 = Tx_n, which shows that x_n is a fixed point.

Case 2: If x_n+1 ≠ x_n for all $n \in ℕ^{+}$ , then $p_{k}^{w} (x_{n}, x_{n + 1}) = p_{k}^{w} ({Tx}_{n - 1}, {Tx}_{n}) \leq λ p_{k}^{w} (x_{n - 1}, x_{n})$ . By repetition of this process, we have $p_{k}^{w} (x_{n}, x_{n + 1}) \leq λ^{n} p_{k}^{w} (x_{0}, x_{1})$ . Similarly, we have $p_{k}^{w} (x_{n + 1}, x_{n}) \leq λ^{n} p_{k}^{w} (x_{1}, x_{0})$ for all $n \in ℕ^{+}$ .

Case 1: $p_{k}^{w} (x_{0}, x_{1}) = 0$ , namely $p_{k}^{w} (x_{0}, {Tx}_{0}) = 0$ . Then we have Tx₀ = x₀ by Remark 2.9 (2), which shows that x₀ is a fixed point.

Case 2: $p_{k}^{w} (x_{0}, x_{1}) \neq 0$ , namely $p_{k}^{w} (x_{0}, {Tx}_{0}) > 0$ . To prove the existence of the fixed point, we consider the following steps:

Step 1: By (WPK3), We have $\begin{matrix} p_{k}^{w} (x_{n}, x_{n + p}) \\ \leq k [p_{k}^{w} (x_{n}, x_{n + 1}) + p_{k}^{w} (x_{n + 1}, x_{n + p}) - p_{k}^{w} (x_{n + 1}, x_{n + 1})] \\ - (k - 1) p_{k}^{w} (x_{n}, x_{n}) \land p_{k}^{w} (x_{n + p}, x_{n + p}) \\ \leq {kp}_{k}^{w} (x_{n}, x_{n + 1}) + {kp}_{k}^{w} (x_{n + 1}, x_{n + p}) \\ \leq {kp}_{k}^{w} (x_{n}, x_{n + 1}) + k [{kp}_{k}^{w} (x_{n + 1}, x_{n + 2}) + {kp}_{k}^{w} (x_{n + 2}, x_{n + p})] \\ \leq {kp}_{k}^{w} (x_{n}, x_{n + 1}) + k^{2} p_{k}^{w} (x_{n + 1}, x_{n + 2}) + k^{2} [{kp}_{k}^{w} (x_{n + 2}, x_{n + p}) + {kp}_{k}^{w} (x_{n + 3}, x_{n + p})] \end{matrix}$ for all $p \in ℕ^{+}$ . By continuing with above process, we have

$p_{k}^{w} (x_{n}, x_{n + p}) \leq {kp}_{k}^{w} (x_{n}, x_{n + 1}) + k^{2} p_{k}^{w} (x_{n + 1}, x_{n + 2}) + \dots + k^{p} p_{k}^{w} (x_{n + p - 1}, x_{n + p}) .$ Since $p_{k}^{w} (x_{n}, x_{n + 1}) \leq λ^{n} p_{k}^{w} (x_{0}, x_{1})$ , we have $\begin{matrix} p_{k}^{w} (x_{n}, x_{n + p}) \\ \leq k λ^{n} p_{k}^{w} (x_{0}, x_{1}) + k^{2} λ^{n + 1} p_{k}^{w} (x_{0}, x_{1}) + \dots \\ + k^{p} λ^{n + p - 1} p_{k}^{w} (x_{0}, x_{1}) \\ \leq (k λ)^{n} p_{k}^{w} (x_{0}, x_{1}) + (k λ)^{n + 1} p_{k}^{w} (x_{0}, x_{1}) + \dots \\ + (k λ)^{n + p - 1} p_{k}^{w} (x_{0}, x_{1}) . \end{matrix}$ Since coefficient k ≥ 1 and $0 < λ < \frac{1}{k}$ , we have 0 < kλ < 1. Thus $Σ_{n = 1}^{\infty} t^{n} p_{k}^{w} (x_{0}, x_{1})$ is convergent, where t = kλ.

Set $S_{n} (t) = Σ_{m = 1}^{n} t^{m} p_{k}^{w} (x_{0}, x_{1})$ . We have $p_{k}^{w} (x_{n}, x_{n + p}) \leq S_{n + p - 1} (t) - S_{n - 1} (t)$ . Then $lim_{n \to + \infty} p_{k}^{w} (x_{n}, x_{n + p}) \leq lim_{n \to + \infty} [S_{n + p - 1} (t) - S_{n - 1} (t)],$ thus $lim_{n \to + \infty} p_{k}^{w} (x_{n}, x_{n + p}) = 0$ . Hence, we have $lim_{n, m \to + \infty} p_{k}^{w} (x_{n}, x_{m}) = 0$ , and then {x_n} is a Cauchy sequence.

Step 2: Since $(X, p_{k}^{w})$ is complete, there exists some point x^* ∈ X, such that $lim_{n \to + \infty} x_{n} = x^{*}$ . This implies $\begin{matrix} lim_{n \to + \infty} p_{k}^{w} (x_{n}, x^{*}) = lim_{n \to + \infty} p_{k}^{w} (x^{*}, x_{n}) \\ = lim_{n \to + \infty} p_{k}^{w} (x_{n}, x_{n}) = p_{k}^{w} (x^{*}, x^{*}) = 0 . \end{matrix}$ We claim that x^* is a fixed point. In fact, we have $\begin{matrix} p_{k}^{w} (x^{*}, {Tx}^{*}) \\ \leq k [p_{k}^{w} (x^{*}, x_{n + 1}) + p_{k}^{w} (x_{n + 1}, {Tx}^{*}) - p_{k}^{w} (x_{n + 1}, x_{n + 1})] \\ - (k - 1) p_{k}^{w} (x^{*}, x^{*}) \land p_{k}^{w} ({Tx}^{*}, {Tx}^{*}) \\ \leq k [p_{k}^{w} (x^{*}, x_{n}) + p_{k}^{w} (x_{n + 1}, {Tx}^{*})] \\ = k [p_{k}^{w} (x^{*}, x_{n}) + p_{k}^{w} ({Tx}_{n}, {Tx}^{*})] \\ \leq k [p_{k}^{w} (x^{*}, x_{n}) + λ p_{k}^{w} (x_{n}, x^{*})] . \end{matrix}$ Then $p_{k}^{w} (x^{*}, {Tx}^{*}) \leq lim_{n \to + \infty} k [p_{k}^{w} (x^{*}, x_{n}) + λ p_{k}^{w} (x_{n}, x^{*})]$ , which implies that

$p_{k}^{w} (x^{*}, {Tx}^{*}) = 0$ . By Remark 2.9 (2), we have x^* = Tx^*.

Now we prove the uniqueness of the fixed point. Suppose x^* ≠ y^*, we have $p_{k}^{w} (x^{*}, y^{*}) = p_{k}^{w} ({Tx}^{*}, {Ty}^{*}) \leq λ p_{k}^{w} (x^{*}, y^{*}) < p_{k}^{w} (x^{*}, y^{*}),$ which is a contradiction. Hence x^* = y^*. □

Corollary 3.4. Let $(X, p_{k}^{w})$ be a complete weak partial-quasi k-metric space with coefficient k ≥ 1, and T : X → X a function satisfying $p_{k}^{w} (Tx, Ty) \leq φ (t) p_{k}^{w} (x, y)$ for all x, y ∈ X, where φ : [0, + ∞) → [0, + ∞) is a continuous function, which satisfies

(1) φ (t) =0 if and only if t = 0;

(2) φ (t) >0 for all t > 0.

If $Σ_{n = 1}^{\infty} k^{n} φ^{n}$ is convergent for all t > 0, where we use φⁿ to denote n^th iterate of φ, then T has a unique fixed point x^* ∈ X, and $p_{k}^{w} (x^{*}, x^{*}) = 0$ .

Proof. It is similar to the proof of Theorem 3.3. □

4 Conclusion

In this paper, we have introduced the notion of weak partial-quasi k-metric spaces, which is a new generalized type of partial quasi-metric spaces given by Kunzi, Pajoohesh and Schellekens (2006). Also, this concept is a generalization and unification of both weak partial metric spaces and partial k-metric spaces. On one hand, we have provided some examples of weak partial-quasi k-metric spaces, and studied the relationships between weak partial-quasi k-metric spaces and weak partial metric spaces. On the other hand, we have obtained a constitution of k-metric in weak partial-quasi k-metric spaces. Moreover, we have extended the work of Heckmann and Shukla on fixed point theorems.

Footnotes

Acknowledgment

The authors thank the editor and the referees for constructive and pertinent suggestions, which have improved the quality of the manuscript greatly.

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