On partial fuzzy k-(pseudo-)metric spaces

1.   Introduction and preliminaries

Since Zadeh ^[24] introduced the notion of fuzzy sets in 1965, several different versions of fuzzy metric spaces were studied by topological researchers. In particular, Kramosi and Michalek ^[11] introduced the fuzzy metric spaces in 1975, basing on statistical metric spaces which introduced by Menger ^[14] in 1942, Schweizer and Sklar ^[19] in 1960, respectively. Kaleva and Seikkala ^[10] introduced the concept of fuzzy metric spaces and studied some fixed point theorems in these spaces in 1984. Later, by modifying the fuzzy metric spaces concept of Kramosi and Mich$\acute{\rm{a}}$lek (usually called KM-fuzzy metric spaces), George and Veeramani ^[4] introduced the notion of fuzzy metric spaces in 1994 (usually called GV-fuzzy metric spaces), and defined Hausdorff topologies on these spaces. In the last years, many authors devoted to study various types of generalizing fuzzy metric spaces by different approaches. For instance, fuzzy pseudo-metric spaces ^[1], fuzzy quasi-metric spaces ^[7,8], fuzzy partial (pseudo-)metric spaces ^[6,23], fuzzy cone metric spaces ^[17], fuzzy b-metric spaces ^[16], fuzzy k-pseudometric spaces ^[22] (also called fuzzy metric type spaces in ^[18]), etc.

Furthermore, since Grabiec extended fixed point theorems of Banach and Edelstein to KM-fuzzy metric spaces ^[5], many researchers investigated the contractive mappings and obtained some interesting fixed point theorems concerning fuzzy metric spaces ^{[2,3,9,15,20,25,26,27,28,29,30]}.

In this paper, firstly, we introduce the concept of partial k-(pseudo-)metric spaces. By generalizing the notion of fuzzy k-pseudo-metric spaces, we introduce the concept of partial fuzzy k-(pseudo-) metric spaces and give some examples to support our results in these spaces. Moreover, we investigate the relationships between (partial) k-pseudo-metric spaces and partial fuzzy k-pseudo-metric spaces. Finally, we provide several fixed pointed theorems in partial fuzzy k-metric spaces.

We recall some basic notions and results that will be used in the following sections (see more details in ^[12,13,21]). Throughout this paper, the letters $\mathbb{R}$, $\mathbb{R^{+}}$, $\mathbb{N^{+}}$ always denote the set of real numbers, of positive real numbers and of positive integers, respectively.

Definition 1.1. ^[13,21] Let $X$ be a nonempty set and the mapping $p_k: X \times X \rightarrow [0, +\infty)$ satisfying the following conditions for some number $k\ge 1$: $\forall x, y, z\in X$,

(PK1) $p_{k}(x, x)\le p_{k}(x, y)$;

(PK2) $p_{k}(x, y) = p_{k}(y, x)$;

(PK3) $p_{k}(x, z)\le k[p_{k}(x, y)+p_{k}(y, z)]-p_{k}(y, y)$;

(KP3) $p_{k}(x, z)\le k[p_{k}(x, y)+p_{k}(y, z)]$.

If $p_k$ satisfies the conditions (PK1)–(PK3), then $p_k$ is called a partial $k$-pseudo-metric. If $p_k$ satisfies the conditions (PK1)–(KP3), then $p_k$ is called a $k$-pseudo-metric.

A (partial) k-pseudo-metric space with coefficient $k\ge1$ is a pair $(X, p_{k})$ such that $p_{k}$ is a (partial) k-pseudo-metric on $X$.

Furthermore, if $p_k$ satisfies (PK1)–(PK3) and the following condition:

(PK4) $x = y$ $\Leftrightarrow$ $p_{k}(x, x) = p_{k}(x, y) = p_{k}{(y, y)}$ for all $x, y \in X$;

Then it is called partial k-metric ^[21] and the pair $(X, p_{k})$ is called a partial k-metric space with a coefficient $k\ge1$.

Particularly, a partial $1$-metric (i.e. $k$ = 1) is called partial metric ^[13].

Example 1.2. Let $X = \{{a, b, c}\}$. Define $p_{k}$: $X \times X \rightarrow [0, +\infty)$ as follows:

$p_{k}(a, a) = p_{k}(a, b) = p_{k}(b, a) = p_{k}(b, c) = p_{k}(c, b) = 2, p_{k}(b, b) = p_{k}(c, c) = 0, p_{k}(c, a) = p_{k}(a, c) = 6$. It is trivial to verify $(X, p_{k})$ is a partial $k$-pseudo-metric space with a coefficient $k = 2$.

Definition 1.3. ^[19] A binary operation $*: [0, 1] \times [0, 1] \rightarrow [0, 1]$ is called a continuous triangular norm (briefly $t$-norm) if it satisfies the following conditions:

(1) $*$ is associative and commutative;

(2) $*$ is continuous;

(3) $a * 1 = a$ for all $a\in [0, 1]$;

(4) $a * b \le c * d$ whenever $a \le c$ and $b \le d$ for all $a, b, c, d\in [0, 1]$.

The following are the three basic $t$-norms: minimum, usual product and Lukasiewicz $t$-norm, which are given by, respectively: $a\wedge b = \min\{{a, b}\}$, $a \cdot b = ab$ and $a *_{L} b = \max\{{0, a+b-1}\}$, $\forall a, b\in [0, 1]$.

Definition 1.4. ^[11] A triple $(X, M, *)$ is called a KM-fuzzy metric space if $X$ is an arbitrary nonempty set, $*$ is a continuous t-norm and $M$ is a fuzzy set on $X \times X\times[0, +\infty)$, satisfying the following conditions: $\forall x, y, z \in X$ and $t, s > 0$,

(1) $M(x, y, 0) = 0$;

(2) $M(x, y, t) = 1$ if and only if $x = y$;

(3) $M(x, y, t) = M(y, x, t)$;

(4) $M(x, y, t) * M(y, z, s) \le M(x, z, t + s)$;

(5) The function $M(x, y, \cdot) : [0, +\infty)\rightarrow [0, 1]$ is left-continuous.

We note that A. George and P. Veeramani ^[4] modified the concept of KM-fuzzy metric spaces and defined a Hausdorff topology on this fuzzy space.

2.   Partial fuzzy k-(pseudo)metric spaces

Definition 2.1. A triple $(X, M_{p_k}, *)$ is called a partial fuzzy k-metric space if $X$ is an arbitrary nonempty set, $*$ is a continuous t-norm and $M_{p_k}$ is a fuzzy set on $X \times X\times[0, +\infty)$, satisfying the following conditions for some number $k\ge 1$: $\forall x, y, z \in X$ and $t, s > 0$,

(PFK1) $M_{p_k}(x, y, 0) = 0$;

(PFK2) $M_{p_k}(x, x, t)\ge M_{p_k}(x, y, t)$;

(PFK3) $M_{p_k}(x, y, t) = M_{p_k}(y, x, t)$;

(PFK4) $M_{p_k}(x, x, t) = M_{p_k}(x, y, t) = M_{p_k}(y, y, t)$ if and only if $x = y$;

(PFK5) $M_{p_k}(x, y, t) * M_{p_k}(y, z, s) \le M_{p_k}(x, z, k(t + s))$;

(PFK6) The function $M_{p_k}(x, y, \cdot) : (0, +\infty)\rightarrow [0, 1]$ is left-continuous.

If it only satisfies (PFK1)–(PFK3) and (PFK5)–(PFK6), then it is called a partial fuzzy k-pseudometric space.

Example 2.2. Let $X = X_a\cup X_b\cup X_c$, where $X_a = \{a\}\times[0, 1], X_b = \{b\}\times[0, 1], X_c = \{c\}\times[0, 1]$. We denote $x\in \{a, b, c\} \times\bar{x}$, where $\bar{x}\in [0, 1]$. Define a fuzzy set on $X \times X\times[0, +\infty)$ as follows:

where

Then $(X, M_{p_k}, *)$ is a partial fuzzy $k$-metric space with a coefficient $k = 3$, where $x*y = x\wedge y$.

To verify this result, we have to check (PFK5).

(PFK5): Since the authors showed that $d(x, z)\le 3[d(x, y)+d(y, z)]$ (see Example 7 in ^[26]). For any $x, y, z\in X$ and $t, s > 0$, without loss of generality, we assume that $M_{p_k}(x, y, t) \le M_{p_k}(y, z, s)$. Namely, $\frac{t}{t+d(x, y)}\le \frac{s}{s+d(y, z)}$, which implies that $sd(x, y)\ge td(y, z)$. Furthermore, we can deduce that$(t+s)d(x, y)\ge t[d(x, y)+d(y, z)].$

Thus, we have$(t+s)[t+d(x, y)]\ge t[(t+s)+(d(x, y)+d(y, z))].$ Then

for all $x, y\in X$, $t, s > 0$.

Remark 2.3. Let $X$ be a nonempty set and $p_{k}$ be a partial k-metric with a coefficient $k\ge 1$. Define a fuzzy set on $X \times X\times(0, +\infty)$ as follows:

for all $x, y\in X$, $l, m > 0$ and $n\in \mathbb{N^{+}}$. Then $(X, M_{p_k}, *)$ is a partial fuzzy k-metric space with a coefficient $k$, where $x*y = x\wedge y$.

Here we only check (PFK5).

In fact, $\frac{a}{a+c}\ge\frac{b}{b+c}$ if $a\ge b$ for all $a, b, c > 0$, and $(t+s)^n\ge t^n+s^n$ for all $s, t > 0$ where $n\in \mathbb{N^{+}}$. Then, for some number $k\ge1$, we have

for all $x, y\in X, t, s, l, m > 0$, $n\in \mathbb{N^{+}}$. It is similar to the proof of Example 2.2.

Example 2.4. Let $X$ be a nonempty set and $p_{k}$ be a partial k-metric with a coefficient $k\ge 1$. Define a fuzzy set on $X \times X\times[0, +\infty)$ as follows:

for all $x, y\in X$, $t > 0$ and $q\ge 1$, where $x*y = x\cdot y$. Then $(X, M_{p_k}, *)$ is a partial fuzzy k-metric space with a coefficient $k(2k)^{(q-1)}$.

To verify this result, we have to check (PFK5). First, we claim that $(a+b)^q\le2^{q-1}(a^q+b^q)$ for all $a, b \ge 0$ and $q\ge1$.

(PFK5): For all $x, y, z\in X$, $t, s > 0$, and some real number $k\ge1$, we can deduce

for all $q\ge 1$. Therefore, we have

for all $x, y, z\in X$, $t, s > 0$, $q\ge 1$.

Example 2.5. Let $X$ be a nonempty set and $p$ be a partial metric. Define a fuzzy set on $X \times X\times(0, +\infty)$ as follows:

for all $x, y\in X$. Then $(X, M_{p_k}, *)$ is a partial fuzzy k-metric space with a coefficient $k\ge 1$, where $x*y = x\wedge y$.

Indeed, $\frac{k(t+s)}{k(t+s)+p(x, z)}\ge\frac{t+s}{t+s+[p(x, y)+p(y, z)]}$ for all $x, y\in X$, $t, s > 0, k\ge1$. It is similar to the proof of Example 2.2.

Apparently, if $(X, M_{p_k}, *)$ is a partial fuzzy k-metric space, then $(X, M_{p_k}, *)$ is not a KM-fuzzy metric space. In fact, it can be illustrated by Example 2.4 for $q = 2$.

Proposition 2.6. Let $X$ be a nonempty set. If $(X, M_{p_k}, *)$ is a partial fuzzy k-metric space with a coefficient $k\ge 1$ and $(X, P, *)$ is a KM-fuzzy metric space, respectively. Define a fuzzy set on $X \times X\times[0, +\infty)$ as follows:

for all $x, y\in X$ and $t\ge0$. Then $(X, M, *)$ is a partial fuzzy k-metric space with a coefficient $k$.

Proof. It is trivial to prove that $(X, M, *)$ satisfied (PFK1)–(PFK3) and (PFK6). We verify condition (FPK4) and (PFK5) in the following.

(PFK4): ($\Rightarrow$) Suppose that $M(x, x, t) = M(x, y, t) = M(y, y, t)$ for all $x, y\in X$ and $t > 0$. Then we have

By Definition 1.4 (2), we have

Since $0\le P(x, y, t)\le 1$, it follows that $P(x, y, t)*M_{p_k}(x, y, t)\le M_{p_k}(x, y, t)$. Thus $M_{p_k}(x, x, t)\le M_{p_k}(x, y, t)$ and $M_{p_k}(y, y, t)\le M_{p_k}(x, y, t)$. By (PFK2), we have $M_{p_k}(x, x, t)\ge M_{p_k}(x, y, t)$ and $M_{p_k}(y, y, t)\ge M_{p_k}(x, y, t)$. Therefore, $M_{p_k}(x, x, t) = M_{p_k}(x, y, t) = M_{p_k}(x, y, t)$. So $x = y$ by Definition 2.1.

($\Leftarrow$) First, we claim that $P(x, x, t) = P(y, y, t) = 1$ if $P(x, y, t) = 1$ for all $x, y\in X$ and $t > 0$, where $(X, P, *)$ is a KM-fuzzy metric space. Suppose $x = y$. It is obvious that $M_{p_k}(x, x, t) = M_{p_k}(x, y, t) = M_{p_k}(y, y, t)$ by the Definition 2.1, and $P(x, y, t) = 1$ by Definition 1.4 (2). Hence, $P(x, x, t)*M_{p_k}(x, x, t) = P(x, y, t)*M_{p_k}(x, y, t) = P(y, y, t)*M_{p_k}(y, y, t)$. Namely, $M(x, x, t) = M(x, y, t) = M(y, y, t)$ for all $x, y\in X$ and $t > 0$.

(PFK5): First, we claim that $P(x, y, \cdot)$ is non-decreasing for all $x, y\in X$ (see Lemma ^[15]). By (PFK5), Definition 1.3 (4) and Definition 1.4 (4), we have

for all $x, y, z\in X$, $t, s > 0$.

Proposition 2.7. Let $X$ be a nonempty set. If $(X, M_{p_k}, *)$ is a partial fuzzy k-metric space with a coefficient $k\ge 1$, where $x*y = xy$. Define a function $\hat p_{k}: X \times X \rightarrow [0, +\infty)$ as follows:

for all $x, y\in X$.

Then $\hat p_{k}$ is a partial k-pseudo metric with a coefficient $3k$.

Proof. We verify the conditions (PK1)–(PK3) step by step.

(PK1) We prove it in the following two cases.

Case 1: If $t = 0$, then $\hat{p}_{k}(x, y) = 0$. It is not difficult to show that $\hat{p}_{k}(x, x) = \hat{p}_{k}(x, y)$, for all $x, y\in X$.

Case 2: If $t > 0$ by the assumption, then we have $\lim_{\varepsilon \to 0^+}\int_{\varepsilon}^{1}\ln M_{p_k}(x, y, t)dt$. Moreover, by (PFK2), we have $\ln M_{p_k}(x, x, t)\ge \ln M_{p_k}(x, y, t)$, which implies that

for all $x, y\in X$, $t > 0$. Namely, $\hat{p}_{k}(x, x)\le\hat{p}_{k}(x, y)$.

(PK2): By (PFK3), it is clear that $\hat{p}_{k}(x, y) = \hat{p}_{k}(y, x)$ for all $x, y\in X$.

(PK3): First, by (PK5), we claim that

for all $x, y, z\in X$, $t > 0$, which implies that

Furthermore, set $u = \frac {t}{2k}$. We can deduce that

and

On the other hand, since $\hat{p}_{k}(y, z)\ge\hat{p}_{k}(y, y)$ by (PK1), we have

for all $x, y, z\in X$, $u > 0$. Hence, $\hat{p}_{k}$ is a partial k-metric with a coefficient $3k$.

Lemma 2.8. Let $X$ be a nonempty set and $(X, M_{p_k}, *)$ be a partial fuzzy k-metric space with a coefficient $k\ge 1$. If $M_{p_k}(x, y, t) = 1$ for all $x, y\in X$ and $t > 0$, then $x = y$. But the converse may not be true.

Proof. By (PFK2), we have $M_{p_k}(x, x, t)\ge M_{p_k}(x, y, t)$ and $M_{p_k}(y, y, t)\ge M_{p_k}(x, y, t)$. Suppose that $M_{p_k}(x, y, t) = 1$. It follows that $M_{p_k}(x, x, t)\ge 1$ and $M_{p_k}(y, y, t)\ge 1$, we can deduce that $M_{p_k}(x, x, t) = M_{p_k}(y, y, t) = 1$. Namely, $M_{p_k}(x, x, t) = M_{p_k}(y, y, t) = M_{p_k}(x, y, t)$. Thus, we have $x = y$ by (PFK5).

In addition, from Example 2.5, if $x = y$, then we have $M_{p_k}(x, x, t) = \frac{t}{t+p(x, x)}$. Since the distance of a point to itself may not be zero in partial metric spaces, so $M_{p_k}(x, x, t)$ may not be 1.

3.   Topological structures of partial fuzzy k-pseudo-metric spaces

In this section we begin by giving some basic notions that will be used in the following. First, we know that each fuzzy metric $M$ on $X$ generates a topology $\mathcal{T}_{M}$ on $X$ with the basis $\mathcal{B} = \{{B(x, r, t): x\in X, 0 < r < 1, t > 0}\}$, where the open ball $B(x, r, t) = \{{y\in X: M(x, y, t) > 1-r}\}$ for all $0 < r < 1$ and $t > 0$. Also, we call that $\mathcal{T}_{M}$ is induced by the fuzzy metric $M$ (see more details in ^[4]).

Theorem 3.1. Let $X$ be a nonempty set and $(X, M_{p_k}, *)$ be a partial fuzzy k-metric space with a coefficient $k\ge 1$. For any $x\in X$, $0 < r < 1$ and $t > 0$, we define the open ball as follows:

Then $\mathcal{T}_{M_{p_k}} = \{V\subset X:$ $\rm {for\ each }$ $x\in V,$ $\rm {there \ exist }$ $0 < r < 1, t > 0$ $\rm { such\ that}$ $B(x, r, t)\subset V\}$ is a topology on $X$.

Proof. It is similar to the proof of Theorem 2.1 ^[16].

Furthermore, we can define another type topological structure on $X$ as follows: $\mathcal{S}_{M_{p_k}} = \{V\subset X:$ $\rm {for\ each }$ $i\in I,$ $\rm {there \ exist }$ $B_{p_k}(x_i, r_i, t_i)$ $\rm {such\ that }$ $V = \bigcup_{i\in I} B_{p_k}(x_i, r_i, t_i)\}$. Then we can deduce that $\mathcal{S}_{M_{p_k}}$ is a suprartopology (see more details in ^[12]).

Theorem 3.2. Let $X$ be a nonempty set and $(X, M_{p_k}, \wedge)$ be a partial fuzzy k-pseudo-metric space with a coefficient $k\ge 1$. Define a function $d_{\alpha}: X \times X \rightarrow [0, +\infty)$ as follows:

Then the following statements hold:

(1) $\{{ d_{\alpha}: \alpha\in (0, 1)}\}$ is non-increasing with respect to $\alpha$.

(2) If $d_{\alpha}(x, y) > t$, then $M_{P_{k}}(x, y, t) < \alpha$.

(3) $\{{ d_{\alpha}: \alpha\in (0, 1)}\}$ is a k-pseudo-metric family on $X$.

Proof. By the Definition of $d_{\alpha}$, it is not difficult to prove (1) and (2).

(3) We verify the conditions (PK1), (PK2) and (KP3) step by step.

(PK1): By (PFK2), we have $M_{p_k}(x, x, t)\ge M_{p_k}(x, y, t)$ for all $x, y\in X$. Then $\{{t > 0:M_{p_k}(x, x, t)\ge\alpha}\}\subset \{{t > 0:M_{p_k}(x, y, t)\ge\alpha}\}$, which implies that $\bigwedge\{{t > 0: M_{p_k}(x, x, t)\ge\alpha}\}\ge\bigwedge \{{t > 0: M_{p_k}(x, y, t)\ge\alpha}\}$. Namely, $d_{\alpha}(x, x)\ge d_{\alpha}(x, y)$.

(PK2): It is trivial by (PFK3).

(KP3): Since $M_{p_k}(x, y, t) \wedge M_{p_k}(y, z, s) \le M_{p_k}(x, z, k(t + s))$ by (PFK5), we have

for all $x, y, z\in X$, $t, s > 0$.

Theorem 3.3. Let $X$ be a nonempty set and $(X, d_{\alpha})$ be a generating space of k-pseudo-metric family for all $\alpha\in (0, 1)$ and some number $k\ge 1$, where $\{{d_{\alpha}: \alpha\in (0, 1)}\}$ is a family of mapping from $X \times X \rightarrow [0, +\infty)$ and $(X, d_{\alpha})$ satisfies the following conditions: $\forall x, y, z \in X$ and for any $\alpha, \beta\in (0, 1)$,

(GPKP1) $d_{\alpha}(x, x)\le d_{\alpha}(x, y)$;

(GPKP2) $d_{\alpha}(x, y) = d_{\alpha}(y, x)$;

(GPKP3) $d_{\alpha\wedge\beta}(x, z)\le k[d_{\alpha}(x, y)+d_{\beta}(y, z)]$;

(GPKP4) $d_{\alpha}(x, y)$ non-increasing with respect to $\alpha$.

Define a function $M_{D}: X \times X \times [0, \infty)\rightarrow [0, 1]$ as follows:

Then $(X, M_{D}, \wedge)$ is a partial fuzzy k-pseudo-metric space with a coefficient $k\ge 1$.

Proof. We verify the conditions (PFK1)–(PFK3), (PFK5) and (PFK6) step by step.

(PFK1): It is clear that $M_{D}(x, y, 0) = 0$.

(PFK2): By (GPKP1), we have $d_{\alpha}(x, x)\le d_{\alpha}(x, y)$. Then

which implies that $\bigvee\{{\alpha\in (0, 1):d_{\alpha}(x, x) < t}\}\ge\bigvee\{{\alpha\in (0, 1):d_{\alpha}(x, y) < t}\}$. Namely, $M_{D}(x, x, t)\ge M_{D}(x, y, t)$.

(PFK3): By (GPKP2), it is easy to show $M_{D}(x, y, t) = M_{D}(y, x, t)$.

(PFK5): We prove $M_{D}(x, z, k(t + s)) \ge M_{D}(x, y, t) \wedge M_{D}(y, z, s)$ as follows:

Case 1: If $M_{D}(x, y, t) = 0$ or $M_{D}(y, z, s) = 0$ for all $t, s > 0$, then $M_{D}(x, y, t) \wedge M_{D}(y, z, s) = 0$. It is easy to see that the above relation holds.

Case 2: Suppose that $M_{D}(x, y, t) \wedge M_{D}(y, z, s) > 0$ for all $s, t > 0$, i.e., $M_{D}(x, y, t) > 0$ and $M_{D}(y, z, s) > 0$. Set $M_{D}(x, y, t) = \beta$ and $M_{D}(y, z, s) = \gamma$. For any $\varepsilon > 0$, where $\varepsilon < \beta\wedge\gamma$. Then there exist $\alpha_{1}, \alpha_{2}\in (0, 1)$, such that $\alpha_{1} > \beta-\varepsilon$, $\alpha_{2} > \gamma-\varepsilon$, and $d_{\alpha_{1}}(x, y) < t$, $d_{\alpha_{2}}(y, z) < s$. By (GPKP4), we have $d_{\beta-\varepsilon}(x, y) < t$, $d_{\gamma-\varepsilon }(y, z) < s$. Furthermore, by (GPKP3), we can deduce

Therefore, it follows that $M_{D}(x, z, k(t+s))\ge (\beta-\varepsilon)\wedge(\gamma-\varepsilon)$. From the arbitrariness of $\alpha$, $\beta$ and the continuity of $\wedge$, it follows that $M_{D}(x, z, k(t+s))\ge\beta\wedge\gamma$, namely, $M_{D}(x, z, k(t+s))\ge M_{D}(x, y, t) \wedge M_{D}(y, z, s)$.

(PFK6): For any $\varepsilon > 0$, we have $M_{D}(x, y, t)-\varepsilon < M_{D}(x, y, t)$. Then there exists $\alpha_{0}\in (0, 1)$, such that $d_{\alpha_{0}}(x, y) < t$ and $M_{D}(x, y, t)-\varepsilon < \alpha_{0}$. Furthermore, we have $M_{D}(x, y, t_{0})\ge\alpha_{0}$ whenever $d_{\alpha_{0}}(x, y) < t_{0} < t$. Therefore, it follows that $M_{D}(x, y, t)-M_{D}(x, y, t_{0})\le M_{D}(x, y, t)-\alpha_{0} < \varepsilon.$ Hence, $M_{D}(x, y, \cdot)$ is left-continuous.

4.   Fixed point theorem on partial fuzzy k-metric spaces

In this section, we investigate fixed point theorems for a self-mappings in partial fuzzy $k$-metric spaces, following the method given by Shen Yonghong et al. ^[20].

Definition 4.1. Let $X$ be a nonempty set and $(X, M_{p_k}, *)$ be a partial fuzzy k-metric space with a coefficient $k\ge 1$.

$(1)$ A sequence $\{{x_{n}}\}_{n\in\mathbb{N^{+}}}$ in $(X, M_{p_k}, *)$ converges to a point $x\in X$ if for any $0 < \varepsilon < 1$ and $t > 0$, there exists $n_{0}\in\mathbb{N^{+}}$ such that $M_{p_k}(x_{n}, x, t) > 1-\varepsilon$ for all $n > n_{0}$ (or equivalently for any open ball $B(x, r, t)$, there exists $n_{0}\in\mathbb{N^{+}}$ such that $x_{n}\in B(x, r, t)$ for all $n\ge n_{0}$), we denote $\lim_{n\to +\infty}x_{n} = x$.

$(2)$ A sequence $\{{x_{n}}\}_{n\in\mathbb{N^{+}}}$ is called a Cauchy sequence if for any $0 < \varepsilon < 1$ and $t > 0$, there exists $n_{0}\in\mathbb{N^{+}}$ such that $M_{p_k}(x_{n}, x_{m}, t) > 1-\varepsilon$ for all $n, m\ge n_{0}$.

$(3)$ $(X, M_{p_k}, *)$ is said to be complete if every Cauchy sequence $\{{x_{n}}\}_{n\in\mathbb{N^{+}}}$ in $X$ converges to a point $x\in X$.

Indeed, we can give another definition type of sequence convergence as follows: a sequence $\{{x_{n}}\}$ in $(X, M_{p_k}, *)$ converges to a point $x\in X$ if for any open set $V$ containing $x$ there exists $n_{0}\in\mathbb{N^{+}}$ such that $x_{n}\in V$ for all $n\ge n_{0}$ in $\mathcal{S}_{M_{p_k}}$, we denote ${\rm Lim}_{n\to +\infty}x_{n} = x$.

Theorem 4.2. Let $X$ be a nonempty set, $(X, M_{p_k}, *)$ be a partial fuzzy k-metric space with a coefficient $k\ge 1$ and $\{x_n\}_{n\in\mathbb{N^{+}}}$ be a sequence in $X$. Then $\lim_{n\to +\infty}x_{n} = x$ if and only if $\lim_{n\to +\infty}M_{p_k}(x_{n}, x, t) = 1$ for all $t > 0$.

Proof. ($\Rightarrow$) Suppose that $\lim_{n\to +\infty}x_{n} = x$. Then for any open ball $B(x, r, t)$, there exists $n_{0}\in\mathbb{N^{+}}$ such that $x_{n}\in B(x, r, t)$ for all $n > n_{0}$. Thus $M_{p_k}(x_{n}, x, t) > 1-r$ for all $n > n_{0}$ and $t, r > 0$, namely, $1-M_{p_k}(x_{n}, x, t) < r$. Hence $\lim_{n\to +\infty}M_{p_k}(x_{n}, x, t) = 1$.

($\Leftarrow$) Suppose that $\lim_{n\to +\infty}M_{p_k}(x_{n}, x, t) = 1$. Then for each $t > 0$, there exists $n_0\in\mathbb{N^{+}}$ such that $1-M_{p_k}(x_{n}, x, t) < r$ for all $n\ge n_0$. Namely, $M_{p_k}(x_{n}, x, t) > 1-r$ for all $n > n_{0}$. Therefore, $x_n\in B(x, r, t)$ for all $n > n_{0}$. Thus $\lim_{n\to +\infty}x_{n} = x$.

Corollary 4.3. Let $X$ be a nonempty set, $(X, M_{p_k}, *)$ be a partial fuzzy k-metric space with a coefficient $k\ge 1$ and $\{x_n\}_{n\in\mathbb{N^{+}}}$ be a sequence in $X$. If $\lim_{n\to +\infty}M_{p_k}(x_{n}, x, t) = 1$ for all $t > 0$, then ${\rm Lim}_{n\to +\infty}x_{n} = \lim_{n\to +\infty}x_{n}$.

Proof. Indeed, by Definition 4.1 and Theorem 3.1, it is not difficult to see that ${\rm Lim}_{n\to +\infty}x_{n} = x$ if $\lim_{n\to +\infty}x_{n} = x$. Then it is trivial by Theorem 4.2.

Theorem 4.4. Let $X$ be a nonempty set, $(X, M_{p_k}, *)$ be a partial fuzzy k-metric space with a coefficient $k\ge 1$ and $\{x_n\}_{n\in\mathbb{N^{+}}}$ be a sequence in $X$. If $\{x_n\}$ is convergent, then it is a Cauchy sequence.

Proof. Suppose that $\{x_n\}$ is convergent. By Definition 4.1 and Theorem 4.2, there exists $n_{0}\in\mathbb{N^{+}}$ such that $M_{p_k}(x_{n}, x, t) > 1-r$ for all $n\ge n_{0}$, $t > 0$ and some number $k\ge 1$. Set $s = \frac{t}{2k}$. By (PFK5), we have

for all $n, m\ge n_{0}$. Furthermore, since $0 < \frac{t}{2k} < t$, we have $M_{p_k}(x, x_{m}\frac {t}{2k}) > 1-r$. Set $r_{0} = M_{p_k}(x, x_{m}\frac {t}{2k})$. Then $r_{0} > 1-r$. By continuity of the $t$-norm, we can find some $s > 0$, such that $r_{0} > 1-s > 1-r$. Thus, there exists $0 < r_{1} < 1$, such that $r_{1}*r_{1}\ge 1-s$, from which, we can deduce that $M_{p_k}(x_{n}, x_{m}, t)\ge r_{1}*r_{1}\ge 1-s > 1-r$, for all $n, m\ge n_{0}$, $t > 0$. Therefore, $\{x_n\}$ is a Cauchy sequence.

Theorem 4.5. Let $X$ be a nonempty set and $(X, M_{p_k}, *)$ be a complete partial fuzzy k-metric space with a coefficient $k\ge 1$, and let $T: X\rightarrow X$ be a function satisfying the following conditions:

(1) $\varphi (M_{p_k}(Tx, Ty, t))\le\lambda\varphi (M_{p_k}(x, y, t))$ for all $x, y\in X$ and $x\ne y$, where $t > 0, \lambda\in (0, 1)$, and $\varphi: [0, 1]\rightarrow [0, 1]$ is a strictly decreasing and continuous mapping;

(2) $\varphi (M_{p_k}(x, y, t)) = 0$ if and only if $M_{p_k}(x, y, t) = 1$.

Then $T$ has a unique fixed point.

Proof. We define a sequence in the following way : $x_{0} = x$, and $x_{n+1} = Tx_{n}$, $f_{n}(t) = M_{p_k}(x_{n}, x_{n+1}, t)$ for all $n\in\mathbb{N^{+}}$, $t > 0, x\in X$, and some number $k\ge 1$.

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

Case 2: If $x_{n+1}\ne x_{n}$, then we have $\varphi(f_{n}(t)) = \varphi (M_{p_k}(x_{n}, x_{n+1}, t)) = \varphi (M_{p_k}(T_{x_{n-1}}, T_{x_{n}}, t))$. Since $\varphi (M_{p_k}(T_{x_{n-1}}, T_{x_{n}}, t))\le \lambda \varphi (M_{p_k}(x_{n-1}, x_{n}, t)) = \varphi(f_{n-1}(t))$ by the condition (1), this follows that $\varphi(f_{n}(t)) < \varphi(f_{n-1}(t))$ for all $n\in\mathbb{N^{+}}$ and $t > 0$. By assumption, $\varphi$ is strictly decreasing, which implies that $\{{ f_{n}(t)}\}$ is an increasing sequence with respect to $n$ for all $t > 0$. Furthermore, since $0\le f_{n}(t)\le 1$ for all $t > 0$, $\{{ f_{n}(t)}\}$ is bounded. Therefore, $\{{ f_{n}(t)}\}$ is convergent. We denote $\lim_{n\to +\infty}f_{n}(t) = f(t)$. Namely, there exists $n_{0} \in \mathbb{N^{+}}$, such that $f_{n}(t)\le f(t)$ for all $n\ge n_{0}$ and $t > 0$. On the other hand, for all $n\in\mathbb{N^{+}}$ and $t > 0$, we have $\varphi(f_{n+1}(t))\le\lambda\varphi(f_{n}(t))$. It follows that $\lim_{n\to +\infty}\varphi(f_{n+1}(t))\le\lim_{n\to +\infty}\lambda\varphi(f_{n}(t))$, which implies that $\varphi(f(t))\le\lambda\varphi(f(t))$. Thus $(1-\lambda)\varphi(f(t))\le0$. So $\varphi(f(t)) = 0$ for all $t > 0$. Namely, $\lim_{n\to +\infty}M_{p_k}(x_{n}, x_{n+1}, t) = 1$.

To prove the existence and uniqueness of the fixed point, we consider the following steps:

Step 1: We claim that $\{{x_{n}}\}$ is a Cauchy sequence in $(X, M_{P_{k}}, *)$. Otherwise, for some $0 < \varepsilon < 1$, we can find two sequences $\{{i_{n}}\}$ and $\{{j_{n}}\}$, such that $M_{p_k}(x_{i_{n}}, x_{j_{n}}, t)\le 1-\varepsilon$, $M_{p_k}(x_{i_{n}-1}, x_{j_{n}-1}, t) > 1-\varepsilon$ and $M_{p_k}(x_{i_{n}-1}, x_{j_{n}}, t) > 1-\varepsilon$ for all $n\in \mathbb{N^{+}}$ and $t > 0$, where $i_{n} > j_{n}\ge n$. Set $g_{(i_{n}, j_{n})}(t) = M_{p_k}(x_{i_{n}}, x_{j_{n}}, t)$. It is not difficult to show that $g_{(i_{n}, j_{n})}(t)\le1-\varepsilon$ by (PFK5). Moreover, we have $\varphi (g_{(i_{n}, j_{n})}(t))\le \lambda\varphi (g_{(i_{n}-1, j_{n}-1)}(t)) < \varphi (g_{(i_{n}-1, j_{n}-1)}(t))$. Thus, $g_{(i_{n}-1, j_{n}-1)}(t) < g_{(i_{n}, j_{n})}(t)$ by the monotonicity of $\varphi$, it follows that

which is a contradiction. In addition, suppose for some $n_{0}\in \mathbb{N^{+}}$, any $p\in\mathbb{N^{+}}$ and $t > 0$. We can deduce that $\{{f_{n_{0}+p}(t)}\}$ is convergent by the monotonicity of $f$. We denote $\lim_{p\to +\infty}f_{n_{0}+p}(t) = f_{n_{0}}(t)$. By assumption, we have $\varphi(f_{n_{0}+p}(t)) \le \lambda \varphi(f_{n_{0}+p-1}(t))$. By repeating the above process, it follows that $\varphi (f_{n_{0}+p}(t))\le(\lambda)^{p} f_{n_{0}}(t)$. Thus, we can deduce $\varphi (f_{n_{0}}(t)) = 0$. Namely, $\varphi (M_{p_k}(x_{n_{0}}, x_{n_{0}+1}, t) = 0$. By the condition (2), we have $M_{p_k}(x_{n_{0}}, x_{n_{0+1}}, t) = 1$. It is clear to see that $x_{n_{0}} = x_{n_{0}+1}$ by Lemma 2.8, which is a contradiction.

Therefore, $\{{x_{n}}\}$ is a Cauchy sequence in $(X, M_{P_{k}}, *)$. By the completeness of $(X, M_{P_{k}}, *)$, there exists a point $x^*\in X$, such that $\lim_{n\to +\infty}x_{n} = x^{*}$.

Step 2: By Step 1, there exists a subsequence $\{{x_{n_{k}}}\}$, where $x_{n_{k}}\ne x_{n}$ for all $n\in\mathbb{N^{+}}$. Then we have

for all $n_{k}\in\mathbb{N^{+}}$, $t > 0$. We can deduce that $\varphi (M_{p_k}(x^{*}, Tx^{*}, t)) = 0$. By the condition (2), we have $M_{p_k}(x^{*}, Tx^{*}, t) = 1$. It is clear to see $x^{*} = Tx^{*}$ by Lemma 2.8.

Step 3: Suppose that $x^{*}\ne y^{*}$, where $Ty^{*} = y^{*}$. We have $\varphi (M_{p_k}(x^{*}, y^{*}, t)) = \varphi (M_{p_k}(Tx^{*}, Ty^{*}, t))\le\lambda\varphi (M_{p_k}(x^{*}, y^{*}, t)) < \varphi (M_{p_k}(x^{*}, y^{*}, t)),$ which is a contradiction. Hence, $x^{*} = y^{*}$.

To conclude this section, we illustrate our result by the following examples.

Example 4.6. Let $X = {\{1, 2, 3, \cdots \}}$. Define a fuzzy set set on $X \times X\times[0, +\infty)$ by $M_{p_k}(x, y, t) = \frac{x\wedge y}{x\vee y}$ for all $x, y\in X$, $t > 0$, and $M_{p_k} = 0$ when $t = 0$. It is trivial to verify that $(X, M_{p_k}, *)$ is a partial fuzzy k-metric space with $a*b = ab$ for all $a, b\in [0, 1]$. Define mappings $T:[0, +\infty)\rightarrow [0, +\infty)$, $\varphi: [0, 1]\rightarrow [0, 1]$, respectively, where $T(v) = \sqrt{v}$ and $\varphi(u) = 1-\sqrt{u}$. Set $\lambda(t) = \frac{1}{1+t^2}$ for all $t > 0$. Thus, all the conditions of Theorem 4.5 are satisfied and obviously $x = 1$ is a fixed point of $T$.

Now, similar to the basic form of the functional equations in dynamic programming, which investigated by Bellman and Lee ^[31], the existence and uniqueness of solution and common solution for a functional equation and system of functional equations are discussed by using Theorem 4.5 as follows.

Let $X$ and $Y$ be Banach spaces, $S\subset X$ be the state space and $D\subset Y$ be the decision space. $B(S)$ denotes the set of all real-valued bounded functions on $S$. Define $u: S\times D\rightarrow \mathbb{R}$, $T: S\times D\rightarrow S$, $H: S\times D\times \mathbb{R}\rightarrow \mathbb{R}$. Moreover, define a fuzzy set set on $B(S) \times B(S)\times[0, +\infty)$ as follows: $M_{p_k}(x, y, t) = e^{-\frac{d(h, k)}{t}}$, for all $h, k\in B(S), t > 0$ and $M_{p_k}(x, y, t) = 0$ when $t = 0$, where $d(h, k) = \bigvee _{x\in S}|h(x)-k(x)|, \forall a, b, \in S$, $a*b = a\cdot b$ $, \forall a, b\in[0, 1]$. It is obvious that $(X, M_{p_k}, *)$ is a complete partial fuzzy k-metric space by Example 2.4. Suppose that the following conditions hold: $\varphi(e^{-\frac{|H(x, y, g(\xi))-H(x, y, h(\xi))|}{t}})\le\lambda\varphi(e^{-\frac{|g(\xi)-h(\xi)|}{t}})$, for $t > 0, \lambda\in (0, 1)$, where $\varphi: [0, 1]\rightarrow [0, 1]$ is a strictly decreasing and continuous mapping. In fact, set the system of functional equations $Ag(x) = opt_{y\in D}\{u(x, y)+H(x, y, g(T(x, y))\}$, for all $x\in S, g\in B(S)$, where $opt$ represents $\wedge$ or $\vee$. For all $g, h\in B(S), x\in S$, there exist $y, z\in D$ such that $d(Ag, Ah)\le\max\{|H(x, y, g(T(x, y)))-H(x, y, h(T(x, y)))|, |H(x, z, g(T(x, z)))-H(x, z, h(T(x, z)))| \}$. It implies that $\varphi(e^{-\frac{|Ag(\xi)-Ah(\xi)|}{t}})\le\varphi(e^{-\frac{|H(x, y, g(\xi))-H(x, y, h(\xi))|}{t}})$. By the above condition, we have that $\varphi(e^{-\frac {d(Ag, Ah)}{t}})\le\lambda\varphi(e^{-\frac{d(g, h)}{t}})$. Hence, $\varphi (M_{p_k}(Ag, Ah, t))\le\lambda\varphi (M_{p_k}(g, h, t))$. Thus, Theorem 4.5 ensures that $A$ has a unique common fixed pointed $w\in B(S)$. That is, the system of functional equations $q(x) = opt_{y\in D}\{u(x, y)+H(x, y, g(T(x, y))\}$ possesses a unique common solution $w\in B(S)$.

5.   Conclusions

In this paper, by introducing the notion of weak partial-quasi k-metric spaces, we generalized and unified weak partial metric spaces and partial k-metric spaces. Moreover, we provided some examples of weak partial-quasi k-metric spaces, and illustrated the relationships between weak partial-quasi k-metric spaces and weak partial metric spaces. Additionally, another purpose of this paper to obtain the constitution of k-metric in weak partial-quasi k-metric spaces. In Section 4, we discussed the existence of fixed point on partial fuzzy k-metric spaces, and presented application of the revealed fixed point theorems.

Acknowledgments

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

Conflict of interest

The authors declare that they have no competing interest.