Fixed Point Theorems in Fuzzy Partial Metric Spaces

Abstract

Inspired by the work of Gregori et al. and guided by some open direction, we propose the concepts of Cauchy sequence and convergent sequence in a fuzzy partial metric space by the residuum operator associated to a continuous t-norm. Based on these notions, we introduce the concepts of two kinds of fuzzy $\eta$-contractive mappings in fuzzy partial metric spaces and present related fixed point theorems.

1. Introduction

A partial metric space is an important generalization of a metric space, in which the distance between a point and itself is not required to be zero. It has been proved that a partial metric p can generate a topology ${\tau }_p$ on X which is $T_0$. At the same time, for a given partial metric p, there is always a consistent metric $d_p$ such that a sequence $\left\{x_n\right\}$ is Cauchy (or convergent to x) in the partial metric space $(X,P)$ if and only if $\left\{x_n\right\}$ is Cauchy (or convergent to x) in the metric space $(X,d_p)$ [1]. Many authors have studied and obtained fixed point results in partial metric spaces (see [2,3,4] and references therein). On the other hand, based on the concept of probabilistic metric space, Kramosil and Michalek [5] introduced the concept of fuzzy metric space, which is a generalization of metric space in the fuzzy setting. Later, George and Veeramani [6] modified the concept of fuzzy metric space in the sense of Kramosil and Michalek and gave a new concept of fuzzy metric space, which we call GV-fuzzy metric space. It has been shown that a GV-fuzzy metric M can generate a topology ${\tau }_M$ on X which is Hausdorff. Several authors have defined various fuzzy contractive mappings and obtained many fixed point theorems in GV-fuzzy metric spaces [7,8,9,10,11,12,13]. In addition, fuzzy metric has also been successfully applied in color image filtering, perceptual color difference, and other engineering applications [14,15,16,17].

In recent years, some scholars have tried to unify both partial metric and fuzzy metric into a single concept since they are both generalizations of classical metric space. For instance, in [18], Yue and Gu generalized partial metric and fuzzy metric to a more general framework by using the minimum t-norm.

In [19], Gregori et al. approached the concept of (GV-)fuzzy partial metric as an extension of the concept of partial metric to the fuzzy setting in the sense of Kramosil and Michalek and in George and Veeramani’s one. Those extensions had been made in a natural way, but for establishing the triangle inequality, they had used the residuum operator associated with a continuous t-norm.

In the conclusions of [19], Gregori et al. pointed out the introduced notion of (GV-)fuzzy partial metric opens several lines of research in the fuzzy setting, one of which is the study of fixed point theory in (GV-)fuzzy partial metrics.

Fixed point theory is widely used in many disciplines, and its core value lies in solving practical problems through mathematical tools. The main application areas are listed below.

Economics and game theory: Prove the existence of an equilibrium state of an economic system. Analyze the mathematical existence of equilibrium points in cooperative and non-cooperative games.

Computer science and algorithm analysis: Prove the existence of the optimal solution of the algorithm. Analyze the complexity of the algorithm and determine the convergence of the optimal solution.

Physics and quantum mechanics: Describe the relationship between particle wave function and position. Study the stability of space-time geometric structure.

Differential equation and dynamic system: Prove the existence of the solution of the initial value problem by the existence and uniqueness theorem. The equilibrium point and stability of the dynamic system are analyzed.

Engineering and science field: Iterative algorithm design in image restoration and signal processing. Stability analysis of neural networks.

Geometry: Fixed points of principal bundles over algebraic curves, vector bundles, and Higgs pairs over smooth projective curves.

The importance of fixed point theory and the blank of fixed point theorem in fuzzy partial metric space urge us to study fixed point theory in it. In order to study the fixed point theory, we must introduce some new concepts.

In this paper, we give the concepts of Cauchy sequence, convergent sequence, and completeness in a GV-fuzzy partial metric space by the residuum operator associated to a continuous t-norm, and based on these notions, we can give the concepts of two kinds of fuzzy contractive mappings. As the main content of this paper, we prove two fixed-point theorems in GV-fuzzy partial metric spaces.

2. Preliminaries

In this section, we give some fundamental definitions and results that will be used in the sequel.

Definition 1

(See Klement et al. [20]). Let $(L,⪯)$ be a lattice and $(L,\ast )$ a semigroup with neutral element.

(i) The triple $(L,⪯,\ast )$ is called a lattice-ordered monoid (or an l-monoid) if for all $x,y,z\in L$ we have

$$x\ast (y\vee z)=(x\ast y)\vee (x\ast z\mathrm{and}(x\vee y)\ast z=(x\vee z)\ast (y\vee z).$$

(ii) An l-monoid $(L,⪯,\ast )$ is said to be commutative if the semigroup $(L,\ast )$ is commutative.

(iii) A commutative l-monoid $(L,⪯,\ast )$ is called a commutative, residuated l-monoid if there exists a further binary operation ${\to }_{\ast }$ on L such that for each $x,y,z\in L$ we have

$$x\ast y⪯z\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}x⪯y{\to }_{\ast }z.$$

In this case, ${\to }_{\ast }$ is called the ∗-residuum. And in each commutative, residuated l-monoid $(L,⪯)$, the ∗-residuum operator ${\to }_{\ast }$ is uniquely determined by the formula

$$x{\to }_{\ast }y=sup\{z\in L:x\ast y⪯z\}$$

(1)

(iv) An l-monoid $(L,⪯,\ast )$ is called integral if there is a greatest element in the lattice $(L,⪯)$, which coincides with the neutral element of the semigroup $(L,\ast )$.

(v) A commutative integral l-monoid $(L,⪯,\ast )$ is called divisible if for each $x,y\in L$ with $y⪯x$: there exists $z\in L$ such that $x\ast z=y$.

Definition 2

(See Schweizer and Sklar [21]). A binary operation $T:[0,1]\to [0,1]$ is called a continuous t-norm if the following conditions are satisfied:

• (T1) $T(a,b)=T(b,a)$;
• (T2) $T(a,b)\le T(c,d)$ for $a\le c,b\le d$;
• (T3) $T(T(a,b),c)=T(a,T(b,c))$;
• (T4) $T(a,0)=0,T(a,1)=1$;
• (T5) T is continuous;

for all $a,b,c,d\in [0,1].$

For $a,b\in [0,1]$, instead of $T(a,b)$ we will use the notation $a\ast b$. The most commonly used t-norm are minimum ∧, the usual product ${\ast }_P$, and Łukasiewicz t-norm ${\ast }_L$ given by the formulae, respectively: $a\wedge b=min\{a,b\},a{\ast }_{P}b=ab$ and $a{\ast }_{L}b=max\{0,a+b-1\}$.

Proposition 1

(See Klement et al. [20]). ∗ is a continuous t-norm if and only if $([0,1],\ast ,\le )$ is a commutative residuated divisible integral l-monoid.

Taking into account the Formula (1), an immediate consequence of the proposition is that the ∗-residuum operator ${\to }_{\ast }$ of a continuous t-norm ∗ is uniquely determined by the formula

$$x{\to }_{\ast }y=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}x\le y;\hfill \\ sup\{z\in L:x\ast z=y\},\hfill & \mathrm{if}x>y.\hfill \end{array}\right.$$

(2)

Paying attention to (2), the ∗-residuum operator of the minimum, the usual product and the Łukasievicz t-norms, respectively, are the following:

$$x{\to }_{\wedge }y=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}x\le y;\hfill \\ y,\hfill & \mathrm{if}x>y.\hfill \end{array}\right.$$

$$x{\to }_{{\ast }_P}=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}x\le y;\hfill \\ {\displaystyle \frac{y}{x}},\hfill & \mathrm{if}x>y.\hfill \end{array}\right.$$

$$x{\to }_{{\ast }_L}y=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}x\le y;\hfill \\ 1-x+y,\hfill & \mathrm{if}x>y.\hfill \end{array}\right.$$

Definition 3

(See George and Veeramani [6]). A fuzzy metric space is an ordered triple $(X,M,\ast )$ such that X is a (non-empty) set, ∗ is a continuous t-norm, and M is a fuzzy set on $X\times X\times (0,+\infty )$ satisfying the following conditions: for all $x,y,z\in X,s,t>0$:

(GV1) $M(x,y,t)>0$;

(GV2) $M(x,y,t)=1$ if and only if $x=y$;

(GV3) $M(x,y,t)=M(y,x,t)$;

(GV4) $M(x,y,t)\ast M(y,z,s)\le M(x,z,t+s)$;

(GV5) $M(x,y,\_):(0,+\infty )\to (0,1]$ is continuous.

If $(X,M,\ast )$ is a fuzzy metric space, we will say that $(M,\ast )$, or simply M, is a fuzzy metric on X.

Definition 4

(See Gregori et al. [19]). A fuzzy partial metric space is an ordered triple $(X,P,\ast )$ such that X is a (non-empty) set, ∗ is a continuous t-norm, and P is a fuzzy set on $X\times X\times (0,\infty )$ satisfying the following conditions for all $x,y,z\in X$ and $s,t\in (0,\infty )$:

$(FPGV1)$ $0<P(x,y,t)\le P(x,x,t)$;

$(FPGV2)$ $P(x,x,t)=P(y,y,t)=P(x,y,t)$ if and only if $x=y$;

$(FPGV3)$ $P(x,y,t)=P(y,x,t)$;

$(FPGV4)$ $P(x,x,t+s){\to }_{\ast }P(x,z,t+s)\ge (P(x,x,t){\to }_{\ast }P(x,y,t))\ast (P(y,y,s){\to }_{\ast }P(y,z,s))$;

$(FPGV5)$ The assignment $P_{x,x,y}:(0,\infty )\to [0,1]$, given by $P_{x,x,y}(t)=P(x,x,t){\to }_{\ast }P(x,y,t)$, is a continuous function.

If $(X,P,\ast )$ is a fuzzy partial metric space, we will say that $(P,\ast )$, or simply P, is a fuzzy partial metric on X.

Remark 1.

(i) In [19], Gregori et al. defined fuzzy partial metric in the sense of Kramosil and Michalek and the one in the sense of George and Veeramani, which is called GV-fuzzy partial metric. This paper only discusses GV-fuzzy partial metric; for convenience, we still call it fuzzy partial metric.

(ii) Comparing our definition with the original one, one can see that we have strengthened the fifth axiom. In the original definition, the fifth axiom is: The assignment $P_{x,y}:(0,\infty )\to [0,1]$, given by $P_{x,y}(t)=P(x,y,t)$, is a continuous function. There are two reasons for the modification of the fifth axiom: the first is that we can give a natural topology ${\tau }_P$ so that a sequence $\left\{x_n\right\}$ convergenting to x with respect to ${\tau }_P$ and convergenting to x with respect to the partial metric P are compatible (See Remark 2); the second is that we can establish the relationship between a fuzzy partial metric and an associated fuzzy metric (See Proposition 2). Although we have made some modifications, our definition is still a natural generalization of partial metric in the fuzzy setting and also a natural generalization of fuzzy metric.

Let $(X,P,\ast )$ be a fuzzy partial metric space. Then P generates a topology ${\tau }_L$ on X that has as a base the family of open sets of the form $\{B_L(x,ϵ,t):x\in X,0<ϵ<1,t>0\}$, where $B_L(x,ϵ,t)=\{y\in X:P(x,x,t){\to }_{\ast }P(x,y,t)>1-ϵ\}$ for all $x\in X,ϵ\in (0,1)$ and $t>0$ (See [19] for details).

Similarly, P generates another topology ${\tau }_R$ on X, which has as a base the family of open sets of the form $\{B_R(x,ϵ,t):x\in X,0<ϵ<1,t>0\}$, where $B_R(x,ϵ,t)=\{y\in X:P(y,y,t){\to }_{\ast }P(y,x,t)>1-ϵ\}$ for all $x\in X,ϵ\in (0,1)$ and $t>0$. In general, ${\tau }_L$ and ${\tau }_R$ are different topologies on X, but they are both $T_0$.

In the sequel we need a fine topology ${\tau }_P$ on X that has as a base the family of open sets of the form $\{B_P(x,ϵ,t):x\in X,0<ϵ<1,t>0\}$, where $B_P(x,ϵ,t)=\{y\in X:(P(x,x,t){\to }_{\ast }P(x,y,t))\bigwedge (P(y,y,t){\to }_{\ast }P(y,x,t))>1-ϵ\}$ for all $x\in X,ϵ\in (0,1)$ and $t>0$. Then, ${\tau }_P$ is Hausdorff (See Proposition 2 and [6] for details).

Example 1

([19]). Let $(X,p)$ be a partial metric space. Define the fuzzy set on $X\times X\times (0,\infty )$ as follows

$$P_e(x,y,t)=e^{-{\displaystyle \frac{p(x,y)}{t}}}.$$

Then $(X,P_e,{\ast }_P)$ is a fuzzy partial metric space.

Example 2.

Let $(X,p)$ be a partial metric space. Define the fuzzy set on $X\times X\times (0,\infty )$ as follows

$$P(x,y,t)=e^{-{\displaystyle \frac{p(x,y)}{t+1}}}.$$

Then $(X,P,\ast )$ is a fuzzy partial metric space, where $\ast ={\ast }_P$.

Proof. 

We will only check that P satisfies axioms $(FPGV4)$ and $(FPGV5)$ of Definition 4.

$(FPGV4)$  $P(x,x,t+s){\to }_{\ast }P(x,y,t+s)=e^{-{\displaystyle \frac{p(x,y)-p(x,x)}{t+s+1}}}$

$\ge e^{-{\displaystyle \frac{p(x,z)-p(x,x)+p(z,y)-p(z,z)}{t+s+1}}}\ge e^{-{\displaystyle \frac{p(x,z)-p(x,x)+p(z,y)-p(z,z)}{max\{t,s\}+1}}}\ge e^{-{\displaystyle \frac{p(x,z)-p(x,x)}{t+1}}}\ast e^{-{\displaystyle \frac{p(z,y)-p(z,z)}{s+1}}}$

$=(P(x,x,t){\to }_{\ast }P(x,z,t))\ast (P(z,z,s){\to }_{\ast }{P}_d(z,y,s)).$

$(FPGV5)$ In fact, $P(x,x,t){\to }_{\ast }P(x,y,t)=e^{-{\displaystyle \frac{p(x,y)-p(x,x)}{t+1}}}$. It is easy to see that the assignment $P_{x,x,y}:(0,\infty )\to [0,1]$, given by $P_{x,x,y}(t)=P(x,x,t){\to }_{\ast }P(x,y,t)=e^{-{\displaystyle \frac{p(x,y)-p(x,x)}{t+1}}}$, is a continuous function. □

Example 3.

Let $(X,d)$ be a metric space and $L\ge 1$. Define the fuzzy set on $X\times X\times (0,\infty )$ as follows

$$P_{d_L}(x,y,t)={\displaystyle \frac{t}{Lt+d(x,y)}}.$$

Then $(X,P_{d_L},\ast )$ is a fuzzy partial metric space, where $\ast ={\ast }_P$. And if $L=1$, then $(X,P_{d_1},\ast )$ is a fuzzy metric space.

Proof. 

We will only check that P satisfies the axioms $(FPGV4)$ and $(FPGV5)$ of Definition 4.

$(FPGV4)$  $P_{d_L}(x,x,t+s){\to }_{\ast }{P}_{d_L}(x,y,t+s)={\displaystyle \frac{1}{L}}{\to }_{\ast }{\displaystyle \frac{t+s}{L(t+s)+d(x,y)}}={\displaystyle \frac{1}{1+\frac{d(x,y)}{L(t+s)}}}.$

$$P_{d_L}(x,x,t){\to }_{\ast }{P}_{d_L}(x,z,t)={\displaystyle \frac{1}{L}}{\to }_{\ast }{\displaystyle \frac{t}{Lt+d(x,z)}}={\displaystyle \frac{1}{1+\frac{d(x,z)}{Lt}}}.$$

$$P_{d_L}(z,z,s){\to }_{\ast }{P}_{d_L}(z,z,s)={\displaystyle \frac{1}{L}}{\to }_{\ast }{\displaystyle \frac{s}{Ls+d(z,y)}}={\displaystyle \frac{1}{1+\frac{d(z,y)}{Ls}}}.$$

It is easy to show that ${\displaystyle \frac{d(x,y)}{t+s}}\le {\displaystyle \frac{d(x,z)}{t}}+{\displaystyle \frac{d(z,y)}{s}}$, so we have

$${\displaystyle \frac{1}{1+\frac{d(x,y)}{L(t+s)}}}\ge {\displaystyle \frac{1}{1+\frac{d(x,z)}{Lt}+\frac{d(z,y)}{Ls}}}\ge {\displaystyle \frac{1}{1+\frac{d(x,z)}{Lt}+\frac{d(z,y)}{Ls}+\frac{d(x,z)}{Lt}\frac{d(z,y)}{Ls}}}={\displaystyle \frac{1}{1+\frac{d(x,z)}{Lt}}}{\displaystyle \frac{1}{1+\frac{d(z,y)}{Ls}}}.$$

Thus, $P_{d_L}(x,x,t+s){\to }_{\ast }{P}_{d_L}(x,y,t+s)\ge (P_{d_L}(x,x,t){\to }_{\ast }{P}_{d_L}(x,z,t))\ast (P_{d_L}(z,z,s){\to }_{\ast }{P}_{d_L}(z,y,s))$.

$(FPGV5)$ In fact, $P_{d_L}(x,x,t){\to }_{\ast }{P}_{d_L}(x,y,t)={\displaystyle \frac{1}{L}}{\to }_{\ast }{\displaystyle \frac{t}{Lt+d(x,y)}}={\displaystyle \frac{Lt}{Lt+d(x,y)}}.$ It is easy to see that the assignment $P_{x,x,y}:(0,\infty )\to [0,1]$, given by $P_{x,x,y}(t)=P(x,x,t){\to }_{\ast }P(x,y,t)$, is a continuous function. □

Definition 5.

(i) A sequence $\left\{x_n\right\}$ in a fuzzy partial metric space $(X,P,\ast )$ is said to be P-Cauchy, or simply Cauchy, if ${lim}_{n,m\to \infty }P(x_n,x_n,t){\to }_{\ast }P(x_n,x_m,t)=1$ and ${lim}_{n,m\to \infty }P(x_m,x_m,t){\to }_{\ast }P(x_m,x_n,t)=1$ for all $t>0$.

And a sequence $\left\{x_n\right\}$ in a fuzzy partial metric space $(X,P,\ast )$ is said to be $1-P$-Cauchy, or simply 1-Cauchy, if ${lim}_{n,m\to \infty }P(x_n,x_m,t)=1$ for all $t>0$.

(ii) A sequence $\left\{x_n\right\}$ in a fuzzy partial metric space $(X,P,\ast )$ is said to be convergent to x with respect to P if ${lim}_{n\to \infty }P(x,x,t){\to }_{\ast }P(x,x_n,t)=1$ and ${lim}_{n\to \infty }P(x_n,x_n,t){\to }_{\ast }P(x_n,x,t)=1$ for all $t>0$.

And a sequence $\left\{x_n\right\}$ in a fuzzy partial metric space $(X,P,\ast )$ is said to be 1-convergent to x if ${lim}_{n\to \infty }P(x,x_n,t)=1$ for all $t>0$.

(iii) $(X,P,\ast )$ is said to be complete if every Cauchy sequence in X is convergent.

And $(X,P,\ast )$ is said to be 1-complete if every 1-Cauchy sequence in X is 1-convergent.

Obviously, if $(X,P,\ast )$ is complete, then $(X,P,\ast )$ is 1-complete. But the converse is not true.

Remark 2.

A sequence $\left\{x_n\right\}$ in a fuzzy partial metric space $(X,P,\ast )$ converges to x with respect to ${\tau }_P$ if and only if $\left\{x_n\right\}$ converges to x with respect to P.

According to [13], denote by $\mathcal{H}$ the family of mappings $\eta :(0,1]\to (0,1]$ satisfying the following two conditions:

(H1) $\eta$ transforms $(0,1]$ onto $(0,\infty ]$;

(H2) $\eta$ is strictly decreasing.

3. Relationship of Fuzzy Metric Spaces and Fuzzy Partial Metric Spaces

Proposition 2.

Let $(X,P,\ast )$ be a fuzzy partial metric space. Define $M_P(x,y,t)=(P(x,x,t){\to }_{\ast }P(x,y,t))\bigwedge (P(y,y,t){\to }_{\ast }P(y,x,t))$, then $(X,M_P,\ast )$ is a fuzzy metric space.

Proof. 

Let $x,y,z\in X$ and $s,t\in (0,\infty )$. We will prove that $(X,M_P,\ast )$ satisfies the axioms of Definition 3.

$\begin{array}{cc}\hfill (GV1)M_P(x,y,t)& =(P(x,x,t){\to }_{\ast }P(x,y,t))\bigwedge (P(y,y,t){\to }_{\ast }P(y,x,t))\hfill \\ & \ge P(x,y,t)\bigwedge P(y,x,t)=P(x,y,t)>0;\hfill \end{array}$

$\begin{array}{cc}\hfill (GV2)M_P(x,y,t)=1& \iff P(x,x,t){\to }_{\ast }P(x,y,t)=1,P(y,y,t){\to }_{\ast }P(y,x,t)=1\hfill \\ & \iff P(x,x,t)\le P(x,y,t)\mathrm{and}P(y,y,t)\le P(y,x,t)\hfill \\ & \iff P(x,x,t)=P(y,y,t)=P(x,y,t)\iff x=y;\hfill \end{array}$

(GV3) It is obvious.

(GV4) $M_P(x,z,t+s)$

$=(P(x,x,t+s){\to }_{\ast }P(x,z,t+s))\bigwedge (P(z,z,t+s){\to }_{\ast }P(z,x,t+s))$

$\ge ((P(x,x,t){\to }_{\ast }P(x,y,t))\ast (P(y,y,s){\to }_{\ast }P(y,z,s)))$

$\bigwedge ((P(z,z,s){\to }_{\ast }P(z,y,s))\ast (P(y,y,t){\to }_{\ast }P(y,x,t)))$

$\ge ((P(x,x,t){\to }_{\ast }P(x,y,t))\bigwedge (P(y,y,t){\to }_{\ast }P(y,x,t)))$

$\ast ((P(y,y,s){\to }_{\ast }P(y,z,s))\bigwedge (P(z,z,s){\to }_{\ast }P(z,y,s)))$

$=M_P(x,y,t)\ast M_P(y,z,s)$

(GV5) It is obvious that $M_P(x,y,\_):(0,+\infty )\to (0,1]$ is continuous since $P(x,x,t){\to }_{\ast }P(x,y,t)$, $P(y,y,t){\to }_{\ast }P(y,x,t)$ and ⋀ are continuous. □

Proposition 3.

Let $(X,P,\ast )$ be a fuzzy partial metric space and $M_P(x,y,t)=(P(x,x,t){\to }_{\ast }P(x,y,t))\bigwedge (P(y,y,t){\to }_{\ast }P(y,x,t))$, then

(i) A sequence $\left\{x_n\right\}$ is P-Cauchy in the fuzzy partial metric space $(X,P,\ast )$ if and only if $\left\{x_n\right\}$ is M-Cauchy in fuzzy metric space $(X,M_P,\ast )$.

(ii) A sequence $\left\{x_n\right\}$ converges to x in the fuzzy partial metric space $(X,P,\ast )$ if and only if $\left\{x_n\right\}$ if $\left\{x_n\right\}$ converges to x in fuzzy metric space $(X,M_P,\ast )$.

(iii) $(X,P,\ast )$ is complete if and only if $(X,M_P,\ast )$ is complete.

4. Main Results

Definition 6.

A mapping $f:X\to X$ is said to be fuzzy 1-$\mathcal{H}$-contractive with respect to $\eta \in \mathcal{H}$ if there exists $k\in (0,1)$ satisfying the following condition: for all $x,y\in X,t>0$

$$\eta (P(fx,fy,t))\le k\eta (P(x,y,t)).$$

(3)

Theorem 1.

Let $(X,P,\ast )$ be a 1-complete fuzzy partial metric space and f be a fuzzy 1-$\mathcal{H}$-contractive with respect to $\eta \in \mathcal{H}$ such that the following assertions hold:

(i) $\tau \ge r\ast s\Rightarrow \eta (\tau )\le \eta (r)+\eta (s)$, for all $r,s,\tau \in \{P(f^{i}x,f^{j}x,t):x\in X,t>0,i,j\in N\}$;

(ii) there exists $x_0\in X$ such that $P(x_0,x_0,t)=1$ for all $t>0$ and ${\bigwedge }_{t>0}P(x_0,f{x}_0,t)>0$. Then f has a unique fixed point.

Proof. 

Suppose that there exists $x_0\in X$ such that $P(x_0,x_0,t)=1$ for all $t>0$ and ${\bigwedge }_{t>0}P(x_0,f{x}_0,t)>0$. Take $x_n=f^n(x_0)$ for each $n\ge 1$. Firstly, we can see that $P(x_n,x_n,t)=1$ for all $n\in N,t>0$ by the contractive condition (3). Again by (3), we have

$$\eta (P(x_n,x_{n+1},t))\le k\eta (P(x_n,x_{n+1},t))\le ······\le k^n\eta (P(x_0,x_1,t)).$$

(4)

For $n,m\in N,m>n,t>0$, let ${(a_i)}_{i\in N}$ be a strictly decreasing sequence of positive numbers such that ${\sum }_{i=1}^{\infty }{a}_i=1$. Since $P(x_n,x_n,t)=1$, then

$$\begin{array}{cc}\hfill P(x_n,x_m,t)& =1{\to }_{\ast }P(x_n,x_m,t)=P(x_n,x_n,t){\to }_{\ast }P(x_n,x_m,t)\hfill \\ \hfill & \ge P(x_n,x_n,{\Sigma }_{i=n}^{m-1}{a}_{i}t){\to }_{\ast }P(x_n,x_m,{\Sigma }_{i=n}^{m-1}{a}_{i}t)\hfill \\ \hfill & \ge {\Pi }_{i=n}^{m-1}P(x_i,x_i,a_{i}t){\to }_{\ast }P(x_i,x_{i+1},a_{i}t)\hfill \\ \hfill & ={\Pi }_{i=n}^{m-1}1{\to }_{\ast }P(x_i,x_{i+1},a_{i}t)\hfill \\ \hfill & ={\Pi }_{i=n}^{m-1}P(x_i,x_{i+1},a_{i}t).\hfill \end{array}$$

Thus, by (i) (4) and the above inequality, we have

$$\eta (P(x_n,x_m,t))\le {\Sigma }_{i=n}^{m-1}\eta (P(x_i,x_{i+1},a_{i}t))\le \eta (\underset{t>0}{\bigwedge }P(x_0,f{x}_0,t)){\Sigma }_{i=n}^{m-1}{k}^i.$$

It is easy to show that the series $\eta ({\bigwedge }_{t>0}P(x_0,f{x}_0,t)){\Sigma }_{i=n}^{m-1}{k}^i$ is convergent. Which means ${lim}_{n,m\to \infty }\eta (P(x_n,x_m,t))=0$, and which hence implies that

$$\underset{n,m\to \infty }{lim}P(x_n,x_m,t)=1.$$

Therefore, $\left\{x_n\right\}$ is a 1-Cauchy sequence in X.

Since X is 1-complete and $\left\{x_n\right\}$ is a 1-Cauchy sequence in X, then there exists $x^{\ast }\in X$ such that ${lim}_{n\to \infty }P(x^{\ast },x_n,t)=1$ for all $t>0$. By $(FPGV1)$, $P(x^{\ast },x^{\ast },t)=1$ for all $t>0$.

Next, we show that $P(x^{\ast },f{x}^{\ast },t)=1$ for all $t>0$. For a given $t\in [0,\infty ]$,

$$\begin{array}{cc}\hfill P(x^{\ast },f{x}^{\ast },t)& =1{\to }_{\ast }P(x^{\ast },f{x}^{\ast },t)=P(x^{\ast },x^{\ast },t){\to }_{\ast }P(x^{\ast },f{x}^{\ast },t)\hfill \\ \hfill & \ge (P(x^{\ast },x^{\ast },{\displaystyle \frac{t}{2}}){\to }_{\ast }P(x^{\ast },x_n,{\displaystyle \frac{t}{2}}))\ast (P(x_n,x_n,{\displaystyle \frac{t}{2}}){\to }_{\ast }P(x_n,f{x}^{\ast },{\displaystyle \frac{t}{2}}))\hfill \\ \hfill & =P(x^{\ast },x_n,{\displaystyle \frac{t}{2}})\ast P(x_n,f{x}^{\ast },{\displaystyle \frac{t}{2}}).\hfill \end{array}$$

By the contractive condition,

$$\eta (P(x_n,f{x}^{\ast },{\displaystyle \frac{t}{2}}))\le k\eta (P(x_{n-1},x^{\ast },{\displaystyle \frac{t}{2}})).$$

Since ${lim}_{n\to \infty }P(x_{n-1},x^{\ast },{\displaystyle \frac{t}{2}})=1$, we have

$$\underset{n\to \infty }{lim}\eta (P(x_n,f{x}^{\ast },{\displaystyle \frac{t}{2}}))\le k\eta (1)=0.$$

That is, ${lim}_{n\to \infty }P(x_n,f{x}^{\ast },{\displaystyle \frac{t}{2}})=1$. Passing the limit as $n\to \infty$, we have $P(x^{\ast },f{x}^{\ast },t)=1$. Thus,

$$P(x^{\ast },f{x}^{\ast },t)=P(x^{\ast },x^{\ast },t)=P(f{x}^{\ast },f{x}^{\ast },t)=1.$$

By $(FPGV2)$, $f{x}^{\ast }=x^{\ast }$.

Finally, we show that f has at most one fixed point. Suppose, on the contrary, there exists another fixed point $y^{\ast }$. Then, by (3),

$$\eta (P(x^{\ast },y^{\ast },t))=\eta (P(f{x}^{\ast },f{y}^{\ast },t))\le k\eta (P(x^{\ast },y^{\ast },t)).$$

Which means $\eta (P(x^{\ast },y^{\ast },t))=0$, that is $P(x^{\ast },y^{\ast },t)=1$. By $(FPGV1)$ and $(FPGV2)$, then $x^{\ast }=y^{\ast }$. Therefore, the fixed point of f is unique. □

Definition 7.

A mapping $f:X\to X$ is said to be fuzzy $\mathcal{H}$-contractive with respect to $\eta \in \mathcal{H}$ if there exists $k\in (0,1)$ satisfying the following condition: for all $x,y\in X,t>0$

$$\eta (P(fx,fx,t){\to }_{\ast }P(fx,fy,t))\le k\eta (P(x,x,t){\to }_{\ast }P(x,y,t)).$$

(5)

Theorem 2.

Let $(X,P,\ast )$ be a complete fuzzy partial metric space and f be a fuzzy $\mathcal{H}$-contractive with respect to $\eta \in \mathcal{H}$ such that

$$\tau \ge r\ast s\Rightarrow \eta (\tau )\le \eta (r)+\eta (s)$$

for all $r,s,\tau \in \{P(f^{i}x,f^{i}x,t){\to }_{\ast }P(f^{i}x,f^{j}x,t):x\in X,t>0,i,j\in N\}$. Then f has a unique fixed point if and only if there exists $x_0\in X$ such that

$$\underset{t>0}{\bigwedge }(P(x_0,x_0,t){\to }_{\ast }P(x_0,f{x}_0,t))>0\mathrm{and}\underset{t>0}{\bigwedge }(P(f{x}_0,f{x}_0,t){\to }_{\ast }P(f{x}_0,x_0,t))>0.$$

Proof. 

Suppose that f has a unique fixed point; then there exists $x_0\in X$ such that $f(x_0)=x_0$. Thus, $P(x_0,x_0,t){\to }_{\ast }P(x_0,f{x}_0,t)=1$ and $P(f{x}_0,f{x}_0,t){\to }_{\ast }P(f{x}_0,x_0,t)=1$ for each $t>0$. Then

$$\underset{t>0}{\bigwedge }(P(x_0,x_0,t){\to }_{\ast }P(x_0,f{x}_0,t))=1>0\mathrm{and}\underset{t>0}{\bigwedge }(P(f{x}_0,f{x}_0,t){\to }_{\ast }P(f{x}_0,x_0,t))=1>0.$$

Conversely, suppose that there exists $x_0\in X$ such that

$$\underset{t>0}{\bigwedge }(P(x_0,x_0,t){\to }_{\ast }P(x_0,f{x}_0,t))>0\mathrm{and}\underset{t>0}{\bigwedge }(P(f{x}_0,f{x}_0,t){\to }_{\ast }P(f{x}_0,x_0,t))>0.$$

For convenience, let $a={\bigwedge }_{t>0}(P(x_0,x_0,t){\to }_{\ast }P(x_0,f{x}_0,t))$ and $b={\bigwedge }_{t>0}(P(f{x}_0,f{x}_0,t){\to }_{\ast }P(f{x}_0,x_0,t))$, then $a,b>0$. Take $x_n=f^n(x_0)$ for each $n\ge 1$. By the contractive condition (7),

$$\begin{array}{cc}\hfill \eta (P(x_n,x_n,t){\to }_{\ast }P(x_n,x_{n+1},t))& \le k\eta (P(x_{n-1},x_{n-1},t){\to }_{\ast }P(x_{n-1},x_n,t))\hfill \\ \hfill & \le k^2\eta (P(x_{n-2},x_{n-2},t){\to }_{\ast }P(x_{n-2},x_{n-1},t))\hfill \\ \hfill & ······\hfill \\ \hfill & \le k^n\eta (P(x_0,x_0,t){\to }_{\ast }P(x_0,x_1,t)).\hfill \end{array}$$

For $n,m\in N,m>n,t>0$, let ${(a_i)}_{i\in N}$ be a strictly decreasing sequence of positive numbers such that ${\sum }_{i=1}^{\infty }{a}_i=1$. Then from $(FPGV4)$,

$$\begin{array}{cc}\hfill P(x_n,x_n,t){\to }_{\ast }P(x_n,x_m,t)& \ge P(x_n,x_n,{\Sigma }_{i=n}^{m-1}{a}_{i}t){\to }_{\ast }P(x_n,x_m,{\Sigma }_{i=n}^{m-1}{a}_{i}t)\hfill \\ \hfill & \ge {\Pi }_{i=n}^{m-1}P(x_i,x_i,a_{i}t){\to }_{\ast }P(x_i,x_{i+1},a_{i}t).\hfill \end{array}$$

Similarly, $P(x_m,x_m,t){\to }_{\ast }P(x_m,x_n,t)\ge {\Pi }_{i=n}^{m-1}P(x_{i+1},x_{i+1},a_{i}t){\to }_{\ast }P(x_{i+1},x_i,a_{i}t)$.

From the definition and known conditions of $\eta$, we have

$$\begin{array}{cc}\hfill \eta (P(x_n,x_n,t){\to }_{\ast }P(x_n,x_m,t))& \le {\Sigma }_{i=n}^{m-1}\eta (P(x_i,x_i,a_{i}t){\to }_{\ast }P(x_i,x_{i+1},a_{i}t))\hfill \\ \hfill & \le \eta (\underset{t>0}{\bigwedge }(P(x_0,x_0,t){\to }_{\ast }P(x_0,f{x}_0,t))){\Sigma }_{i=n}^{m-1}{k}^i\hfill \\ \hfill & =\eta (a){\Sigma }_{i=n}^{m-1}{k}^i.\hfill \end{array}$$

Similarly, we have $\eta (P(x_m,x_m,t){\to }_{\ast }P(x_m,x_n,t))\le \eta (b){\Sigma }_{i=n}^{m-1}{k}^i$.

Since $a,b>0$, it is easy to show that both series $\eta (a){\Sigma }_{i=n}^{m-1}{k}^i$ and $\eta (b){\Sigma }_{i=n}^{m-1}{k}^i$ are convergent. Which means

$$\underset{n,m\to \infty }{lim}\eta (P(x_n,x_n,t){\to }_{\ast }P(x_n,x_m,t))=0\mathrm{and}\underset{n,m\to \infty }{lim}\eta (P(x_m,x_m,t){\to }_{\ast }P(x_m,x_n,t))=0.$$

And thus

$$\underset{n,m\to \infty }{lim}P(x_n,x_n,t){\to }_{\ast }P(x_n,x_m,t)=1\mathrm{and}\underset{n,m\to \infty }{lim}P(x_m,x_m,t){\to }_{\ast }P(x_m,x_n,t)=1.$$

Therefore, $\left\{x_n\right\}$ is a Cauchy sequence in X.

Since X is complete and $\left\{x_n\right\}$ is a Cauchy sequence in X, then there exists $x^{\ast }\in X$ such that

$$\underset{n\to \infty }{lim}P(x^{\ast },x^{\ast },t){\to }_{\ast }P(x^{\ast },x_n,t)=1$$

(6)

and

$$\underset{n\to \infty }{lim}P(x_n,x_n,t){\to }_{\ast }P(x_n,x^{\ast },t)=1$$

(7)

for all $t>0$.

Fix a given $t\in (0,\infty )$. Since $\eta (P(x_n,x_n,{\displaystyle \frac{t}{2}}){\to }_{\ast }P(x_n,f{x}^{\ast },{\displaystyle \frac{t}{2}}))\le k\eta (P(x_{n-1},x_{n-1},{\displaystyle \frac{t}{2}}){\to }_{\ast }P(x_{n-1},x^{\ast },{\displaystyle \frac{t}{2}}))$, by (7), then we have

$$\underset{n\to \infty }{lim}\eta (P(x_n,x_n,{\displaystyle \frac{t}{2}}){\to }_{\ast }P(x_n,fx,{\displaystyle \frac{t}{2}}))\le k\eta (1)=0.$$

Hence,

$$\underset{n\to \infty }{lim}(P(x_n,x_n,{\displaystyle \frac{t}{2}}){\to }_{\ast }P(x_n,f{x}^{\ast },{\displaystyle \frac{t}{2}}))=1.$$

(8)

By $(FPGV4)$,

$$P(x^{\ast },x^{\ast },t){\to }_{\ast }P(x^{\ast },f{x}^{\ast },t)\ge (P(x^{\ast },x^{\ast },{\displaystyle \frac{t}{2}}){\to }_{\ast }P(x^{\ast },x_n,{\displaystyle \frac{t}{2}}))\ast (P(x_n,x_n,{\displaystyle \frac{t}{2}}){\to }_{\ast }P(x_n,fx,{\displaystyle \frac{t}{2}})).$$

Then, from (7), (8), $P(x^{\ast },x^{\ast },t){\to }_{\ast }P(x^{\ast },f{x}^{\ast },t)=1$, which means $P(x^{\ast },x^{\ast },t)\le P(x^{\ast },f{x}^{\ast },t)$. Then by $(FPGV1)$,

$$P(x^{\ast },x^{\ast },t)=P(x^{\ast },f{x}^{\ast },t).$$

(9)

On the other hand, since $\eta (P(f{x}^{\ast },f{x}^{\ast },{\displaystyle \frac{t}{2}}){\to }_{\ast }P(f{x}^{\ast },x_n,{\displaystyle \frac{t}{2}}))\le k\eta (P(x^{\ast },x^{\ast },{\displaystyle \frac{t}{2}}){\to }_{\ast }P(x^{\ast },x_{n-1},{\displaystyle \frac{t}{2}}))$, by (6), then we have

$$\underset{n\to \infty }{lim}\eta (P(f{x}^{\ast },f{x}^{\ast },{\displaystyle \frac{t}{2}}){\to }_{\ast }P(f{x}^{\ast },x_n,{\displaystyle \frac{t}{2}}))\le k\eta (1)=0.$$

Hence,

$$\underset{n\to \infty }{lim}(P(f{x}^{\ast },f{x}^{\ast },{\displaystyle \frac{t}{2}}){\to }_{\ast }P(f{x}^{\ast },x_n,{\displaystyle \frac{t}{2}}))=1.$$

(10)

By $(FPGV4)$,

$$P(f{x}^{\ast },f{x}^{\ast },t){\to }_{\ast }P(f{x}^{\ast },x^{\ast },t)\ge (P(f{x}^{\ast },f{x}^{\ast },{\displaystyle \frac{t}{2}}){\to }_{\ast }P(fx,x_n,{\displaystyle \frac{t}{2}}))\ast (P(x_n,x_n,{\displaystyle \frac{t}{2}}){\to }_{\ast }P(x_n,x^{\ast },{\displaystyle \frac{t}{2}})).$$

Then, from (7), (10), $P(f{x}^{\ast },f{x}^{\ast },t){\to }_{\ast }P(f{x}^{\ast },x^{\ast },t)=1$, which means $P(f{x}^{\ast },f{x}^{\ast },t)\le P(x^{\ast },f{x}^{\ast },t)$. Then by $(FPGV1)$,

$$P(f{x}^{\ast },f{x}^{\ast },t)=P(x^{\ast },f{x}^{\ast },t).$$

(11)

Thus, by (9) and (11), $P(x^{\ast },x^{\ast },t)=P(x^{\ast },f{x}^{\ast },t)=P(f{x}^{\ast },f{x}^{\ast },t)$. Therefore, by $(FPGV2)$, $f{x}^{\ast }=x^{\ast }$.

Finally, we show that f has at most one fixed point. Suppose, on the contrary, there exists another fixed point $y^{\ast }$. Then $\eta (P(x,x,t){\to }_{\ast }P(x^{\ast },y^{\ast },t))=\eta (P(f{x}^{\ast },f{x}^{\ast },t){\to }_{\ast }P(f{x}^{\ast },f{y}^{\ast },t))\le k\eta (P(x^{\ast },x^{\ast },t){\to }_{\ast }P(x^{\ast },y^{\ast },t)).$ Which means $\eta (P(x^{\ast },x^{\ast },t){\to }_{\ast }P(x^{\ast },y^{\ast },t))=0$, that is $P(x^{\ast },x^{\ast },t){\to }_{\ast }P(x^{\ast },y^{\ast },t)=1$. Similarly, $P(y^{\ast },y^{\ast },t){\to }_{\ast }P(x^{\ast },y^{\ast },t)=1$. Then by $(FPGV2)$, $x^{\ast }=y^{\ast }$. Therefore, the fixed point of f is unique. □

Corollary 1.

Let $(X,P,\ast )$ be a complete fuzzy partial metric space and f be a fuzzy $\mathcal{H}$-contractive with respect to $\eta \in \mathcal{H}$ such that

$$\tau \ge r\ast s\Rightarrow \eta (\tau )\le \eta (r)+\eta (s)$$

for all $r,s,\tau \in \{P(f^{i}x,f^{i}x,t){\to }_{\ast }P(f^{i}x,f^{j}x,t):x\in X,t>0,i,j\in N\}$. Then f has a unique fixed point if there exists $x_0\in X$ such that ${\bigwedge }_{t>0}P(x_0,f{x}_0,t)>0$.

Example 4.

Let $X=[0,+\infty )$ and $p=max\{x,y\}$. Then $(X,P,{\ast }_P)$ is a complete fuzzy partial metric space, where $P(x,y,t)=e^{-{\displaystyle \frac{p(x,y)}{t+1}}}=e^{-{\displaystyle \frac{max\{x,y\}}{t+1}}}.$

Now, let $\eta (t)={\displaystyle \frac{1}{t}}-1$ and $fx={\displaystyle \frac{x}{2}}$. We shall show that f satisfying the following condition:

$$\eta (P(fx,fx,t){\to }_{\ast }P(fx,fy,t))\le {\displaystyle \frac{1}{2}}\eta (P(x,x,t){\to }_{\ast }P(x,y,t)).$$

We distinguish two cases:

(i) If $x\ge y$, then

$$P(x,x,t){\to }_{\ast }P(x,y,t)=e^{-{\displaystyle \frac{x}{t+1}}}{\to }_{\ast }{e}^{-{\displaystyle \frac{x}{t+1}}}=1$$

$$P(fx,fx,t){\to }_{\ast }P(fx,fy,t)=e^{-{\displaystyle \frac{x}{2(t+1)}}}{\to }_{\ast }{e}^{-{\displaystyle \frac{x}{2(t+1)}}}=1.$$

Since $\eta (1)=0$, then

$$\eta (P(fx,fx,t){\to }_{\ast }P(fx,fy,t))=0\le {\displaystyle \frac{1}{2}}\eta (P(x,x,t){\to }_{\ast }P(x,y,t))=0.$$

(ii) If $x<y$, then

$$P(x,x,t){\to }_{\ast }P(x,y,t)=e^{-{\displaystyle \frac{x}{t+1}}}{\to }_{\ast }{e}^{-{\displaystyle \frac{y}{t+1}}}=e^{-{\displaystyle \frac{y-x}{t+1}}}$$

$$P(fx,fx,t){\to }_{\ast }P(fx,fy,t)=e^{-{\displaystyle \frac{x}{2(t+1)}}}{\to }_{\ast }{e}^{-{\displaystyle \frac{y}{2t}}}=e^{-{\displaystyle \frac{y-x}{2(t+1)}}}.$$

and $\eta (P(x,x,t){\to }_{\ast }P(x,y,t))=e^{{\displaystyle \frac{y-x}{t+1}}}-1$, $\eta (P(fx,fx,t){\to }_{\ast }P(fx,fy,t))=e^{{\displaystyle \frac{y-x}{2(t+1)}}}-1.$ The fact that $e^{{\displaystyle \frac{1}{2}}z}-1\le {\displaystyle \frac{1}{2}}(e^z-1)$ for all $z\ge 0$ is very easy to verify.

Now since $z={\displaystyle \frac{y-x}{t+1}}>0$, then

$$\eta (P(fx,fx,t){\to }_{\ast }P(fx,fy,t))\le {\displaystyle \frac{1}{2}}\eta (P(x,x,t){\to }_{\ast }P(x,y,t)).$$

Let $x_0=1$, then

$$\underset{t>0}{\bigwedge }(P(x_0,x_0,t){\to }_{\ast }P(x_0,f{x}_0,t))=1>0\mathrm{and}\underset{t>0}{\bigwedge }(P(f{x}_0,f{x}_0,t){\to }_{\ast }P(f{x}_0,x_0,t))=e^{-{\displaystyle \frac{1}{2}}}>0.$$

So all the hypotheses of Theorem 2 are satisfied, and f has a unique fixed point. In this example $x=0$ is the fixed point.

Example 5.

Let $X=\{0,1,2,3\}$, and $P(x,y,t)=e^{-{\displaystyle \frac{max\{x,y\}}{t}}}$ for all $x,y\in X,t>0$. It is straightforward to verify that $(X,P,{\ast }_L)$ is a complete fuzzy partial metric space.

Let $\eta (t)=-lnt$ and $fx$ is a floor function as $fx=\lfloor {\displaystyle \frac{x}{2}}\rfloor$. We will show that f satisfying the following condition:

$$\eta (P(fx,fx,t){\to }_{\ast }P(fx,fy,t))\le {\displaystyle \frac{1}{2}}\eta (P(x,x,t){\to }_{\ast }P(x,y,t)).$$

We select three cases for verification, others follow similarly and are omitted.

(i) When $x=0,y=1$,

$P(fx,fx,t){\to }_{{\ast }_L}P(fx,fy,t)=P(0,0,t){\to }_{{\ast }_L}P(0,0,t)=1{\to }_{{\ast }_L}1=1$,

$P(x,x,t){\to }_{{\ast }_L}P(x,y,t)=P(0,0,t){\to }_{{\ast }_L}P(0,1,t)=1{\to }_{{\ast }_L}{e}^{-{\displaystyle \frac{1}{t}}}=e^{-{\displaystyle \frac{1}{t}}}$.

Then, we have

$\eta (P(fx,fx,t){\to }_{{\ast }_L}P(fx,fy,t))=0\le {\displaystyle \frac{1}{2}}\eta (P(x,x,t){\to }_{{\ast }_L}P(x,y,t))={\displaystyle \frac{1}{2}}\eta (e^{-{\displaystyle \frac{1}{t}}})={\displaystyle \frac{1}{2t}}$.

(ii) When $x=2,y=3$,

$P(fx,fx,t){\to }_{{\ast }_L}P(fx,fy,t)=P(1,1,t){\to }_{{\ast }_L}P(1,1,t)=e^{-{\displaystyle \frac{1}{t}}}{\to }_{{\ast }_L}{e}^{-{\displaystyle \frac{1}{t}}}=1$,

$P(x,x,t){\to }_{{\ast }_L}P(x,y,t)=P(2,2,t){\to }_{{\ast }_L}P(2,3,t)=e^{-{\displaystyle \frac{2}{t}}}{\to }_{{\ast }_L}{e}^{-{\displaystyle \frac{3}{t}}}=1-e^{-{\displaystyle \frac{2}{t}}}+e^{-{\displaystyle \frac{3}{t}}}$.

Since $\eta (1-e^{-{\displaystyle \frac{2}{t}}}+e^{-{\displaystyle \frac{3}{t}}})=-ln(1-e^{-{\displaystyle \frac{2}{t}}}+e^{-{\displaystyle \frac{3}{t}}})>0$, then we have

$\eta (P(fx,fx,t){\to }_{{\ast }_L}P(fx,fy,t))=0\le {\displaystyle \frac{1}{2}}\eta (P(x,x,t){\to }_{{\ast }_L}P(x,y,t))$.

(iii) When $x=3,y=0$,

$P(fx,fx,t){\to }_{{\ast }_L}P(fx,fy,t)=P(1,1,t){\to }_{{\ast }_L}P(1,0,t)=e^{-{\displaystyle \frac{1}{t}}}{\to }_{{\ast }_L}{e}^{-{\displaystyle \frac{1}{t}}}=1$,

$P(x,x,t){\to }_{{\ast }_L}P(x,y,t)=P(3,3,t){\to }_{{\ast }_L}P(3,0,t)=e^{-{\displaystyle \frac{3}{t}}}{\to }_{{\ast }_L}{e}^{-{\displaystyle \frac{3}{t}}}=1$.

Then we have

$\eta (P(fx,fx,t){\to }_{{\ast }_L}P(fx,fy,t))=0\le {\displaystyle \frac{1}{2}}\eta (P(x,x,t){\to }_{{\ast }_L}P(x,y,t))=0$.

Let $x_0=0$, then

$P(x_0,x_0,t){\to }_{{\ast }_L}P(x_0,f{x}_0,t))=P(0,0,t){\to }_{{\ast }_L}P(0,0,t))=1$,

$P(f{x}_0,f{x}_0,t){\to }_{{\ast }_L}P(f{x}_0,x_0,t))=P(0,0,t){\to }_{{\ast }_L}P(0,0,t))=1$.

Since ${\bigwedge }_{t>0}1=1>0$, all the hypotheses of Theorem 2 are satisfied. Therefore f has a unique fixed point $x=0$.

5. Conclusions

In the original paper of Gregori et al., it is difficult to define concepts such as Cauchy sequence and convergent sequence in fuzzy partial metric space, while these concepts are the basis of studying fixed points. In the present work, we give the concepts of Cauchy sequence, convergent sequence, and completeness in a fuzzy partial metric space by the residuum operator associated to a continuous t-norm. At the same time, we establish the relationship between a fuzzy partial metric space and a fuzzy metric space, which provides the corresponding basis for the application of fuzzy partial metric space. As the main content of this paper, we prove two fixed-point theorems in fuzzy partial metric spaces. Whether it can weaken the contractive conditions in this paper is one of our research directions, and it is also an open research direction to explore more other types of contractive conditions and get more fixed-point theorems.