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# Weakly compatible and quasi-contraction results in fuzzy cone metric spaces with application to the Urysohn type integral equations

*Advances in Difference Equations*
**volume 2020**, Article number: 280 (2020)

## Abstract

In this paper, we present some weakly compatible and quasi-contraction results for self-mappings in fuzzy cone metric spaces and prove some coincidence point and common fixed point theorems in the said space. Moreover, we use two Urysohn type integral equations to get the existence theorem for common solution to support our results. The two Urysohn type integral equations are as follows:

where \(l\in [0,1]\) and \(x,y,g\in \mathbf{E}\), where **E** is a real Banach space and \(K_{1},K_{2}:[0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R}\).

## Introduction

In 2007, Huang et al. [1] introduced the concept of cone metric space and proved some fixed point theorems for the underlying cone. In [2] Abbas et al. presented some noncommuting mapping results in cone metric spaces without continuity. After that, a series of authors (see [3–11]) contributed their ideas to the problems on cone metric spaces.

The initial version of fuzzy set theory was given by Zadeh [12], while Kramosil et al. in [13] introduced the fuzzy metric space or (shortly *FM*-space). Later on, a stronger form of the metric fuzziness was given by George et al. [14]. Some more related results in the context of fuzzy metric space can be found (e.g., see [15–19]).

Recently, Oner et al. in [20] introduced the concept of fuzzy cone metric space or shortly *FCM*-space. They presented some basic properties and a fuzzy cone Banach contraction theorem in a fuzzy cone metric space with the assumption that fuzzy cone contractive sequences are Cauchy. Some more properties and fixed point results in *FCM*-spaces can be found (e.g., see[21–26] and the references therein).

The aim of this paper is to obtain some coincidence point and common fixed point results for weakly compatible self-mappings in *FCM*-spaces. We also give the concept of quasi-contraction for weakly compatible self-mappings and establish some common fixed point theorems. Moreover, we present an integral type application from which we obtained the existence of fixed point results. The application of integral equations in fuzzy cone metric spaces is the new direction in the theory of fixed point. This new concept of application will be very fruitful for finding the existence solution of integral value problems on *FCM*-spaces. For this purpose, we use the two Urysohn integral type equations for common solution to support our results. We also present some illustrative examples to support our work.

## Preliminaries

In this section, we present some basic definitions and a helpful concept for our main results.

### Definition 2.1

([27])

An operation \(\ast :[0,1]\times [0,1]\to [0,1]\) is a continuous *s*-norm if the following hold:

- (1)
∗ is commutative, associative, and continuous.

- (2)
\(1\ast \beta =\beta \) and \(\beta \ast \beta _{1}\le \delta \ast \delta _{1}\), whenever, \(\beta \le \delta \) and \(\beta _{1}\le \delta _{1}\), for each \(\beta ,\beta _{1},\delta ,\delta _{1}\in [0,1]\).

The basic continuous *s*-norms of product, *L*ukasiewicz, and minimum are defined respectively as follows (see [27]):

### Definition 2.2

([14])

A three-tuple \((X,M,\ast )\) is said to be a fuzzy metric space if *X* is an arbitrary set, ∗ is a continuous *s*-norm, and *M* is a fuzzy set on \(X^{2}\times (0,\infty )\) satisfying the following conditions:

- (i)
\(M(x,y,s)>0\) and \(M(x,y,s)=1 \Leftrightarrow x=y\);

- (ii)
\(M(x,y,s)=M(y,x,s)\);

- (iii)
\(M(x,y,t+s)\geq M(x,z,t)\ast M(z,y,s) \);

- (iv)
\(M(x,y,\cdot ):(0,\infty )\to [0,1]\) is continuous;

\(\forall x,y,z\in X\) and \(s,t>0\).

For more details, we shall refer the readers to study [14].

### Definition 2.3

([1])

A subset *P* of a real Banach space **E** is called a cone if

- (1)
*P*is closed, nonempty and \(P\ne \{\vartheta \}\), where*ϑ*is the zero element of**E**. - (2)
If \(x, y \in P\) and \(\beta ,\delta \in [0, \infty )\), then \(\beta x + \delta y \in P\).

- (3)
If both \(x,-x \in P\), then \(x =\vartheta \).

A partial ordering “⪯” for a given cone *P* on **E** is defined as \(y \preceq x\) iff \(x - y \in P\). \(y \prec x\) stands for \(y \preceq x\) and \(y \ne x\), while \(y\ll x\) stands for \(x - y \in \operatorname{int}(P)\). Throughout this paper, all the cones have nonempty interior.

### Definition 2.4

([20])

A three-tuple \((X,M,\ast )\) is known as a fuzzy cone metric space (*FCM*-space) if *P* is a cone of **E**, *X* is an arbitrary set, ∗ is a continuous *s*-norm, and a fuzzy set *M* on \(X^{2}\times \operatorname{int}(P)\) satisfies the following:

- (1)
\(M(x,y,s)>0\) and \(M(x,y,s)=1\Leftrightarrow x=y\);

- (2)
\(M(x,y,s)=M(y,x,s)\);

- (3)
\(M(x,y,s)\ast M(y,z,t)\le M(x,z,s+t)\);

- (4)
\(M(x,y,\cdot ):\operatorname{int}(P)\to [0,1]\) is continuous;

\(\forall x,y,z\in X\) and \(s,t\in \operatorname{int}(P)\).

### Remark 2.5

Every *FM*-space becomes an *FCM*-space if \(\mathbf{E}=\mathbb{R}\), \(P=[0,\infty )\), and \(\beta *\delta =\beta \delta \) [20–22].

### Definition 2.6

([20])

Let \((X,M,\ast )\) be an *FCM*-space, \(x\in X\), and \((x_{i})\) be a sequence in *X*. Then,

- (i)
\((x_{i})\) converges to

*x*, if for \(s\gg \vartheta \) and \(0< r<1\), \(\exists i_{1}\in \mathbf{{N}}\), s.t. \(M(x_{i},x,s)>1-r\), \(\forall i\ge i_{1}\). We denote this by \(\lim_{i\to \infty }x_{i}=x\) or \(x_{i}\to x\) as \(i\to \infty \). - (ii)
\((x_{i})\) is said to be a Cauchy sequence if, for \(0< r<1\) and \(s\gg \vartheta \), \(\exists i_{1}\in \mathbf{{N}}\), s.t. \(M(x_{i},x_{j},s)>1-r\), \(\forall i,j\ge i_{1}\).

- (iii)
\((X,M,*)\) is said to be complete if every Cauchy sequence is convergent in

*X*. - (iv)
\((x_{i})\) is said to be a fuzzy cone contractive if \(\exists \beta \in (0,1)\) such that

$$ \frac{1}{M(x_{i},x_{i+1},s)}-1\le \beta \biggl( \frac{1}{M(x_{i-1},x_{i},s)}-1 \biggr) $$for \(s\gg \vartheta \), \(i\ge 1\).

### Definition 2.7

([25])

Let \((X,M,\ast )\) be an *FCM*-space. The fuzzy cone metric *M* is triangular if

\(\forall x,y,z\in X\) and each \(s\gg \vartheta \).

### Lemma 2.8

([20])

*Let*\(x\in X\)*in an**FCM*-*space*\((X,M,\ast )\)*and*\((x_{i})\)*be a sequence in**X*. *Then*\((x_{i})\)*converges to**x**if and only if*\(M(x_{i},x,s)\to 1\)*as*\(i\to \infty \)*for each*\(s\gg \vartheta \).

### Definition 2.9

([20])

Let \((X,M,\ast )\) be an *FCM*-space, and a mapping \(F_{1}:X\to X\) is said to be fuzzy cone contractive if \(\exists \beta \in (0,1)\) such that

for each \(x,y\in X\) and \(s\gg \vartheta \). *β* is called the contraction constant of \(F_{1}\).

### Definition 2.10

([2])

Let \(F_{1}\) and *ℓ* be two self-mappings on a set *X* (i.e., \(F_{1},\ell :X\to X\)). If \(u=F_{1}v=\ell v\) for some \(v\in X\), then *v* is called a coincidence point of \(F_{1}\) and *ℓ*, and *u* is called a point of coincidence of \(F_{1}\) and *ℓ*. The self-mappings \(F_{1}\) and *ℓ* are said to be weakly compatible if they commute at their coincidence point, i.e., \(F_{1}v=\ell v\) for some \(v\in X\), then \(F_{1}\ell v=\ell F_{1}v\).

### Proposition 2.11

([2])

*Let*\(F_{1}\)*and**ℓ**be weakly compatible self*-*maps of a set**X*. *If*\(F_{1}\)*and**ℓ**have a unique point of coincidence*\(u=F_{1}v=\ell v\), *then**u**is the unique common fixed point of*\(F_{1}\)*and**ℓ*.

### Definition 2.12

([28])

A pair \((\ell ,F_{1})\) of self-maps on *X* is called occasionally weakly compatible if \(\exists v\in X\) such that \(\ell v=F_{1}v\) and \(F_{1}\ell v=\ell F_{1}v\).

### Lemma 2.13

([28])

*Let*\(F_{1}\)*and**ℓ**be occasionally weakly compatible self*-*maps of a set**X*. *If*\(F_{1}\)*and**ℓ**have a unique point of coincidence*, \(F_{1}v=\ell v=u\), *then**u**is a unique common fixed point of**ℓ**and*\(F_{1}\).

“A self-mapping \(F_{1}\) in a complete *FCM*-space in which the contractive sequences are Cauchy and hold (2.1), then \(F_{1}\) has a unique fixed point in *X*” is a Banach contraction principle, which has been obtained in [20].

We note that fuzzy cone contractive sequences can be proved to be Cauchy sequences for weakly compatible self-mappings in *FCM*-spaces (see the proof of Theorem 3.1). In this paper we use the concept of complete *FCM*-spaces given by Rehman and Li [25] and prove some coincidence point and common fixed point theorems for weakly compatible three self-mappings and some quasi-contraction results in *FCM*-spaces. Moreover, we present some illustrative examples and the application of two Urysohn’s integral type equations for the existence of common solution to support our work.

## Weakly compatible mapping results in *FCM*-space

### Theorem 3.1

*Let*\(F_{1},F_{2},\ell :X\to X\)*be three self*-*maps and**M**be triangular in a complete**FCM*-*space*\((X,M,\ast )\)*satisfying*\(\forall x,y\in X\),

*for every*\(s\gg \vartheta \)*and*\(\beta ,\gamma ,\delta \in [0,\infty )\)*with*\(\beta +2\gamma +2\delta <1\). *If*\(F_{1}(X)\cup F_{2}(X)\subset \ell (X)\)*and*\(\ell (X)\)*is a complete subspace of**X*, *then*\(F_{1}\), \(F_{2}\), *and**ℓ**have a unique point of coincidence*. *Moreover*, *if*\((F_{1}, \ell )\)*and*\((F_{2}, \ell )\)*are weakly compatible*. *Then*\(F_{1}\), \(F_{2}\), *and**ℓ**have a unique common fixed point in**X*.

### Proof

Fix \(x_{0}\in X\) and use the condition \(F_{1}(X)\cup F_{2}(X)\subset \ell (X)\). We define some iterative sequences in *X* such that

Now, by (3.1) for \(s\gg \vartheta \),

Then

where \(\alpha =\frac{\beta +\gamma +\delta }{1-(\gamma +\delta )}<1\). Similarly,

which shows that a sequence \((\ell x_{i})_{i\geq 0}\) is fuzzy cone contractive. Hence,

Since *M* is triangular, for all \(j>i>i_{0}\),

which shows that a sequence \((\ell x_{i})\) is Cauchy sequence and \(\ell (X)\) is a complete subspace of *X*. Hence \(\exists u,v\in X\) such that \(\ell x_{i}\to u=\ell v\) as \(i\to \infty \), i.e.,

Since *M* is triangular, we have

Now by (3.1), (3.2), and (3.3), for \(s\gg \vartheta \),

Then,

Thus, from (3.3) and (3.4), we have

\(\gamma +\delta <1\), since \(\beta +2\gamma +2\delta <1\), then \(M(\ell v,F_{1}v,s)=M(u,F_{1}v,s)=1\), i.e., \(u=\ell v=F_{1}v\).

Similarly, we can prove that \(u=\ell v=F_{2}v\). It follows that *u* is a common coincidence point of the mappings *ℓ*, \(F_{1}\), and \(F_{2}\) in *X* such that \(u=\ell v=F_{1}v=F_{2}v\).

Now we prove the uniqueness of the point of coincidence in *X* for the mappings \(F_{1}\), \(F_{2}\), and *ℓ*. Let \(u^{*}\) be the other point in *X* such that

for some \(v^{*}\in X\). Then, by using (3.1) for \(s\gg \vartheta \),

\(\beta +2\delta <1\), since \(\beta +2\gamma +2\delta <1\). Thus we get that \(M(u,u^{*},t)=1\), that is, \(u=u^{*}\). By using the weak compatibility of \((F_{1},\ell )\), \((F_{2},\ell )\) and Proposition 2.11, we can get a unique common fixed point of \(F_{1}\), \(F_{2}\), and *ℓ*, that is, \(\ell v=F_{1}v=F_{2}v=v\). □

By using the map \(\ell =I_{x}\) and by taking into account that every self-mapping is weakly compatible with identity map, i.e., \(I_{x}\), we can get the following corollary.

### Corollary 3.2

*Let*\((X,M,\ast )\)*be a complete fuzzy cone metric space in which**M**is triangular and the mappings*\(F_{1},F_{2}:X\to X\)*satisfy*

*for all*\(x,y\in X, s\gg \vartheta \), *and*\(\beta ,\gamma ,\delta \in [0,\infty )\)*with*\(\beta +2\gamma +2\delta <1\). *Then*\(F_{1}\)*and*\(F_{2}\)*have a unique common fixed point in**X*. *Moreover*, *the fixed point of*\(F_{1}\)*is to be a fixed point of*\(F_{2}\)*and conversely*.

### Example 3.3

Let \(X=[0,1]\), ∗ be a continuous *t*-norm, and \(M:X^{2}\times (0,\infty )\to [0,1]\) be written as

\(\forall x,y\in X\) and \(s>0\). Then easily one can verify that *M* is triangular and \((X,M,\ast )\) is a complete *FCM*-space. Now we can define the mappings \(F_{1},F_{2},\ell :X\to X\) as

for every \(z\in X\). Then from (3.1) we have that

Hence all the conditions of Theorem 3.1 are satisfied with \(\beta =1/2\), \(\gamma =2/15\), and \(\delta =1/9\). Thus, \(F_{1}\), \(F_{2}\), and *ℓ* have a unique common fixed point in *X*, that is, 0.

## Quasi-contraction results in *FCM*-spaces

### Definition 4.1

Let \((X,M,\ast )\) be an *FCM*-space, and let *ℓ*, \(F_{1}\) be two self-maps on *X*. Then \(F_{1}\) is called a fuzzy cone quasi-contraction (resp; *ℓ*-quasi-contraction) if, for some \(q_{c}\in [0,1)\), for all \(x,y\in X\) and \(s\gg \vartheta \), there exists

such that

### Theorem 4.2

*Let*\(F_{1},\ell :X\to X\)*be two self*-*maps and**M**be triangular in a complete**FCM*-*space*\((X,M,\ast )\)*such that*\(F_{1}(X)\subset \ell (X)\)*and*\(\ell (X)\)*is closed*. *If*\(F_{1}\)*is an**ℓ*-*quasi*-*contraction with constant*\(q_{c}\in [0,1)\), *then**ℓ**and*\(F_{1}\)*have a unique point of coincidence*. *Moreover*, *if a pair*\((\ell , F_{1})\)*is occasionally weakly compatible*, *then*\(F_{1}\)*and**ℓ**have a unique common fixed point in**X*.

### Proof

Fix \(x_{0}\in X\) and use the condition \(F_{1}(X)\subset \ell (X)\). We construct a sequence \((y_{i})\) in *X* such that

Now, we have to show that \((y_{i})\) is a Cauchy sequence. First, we prove that

for all \(i\geq 1\) and \(s\gg \vartheta \). Indeed,

where

Then we may have the following four cases:

- (i)
First,

$$\begin{aligned} \frac{1}{M(y_{i},y_{i+1},s)}-1&\leq q_{c} \biggl( \frac{1}{M(y_{i-1},y_{i},s)}-1 \biggr) \\ &\leq \frac{q_{c}}{1-q_{c}} \biggl( \frac{1}{M(y_{i-1},y_{i},s)} -1 \biggr), \quad \text{for } s\gg \vartheta . \end{aligned}$$Thus (4.4) holds as \(q_{c}< q_{c}/(1-q_{c})\) since \(q_{c}\in [0,1)\).

- (ii)
Second, by using the

*M*triangular property, we have$$\begin{aligned} \frac{1}{M(y_{i},y_{i+1},s)}-1&\leq q_{c} \biggl( \frac{1}{M(y_{i-1},y_{i+1},s)}-1 \biggr) \\ &\leq q_{c} \biggl(\frac{1}{M(y_{i-1},y_{i},s)}-1+ \frac{1}{M(y_{i},y_{i+1},s)}-1 \biggr) \\ &\leq \frac{q_{c}}{1-q_{c}} \biggl(\frac{1}{M(y_{i-1},y_{i},s)} -1 \biggr),\quad \text{for } s\gg \vartheta . \end{aligned}$$It follows that (4.4) holds.

- (iii)
Third,

$$ \frac{1}{M(y_{i},y_{i+1},s)}-1\leq q_{c}.0, \quad \text{which implies that } M(y_{i},y_{i+1},s)=1 \text{ for } s\gg \vartheta . $$Hence (4.4) holds.

- (iv)
Fourth,

$$\begin{aligned}& \frac{1}{M(y_{i},y_{i+1},s)}-1\leq q_{c} \biggl( \frac{1}{M(y_{i},y_{i+1},s)}-1 \biggr), \\& \text{which implies } M(y_{i},y_{i+1},s)=1 \text{ for } s\gg \vartheta . \end{aligned}$$In this case, immediately (4.4) follows since \(q_{c}\in [0,1)\).

Now, we may assume that \(\delta =\frac{q_{c}}{1-q_{c}}<1\), then we have that

In view of (4.4),

for all \(i\geq 1\) and \(s\gg \vartheta \), which shows that \((y_{i})\) is a fuzzy cone contractive sequence in *X* such that

Since *M* is triangular, then for all \(j>i\geq i_{0}\),

which shows that \((y_{i})\) is a Cauchy sequence in *X*. Since \((X,M,\ast )\) is complete and \(\ell (X)\) is closed, \(\exists v\in X\) such that \(y_{i}=F_{1}x_{i}=\ell x_{i+1}\to \ell v\) as \(i\to \infty \), i.e.,

Now we have to show that \(\ell v=F_{1}v\). By using the triangularity of *M*, we have

By the definition of *ℓ*-quasi-contraction, we have that

where

This implies

for \(s\gg \vartheta \). Then we have the following two cases:

*Case* i: If \(\mathcal{U}_{i}\to 1\) as \(i\to \infty \). Then from (4.8), (4.9), and (4.10), we get that \(M(\ell v,F_{1}v,s)=1\) as \(i\to \infty \) for \(s\gg \vartheta \). That is, \(\ell v=F_{1}v=u\).

*Case* ii: If \(\mathcal{U}_{i}\to M(\ell v,F_{1}v,s)\) as \(i\to \infty \). Then from (4.10) we have that

Now, this together with (4.8) and (4.9) gives,

Since \(q_{c}<1\), therefore \(M(\ell v,F_{1}v,s)=1\), i.e., \(\ell v=F_{1}v=u\). Thus from both cases we get that \(\ell v=F_{1}v=u\). Hence, the same as in Theorem 3.1, *v* is the coincidence point of \((\ell ,F_{1})\) and *u* is its coincidence point in *X*. The uniqueness of the coincidence point can be shown by the standard way. By using Lemma 2.13, one can readily obtain that, when \((\ell ,F_{1})\) is occasionally weakly compatible, then *u* is a unique common fixed point of *ℓ* and \(F_{1}\) in *X*. □

### Theorem 4.3

*Let**ℓ**be a self*-*map on**X**and**M**be triangular in a complete**FCM*-*space*\((X,M,\ast )\)*such that*\(\ell ^{2}\)*is continuous*. *Let the self*-*map*\(F_{1}:X\to X\)*that commutes with**ℓ*. *Further*, *we assume that*\(F_{1}\)*and**ℓ**satisfy*

*and let*\(F_{1}\)*be an**ℓ*-*quasi*-*contraction*. *Then*\(F_{1}\)*and**ℓ**have a unique common fixed point in X*.

### Proof

By condition (4.11), starting with fix \(x_{0}\in \ell (X)\), define a sequence \((x_{i})\) in *X* such that

as in Theorem 4.2. Now

The same as in Theorem 4.2, we can get that \((v_{i})\) is a Cauchy sequence and convergent to some point \(v\in X\) such that

Further, we have to show that \(\ell ^{2}v=F_{1}\ell v\). Since,

by the continuity of \(\ell ^{2}\), it follows that

Now, by the triangular property of *M*, we have

and

where

Now, by using (4.13) for \(s\gg \vartheta \), we can get the following:

Equation (4.16) can be written as

as \(i\to \infty \). Then we have the following two cases:

*Case* i: If \(\mathcal{U}_{i}\to 1\) as \(i\to \infty \), then from (4.13), (4.14), and (4.15), we can get \(M(\ell ^{2}v,F_{1}\ell v, s)=1\), for \(s\gg \vartheta \). This implies that \(F_{1}\ell v=\ell ^{2}v\).

*Case* ii: If \(\mathcal{U}_{i}\to M(\ell ^{2}v,F_{1}\ell v,s)\) as \(i\to \infty\), for \(s\gg \vartheta \). Then we have

This together with (4.13) and (4.14) leads to

Since \(0\le q_{c}<1\), this implies that \(M(\ell ^{2}v,F_{1}\ell v,s)=1\), that is, \(F_{1}\ell v=\ell ^{2}v\). Thus from both cases we get that \(F_{1}\ell v=\ell ^{2}v\). This implies that *ℓv* is the common fixed point of *ℓ* and \(F_{1}\).

Now we prove the uniqueness. Assume that \(\ell v=w\) such that \(F_{1}w=\ell w\), and let \(w^{*}\) be the other common fixed point of the mappings *ℓ* and \(F_{1}\) such that \(F_{1}w^{*}=\ell w^{*}\). Then, by the standard way of *ℓ*-quasi-contraction, easily we can get that \(w=w^{*}\). This completes the proof. □

## Application

In this section, we present an integral type application, which is the new direction in *FCM*-spaces. For this purpose, we present the two Urysohn integral type equations, or shortly UITEs, to prove the existence result for common solution. Assume that \(X=[0,1]\), and let **E** be the real-valued functions on *X*. Then **E** is a vector space over \(\mathbb{R}\) under the following operations:

for all \(x,y\in {\mathbf{E}}\) and \(\beta \in \mathbb{R}\), and

∗ is a continuous *s*-norm and an *FM*-space \(M:\mathbf{E}\times \mathbf{E}\times (0,\infty )\to [0,1]\) can be expressed as

for all \(x,y\in \mathbf{E}\) and \(s>0\). Then easily we can show that *M* is triangular and \((\mathbf{E},M,\ast )\) is a complete *FCM*-space.

### Theorem 5.1

*The two UITEs are*

*where*\(l\in [0,1]\)*and*\(x,y,g\in \mathbf{E}\).

*Assume that*\(K_{1},K_{2}:[0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R}\)*are such that*\(A_{x},B_{y}\in \mathbf{E}\)*for every*\(x,y\in \mathbf{E}\), *where*

*where*\(l\in [0,1]\). *If there exists*\(\lambda \in (0,1)\)*such that*, *for all*\(x,y\in X\),

*where*

*Then the two UITEs* (5.1) *have a unique common solution*.

### Proof

Define the mappings \(F_{1},F_{2},\ell : \mathbf{E}\to \mathbf{E}\):

If

then

\(\forall x,y\in \mathbf{E}\), by Theorem 3.1 with \(\lambda =\beta \) and \(\gamma =\delta =0\) in Theorem 3.1. Then the two UITEs (5.1) have a unique common solution. If

then

\(\forall x,y\in \mathbf{E}\), by Theorem 3.1 with \(\lambda =\gamma \) and \(\beta =\delta =0\). Then the two UITEs (5.1) have a unique common solution. Again, if

then

\(\forall x,y\in \mathbf{E}\), by Theorem 3.1 with \(\lambda =\delta \) and \(\beta =\gamma =0\). Then from the two UITEs (5.1), we have a unique common solution. □

Now, we present a special type of example for UITEs.

### Example 5.2

Let \(X=[0,1]\) and the following integral equation be of the form

The problem system of equations (5.4) is a special kind of problem system of equations (5.1), where \(g(l)=\frac{l}{3}\) and \(l\in [0,1]\), and

Then we have

where \(N(x,y)=\|x(v)-y(v)\|\). Now, we have to show that \(\|A_{x}(l)-B_{y}(l)\|\leq \lambda N(x,y)\), from the system of equations (5.2), we have

Hence, all the conditions of Theorem 5.1 with \(\lambda =\frac{1}{3}<1\) hold. The problem system of equations (5.4) has a unique common solution by using Theorem 5.1.

## Conclusion

We defined weakly compatible self-mappings in fuzzy cone metric spaces and proved some coincidence point and common fixed point theorems under the fuzzy cone contraction condition without the assumption that fuzzy cone contractive sequences are Cauchy by using the “*M* triangular condition”. This change, to use “*M* triangular condition” to weaken the “fuzzy cone contractive sequences are Cauchy”, is expected to bring a wider range of applications of fixed point theorems in fuzzy cone metric spaces. We also gave the concept of quasi-contraction and proved some common fixed point theorems in fuzzy cone metric spaces. Moreover, we presented an application of the two Urysohn integral type equations for common solution to support our result. We also presented some illustrative examples to support our theoretical work.

## References

- 1.
Huang, L., Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl.

**332**, 1468–1476 (2007) - 2.
Abbas, M., Jungck, G.: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. J. Math. Anal. Appl.

**341**(1), 416–420 (2008) - 3.
Abbas, M., Rhoades, B., Nazir, T.: Common fixed points for four maps in cone metric spaces. Appl. Math. Comput.

**216**, 80–86 (2010) - 4.
Altun, I., Damjanovic, B., Djoric, D.: Fixed point and common fixed point theorems on ordered cone metric spaces. Appl. Math. Lett.

**23**, 310–316 (2010) - 5.
Altun, I., Durmaz, G.: Some fixed point theorems on ordered cone metric space. Rend. Circ. Mat. Palermo

**58**, 319–325 (2009) - 6.
Ilic, D., Rakocevic, V.: Common fixed point for maps on cone metric space. J. Math. Anal. Appl.

**341**, 876–882 (2008) - 7.
Ilic, D., Rakocevic, V.: Quasi-contraction on a cone metric space. Appl. Math. Lett.

**22**, 728–731 (2009) - 8.
Jungck, G., Radenovic, S., Radojevic, S., Rakocevic, V.: Common fixed point theorems for weakly compatible pairs on cone metric spaces. Fixed Point Theory Appl.

**2009**, Article ID 643840 (2009) - 9.
Kadelburg, Z., Radenovic, S., Rosic, B.: Strict contractive conditions and common fixed point theorems in cone metric spaces. Fixed Point Theory Appl.

**2019**, Article ID 173838 (2009) - 10.
Radenovic, S., Rhoades, B.E.: Fixed point theorems for two non-self maps in cone metric spaces. Comput. Math. Appl.

**57**, 1701–1707 (2009) - 11.
Vetro, P.: Common fixed points in cone metric spaces. Rend. Circ. Mat. Palermo, Ser. II

**56**(3), 464–468 (2007) - 12.
Zadeh, L.A.: Fuzzy sets. Inf. Control

**8**, 338–353 (1965) - 13.
Kramosil, O., Michalek, J.: Fuzzy metric and statistical metric spaces. Kybernetika

**11**, 336–344 (1975) - 14.
George, A., Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy Sets Syst.

**64**, 395–399 (1994) - 15.
Di Bari, C., Vetro, C.: Fixed points, attractors and weak fuzzy contractive mappings on fuzzy metric spaces. J. Fuzzy Math.

**13**, 973–982 (2005) - 16.
Grabiec, M.: Fixed point in fuzzy metric spaces. Fuzzy Sets Syst.

**27**, 385–389 (1988) - 17.
Hadzic, O., Pap, E.: A fixed point theorem for multivalued mappings in probabilistic metric spaces and an application in fuzzy metric spaces. Fuzzy Sets Syst.

**127**, 333–344 (2002) - 18.
Kiani, F., Amini-Haradi, A.: Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces. Fixed Point Theory Appl.

**2011**, 94 (2011) - 19.
Sadeghi, Z., Vaezpour, S.M., Park, C., Saadati, R., Vetro, C.: Set-valued mappings in partially ordered fuzzy metric spaces. J. Inequal. Appl.

**2014**, 157 (2014) - 20.
Oner, T., Kandemire, M.B., Tanay, B.: Fuzzy cone metric spaces. J. Nonlinear Sci. Appl.

**8**, 610–616 (2015) - 21.
Oner, T.: Some topological properties of fuzzy cone metric spaces. J. Nonlinear Sci. Appl.

**9**, 799–805 (2016) - 22.
Oner, T.: On some results in fuzzy cone metric spaces. Int. J. Adv. Comput. Eng. Netw.

**4**(2), 37–39 (2016) - 23.
Oner, T.: On the metrizibility of fuzzy cone metric spaces. Int. J. Manag. Appl. Sci.

**2**(5), 133–135 (2016) - 24.
Priyobarta, N., Rohen, Y., Upadhyay, B.B.: Some fixed point results in fuzzy cone metric spaces. Int. J. Pure Appl. Math.

**109**(3), 573–582 (2016) - 25.
Rehman, S.U., Li, H.X.: Fixed point theorems in fuzzy cone metric spaces. J. Nonlinear Sci. Appl.

**10**, 5763–5769 (2017) - 26.
Rehman, S.U., Li, Y., Jabeen, S., Mahmood, T.: Common fixed point theorems for a pair of self-mappings in fuzzy cone metric spaces Abstr. Appl. Anal.

**2019**, Article ID 2841606 (2019) - 27.
Schweizer, B., Sklar, A.: Statistical metric spaces. Pac. J. Math.

**10**(1960), 313–334 (1960) - 28.
Jungck, G., Rhoades, B.E.: Fixed point theorems for occasionally weakly compatible mappings. Fixed Point Theory

**7**(2), 287–296 (2006)

### Acknowledgements

The authors would like to express their gratitude to the anonymous referee for very helpful suggestions and comments which led to improvements of our original manuscript.

### Availability of data and materials

Data sharing is not applicable to this article as no dataset were generated or analysed during the current study.

## Funding

This work is supported by the Fundamental Research Funds for the Central Universities, the National Natural Science Foundation of China (No.11201019), the International Cooperation Project No. 2010DFR00700, Fundamental Research of Civil Aircraft No. MJ-F-2012-04 and Beijing Natural Science Foundation (No. 1192012, Z180005).

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### Contributions

All the authors have equally contributed to the final manuscript. All authors read and approved the final manuscript.

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## Ethics declarations

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The authors declare that they have no competing interests.

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### Cite this article

Jabeen, S., Ur Rehman, S., Zheng, Z. *et al.* Weakly compatible and quasi-contraction results in fuzzy cone metric spaces with application to the Urysohn type integral equations.
*Adv Differ Equ* **2020, **280 (2020). https://doi.org/10.1186/s13662-020-02743-5

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### MSC

- 47H10
- 54H25

### Keywords

- Coincidence point
- Common fixed point
- Fuzzy cone metric space
- Weakly compatible mappings
- Contraction conditions