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

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:

$$\begin{aligned} &x(l)= \int _{0}^{1}K_{1}\bigl(l,v,x(v) \bigr)\,dv+g(l), \\ &y(l)= \int _{0}^{1}K_{2}\bigl(l,v,y(v) \bigr)\,dv+g(l), \end{aligned}$$

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 [311]) 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 [1519]).

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[2126] 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. (1)

    is commutative, associative, and continuous.

  2. (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, Lukasiewicz, and minimum are defined respectively as follows (see [27]):

$$ \beta \ast \delta =\beta \delta ,\qquad \beta \ast \delta = \max \{\beta + \delta -1,0\},\quad \text{and}\quad \beta \ast \delta = \min\{\beta , \delta \}. $$

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:

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (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. (1)

    P is closed, nonempty and \(P\ne \{\vartheta \}\), where ϑ is the zero element of E.

  2. (2)

    If \(x, y \in P\) and \(\beta ,\delta \in [0, \infty )\), then \(\beta x + \delta y \in P\).

  3. (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. (1)

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

  2. (2)

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

  3. (3)

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

  4. (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 \(\beta *\delta =\beta \delta \) [2022].

Definition 2.6

([20])

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

  1. (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 \(i\to \infty \).

  2. (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. \(M(x_{i},x_{j},s)>1-r\), \(\forall i,j\ge i_{1}\).

  3. (iii)

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

  4. (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

$$ \frac{1}{M(x,z,s)}-1\le \biggl(\frac{1}{M(x,y,s)}-1 \biggr) + \biggl( \frac{1}{M(y,z,s)}-1 \biggr), $$

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

Lemma 2.8

([20])

Let\(x\in X\)in anFCM-space\((X,M,\ast )\)and\((x_{i})\)be a sequence inX. Then\((x_{i})\)converges toxif 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

$$ \frac{1}{M(F_{1}x,F_{1}y,s)}-1\le \beta \biggl(\frac{1}{M(x,y,s)}-1 \biggr) $$
(2.1)

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 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 , and u is called a point of coincidence of \(F_{1}\) and . The self-mappings \(F_{1}\) and 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\(F_{1}\)andbe weakly compatible self-maps of a setX. If\(F_{1}\)andhave a unique point of coincidence\(u=F_{1}v=\ell v\), thenuis the unique common fixed point of\(F_{1}\)and.

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\(F_{1}\)andbe occasionally weakly compatible self-maps of a setX. If\(F_{1}\)andhave a unique point of coincidence, \(F_{1}v=\ell v=u\), thenuis a unique common fixed point ofand\(F_{1}\).

“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 andMbe triangular in a completeFCM-space\((X,M,\ast )\)satisfying\(\forall x,y\in X\),

$$\begin{aligned} \frac{1}{M(F_{1}x,F_{2}y,s)}-1&\leq \beta \biggl( \frac{1}{M(\ell x,\ell y,s)}-1 \biggr) +\gamma \biggl( \frac{1}{M(\ell x,F_{1}x,s)}-1 + \frac{1}{M(\ell y,F_{2}y,s)}-1 \biggr) \\ &\quad {}+\delta \biggl(\frac{1}{M(\ell y,F_{1}x,s)}-1+ \frac{1}{M(\ell x,F_{2}y,s)}-1 \biggr) \end{aligned}$$
(3.1)

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 ofX, then\(F_{1}\), \(F_{2}\), andhave a unique point of coincidence. Moreover, if\((F_{1}, \ell )\)and\((F_{2}, \ell )\)are weakly compatible. Then\(F_{1}\), \(F_{2}\), andhave a unique common fixed point inX.

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

$$ \ell x_{2i+1}=F_{1} x_{2i}\quad \text{and}\quad \ell x_{2i+2}=F_{2} x_{2i+1}, \quad \text{for all } i\geq 0. $$

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

$$\begin{aligned} &\frac{1}{M(\ell x_{2i+1},\ell x_{2i+2},s)}-1 = \frac{1}{M(F_{1} x_{2i},F_{2} x_{2i+1},s)}-1 \\ &\quad \leq \beta \biggl(\frac{1}{M(\ell x_{2i},\ell x_{2i+1},s)}-1 \biggr) + \gamma \biggl( \frac{1}{M(\ell x_{2i},F_{1} x_{2i},s)}-1 + \frac{1}{M(\ell x_{2i+1},F_{2} x_{2i+1},s)}-1 \biggr) \\ &\qquad{} +\delta \biggl(\frac{1}{M(\ell x_{2i+1},F_{1}x_{2i},s)}-1+ \frac{1}{M(\ell x_{2i},F_{2} x_{2i+1},s)}-1 \biggr) \\ &\quad \leq \beta \biggl(\frac{1}{M(\ell x_{2i},\ell x_{2i+1},s)}-1 \biggr) + \gamma \biggl( \frac{1}{M(\ell x_{2i},\ell x_{2i+1},s)}-1 + \frac{1}{M(\ell x_{2i+1},\ell x_{2i+2},s)}-1 \biggr) \\ &\qquad {}+\delta \biggl(\frac{1}{M(\ell x_{2i},\ell x_{2i+1},s)}-1+ \frac{1}{M(\ell x_{2i+1},\ell x_{2i+2},s)}-1 \biggr). \end{aligned}$$

Then

$$ \frac{1}{M(\ell x_{2i+1},\ell x_{2i+2},s)}-1\le \alpha \biggl( \frac{1}{M(\ell x_{2i},\ell x_{2i+1},s)}-1 \biggr),\quad \text{for } s \gg \vartheta , $$

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

$$\begin{aligned} &\frac{1}{M(\ell x_{2i+2},\ell x_{2i+3},s)}-1 \\ &\quad \le \alpha \biggl(\frac{1}{M(\ell x_{2i+1},\ell x_{2i+2},s)}-1 \biggr) \leq \alpha ^{2} \biggl(\frac{1}{M(\ell x_{2i},\ell x_{2i+1},s)}-1 \biggr)\leq \cdots \\ &\quad \leq \alpha ^{2i+2} \biggl( \frac{1}{M(\ell x_{0},\ell x_{1},s)}-1 \biggr), \end{aligned}$$

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

$$ \lim_{i\to \infty }M(\ell x_{i},\ell x_{i+1},s)=1 \quad \text{for } s \gg \vartheta . $$
(3.2)

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

$$\begin{aligned} &\frac{1}{M(\ell x_{i},\ell x_{j},s)}-1 \\ &\quad \leq \biggl(\frac{1}{M(\ell x_{i},\ell x_{i+1},s)}-1 \biggr) + \biggl( \frac{1}{M(\ell x_{i+1},\ell x_{i+2},s)}-1 \biggr) +\cdots + \biggl( \frac{1}{M(\ell x_{j-1},\ell x_{j},s)}-1 \biggr) \\ &\quad \leq \bigl(\alpha ^{i}+\alpha ^{i+1}+\cdots +\alpha ^{j-1}\bigr) \biggl( \frac{1}{M(\ell x_{1},\ell x_{0},s)}-1 \biggr) \\ &\quad \leq \frac{\alpha ^{i}}{1-\alpha } \biggl( \frac{1}{M(\ell x_{1},\ell x_{0},s)}-1 \biggr) \\ &\quad \to 0 \quad \text{as } i\to \infty , \end{aligned}$$

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.,

$$ \lim_{i\to \infty }M(u,\ell x_{i},s)=1\quad \text{for } s\gg \vartheta . $$
(3.3)

Since M is triangular, we have

$$ \frac{1}{M(\ell v,F_{1} v,s)}-1\leq \biggl( \frac{1}{M(\ell v,\ell x_{2i+2},s)}-1 \biggr)+ \biggl( \frac{1}{M(\ell x_{2i+2},F_{1}v,s)}-1 \biggr), \quad \text{for } s\gg \vartheta . $$
(3.4)

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

$$\begin{aligned} &\frac{1}{M(\ell x_{2i+2},F_{1}v,s)}-1=\frac{1}{M(F_{2}x_{2i+1},F_{1}v,s)}-1 \\ &\quad \leq \beta \biggl(\frac{1}{M(\ell v,\ell x_{2i+1},s)}-1 \biggr) + \gamma \biggl( \frac{1}{M(\ell v,F_{1}v,s)}-1+ \frac{1}{M(\ell x_{2i+1},F_{2}x_{2i+1},s)}-1 \biggr) \\ &\qquad {}+\delta \biggl(\frac{1}{M(\ell x_{2i+1},F_{1}v,s)}-1+ \frac{1}{M(\ell v,F_{2}x_{2i+1},s)}-1 \biggr) \\ &\quad = \beta \biggl(\frac{1}{M(\ell v,\ell x_{2i+1},s)}-1 \biggr) +\gamma \biggl( \frac{1}{M(\ell v,F_{1}v,s)}-1+ \frac{1}{M(\ell x_{2i+1},\ell x_{2i+2},s)}-1 \biggr) \\ &\qquad {}+\delta \biggl(\frac{1}{M(\ell x_{2i+1},F_{1}v,s)}-1+ \frac{1}{M(\ell v,\ell x_{2i+2},s)}-1 \biggr) \\ &\quad \to (\gamma +\delta ) \biggl(\frac{1}{M(u,F_{1}v,s)}-1 \biggr) \quad \text{as } i \to \infty . \end{aligned}$$

Then,

$$ \limsup_{i\to \infty } \biggl(\frac{1}{M(\ell x_{2i+2},F_{1}v,s)}-1 \biggr)\leq ( \gamma +\delta ) \biggl(\frac{1}{M(u,F_{1}v,s)}-1 \biggr), \quad \text{for } s\gg \vartheta . $$

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

$$ \biggl(\frac{1}{M(u,F_{1}v,s)}-1 \biggr)\leq (\gamma +\delta ) \biggl( \frac{1}{M(u,F_{1}v,s)}-1 \biggr),\quad \text{for } s\gg \vartheta . $$

\(\gamma +\delta <1\), 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 , \(F_{1}\), 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 . Let \(u^{*}\) be the other point in X such that

$$ u^{*}=\ell v^{*}=F_{1}v^{*}=F_{2}v^{*} $$

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

$$\begin{aligned} \frac{1}{M(u,u^{*},s)}-1&=\frac{1}{M(F_{1}v,F_{2}v^{*},s)}-1 \\ &\leq \beta \biggl(\frac{1}{M(\ell v,\ell v^{*},s)}-1 \biggr) +\gamma \biggl( \frac{1}{M(\ell v,F_{1}v,s)}-1+ \frac{1}{M(\ell v^{*},F_{2}v^{*},s)}-1 \biggr) \\ &\quad {}+\delta \biggl(\frac{1}{M(\ell v^{*},F_{1}v,s)}-1+ \frac{1}{M(\ell v,F_{2}v^{*},s)}-1 \biggr) \\ &= \beta \biggl(\frac{1}{M(u,u^{*},s)}-1 \biggr)+\gamma \biggl( \frac{1}{M(u,u,s)}-1+\frac{1}{M(u^{*},u^{*},s)}-1 \biggr) \\ &\quad {}+\delta \biggl(\frac{1}{M(u^{*},u,s)}-1+\frac{1}{M(u,u^{*},s)}-1 \biggr) \\ &=(\beta +2\delta ) \biggl(\frac{1}{M(u^{*},u,s)}-1 \biggr), \end{aligned}$$

\(\beta +2\delta <1\), 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 , 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 whichMis triangular and the mappings\(F_{1},F_{2}:X\to X\)satisfy

$$\begin{aligned} \frac{1}{M(F_{1}x,F_{2}y,s)}-1&\leq \beta \biggl(\frac{1}{M(x,y,s)}-1 \biggr)+\gamma \biggl(\frac{1}{M(x,F_{1}x,s)}-1+\frac{1}{M(y,F_{2}y,s)}-1 \biggr) \\ &\quad {}+\delta \biggl(\frac{1}{M(y,F_{1}x,s)}-1+\frac{1}{M(x,F_{2}y,s)}-1 \biggr) \end{aligned}$$
(3.5)

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 inX. 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

$$ M(x,y,s)=\frac{s}{s+ \vert x-y \vert } $$

\(\forall x,y\in X\) 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

$$ F_{1}z=F_{2}z=\frac{z}{z+6}\quad \text{and}\quad \ell z=\frac{z}{3} $$

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

$$\begin{aligned} \frac{1}{M(F_{1}x,F_{2}y,s)}-1&= \biggl\vert \frac{F_{1}x-F_{2}y}{s} \biggr\vert = \frac{1}{s} \biggl\vert \frac{x}{x+6}-\frac{y}{y+6} \biggr\vert \\ &=\frac{1}{s} \biggl\vert \frac{x(y+6)-y(x+6)}{(x+6)(y+6)} \biggr\vert \\ &\leq \frac{1}{s} \biggl\vert \frac{6x-6y}{36} \biggr\vert = \frac{1}{2} \biggl( \frac{1}{M(\ell x,\ell y,s)}-1 \biggr) \\ &\le \beta \biggl(\frac{1}{M(\ell x,\ell y,s)}-1 \biggr) +\gamma \biggl( \frac{1}{M(\ell x,F_{1}x,s)}-1+\frac{1}{M(\ell y,F_{2}y,s)}-1 \biggr) \\ &\quad {}+\delta \biggl(\frac{1}{M(\ell y,F_{1}x,s)}-1+ \frac{1}{M(\ell x,F_{2}y,s)}-1 \biggr) . \end{aligned}$$

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

$$\begin{aligned}& \mathcal{U}\in C(x,y,s)=\left \{ \textstyle\begin{array}{l} M(x,y,s),M(x,F_{1}x,s),M(x,F_{1}y,s), \\ M(y,F_{1}x,s),M(y,F_{1}y,s) \end{array}\displaystyle \right \} \end{aligned}$$
(4.1)
$$\begin{aligned}& \bigl(\text{resp}; \mathcal{U}\in C(\ell ; x, y, s) \bigr)=\left \{ \textstyle\begin{array}{l} M(\ell x,\ell y,s),M(\ell x,F_{1}x,s),M(\ell x,F_{1}y,s), \\ M(\ell y,F_{1}x,s), M(\ell y,F_{1}y,s) \end{array}\displaystyle \right \} \end{aligned}$$
(4.2)

such that

$$ \frac{1}{M(F_{1}x,F_{1}y,s)}-1\leq q_{c} \biggl( \frac{1}{\mathcal{U}}-1 \biggr). $$
(4.3)

Theorem 4.2

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

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

$$ y_{i}=F_{1}x_{i}=\ell x_{i+1}\quad \text{for all } i\geq 0. $$

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

$$ \frac{1}{M(y_{i},y_{i+1},s)}-1\leq \frac{q_{c}}{1-q_{c}} \biggl( \frac{1}{M(y_{i-1},y_{i},s)}-1 \biggr) $$
(4.4)

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

$$ \frac{1}{M(y_{i},y_{i+1},s)}-1=\frac{1}{M(F_{1}x_{i},F_{1}x_{i+1},s)}-1 \leq q_{c} \biggl(\frac{1}{\mathcal{U}_{i}}-1 \biggr), $$
(4.5)

where

$$\begin{aligned} \mathcal{U}_{i} &\in \left \{ \textstyle\begin{array}{l} M(\ell x_{i},\ell x_{i+1},s),M(\ell x_{i},F_{1}x_{i},s),M(\ell x_{i},F_{1}x_{i+1},s), \\ M(\ell x_{i+1},F_{1}x_{i},s), M(\ell x_{i+1},F_{1}x_{i+1},s) \end{array}\displaystyle \right \} \\ &=\left \{ \textstyle\begin{array}{l} M(y_{i-1},y_{i},s),M(y_{i-1},y_{i},s),M(y_{i-1},y_{i+1},s), \\ M(y_{i},y_{i},s), M(y_{i},y_{i+1},s) \end{array}\displaystyle \right \} \\ &=\bigl\{ M(y_{i-1},y_{i},s),M(y_{i-1},y_{i+1},s),1,M(y_{i},y_{i+1},s) \bigr\} . \end{aligned}$$
(4.6)

Then we may have the following four cases:

  1. (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 \(q_{c}\in [0,1)\).

  2. (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.

  3. (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.

  4. (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

$$ \frac{1}{M(y_{i},y_{i+1},s)}-1\leq \delta \biggl( \frac{1}{M(y_{i-1},y_{i},s)} \biggr), \quad \text{for } s\gg \vartheta . $$

In view of (4.4),

$$ \frac{1}{M(y_{i},y_{i+1},s)}-1\leq \delta \biggl( \frac{1}{M(y_{i-1},y_{i},s)} \biggr)\leq \cdots \leq \delta ^{i} \biggl( \frac{1}{M(y_{0},y_{1},s)} \biggr) $$

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

$$ \lim_{i\to \infty }M(y_{i},y_{i+1},s)=1 \quad \text{for } s\gg \vartheta . $$
(4.7)

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

$$\begin{aligned} &\frac{1}{M(y_{i},y_{j},s)}-1 \\ &\quad \leq \biggl(\frac{1}{M(y_{i},y_{i+1},s)}-1 \biggr)+ \biggl( \frac{1}{M(y_{i+1},y_{i+2},s)}-1 \biggr) +\cdots + \biggl( \frac{1}{M(y_{j-1},y_{j},s)}-1 \biggr) \\ &\quad \leq \bigl(\delta ^{i}+\delta ^{i+1}+\cdots +\delta ^{j-1} \bigr) \biggl(\frac{1}{M(y_{0},y_{1},s)}-1 \biggr) \\ &\quad \leq \frac{\delta ^{i}}{1-\delta } \biggl( \frac{1}{M(y_{0},y_{1},s)}-1 \biggr) \\ &\quad \to 0\quad \text{as } i\to \infty , \end{aligned}$$

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.,

$$ \lim_{i\to \infty }M(y_{i},\ell v,s)=1\quad \text{for } s\gg \vartheta . $$
(4.8)

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

$$ \frac{1}{M(\ell v,F_{1}v,s)}-1\leq \biggl(\frac{1}{M(\ell v,y_{i},s)}-1 \biggr) + \biggl(\frac{1}{M(y_{i},F_{1}v,s)}-1 \biggr),\quad \text{for } s \gg \vartheta . $$
(4.9)

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

$$ \frac{1}{M(y_{i},F_{1}v,s)}-1=\frac{1}{M(F_{1}x_{i},F_{1}v,s)}-1\leq q_{c} \biggl(\frac{1}{\mathcal{U}_{i}}-1 \biggr), \quad \text{for } s\gg \vartheta , $$
(4.10)

where

$$\begin{aligned} \mathcal{U}_{i} &\in \left \{ \textstyle\begin{array}{l} M(\ell x_{i},\ell v,s),M(\ell x_{i},F_{1}x_{i},s),M(\ell x_{i},F_{1}v,s), \\ M(\ell v,F_{1}x_{i},s), M(\ell v,F_{1}v,s) \end{array}\displaystyle \right \} \\ &=\left \{ \textstyle\begin{array}{l} M(\ell x_{i},\ell v,s),M(\ell x_{i},\ell x_{i+1},s),M(\ell x_{i},F_{1}v,s), \\ M(\ell v,\ell x_{i+1},s), M(\ell v,F_{1}v,s) \end{array}\displaystyle \right \} \\ &\to \bigl\{ 1,1,M(\ell v,F_{1}v,s),1,M(\ell v,F_{1}v,s) \bigr\} \quad \text{as } i\to \infty . \end{aligned}$$

This implies

$$ \mathcal{U}_{i}\to \bigl\{ 1,M(\ell v,F_{1}v,s)\bigr\} \quad \text{as } i\to \infty $$

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

$$ \limsup_{i\to \infty } \biggl(\frac{1}{M(y_{i},F_{1}v,s)}-1 \biggr) \leq q_{c} \biggl(\frac{1}{M(\ell v,F_{1}v,s)}-1 \biggr), \quad \text{for } s\gg \vartheta . $$

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

$$ \frac{1}{M(\ell v,F_{1}v,s)}-1\leq q_{c} \biggl( \frac{1}{M(\ell v,F_{1}v,s)}-1 \biggr),\quad \text{for } s\gg \vartheta . $$

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 and \(F_{1}\) in X. □

Theorem 4.3

Letbe a self-map onXandMbe triangular in a completeFCM-space\((X,M,\ast )\)such that\(\ell ^{2}\)is continuous. Let the self-map\(F_{1}:X\to X\)that commutes with. Further, we assume that\(F_{1}\)andsatisfy

$$ F_{1}\ell (X)\subset \ell ^{2}(X), $$
(4.11)

and let\(F_{1}\)be an-quasi-contraction. Then\(F_{1}\)andhave 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

$$ y_{i}=F_{1}x_{i}=\ell x_{i+1}\quad \text{for } i\geq 0, $$

as in Theorem 4.2. Now

$$ \ell y_{i+1}=\ell F_{1}x_{i+1}=F_{1} \ell x_{i+1}=F_{1}y_{i}=v_{i}\quad \text{for } i\geq 1. $$

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

$$ \lim_{i\to \infty }M(\ell y_{i+1},v,s)=1 \quad \text{for } s \gg \vartheta . $$

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

$$ \lim_{i\to \infty }\ell y_{i}=\lim _{i\to \infty }\ell F_{1}x_{i}= \lim _{i\to \infty }F_{1}\ell x_{i}=\lim _{i\to \infty }F_{1}y_{i-1}= \lim _{i\to \infty }v_{i-1}=v, $$
(4.12)

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

$$ \lim_{i\to \infty }\ell ^{4}x_{i}= \lim_{i\to \infty }\ell ^{3}F_{1}x_{i-1}= \ell ^{2}v. $$
(4.13)

Now, by the triangular property of M, we have

$$ \begin{aligned}[b] \frac{1}{M(\ell ^{2}v,F_{1}\ell v,s)}-1&\leq \biggl( \frac{1}{M(\ell ^{2}v,\ell ^{3}F_{1}x_{i},s)}-1 \biggr) \\ &\quad {}+ \biggl( \frac{1}{M(\ell ^{3}F_{1}x_{i},F_{1}\ell v,s)}-1 \biggr),\quad \text{for } s\gg \vartheta \end{aligned} $$
(4.14)

and

$$ \frac{1}{M(\ell ^{3}F_{1}x_{i},F_{1}\ell v,s)}-1= \frac{1}{M(F_{1}\ell ^{3}x_{i},F_{1}\ell v,s)}-1\leq q_{c} \biggl( \frac{1}{\mathcal{U}_{i}}-1 \biggr), $$
(4.15)

where

$$ \mathcal{U}_{i}\in \left \{ \textstyle\begin{array}{l} M\bigl(\ell ^{4}x_{i},\ell ^{2}v,s\bigr),M\bigl(\ell ^{4}x_{i},F_{1}\ell ^{3}x_{i},s\bigr),M\bigl( \ell ^{2}v,F_{1}\ell v,s\bigr), \\ M\bigl(\ell ^{4}x_{i},F_{1}\ell v,s\bigr), M\bigl(\ell ^{2}v,F_{1}\ell ^{3}x_{i},s\bigr) \end{array}\displaystyle \right \} . $$
(4.16)

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

$$\begin{aligned}& M\bigl(\ell ^{4}x_{i},\ell ^{2}v,s\bigr)\to M\bigl(\ell ^{2}v,\ell ^{2}v,s\bigr)=1\quad \text{as } i \to \infty , \\& M\bigl(\ell ^{4}x_{i},F_{1}\ell ^{3}x_{i},s\bigr)\to M\bigl(\ell ^{2}v,\ell ^{2}v,s\bigr)=1 \quad \text{as } i\to \infty , \\& M\bigl(\ell ^{4}x_{i},F_{1}\ell v,s\bigr)\to M\bigl(\ell ^{2}v,F_{1}\ell v,s\bigr)\quad \text{as } i \to \infty , \\& M\bigl(\ell ^{2}v,F_{1}\ell ^{3}x_{i},s \bigr)\to M\bigl(\ell ^{2}v,\ell ^{2}v,s\bigr)=1 \quad \text{as } i\to \infty . \end{aligned}$$

Equation (4.16) can be written as

$$ \mathcal{U}_{i}\in \bigl\{ 1,1,M\bigl(\ell ^{2}v,F_{1} \ell v,s\bigr),M\bigl(\ell ^{2}v,F_{1} \ell v,s\bigr),1 \bigr\} =\bigl\{ 1,M\bigl(\ell ^{2}v,F_{1}\ell v,s\bigr) \bigr\} $$

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

$$ \limsup_{i\to \infty } \biggl( \frac{1}{M(\ell ^{3}F_{1}x_{i},F_{1}\ell v,s)}-1 \biggr)\leq q_{c} \biggl(\frac{1}{M(\ell ^{2}v,F_{1}\ell v,s)}-1 \biggr), \quad \text{for } s \gg \vartheta . $$

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

$$ \frac{1}{M(\ell ^{2}v,F_{1}\ell v,s)}-1\leq q_{c} \biggl( \frac{1}{M(\ell ^{2}v,F_{1}\ell v,s)}-1 \biggr),\quad \text{for } s\gg \vartheta . $$

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 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 and \(F_{1}\) such that \(F_{1}w^{*}=\ell w^{*}\). Then, by the standard way of -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:

$$ (x+y) (l)=x(l)+y(l),\qquad (\beta x) (l)=\beta x(l) $$

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

$$ P=\bigl\{ x\in \mathbf{E}| x(l)\geq 0, \forall l\in [0,1] \bigr\} . $$

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

$$ M(x,y,s)=\frac{s}{s+d(x,y)}, \quad \text{where } d(x,y)= \Vert x-y \Vert $$

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

$$ \begin{aligned} &x(l)= \int _{0}^{1}K_{1}\bigl(l,v,x(v) \bigr)\,dv+g(l), \\ &y(l)= \int _{0}^{1}K_{2}\bigl(l,v,y(v) \bigr)\,dv+g(l), \end{aligned} $$
(5.1)

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

$$ \begin{aligned} &A_{x}(l)= \int _{0}^{1}K_{1}\bigl(l,v,x(v) \bigr)\,dv+g(l), \\ &B_{y}(l)= \int _{0}^{1}K_{2}\bigl(l,v,y(v) \bigr)\,dv+g(l), \end{aligned} $$
(5.2)

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

$$ \Vert A_{x}-B_{y} \Vert \leq \lambda N(x,y), $$
(5.3)

where

$$ N(x,y)= \max \bigl\{ \Vert x-y \Vert , \bigl( \Vert A_{x}-x \Vert + \Vert B_{y}-y \Vert \bigr), \bigl( \Vert A_{x}-y \Vert + \Vert B_{y}-x \Vert \bigr) \bigr\} . $$

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

Proof

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

$$ \ell (x)=x, \qquad F_{1}(x)=A_{x} \quad \text{and}\quad F_{2}(y)=B_{y}. $$

If

$$ N(x,y)= \Vert x-y \Vert , $$

then

$$ \bigl\Vert F_{1}(x)-F_{2}(y) \bigr\Vert \leq \delta \Vert x-y \Vert , $$

\(\forall x,y\in \mathbf{E}\), 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

$$ N(x,y)= \Vert A_{x}-x \Vert + \Vert B_{y}-y \Vert , $$

then

$$ \bigl\Vert F_{1}(x)-F_{2}(y) \bigr\Vert \leq \lambda \bigl( \bigl\Vert F_{1}(x)-x \bigr\Vert + \bigl\Vert F_{2}(y)-y \bigr\Vert \bigr), $$

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

$$ N(x,y)= \Vert A_{x}-y \Vert + \Vert B_{y}-x \Vert , $$

then

$$ \bigl\Vert F_{1}(x)-F_{2}(y) \bigr\Vert \leq \lambda \bigl( \bigl\Vert F_{1}(x)-y \bigr\Vert + \bigl\Vert F_{2}(y)-x \bigr\Vert \bigr), $$

\(\forall x,y\in \mathbf{E}\), 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

$$ \begin{aligned} &x(l)= \int _{0}^{1}\frac{1}{3(l+1+x(v))}\,dv+ \frac{l}{3}, \\ &y(l)= \int _{0}^{1}\frac{1}{3(l+1+y(v))}\,dv+ \frac{l}{3}. \end{aligned} $$
(5.4)

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

$$ K_{i}\bigl(l,v,w_{i}(v)\bigr)=\frac{1}{3(l+1+w_{i}(v))},\quad \text{where } i=1,2. $$

Then we have

$$\begin{aligned} \bigl\Vert K_{1}\bigl(l,v,x(v)\bigr)-K_{2} \bigl(l,v,y(v)\bigr) \bigr\Vert &= \biggl\Vert \frac{1}{3(l+1+x(v))}- \frac{1}{3(l+1+y(v))} \biggr\Vert \\ &=\frac{1}{3} \biggl\Vert \frac{x(v)-y(v)}{(l+1+x(v))(l+1+y(v))} \biggr\Vert \\ &\leq \frac{1}{3} \bigl\Vert x(v)-y(v) \bigr\Vert \\ &=\frac{1}{3} N(x,y), \end{aligned}$$

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

$$\begin{aligned} \bigl\Vert A_{x}(l)-B_{y}(l) \bigr\Vert &= \biggl\Vert \int _{0}^{1}K_{1}\bigl(l,v,x(v) \bigr)\,dv- \int _{0}^{1}K_{2}\bigl(l,v,y(v) \bigr)\,dv \biggr\Vert \\ &= \int _{0}^{1} \bigl\Vert K_{1} \bigl(l,v,x(v)\bigr)-K_{2}\bigl(l,v,y(v)\bigr) \bigr\Vert \,dv \\ &\leq \int _{0}^{1}\frac{1}{3} \bigl\Vert x(v)-y(v) \bigr\Vert \,dv \\ &= \int _{0}^{1}\frac{1}{3}N(x,y)\,dv \\ &=\frac{1}{3}N(x,y) \int _{0}^{1}dv \\ &=\frac{1}{3}N(x,y). \end{aligned}$$

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.

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

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Correspondence to Saif Ur Rehman.

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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