An Application to Nonlinear Fractional Differential Equation via α-ΓF-Fuzzy Contractive Mappings in a Fuzzy Metric Space

Abstract

In this paper, we first introduce a new family of functions like an implicit function called $\Gamma$-functions. Furthermore, we introduce a new concept of $\alpha$-$\Gamma F$-fuzzy contractive mappings, which is weaker than the class of fuzzy F-contractive mappings. Then, the existence and uniqueness of the fixed point are established for a new type of fuzzy contractive mappings in the setting of fuzzy metric spaces. Moreover, some examples and an application to nonlinear fractional differential equation are given, and these show the importance of the introduced theorems in fuzzy settings.

1. Introduction

The notion of fuzzy metric spaces was first introduced by Kramosil et al. [1]. George et al. [2] modified the notion of Kramosil et al. [1] by obtaining Hausdorff topology in fuzzy metric spaces and said that every metric induces a fuzzy metric. Later, fixed point theory was first introduced by Grabic [3] in fuzzy metric spaces by extending Banach contraction conditions and Edelstein contraction conditions [4] in terms of fuzzy, in the sense of Kramosil et al. [1]. Grabic’s [3] fixed point results were based on strong conditions associated with completeness of the fuzzy metric spaces called G-completeness. After that, George and Veeramani weakened Grabic [3] conditions and introduced M-completeness of fuzzy metric spaces.

Later, Tirado [5], Gregori et al. [6] and Mihet [7] defined different classes of fuzzy contractive conditions. In 2012, Wardowski [8] introduced a new contraction called F-contraction in a metric space and proved some fixed point theorems in complete metric spaces. Recently, H. Hung et al. [9] introduced a new type of condition called fuzzy F-contraction in a fuzzy metric space. As compared to the F-contraction, this is much simpler and more straightforward as it contains only one condition—that is, the function F is strictly increasing and proves some fixed point theorems for fuzzy F-contraction conditions. On the other hand, Hussain et al. [10] introduced the concept of $\alpha$-$GF$-contraction conditions in a metric space as a generalization of F-contraction and obtained some interesting fixed point results.

In this paper, we first introduce a family of function $\Gamma$, such as an implicit function, and give $\alpha$-$\Gamma F$-fuzzy contractive conditions depending on the class of functions (for instance, $\Gamma$-function in fuzzy metric spaces). We give here some fixed point theorems using the concepts of $\alpha$-admissible and weaker conditions of continuity of the function. Finally, we obtain the fuzzy F-contraction theorem given by H. Hung et al. [9] as in the form of corollary of our main result, which proves that our generalization is fruitful. At the end, as an application of our result, we produce the existence of a solution of nonlinear fractional differential equations via the introduced fuzzy contractive conditions.

2. Preliminaries

We require some basic concepts before coming to the main results. Throughout the article, $\mathbb{N}$, ${\mathbb{R}}^{+}$ and $\mathbb{R}$ will denote the set of natural numbers, non-negative real numbers and real numbers, respectively.

Definition 1

([11]). A mapping $\ast :[0,1]\times [0,1]\to [0,1]$ is called a continuous triangular norm (t-norm for short) if ∗ satisfies the following conditions:

 1. 

∗ is commutative and associative—that is, $a\ast b=b\ast a$ and $a\ast (b\ast c)=(a\ast b)\ast c$, for all $a,b,c\in [0,1]$;

 2. 

∗ is continuous;

 3. 

$1\ast a=a$, for all $a\in [0,1]$;

 4. 

$a\ast b\le c\ast d$, whenever $a\le c$ and $b\le d$, with $a,b,c,d\in [0,1]$.

Definition 2

([2]). 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 $X\times X\times (0,+\infty )$ satisfying the following conditions, for all $x,y,z\in X$ and $s,t>0$,

 1. 

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

 2. 

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

 3. 

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

 4. 

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

 5. 

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

For the topological properties of fuzzy metric space, reader can refer to [2].

Definition 3

([2,11]). Let $(X,M,\ast )$ be a fuzzy metric space and $\left\{x_n\right\}$ be a sequence in X. Then, $\left\{x_n\right\}$ is called an M-Cauchy sequence of X, if for each $ϵ\in (0,1)$ and each $t>0$, there is $n_0\in \mathbb{N}$ such that $M(x_n,x_m,t)>1-ϵ$, for all $n,m\ge n_0$. On the other hand, $\left\{x_n\right\}$ is called a G-Cauchy sequence if $\underset{n\to +\infty }{lim}M(x_n,x_{n+m},t)=1$ for each $m\in \mathbb{N}$ and $t>0$ or, equivalently, $\underset{n\to +\infty }{lim}M(x_n,x_{n+1},t)=1$ for all $t>0$.

The sequence $\left\{x_n\right\}$ is called convergent and converges to x if, for each $ϵ\in (0,1)$ and each $t>0$, there exists $n_0\in N$ such that $M(x_n,x,t)>1-ϵ$, for all $n\ge n_0$.

In 2012, Wardowski [8] introduced a new strong contraction condition called F-contraction in a metric space $(X,d)$. Here, he introduced the following class of function, where F denotes the family of all functions F that satisfy conditions (F1)–(F3).

Definition 4.

Let $F:(0,+\infty )\to R$ be a mapping satisfying:

 (F1) 

F is strictly increasing—that is, $s<t$ implies $F(s)<F(t)$ for all $s,t>0$,

 (F2) 

for every sequence $\left\{s_n\right\}$ in $R^{+}$ we have $\underset{n\to +\infty }{lim}{s}_n=0$ if and only if $\underset{n\to +\infty }{lim}F(s_n)=-\infty$,

 (F3) 

there exists a number $k\in (0,1)$ such that $\underset{s\to 0^{+}}{lim}{s}^k·F(s)=0$.

Definition 5

([8]). A mapping $T:X\to X$ is called an F-contraction on X if there exists $F\in \mathit{F}$ and $\tau >0$ such that for all $x,y\in X$ with $d(Tx,Ty)>0$, we have

$$\tau +F(d(Tx,Ty))\le F(d(x,y)).$$

Recently, Huang et al. [9] introduced the concept of fuzzy F-contraction in a fuzzy metric space and proved fixed-point theorems in a complete fuzzy metric space. In the definition of Huang et al. [9], ${\Delta }_F$ denote by the class of all mappings $F:[0,1]\to (0,+\infty )$ satisfying the following condition: for all $x,y\in [0,1]$, $x<y$ implies $F(x)<F(y)$. Thus, F is strictly increasing on $[0,1]$.

Definition 6.

Let $(X,M,\ast )$ be a fuzzy metric space and $F\in {\Delta }_F$. The mapping $T:X\to X$ is said to be a fuzzy F-contraction if there exists $\tau \in (0,1)$ such that

$$\begin{array}{c}\hfill \tau ·F(M(Tx,Ty,t))\ge F(M(x,y,t)).\end{array}$$

(1)

Now, we recall the concept of α-admissible mappings introduced by Samet et al. [12].

Definition 7

([12]). Let T be a self-mapping on X and $\alpha :X\times X\to [0,\infty )$ be a function. We say that T is an α-admissible mapping if

$$x,y\in X,\alpha (x,y)\ge 1 implies\hspace{1em}\alpha (Tx,Ty)\ge 1.$$

Definition 8

([13]). Let T be a self-mapping on X and $\alpha ,\eta :X\times X\to [0,\infty )$ be two functions. We say that T is an α-admissible mapping with respect to η if

$$x,y\in X,\alpha (x,y)\ge \eta (x,y) implies \alpha (Tx,Ty)\ge \eta (Tx,Ty).$$

If we take $\eta (x,y)=1$ then Definition 8 reduces into Definition 7. Furthermore, if we take $\alpha (x,y)=1$ in Definition 8, then we say that T is an $\eta$-subadmissible mapping.

3. Fixed Point Theorems for α-ΓF-Fuzzy Contractions

In this section, we first introduce a new class of function called $\Gamma$-function and the concept of $\alpha$-$\Gamma F$-fuzzy contractions and prove some fixed point theorems in a fuzzy metric space. We begin with the following definition:

Let ${\Delta }_{\Gamma }$ denote the set of all continuous functions $\Gamma :{({\mathbb{R}}^{+})}^4\to \mathbb{R}$ satisfying:

1. For all $t_1,t_2,t_3,t_4\in {\mathbb{R}}^{+}$ with $\max (t_1,t_2,t_3,t_4)=1$, there exists $\tau \in (0,1)$ such that $\Gamma (t_1,t_2,t_3,t_4)=\tau .$

We have the following examples:

• $\Gamma (t_1,t_2,t_3,t_4)=\tau +L·{\log }_e\max (t_1,t_2,t_3,t_4)$, where $L\in {\mathbb{R}}^{+}$.
• $\Gamma (t_1,t_2,t_3,t_4)=\frac{\tau }{\max (t_1,t_2,t_3,t_4)}$.
• $\Gamma (t_1,t_2,t_3,t_4)=\frac{e·\tau }{e^{\max (t_1,t_2,t_3,t_4)}}$.

Here, $\tau \in (0,1)$, and then $\Gamma \in {\Delta }_{\Gamma }$.

Now, we define a new class of fuzzy contractive conditions depending on the class of $\Gamma$ functions.

Definition 9.

Let $(X,M,\ast )$ be a fuzzy metric space and a mapping $T:X\to X$. Furthermore, suppose that $\alpha ,\eta :X\times X\to [0,+\infty )$ be two functions. T is said to be an α-η-$\Gamma F$-fuzzy contractive mapping on X, if for $x,y\in X$ with $\eta (x,Tx)\le \alpha (x,y)$ and $M(Tx,Ty,t)>0$, we have

$$\begin{array}{c}\Gamma (M(x,Tx,t),M(y,Ty,t),M(x,Ty,t),M(y,Tx,t))·F(M(Tx,Ty,t))\ge F(M(x,y,t)),\end{array}$$

(2)

where $\Gamma \in {\Delta }_{\Gamma }$ and $F\in {\Delta }_F$.

Next, we give the concept of $\alpha$-$\eta$-continuous mapping on a fuzzy metric space.

Definition 10.

Let $(X,M,\ast )$ be a fuzzy metric space and $\alpha ,\eta :X\times X\to [0,\infty )$ and $T:X\to X$ be a function. We say T is an α-η-continuous mapping on a fuzzy metric space. If, for a given $x\in X$ and sequence $\left\{x_n\right\}$ with

$$\begin{array}{c}\underset{n\to +\infty }{lim}M(x_n,x,t)\to 1,\alpha (x_n,x_{n+1})\ge \eta (x_n,x_{n+1})\end{array}$$

for all $n\in \mathbb{N}$. This implies that $\underset{n\to +\infty }{lim}M(T{x}_n,Tx,t)\to 1$.

Example 1.

Let $X=[0,+\infty )$ and t-norm be defined by $t(a,b)=ab$ for all $a,b\in [0,1]$, define a fuzzy set $M:X^2\times (0,+\infty )\to [0,1]$ such that $M(x,y,t)=\frac{t}{t+|x-y|}$ for all $x,y\in X$ and $t\in (0,+\infty )$ is a fuzzy metric space. Let $T:X\to X$ and $\alpha ,\eta :X\times X\to [0,+\infty )$ be defined by

$$T(x)=\left\{\begin{array}{cc}{x}^5,\hfill & if x\in [0,1]\hfill \\ sin3\pi ·x+3,\hfill & if x\in (1,\infty )\hfill \end{array}\right.,\alpha (x,y)=\left\{\begin{array}{cc}{x}^2+y+2,\hfill & if x,y\in [0,1]\hfill \\ 0,\hfill & if otherwise\hfill \end{array}\right.$$

and $\eta (x)=x^2+1$. Clearly, T is not continuous, but T is α-η-continuous mapping on $(X,M,\ast )$.

We need the following lemma to prove our main results.

Lemma 1

([9]). Let $(X,M,\ast )$ be a fuzzy metric space and $\left\{x_n\right\}$ be a sequence in X such that for each $n\in \mathbb{N}$,

$$\underset{t\to 0^{+}}{lim}M(x_n,x_{n+1},t)>0,$$

and for any $t>0$,

$$\underset{n\to +\infty }{lim}M(x_n,x_{n+1},t)=1.$$

If $\left\{x_n\right\}$ is not a Cauchy sequence in X, then there exists $ϵ\in (0,1)$, $t_0>0$, and two sequences of positive integers $\left\{n_k\right\}$, $\left\{m_k\right\}$, $n_k>m_k>k$, $k\in \mathbb{N}$, such that the following sequences

$$\begin{gathered} \left\{M(x_{m_k},x_{n_k},t_0)\right\},\left\{M(x_{m_k},x_{n_k+1},t_0)\right\},\{M(x_{m_k-1},x_{n_k},t_0) \\ \left\{M(x_{m_k-1},x_{n_k+1},t_0)\right\},\left\{M(x_{m_k+1},x_{n_k+1},t_0)\right\} \end{gathered}$$

tend to $1-ϵ$ as $k\to +\infty$.

Now, we are ready to prove our main results.

Theorem 1.

Let $(X,M,\ast )$ be a complete fuzzy metric space. Let $T:X\to X$ be a self mapping satisfying the following assertions:

 1. 

T is α-admissible mapping with respect to η;

 2. 

T is an α-η-$\Gamma F$-fuzzy contractive mapping;

 3. 

there exists $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge \eta (x_0,T{x}_0)$;

 4. 

T is an α-η-continuous map.

Then, T has a fixed point. Moreover, T has a unique fixed point whenever $\alpha (x,y)\ge \eta (x,x)$ for all $x,y\in Fix(T)$.

Proof. 

Let $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge \eta (x_0,T{x}_0)$. For $x_0\in X$, we define the sequence $\left\{x_n\right\}$ by $x_{n+1}=T^n{x}_0=T{x}_n$ for all $n\in \mathbb{N}$. Now, since T is an $\alpha$-admissible mapping with respect to $\eta$, then

$$\begin{array}{c}\hfill \alpha (x_0,x_1)=\alpha (x_0,T{x}_0)\ge \eta (x_0,T{x}_0)=\eta (x_0,x_1),\end{array}$$

by continuing this process, we have

$$\begin{array}{c}\eta (x_{n-1},x_n)\le \alpha (x_{n-1},x_n)\end{array}$$

for all $n\in \mathbb{N}$.

Furthermore, let $n_0\in \mathbb{N}$ such that $x_{n_0}=T{x}_{n_0}$ then $x_{n_0}$ is fixed point of T and nothing to prove.

Let us assume $x_n\ne x_{n+1}$ or $M(x_n,x_{n+1},t)\in [0,1)$ for all $n\in \mathbb{N}$. Since, T is an $\alpha$-$\eta$-$\Gamma F$-contractive mapping, and thus we use $x=x_{n-1}$, $y=x_n$ in (2), we obtain

$$\begin{array}{c}\hfill \Gamma (M(x_{n-1},T{x}_{n-1},t),M(x_n,T{x}_n,t),M(x_{n-1},T{x}_n,t),M(x_n,T{x}_{n-1},t))·\\ \hfill F(M(T{x}_{n-1},T{x}_n,t))\ge F(M(x_{n-1},x_n,t)),\end{array}$$

which implies

$$\begin{array}{c}\hfill \Gamma (M(x_{n-1},x_n,t),M(x_n,x_{n+1},t),M(x_{n-1},x_{n+1},t),M(x_n,x_n,t))·\\ \hfill F(M(x_n,x_{n+1},t))\ge F(M(x_{n-1},x_n,t)).\end{array}$$

Since, $\max (M(x_{n-1},x_n,t),M(x_n,x_{n+1},t),M(x_{n-1},x_n,t),M(x_n,x_n,t))=1$, by definition of $\Gamma$-function, there exists $\tau \in (0,1)$ such that

$$\begin{array}{c}\Gamma (M(x_{n-1},x_n,t),M(x_n,x_{n+1},t),M(x_{n-1},x_n,t),M(x_n,x_n,t))=\tau .\end{array}$$

Therefore,

$$\begin{array}{c}\tau ·F(M(x_n,x_{n+1},t))\ge F(M(x_{n-1},x_n,t)).\end{array}$$

We have

$$\begin{array}{c}F(M(x_n,x_{n+1},t))>\tau ·F(M(x_n,x_{n+1},t))\ge F(M(x_{n-1},x_n,t)).\end{array}$$

(3)

Since F is a strictly increasing function

$$\begin{array}{c}M(x_n,x_{n+1},t)>M(x_{n-1},x_n,t).\end{array}$$

Thus, the sequence $\left\{M(x_n,x_{n+1},t)\right\}$$(t>0)$ is a strictly increasing bounded from above, and thus sequence $\left\{M(x_n,x_{n+1},t)\right\}$$(t>0)$ is convergent. In other words, there exists $a(t)\in [0,1]$ such that

$$\underset{n\to +\infty }{lim}M(x_n,x_{n+1},t)=a(t).$$

(4)

for any $t>0$ and $n\in \mathbb{N}$. It follows that

$$M(x_n,x_{n+1},t)<a(t).$$

(5)

by (4) and (5), for any $t>0$, we have

$$\underset{n\to +\infty }{lim}F(M(x_n,x_{n+1},t))=F(a(t)-0).$$

(6)

We have to show that $a(t)=1$. Assume that $a(t)<1$ for some $t>0$ and by taking limit as $n\to +\infty$ in (3) and using (6), we obtain

$$\begin{array}{c}\hfill F(a(t)-0)\ge \tau ·F(a(t)-0)\ge F(a(t)-0).\end{array}$$

This is a contradiction with $F(a(t)-0)>0$. Therefore,

$$\underset{n\to +\infty }{lim}M(x_n,x_{n+1},t)=1.$$

(7)

Next, we have to prove that $\left\{x_n\right\}$ is a Cauchy sequence. Suppose that $\left\{x_n\right\}$ is not a Cauchy sequence. By using the Lemma 1, then there exists $ϵ\in (0,1)$, $t_0>0$ and sequence $\left\{x_{m_k}\right\}$ and $\left\{x_{n_k}\right\}$ such that

$$\begin{array}{c}\underset{k\to +\infty }{lim}M(x_{m_k},x_{n_k},t_0)=1-ϵ.\end{array}$$

(8)

Again, with $x=x_{m_k}$ and $y=x_{n_k}$ in (2), we have

$$\begin{array}{c}\hfill \Gamma (M(x_{m_k},x_{m_k+1},t),M(x_{n_k},x_{n_k+1},t),M(x_{m_k},x_{n_k+1},t),M(x_{n_k},x_{m_k+1},t))\\ \hfill ·F(M(x_{m_k+1},x_{n_k+1},t))\ge F(M(x_{m_k},x_{n_k},t)).\end{array}$$

Letting limit as $k\to +\infty$, we have

$$\begin{array}{c}\hfill \underset{k\to +\infty }{lim}[\Gamma (M(x_{m_k},x_{m_k+1},t),M(x_{n_k},x_{n_k+1},t),M(x_{m_k},x_{n_k+1},t),\\ \hfill M(x_{n_k},x_{m_k+1},t))·F(M(x_{m_k+1},x_{n_k+1},t))]\ge \underset{k\to +\infty }{lim}F(M(x_{m_k},x_{n_k},t))\end{array}$$

(9)

this implies

$$\begin{array}{c}\Gamma (1,1,\underset{k\to +\infty }{lim}M(x_{m_k},x_{n_k+1},t),\underset{k\to +\infty }{lim}M(x_{n_k},x_{m_k+1},t))·\\ \hfill \underset{k\to +\infty }{lim}F(M(x_{m_k+1},x_{n_k+1},t))]\ge \underset{k\to +\infty }{lim}F(M(x_{m_k},x_{n_k},t)).\end{array}$$

Since $\max (1,1,\underset{k\to +\infty }{lim}M(x_{m_k},x_{n_k+1},t),\underset{k\to +\infty }{lim}M(x_{n_k},x_{m_k+1},t))=1$, there exists $\tau \in (0,1)$ such that

$$\Gamma (1,1,\underset{k\to +\infty }{lim}M(x_{m_k},x_{n_k+1},t),\underset{k\to +\infty }{lim}M(x_{n_k},x_{m_k+1},t))=\tau .$$

Using (8) and (9), implies that

$$\begin{array}{c}\tau ·F((1-ϵ)-0)\ge F((1-ϵ)-0).\end{array}$$

Additionally,

$$\begin{array}{c}\hfill F((1-ϵ)-0)\ge \tau ·F((1-ϵ)-0)\ge F((1-ϵ)-0).\end{array}$$

This is a contraction with $F((1-ϵ)-0)>0$. Thus, the sequence $\left\{x_n\right\}$ is a Cauchy sequence in X. Since fuzzy metric space $(X,M,\ast )$ is complete, then there exists $x^{\ast }\in X$ such that

$$\begin{array}{c}\underset{n\to +\infty }{lim}{x}_n=x^{\ast }.\end{array}$$

Let us prove that $x^{\ast }$ is a fixed point of T. Since T is an $\alpha$-$\eta$-continuous and $\eta (x_{n-1},x_n)\le \alpha (x_{n-1},x_n)$ for all $n\in \mathbb{N}$. Then, $\underset{n\to +\infty }{lim}M(T{x}_n,T{x}^{\ast },t)=1$ implies $M(x^{\ast },T{x}^{\ast },t)=1$— that is, $x^{\ast }=T{x}^{\ast }$. □

Let $x,y\in Fix\left[T\right]$ such that $x\ne y$, by Equation (2),

$$\begin{array}{c}\Gamma (M(x,x,t),M(y,y,t),M(x,y,t),M(y,x,t))·F(M(x,y,t))\ge F(M(x,y,t))\end{array}$$

$$\begin{array}{c}\Gamma (1,1,M(x,y,t),M(y,x,t))·F(M(x,y,t))\ge F(M(x,y,t)).\end{array}$$

Since $\max (1,1,M(x,y,t),M(y,x,t))=1$, there exists $\tau \in (0,1)$ such that $\Gamma (1,1,M(x,y,t),M(y,x,t))=\tau$. Thus, we can deduce above

$$\begin{array}{c}\tau ·F(M(x,y,t))\ge F(M(x,y,t)).\end{array}$$

This implies that

$$\begin{array}{c}F(M(x,y,t))>\tau ·F(M(x,y,t))\ge F(M(x,y,t)),\end{array}$$

which is a contradiction. Thus, T has a unique fixed point.

We can deduce the following Corollary.

Corollary 1.

Let $(X,M,\ast )$ be a complete fuzzy metric space. Let $T:X\to X$ be a self mapping satisfying the following assertions:

 1. 

T is α-admissible mapping with respect to η;

 2. 

if, for $x,y\in X$ with $\alpha (x,y)\ge \eta (x,Tx)$ and $M(Tx,Ty,t)>0$, we have

$$\begin{array}{c}\tau ·F(M(Tx,Ty,t))\ge F(M(x,y,t)),\end{array}$$

where $x\ne y$, $\tau \in (0,1)$ and $F\in {\Delta }_F$;

 3. 

there exists $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge \eta (x_0,T{x}_0)$;

 4. 

T is an α-η-continuous map.

Then, T has a fixed point. Moreover, T has a unique fixed point whenever $\alpha (x,y)\ge \eta (x,x)$ for all $x,y\in Fix(T)$.

Example 2.

Suppose $X=[0,\infty )$ and t-norm is defined by $t(a,b)=ab$ for all $a,b\in [0,1]$. Define a fuzzy set $M:X^2\times (0,+\infty )\to [0,1]$ such that

$$M(x,y,t)=e^{\frac{-|x-y|}{t}}$$

for all $x,y\in X$ and for all $t>0$. Thus, $(X,M,\ast )$ is a complete fuzzy metric space.

Define $T:X\to X$ such that

$$T(x)=\left\{\begin{array}{cc}\frac{1}{2}{x}^2,\hfill & if x\in [0,1]\hfill \\ 3x,\hfill & if x\in (1,\infty ).\hfill \end{array}\right.$$

(10)

Let $\eta ,\alpha :X\times X\to [0,\infty ]$ defined by $\eta (x,y)=\frac{1}{4}$ for all $x,y\in X$ and

$$\alpha (x,y)=\left\{\begin{array}{cc}\frac{1}{2},\hfill & if x,y\in [0,1]\hfill \\ \frac{1}{9},\hfill & otherwise.\hfill \end{array}\right.$$

(11)

Furthermore, let $F(x)={\log }_{e}x$ be any strictly increasing function and consider $\Gamma$ function $\Gamma :{(R^{+})}^4\to R$ defined by $\Gamma (t_1,t_2,t_3,t_4)=\tau$, where $\tau \in (0,1)$.

• Let $\alpha (x,y)\ge \eta (x,y)$ then $x,y\in [0,1]$, on the other hand $T(x)\in [0,1]$ for all $x,y\in [0,1]$, then $\alpha (Tx,Ty)\ge \eta (Tx,Ty)$ (or $\alpha (Tx,Ty)=\alpha (\frac{1}{2}{x}^2,\frac{1}{2}{y}^2)=\frac{1}{2}>\frac{1}{4}$). This means that T is an $\alpha$-admissible mapping with respect to $\eta$.
• There exist $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge \eta (x_0,T{x}_0)$.
• Let $x_n=\frac{1}{n}\to 0$, $\alpha (x_n,x_{n+1})=\alpha (\frac{1}{n},\frac{1}{n+1})=\frac{1}{2}\ge \eta (\frac{1}{n},\frac{1}{n+1})=\frac{1}{4}$ for all $n\in \mathbb{N}$. This implies $T(x_n)=T(\frac{1}{n})\to 0=T(0)$. Thus, T is an $\alpha$-$\eta$ continuous map.

Now,

$$\begin{array}{c}\hfill \tau ·F(Tx,Ty,t)=\tau ·F(e^{-(\frac{1}{2t}|x^2-y^2|)})\\ \hfill =\tau ·\log \{e^{-\frac{1}{2t}|x^2-y^2|}\}\\ \hfill =-\tau ·\frac{1}{2t}|x^2-y^2|\\ \hfill \ge -\frac{|x-y|}{t}\\ \hfill =\log e^{\frac{-|x-y|}{t}}\\ \hfill =F(M(x,y,t)).\end{array}$$

Thus, T is an $\alpha$-$\eta$-$\Gamma F$-fuzzy contractive mapping. Thus, $x=0$ is a fixed point for self map T. Now, consider $x=0$, $y=3$ and $\tau \in (0,1)$

$$\begin{array}{c}\hfill \tau ·F(Tx,Ty,t)=\tau ·F(M(0,T3,t))=\tau ·F(M(0,9,t))=\\ \hfill \tau ·\log e^{-\frac{9}{t}}=-\tau ·\frac{9}{t}<\log e^{-\frac{3}{t}}=-\frac{3}{t}=F(M(x,y,t)).\end{array}$$

Hence, this example can not hold the Theorem 1 proved in [9], such as $\tau ·F(M(Tx,Ty,t))\ge F(M(x,y,t))$ does not hold.

When we use $\eta (x,y)=1$ in Definition 9, Theorem 1 and Corollary 1, we obtain the following.

Definition 11.

Let $(X,M,\ast )$ be a fuzzy metric space and a mapping $T:X\to X$. Furthermore, suppose that $\alpha :X\times X\to [0,+\infty )$ be a function. We say T is said to be an α-$\Gamma F$-fuzzy contractive mapping on X if, for $x,y\in X$ with $\alpha (x,y)\ge 1$ and $M(Tx,Ty,t)>0$, we have

$$\begin{array}{c}\Gamma (M(x,Tx,t),M(y,Ty,t),M(x,Ty,t),M(y,Tx,t))·F(M(Tx,Ty,t))\ge F(M(x,y,t)),\end{array}$$

where $\Gamma \in {\Delta }_{\Gamma }$ and $F\in {\Delta }_F$.

Theorem 2.

Let $(X,M,\ast )$ be a complete fuzzy metric space. Let $T:X\to X$ be a self mapping satisfying the following assertions:

 1. 

T is α-admissible mapping;

 2. 

T is an α-$\Gamma F$-fuzzy contractive mapping;

 3. 

there exists $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge 1$;

 4. 

T is α-continuous mapping.

Then, T has a fixed point. Moreover, T has a unique fixed point in X whenever $\alpha (x,y)\ge 1$ for all $x,y\in Fix(T)$.

Proof. 

Similar to the Proof of Theorem 1. □

Corollary 2.

Let $(X,M,\ast )$ be a complete fuzzy metric space. Let $T:X\to X$ be a self mapping satisfying the following assertions:

 1. 

T is α-admissible mapping;

 2. 

if, for $x,y\in X$ with $\alpha (x,y)\ge 1$ and $M(Tx,Ty,t)>0$, we have

$$\begin{array}{c}\tau ·F(M(Tx,Ty,t))\ge F(M(x,y,t))\end{array}$$

where $x\ne y$, $\tau \in (0,1)$ and $F\in {\Delta }_F$;

 3. 

there exists $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge 1$;

 4. 

T is α-continuous map.

Then, T has a fixed point. Moreover, T has a unique fixed point in X whenever $\alpha (x,y)\ge 1$ for all $x,y\in Fix(T)$.

Taking $\alpha (x,y)=1$ in Corollary 2 for all $x,y\in X$. We deduce the following fixed point result.

Corollary 3

([9]). Let $(X,M,\ast )$ be a complete fuzzy metric space such that

$$\underset{t\to 0^{+}}{lim}M(x,y,t)>0$$

for all $x,y\in X$. If $T:X\to X$ is a continuous fuzzy F-contraction, then T has a unique fixed-point in X.

In the next theorem, we omit the continuity hypothesis of T.

Theorem 3.

Let $(X,M,\ast )$ be a complete fuzzy metric space. Let $T:X\to X$ be a self mapping satisfying the following assertions:

 1. 

T is α-admissible mapping with respect to η;

 2. 

T is an α-η-$\Gamma F$-fuzzy contractive mapping;

 3. 

there exists $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge \eta (x_0,T{x}_0)$;

 4. 

if $\left\{x_n\right\}$ is a sequence in X such that $\alpha (x_n,x_{n+1})\ge \eta (x_n,x_{n+1})$ with $x_n\to x$ as $n\to +\infty$, then

$$\eta (T{x}_n,T^2{x}_n)\le \alpha (T{x}_n,x) or \eta (T^2{x}_n,T^3{x}_n)\le \alpha (T^2{x}_n,x)$$

holds for all $n\in \mathbb{N}$.

Then, T has a fixed point. Moreover, T has a unique fixed point whenever $\alpha (x,y)\ge \eta (x,x)$ for all $x,y\in Fix(T)$.

Proof. 

Let $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge \eta (x_0,T{x}_0)$. Similar to the proof of the Theorem 1, we can conclude that

$$\alpha (x_n,x_{n+1})\ge \eta (x_n,x_{n+1}) and x_n\to x^{\ast } as n\to +\infty$$

where, $x_{n+1}=T{x}_n$. By assumption 4, either

$$\eta (T{x}_n,T^2{x}_n)\le \alpha (x_{n+1},x^{\ast }) or \eta (T^2{x}_n,T^3{x}_n)\le \alpha (T^2{x}_n,x^{\ast })$$

holds for all $n\in \mathbb{N}$. This implies that

$$\eta (x_{n+1},x_{n+2})\le \alpha (x_{n+1},x^{\ast }) or \eta (x_{n+2},x_{n+3})\le \alpha (x_{n+2},x^{\ast })$$

holds for all $n\in \mathbb{N}$. Equivalently, there exists a subsequence $\left\{x_{n_k}\right\}$ of $\left\{x_n\right\}$ such that

$$\eta (x_{n_k},x_{n_k+1})\le \alpha (x_{n_k},x^{\ast }),$$

and by (2), we obtain

$$\begin{array}{c}\hfill \Gamma (M(x_{n_k},T{x}_{n_k},t),M(x^{\ast },T{x}^{\ast },t),M(x_{n_k},T{x}^{\ast },t),M(x^{\ast },T{x}_{n_k},t))\\ \hfill ·F(M(T{x}_{n_k},T{x}^{\ast },t))\ge F(M(x_{n_k},x^{\ast },t)),\end{array}$$

which implies for any $t>0$,

$$\begin{array}{c}F(M(T{x}_{n_k},T{x}^{\ast },t))>\tau ·F(M(T{x}_{n_k},T{x}^{\ast },t))\ge F(M(x_{n_k},x^{\ast },t)).\end{array}$$

Since F is a strictly increasing function,

$$\begin{array}{c}M(T{x}_{n_k},T{x}^{\ast },t)>M(x_{n_k},x^{\ast },t).\end{array}$$

Taking limit as $k\to +\infty$ in the above inequality, we obtain $M(x^{\ast },T{x}^{\ast },t)=1$—that is, $x^{\ast }=T{x}^{\ast }$. The uniqueness of the fixed point is similar to Theorem 1. □

Corollary 4.

Let $(X,M,\ast )$ be a complete fuzzy metric space. Let $T:X\to X$ be a self mapping satisfying the following assertions:

 1. 

T is α-admissible mapping with respect to η;

 2. 

if, for $x,y\in X$ with $\alpha (x,y)\ge \eta (x,Tx)$ and $M(Tx,Ty,t)>0$, we have

$$\begin{array}{c}\tau ·F(M(Tx,Ty,t))\ge F(M(x,y,t))\end{array}$$

where $x\ne y$, $\tau \in (0,1)$ and $F\in {\Delta }_F$;

 3. 

there exists $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge \eta (x_0,T{x}_0)$;

 4. 

if $\left\{x_n\right\}$ is a sequence in X such that $\alpha (x_n,x_{n+1})\ge \eta (x_n,x_{n+1})$ with $x_n\to x$ as $n\to +\infty$, then

$$\eta (Tx,T^2{x}_n)\le \alpha (T{x}_n,x) or \eta (T^2{x}_n,T^3{x}_n)\le \alpha (T^2{x}_n,x)$$

holds for all $n\in N$.

Then, T has a fixed point. Moreover, T has a unique fixed point whenever $\alpha (x,y)\ge \eta (x,x)$ for all $x,y\in Fix(T)$.

When we consider $\eta (x,y)=1$ in Theorem 3 and Corollary 4, we obtain the following.

Theorem 4.

Let $(X,M,\ast )$ be a complete fuzzy metric space. Let $T:X\to X$ be a self mapping satisfying the following assertions:

 1. 

T is an α-admissible mapping;

 2. 

T is an α-$\Gamma F$-fuzzy contractive mapping;

 3. 

there exists $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge 1$;

 4. 

if $\left\{x_n\right\}$ is a sequence in X such that $\alpha (x_n,x_{n+1})\ge 1$ with $x_n\to x$ as $n\to \infty$, then $\alpha (x_n,x)\ge 1$ or $\alpha (x_{n+1},x)\ge 1$ holds for all $n\in \mathbb{N}$.

Then, T has a fixed point. Moreover, T has a unique fixed point whenever $\alpha (x,y)\ge 1$ for all $x,y\in Fix(T)$.

Proof. 

Let $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge 1$. Similarly to Theorem 3, we can conclude that

$$\begin{array}{c}\alpha (x_n,x_{n+1})\ge 1 and x_n\to x^{\ast } as n\to +\infty \end{array}$$

where $x_{n+1}=T{x}_n$. By assumption 4, $\alpha (T{x}_n,x)\ge 1$ holds for all $n\in \mathbb{N}$.

Equivalently, there exists a subsequence $\left\{x_{n_k}\right\}$ of $\left\{x_n\right\}$ such that $\alpha (x_{n_k},x)\ge 1$ and by definition of $\alpha$-$\Gamma F$-fuzzy contractive mapping, we deduce that

$$\begin{array}{c}\hfill \Gamma (M(x_{n_k},T{x}_{n_k},t),M(x^{\ast },T{x}^{\ast },t),M(x_{n_k},T{x}^{\ast },t),M(x^{\ast },T{x}_{n_k},t))\\ \hfill ·F(M(T{x}_{n_k},T{x}^{\ast },t))\ge F(M(x_{n_k},x^{\ast },t)).\end{array}$$

This implies that

$$\begin{array}{c}F(M(T{x}_{n_k},T{x}^{\ast },t))>\tau ·F(M(T{x}_{n_k},T{x}^{\ast },t))\ge F(M(x_{n_k},x^{\ast },t)).\end{array}$$

Since F is strictly increasing function,

$$\begin{array}{c}M(T{x}_{n_k},T{x}^{\ast },t)>M(x_{n_k},x^{\ast },t).\end{array}$$

Taking limit $k\to +\infty$ in above inequality, we find

$$M(x^{\ast },T{x}^{\ast },t)=1—\mathrm{that} \mathrm{is}, x^{\ast }=T{x}^{\ast }$$

.

Uniqueness follows from the above Theorem 1. □

Corollary 5.

Let $(X,M,\ast )$ be a complete fuzzy metric space. Let $T:X\to X$ be a self mapping satisfying the following assertions:

 1. 

T is α-admissible;

 2. 

if, for $x,y\in X$ with $\alpha (x,y)\ge 1$ and $M(Tx,Ty,t)>0$, we have

$$\tau ·F(M(Tx,Ty,t))\ge F(M(x,y,t))$$

where $x\ne y$, $\tau \in (0,1)$ and $F\in {\Delta }_F$;

 3. 

there exists $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge 1$;

 4. 

if $\left\{x_n\right\}$ is a sequence in X such that $\alpha (x_n,x_{n+1})\ge 1$ with $x_n\to x$ as $n\to \infty$, then either $\alpha (T{x}_n,x)\ge 1$ or $\alpha (T^2{x}_n,x)\ge 1$ holds for all $n\in \mathbb{N}$.

Then, T has a fixed point Moreover, T has a unique fixed point, whenever $\alpha (x,y)\ge 1$, for all $x,y\in Fix(T)$.

4. Application

As an application of the Corollary 5, we established here the existence theorem of solutions for a nonlinear fractional differential equation.

We study the problem considered in [14] for the existence of solutions for the nonlinear fractional differential equation.

$$\begin{array}{c}{}^C{D}^{\beta }(x(t))=f(t,x(t))\end{array}$$

(12)

where $(0<t<1,1<\beta \le 2)$ via the integral boundary conditions

$$\begin{array}{c}x(0)=0,x(1)={\int }_0^{\eta }x(s)ds\hspace{1em}(0<\eta <1)\end{array}$$

(13)

where ${}^C{D}^{\beta }$ denote the Caputo fractional derivative of order $\beta$ and $f:[0,1]\times R\to R$ is a continuous function. Here, $(X,||.|{|}_{\infty })$, where $X=C([0,1],R)$, is the Banach space of continuous function from $[0,1]$ into R endow with the supremum norm

$${||x||}_{\infty }=\underset{t\in [0,1]}{sup}|x(t)|.$$

(14)

Let $(X,M,\ast )$ be any complete fuzzy metric space. The triplet $(X,M,{\ast }_p)$ is a fuzzy metric space, where the set M is defined by

$$\begin{array}{c}M(x,y,t)=e^{\frac{-|x-y|}{t}}\end{array}$$

(15)

for all $x,y\in X$ and $t>0$. For a continuous function $g:R^{+}\to R$, the Caputo derivative of fractional order $\beta$ is defined as

$$\begin{array}{c}{}^C{D}^{\beta }g(t)=\frac{1}{\Gamma (n-\beta )}{\int }_0^t{(t-s)}^{n-\beta -1}{g}^n(s).ds\end{array}$$

(16)

$(n-1<\beta <n,n=[\beta ]+1)$, where $\left[\beta \right]$ denote the integer part of the real number $\beta$.

Now, for continuous function $g:R^{+}\to R$, the Reimann–Liouville fractional derivatives of order $\beta$ is defined by

$$\begin{array}{c}{D}^{\beta }g(t)=\frac{1}{\Gamma (n-\beta )}\frac{d^n}{d{t}^n}{\int }_0^t\frac{g(s)}{{(t-s)}^{\beta -n-1}}.ds\end{array}$$

(17)

$n=\left[\beta \right]+1$, the right hand side is point-wise defined on $(0,+\infty )$.

Now, we give the following existence theorem.

Theorem 5.

Suppose that

 1. 

there exists a function $\xi :R\times R\to R$ and $\tau \in (0,1)$ such that

$$\begin{array}{c}|f(t,a)-f(t,b)|\le \frac{\Gamma (\beta +1)}{5}\frac{|a-b|}{\tau }\end{array}$$

(18)

for all $t\in [0,1]$ and $a,b\in R$ with $\xi (a,b)>0$;

 2. 

there exists $x_0\in X$ such that $\xi (x_0(t),T{x}_0(t))>0$ for all $t\in [0,1]$, where the operater $T:X\to X$ is defined by

$$\begin{array}{c}\hfill Tx(t)=\frac{1}{\Gamma \beta }{\int }_0^t{(t-s)}^{\beta -1}f(s,x(s))ds-\frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^t{(1-s)}^{\beta -1}f(s,x(s))ds\\ \hfill +\frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^{\eta }\left({\int }_0^s{(s-m)}^{\beta -1}f(m,x(m))dm\right)ds\hspace{1em}\hspace{1em} (t\in [0,1]);\end{array}$$

 3. 

for each $t\in [0,1]$ and $x,y\in X$, $\xi (x(t),y(t))>0$ implies $\xi (Tx(t),Ty(t))>0$;

 4. 

If $\left\{x_n\right\}$ is a sequence in X such that $x_n\to x$ in X and $\xi (x_n,x_{n+1})>0$ for all $n\in \mathbb{N}$, then $\xi (x_n,x)>0$ for all $n\in \mathbb{N}.$

Then, (12) has at least one solution.

Proof. 

It is well-known that $x\in X$ is a solution of (12) if and only if $x\in X$ is a solution of the integral equation

$$\begin{array}{c}\hfill x(t)=\frac{1}{\Gamma \beta }{\int }_0^t{(t-s)}^{\beta -1}f(s,x(s))ds-\frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^t{(1-s)}^{\beta -1}f(s,x(s))ds\\ \hfill +\frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^{\eta }({\int }_0^s{(s-m)}^{\beta -1}f(m,x(m))dm)ds\hspace{1em}\hspace{1em}(t\in [0,1]).\end{array}$$

Then, problem (12) is equivalent to find $x^{\ast }\in X$, which is a fixed point of T.

Now, let $x,y\in X$ such that $\xi (x(t),y(t))>0$ for all $t\in [0,1].$ By (i), we find $|Tx(t)-Ty(t)|=$

$$\begin{array}{c}\hfill |\frac{1}{\Gamma \beta }{\int }_0^t{(t-s)}^{\beta -1}f(s,x(s))ds-\frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^t{(1-s)}^{\beta -1}f(s,x(s))ds\\ \hfill +\frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^{\eta }({\int }_0^s{(s-m)}^{\beta -1}f(m,x(m))dm)ds\\ \hfill -\frac{1}{\Gamma \beta }{\int }_0^t{(t-s)}^{\beta -1}f(s,y(s))ds+\frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^t{(1-s)}^{\beta -1}f(s,y(s))ds\\ \hfill -\frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^{\eta }({\int }_0^s{(s-m)}^{\beta -1}f(m,y(m))dm)ds|\end{array}$$

$$\begin{array}{c}\hfill \le |\frac{1}{\Gamma \beta }{\int }_0^t{|t-s|}^{\beta -1}|f(s,x(s))-f(s,y(s))|ds+\\ \hfill \frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^t{(1-s)}^{\beta -1}|f(s,x(s))-f(s,y(s))|ds\\ \hfill +\frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^{\eta }|{\int }_0^s{(s-m)}^{\beta -1}f(m,x(m))-f(m,y(m))dm|ds\end{array}$$

$$\begin{array}{c}\hfill \le |\frac{1}{\Gamma \beta }{\int }_0^t{|t-s|}^{\beta -1}\frac{\Gamma (\beta +1)}{5}\frac{|y(s)-x(s)|}{\tau }ds+\\ \hfill \frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^t{(1-s)}^{\beta -1}\frac{\Gamma (\beta +1)}{5}\frac{|y(s)-x(s)|}{\tau }ds\\ \hfill +\frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^{\eta }({\int }_0^s{|s-m|}^{\beta -1}\frac{\Gamma (\beta +1)}{5}\frac{|y(m)-x(m)|}{\tau }dm)ds\end{array}$$

$$\begin{array}{c}\hfill \le \Gamma \frac{(\beta +1)}{5}\frac{1}{\tau }||x-{y||}_{\infty }.\underset{t\in (0,1)}{sup}(\frac{1}{\Gamma \beta }{\int }_0^1{(t-s)}^{\beta -1}ds+\\ \hfill \frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^1{(1-s)}^{\beta -1}ds+\frac{2t}{(2-{\eta }^2)\Gamma \beta }{\int }_0^{\eta }{\int }_0^s{|s-m|}^{\beta -1}dmds)\end{array}$$

$$\begin{array}{c}\le \frac{||x-{y||}_{\infty }}{\tau }.\end{array}$$

Thus, for each $x,y\in X$ with $\xi (x(t)-y(t))>0$ for all $t\in [0,1]$, we have

$$\begin{array}{c}\hfill ||Tx-{Ty||}_{\infty }\le \frac{||x-{y||}_{\infty }}{\tau }\\ \hfill \frac{||Tx-{Ty||}_{\infty }}{t}\le \frac{||x-{y||}_{\infty }}{t·\tau }\\ \hfill -\frac{||x-{y||}_{\infty }}{t}\le -\tau ·\frac{||Tx-{Ty||}_{\infty }}{t}\\ \hfill {\log }_e{e}^{-\frac{||x-{y||}_{\infty }}{t}}\le {\log }_e{e}^{-\tau ·\frac{||Tx-{Ty||}_{\infty }}{t}}\\ \hfill {\log }_e{e}^{-\frac{||x-{y||}_{\infty }}{t}}\le \tau ·{\log }_e{e}^{-\frac{||Tx-{Ty||}_{\infty }}{t}}\end{array}$$

Now, consider the function $F:R^{+}\to R^{+}$ defined by $F(t)={\log }_{e}t$ for each $t>0$ such that $F\in \mathit{F}$. The above inequality implies that

$$F(M(x,y,t))\le \tau ·F(M(Tx,Ty,t))$$

for all $x,y\in X$ with $M(Tx,Ty,t)>0.$ Therefore, T is an $\alpha$-$\Gamma F$-contractive mapping.

Next, by using assumption 3 of Theorem 5, $\alpha (x,y)\ge 1$ implies $\xi (x(t),y(t))>0$, which implies $\xi (Tx(t,Ty(t)))>0$, which implies $\alpha (Tx,Ty)\ge 1$ for all $x,y\in X$. Hence, T is $\alpha$-admissible.

From assumption 2 of Theorem 5, there exists $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge 1$.

Finally, from assumption 4 of Theorem 5, if $\left\{x_n\right\}$ be a sequence in X such that $\alpha (x_n,x_{n+1})\ge 1$ for all $n\in \mathbb{N}$ implies $\xi (x_n,x_{n+1})>0$ for all $n\in \mathbb{N}$, then $\xi (x_n,x)>0$ for all $n\in \mathbb{N}$ implies $\alpha (x_n,x)\ge 1$ for all $n\in \mathbb{N}$. Therefore, condition 4 of Corollary 5 holds true.

With this as an application of our Corollary 5, we deduce that the existence of $x^{\ast }\in X$ such that $x^{\ast }=T{x}^{\ast }$ and $x^{\ast }$ is a solution of the problem (12). □

5. Conclusions

In this manuscript, we introduced a family of functions called a class of $\Gamma$-functions, such as an implicit function, which is an essential tool for generalizing the existing contraction condition given in [9]. Furthermore, the fuzzy contractive condition (2) is a direct generalization of the fuzzy F-contraction introduced in [9]. Here, we proved fixed-point theorems by using the weaker condition of continuity, the $\alpha$ admissible property of the map and by considering $\alpha$-$\Gamma F$-fuzzy contractive conditions in a complete fuzzy metric space.

At the end of the main section, an application existence of the solution of fractional differential equation via $\alpha$-$\Gamma F$-fuzzy contractive conditions was discussed by considering nonlinear fractional differential Equation (12) with some boundary conditions. These new concepts will lead to further investigations and applications. By using the recent ideas in the literature, it is possible to extend our results to periodic points, best proximity points, n-tuple fixed points and cyclical fixed points in fuzzy metric spaces as well as fuzzy metric-like spaces, etc. (see [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]).