Study of a Dynamical Problem under Fuzzy Conformable Differential Equation

Abstract

The notion of inclusion by generalized conformable differentiability is used to analyze fuzzy conformable differential equations (FCDE). This idea is based on expanding the class of conformable differentiable fuzzy mappings, and we use generalized lateral conformable derivatives to do so. We’ll see that both conformable derivatives are distinct and that they lead to different FCDE solutions. The approach’s utility and efficiency are demonstrated with an example.

1. Introduction

Aubin and Cellina [1] established the notion of differential inclusions systemically. They looked at the existence and qualities of differential inclusion solutions of the form [2].

$$u^{\text{'}}\left(t\right)\in \mathrm{\Phi }\left(u\left(t\right)\right)\mathit{\text{or}}{u}^{\text{'}}\left(t\right)\in \mathrm{\Phi }\left(t,u\left(t\right)\right).$$E1

In this paper we will consider the conformable fractional differential equation.

$$\begin{array}{ll}{u}^{\left(q\right)}\left(t\right)& =\mathrm{\Phi }\left(t,u\left(t\right)\right)\\ u^{\kappa }\left(0\right)& \in {\left[u_0\right]}^{\kappa },\kappa \in \left[0,1\right]\end{array}$$E2

where $t\in \left(0,a\right)$ and $u_0$ is a fuzzy number. $u^{\left(q\right)}$ is the conformable fractional derivative of $u$ of order $\gamma \in \left(0,1\right]$ [3, 4, 5]. There are numerous options for defining a fuzzy fractional derivatives and, as a result, see [6, 7, 8, 9], studying Eq. (2). [10, 11, 12, 13, 14, 15, 16] constructed the generalized derivative of a set value function and investigated it, while [17, 18, 19, 20] explored the generalized conformable fractional derivative.

The objective of this research is to see if fuzzy solutions exist using conformable differential inclusion, using the generalized conformable differentiability concept This idea is based on expanding the class of differentiable fuzzy mappings, and we use lateral conformable derivatives to do so. We will see that both derivatives are different and they lead us to different solutions from an FCDE.

2. Preliminaries

We’ll go through a few definitions now that will come in handy later in the paper. Let us start with a definition. ${\mathrm{R}}_{\mathcal{F}}$ the class of fuzzy subsets of the real axis $\left\{\eta :\mathrm{R}\to \left[0,1\right]\right\}$ satisfying the following properties:

1. $\eta$ is normal,

2. $\eta$ is convex fuzzy set,

3. $\eta$ is upper semicontinuous,

4. ${\left[\eta \right]}^0=\mathit{cl}\left\{x\in \mathrm{R}|\eta \left(x\right)>0\right\}$ is compact.

Then ${\mathrm{R}}_{\mathcal{F}}$ is called the space of fuzzy numbers [21].

If $\eta$ is a fuzzy set, we define ${\left[\eta \right]}^{\kappa }=\left\{x\in \mathrm{R}|\eta \left(x\right)\ge \kappa \right\}$ the $\kappa$-level sets of $\eta$, with $0<\kappa \le 1$. Also, if $\eta \in {\mathrm{R}}_{\mathcal{F}}$ then $\kappa$-cut of $\eta$ denoted by ${\left[\eta \right]}^{\kappa }=\left[{\eta }_1^{\kappa },{\eta }_2^{\kappa }\right].$

For $\eta ,\nu \in {\mathrm{R}}_{\mathcal{F}}$ and $\lambda \in \mathrm{R}$ the sum $\eta +\nu$ and the product $\mathit{\lambda \eta }$ are defined by $\forall \kappa \in \left[0,1\right],$

$${\left[\eta +\nu \right]}^{\kappa }=\left[{\eta }_1^{\kappa }+{\nu }_1^{\kappa },{\eta }_2^{\kappa }+{\nu }_2^{\kappa }\right],$$E3

$${\left[\mathit{\lambda \eta }\right]}^{\kappa }=\lambda {\left[\eta \right]}^{\kappa }=\left\{\begin{array}{l}\left[{\mathit{\lambda \eta }}_1^{\kappa },{\mathit{\lambda \eta }}_2^{\kappa }\right],\lambda \ge 0;\\ \left[{\mathit{\lambda \eta }}_2^{\kappa },{\mathit{\lambda \eta }}_1^{\kappa }\right],\lambda <0,\end{array}\right.$$E4

Let $d:{\mathrm{R}}_{\mathcal{F}}\times {\mathrm{R}}_{\mathcal{F}}\to {\mathrm{R}}_{+}\cup \left\{0\right\}$ by the following equation [22].

$$d\left(\eta ,\nu \right)=\underset{\kappa \in \left[0,1\right]}{sup}{d}_H\left({\left[\eta \right]}^{\kappa },{\left[\nu \right]}^{\kappa }\right),\mathit{\text{for}}\mathit{all}\eta ,\nu \in {\mathrm{R}}_{\mathcal{F}},$$E5

$$=\underset{\kappa \in \left[0,1\right]}{sup}max\left\{\left|,{\eta }_1^{\kappa }-{\nu }_1^{\kappa },\right|,,,,\left|,{\eta }_2^{\kappa }-{\nu }_2^{\kappa },\right|\right\}$$E6

where $d_H$ is the Hausdorff metric.

The following properties are well-known see [22, 23]: $\forall \eta ,\nu ,\omega ,\rho \in {\mathrm{R}}_{\mathcal{F}}$ and $\lambda \in \mathrm{R}$.

$$\begin{array}{c}d\left(\eta +\omega ,\nu +\omega \right)=d\left(\eta ,\nu \right)\mathit{\text{and}}d\left(\eta ,\nu \right)=d\left(\nu ,\eta \right),\\ d\left(\mathit{\lambda \eta },\mathit{\lambda \nu }\right)=\mid \lambda \mid d\left(\eta ,\nu \right),\\ d\left(\eta +\nu ,\omega +\rho \right)\le d\left(\eta ,\omega \right)+d\left(\nu ,\rho \right).\end{array}$$E7

And $\left({\mathrm{R}}_{\mathcal{F}},d\right)$ is a complete metric space.

Theorem 1. [1, 22] Let $\eta :\left[0,a\right]\to {\mathrm{R}}_{\mathcal{F}}$ and ${\left[\eta \left(t\right)\right]}^{\kappa }=\left[{\eta }_1^{\kappa }\left(t\right),{\eta }_2^{\kappa }\left(t\right)\right]$ be Seikkala differentiable. Then, ${\eta }_1^{\kappa }\left(t\right)$ and ${\eta }_2^{\kappa }\left(t\right)$ are differentiable and

$${\left[{\eta }^{\prime }\left(t\right)\right]}^{\kappa }=\left[{\left({\eta }_1^{\kappa }\right)}^{\prime }\left(t\right),{\left({\eta }_2^{\kappa }\right)}^{\prime }\left(t\right)\right],\kappa \in \left[0,1\right].$$E8

Definition 1. [24] Let $\eta :\left[0,a\right]\to {\mathrm{R}}_{\mathcal{F}}$. ${\int }_b^c\eta \left(t\right)\mathit{dt},b,c\in \left[0,a\right]$ is the fuzzy integral, defined by

$${\left[{\int }_b^c\eta \left(t\right)\mathit{dt}\right]}^{\kappa }=\left[{\int }_b^c{\eta }_1^{\kappa }\left(t\right)\mathit{dt},{\int }_b^c{\eta }_2^{\kappa }\left(t\right)\mathit{dt}\right],$$E9

for all $0\le \kappa \le 1$. In [24], if $\eta :\left[0,a\right]\to {\mathrm{R}}_{\mathcal{F}}$ is continuous, it is fuzzy integrable.

Theorem 2. [22, 25] If $\eta \in {\mathrm{R}}_{\mathcal{F}}$, then:

1. $${\left[\eta \right]}^{{\kappa }_2}\subset {\left[\eta \right]}^{{\kappa }_1},\mathit{\text{if}}0\le {\kappa }_1\le {\kappa }_2\le 1;$$E10

2. $\left\{{\kappa }_k\right\}\subset \left[0,1\right]$ is a increasing sequence which converges to $\kappa$,

   $${\left[\eta \right]}^{\kappa }=\underset{k\ge 1}{\cap }{\left[\eta \right]}^{{\kappa }_k}.$$E11

Alternatively, if ${\mathrm{\Upsilon }}_{\kappa }=\{\left[{\eta }_1^{\kappa },{\eta }_2^{\kappa }\right];\kappa \in (0,1\left]\right\}$ is a closed real intervals $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$, then $\left\{{\mathrm{\Upsilon }}_{\kappa }\right\}$ defined a fuzzy number $\eta \in {\mathrm{R}}_{\mathcal{F}}$ such that ${\left[\eta \right]}^{\kappa }={\mathrm{\Upsilon }}_{\kappa }.$.

3. Fuzzy conformable differentiability and integral

The funcion $\mathrm{\Phi }:\left[a,b\right]\to {\mathbb{R}}_{\mathcal{F}}$ is called fuzzy function. The $\kappa$-level representation of fuzzy function $\mathrm{\Phi }$ given by ${\left[\mathrm{\Phi }\left(t\right)\right]}^{\kappa }=\left[{\varphi }_1^{\kappa }\left(t\right),{\varphi }_2^{\kappa }\left(t\right)\right],\forall t\in \left[a,b\right],\forall \kappa \in \left[0,1\right]$.

Definition 2. [17] Let $\mathrm{\Phi }:\left(0,a\right)\to {\mathbb{R}}_{\mathcal{F}}$ be a fuzzy function. ${\gamma }^{\mathit{th}}$ order” fuzzy conformable derivative” of $\mathrm{\Phi }$ is defined by

$$T_{\gamma }\left(\mathrm{\Phi }\right)\left(t\right)=\underset{\epsilon \to 0^{+}}{lim}\frac{\mathrm{\Phi }\left(t+\epsilon t^{1-\gamma }\right)\ominus \mathrm{\Phi }\left(t\right)}{\epsilon }=\underset{\epsilon \to 0^{+}}{lim}\frac{\mathrm{\Phi }\left(t\right)\ominus \mathrm{\Phi }\left(t-\epsilon t^{1-\gamma }\right)}{\epsilon }.$$E12

for all $t>0,\gamma \in \left(0,1\right)$. Let ${\mathrm{\Phi }}^{\left(\gamma \right)}\left(t\right)$ stands for $T_{\gamma }\left(\mathrm{\Phi }\right)\left(t\right)$. Hence

$${\mathrm{\Phi }}^{\left(\gamma \right)}\left(t\right)=\underset{\epsilon \to 0^{+}}{lim}\frac{\mathrm{\Phi }\left(t+\epsilon t^{1-\gamma }\right)\ominus \mathrm{\Phi }\left(t\right)}{\epsilon }=\underset{\epsilon \to 0^{+}}{lim}\frac{\mathrm{\Phi }\left(t\right)\ominus \mathrm{\Phi }\left(t-\epsilon t^{1-\gamma }\right)}{\epsilon }.$$E13

If $\mathrm{\Phi }$ is $\gamma$-differentiable in some $\left(0,a\right)$, and ${lim}_{t\to 0^{+}}{\mathrm{\Phi }}^{\left(\gamma \right)}\left(t\right)$ exists, then

$${\mathrm{\Phi }}^{\left(\gamma \right)}\left(0\right)=\underset{t\to 0^{+}}{lim}{\mathrm{\Phi }}^{\left(\gamma \right)}\left(t\right)$$E14

and the limits (in the metric d).

Remark 1. [26]

1. — If $\mathrm{\Phi }$ is $\gamma$-differentiable then the multivalued mapping ${\mathrm{\Phi }}_{\kappa }$ is $\gamma$-differentiable for all $\kappa \in \left[0,1\right]$ and

$$T_{\gamma }{\mathrm{\Phi }}_{\kappa }={\left[{\mathrm{\Phi }}^{\left(\gamma \right)}\left(t\right)\right]}^{\kappa },$$E15

where $T_{\gamma }{\mathrm{\Phi }}_{\kappa }$ is denoted from the conformable fractional derivative of ${\mathrm{\Phi }}_{\kappa }$ of order $\gamma$.

1. — ${\left[\eta \right]}^{\kappa }\ominus {\left[\nu \right]}^{\kappa },\kappa \in \left[0,1\right]$ does not imply the existence of Hukuhara difference (H-difference) $\eta \ominus \nu .$

Theorem 3. [26]

Let $\mathrm{\Phi }:\left(0,a\right)\to {\mathbb{R}}_{\mathcal{F}}$, ${\left[\mathrm{\Phi }\left(t\right)\right]}^{\kappa }=\left[{\varphi }_1^{\kappa }\left(t\right),{\varphi }_2^{\kappa }\left(t\right)\right],\kappa \in \left[0,1\right]$.

1. If $\mathrm{\Phi }$ is ${\gamma }_{\left(1\right)}$-differentiable, then ${\varphi }_1^{\kappa }\left(t\right)$ and ${\varphi }_2^{\kappa }\left(t\right)$ are $\gamma$-differentiable and

   $${\left[{\mathrm{\Phi }}^{\left({\gamma }_{\left(1\right)}\right)}\left(t\right)\right]}^{\kappa }=\left[{\left({\varphi }_1^{\kappa }\right)}^{\left(\gamma \right)}\left(t\right),{\left({\varphi }_2^{\kappa }\right)}^{\left(\gamma \right)}\left(t\right)\right]$$E16

2. If $\mathrm{\Phi }$ is ${\gamma }_{\left(2\right)}$-differentiable, then ${\varphi }_1^{\kappa }\left(t\right)$ and ${\varphi }_2^{\kappa }\left(t\right)$ are $\gamma$-differentiable and

   $${\left[{\mathrm{\Phi }}^{\left({\gamma }_{\left(2\right)}\right)}\left(t\right)\right]}^{\kappa }=\left[{\left({\varphi }_2^{\kappa }\right)}^{\left(\gamma \right)}\left(t\right),{\left({\varphi }_1^{\kappa }\right)}^{\left(\gamma \right)}\left(t\right)\right].$$E17

Theorem 4. [17] Let $\gamma \in \left(0,1\right]:$.

1. If $\mathrm{\Phi }$ is $\left(1\right)$-differentiable and $\mathrm{\Phi }$ is ${\gamma }_{\left(1\right)}$-differentiable, then

   $$T_{{\gamma }_{\left(1\right)}}\mathrm{\Phi }\left(t\right)=t^{1-\gamma }{D}_1^1\mathrm{\Phi }\left(t\right)$$E18

2. If $\mathrm{\Phi }$ is $\left(2\right)$-differentiable and $\mathrm{\Phi }$ is ${\gamma }_{\left(2\right)}$-differentiable, then

   $$T_{{\gamma }_{\left(2\right)}}\mathrm{\Phi }\left(t\right)=t^{1-\gamma }{D}_2^1\mathrm{\Phi }\left(t\right)$$E19

Theorem 5. If $\mathrm{\Phi }:\left(0,a\right)\to {\mathbb{R}}_{\mathcal{F}}$ is $\gamma$-differentiable then it is continuous.

Proof. Denote ${\mathrm{\Phi }}_{\kappa }\left(t\right)=\left[{\varphi }_1^{\kappa }\left(t\right),{\varphi }_2^{\kappa }\left(t\right)\right],\kappa \in \left[0,1\right].$ Then ${\varphi }_1^{\kappa }\left(t\right)$ and ${\varphi }_2^{\kappa }\left(t\right)$ are continuous at $t_0$, so $\mathrm{\Phi }$ is continuous at $t_0.$

If $\epsilon >0$ and $\kappa \in \left[0,1\right],$ we have:

$${\left[\mathrm{\Phi }\left(t_0+\epsilon t_0^{1-\gamma }\right)\ominus \mathrm{\Phi }\left(t_0\right)\right]}^{\kappa }=\left[{\varphi }_1^{\kappa }\left(t_0+\epsilon t_0^{1-\gamma }\right)-{\varphi }_1^{\kappa }\left(t_0\right),{\varphi }_2^{\kappa }\left(t_0+\epsilon t_0^{1-\gamma }\right)-{\varphi }_2^{\kappa }\left(t_0\right)\right]$$

Dividing and multiplying by $\epsilon ,$ we have:

$${\left[\mathrm{\Phi }\left(t_0+\epsilon t_0^{1-\gamma }\right)\ominus \mathrm{\Phi }\left(t_0\right)\right]}^{\kappa }=\left[\frac{{\varphi }_1^{\kappa }\left(t_0+\epsilon t_0^{1-\gamma }\right)-{\varphi }_1^{\kappa }\left(t_0\right)}{\epsilon }\cdot \epsilon ,\frac{{\varphi }_2^{\kappa }\left(t_0+\epsilon t_0^{1-\gamma }\right)-{\varphi }_2^{\kappa }\left(t_0\right)}{\epsilon }\cdot \epsilon \right]$$

Similarly, we obtain:

$${\left[\mathrm{\Phi }\left(t_0\right)\ominus \mathrm{\Phi }\left(t_0-\epsilon t_0^{1-\gamma }\right)\right]}^{\kappa }=\left[\frac{{\varphi }_1^{\kappa }\left(t_0\right)-{\varphi }_1^{\kappa }\left(t_0-\epsilon t_0^{1-\gamma }\right)}{\epsilon }\cdot \epsilon ,\frac{{\varphi }_2^{\kappa }\left(t_0\right)-{\varphi }_2^{\kappa }\left(t_0-\epsilon t_0^{1-\gamma }\right)}{\epsilon }\cdot \epsilon \right]$$

Then

$$\begin{array}{lll}\underset{\epsilon \to 0^{+}}{lim}{\left[\mathrm{\Phi }\left(t_0+\epsilon t_0^{1-\gamma }\right)\ominus \mathrm{\Phi }\left(t_0\right)\right]}^{\kappa }& =& \left[\underset{\epsilon \to 0^{+}}{lim}\frac{{\varphi }_1^{\kappa }\left(t_0+\epsilon t_0^{1-\gamma }\right)-{\varphi }_1^{\kappa }\left(t_0\right)}{\epsilon }\cdot \underset{\epsilon \to 0^{+}}{lim}\epsilon \right.,\\ & & \left.\underset{\epsilon \to 0^{+}}{lim}\frac{{\varphi }_2^{\kappa }\left(t_0+\epsilon t_0^{1-\gamma }\right)-{\varphi }_2^{\kappa }\left(t_0\right)}{\epsilon }\cdot \underset{\epsilon \to 0^{+}}{lim}\epsilon \right]\end{array}$$

Similarly, we obtain:

$$\begin{array}{lll}\underset{\epsilon \to 0^{+}}{lim}{\left[\mathrm{\Phi }\left(t_0\right)\ominus \mathrm{\Phi }\left(t_0-\epsilon t_0^{1-\gamma }\right)\right]}^{\kappa }& =& \left[\underset{\epsilon \to 0^{+}}{lim}\frac{{\varphi }_1^{\kappa }\left(t_0\right)-{\varphi }_1^{\kappa }\left(t_0-\epsilon t_0^{1-\gamma }\right)}{\epsilon }\cdot \underset{\epsilon \to 0^{+}}{lim}\epsilon \right.,\\ & & \left.\underset{\epsilon \to 0^{+}}{lim}\frac{{\varphi }_2^{\kappa }\left(t_0\right)-{\varphi }_2^{\kappa }\left(t_0-\epsilon t_0^{1-\gamma }\right)}{\epsilon }\cdot \underset{\epsilon \to 0^{+}}{lim}\epsilon \right]\end{array}$$

Let $h=\epsilon t_0^{1-\gamma }.$ Then

$$\underset{h\to 0^{+}}{lim}{\left[\mathrm{\Phi }\left(t_0+h\right)\ominus \mathrm{\Phi }\left(t_0\right)\right]}^{\kappa }=\left[{\left({\varphi }_1^{\kappa }\right)}^{\left(\gamma \right)}\left(t_0\right)\cdot 0,{\left({\varphi }_2^{\kappa }\right)}^{\left(\gamma \right)}\left(t_0\right)\cdot 0\right]$$

Similarly, we obtain:

$$\underset{h\to 0^{+}}{lim}{\left[\mathrm{\Phi }\left(t_0-h\right)\right]}^{\kappa }={\left[\mathrm{\Phi }\left(t_0\right)\right]}^{\kappa }$$

which implies that

$$\underset{h\to 0^{+}}{lim}{\left[\mathrm{\Phi }\left(t_0+h\right)\right]}^{\kappa }={\left[\mathrm{\Phi }\left(t_0\right)\right]}^{\kappa }$$

Similary, we obtain:

$$\underset{h\to 0^{+}}{lim}{\left[\mathrm{\Phi }\left(t_0-h\right)\right]}^{\kappa }={\left[\mathrm{\Phi }\left(t_0\right)\right]}^{\kappa }$$

Hence, $\mathrm{\Phi }$ is continuous at $t_0$. □

Remark 2. If $\mathrm{\Phi }:\left(0,a\right)\to {\mathbb{R}}_{\mathcal{F}}$ is $\gamma$-differentiable and ${\mathrm{\Phi }}^{\left(\gamma \right)}$ for all $\gamma \in \left(0,1\right]$ is continuous, then we denote $\mathrm{\Phi }\in C^1\left(\left(0,a\right),{\mathbb{R}}_{\mathcal{F}}\right)$.

Theorem 6. Let $\gamma \in \left(0,1\right]$ and if $\mathrm{\Phi },\mathrm{\Psi }:\left(0,a\right)\to {\mathbb{R}}_{\mathcal{F}}$ are $\gamma$-differentiable and $\lambda \in \mathbb{R}$ then

1. $$T_{\gamma }\left(\mathrm{\Phi }+\mathrm{\Psi }\right)\left(t\right)=T_{\gamma }\left(\mathrm{\Phi }\right)\left(t\right)+T_{\gamma }\left(\mathrm{\Psi }\right)\left(t\right)$$E20

2. $$T_{\gamma }\left(\lambda \mathrm{\Phi }\right)\left(t\right)=\lambda T_{\gamma }\left(\mathrm{\Phi }\right)\left(t\right).$$E21

proof. We present the details only for case $\left(\mathrm{i}\right),$ since the other case is anlogous. Since $\mathrm{\Phi }$ is ${\gamma }_{\left(1\right)}$-differentiable it follows that $\mathrm{\Phi }\left(t+\epsilon t^{1-\gamma }\right)\ominus \mathrm{\Phi }\left(t\right)$ exists i.e. there exists $u_1\left(t,\epsilon t^{1-\gamma }\right)$ such that

$$\mathrm{\Phi }\left(t+\epsilon t^{1-\gamma }\right)=\mathrm{\Phi }\left(t\right)+u_1\left(t,\epsilon t^{1-\gamma }\right)$$E22

Analogously since $\mathrm{\Psi }$ is ${\gamma }_{\left(1\right)}$-differentiable there exists $v_1\left(t,\epsilon t^{1-\gamma }\right)$ such that

$$\mathrm{\Psi }\left(t+\epsilon t^{1-\gamma }\right)=\mathrm{\Psi }\left(t\right)+v_1\left(t,\epsilon t^{1-\gamma }\right)$$

and we get

$$\mathrm{\Phi }\left(t+\epsilon t^{1-\gamma }\right)+\mathrm{\Psi }\left(t+\epsilon t^{1-\gamma }\right)=\mathrm{\Phi }\left(t\right)+\mathrm{\Psi }\left(t\right)+u_1\left(t,\epsilon t^{1-\gamma }\right)+v_1\left(t,\epsilon t^{1-\gamma }\right)$$E23

that is the H-difference

$$\left(\mathrm{\Phi }\left(t+\epsilon t^{1-\gamma }\right)+\mathrm{\Psi }\left(t+\epsilon t^{1-\gamma }\right)\right)\ominus \left(\mathrm{\Phi }\left(t\right)+\mathrm{\Psi }\left(t\right)\right)=u_1\left(t,\epsilon t^{1-\gamma }\right)+v_1\left(t,\epsilon t^{1-\gamma }\right)$$E24

By similar reasoning we get that there exist $u_2\left(t,\epsilon t^{1-\gamma }\right)$ and $v_2\left(t,\epsilon t^{1-\gamma }\right)$ such that

$$\begin{array}{l}\mathrm{\Phi }\left(t\right)=\mathrm{\Phi }\left(t-\epsilon t^{1-\gamma }\right)+u_2\left(t,\epsilon t^{1-\gamma }\right)\\ \mathrm{\Psi }\left(t\right)=\mathrm{\Psi }\left(t-\epsilon t^{1-\gamma }\right)+v_2\left(t,\epsilon t^{1-\gamma }\right)\end{array}$$

and so

$$\left(\mathrm{\Phi }\left(t\right)+\mathrm{\Psi }\left(t\right)\right)=\left(\mathrm{\Phi }\left(t-\epsilon t^{1-\gamma }\right)+\mathrm{\Psi }\left(t-\epsilon t^{1-\gamma }\right)\right)+u_2\left(t,\epsilon t^{1-\gamma }\right)+v_2\left(t,\epsilon t^{1-\gamma }\right)$$

that is the H-difference

$$\left(\mathrm{\Phi }\left(t\right)+\mathrm{\Psi }\left(t\right)\right)\ominus \left(\mathrm{\Phi }\left(t-\epsilon t^{1-\gamma }\right)+\mathrm{\Psi }\left(t-\epsilon t^{1-\gamma }\right)\right)=u_2\left(t,\epsilon t^{1-\gamma }\right)+v_2\left(t,\epsilon t^{1-\gamma }\right)$$E25

We observe that

$$\begin{array}{ll}\underset{\epsilon \to 0^{+}}{lim}\frac{u_1\left(t,\epsilon t^{1-\gamma }\right)}{\epsilon }& =\underset{\epsilon \to 0^{+}}{lim}\frac{u_2\left(t,\epsilon t^{1-\gamma }\right)}{\epsilon }={\mathrm{\Phi }}^{\left(\gamma \right)}\left(t\right)\text{and}\\ \underset{\epsilon \to 0^{+}}{lim}\frac{v_1\left(t,\epsilon t^{1-\gamma }\right)}{\epsilon }& =\underset{\epsilon \to 0^{+}}{lim}\frac{v_2\left(t,\epsilon t^{1-\gamma }\right)}{\epsilon }={\mathrm{\Psi }}^{\left(\gamma \right)}\left(t\right).\end{array}$$

Finally, by multiplying (24) and (25) with $\frac{1}{\epsilon }$ and passing to limit with ${lim}_{\epsilon \to 0^{+}}$ we get that $\mathrm{\Phi }+\mathrm{\Psi }$ is ${\gamma }_{\left(1\right)}$-differentiable and $T_{\gamma }\left(\mathrm{\Phi }+\mathrm{\Psi }\right)\left(t\right)=T_{\gamma }\mathrm{\Phi }\left(t\right)+T_{\gamma }\mathrm{\Psi }\left(t\right)$ The case when $\mathrm{\Phi }$ and $\mathrm{\Psi }$ are ${\gamma }_{\left(2\right)}$-differentiable is similar to the previous one. □

Definition 3. Let $\mathrm{\Phi }\in C\left(\left(0,a\right),{\mathbb{R}}_{\mathcal{F}}\right)\cap L^1\left(\left(0,a\right),{\mathbb{R}}_{\mathcal{F}}\right),$ Define the fuzzy fractional integral for $\gamma \in \left(0,1\right].$

$$I_{\gamma }\left(\mathrm{\Phi }\right)\left(t\right)=I_1\left(t^{\gamma -1}\mathrm{\Phi }\right)\left(t\right)={\int }_0^t\frac{\mathrm{\Phi }}{s^{1-\gamma }}\left(s\right)\mathit{ds},$$E26

where the integral is the usual Riemann improper integral.

Theorem 7. $T_{\gamma }{I}_{\gamma }\left(\mathrm{\Phi }\right)\left(t\right)$, for $t\ge 0$, where $\mathrm{\Phi }$ is any continuous function in the domain of $I_{\gamma }$.

Proof. Since $\mathrm{\Phi }$ is continuous, then $I_{\gamma }\left(\mathrm{\Phi }\right)\left(t\right)$ is clearly conformable differentiable. Hence,

$$\begin{array}{c}{\left[T_{\gamma }{I}_{\gamma }\left(\mathrm{\Phi }\right)\left(t\right)\right]}^{\kappa }={\left[t^{1-\gamma }\frac{d}{\mathit{dt}}{I}_{\gamma }\left(\mathrm{\Phi }\right)\left(t\right)\right]}^{\kappa }\\ =\left[t^{1-\gamma }\frac{d}{\mathit{dt}}{\int }_0^t\frac{{\varphi }_1^{\kappa }\left(x\right)}{x^{1-\gamma }}\mathit{dx},t^{1-\gamma }\frac{d}{\mathit{dt}}{\int }_0^t\frac{{\varphi }_2^{\kappa }\left(x\right)}{x^{1-\gamma }}\mathit{dx}\right]\\ =\left[t^{1-\gamma }\frac{{\varphi }_1^{\kappa }\left(t\right)}{t^{1-\gamma }},t^{1-\gamma }\frac{{\varphi }_2^{\kappa }\left(t\right)}{t^{1-\gamma }}\right]\\ ={\left[\mathrm{\Phi }\left(t\right)\right]}^{\kappa }\end{array}$$

□

Theorem 8. Let $\gamma \in \left(0,1\right]$ and $\mathrm{\Phi }$ be $\gamma$-differentiable in $\left(0,a\right)$ and assume that the conformable derivative ${\mathrm{\Phi }}^{\left(\gamma \right)}$ is integrable over $\left(0,a\right)$. Then $\forall s\in \left(0,a\right)$ we have.

$$\mathrm{\Phi }\left(s\right)=\mathrm{\Phi }\left(a\right)+I_{\gamma }{\mathrm{\Phi }}^{\left(\gamma \right)}\left(t\right)$$E27

Proof. Let $\gamma \in \left(0,1\right]$ and $\kappa \in \left[0,1\right]$ be fixed. We will demonstrate this.

$${\mathrm{\Phi }}_{\kappa }\left(s\right)={\mathrm{\Phi }}_{\kappa }\left(a\right)+I_{\gamma }{\mathrm{\Phi }}_{\kappa }^{\left(\gamma \right)}$$E28

where ${\mathrm{\Phi }}_{\kappa }^{\left(\gamma \right)}$ is conformable derivative of ${\mathrm{\Phi }}_{\kappa },$ the equation is then obtained by applying Theorems 3 and 4.

$$\begin{array}{ll}{\mathrm{\Phi }}_{\kappa }\left(s\right)& ={\mathrm{\Phi }}_{\kappa }\left(a\right)+I_{\gamma }{\mathrm{\Phi }}_{\kappa }^{\left(\gamma \right)}\\ & ={\mathrm{\Phi }}_{\kappa }\left(a\right)+I_{\gamma }\left(t^{1-\gamma }{\mathrm{\Phi }}_{\kappa }^{\text{'}}\right)\end{array}$$

by (26) we have

$$\begin{array}{ll}{\mathrm{\Phi }}_{\kappa }\left(s\right)& ={\mathrm{\Phi }}_{\kappa }\left(a\right)+I_{\gamma }\left(t^{1-\gamma }{\mathrm{\Phi }}_{\kappa }^{\text{'}}\right)\\ & ={\mathrm{\Phi }}_{\kappa }\left(a\right)+{\int }_0^s{t}^{\gamma -1}\left(t^{1-\gamma }{\mathrm{\Phi }}_{\kappa }^{\text{'}}\right)\end{array}$$

So

$${\mathrm{\Phi }}_{\kappa }\left(s\right)={\mathrm{\Phi }}_{\kappa }\left(a\right)+{\int }_0^s{\mathrm{\Phi }}_{\kappa }^{\text{'}}$$E29

where ${\mathrm{\Phi }}_{\kappa }^{\text{'}}$ is the derivative of ${\mathrm{\Phi }}_{\kappa }$, For a fuzzy mapping, the (29) is likewise true $\mathrm{\Phi }:\left(0,a\right)\to {\mathbb{R}}_{\mathcal{F}}$. In [1], the equality (28) now follows Theorem (8).

4. Solutions via conformable differential inclusions

We consider the fuzzy conformable differential equation.

$$u^{\left(\gamma \right)}\left(t\right)=\mathrm{\Phi }\left(t,u\left(t\right)\right),u\left(0\right)=u_0,\gamma \in \left(0,1\right],$$E30

where $\mathrm{\Phi }:\left(0,a\right)\times {\mathrm{R}}_{\mathcal{F}}\to {\mathrm{R}}_{\mathcal{F}}$ is generated from a continuous function using Zadeh’s extension principle. $\psi :\left(0,a\right)\times \mathrm{R}\to \mathrm{R}.$

Let $\mathrm{\Phi }\left(t,u\right)$ can be calculated at the level, i.e., $\forall \kappa \in \left[0,1\right]$.

$${\left[\mathrm{\Phi }\left(t,u\right)\right]}^{\kappa }=\psi \left(t,{\left[u\right]}^{\kappa }\right)$$

for all $t\in \left(0,a\right),u\in {\mathrm{R}}_{\mathcal{F}}$ and $\kappa \in \left[0,1\right].$ We interpret the fuzzy initial value problem (30) as a set of differential inclusions, following Diamonds [7, 10].

$${\left(v^{\left(\gamma \right)}\right)}^{\kappa }\left(t\right)=\psi \left(t,v^{\kappa }\left(t\right)\right),v^{\kappa }\left(0\right)\in {\left[u_0\right]}^{\kappa }$$E31

The reachable sets, under reasonable assumptions.

$${\mathrm{\Upsilon }}_{\kappa }\left(t\right)=\left\{v^{\kappa }\left(t\right)|v^{\kappa }\mathit{\text{is}}a\mathit{\text{solution}}\mathit{\text{of}}\left(31\right)\right\},$$

are $\kappa$-cuts of a fuzzy set $u\left(t\right),$ which we call a solution of (30). If we assume that the solutions to the initial value problem are unique, ${\left(v^{\left(\gamma \right)}\right)}^{\kappa }\left(t\right)=\psi \left(t,v^{\kappa }\left(t\right)\right),v^{\kappa }\left(0\right)=v_0,$ it follows that ${\mathrm{\Upsilon }}_{\kappa }\left(t\right)=\left[w_1\left(t\right),w_2\left(t\right)\right],$ where.

$$\begin{array}{l}{\left(w^{\left(\gamma \right)}\right)}_1\left(t\right)=\psi \left(t,w_1\left(t\right)\right),w_1\left(0\right)=u_{01}^{\kappa }\text{and}\\ {\left(w^{\left(\gamma \right)}\right)}_2\left(t\right)=\psi \left(t,w_2\left(t\right)\right),w_2\left(0\right)=u_{02}^{\kappa }\end{array}$$

Theorem 9. The fuzzy solution and solution by differential inclusions solution using the first form are equivalent if $\psi$ is nondecreasing with respect to the second argument.

proof. For each $\kappa \in \left[0,1\right]$ and $\forall \gamma \in \left(0,1\right],$ we think $\left[u\left(t\right)\right)]{}^{\kappa }=\left[u_1^{\kappa }\left(t\right),u_2^{\kappa }\left(t\right)\right]$ and $\left[u\left(0\right)\right)]{}^{\kappa }=\left[u_{01}^{\kappa },u_{02}^{\kappa }\right].$ Since $g$ is continuous and

$${\left[\mathrm{\Phi }\left(t,u\left(t\right)\right)\right]}^{\kappa }=\psi \left(t,{\left[u\left(t\right))\right]}^{\kappa }\right)=\psi \left(t,\left[u_1^{\kappa }\left(t\right),u_2^{\kappa }\left(t\right)\right]\right),$$

then $\psi (t,\left[u_1^{\kappa }\left(t\right),u_2^{\kappa }\left(t\right))\right]$ is compact and connected i.e. a closed bounded interval.

As $\psi$ is nondecreasing, we have that

$$\psi \left(t,\left[u_1^{\kappa }\left(t\right),u_2^{\kappa }\left(t\right)\right]\right)=\left[\psi \left(t,u_1^{\kappa }\left(t\right)\right),\psi \left(t,u_2^{\kappa }\left(t\right)\right)\right]$$

As a result, the conformable differential system for boundary functions of fuzzy solution is uncoupled into two initial value problems:

$$\begin{array}{l}{\left(u^{\left(\gamma \right)}\right)}_1^{\kappa }\left(t\right)=\psi \left(t,u_1^{\kappa }\left(t\right)\right),u_1^{\kappa }\left(0\right)=u_{01}^{\kappa },\\ {\left(u^{\left(\gamma \right)}\right)}_2^{\kappa }\left(t\right)=\psi \left(t,u_2^{\kappa }\left(t\right)\right),u_2^{\kappa }\left(0\right)=u_{02}^{\kappa },\end{array}$$

This results in $u_1^{\kappa }=w_1$ and $u_2^{\kappa }=w_2$.

Now, we offer the following result as an extension of the preceding theorem to the class of differentiable functions with regard to the second form (17).

Theorem 10. If $\psi$ is nonincreasing with respect to the second argument then, using the derivative in the second form (17), the fuzzy solution of (30) and the solution via differential inclusions are identical.

proof. Let $\kappa \in \left[0,1\right]$ and $\gamma \in \left(0,1\right],$ we consider.

$$\left[u\left(t\right)\right)]{}^{\kappa }=\left[u_1^{\kappa }\left(t\right),u_2^{\kappa }\left(t\right)\right]\mathit{\text{and}}{\left[u\left(0\right))\right]}^{\kappa }=\left[u_{01}^{\kappa },u_{02}^{\kappa }\right].$$

So

$${\left[\mathrm{\Phi }\left(t,u\left(t\right)\right)\right]}^{\kappa }=\psi \left(t,{\left[u\left(t\right))\right]}^{\kappa }\right)=\psi \left(t,\left[u_1^{\kappa }\left(t\right),u_2^{\kappa }\left(t\right)\right]\right)$$

and $\psi$ is continuous, and $\psi \left(t,\left[u_1^{\kappa }\left(t\right),u_2^{\kappa }\left(t\right)\right]\right)$ is a closed bounded interval. Since $\psi$ is nonincreasing, it follows that

$$\psi \left(t,\left[u_1^{\kappa }\left(t\right),u_2^{\kappa }\left(t\right)\right]\right)=\left[\psi \left(t,u_2^{\kappa }\left(t\right)\right),\psi \left(t,u_1^{\kappa }\left(t\right)\right)\right]$$

Consequently, from (31), we have the conformable differential system.

$$\begin{array}{l}{\left(u^{\left(\gamma \right)}\right)}_2^{\kappa }\left(t\right)=\psi \left(t,u_2^{\kappa }\left(t\right)\right),u_2^{\kappa }\left(0\right)=u_{02}^{\kappa },\\ {\left(u^{\left(\gamma \right)}\right)}_1^{\kappa }\left(t\right)=\psi \left(t,u_1^{\kappa }\left(t\right)\right),u_1^{\kappa }\left(0\right)=u_{01}^{\kappa },\end{array}$$E32

If $u^{\left(\gamma \right)}\left(t\right)$ is consider in the form (2), we have that

$${\left[\left(u^{\left(\gamma \right)}\right)\left(t\right)\right]}^{\kappa }=\left[{\left(u^{\left(\gamma \right)}\right)}_2^{\kappa }\left(t\right),{\left(u^{\left(\gamma \right)}\right)}_1^{\kappa }\left(t\right)\right]=\left[\psi \left(t,u_2^{\kappa }\left(t\right)\right),\psi \left(t,u_1^{\kappa }\left(t\right)\right)\right]$$

The proof is complete after obtaining the differential system (32).

Example 1. Consider the fuzzy initial value problem.

$$u^{\left(\gamma \right)}\left(t\right)=u^2\left(t\right),u\left(0\right)=u_0$$E33

Where $u_0$ is a traingular fuzzy number ${\left[u_0\right]}^{\kappa }=\left[1+\kappa ,3-\kappa \right].$ Since $u^2$ is continuous and we are operating on ${\mathrm{R}}_{\mathcal{F}}$, we can solve the equation levelwise.

Since $u^2$ is increasing when $x>0$, we have to solve a conformable differential system for $\gamma \in \left(0,1\right]$.

$${\left(u_1^{\kappa }\right)}^{\left(\gamma \right)}\left(t\right)={\left(u_1^{\kappa }\right)}^2\left(t\right),u_1^{\kappa }\left(0\right)=1+\kappa ,$$E34

$${\left(u_2^{\kappa }\right)}^{\left(\gamma \right)}\left(t\right)={\left(u_2^{\kappa }\right)}^2\left(t\right),u_2^{\kappa }\left(0\right)=3-\kappa ,$$E35

where ${\left[u\left(t\right)\right]}^{\kappa }=\left[u_1^{\kappa },u_2^{\kappa }\right].$

The solutions are

$$u_1^{\kappa }\left(t\right)=\frac{-1-\kappa }{\left(\frac{t^{\gamma }}{\gamma }+\frac{t^{\gamma }}{\gamma }\kappa \right)-1}\text{and}{u}_2^{\kappa }\left(t\right)=\frac{-3+\kappa }{\left(3\frac{t^{\gamma }}{\gamma }-\frac{t^{\gamma }}{\gamma }\kappa \right)-1}$$

We can see this $u_2^{\kappa }\left(t\right)<\infty$ for $t<\frac{1}{3}$ and

$$0\le u_1^{\kappa }\left(t\right)\le u_2^{\kappa }\left(t\right)$$

for these values of $t.$

As a result, there is a fuzzy solution to the fuzzy initial value problem. $u\left(t\right)$ for $0\le t<\frac{1}{3}.$

5. Conclusion

The fuzzy conformable differential inclusions (FCDI) are introduced, which have been used by various authors to solve FDE for $\gamma =1$ [2, 6]. It also has the advantage that the solutions derived using FCDI appear to be more intuitive than other conformable derivative solutions first form (9), [19]. It’s also worth noting that this interpretation has a drawback in that we cannot discuss the fuzzy conformable derivative. Instead, we address this obstacle by utilizing the fuzzy conformable derivative in the second form (17), and the fuzzy solution and the solution via conformable differential inclusions are identical.

Conflicts of interest

The authors declare that they have no conflicts of interest.

Data availability

The data used to support the findings of this study are available from the corresponding author upon request.