## 2 Preliminaries  
In this section, we give some basic definitions and results about fuzzy numbers as well as fractional derivatives and integrals which are needed in the article. **Definition 1**. : _Let \(\mathbb{R}\) be the set of real numbers and \(C=\left\{(z,\phi_{C(z)}):z\in\mathbb{R}\right\}\) be the fuzzy set on \(\mathbb{R}\) where \(\phi_{C(z)}\in[0,1]\), that is,_  
\[\phi_{C}:\mathbb{R}\to[0,1]\] \[z\mapsto\phi_{C}(z).\]  
**Definition 2**. : _A fuzzy set \(C\) with its (membership) function \(\phi:\mathbb{R}\to\) [0,1] is a fuzzy number if it is normal (there exists \(z_{0}\in\mathbb{R}\), such that \(\phi(z_{0})=1\)), fuzzy convex, or upper semi-continuous, and the closure of its r-level \([\phi]_{0}=\) i.e., \(\{x\in\mathbb{R}:\phi(z)>0\}\) is a compact set. The space of all real fuzzy numbers is denoted by \(\mathbb{R}_{\mathbb{F}}\)._  
**Remark 1** ([11]). : _For any \(\phi\in\mathbb{R}_{\mathbb{F}}\), we have the lower and upper functions \(\underline{\phi},\overline{\phi}_{r}\) that satisfy_  
\[\phi_{r}=[\underline{\phi},\overline{\phi}_{r}],\]  
_for any \(r\in[0,1]\), where functions \(\underline{\phi},\overline{\phi}_{r}:[0,1]\to\mathbb{R}\) are bounded (non-decreasing and non-increasing, respectively), with a left continuous function in (0,1] and a right continuous function at 0. Also, we have \(\underline{\phi}_{r}\leq\overline{\phi}_{r}\)._  
**Definition 3**. : _The fuzzy metric \(d_{\infty}:\mathbb{R}_{\mathbb{F}}\times\mathbb{R}_{\mathbb{F}}\to\mathbb{R}^{+ }\bigcup\{0\}\) is defined by_  
\[d_{\infty}(\phi,\psi)=\sup_{0\leq r\leq 1}\max\Bigl{\{}|\underline{\phi}_{r}- \underline{\psi}_{r}|,|\overline{\phi}_{r}-\overline{\psi}_{r}|\Bigr{\}}=\sup_ {0\leq r\leq 1}d_{H}([\phi_{r}],[\psi_{r}]),\]  
_where \(d_{H}\) is the classical Hausdorff distance between real intervals._  
**Proposition 1** ([11]). : _Suppose that \(\phi,\psi,\eta,\phi\in\mathbb{R}_{\mathbb{F}}\) and \(\lambda\in\mathbb{R}\). Then, we have the following results:_  
_(1)_ \(d_{\infty}(\phi\ominus\psi,\eta\ominus\theta)\leq d_{\infty}(\phi,\eta)+d_{ \infty}(\psi,\theta)\)_, and the_ \(H\)_-differences_ \(\phi\ominus\psi,\eta\ominus\theta\) _exist._  
_(2)_ \(d_{\infty}(\theta\ominus\psi,\eta\oplus\theta)\leq d_{\infty}(\phi,\eta)+d_{ \infty}(\psi,\theta)\)_._  
_(3)_ \(d_{\infty}(\lambda\odot\phi,\lambda\odot\psi)=|\lambda|d_{\infty}(\phi,\psi)\)_._  
_(4)_ \(d_{\infty}(\phi\ominus\eta,\psi\oplus\eta)\leq d_{\infty}(\phi,\psi)\)_._  
**Definition 4**. : _Let \(\xi_{0}\in J\), \(h\in\mathbb{R}\) such that \(\xi_{0}+h\in J\) and \(f\in C(J,\mathbb{R}_{\mathbb{F}})\). If there exists \(f^{\prime}(\xi_{0})\in\mathbb{R}_{\mathbb{F}}\) such that_  
\[f^{\prime}(\xi_{0})=\lim_{h\to 0}\frac{1}{h}\otimes[f(\xi_{0}+h)\ominus_{gH} f(\xi_{0})],\]  
_then we say that \(f\) is \(gH\)-differentiable at \(\xi_{0}\). Let \(C^{1}(J,\mathbb{R}_{\mathbb{F}})\) be the space of continuously \(gH\)-differentiable functions on \(J\)._  
_Suppose that \(f\in C(J,\mathbb{R}_{\mathbb{F}})\) is \(gH\)-differentiable at \(\xi_{0}\in J\), \([f(\xi)]_{\alpha}=[f_{\alpha}^{-}(\xi),f_{\alpha}^{+}(\xi)]\) for all \(0\leq\alpha\leq 1\), \(\xi\in J\). Then, we say that \(f\) is (i)-\(gH\) differentiable at \(\xi_{0}\) if \([f^{\prime}(\xi_{0})]_{\alpha}=[(f_{\alpha}^{-})^{\prime}(\xi_{0}),(f_{\alpha} ^{+})^{\prime}(\xi_{0})]\), for all \(\alpha\in[0,1]\)._  
**Remark 2**. : _The space \(C^{1}(J,\mathbb{R}_{\mathbb{F}})\) denotes the set of fuzzy-valued functions on \(J\) that are continuous and generalized-Hukuhara-differentiable, with continuous derivatives in the fuzzy sense. Equipped with the metric \(d_{\infty}\) defined in Definition 3, the space \(C^{1}(J,\mathbb{R}_{\mathbb{F}})\) is a complete metric space, ensuring the applicability of fixed-point theorems and other analytical tools._  
**Definition 5**. :
1. _The subset_ \(C\) _of_ \(\mathbb{R}_{\mathbb{F}}\) _is compact-supported if there is a compact set_ \(U\in\mathbb{R}\) _such that_ \(U\) _consists of all sets_ \([\phi]_{0}\)_, where_ \(\phi\in C\)_._
2.