## 3 Main Theoretical Results  
### Mild Solutions of the Generalized Proportional Caputo Fractional Differential Equation  
First, we will investigate the linear case and we will provide an explicit solution to the corresponding initial value problem. Consider the linear scalar fractional differential equation with the PFD and an initial condition (IVPL)  
\[ \begin{array}{l}(\overset{\mathbb{C}}{0}\mathcal{D}^{\gamma,\hat{\xi}} \varphi)(\sigma)=-\lambda\,\varphi(t)+\Phi(t),\ \ \ \sigma\in(0,a],\\ \varphi(0)=\varphi_{0}\end{array}  \tag{2}\]  
where \(\varphi_{0}\in\mathbb{R}\), \(\xi\in(0,1]\), \(\gamma\in(0,1)\), \(\lambda\in\mathbb{R}\) is a constant, \(\Phi\in C([0,a],\mathbb{R})\). **Lemma 3** (see Example 5.7 [10]). : _Let the function \(\Phi\in I^{\gamma,\xi}[0,a]\). Then the IVPL (2) has a unique solution \(\varphi\in C^{\gamma,\xi}[0,a]\) given by the equality_  
\[ \begin{split}&\varphi(\sigma)=\varphi_{0}e^{(\xi-1)\frac{\varphi}{ \xi}}E_{\gamma}\Big{(}-\lambda\big{(}\frac{\sigma}{\xi}\big{)}^{\gamma}\Big{)} \\ &\quad+\frac{1}{\xi^{\gamma}\Gamma(\gamma)}\int_{0}^{\sigma}( \sigma-\tau)^{\gamma-1}e^{(\xi-1)(\frac{\sigma-\tau}{\xi})}E_{\gamma,\gamma}( -\lambda\big{(}\frac{\sigma-\tau}{\xi}\big{)}^{\gamma})\Phi(\tau)\ d\tau,\ \ \sigma\in(0,a],\end{split}  \tag{3}\]  
_where the notations \(E_{\gamma}(. )\) and \(E_{\gamma,\gamma}(. )\) are used for the Mittag-Leffler functions with one and two parameters, respectively._  
Consider the nonlinear scalar differential equation with PFD and a nonlinear boundary condition (GPDE):  
\[ \begin{split}&(\zeta_{0}^{C}\mathcal{D}^{\gamma,\xi}y)(\sigma)= \vartheta(\sigma,y(\sigma)),\quad\sigma\in(0,a],\\ &\Delta(y(0),y(a))=0,\end{split}  \tag{4}\]  
where \(\xi\in(0,1],\gamma\in(0,1)\), \(\vartheta:[0,a]\times\mathbb{R}\to\mathbb{R}\) and \(\Delta:\mathbb{R}\times\mathbb{R}\to\mathbb{R}\). **Remark 2**. : _Any function \(y\in C^{\gamma,\xi}[0,a]\) satisfying both equalities (4) is called a solution of GPDE (4)._  
Let \(\lambda,\mu>0\) be constants (to be determined later). For any solution \(y\in C^{\gamma,\xi}[0,T]\) of GPDE (4) we define \(\psi_{y}=y(0)-\frac{1}{\mu}\Delta(y(0),y(a))\). Then GPDE (4) can be equivalently written in the form  
\[ \begin{split}&(\zeta_{0}^{C}\mathcal{D}^{\gamma,\xi}y)(\sigma)=- \lambda y(\sigma)+\mathrm{Y}(\sigma,y(\sigma)),\quad\sigma\in[0,a],\\ & y(0)=\psi_{y},\end{split}  \tag{5}\]  
where  
\[\mathrm{Y}(\sigma,y)=\vartheta(\sigma,y)+\lambda y,\ \ y\in\mathbb{R},\ \sigma\in(0,a]. \tag{6}\]  
Note the problem (5) is an initial value problem for the nonlinear differential equation with PFD. Based on the explicit solution (3) of the IVPL (2) we will define a mild solution of (5) and its equivalent GPDE (4). Define the integral fractional operator \(\Omega:C[0,a]\to C[0,a]\) by  
\[ \begin{split}&(\Omega\varphi)(\sigma)=\Big{(}\varphi(0)-\frac{1}{ \mu}\Delta(\varphi(0),\varphi(a))\Big{)}e^{(\xi-1)\frac{\varphi}{\xi}}E_{ \gamma}(-\lambda\big{(}\frac{\sigma}{\xi}\big{)}^{\gamma})\\ &\quad+\frac{1}{\xi^{\gamma}\Gamma(\gamma)}\int_{0}^{\sigma}( \sigma-\tau)^{\gamma-1}e^{(\xi-1)(\frac{\sigma-\tau}{\xi})}E_{\gamma,\gamma}( -\lambda\big{(}\frac{\sigma-\tau}{\xi}\big{)}^{\gamma})\Big{(}\vartheta( \tau,\varphi(\tau))+\lambda\varphi(\tau)\Big{)}d\tau,\\ &\qquad\qquad\sigma\in[0,a],\end{split}  \tag{7}\]  
where \(\lambda,\mu\) are given positive constants (we will define them later). The mild solutions of fractional differential equations play an important role in the study of various qualitative properties of the solutions. Mild solutions are defined by the application of an appropriate integral operator deeply connected with the studied fractional differential equation (see, for example, Definition 3.1. Ref. [27] for fractional neutral evolution equations, Ref. [28] for fractional evolution equation, Ref. [29] for Caputo-Hadamard fractional differential equations). **Definition 1**. : _Any fixed point \(\varphi\in C[0,a]\) (if any) of the operator \(\Omega\) is called a mild solution of GPDE (4)._  
**Theorem 1**.