## 2 Preliminary Notes  
In this paper we apply the generalized proportional fractional derivative of Caputo type to differential equations. For better understanding of the results we will give some known-in-the-literature definitions and results about this type of derivative. \[({}_{0}\mathcal{I}^{\gamma\xi}y)(\sigma)=\frac{1}{\xi^{\gamma}\Gamma(\gamma)} \int_{0}^{\sigma}e^{\frac{\xi-1}{\xi}(\sigma-\tau)}(\sigma-\tau)^{\gamma-1}y( \tau)\;d\tau,\;\;\;\sigma\in(0,a],\;\;\gamma\geq 0,\;\xi\in(0,1],\]  
Suppose we have a point \(a\in\mathbb{R}:\;a<\infty\) and the function \(y:[0,a]\to\mathbb{R}\). The _generalized proportional fractional integral_ (GPFI) of the function \(y\) is defined by [10; 11]. \[({}_{0}\mathcal{I}^{\gamma\xi}y)(\sigma)=\frac{1}{\xi^{\gamma}\Gamma(\gamma)} \int_{0}^{\sigma}e^{\frac{\xi-1}{\xi}(\sigma-\tau)}(\sigma-\tau)^{\gamma-1}y( \tau)\;d\tau,\;\;\;\sigma\in(0,a],\;\;\gamma\geq 0,\;\xi\in(0,1],\]  
and the _generalized proportional Caputo fractional derivative_ (PFD) of function \(y\) is defined by the equality [10; 11]. \[ \begin{split}&({}_{0}^{C}\mathcal{D}^{\gamma\xi}y)(\sigma)=\frac{1 -\xi}{\xi^{1-\gamma}\Gamma(1-\gamma)}\int_{0}^{\sigma}e^{\frac{\xi-1}{\xi}( \sigma-\tau)}(\sigma-\tau)^{-\gamma}y(\tau)\;d\tau\\ &+\frac{\xi^{\gamma}}{\Gamma(1-\gamma)}\int_{0}^{\sigma}e^{\frac{ \xi-1}{\xi}(\sigma-\tau)}(\sigma-\tau)^{-\gamma}y^{\prime}(\tau)\;d\tau,\;\;\; \sigma\in(0,a],\;\;\gamma\in(0,1),\;\xi\in(0,1].\end{split}  \tag{1}\]The defined above PFD and GPFI are generalizations of the classical Caputo fractional derivative of order \(\gamma\in(0,1)\) and Riemann-Liouville fractional integral of order \(\gamma>0\), respectively (for more information, see, for example, the classical book [25]). **Remark 1** (see [10] [Remark 3.2]). : _The relations \((\overset{\mathbb{C}}{0}\mathcal{D}^{\gamma,\hat{\xi}}K)(\sigma)\
eq 0\) with \(K\
eq 0\) being a constant and \((\overset{\mathbb{C}}{0}\mathcal{D}^{\gamma,\hat{\xi}}e^{\frac{\xi-1}{\xi}( \cdot)})(\sigma)=0\) for \(\sigma>0\) hold. Note that these properties of PFD are totally different than the well known properties of ordinary derivatives of a constant. These properties, combined with the memory properties of PFD, gives us additionally better opportunities in the application for modeling real world phenomena._  
**Lemma 1** (Proposition 5.2 [10]). : _For \(\xi\in(0,1]\) and \(\gamma\in(0,1)\) we have the equality for PFD_  
\[\overset{\mathbb{C}}{0}\mathcal{D}^{\gamma,\hat{\xi}}(e^{\frac{\xi-1}{\xi}t} t^{\beta-1})(\sigma)=\frac{\xi^{\gamma}\Gamma(\beta)}{\Gamma(\beta-\gamma)}e^{ \frac{\xi-1}{\xi}\sigma}\sigma^{\beta-1-\gamma},\ \ \beta>0.\]  
**Lemma 2** (Lemma 5 [26]). : _If we have a function \(\varphi\in C([0,a],\mathbb{R})\), and there exists a number \(\eta\in(0,a):\ \varphi(\eta)=0\), and \(\varphi(\sigma)<0\) for \(0\leq\sigma<\eta\). Then if there exists \((\,\overset{\mathbb{C}}{0}\mathcal{D}^{\gamma,\hat{\xi}}\varphi)(\eta)\), then the inequality \((\,\overset{\mathbb{C}}{0}\mathcal{D}^{\gamma,\hat{\xi}}\varphi)(\eta)>0\) holds._  
Consider the classes of functions:  
\[ \begin{array}{l}C^{\gamma,\hat{\xi}}[0,a]=\{\varphi:[0,a]\to\mathbb{R}:\ \ ( \overset{\mathbb{C}}{0}\mathcal{D}^{\gamma,\hat{\xi}}\varphi)(\sigma)\ \ \text{exists for all}\ \ \sigma\in(0,a]\},\\ I^{\gamma,\hat{\xi}}[0,a]=\{\varphi:[0,a]\to\mathbb{R}:\ \ (\overset{ \mathbb{R}}{0}\mathcal{D}^{\gamma,\hat{\xi}}\varphi)(\sigma)\ \ \text{exists for all}\ \sigma\in(0,a]\}.\end{array} \]  
For any function \(\varphi\in C^{\gamma,\hat{\xi}}[0,a]\) and \(\,\overset{\mathbb{C}}{0}\mathcal{D}^{\gamma,\hat{\xi}}\varphi(. )\in\Gamma^{ \gamma,\hat{\xi}}[0,a]\) with \(\xi\in(0,1],\gamma\in(0,1)\) we have the following result:  
**Proposition 1** (Theorem 5.3 [10]). : _Let \(\xi\in(0,1]\) and \(\gamma\in(0,1)\).