On the oscillation of differential equations in frame of generalized proportional fractional derivatives

Abstract

In this paper, sufficient conditions are established for the oscillation of all solutions of generalized proportional fractional differential equations of the form $ \begin{equation} \left\{ \begin{array}{l} {_{a}D}^{\alpha, \rho}x(t) + \xi_1(t,x(t)) = \mu(t) + \xi_2(t,x(t)),\quad t \gt a \ge 0,\\[0.3cm] \lim\limits_{t\to a^{+}} {_{a}I}^{j-\alpha, \rho}x(t) = b_j,\quad j = 1,2,\ldots,n, \end{array} \right. \end{equation} $ where $n = \lceil \alpha \rceil$, ${_{a}D}^{\alpha, \rho}$ is the generalized proportional fractional derivative operator of order $\alpha\in \mathbb{C}$, $Re(\alpha)\ge 0$, $0 \lt \rho\le 1$ in the Riemann-Liouville setting and ${_{a}I}^{\alpha, \rho}$ is the generalized proportional fractional integral operator. The results are also obtained for the generalized proportional fractional differential equations in the Caputo setting. Numerical examples are provided to illustrate the applicability of the main results.