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

1 Department of General Education, Faculty of Science and Health Technology, Navamindradhiraj University, Bangkok 10300, Thailand
2 Department of Mathematics and General Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia
3 Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand

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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>a \ge 0,\\[0.3cm] \lim_{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 $\alpha\in \mathbb{C}$, $Re(\alpha)\ge 0$, $0<\rho\le 1$ 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.
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