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

Weerawat Sudsutad; Jehad Alzabut; Chutarat Tearnbucha; Chatthai Thaiprayoon; Weerawat Sudsutad; Jehad Alzabut; Chutarat Tearnbucha; Chatthai Thaiprayoon

doi:10.3934/math.2020058

AIMS Mathematics

2020, Volume 5, Issue 2: 856-871. doi: 10.3934/math.2020058

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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

Received: 05 October 2019 Accepted: 30 December 2019 Published: 02 January 2020
MSC : 26A33, 34A08, 34C10

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.
Citation: Weerawat Sudsutad, Jehad Alzabut, Chutarat Tearnbucha, Chatthai Thaiprayoon. On the oscillation of differential equations in frame of generalized proportional fractional derivatives[J]. AIMS Mathematics, 2020, 5(2): 856-871. doi: 10.3934/math.2020058

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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.

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