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

Weerawat Sudsutad; Jehad Alzabut; Chutarat Tearnbucha; Chatthai Thaiprayoon

doi:10.3934/math.2020058

AIMS Mathematics, 2020, 5(2): 856-871. doi: 10.3934/math.2020058. Research article Special Issues

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

Weerawat Sudsutad, Jehad Alzabut, Chutarat Tearnbucha, Chatthai Thaiprayoon

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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Special Issues: 2nd International Conference on Mathematical Modeling, Applied Analysis and Computation (ICMMAAC-19), August 8–10, 2019, JECRC University, Jaipur, India

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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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Keywords: proportional fractional integral; proportional fractional derivative; proportional fractional operator; fractional differential equation; oscillation theory

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

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This article has been cited by:

1. Jehad Alzabut, James Viji, Velu Muthulakshmi, Weerawat Sudsutad, Oscillatory Behavior of a Type of Generalized Proportional Fractional Differential Equations with Forcing and Damping Terms, Mathematics, 2020, 8, 6, 1037, 10.3390/math8061037

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