Some qualitative properties of solutions to a nonlinear fractional differential equation involving two $ \Phi $-Caputo fractional derivatives

Choukri Derbazi; Qasem M. Al-Mdallal; Fahd Jarad; Zidane Baitiche; Choukri Derbazi; Qasem M. Al-Mdallal; Fahd Jarad; Zidane Baitiche

doi:10.3934/math.2022552

AIMS Mathematics

2022, Volume 7, Issue 6: 9894-9910. doi: 10.3934/math.2022552

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Research article

Some qualitative properties of solutions to a nonlinear fractional differential equation involving two $ \Phi $-Caputo fractional derivatives

1.
Laboratoire Equations Différentielles, Department of Mathematics, Faculty of Exact Sciences, Frères Mentouri University, P. O. Box 325, Constantine 25017, Algeria
2.
Department of Mathematical Sciences, United Arab Emirates University, P. O. Box 15551, Al-Ain, United Arab Emirates
3.
Department of Mathematics, Çankaya University, Ankara 06790, Turkey
4.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 12 October 2021 Revised: 20 February 2022 Accepted: 01 March 2022 Published: 18 March 2022
MSC : 26A33, 34A08

The momentous objective of this work is to discuss some qualitative properties of solutions such as the estimate of the solutions, the continuous dependence of the solutions on initial conditions and the existence and uniqueness of extremal solutions to a new class of fractional differential equations involving two fractional derivatives in the sense of Caputo fractional derivative with respect to another function $ \Phi $. Firstly, using the generalized Laplace transform method, we give an explicit formula of the solutions to the aforementioned linear problem which can be regarded as a novelty item. Secondly, by the implementation of the $ \Phi $-fractional Gronwall inequality, we analyze some properties such as estimates and continuous dependence of the solutions on initial conditions. Thirdly, with the help of features of the Mittag-Leffler functions (MLFs), we build a new comparison principle for the corresponding linear equation. This outcome plays a vital role in the forthcoming analysis of this paper especially when we combine it with the monotone iterative technique alongside facet with the method of upper and lower solutions to get the extremal solutions for the analyzed problem. Lastly, we present some examples to support the validity of our main results.
- $ \Phi $-Caputo fractional derivative,
- multi-terms,
- generalized Laplace transforms,
- extremal solutions,
- monotone iterative style,
- upper (lower) solutions
Citation: Choukri Derbazi, Qasem M. Al-Mdallal, Fahd Jarad, Zidane Baitiche. Some qualitative properties of solutions to a nonlinear fractional differential equation involving two $ \Phi $-Caputo fractional derivatives[J]. AIMS Mathematics, 2022, 7(6): 9894-9910. doi: 10.3934/math.2022552

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Abstract

The momentous objective of this work is to discuss some qualitative properties of solutions such as the estimate of the solutions, the continuous dependence of the solutions on initial conditions and the existence and uniqueness of extremal solutions to a new class of fractional differential equations involving two fractional derivatives in the sense of Caputo fractional derivative with respect to another function $ \Phi $. Firstly, using the generalized Laplace transform method, we give an explicit formula of the solutions to the aforementioned linear problem which can be regarded as a novelty item. Secondly, by the implementation of the $ \Phi $-fractional Gronwall inequality, we analyze some properties such as estimates and continuous dependence of the solutions on initial conditions. Thirdly, with the help of features of the Mittag-Leffler functions (MLFs), we build a new comparison principle for the corresponding linear equation. This outcome plays a vital role in the forthcoming analysis of this paper especially when we combine it with the monotone iterative technique alongside facet with the method of upper and lower solutions to get the extremal solutions for the analyzed problem. Lastly, we present some examples to support the validity of our main results.

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