TY - JOUR

T1 - Some qualitative properties of solutions to a nonlinear fractional differential equation involving two Φ-Caputo fractional derivatives

AU - Derbazi, Choukri

AU - Al-Mdallal, Qasem M.

AU - Jarad, Fahd

AU - Baitiche, Zidane

N1 - Publisher Copyright:
© 2022 the Author(s), licensee AIMS Press.

PY - 2022

Y1 - 2022

N2 - 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 Φ. 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 Φ-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.

AB - 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 Φ. 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 Φ-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.

KW - extremal solutions

KW - generalized Laplace transforms

KW - monotone iterative style

KW - multi-terms

KW - upper (lower) solutions

KW - Φ-Caputo fractional derivative

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U2 - 10.3934/math.2022552

DO - 10.3934/math.2022552

M3 - Article

AN - SCOPUS:85126991694

VL - 7

SP - 9894

EP - 9910

JO - AIMS Mathematics

JF - AIMS Mathematics

SN - 2473-6988

IS - 6

ER -