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

The momentous objective of this work is to discuss some qualitative properties of solutions such as the estimate on the solutions, the continuous dependence of the solutions on initial conditions as well as the existence and uniqueness of extremal solutions for a new class of fractional differential equations involving two fractional derivatives in the sense of Caputo fractional derivative with respect to a new function $\Phi$. Firstly, by using the generalized Laplace transform method, we give an explicit formula of the solutions for 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 (M-LFs) 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 offer some examples to confirm the validity of our main results.