  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.
