The first aim of this work is to establish a Peano's type existence theorem for an initial value problem involving complex fractional derivative, and the second is, as a consequence of this theorem, to give a partial answer to the local existence of the continuous solution for the following problemMoreover, in the special cases of considered problem, we investigate the some geometric properties of the solutions.

In this note, an extensive result consisting of several relations between certain inequalities and normalized analytic functions is first stated and some consequences of the result together with some examples are next presented. For the proof of the presented result, some of the assertions indicated in [5], [8] and [11] along with the results in [3] and [4] are also considered.

The uterine electromyogram, also called electrohysterogram (EHG), is the electrical signal generated by uterine contractile activity. The EHG has been considered an expanding technique for pregnancy monitoring and preterm risk evaluation. Data were collected on the abdominal surface. It has been speculated the effect of the placenta location on the characteristics of the EHG. In this work, a preliminary exploration method is proposed using the average spectra of Alvarez waves contractions of subjects with anterior and non-anterior placental position as a basis for the triple-dispersion Cole model that provides a best fit for these two cases. This leads to the uterine impedance estimation for these two study cases. Non-linear least square fitting (NLSF) was applied for this modelling process, which produces electric circuit fractional models’ representations. A triple-dispersion Cole-impedance model was used to obtain the uterine impedance curve in a frequency band between 0.1 and 1 Hz. A proposal for the interpretation relating the model parameters and the placental influence on the myometrial contractile action is provided. This is the first report regarding in silico estimation of the uterine impedance for cases involving anterior or non-anterior placental positions.

<abstract><p>The first objective of this article is to discuss the local existence of the solution to an initial value problem involving a non-linear differential equation in the sense of Riemann-Liouville fractional derivative of order $ \sigma\in(1, 2), $ when the nonlinear term has a singularity at zero of its independent argument. Hereafter, by using some tools of Lebesgue spaces such as Hölder inequality, we obtain Nagumo-type, Krasnoselskii-Krein-type and Osgood-type uniqueness theorems for the problem.</p></abstract>

We review the direct and inverse Laplace transforms on non-uniform time scales. We introduce full backward-compatible unilateral Laplace transforms and studied their properties. We also present the corresponding inverse integrals and some examples.