Cícero L. Frota | Searcharxiv

arXiv2025

A General Version of Carathéodory's Existence and Uniqueness Theorem

In this paper, we establish a general version of Carathéodory's existence and uniqueness theorem for a semilinear system of integro-differential equations arising from differential equations with distinct orders of Caputo fractional derivative. The main result of our work demonstrates that the integrability order of the Carathéodory function $f$ must be at least greater than the maximum of the reciprocals of all differentiation orders in the system; otherwise, even the existence of a solution cannot be guaranteed.

math.CA↗

arXiv2023

On the Initial Boundary Value Problem to the Time-Fractional Wave Equation with Acoustic Boundary Conditions

This paper is concerned with the study of the well-posedeness for the initial boundary value problem to the time-fractional wave equation with acoustic boundary conditions. The problem is considered in a bounded and connected domain $Ω\subset {\mathbb{R}^{n}}$, $n \geq 2$, which includes simply connected regions. The boundary of $Ω$ is made up of two disjoint pieces $Γ_{0}$ and $Γ_{1}.$ Homogeneous Dirichlet conditions are enforced on $Γ_0$, while acoustic boundary conditions are considered on $Γ_1$. To establish our main result, we employ the Faedo-Galerkin method and successfully solve a general system of time-fractional ordinary differential equations which extends the scope of the classical Picard-Lindelöf theorem.

math.AP↗