On the Cauchy-Kovalevskaya theorem for Caputo fractional differential equations - Jornet, Marc (Physica. D, 2024)
We aim at proving the Cauchy-Kovalevskaya theorem for systems of nonlinear fractional differential equations in the Caputo sense, not necessarily polynomial or compartmental. Essentially, the theorem states that if the input function has a Taylor series, then the solution can be locally expressed as a fractional power series. We use, in the real field, the method of majorants and the analytic version of the implicit-function theorem, in a way that circumvents difficulties associated to fractional calculus. Some corollaries on continuity are derived, with computational examples for illustration, and a discussion on fractional partial differential equations is included with a case study and counterexamples. Open problems are raised at the end. •We investigate systems of nonlinear Caputo fractional differential equations.•We prove the Cauchy-Kovalevskaya theorem on convergence of fractional power series.•We use majorants and the implicit-function theorem adequately.•Corollaries on continuity and computational examples (e.g. Gompertz) are included.•We address and discuss Caputo fractional PDEs (quasilinear, heat and Schrödinger).