Filippov’s theorem implies that, given an absolutely continuous function y: [t 0; T] → ℝd and a set-valued map F(t, x) measurable in t and l(t)-Lipschitz in x, for any initial condition x 0, there exists a solution x(·) to the differential inclusion x′(t) ∈ F(t, x(t)) starting from x 0 at the time t 0 and satisfying the estimation $$\left|x\left(t\right)-y\left(t\right)\right|⩽r\left(t\right)=\left|x_0-y\left(t_0\right)\right|e^{{\int }_{t_0}^{t}l\left(s\right)ds}+{\int }_{t_0}^t\gamma \left(s\right)e^{{\int }_s^{t}l\left(\tau \right)d\tau }ds,$$ where the function γ(·) is the estimation of dist(y′(t), F(t, y(t))) ≤ γ(t). Setting P(t) = x ∈ ℝn: |x −y(t)| ≤ r(t), we may formulate the conclusion in Filippov’s theorem...

We give a new proof of multisummability of formal power series solutions of a non linear meromorphic differential equation. We use the recent Malgrange-Ramis definition of multisummability. The first proof of the main result is due to B. Braaksma. Our method of proof is very different: Braaksma used Écalle definition of multisummability and Laplace transform. Starting from a preliminary normal form of the differential equation$$x\frac{d\overrightarrow{y}}{dx}={\overrightarrow{G}}_0\left(x\right)+\left[\lambda \left(x\right)+A_0\right]\overrightarrow{y}+x^{\mu }\overrightarrow{G}(x,\overrightarrow{y}),$$the idea of our proof is to interpret a formal power series solution...

Specialized literature concerning studies on Orbital Dynamics usually mentions the Gauss-Jackson or sum squared (∑2) method for the numerical integration of second order differential equations. However, as far as we know, no detailed description of this code is available and there is some confusion about the order of the method and its relation with the Störmer method. In this paper we present a simple way of deriving this algorithm and its corresponding analog for first order equations from the...

Let $p,q$ be analytic functions in the unit disk $𝒰$. For $\alpha \in [0,1)$ the authors consider the differential subordination and the differential equation of the Briot-Bouquet type: $$p^{1-\alpha }\left(z\right)+\frac{z{p}^{\text{'}}\left(z\right)}{\delta p^{\alpha }\left(z\right)+\lambda p\left(z\right)}\prec h\left(z\right),z\in 𝒰,$$