We study differential equations $F(y,...,y^{\left(n\right)})=0$ where $F$ is a formal series in $y,y^{\prime },...,y^{\left(n\right)}$ with coefficients in some field of generalized power series${𝕂}_r$ with finite rank $r\in {\mathbb{N}}^{*}$. Our purpose is to express the support $\mathrm{Supp}{y}_0$, i.e. the set of exponents, of the elements $y_0\in {𝕂}_r$ that are solutions, in terms of the supports of the coefficients of the equation, namely $\mathrm{Supp}F$.

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...

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: $$p^{1-\alpha }\left(z\right)+\frac{z{p}^{\text{'}}\left(z\right)}{\delta p^{\alpha }\left(z\right)+\lambda p\left(z\right)}\prec h\left(z\right),z\in 𝒰,$$