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Register a new userAn analytic proof of the Malgrange-Sibuya theorem on the convergence of formal solutions of an ODE
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We propose an analytic proof of the Malgrange-Sibuya theorem concerning a sufficient condition of the convergence of a formal power series satisfying an ordinary differential equation. The proof is based on the majorant method and allows to estimate the radius of convergence of such a series.
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We propose a sufficient condition of the convergence of a complex power type formal series of the form $varphi=sum_{k=1}^{infty}alpha_k(x^{{rm i}gamma}),x^k$, where $alpha_k$ are functions meromorphic at the origin and $gammain{mathbb R}setminus{0}$, that satisfies an analytic ordinary differential equation (ODE) of a general type. An example of a such type formal solution of the third Painleve equation is presented and the proposed sufficient condition is applied to check its convergence.
A sufficient condition of the convergence of an exotic formal series (a kind of power series with complex exponents) solution to an ODE of a general form is proposed.
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Here we present some compliments to theorems of Gerard and Sibuya, on the convergence of multivariate formal power series solutions of nonlinear meromorphic Pfaffian systems. Their the most known results concern completely integrable systems with nondegenerate linear parts, whereas we consider some cases of non-integrability and degeneracy.
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