Painlevé ’ s theorem extended

we extend Painlevé’s determinateness theorem from the theory of ordinary differential equations in the complex domain allowing more general ’multiple-valued’ Cauchy’s problems. We study C0−continuability (near singularities) of solutions. Foreword and preliminaries In this paper we slightly improve Painlevé’s determinateness theorem (see [HIL], th.3.3.1), investigating the C0−continuability of the solutions of finitely ’multiplevalued’, meromorphic Cauchy’s problems. In particular, will shall be interested in phenomena taking place when the attempt of continuating a solution along an arc leads to singularities of the known terms: we shall see that, under not too restricting hypotheses, this process will converge to a limit. Of course we shall formalize ’multiple valuedness’ by means of Riemann domains over regions in C (see also [GRO], p.43 ff). Branch points will be supposed to lie on algebraic curves. We recall that in the classical statement of the theorem ’multiple valuedness’ in the known term is allowed with respect to the independent variable only. The following theorem extends to the complex domain the so called ’singlesequence criterion’ from the theory of real o.d.e.’s (see e.g.[GIU], th. 3.2); a technical lemma ends the section; the local existence-and-uniqueness theorem is reported in the appendix. Theorem 1 Let W be a C -valued holomorphic mapping, solution of the equation W (z) = F (W (z), z) in V ⊂ C , where F is a C -valued holomorphic mapping in a neighbourhood of graph(W ). Let z∞ ∈ ∂V, suppose that there exists a sequence {zn} → z∞, such that, set W (zn) := Wn, limn→∞Wn = W∞ ∈ C and that F is holomorphic at (W∞, z∞): then W admits analytical continuation up to z∞. Proof: we deal only with the case N = 1: we can find a > 0 and b > 0 such that the Taylor’s developments ∑ ∞ k,l=0 ckln(W − Wn)(z − zn) at (Wn, zn) of F are absolutely and uniformly convergent in all closed bidiscs D((Wn, zn), a, b). By means of Cauchy’s estimates we can find an upper bound T for ∑ ∞ k,l=0 |ckln|ab; by theorem 2.5.1 of [HIL] the solutions Sn of W ′ = F (W, z), W (zn) = Wn have radius of convergence at least a(1−e−b/2aT ) := σ. Thus there exists M such that z∞ ∈ D(zM , σ); by continuity, SM(z∞) = W0, and, by uniqueness, SM = S∞ in D(zM , σ) ∩ D(z∞, σ), i.e. W admits analytical continuation up to z∞.