Multisummability of formal power series solutions of nonlinear meromorphic differential equations

A new proof of multisummability of formal solutions of non linear meromorphic differential equations

“…This definition is slightly different from those of [I], [2], [3], [8] : one replaces the notion of multisummability in one direction by the notion of multisummability on a family {Ji,..., In} of nested closed arcs of the unit circle S (or more generally of its universal covering 5'). In this paragraph we will use the definitions and notations of [7].…”

Analytic continuation of solutions of some nonlinear convolution partial differential equations

“…In order to overcome the difficulty of the case where λ i (0) = 0 occurs for some i, we will employ the same method as in Braaksma [3] and Ouchi [9]. We note that if I = p i=1 I i for some open intervals I i (i = 1, 2, .…”

“…We denote by H(S I (r] ∪ ∆ t0 (r)) the set of all functions f (t) which are continuous on S I (r]∪∆ t0 (r) and holomorphic in S I (r)∪∆ t0 (r). In order to prove the analytic continuation in a local region, Braaksma [3] and Ouchi [9] have used a new convolution (f * k g)(t) of two functions f (t) and g(t) in H(S I (r] ∪ ∆ t0 (r)). Let us recall its definition.…”

“…The multisummability of formal solutions of general ordinary differential equations was first proved by Braaksma [3]; different proofs were given by many authors (see Balser [1,2], Ramis-Sibuya [10] and their references). In the proof of Braaksma [3], the key point of the proof is that he proved an analytic continuation property of a solution of the convolution equation which is obtained by Borel transformation of the ordinary differential equation.…”

“…In the proof of Braaksma [3], the key point of the proof is that he proved an analytic continuation property of a solution of the convolution equation which is obtained by Borel transformation of the ordinary differential equation.…”