Monomial summability and doubly singular differential equations

Abstract

International audienceIn this work, we consider systems of differential equations that are doubly singular, i.e. that are both singularly perturbed and exhibit an irregular singular point. Under a condition that also guanrantees the existence of a unique formal solution, we show that this formal solution is monomially summable, i.e. summable with respect to the monomial in the independent variable and in the parameter in a (new) sense that will be defined. As a preparation, Poincaré asymptotics and Gevrey asymptotics in a monomial are studied

Tauberian properties for monomial summability with applications to Pfaffian systems

“…In our context, the main result obtained by the authors in [2] is the following statement on summability of formal solutions of doubly singular equations. We shall expect that this result joint with Theorem 3.8 lead to a direct generalization of Theorem 4.2.…”

“…By induction on n. If n = 1 the statement is just Theorem 3.8, (2). Suppose the statement is true for n − 1 and let us prove it for n. If p j /p 0 = q j /q 0 = k 0 /k j hold for all j = 1, ..., n thenf 0 is k 0 −summable in x p0 1 x q0 2 .…”

“…We recall briefly the basic notions of asymptotic expansions, Gevrey asymptotic expansions and summability in the case of one variable and their extensions to the monomial case in two variables, as introduced in the paper [2].…”

“…The proofs of (2) and (3) are analogous so we only prove (2). If ∂f1 ∂y (0, 0, 0) is invertible then (15a) has a unique solutionŷ, 1-summable in x p 1 x q 2 .…”

“…Monomial summability was introduced in [2] and used in order to study summability properties of solutions of a class of singularly perturbed differential systems (doubly singular differential systems), that are in some sense a generalization of the singularly perturbed differential equations studied by Canalis-Durand, Ramis, Schäfke and Sibuya in [3]. It is a notion that needs further development in order to treat other problems.…”