Frobenius structure for rank one p-adic differential equations

We generalize to all rank one p-adic differential equations over the Robba ring R the theorem 2.3.1 of [Ch-Ch] which provides the existence of a Frobenius structure of order h for solvable rank one operators of the form (d)/(dx) +g(x), g(x) is an element of x(-2) K[x(-1)]. It follows a generalization of a theorem of Matsuda which asserts that the Robba's exponential exp(Sigma(m)(i=0) pi(m-i)x(pi)/p(i)) has a Frobenius structure. Namely our theorem works in the case p = 2. In the appendix we describe the variation of the radius of convergence of a differential module by pull-back by a Kummer ramification.