The differential equation $y'=fy$ in the algebras $H(D)$

Let $D$ be an a clopen bounded infraconnected set in an algebraically closed complete ultrametric valued field, and $H(D)$ the Banach algebra of the analytic elements in $D$[10,11,3]. Let $f$ be an element of $H(D)$; we show that if the differential equation $f'=fy$ has a solution $g$ invertible in $H(D)$, then the space of the solutions in $H(D)$ has dimension 1. We prove that a solution $g$ has no zero isolated in $D$ and that if $g$ is not invertible, it is strictly annulled by a $T$-filter [6]. At last we prove that if $H(D)$ has no divisor of zero the space has dimension 0 or 1.