ODEs involving the derivative of a Preisach operator with a degenerated density

This paper examines a class of operator-differential equations that contain derivatives of the Preisach operator. We offer a solution for situations where the density function of the Preisach operator is zero in a region bounded by the Preisach plane bi-sector. This solution is relevant for systems in which output remains constant following a change in direction of input. We apply the solution to problems in terrestrial hydrology.