The hysteresis limit in relaxation oscillation problems

A singularly perturbed differential equation with a small coefficient multiplying the derivative is shown to exhibit a limiting hysteresis behavior as the singular parameter tends to zero. The convergence takes place in the space of left-continuous regulated functions and is related to the generalized Helly selection principle for regulated functions established by Franková. Examples show that convergence cannot be expected in general if no regularity is assumed either for the forcing term or for the equilibrium set.