On first order discontinuous implicit boundary value problems

In this paper, we shall consider the existence of extremal solutions to boundary value problems of first order discontinuous implicit differential equations with discontinuous implicit boundary conditions, in ordered Banach spaces. The main tools are a fixed point theorem based on a generalized iteration method and a method of upper and lower solutions. We shall also study dependence of the extremal solutions on data, the existence of upper and lower solutions, and cases when the extremal solutions are obtained as limits of successive approximations. Examples are given to illustrate the obtained results. AMS (MOS) subject classification: 34A09, 34B15, 34G20.