On estimates of solutions of boundary-value problems for implicit differential equations with deviating argument

A two-point boundary-value problem for an implicit differential equation with a deviating argument is examined. An existence theorem and an estimate for the solution are obtained, which is similar to the Chaplygin theorem on differential inequalities. We use results on equations with covering and monotonic mappings in partially ordered spaces and conditions for ordered covering of the Nemytskii operator in the space of measurable essentially bounded functions.