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We study boundary value and control problems using methods based on results on operator equations in partially ordered spaces. Sufficient conditions are obtained for the existence of a coincidence point for two mappings acting from a partially ordered space into an arbitrary set, an estimate for such a point is found, and corollaries about a fixed point for a mapping that acts in a partially ordered space and is not monotone are derived. The established results are applied to the study of functional and differential equations. For the Nemytskii operator in the space of measurable vector functions, sufficient conditions for the existence of a fixed point are obtained and it is shown that these conditions do not follow from the classical fixed point theorems. Assertions on the existence and estimates of the solution of the Cauchy problem are proved, and the solutions are given to a periodic boundary value problem and a control problem for systems of ordinary differential equations of the first order unsolved for the derivative of the desired vector function. [ABSTRACT FROM AUTHOR]