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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]

A coincidence point of a pair of mappings is an element, at which these mappings take on the same value. Coincidence points of mappings of partially ordered spaces were studied by A.V. Arutyunov, E.S. Zhukovskiy, and S.E. Zhukovskiy (see Topology and its Applications, 2015, V. 179, No. 1, p. 13–33); they proved, in particular, that an orderly covering mapping and a monotone one, both acting from a partially ordered space to a partially ordered one, possess a coincidence point. In this paper, we study the existence of a coincidence point of a pair of mappings that act from a partially ordered space to a set, where no binary relation is defined, and, consequently, it is impossible to define covering and monotonicity properties of mappings. In order to study the mentioned problem, we define the notion of a "quasi-coincidence" point. We understand it as an element, for which there exists another element, which does not exceed the initial one and is such that the value of the first mapping at it equals the value of the second mapping at the initial element. It turns out that the following condition is sufficient for the existence of a coincidence point: any chain of "quasi-coincidence" points is bounded and has a lower bound, which also is a "quasi-coincidence" point. We give an example of mappings that satisfy the above requirements and do not allow the application of results obtained for coincidence points of orderly covering and monotone mappings. In addition, we give an interpretation of the stability notion for coincidence points of mappings that act in partially ordered spaces with respect to their small perturbations and establish the corresponding stability conditions. [ABSTRACT FROM AUTHOR]