Abstract

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.

中文翻译：

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泛函和微分不等式及其在控制问题中的应用

摘要

我们使用基于部分有序空间中算子方程式结果的方法研究边值问题和控制问题。对于存在从局部有序空间到任意集合的两个映射的重合点的存在，获得了充分的条件，找到了该点的估计，并确定了在局部有序空间中起作用的映射的固定点的推论并且不是单调的。所建立的结果将用于功能方程和微分方程的研究。对于在可测向量函数空间中的Nemytskii算符，获得了存在不动点的充分条件，并且表明这些条件不遵循经典不动点定理。证明了柯西问题解的存在性和估计，