On ordered-covering mappings and implicit differential inequalities

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• 作者：E. S. Zhukovskiy
• 刊名：Differential Equations
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：52
• 期：12
• 页码：1539-1556
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Ordinary Differential Equations; Partial Differential Equations; Difference and Functional Equations;
• 出版者：Pleiades Publishing
• ISSN：1608-3083
• 卷排序：52

文摘

We define the set of ordered covering of a mapping that acts in partially ordered spaces; we suggest a method for finding the set of ordered covering of vector functions of several variables and the Nemytskii operator acting in Lebesgue spaces. We prove assertions on operator inequalities in arbitrary partially ordered spaces. We obtain conditions that use a set of ordered covering of the corresponding mapping and ensure that the existence of an element u such that f(u) ≥ y implies the solvability of the equation f(x) = y and the estimate x ≤ u for its solution. We study the problem on the existence of the minimal and least solutions. These results are used for the analysis of an implicit differential equation. For the Cauchy problem, we prove a theorem on an inequality of the Chaplygin type.