On Uniqueness of a Solution to the Boundary Initial Value Problem

Abstract

A first-order ordinary differential equation, solved with respect to derivative, is considered. It's right-hand side is defined and continuous on the set, consisting of a connected open subset of a two-dimensional Euclidean space and a part of its boundary. In the papers, dated by 2020, problems related to the existence or absence of the solution to the BIVP -- Initial Value Problem set at the boundary point -- were researched using different approaches. Such formulation of the Initial Value Problem is different from the formulation, established in the classic theory, where it is set at an interior point. This paper is devoted to solving problems related to uniqueness or non-uniqueness of the BIVP solutions. New definitions, related to uniqueness, absent in the formulation of the IIVP -- Initial Value Problem set at the interior point of the equation's domain -- are introduced. The theorems about the formal, local and global uniqueness of the BIVP solutions are proven. The differences between BIVP and IIVP are shown. For example, non-equivalence of the definitions of the formal, local and global uniqueness' for BIVP and IIVP is demonstrated, This non-equivalence leads to the appearance of hidden non-uniqueness points along with uniqueness and non-uniqueness points. Suggested theory is supposed to fill in the blanks in an existing literature, related to the problems of existence and uniqueness of the BIVP solutions.