On existence, continuability and uniqueness of a solution to the boundary initial value problem of a first-order ordinary differential equation

A first-order ordinary differential equation, solved with respect to the derivative, is considered. It is assumed that its right-hand side is continuous on a set consisting of a connected open subset of a two-dimensional Euclidean space and some part of its boundary. A theory is presented devoted to solving the questions of existence, continuability and uniqueness of solutions of the BIVP that is the initial value problem posed at a boundary point. This theory will allow to supplement the existing theory of first-order ODEs, in which the Cauchy problem is posed at an interior point (IIVP). The main results on existence or non-existence of a BIVP solution were obtained in 2020, therefore in this paper they are only systematized and supplemented. The results related to continuability, as well as the uniqueness or non-uniqueness of solutions to the BIVP are new. Theorems on the formal and local uniqueness of solutions to the Cauchy problem are proved. The differences between BIVP and IIVP are shown. For example, the non-equivalence of the concepts of formal and local uniqueness for BIVP and IIVP is demonstrated. This non-equivalence leads to the appearance of so-called hidden points of non-uniqueness along with points of non-uniqueness and uniqueness.