Topological Method for the Existence of Weak Solution of Boundary Value Problem of Ordinary Differential Equations

This paper investigates the existence of weak solutions of boundary value problems associated with ordinary differential equations of order 2k by means of topological method. The approach adopted is purely analytical and does not rely on numerical computations; instead, it focuses on the qualitative behavior of the solution. The primary aim is to establish the existence of weak solutions of even-order boundary value problems and to examine their relationship with classical solutions. The principal contribution of this work is the proof of a modified version of Félix Browder’s existence theorem, which provides sufficient conditions for the existence of weak solution to these problems. Several examples are presented to demonstrate the effectiveness and applicability of the main theorem.