On isolated sets of solutions of some two-point boundary value problems Original Research Article

We present a geometric approach to the question of existence of solutions of the two-point boundary value problem View the MathML source where P and Q are submanifolds of the phase space. For an isolated set K of initial values of solutions of the problem, we associate the intersection indexι(f,K), an element of View the MathML source (or of View the MathML source if some of the submanifolds is not orientable) satisfying the solvability (i.e. ι(f,K)≠0 implies K≠∅), additivity and continuation invariance properties. We prove a theorem on calculation of ι(f,K) if K is naturally generated by an isolated segment which is concordant, in some way, with the considered problem. As an application, we provide another proof of the classical Bernstein–Nagumo Theorem on existence of solutions of some second-order boundary value problems. Other applications refer to problems associated with first-order planar equations.