On isolated sets of solutions of some two-point boundary value problems. (Q1395860)

A geometric approach is presented in order to study the existence of solutions to a separated boundary value problem of the form \(u'=f(t,u), \quad u(a)\in P, \; u(b)\in Q\), where \(P, Q\) are submanifolds of the phase space. This paper represents a further step in the author's investigation [ibid. 22, No. 6, 707--737 (1994; Zbl 0801.34041), Topol. Methods Nonlinear Anal. 13, No. 1, 73--89 (1999; Zbl 0935.34017)] on the use of Ważewski-Conley techniques in the study of BVPs. More precisely, it is developed in this paper the concept of intersection number, which arises in the framework of singular homology theory. The intersection number has solvability, additivity and invariance properties that are crucial in the proofs. Using the intersection number and the notion of ``isolating segment'', the author proves existence results on the given problem. Applications provide another proof of the classical Bernstein-Nagumo theorem for a second-order equation as well as existence results on first-order planar systems. For related results, we refer to \textit{R. Manásevich, J. Mawhin} and \textit{F. Zanolin} [J. Differ. Equations 126, No. 2, 355--373 (1996; Zbl 0848.34027)].