Linkage of cycles and boundary value problems for ordinary differential equations (Q5951182)

The paper collects all meaningful information from the proofs in two papers of the author [Sb. Math. 188, No. 6, 933-953 (1997); translation from Mat. Sb. 188, No. 6, 139-160 (1997; Zbl 0895.34007) and Differ Equations 33, No. 8, 1063-1074 (1997); translation from Differ. Uravn. 33, No. 8, 1057-1068 (1997; Zbl 0929.34012)] into a single unified result. Let \(M\) be a closed subset of a metric space \(Y\) and let \(P\) be a compact subset of \([a,b]\times Y\). Under appropriate homomorphism group hypotheses, \(P\cap([a,b]\times M)=\emptyset\) implies \(P_b\cap M\neq\emptyset\). Further, there are discussed some boundary value problems in \(\mathbb{R}^n\), including the Dirichlet problem \(x''(t)\in F(t,x(t),x'(t))\), \(x(t_0)=x(t_1)=\vec{0}\). Although these items are not very brief, they do not represent rigorous proofs, but only deal with an explanation of situations appearing in the analysis of differential equations.