Positive solutions for higher-order Lidstone boundary value problems. A new approach via Spernerʹs Lemma

Consider the higher-order nonlinear scalar differential equation where associated to the Lidstone boundary conditions Existence of a solution of boundary value problems (BVP) (1),(2) such that are given, under superlinear or sublinear growth in f. Similarly, existence for the BVP (1)–(3), under the same assumptions, is proved such that We further prove analogous results for the case when , i.e., derivatives of the obtaining solution satisfy inverse inequalities. The approach is based on an analysis of the corresponding vector field on the face-plane and the well-known, from combinatorial analysis, Knaster-Kuratowski-Mazurkiewiczʹs principle or as it is known, Spernerʹs Lemma.