Positive Solutions for Higher Order Lidstone Boundary Value Problems. a New Approach via Sperner’s Lemma

Consider the higher order nonlinear scalar differential equations (0.1) x(t) = −f(t, x(t), ..., y(t), ...y(2(n−1))(t)), 0 ≤ t ≤ 1 where f ∈ C([0, 1] × R+, R+), R+ = [0,∞) associated to the Lidstone boundary conditions (0.2) x(0) = 0 = x(1), (0.3) x(0) = 0 = x(1). Existence of a solution of boundary value Problems (BVP) (0.1)-(0.2) such that x(t) > 0, 0 < t < 1, i = 0, 1, ..., n− 1 are given, under superlinear or sublinear growth in f . Similarly existence for the BVP (0.1)-(0.3), under the same assumptions, is proved such that x(t) > 0, 0 < t < 1, i = 0, 1, ..., 2n− 1. We further prove analogous results for the case when −f ∈ C([0, 1] × R−, R−), 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 wellknown from combinatorial analysis, Knaster-Kuratowski-Mazurkiewicz’s principle or as it is known, Sperner’s Lemma .