Positive solutions for higher-order Lidstone boundary value problems. A new approach via Sperner's Lemma
Consider the higher-order nonlinear scalar differential equation x((2n))(t) = -f (t,x(t),...,y((2j))(t),...,y((2(n-1)))(t)), 0 less than or equal tot less than or equal to1 where f epsilon C ([0,1] x R-+(n),R+), R+ = [0,infinity) asssociated to the Lidstone boundary conditions x((2i))(0) = 0=x((2i))(1), (2) x((2i))(0) = 0 = x((2i+1))(1). (3) Existance of a solution boundary value problems (BVP) (1),(2) such that x((2i))(t) > 0, 0   0, 0 < t < 1, i = 0,1,...,2n - 1. We further prove analogous results for the case when -f epsilon C([0,1] x R--(n),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 well-known, from combinatorial analysis, Knaster-Kuratowski-Mazurkiewicz's principle or as it is known, Sperner's Lemma. (C) 2001 Elsevier Science Ltd. All rights reserved.
positive solution vector field knaster-kuratowski-mazurkiewicz's principle sturm-liouville boundary value problems sublinear superlinear growth rate sperner's property