Fourth-order m-point boundary value problems on time scales

Let T be a time scale with [a, b] C T. We establish criteria for existence of one or more than one positive solutions of the non-eigenvalue problem (0.1) {y?4(t) y?2 (? (t, y(t)), t?(a,b) ? T, (0.1) { y(a) = ? l=1 m-2 a1y (?i), (y?2(b) = ? l=1 m-2b1y (?i), (0.1) { y?2 (a) = ?? l=1 m-2 a1y?2 (?i), y?2(a2(b)) = ? i=1 m-2 biy?2 (?i), where ? € (a,b) ai, bi ? [0, ?) (for i € { 1, 2, ..., m-2}) are given constants. Later, we consider the existence and multiplicity of positive solutions for the eigenvalue problem y?2(t) - q(t)y?2(?(t)) = ?f (t,y(t)) with the same boundary conditions. We shall also obtain criteria which lead to nonexistence of positive solutions. In both problems, we will use Krasnoselskii fixed point theorem. © Dynamic Publishers, Inc.