FOURTH-ORDER M-POINT BOUNDARY VALUE PROBLEMS ON TIME SCALES

Let T be a time scale with [a, b] ⊂ T. We establish criteria for existence of one or more than one positive solutions of the non-eigenvalue problem (0.1)    y △4 (t) − q(t)y ∆2 (σ(t)) = f(t, y(t)), t ∈ [a, b] ⊂ T, y(a) = Pm−2 i=1 aiy(ξi), y(σ 2 (b)) = Pm−2 i=1 biy(ξi), y ∆2 (a) = Pm−2 i=1 aiy ∆2 (ξi), y∆2 (σ 2 (b)) = Pm−2 i=1 biy ∆2 (ξi), where ξi ∈ (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 △4 (t) − q(t)y ∆2 (σ(t)) = λf(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.