Positive solutions of nonlinear mm-point boundary value problems on time scales

In this paper, by using fixed point theorems in a cone and the associated Green’s function, we study the existence of at least two and three positive solutions to the mm-point boundary value problem (BVP) on time scales, uΔ∇(t)+f(t,u(t))=0,t∈[0,1]⊂T,βu(0)−γuΔ(0)=0,u(1)=∑i=1m−2αiu(εi),m≥3, where TT is a time scale, f∈Cld([0,1]×[0,∞),[0,∞))f∈Cld([0,1]×[0,∞),[0,∞)), β,γ∈[0,∞),αi∈[0,∞)β,γ∈[0,∞),αi∈[0,∞) for i=1,2,…,m−2i=1,2,…,m−2, and εiεi satisfy 0<ε1<ε2<⋯<εm−2<ρ(1)0<ε1<ε2<⋯<εm−2<ρ(1).