Copyright © 2011 Chenghua Gao and Hua Luo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We discuss the existence of solutions for the first-order multipoint BVPs on time scale 𝕋: uΔ(t)+p(t)u(σ(t))=λf(t,u(σ(t))), t∈[0,T]𝕋, u(0)-∑i=1mαiu(ξi)=0, where λ>0 is a parameter, T>0 is a fixed number, 0,T∈𝕋, f:[0,T]𝕋×[0,∞)→[0,∞) is continuous, p is regressive and rd-continuous, αi≥0, ξi∈𝕋, i=1,2,…,m, 0=ξ0<ξ1<ξ2<⋯<ξm-1<ξm=σ(T), and 1-∑i=1m(αi/ep(ξi,0))>0. For suitable λ>0, some existence, multiplicity, and nonexistence criteria of positive solutions are established by using well-known results from the fixed-point index.