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.