Positive Solutions to Nonlinear First-Order Nonlocal BVPs with Parameter on Time Scales

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 − ∑mi 1 αiu 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 rdcontinuous, αi ≥ 0, ξi ∈ T, i 1, 2, . . . , m, 0 ξ0 < ξ1 < ξ2 < · · · < ξm−1 < ξm σ T , and 1 −∑mi 1 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.