Positive Solutions to Nonlinear First-Order Nonlocal BVPs with Parameter on Time Scales. (27th June 2011) - Gao, Chenghua; Luo, Hua
Abstract : We discuss the existence of solutions for the first-order multipoint BVPs on time scale𝕋 :u Δ ( t ) + p ( t ) u ( σ ( t ) ) = λ f ( t, u ( σ ( t ) ) ), t ∈ [ 0, T ] 𝕋, u ( 0 ) - ∑ i = 1 m α i u ( ξ i ) = 0, whereλ > 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, α i ≥ 0, ξ i ∈ 𝕋, i = 1, 2, …, m, 0 = ξ 0 < ξ 1 < ξ 2 < ⋯ < ξ m - 1 < ξ m = σ ( T ), and1 - ∑ i = 1 m ( α i / e p ( ξ i, 0 ) ) > 0 . For suitableλ > 0, some existence, multiplicity, and nonexistence criteria of positive solutions are established by using well-known results from the fixed-point index.