Abstract

Let T be a time scale. In this paper, we study the existence of positive solutions for the following nonlinear second-order m-point boundary value problem with sign changing coefficient on time scales {uΔ∇(t)+f(t,u(t))=0, 0<;t<;T, uΔ(0) = Σi=1m-2 aiuΔ(ξi), u(T)=Σi=1k biu(ξi)- Σi=k+1s biu(ξi) - Σi=s+1m-2 biuΔ(ξi), where 1 ≤ k ≤ s ≤ m - 2, ai, bi ∈ (0, +∞) with 0 <; Σi=1k bi- Σi=k+1s bi<;1, 0<;Σi=1m-2 ai<;1, 0<;ξ1 <;ξ2 ⋯<; ξm-2 <; ρ(T), f ∈ C([0,1] × R,R), R = (-∞,+∞). Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence of nontrivial solution are obtained by using Leray-Schauder nonlinear alternative. As an application, some examples to demonstrate our results are given. In particular, our criteria extend and improve some known results.

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