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Existence of positive solutions for second order m-point boundary value problems

Ruyun Ma (2002)

Annales Polonici Mathematici

Let α,β,γ,δ ≥ 0 and ϱ:= γβ + αγ + αδ > 0. Let ψ(t) = β + αt, ϕ(t) = γ + δ - γt, t ∈ [0,1]. We study the existence of positive solutions for the m-point boundary value problem ⎧u” + h(t)f(u) = 0, 0 < t < 1, ⎨ α u ( 0 ) - β u ' ( 0 ) = i = 1 m - 2 a i u ( ξ i ) γ u ( 1 ) + δ u ' ( 1 ) = i = 1 m - 2 b i u ( ξ i ) , where ξ i ( 0 , 1 ) , a i , b i ( 0 , ) (for i ∈ 1,…,m-2) are given constants satisfying ϱ - i = 1 m - 2 a i ϕ ( ξ i ) > 0 , ϱ - i = 1 m - 2 b i ψ ( ξ i ) > 0 and Δ : = - i = 1 m - 2 a i ψ ( ξ i ) ϱ - i = 1 m - 2 a i ϕ ( ξ i ) ϱ - i = 1 m - 2 b i ψ ( ξ i ) - i = 1 m - 2 b i ϕ ( ξ i ) < 0 . We show the existence of positive solutions if f is either superlinear or sublinear by a simple application of a fixed point theorem in cones. Our result extends a result established by Erbe and Wang for two-point...

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