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

Ruyun Ma

Annales Polonici Mathematici, Tome 79 (2002), p. 265-276 / Harvested from The Polish Digital Mathematics Library

Résumé

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, ⎨$αu(0) - βu^{\prime }(0) = ∑_{i=1}^{m-2} a_{i}u(ξ_{i})$ ⎩$γu(1) + δu^{\prime }(1) = ∑_{i=1}^{m-2} b_{i}u(ξ_{i})$, 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 $Δ:= \begin{vmatrix} -∑_{i=1}^{m-2} a_{i}ψ(ξ_{i}) & ϱ - ∑_{i=1}^{m-2} a_{i}ϕ(ξ_{i}) \\ ϱ - ∑_{i=1}^{m-2} b_{i}ψ(ξ_{i}) & -∑_{i=1}^{m-2} b_{i}ϕ(ξ_{i}) \end{vmatrix} < 0$. 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 BVPs and a result established by the author for three-point BVPs.

Détail

Publié le : 2002-01-01

Zbl 1055.34025

EUDML-ID : urn:eudml:doc:280757