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

Ruyun Ma. "Existence of positive solutions for second order m-point boundary value problems." Annales Polonici Mathematici 79.3 (2002): 265-276. <http://eudml.org/doc/280757>.

@article{RuyunMa2002,
abstract = {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.},
author = {Ruyun Ma},
journal = {Annales Polonici Mathematici},
keywords = {mulit-point boundary value problems; positive solution; fixed-point theorem},
language = {eng},
number = {3},
pages = {265-276},
title = {Existence of positive solutions for second order m-point boundary value problems},
url = {http://eudml.org/doc/280757},
volume = {79},
year = {2002},
}

TY - JOUR
AU - Ruyun Ma
TI - Existence of positive solutions for second order m-point boundary value problems
JO - Annales Polonici Mathematici
PY - 2002
VL - 79
IS - 3
SP - 265
EP - 276
AB - 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.
LA - eng
KW - mulit-point boundary value problems; positive solution; fixed-point theorem
UR - http://eudml.org/doc/280757
ER -