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

• Autorzy: Ruyun Ma
• Języki publikacji: EN

Abstrakty

EN

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,
⎨$αu(0) - βu'(0) = ∑_{i=1}^{m-2} a_{i}u(ξ_{i})$
⎩$γu(1) + δu'(1) = ∑_{i=1}^{m-2} b_{i}u(ξ_{i})$,
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.

Kategorie tematyczne

• 34B18: Positive solutions of nonlinear boundary value problems
• 34B10: Nonlocal and multipoint boundary value problems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2002
• Tom: 79
• Numer: 3
• Strony: 265-276

Daty

wydano

2002

Twórcy

autor

Ruyun Ma

• Department of Mathematics, Northwest Normal University, Lanzhou 730070, Gansu, People's Republic of China

Identyfikatory

DOI

10.4064/ap79-3-4