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Existence of positive solutions for a nonlinear fourth order boundary value problem

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Ma R.

Annales Polonici Mathematici

2003

tom 81

nr 1

79-84

We study the existence of positive solutions of the nonlinear fourth order problem $u^{(4)}(x) = λa(x)f(u(x))$, u(0) = u'(0) = u''(1) = u'''(1) = 0, where a: [0,1] → ℝ may change sign, f(0) < 0, and λ < 0 is sufficiently small. Our approach is based on the Leray-Schauder fixed point theorem.

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

100%

Ma R.

Annales Polonici Mathematici

2002

tom 79

nr 3

265-276

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.

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2 Annales Polonici Mathematici

2 Ma R.

1 2003

1 2002

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Existence of positive solutions for a nonlinear fourth order boundary value problem

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