Abstract
In this paper, we investigate the existence of positive solutions for a class of singular nth-order three-point boundary value problem. The associated Green’s function for the boundary value problem is given at first, and some useful properties of the Green’s function are obtained. The main tool is fixed-point index theory. The results obtained in this paper essentially improve and generalize some well-known results.
Highlights
The existence of positive solutions for higher-order boundary value problems has been studied by many authors using various methods
We study the existence of positive solutions for a singular nth-order three-point boundary value problem as follows u(n)(t) + h(t)f (t, u(t)) = 0, t ∈ [a, b], u(a) = αu(η), u (a) = 0, · · ·, u(n−2)(a) = 0, u(b) = βu(η), (1.2)
No paper has appeared in the literature which discusses the existence of positive solutions for the problem (1.2)
Summary
The existence of positive solutions for higher-order boundary value problems has been studied by many authors using various methods (see [1,2,3,4, 7,8,9,10,11,12,13,14] and the references therein). In paper [3], by using the Krasnosel’skii fixed point theorem, Eloe and Ahmad established the existence of at least one positive solution for the following nth-order three-point boundary value problem u(n)(t) + a(t)f (u(t)) = 0, t ∈ (0, 1), u(0) = 0, u (0) = 0, · · · , u(n−2)(0) = 0, u(1) = αu(η),. We study the existence of positive solutions for a singular nth-order three-point boundary value problem as follows u(n)(t) + h(t)f (t, u(t)) = 0, t ∈ [a, b], u(a) = αu(η), u (a) = 0, · · · , u(n−2)(a) = 0, u(b) = βu(η),. It is noted that our method here is different from that of Eloe and Ahmad [3]
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