Abstract
In this paper, we study the existence of positive solutions for the following nonlinear second-order third-point semi-positive BVP. We derive an explicit interval of positive parameters, which for any l , μ in this interval, the existence of positive solutions to the boundary value problem is guaranteed under the condition that a t , x , b t , x are all superlinear (sublinear), or one is superlinear, the other is sublinear.
Highlights
In the applied mathematical field, three-point BVP can describe many phenomena
Moshinsky [1] introduced the vibrations of a guy wire with a uniform cross-section and composed of N parts of different densities using a multipoint BVP
Timoshenko [2] revealed that the theory of elastic stability can be used by the method of a three-point BVP
Summary
In the applied mathematical field, three-point BVP can describe many phenomena. Moshinsky [1] introduced the vibrations of a guy wire with a uniform cross-section and composed of N parts of different densities using a multipoint BVP. In their paper [7], Ma and Wang obtained the existence of positive solutions for a three-point BVP by Krasnoselskii’s fixed theorem:. ( u′′ðtÞ + aðtÞu′ðtÞ + bðtÞuðtÞ + hðtÞf ðuÞ = 0, 0 ≤ t ≤ 1, uð0Þ = 0, uð1Þ = αuðηÞ, ð1Þ where α is a positive constant, 0 < η < 1, aðtÞ ∈ Cð1⁄20, 1, R+Þ, bðtÞ ∈ Cð1⁄20, 1, R−Þ, f ∈ CðR+, R+Þ,h ∈ Cð1⁄20, 1, R+Þ and there exists x0 ∈ ð0,+∞Þ such that hðx0Þ > 0: In our paper, we study the existence of positive solutions of second-order third-point semipositive BVP:. Suppose operator A is completely continuous and satisfies the following conditions: kTxk ≤ kxk, ∀x ∈ P ∩ ∂Ω1, kTxk ≥ kxk, Ax ≠ x,∀x ∈ P ∩ ∂Ω2, ð3Þ kTxk ≤ kxk, ∀x ∈ P ∩ ∂Ω3: operator T has at least two fixed points x∗ and x∗∗ in P ∩ ðΩ 3/Ω1Þ, and x∗ ∈ P ∩ ðΩ2/Ω1Þ and x∗∗ ∈ P ∩ ðΩ 3/Ω 2Þ
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