Abstract
This paper investigates the eigenvalue problem for a class of singular elastic beam equations where one end is simply supported and the other end is clamped by sliding clamps. Firstly, we establish a necessary and sufficient condition for the existence of positive solutions, then we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from Our nonlinearity may be singular at and/or .
Highlights
Singular differential equations arise in the fields of gas dynamics, Newtonian fluid mechanics, the theory of boundary layer, and so on
We study the existence of positive solutions and the structure of positive solution set for the BVP 1.1
We construct a special cone and present a necessary and sufficient condition for the existence of positive solutions, we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from 0, θ
Summary
Singular differential equations arise in the fields of gas dynamics, Newtonian fluid mechanics, the theory of boundary layer, and so on. We first establish a necessary and sufficient condition for the existence of positive solutions of BVP 1.1 for any λ > 0 by using the following Lemma 1.1. Papers 24, 25 derived the existence of positive solutions of BVPs by the lower and upper solution method, but the nonlinearity f t, u does not depend on the derivatives of the unknown functions, and f t, u is decreasing with respect to u. Ma and An and Ma and Xu discussed the global structure of positive solutions for the nonlinear eigenvalue problems and obtained the existence of an unbounded connected branch of positive solution set by using global bifurcation theorems see 29, 30. C ∩ {λ} × M / ∅, ∀λ ∈ a, b , 1.7 where lim supn → ∞Cn {x ∈ M : there exists a sequence xni ∈ Cni such that xni → x, i → ∞ }
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