Abstract
This paper is concerned with the existence, nonexistence, and uniqueness of convex monotone positive solutions of elastic beam equations with a parameter λ. The boundary conditions mean that the beam is fixed at one end and attached to a bearing device or freed at the other end. By using fixed point theorem of cone expansion, we show that there exists such that the beam equation has at least two, one, and no positive solutions for , and , respectively; furthermore, by using cone theory we establish some uniqueness criteria for positive solutions for the beam and show that such solution depends continuously on the parameter λ. In particular, we give an estimate for critical value of parameter λ. MSC:34B18, 34B15.
Highlights
Introduction and preliminariesIn this paper, we consider the following nonlinear fourth-order two-point boundary value problem (BVP) for elastic beam equation: x( )(t) = λf (t, x(t)), < t
Throughout this paper, we assume that f ∈ C([, ] × R+, R+), q ∈ C(R+, R+), R+ = [, +∞). x ∈ C[, ] is called a positive solution of BVP ( ) if x is a solution of BVP ( ) and x(t) >, < t
For a small sample of such work, we refer the reader to the work of Bai and Wang [ ], Bai [ ], Bonanno and Bellaa [ ], Li [ ], Liu and Li [ ], Liu [ ], Ma and Xu [ ], and Ma and Thompson [ ] on an elastic beam whose two ends are supported, the works of Yang [ ] and Zhang [ ] on an elastic beam of which one end is embedded and another end is fastened with a sliding clamp, and the work of Graef et al [ ] on multipoint boundary value problems
Summary
Let xλn ∈ K \{θ } be a fixed point of Cλn. Arguing as above in case , we can show that {xλn }+ ∞ is a bounded subset in K , that is, there exists a constant M > such that xλn ≤ M, n = , , . There exists a λ∗ ≥ λ∗ > such that BVP ( ) has at least two, one, and no positive solutions for < λ ≤ λ∗, λ∗ < λ ≤ λ∗ and λ > λ∗, respectively. We give some sufficient conditions that BVP ( ) has no positive solutions. When q(x) ≡ , BVP ( ) becomes a cantilever beam problem ( ) In this case, we can delete the conditions on q in Theorems .
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