Positive solutions for nonlinear elastic beam models

Bendong Lou

doi:10.1155/s0161171201004203

Abstract

We give a negative answer to a conjecture of Korman on nonlinear elastic beam models. Moreover, by modifying the main conditions in the conjecture (generalizing the original ones at some points), we get positive results, that is, we obtain the existence of positive solutions for the models.

Highlights

In 1988, Korman [3] studied the nonlinear elastic beam models u = f (x, u), 0 < x < 1, u(0) = α, u (0) = β, u(1) = γ, −u (1) = δ, (1.1)where α, β, γ, δ ≥ 0, f : [0, 1] × R+ → R+ is continuous
Korman [3] studied the existence of positive solutions of (1.1) by using monotone iterations
His results covered in particular the sublinear nonlinearities

Summary

Introduction

Korman [3] studied the existence of positive solutions of (1.1) by using monotone iterations. His results covered in particular the sublinear nonlinearities. Where G(x, y) is given by (2.4), and it is easy to verify by G(x, y) > 0 (x, y ∈ (0, 1)) that the spectral radius of L, r (L) is positive, Krein-Rutman’s theorem (see [4]) shows that there exists φ(x) ∈ P , φ(x) ≡ 0 and φ = 1 such that φ(x) = λ1 G(x, y)φ(y) dy,. Let P be a solid cone in E, A : P → P a completely continuous increasing operator. We will investigate the existence of positive solutions of (1.1) by modifying the conditions of the conjecture. Suppose that v∗(x) is defined by (2.13), φ(x) and λ1 are given by (2.23), and η1 ≡

It is easy to see that

For example