## Abstract and Applied Analysis

### Positive Solutions of a Nonlinear Fourth-Order Dynamic Eigenvalue Problem on Time Scales

#### Abstract

Let $\mathbb{T}$ be a time scale and $a,b\in \mathbb{T}$, $a<{\rho }^{2}(b)$. We study the nonlinear fourth-order eigenvalue problem on $\mathbb{T}$, ${{u}^{\Delta }}^{4}(t)=\lambda h(t)f(u(t),{{u}^{\Delta }}^{2}(t))$, $t\in [a,{\rho }^{2}(b){]}_{\mathbb{T}}$, $u(a)={u}^{\Delta }(\sigma (b))={{u}^{\Delta }}^{2}(a)={{u}^{\Delta }}^{3}(\rho (b))=0$ and obtain the existence and nonexistence of positive solutions when $0<\lambda \le {\lambda }^{*}$ and $\lambda >{\lambda }^{*}$, respectively, for some ${\lambda }^{*}$. The main tools to prove the existence results are the Schauder fixed point theorem and the upper and lower solution method.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 798796, 17 pages.

Dates
First available in Project Euclid: 4 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365099944

Digital Object Identifier
doi:10.1155/2012/798796

Mathematical Reviews number (MathSciNet)
MR2926881

Zentralblatt MATH identifier
1242.34157

#### Citation

Luo, Hua; Gao, Chenghua. Positive Solutions of a Nonlinear Fourth-Order Dynamic Eigenvalue Problem on Time Scales. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 798796, 17 pages. doi:10.1155/2012/798796. https://projecteuclid.org/euclid.aaa/1365099944

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