## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 2014, Special Issue (2014), Article ID 578672, 10 pages.

### The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions

Yanping Guo, Xuefei Lv, Yude Ji, and Yongchun Liang

#### Abstract

We consider the fourth-order difference equation: $\mathrm{\Delta}(z(k+1){\mathrm{\Delta}}^{3}u(k-1))=w(k)f(k,u(k))$, $k\in \{\mathrm{1,2},\dots ,n-1\}$ subject to the boundary conditions: $u(0)=u(n+2)={\sum}_{i=1}^{n+1}g(i)u(i)$, $a{\mathrm{\Delta}}^{2}u(0)-bz(2){\mathrm{\Delta}}^{3}u(0)={\sum}_{i=3}^{n+1}h(i){\mathrm{\Delta}}^{2}u(i-2)$, $a{\mathrm{\Delta}}^{2}u(n)-bz(n+1){\mathrm{\Delta}}^{3}u(n-1)={\sum}_{i=3}^{n+1}h(i){\mathrm{\Delta}}^{2}u(i-2)$, where $a,b>0$ and $\mathrm{\Delta}u(k)=u(k+1)-u(k)$ for $k\in \{\mathrm{0,1},\dots ,n-1\}$, $f:\{\mathrm{0,1},\dots ,n\}\times [0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ is continuous. $h(i)$ is nonnegative $i\in \{\mathrm{2,3},\dots ,n+2\}$; $g(i)$ is nonnegative for $i\in \{\mathrm{0,1},\dots ,n\}$. Using fixed point theorem of cone expansion and compression of norm type and Hölder’s inequality, various existence, multiplicity, and nonexistence results of positive solutions for above problem are derived, which extends and improves some known recent results.

#### Article information

**Source**

Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 578672, 10 pages.

**Dates**

First available in Project Euclid: 6 October 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1412606008

**Digital Object Identifier**

doi:10.1155/2014/578672

**Mathematical Reviews number (MathSciNet)**

MR3240549

**Zentralblatt MATH identifier**

07022645

#### Citation

Guo, Yanping; Lv, Xuefei; Ji, Yude; Liang, Yongchun. The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 578672, 10 pages. doi:10.1155/2014/578672. https://projecteuclid.org/euclid.aaa/1412606008