Abstract and Applied Analysis

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

Abstract

We consider the fourth-order difference equation: $\mathrm{\Delta }(z(k+1){\mathrm{\Delta }}^{3}u(k-1))=w(k)f(k,u(k))$,  $k\in \{1,2,\dots ,n-1\}$ 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 $k\in \{0,1,\dots ,n-1\}$,  $f:\{0,1,\dots ,n\}×[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ is continuous. $h(i)$ is nonnegative $i\in \{2,3,\dots ,n+2\}$; $g(i)$ is nonnegative for $i\in \{0,1,\dots ,n\}$. 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