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2014 The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions
Yanping Guo, Xuefei Lv, Yude Ji, Yongchun Liang
Abstr. Appl. Anal. 2014(SI71): 1-10 (2014). DOI: 10.1155/2014/578672

## 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.

## Citation

Yanping Guo. Xuefei Lv. Yude Ji. Yongchun Liang. "The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions." Abstr. Appl. Anal. 2014 (SI71) 1 - 10, 2014. https://doi.org/10.1155/2014/578672

## Information

Published: 2014
First available in Project Euclid: 6 October 2014

zbMATH: 07022645
MathSciNet: MR3240549
Digital Object Identifier: 10.1155/2014/578672  