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

Abstract

We consider the fourth-order difference equation: $\mathrm{\Delta }(z(k+1){\mathrm{\Delta }}^{3}u(k-1))=w(k)f(k,u(k))$, $k\in \{\mathrm{1,2},\dots ,n-1\}$ 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.