Positive solutions for some discrete semipositone problems via bifurcation theory

Abstract

Abstract Let T > 1 $T > 1$ be an integer, and let T = { 1 , 2 , … , T } $\mathbb{T}=\{1, 2,\ldots ,T\}$ . We show the existence of positive solutions of the Dirichlet boundary value problem with second-order difference operator { − △ 2 u ( j − 1 ) = λ f ( j , u ( j ) ) , j ∈ T , u ( 0 ) = u ( T + 1 ) = 0 , $$ \textstyle\begin{cases} -\triangle ^{2} u(j-1)=\lambda f(j, u(j)), \quad j\in \mathbb{T},\\ u(0)=u(T + 1)=0, \end{cases} $$ where λ > 0 $\lambda >0$ is a parameter, and f : T × R + → R $f:\mathbb{T}\times \mathbb{R} ^{+}\to \mathbb{R}$ is a continuous function satisfying f ( j , 0 ) < 0 $f(j, 0)<0$ for all j ∈ T $j\in \mathbb{T}$ . The proofs of the main results are based upon topological degree and global bifurcation techniques.

Keywords

• Second-order difference operator
• Semipositone problems
• Positive solutions
• Topological degree
• Bifurcation