Multiple Positive Solutions of a Second Order Nonlinear Semipositone m -Point Boundary Value Problem on Time Scales

Abstract

In this paper, we study a general second-order $m$-point boundary value problem for nonlinear singular dynamic equation on time scales $u^{\Delta \nabla }\left(t\right)+a\left(t\right)u^{\Delta }\left(t\right)+b\left(t\right)u\left(t\right)+\lambda q\left(t\right)f(t,u(t\left)\right)=0$, $t\in (\mathrm{0,1}{)}_{\mathbb{𝕋}}$, $u\left(\rho \right(0\left)\right)=0$, $u\left(\sigma \right(1\left)\right)={\sum }_{i=1}^{m-2}{\alpha }_{i}u\left({\eta }_i\right)$. This paper shows the existence of multiple positive solutions if $f$ is semipositone and superlinear. The arguments are based upon fixed-point theorems in a cone.