Solutions of Second-Order m -Point Boundary Value Problems for Impulsive Dynamic Equations on Time Scales

Abstract

We study a general second-order $m$-point boundary value problems for nonlinear singular impulsive dynamic equations on time scales $u^{\Delta \nabla }(t)+a(t)u^{\Delta }(t)+b(t)u(t)+q(t)f(t,u(t))=\mathrm{0},t\in (\mathrm{0,1}),t\ne t_k,u^{\Delta }(t_k^{+})=u^{\Delta }(t_k)-I_k(u(t_k))$, and $k=\mathrm{1,2},\dots ,n,\mathrm{}u(\rho (\mathrm{0}))=\mathrm{0},u(\sigma (\mathrm{1}))={\sum }_{i=\mathrm{1}}^{m-\mathrm{2}}{\alpha }_{i}u({\eta }_i)\mathrm{‍}.$ The existence and uniqueness of positive solutions are established by using the mixed monotone fixed point theorem on cone and Krasnosel’skii fixed point theorem. In this paper, the function items may be singular in its dependent variable. We present examples to illustrate our results.