Positive Solutions for Neumann Boundary Value Problems of Second-Order Impulsive Differential Equations in Banach Spaces

Abstract

The existence of positive solutions for Neumann boundary value problem of second-order impulsive differential equations $-u″\left(t\right)+Mu\left(t\right)=f(t,u\left(t\right)$, $t\in J$, $t\ne t_k$, $-\Delta u\text{'}{|}_{t=t_k}=I_k\left(u\right(t_k\left)\right)$, $k=\mathrm{1,2},\dots ,m$, $u\text{'}\left(0\right)=u\text{'}\left(1\right)=\theta$, in an ordered Banach space $E$ was discussed by employing the fixed point index theory of condensing mapping, where $M>0$ is a constant, $J=\left[\mathrm{0,1}\right]$, $f\in C(J\times K,K)$, $I_k\in C(K,K)$, $k=\mathrm{1,2},\dots ,m$, and $K$ is the cone of positive elements in $E$. Moreover, an application is given to illustrate the main result.