Existence Result for Impulsive Differential Equations with Integral Boundary Conditions

Abstract

We investigate the following differential equations: $-(y^{[1]}(x))\text{'}+q(x)y(x)=\lambda f(x,y(x))$, with impulsive and integral boundary conditions $-\mathrm{\Delta }(y^{[1]}(x_i))=I_i(y(x_i))$, $i=\mathrm{1,2},\dots ,m$, $y(0)-a{y}^{[1]}(0)={\int }_0^{\omega }{g}_0(s)y(s)ds$, $y(\omega )-b{y}^{[1]}(\omega )={\int }_0^{\omega }{g}_1(s)y(s)ds$, where $y^{[1]}(x)=p(x)y\text{'}(x)$. The expression of Green's function and the existence of positive solution for the system are obtained. Upper and lower bounds for positive solutions are also given. When $p(t)$, $I(·)$, $g_0(s)$, and $g_1(s)$ take different values, the system can be simplified to some forms which has been studied in the works by Guo and LakshmiKantham (1988), Guo et al. (1995), Boucherif (2009), He et al. (2011), and Atici and Guseinov (2001). Our discussion is based on the fixed point index theory in cones.