Boundary value problems for nonlinear impulsive integrodifferential equations (Q913187)

Consider the boundary value problem for nonlinear impulsive integrodifferential equations of the form \[ (1)\quad - y''=f(t,Ty,y,y'),\quad t\neq t_ k,0,1,\quad t\in J, \] \[ (2)\quad \Delta y|_{t=t_ k}=I_ k(y(t_ k)),\quad k=1,2,...,p, \] (3) \(\Delta y'|_{t=t_ k}=N_ k(y'(t_ k))\), \(k=1,2,...,p,\) where \(J=[0,1]\), \(0<t_ 1<t_ 2<...<t_ p<1\), f: \(J\times R\times R\times R\to R\), and \(I_ k\), \(N_ k: R\to R\) for each \(k=1,2,...,p\). Here T is a Volterra operator from PC[J,R] into PC[J,R], where \(PC[J,R]=\{y: J\to R:\) y(t) is continuous at \(t\neq t_ k\), \(y(t^-)\) and \(y(t^+)\) exist, and \(y(t^-)=y(t)\) for \(t=t_ k\), \(k=1,2,...,p\}.\) The authors establish existence results in a sector by suitably modifying the notion of upper and lower solutions and then develop the monotone method and prove the existence of extremal solutions of (1)-(3). As an application, existence results for boundary value problem for a third order impulsive differential equation are also derived.