Minimal nonnegative solutions for impulsive integro-differential equations on unbounded domains in Banach spaces (Q1267795)

The second order equation \(x''(t)=f(t,x(t),x'(t),(Tx)(t),(Sx)(t))\) with countably many impulsive conditions \(\Delta x| _{t=t_k}=I_k(x(t_k),x'(t_k))\), \(\Delta x'| _{t=t_k}= \overline I_k(x(t_k),x'(t_k))\) is studied on \([0,\infty]\) in an ordered Banach space; boundary conditions are \(x(0)=x_0\) and \(x'(\infty)=x_\infty\), and \(T,S\) are positive linear integral operators, \(T\) of Volterra type. All given functions are assumed to be continuous, monotone and to satisfy linear growth conditions with small constants. The existence of a unique minimal nonnegative solution is proved by successive approximation; this solution depends continuously from the boundary conditions.