  In this paper we study the existence, localization and multiplicity of
positive solutions for parabolic systems with nonlocal initial conditions. In
order to do this, we extend an abstract theory that was recently developed by
the authors jointly with Radu Precup, related to the existence of fixed points
of nonlinear operators satisfying some upper and lower bounds. Our main tool is
the Granas fixed point index theory. We also provide a non-existence result and
an example to illustrate our theory.
