  In this paper, we study stability properties of nonuniform hyperbolicity for
evolution processes associated with differential equations in Banach spaces. We
prove a robustness result of nonuniform hyperbolicity for linear evolution
processes, that is, we show that the property of admitting a nonuniform
exponential dichotomy is stable under perturbation. Moreover, we provide
conditions to obtain uniqueness and continuous dependence of projections
associated with nonuniform exponential dichotomies. We also present an example
of evolution process in a Banach space that admits nonuniform exponential
dichotomy and study the permanence of the nonuniform hyperbolicity under
perturbation. Finally, we prove persistence of nonuniform hyperbolic solutions
for nonlinear evolution processes under perturbations.
