  In this paper, a new notion called the general nonuniform
$(h,k,\mu,\nu)$-dichotomy for a sequence of linear operators is proposed, which
occurs in a more natural way and is related to nonuniform hyperbolicity. Then,
sufficient criteria are established for the existence of nonuniform
$(h,k,\mu,\nu)$-dichotomy in terms of appropriate Lyapunov exponents for the
sequence of linear operators. Moreover, we investigate the stability theory of
sequences of non uniformly hyperbolic linear operators in Banach spaces, which
admit a nonuniform $(h,k,\mu,\nu)$-dichotomy. In the case of linear
perturbations, we investigate parameter dependence of robustness or roughness
of the nonuniform $(h,k,\mu,\nu)$-dichotomies and show that the stable and
unstable subspaces of nonuniform $(h,k,\mu,\nu)$-dichotomies for the linear
perturbed system are Lipschitz continuous for the parameters. In the case of
nonlinear perturbations, we construct a new version of the Grobman-Hartman
theorem and explore the existence of parameter dependence of stable Lipschitz
invariant manifolds when the nonlinear perturbation is of Lipschitz type.

Dynamical Systems