Lyapunov-type functions are introduced into upper estimates for the Hausdorff dimension of negatively invariant sets of cocycles. Nonautonomous differential equations are studied based on the theory of cocycles and their attractors. For a negatively invariant for a local cocycle, a metric space and an arbitrary subset are considered. The nonautonomous set is found to be compact and negatively invariant for the local cocycle. For the nonautonomous ordinary differential equation, there exists a continuous function with derivatives along a given trajectory. To estimate the Hausdorff dimension, the inequality is verified.