We show the normal hyperbolicity property for the equilibria of the evolution equation $\partial m(r,t)/\partial t=-m(r,t)+g(\beta J\mathrm{*}m(r,t)+\beta h),h,\beta \ge 0,$ and using the normal hyperbolicity property we prove the continuity (upper semicontinuity and lower semicontinuity) of the global attractors of the flow generated by this equation, with respect to functional parameter $J$.

The conjugacy of a discrete semidynamical system and its partially decoupled discrete semidynamical system in a Banach space is proved in a neighbourhood of the nontrivial invariant manifold.