On convex closed bounded bodies without farthest points such that the closure of their complement is antiproximinal

We consider an abstract attainability problem with constraints of asymptotic nature defined in the form of a nonempty family of subsets in the space of ordinary solutions. Various variants of implementing asymptotic effects are considered (convergence in a topological space, cycles, and so on). A rather general method is suggested for presenting the results of action of approximate solutions; this method generalizes constructions based on sequences in the space of ordinary solutions.