Let f be a continuous map on infra-solvmanifold M of type(R)and N∞(f)be the asymptotic Nielsen number of f.In this paper,the sufficient conditions to assure that log N∞(f)is the infimum of topological entropies of the homotopy class of the map f are given by using Nielsen fixed point theory.These conclusions will generalize the similar results on infra-nilmanifolds.