  In this paper, we investigate the relationship between semisolidity and
locally weak quasisymmetry of homeomorphisms in quasiconvex and complete metric
spaces. Our main objectives are to (1) generalize the main result in [X. Huang
and J. Liu, Quasihyperbolic metric and quasisymmetric mappings in metric
spaces, Trans. Amer. Math. Soc. 367 (2015), 6225-6246] together with other
related results, and (2) give a complete answer to the open problem given in
[X. Huang and J. Liu, Quasihyperbolic metric and quasisymmetric mappings in
metric spaces, Trans. Amer. Math. Soc. 367 (2015), 6225-6246]. As an
application, we prove that the composition of two locally weakly quasisymmetric
mappings is a locally weakly quasisymmetric mapping and that it is
quasiconformal.
