Semisolidity and locally weak quasisymmetry of homeomorphisms in metric spaces

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.