Given a quasi-symmetric self-homeomorphism h of the unit circle Sl, let Q(h) be the set of all quasiconformal mappings with the boundary correspondence h. In this paper, it is shown that there exists certain quasi-symmetric homeomorphism h, such that Q(h) satisfies either of the conditions,
(1) Q(h) admits a quasiconformal mapping that is both uniquely locally-extremal and uniquely extremal-non-decreasable instead of being uniquely extremal;
(2) Q(h) contains infinitely many quasiconformal mappings each of which has an extremal non-decreasable dilatation.
An infinitesimal version of this result is also obtained.