Boundaries in hyperspaces

Let C(X)C(X) be the hyperspace of subcontinua of a continuum X. Given a proper subcontinuum A of X  , we study the boundary Bd(C(A))Bd(C(A)) of C(A)C(A) in C(X)C(X). We show that Bd(C(A))Bd(C(A)) is always arcwise connected, and then a subcontinuum of C(X)C(X). We give conditions under which Bd(C(A))Bd(C(A)) coincides with the family of all subcontinua of A intersecting the boundary of A in X  . We also study conditions for the local connectedness and the unicoherence of sets of the form Bd(C(A))Bd(C(A)).