Induced mappings on hyperspaces C(p,X) and K(X)

Given a metric continuum X and p∈X, we denote by C(X) and C(p,X) the hyperspace of all subcontinua of X and the hyperspace of all subcontinua of X containing p, respectively. Thus K(X) is defined as the collection of all subsets of C(C(X)) of the form C(q,X) where q∈X. For a mapping f:X→Y between continua, we consider f¯, f¯p, f˜ and fˇ the natural induced mapping by f at those hyperspaces. In this paper we prove some relationships between the mappings f, f¯, f¯p, f˜ and fˇ for the following classes of mappings: monotone, confluent, weakly confluent, light, open. Moreover, we show that f¯p is a monotone mapping.