Retractions and contractibility in hyperspaces

Given a metric continuum X, let X2 denote the hyperspace of all nonempty closed subsets of X. For each positive integer k let Ck(X) stand for the hyperspace of members of X2 having at most k components. Consider mappings (where B∈Cm(X)) and both defined by A↦A∪B. We give necessary and sufficient conditions under which these mappings are deformation retractions (under a special convention for φB). The conditions are related to the contractibility of the corresponding hyperspaces.