Retractions and contractibility in hyperspaces

Given a metric continuum X, let 2 X denote the hyperspace of all nonempty closed subsets of X. For each positive integer k let C k ( X ) stand for the hyperspace of members of 2 X having at most k components. Consider mappings φ B : C k ( X ) → C k + m ( X ) (where B ∈ C m ( X ) ) and ψ B : 2 X → 2 X both defined by A ↦ A ∪ B . We give necessary and sufficient conditions under which these mappings are deformation retractions (under a special convention for φ B ). The conditions are related to the contractibility of the corresponding hyperspaces.