Abstract: This paper proves the following results: let X be a continuum, let k, m ∈ N, and let B ∈ Cm(X), consider the continuous surjection f_k: C_k(X) → C_k(X). We define the mapping B : C_k(X) → C_k+m(X): by B (A) = f_k(A) B. Then following assertions are equivalent: (1) The hyperspace C_k(X) is g-contractible; (2) For each m ∈ N and for each B ∈ C_m(X) the mapping B is a W-deformation in C_k+m(X); (3) For each m ∈ N there exists B ∈C_m(X) such that the mapping B is a W-deformation in C_k+m(X); (4) There exists m ∈ N such that for each B ∈ C_m(X) the mapping B is a W-deformation in C_k+m(X); (5) There exist m ∈ N and B ∈ C_m(X) such that the mapping B is a W-deformation in C_k+m(X). 

Key words: continuum, hyperspaces, g-contractible, W-deformation mapping