Borsuk-Ulam type theorems for multivalued maps

Let Xn (n > 1) be a finitistic space with cohomology type (0, 0). Let (Xn, E, pi, B) be a fibre bundle and (Rk, E', pi', B) be a k-dimensional real vector bundle with fibre preserving G = Zp, p > 2 a prime, action such that G acts freely on E and E' -{0}, where {0} is the zero section of the vector bundle. We determine a lower bound of the cohomological dimension of the set Ap = {x A E I p(x) n p(gx) n center dot center dot center dot n p(gp-1x) 0 0} for an admissible multivalued fibre preserving map p : E-> E'.