Multivalued pseudo-contractive mappings defined on unbounded sets in Banach spaces

Let X be a real Banach space. A multivalued operator T from K into 2 is said to be pseudo-contractive if for every x, y in K, u ∈ T (x), v ∈ T (y) and all r > 0, ‖x−y‖ ≤ ‖(1+r)(x−y)−r(u−v)‖. Denote by G(z, w) the set {u ∈ K : ‖u−w‖ ≤ ‖u−z‖}. Suppose every bounded closed and convex subset of X has the fixed point property with respect to nonexpansive selfmappings. Now if T is a Lipschitzian and pseudo-contractive mapping from K into the family of closed and bounded subsets of K so that the set G(z,w) is bounded for some z ∈ K and some w ∈ T (z), then T has a fixed point in K.