W. A. Kirk

Abstract

        Let D be a bounded closed convex subset of a Banach space, and let  T : D x  D -----> D  be a continuous mapping which satisfies for all  x,y,z,t Î D ,

with strict inequality holding when ||x-z|| ¹  ||y-t||. Suppose T condensing in the sense that

for subsets U, V of D for which g( U \ V) > 0 (where g denotes the usual Kuratowski set-measure of noncompactness). A projection -iteration method is shown to converge to a solution of x=T(x,x). The significance of this result is that it holds in arbitrary spaces.