We extend the notion of k-strictly pseudononspreading mappings introduced in Nonlinear Analysis 74 (2011) 1814-1822 to the notion of the more general pseudononspreading mappings. It is shown with example that the class of pseudononspreading mappings is more general than the class of k-strictly pseudonon-spreading mappings. Furthermore, it is shown with explicit examples that the class of pseudononspreading mappings and the important class of pseudocontractive mappings are independent. Some fundamental properties of the class of pseudononspreading mappings are proved. In particular, it is proved that the fixed point set of certain class of pseudononspsreading selfmappings of a nonempty closed and convex subset of a real Hilbert space is closed and convex. Demiclosedness property of such class of pseudonon-spreading mappings is proved. Certain weak and strong convergence theorems are then proved for the iterative approximation of fixed points of the class of pseudononspreading mappings.
