We study an implicit predictor-corrector iteration process for finitely many asymptotically quasi-nonexpansive self-mappings on a nonempty closed convex subset of a Banach space. We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. In the case is a uniformly convex Banach space and the mappings are asymptotically nonexpansive, we verify the weak (resp., strong) convergence of this iteration process to a common fixed point of these mappings if Opial's condition is satisfied (resp., one of these mappings is semicompact). Our results improve and extend earlier and recent ones in the literature.

A geometric classification of the compact four-dimensional Einstein-Weyl manifolds with at least four-dimensional symmetry group is given. Our results also sharpen previous results on four-dimensional Einstein metrics and correct Parker's topological classification of cohomogeneity-one four-manifolds.

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In this paper we continue our study of hopficity begun in [1], [2], [3], [4] and [5]. Let$A$be hopfian and let$B$have a cyclic center of prime power order. We improve Theorem 4 of [2] by showing that if$B$has finitely many normal subgroups which form a chain (we say$B$is$n$-normal), then$AxB$is hopfian. We then consider the case when$B$is a$p$-group of nilpotency class 2 and show that in certain cases$AxB$is hopfian.

We prove an explicit formula of ChungYaus Discrete Greens functions as well as hitting times of random walks on graphs. The formula is expressed in terms of two natural counting invariants of graphs. Uniform derivations of Greens functions and hitting times for trees and other special graphs are given.