We introduce a new implicit iterative scheme with perturbation for finding the approximate solutions of a hierarchical variational inequality, that is, a variational inequality over the common fixed point set of a finite family of nonexpansive mappings. We establish some convergence theorems for the sequence generated by the proposed implicit iterative scheme. In particular, necessary and sufficient conditions for the strong convergence of the sequence are obtained.

Let X and Y be locally compact Hausdorff spaces. We give a full description of disjointness preserving Fredholm linear operators T from C0(X) into C0(Y), and show that is continuous if either Y contains no isolated point or T has closed range. Our task is achieved by writing T as a weighted composition operator Tf = h f . Through the relative homeomorphism , the structure of the range space of T can be completely analyzed, and X and Y are homeomorphic after removing finite subsets.

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.