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 A be a C*-algebra, L a closed left ideal of A and p the closed projection related to L. We show that for an xp in A**p ( A**/L**) if pAxp pAp and px*xp pAp then xp Ap ( A/L). The proof goes by interpreting elements of A**p (resp. Ap) as admissible (resp. continuous admissible) vector sections over the base space F(p) = { A* : 0, (p) = 1} in the notions developed by Diximier and Douady, Fell, and Tomita. We consider that our results complement both Kadison function representation and Takesaki duality theorem.

In this paper, using the concept of attractive points of a nonlinear mapping, we obtain a strong convergence theorem of Halperns type [<i>Bull. Amer. Math. Soc</i>. <b>73</b> (1967), 957961] for a wide class of nonlinear mappings which contains nonexpansive mappings, nonspreading mappings and hybrid mappings in a Hilbert space. Using this result, we obtain well-known and new strong convergence theorems of Halperns type in a Hilbert space. In particular, we solve a problem posed by Kurokawa and Takahashi [<i>Nonlinear Anal</i>. <b>73</b> (2010, 15621568].

We define type A, type B, type C as well as C*-semi-finite C*-algebras.

Let T be a locally compact Hausdorff space, called base space. Suppose for each t in T there is a (real or complex) Banach space Et. A vector field x is an element in the product space t T Et, that is, x (t) Et, for all t T.