The hybrid steepest-descent method introduced by Yamada (2001) is an algorithmic solution to the variational inequality problem over the fixed point set of nonlinear mapping and applicable to a broad range of convexly constrained nonlinear inverse problems in real Hilbert spaces. Lehdili and Moudafi (1996) introduced the new prox-Tikhonov regularization method for proximal point algorithm to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in Hilbert spaces. In this paper, motivated by Yamada's hybrid steepest-descent and Lehdili and Moudafi's algorithms, a generalized hybrid steepest-descent algorithm for computing the solutions of the variational inequality problem over the common fixed point set of sequence of nonexpansive-type mappings in the framework of Banach space is proposed. The strong convergence

In this paper, we introduce a new class of generalized co-complementarity problems in Banach spaces. An iterative algorithm for finding approximate solutions of these problems is considered. Some convergence results for this iterative algorithm are derived and several existence results are obtained.

Let A 1, A 2 be standard operator algebras on complex Banach spaces X 1, X 2, respectively. For k 2, let (i 1,, i m) be a sequence with terms chosen from {1,, k}, and define the generalized Jordan product T 1 T k= T i 1 T i m+ T i m T i 1 on elements in A i. This includes the usual Jordan product A 1 A 2= A 1 A 2+ A 2 A 1, and the triple {A 1, A 2, A 3}= A 1 A 2 A 3+ A 3 A 2 A 1. Assume that at least one of the terms in (i 1,, i m) appears exactly once. Let a map : A 1 A 2 satisfy that ( (A 1) (A k))= (A 1 A k) whenever any one of A 1,, A k has rank at most one. It is shown in this paper that if the range of contains all operators of rank at most three, then must be a Jordan isomorphism multiplied by an m th root of unity. Similar results for maps between self-adjoint operators acting on Hilbert spaces are also obtained.

Let A1, A2 be (not necessarily unital or closed) standard operator algebras on locally convex spaces X1, X2, respectively. For k 2, define different kinds of products T1 Tk on elements in Ai, which covers the usual product T1 Tk= T1 Tk, and the Jordan triple product T1 T2= T2T1T2. Let : A1 A2 be a (not necessarily linear) map satisfying that ( (A1) (Ak))= (A1 Ak) whenever any one of Ais is of rank zero or one. It is shown that if the range of contains all rank one and rank two operators then it must be a Jordan isomorphism multiplied by a root of unity. Similar results for self-adjoint operators acting on Hilbert spaces are obtained.

The object of this paper is to establish an existence result for nonlinear inequalities for not necessarily pseudomonotone maps. As a consequence of our result, we give some existence results for variational and variational-like inequalities.

Assume that <i>F</i> is a nonlinear operator on a real Hilbert space <i>H</i> which is -strongly monotone and -Lipschitzian on a nonempty closed convex subset <i>C</i> of <i>H</i>. Assume also that <i>C</i> is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on <i>H</i>. We construct an iterative algorithm with variable parameters which generates a sequence {<i>x</i> _<i>n</i>} from an arbitrary initial point <i>x</i> ₀ <i>H</i>. The sequence {<i>x</i> _<i>n</i>} is shown to converge in norm to the unique solution <i>u</i>  of the variational inequality

In this paper, we prove that into isometries and disjointness preserving linear maps from<i>C</i>₀(<i>X</i>) into<i>C</i>₀(<i>Y</i>) are essentially weighted composition operators<i>Tf</i>=<i>h</i><i>f</i> for some continuous map and some continuous scalar-valued function<i>h</i>.

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points <i>F</i>(<i>S</i>) of a nonexpansive mapping <i>S</i> and the set of solutions _<i>A</i> of the variational inequality for a monotone, Lipschitz continuous mapping <i>A</i>. We introduce a hybrid extragradient-like approximation method which is based on the well-known extragradient method and a hybrid (or outer approximation) method. The method produces three sequences which are shown to converge strongly to the same common element of {F(S)\cap\Omega_{A}}. As applications, the method provides an algorithm for finding the common fixed point of a nonexpansive mapping and a pseudocontractive mapping, or a common zero of a monotone Lipschitz continuous mapping and a maximal monotone mapping.

We prove a mean ergodic theorem for amenable discrete quantum groups. As an application, we prove a Wiener type theorem for continuous measures on compact metrizable groups.

Motivated by reformulating Furstenberg's $\times p,\times q$ conjecture via representations of a crossed product $C^*$-algebra, we show that in a discrete $C^*$-dynamical system $(A,\Gamma)$, the space of (ergodic) $\Gamma$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A\rtimes\Gamma$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $\Gamma$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classification a special type of irreducible representations of $C(X)\rtimes \Gamma$.