Let H: Mm Mm be a holomorphic function of the algebra Mm of complex m m matrices. Suppose that H is orthogonally additive and orthogonally multiplicative on self-adjoint elements. We show that either the range of H consists of zero trace elements, or there is a scalar sequence {n} and an invertible S in Mm such that

The purpose of this paper is to construct two superimposed optimization methods for solving the mixed equilibrium problem and variational inclusion. We show that the proposed superimposed methods converge strongly to a solution of some optimization problem. Note that our methods do not involve any projection.

In this paper, we introduce and study a triple hierarchical variational inequality (THVI) with constraints of minimization and equilibrium problems. More precisely, let Fix ( T ) be the fixed point set of a nonexpansive mapping, let Fix ( T ) be the solution set of a mixed equilibrium problem (MEP), and let <i></i> be the solution set of a minimization problem (MP) for a convex and continuously Frechet differential functional in Hilbert spaces. We want to find a solution Fix ( T ) of a variational inequality with a variational inequality constraint over the intersection of Fix ( T ) , Fix ( T ) , and <i></i>. We propose a hybrid iterative algorithm with regularization to compute approximate solutions of the THVI, and we present the convergence analysis of the proposed iterative algorithm.

Let <i>C</i> be a nonempty closed convex subset of a real Hilbert space <i>H</i>. Let <i>S</i> : <i>C</i> <i>C</i> be an asymptotically nonexpansive map in the intermediate sense with the fixed point set <i>F</i>(<i>S</i>). Let <i>A</i> : <i>C</i> <i>H</i> be a Lipschitz continuous map, and <i>VI</i>(<i>C, A</i>) be the set of solutions <i>u</i> <i>C</i> of the variational inequality

The Opial property of Hilbert spaces and some other special Banach spaces is a powerful tool in establishing fixed point theorems for nonexpansive and, more generally, nonspreading mappings. Unfortunately, not every Banach space shares the Opial property. However, every Banach space has a similar Bregman-Opial property for Bregman distances. In this paper, using Bregman distances, we introduce the classes of Bregman nonspreading mappings and investigate the Mann and Ishikawa iterations for these mappings. We establish weak and strong convergence theorems for Bregman nonspreading mappings.