In this paper we deal with the following generalized vector quasi-equilibrium problem: given a closed convex set K in a normed space K , a subset K in a Hausdorff topological vector space K , and a closed convex cone K in K . Let K , K be two multifunctions and K be a single-valued mapping. Find a point K such that\begin {gather}(\hat x,\hat y)\in\Gamma (\hat x)\times\Phi (\hat x),\,\,{\rm and}\,\,\{f (\hat x,\hat y, z): z\in\Gamma (\hat x)\}\cap (-{\rm Int} C)=\emptyset.\notag\end {gather} We prove some existence theorems for the problem in which K can be discontinuous and K can be unbounded.

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

We study maps of positive operators of the Schatten -classes (), which preserve the -norms of convex combinations, that is, . They are exactly those carrying the form for a unitary or antiunitary . In the case , we have the same conclusion whenever it just holds for all the positive Hilbert-Schmidt class operators of norm . Some examples are demonstrated.

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.

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].