In this paper, we prove a Halpern-type strong convergence theorem for nonexpansive mappings in a Banach space whose norm is uniformly Gâteaux differentiable. Also, we discuss the sufficient and necessary condition about this theorem. This is a partial answer of the problem raised by Reich in 1983.

The Fréchet mean generalizes the idea of averaging in spaces where pairwise addition is not well-defined. In general metric spaces, however, the Fréchet sample mean is not a consistent estimator of the theoretical Fréchet mean. For non-trivial examples, sequences of Fréchet sample mean sets may fail to converge in a set-analytical sense. Hence, it becomes necessary to consider other types of co...

In this paper, we introduce a generalized iteration scheme with errors for convergence to common fixed points of two nonexpansive mappings. This scheme contains a wide variety of existing iteration schemes as its special cases. The main feature of this scheme is that its special cases can handle both strong convergence like Halpern-type and weak convergence like Ishikawatype iteration schemes. ...

1. Introduction. Let (R") is the algebra Q1(Rn) of all real-valued functions of class C1. In other words, for every fEG1(Rn) there is a sequence {pn} of polynomials such that pn—*/ uniformly o...

Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra). We prove a strong convergence theorem that for 0-connected algebras and modules over a (−1)-connected operad, the homotopy completion tower interpolates (in a strong ...

An entangled ergodic theorem was introduced in [1] in connection with the quantum central limit theorem, and clearly formulated in [6]. Namely, let U be a unitary operator on the Hilbert space H, and for m ≥ k, α : {1, . . . , m} 7→ {1, . . . , k} a partition of the set {1, . . . , m} in k parts. The entangled ergodic theorem concerns the convergence in the strong, or merely weak (s–limit, or w...

In this paper we obtain almost sure convergence theorems for vectorvalued uniform amarts and C-sequences without assuming the Radon-Nikodym Property. Specifically, it is shown that if a limit exists in a weak sense for these martingale generalizations, then a.s. scalar and strong convergence follow. These results lead to some new versions of the Ito-Nisio theorem. Convergence results for random...

We will extend Reed's Semi-Strong Perfect Graph Theorem by proving that unbreakable C 5-free graphs diierent from a C 6 and its complement have unique P 4-structure.

V. Chv atal proved that no minimal imperfect graph has a small transversal, that is, a set of vertices of cardinality at most + ! 1 which meets every !-clique and every -stable set. In this paper we prove that a slight generalization of this notion of small transversal leads to a conjecture which is as strong as Berge's Strong Perfect Graph Conjecture for a very large class of graphs, although ...

In order to prove the Strong Perfect Graph Conjecture, the existence of a ”simple” property P holding for any minimal non-quasi-parity Berge graph G would really reduce the difficulty of the problem. We prove here that this property cannot be of type ”G is F-free”, where F is any fixed family of Berge graphs.