We prove that the sequences generate by the Douglas-Rachford method converge weakly to a solution of the inclusion problem 2000 Mathematics Subject Classification: 47H05, 49J52, 47N10.

We provide some new results on the weak convergence of sequences or nets lying in L,((T, & r), X) = L,(p, X), 1 <p < co, i.e., the space of equivalence classes of X-valued (X is a Banach space) Bochner integrable functions on the finite measure space ( r, z, I”). Our theorems generalize in several directions recent resuls on weak sequential convergence in L,(p, X) obtained by M. A. Khan and M. ...

The lack of measurability of the empirical process with respect to the sigma-field generated by the ‘natural’ l∞ metric, as illustrated in the previous notes, needs an extension of the standard weak convergence theory that can handle situations where the converging stochastic processes may no longer be measurable (though the limit will be a tight Borel measurable random element). Of course, an ...

begin{abstract} In this paper, we prove some strong and weak convergence of three step random iterative scheme with errors to common random fixed points of three asymptotically nonexpansive nonself random mappings in a real uniformly convex separable Banach space. end{abstract}

The first part of the paper deals with general features of weak star convergence from a topological point of view. We present basic facts about weak and weak star topologies, dual spaces and representations of linear functionals as Radon measures. A special attention is payed to finitely additive measures and some results regarding the Baire sets. We prove that in a wide class of topological sp...

This mostly expository paper shows how weak convergence methods provide simple, elegant proofs of (i) the stabilization of an inverted pendulum under fast vertical oscillations, (ii) the existence of particle traps induced by rapidly varying electric fields and (iii) the adiabatic invariance of ∫ Γ p dx for slowing varying planar Hamiltonian dynamics. Under an appropriate, but very restrictive,...

We offer an umbrella type result which extends the convergence of classical empirical process on the line to more general processes indexed by functions of bounded variation. This extension is not contingent on the type of dependence of the underlying sequence of random variables. As a consequence we establish the weak convergence for stationary empirical processes indexed by general classes of...

We have not found proofs of these facts in the literature. The aim of this paper is to present such proofs. Moreover, we show that the first statement also holds for the case p = 1. The two facts described above are used in the proof of the reiterated homogenization result for monotone operators, see [14] and [15]. The solution uξh is used to define a sequence of functions similar to the ones i...

We prove a general result on equality of the weak limits of the zero counting measure, dνn, of orthogonal polynomials (defined by a measure dμ) and 1 n Kn(x, x) dμ(x). By combining this with the asymptotic upper bounds of Máté–Nevai and Totik on nλn(x), we prove some general results on ∫ I 1 n Kn(x, x) dμs → 0 for the singular part of dμ and ∫ I |ρE(x) − w(x) n Kn(x, x)| dx → 0, where ρE is the...