This defines an operator MK̄ , which we will call a Minkowski operator. Denote by A(C) the Frechet space of entire functions in n variables with the usual topology of the uniform convergence on compact sets in C, and C(R) the Frechet space of r times differentiable functions on R with the topology of the uniform convergence on compact sets in R of all partial derivatives up to the order r (1 ≤ r...

Let X be a real or complex Banach space and Y be a normed linear space. Suppose that f:X → Y is a Frechet differentiable function and F:X ⇉ 2 is a set-valued mapping with closed graph. Uniform convergence of Chord method for solving generalized equation y ∈ f x + F x ...... . (∗), where y ∈ Y a parameter, is studied in the present paper. More clearly, we obtain the uniform convergence of the se...

Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming Finite Element, Mixed Finite Element and Finite Volume methods. We show here that this framework can be applied to a family of degenerate non-linear parabolic equations (which contain in particular the Ri...

The concepts of I-convergence and I-Cauchy condition are a generalization of statistical convergence and statistical Cauchy conditions and are dependent on the notion of the ideal I of subsets of the set N of positive integers. In this paper, we shall introduce two new notions of I-uniform continuity and I-uniform boundedness of a function with values in R or in a metric space and then study th...

Within the categoryW of archimedean lattice-ordered groups with weak order unit, we show that the objects of the form C (L), the set of continuous real-valued functions on a locale L, are precisely those which are divisible and complete with respect to a variant of uniform convergence, here termed indicated uniform convergence. We construct the corresponding completion of aW-object A purely alg...

We develop and analyze multilevel methods for nonuniformly elliptic operators whose ellipticity holds in a weighted Sobolev space with an A2–Muckenhoupt weight. Using the so-called Xu-Zikatanov (XZ) identity, we derive a nearly uniform convergence result under the assumption that the underlying mesh is quasi-uniform. As an application we also consider the socalled α-harmonic extension to locali...

In this work we are interested in the numerical approximation of 1D parabolic singularly perturbed problems of reaction–diffusion type. To approximate the multiscale solution of this problem we use a numerical scheme combining the classical backward Euler method and central differencing. The scheme is defined on some special meshes which are the tensor product of a uniform mesh in time and a sp...

Let \(\Phi \) be a nuclear space, and let '\) denote its strong dual. In this paper, we introduce sufficient conditions for the almost sure uniform convergence on bounded intervals of time sequence '\)-valued processes having continuous (respectively, càdlàg) paths. The main result is formulated first in general setting cylindrical but later specialized to other situations interest. particular,...

The purpose of this paper is to provide new estimates for certain multilevel algorithms. In particular, we are concerned with the simple additive multilevel algorithm discussed recently together with J. Xu and the standard V-cycle algorithm with one smoothing step per grid. We shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of le...