Peano differentiability is a notion of higher order differentiability in the ordinary sense. H. W. Oliver gave sufficient conditions for the mth Peano derivative to be a derivative in the ordinary sense in the case of functions of a real variable. Here we generalize this theorem to functions of several variables.

We propose a direct method to control the first order fractional difference quotients of solutions to quasilinear subelliptic equations in the Heisenberg group. In this way we implement iteration methods on fractional difference quotients to obtain weak differentiability in the T -direction and then second order weak differentiability in the horizontal directions.

(Thus the subgradients of / correspond to the nonvertical supporting hyperplanes to the convex set consisting of all the points of E®R lying above the graph of /.) The set of subgradients of / at x is denoted by dfix). If d/(x) is not empty, / is said to be subdifferentiable at x. If/actually had a gradient x* = V/(x) at x in the sense of Gateaux (or Frechet), one would in particular have d/(x)...

Suppose that C is a nonempty closed convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. A viscosity iterative process is constructed in this paper. A strong convergence theorem is proved for a common element of the set of fixed points of a finite family of pseudocontractive mappings and the set of solutions of a finite family of monotone mappings....

We summarize recent ...ndings on the construction of designs, to be used in experiments for which regression is the anticipated method of analysis. The designs are to be robust against departures from the assumed linear response and/or departures from the assumption of uncorrelated errors. Included is a discussion of an “in...nitesimal” approach to design. In this approach one aims to minimize ...

A. The von Mises Expansion Before diving into the auxiliary results of Section 5, let us first derive some properties of the von Mises expansion. It is a simple calculation to verify that the Gateaux derivative is simply the functional derivative of in the event that T (F ) = R (f). Lemma 8. Let T (F ) = R (f)dμ where f = dF/dμ is the Radon-Nikodym derivative, is differentiable and let G be som...

Some algorithms for nding common xed point of a family of mappings isconstructed. Indeed, let C be a nonempty closed convex subset of a uniformlyconvex Banach space X whose norm is Gateaux dierentiable and let {Tn} bea family of self-mappings on C such that the set of all common fixed pointsof {Tn} is nonempty. We construct a sequence {xn} generated by the hybridmethod and also we give the cond...