The theory of weighted inequalities for the Hardy operator, acting on monotone functions in R+, was first introduced in [2]. Extensions of these results to higher dimensions have been considered only in very specific cases: in particular, in the diagonal case, for p = 1 only (see [3]). The main difficulty in this context is that the level sets of the monotone functions are not totally ordered, ...

We study a modified Newton’s method with fifth-order convergence for nonlinear equations in Banach spaces. Wemake an attempt to establish the semilocal convergence of this method by using recurrence relations. The recurrence relations for the method are derived, and then an existenceuniqueness theorem is given to establish the R-order of the method to be five and a priori error bounds. Finally,...

We study the approximation problem for C∞ functions f : [0, 1] → R with respect to a Wm p -norm. The optimal order of convergence is infinite, hence excellent, but the problem is still intractable for m ≥ 1. This means that the order of convergence supplies incomplete information concerning the computational difficulty of a problem. For m = 0 and p = 2, we prove that the problem is not polynomi...

A quadrature rule as simple as the classical Gauss formula, with a lower computational cost and having the same convergence order of best weighted polynomial approximation in L is constructed to approximate integrals on unbounded intervals. An analogous problem is discussed in the case of Lagrange interpolation in weighted L norm. The order of convergence in our results is the best in the liter...

where fn : E × E → R, f : E × E → R, φn : E × E → R ∪ {∞}, φ : E × E → R ∪ {∞}. The aim of this paper is to give suitable conditions on the convergence of (fn)n to f and (φn)n to φ in order to obtain a convergence result for the solutions of (1.1)n to solutions of (1.2). This study was motivated by the increasing interest in the topic of generalized quasi-variational inequalites (in short g.q.v...