Boundary value problems for systems of difference equations associated with systems of second-order ordinary differential equations

We study difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order ordinary differential equations. We formulate conditions which guarantee a priori bounds on first differences of solutions to the discretized problem. We establish existence results for solutions to the discretized boundary value problems subject to nonlinear boundary conditions. We apply our results to show that solutions to the discrete problem converge to solutions of the continuous problem in an aggregate sense. (C) 2002 Elsevier Science Ltd. All rights reserved.

Existence of multiple solutions for second-order discrete boundary value problems

We give conditions on f involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two-point boundary value problem yk+1 - 2yk + yk-1 + f (k, yk, vk) = 0, for k = 1,..., n - 1, y0 = 0 = yn,, where f is continuous and vk = yk - yk-1, for k = 1,..., n. In the special case f (k, t, p) = f (t) greater than or equal to 0, we give growth conditions on f and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and Peterson and are in the spirit of our results for the continuous analogue. (C) 2002 Elsevier Science Ltd. All rights reserved.

Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations

We study the continuous problem y"=f(x,y,y'), xc[0,1], 0=G((y(0),y(1)),(y'(0), y'(1))), and its discrete approximation (y(k+1)-2y(k)+y(k-1))/h(2) =f(t(k), y(k), v(k)), k = 1,..., n-1, 0 = G((y(0), y(n)), (v(1), v(n))), where f and G = (g(0), g(1)) are continuous and fully nonlinear, h = 1/n, v(k) = (y(k) - y(k-1))/h, for k =1,..., n, and t(k) = kh, for k = 0,...,n. We assume there exist strict lower and strict upper solutions and impose additional conditions on f and G which are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. We show that the discrete approximation also has solutions which approximate solutions of the continuous problem and converge to the solution of the continuous problem when it is unique, as the grid size goes to 0. Homotopy methods can be used to compute the solution of the discrete approximation. Our results were motivated by those of Gaines.

The nonexistence of spurious solutions to discrete, two-point boundary value problems

We investigate difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations. We formulate conditions under which all solutions to the discrete problem satisfy certain a priori bounds which axe independent of the step-size. As a result, the nonexistence of spurious solutions are guaranteed. Some existence and convergence theorems for solutions to the discrete problem are also presented. (C) 2002 Elsevier Science Ltd. All rights reserved.

Existence of multiple solutions for finite difference approximations to second-order boundary value problems

Error condition detected We consider discrete two-point boundary value problems of the form D-2 y(k+1) = f (kh, y(k), D y(k)), for k = 1,...,n - 1, (0,0) = G((y(0),y(n));(Dy-1,Dy-n)), where Dy-k = (y(k) - Yk-I)/h and h = 1/n. This arises as a finite difference approximation to y" = f(x,y,y'), x is an element of [0,1], (0,0) = G((y(0),y(1));(y'(0),y'(1))). We assume that f and G = (g(0), g(1)) are continuous and fully nonlinear, that there exist pairs of strict lower and strict upper solutions for the continuous problem, and that f and G satisfy additional assumptions that are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. Under these assumptions we show that there are at least three distinct solutions of the discrete approximation which approximate solutions to the continuous problem as the grid size, h, goes to 0. (C) 2003 Elsevier Science Ltd. All rights reserved.

Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C: K--> 2(Y) a point-to-set mapping such that for any x is an element of K, C(x) is a pointed, closed, and convex cone in Y and int C(x) not equal 0. Given a mapping g : K --> K and a vector valued bifunction f : K x K - Y, we consider the implicit vector equilibrium problem (IVEP) of finding x* is an element of K such that f (g(x*), y) is not an element of - int C(x) for all y is an element of K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems. (C) 2003 Elsevier Science Ltd. All rights reserved.

On continuous selection problems for multivalued mappings with the local intersection property in hyperconvex metric spaces

In the paper we present two continuous selection theorems in hyperconvex metric spaces and apply these to study xed point and coincidence point problems as well as variational inequality problems in hyperconvex metric spaces.

The uniqueness of solutions to discrete, victor, two-point boundary value problems

Difference equations which may arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations are investigated and conditions are formulated under which solutions to the discrete problem are unique. Some existence, uniqueness implies existence, and convergence theorems for solutions to the discrete problem are also presented.

O t??nel no fim da luz: a privatiza????o do setor el??trico em Mato Grosso

Este artigo tem por objetivo analisar a forma????o de consensos sociais e pol??ticos para a reforma do Estado e de seu aparelho em n??vel estadual, conforme a governabilidade democr??tica. Optou-se por uma pesquisa sobre a privatiza????o das Centrais El??tricas Mato- Grossenses S.A. (Cemat), considerada representativa tanto da reforma empreendida pelo Estado de Mato Grosso quanto das dificuldades do setor el??trico em n??vel nacional. O referencial metodol??gico dessa pesquisa ?? composto de um problema e de quatro hip??teses de trabalho. Trata-se de estudo de caso da esp??cie ???provas de plausibilidade???, em que, por uma aplica????o iterativa, apresenta-se uma proposi????o te??rica inicial e, em seguida, as conclus??es da pesquisa s??o comparadas, o que permite sua revis??o e compara????o. Para tanto, descrevem-se a privatiza????o da Cemat e os problemas da desestatiza????o do setor el??trico brasileiro no seu conjunto. Por fim, na conclus??o, verifica-se a validade das hip??teses e fazem-se algumas considera????es finais sobre a pesquisa.

Concepções de alfabetização, leitura e escrita que ancoram o Projeto Trilhas

Este trabalho tem como objeto de estudo os impressos do Projeto Trilhas, material pedagógico produzido em parceria entre o Instituto Natura, a Comunidade Educativa CEDAC e o Ministério da Educação. Trata-se de uma análise documental com as quais se procurou envolver as noções de enunciado, texto, gênero e suporte que possibilitaram fundamentar a proposta metodológica, pautada pelo diálogo, que teve como escopo problematizar como esse conjunto de materiais pode contribuir no processo do ensino e da aprendizagem das crianças matriculadas nas turmas do primeiro ano do Ensino Fundamental, com foco na análise das concepções de alfabetização, leitura e escrita, engendradas nos materiais. Para isso, o referencial teórico que balizou as reflexões se fundamentou nas contribuições da perspectiva bakhtiniana de linguagem e lançou ancoragens no conceito de alfabetização proposto criticamente por Gontijo (2008). As análises se constituíram como uma arena, isto é, um palco de alteridade. Logo, buscaram compreender como o conceito e as concepções se materializaram nas atividades produzidas pelos sujeitos-autores e problematizaram como os impressos do Projeto Trilhas podem contribuir para a melhoria do ensino e da aprendizagem das crianças matriculadas no primeiro ano do Ensino Fundamental. Com as análises, sustenta-se que o conceito que solidifica a constituição dos impressos deste projeto se aproxima das contribuições de Ferreiro e Teberosky (1999), isto é, a alfabetização é o processo pelo qual as crianças assimilam o código escrito e compreende os usos que são dados a ele nas culturas do escrito. A leitura se configurou como decodificação dos signos linguísticos e compreensão de significados, e a escrita como codificação.

Reviewing horizontalization: the challenge of analysis in Brazilian foreign policy

Abstract This article presents the increasing demands over the Brazilian Ministry of Foreign Affairs (Itamaraty) for opening its doors to other actors. This discussion will be followed by relevant theoretical and methodological analysis. We will defend the need to overcome problems related to: 1) conceptual vagueness about what the concept of participation means; 2) lack of clarity in the baseline to which comparisons are made; 3) fragile empirical basis; 4) limitations on the use of sources; and 5) how to understand the impact exerted by systemic forces.