Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations

We prove the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution. To this aim, we prove an appropriate fixed point theorem in partially ordered sets. ★ Partially supported by Ministerio de Ciencia y Tecnología/ FEDER, project BFM2001-3884-C02-01; and by Xunta de Galicia/ FEDER, project PGIDIT02PXIC20703PN.     

On Birkhoff integrability for scalar functions and vector measures

The Bartle–Dunford–Schwartz integral for scalar functions with respect to vector measures is characterized by means of Riemann-type sums based on partitions of the domain into countably many measurable sets. In this setting, two natural notions of integrability (Birkhoff integrability and Kolmogoroff integrability) turn out to be equivalent to Bartle–Dunford–Schwartz integrability. A. Fernández, F. Mayoral and F. Naranjo were supported by MEC and FEDER (project MTM2006–11690–C02–02) and La Junta de Andalucía. J. Rodríguez was supported by MEC and FEDER (project MTM2005-08379), Fundación Séneca (project 00690/PI/04) and the Juan de la Cierva Programme (MEC and FSE).     

Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations

We prove some fixed point theorems in partially ordered sets, providing an extension of the Banach contractive mapping theorem. Having studied previously the nondecreasing case, we consider in this paper nonincreasing mappings as well as non monotone mappings. We also present some applications to first–order ordinary differential equations with periodic boundary conditions, proving the existence of a unique solution admitting the existence of a lower solution. Research partially supported by Ministerio de Educación y Ciencia and FEDER, Project MTM2004-06652-C03-01, and by Xunta de Galicia and FEDER, Projects PGIDIT02PXIC20703PN and PGIDIT05PXIC20702PN     

On a mapping approach to investigating the bootstrap accuracy

Summary. A simple mapping approach is proposed to study the bootstrap accuracy in a rather general setting. It is demonstrated that the bootstrap accuracy can be obtained through this method for a broad class of statistics to which the commonly used Edgeworth expansion approach may not be successfully applied. We then consider some examples to illustrate how this approach may be used to find the bootstrap accuracy and show the advantage of the bootstrap approximation over the Gaussian approximation. For the multivariate Kolmogorov–Smirnov statistic, we show the error of bootstrap approximation is as small as that of the Gaussian approximation. For the multivariate kernel type density estimate, we obtain an order of the bootstrap error which is smaller than the order of the error of the Gaussian approximation given in Rio (1994). We also consider an application of the bootstrap accuracy for empirical process to that for the copula process. Received: 23 June 1995 / In revised form: 18 June 1996     

Some approximation results for the compact-open topology

We obtain some Stone-Weierstrass type approximation theorems and a result on the approximation property for the compact-open topology in the non-locally convex setting.     

Approximation of the Fourier Transform and the Dual Gabor Window

Many results and problems in Fourier and Gabor analysis are formulated in the continuous variable case, i.e., for functions on . In contrast, a suitable setting for practical computations is the finite case, dealing with vectors of finite length. We establish fundamental results for the approximation of the continuous case by finite models, namely, the approximation of the Fourier transform and the approximation of the dual Gabor window of a Gabor frame. The appropriate function space for our approach is the Feichtinger space S₀. It is dense in L², much larger than the Schwartz space, and it is a Banach space.     

Lower Semicontinuity in SBH

The aim of this paper consists in giving a first characterization of lower semicontinuity for surface integrals in the space of special functions of bounded hessian, showing at the same time some conditions necessarily verified by the integrands. Partially supported by FCT/POCTI/FEDER.     

Feedback Stackelberg equilibrium strategies when the private label competes with the national brand

We consider a noncooperative differential game where a retailer sells her own private label in addition to the manufacturer’s brand. We assume that each brand’s goodwill evolves according to a modified Nerlove-Arrow dynamics, in such a way that the advertising effort of one brand hurts the competitor’s goodwill stock. We characterize Feedback-Stackelberg pricing and advertising strategies and employ simulations to analyze their sensitivity to the main model parameters. We thank an anonymous Reviewer for very helpful comments. Research supported by NSERC, Canada, and FQRSC, Quebec. Research completed when the second author was a visiting professor at GERAD, HEC Montréal. The second author’s research was partially supported by MEC and JCYL under projects SEJ2005-03858/ECON and VA045A06, co-financed by FEDER funds.     

A general approach to approximation problems of the Bernstein type

The Bernstein approximation problem is treated in a general setting of Banach function spaces. An abstract theory is developed which includes as special cases the solutions of the Bernstein problem given by Mergelyan and Pollard. It also provides a unified setting for related results from polynomial and trigonometric approximation theory and yields as corollaries a number of new results.     

Periodic Orbits for Some Systems of Delay Differential Equations

We provide sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay. We extend Kaplan-Yorke＇s method for finding periodic orbits from a delay differential equation with several delays to a system of delay differential equations with a unique delay.     

Multivariate approximation in the worst case setting over reproducing kernel Hilbert spaces

We study the worst case setting for approximation of d variate functions from a general reproducing kernel Hilbert space with the error measured in the L_∞ norm. We mainly consider algorithms that use n arbitrary continuous linear functionals. We look for algorithms with the minimal worst case errors and for their rates of convergence as n goes to infinity. Algorithms using n function values will be analyzed in a forthcoming paper.We show that the L_∞ approximation problem in the worst case setting is related to the weighted L₂ approximation problem in the average case setting with respect to a zero-mean Gaussian stochastic process whose covariance function is the same as the reproducing kernel of the Hilbert space. This relation enables us to find optimal algorithms and their rates of convergence for the weighted Korobov space with an arbitrary smoothness parameter α>1, and for the weighted Sobolev space whose reproducing kernel corresponds to the Wiener sheet measure. The optimal convergence rates are n^-(α-1)/2 and n^-1/2, respectively.We also study tractability of L_∞ approximation for the absolute and normalized error criteria, i.e., how the minimal worst case errors depend on the number of variables, d, especially when d is arbitrarily large. We provide necessary and sufficient conditions on tractability of L_∞ approximation in terms of tractability conditions of the weighted L₂ approximation in the average case setting. In particular, tractability holds in weighted Korobov and Sobolev spaces only for weights tending sufficiently fast to zero and does not hold for the classical unweighted spaces.     

Necessary Optimality Conditions for a Dead Oil Isotherm Optimal Control Problem

We study a system of nonlinear partial differential equations resulting from the traditional modelling of oil engineering within the framework of the mechanics of a continuous medium. Recent results on the problem provide existence, uniqueness and regularity of the optimal solution. Here we obtain the first necessary optimality conditions. Work supported by the Portuguese Foundation for Science and Technology (FCT) through the Centre for Research in Optimization and Control (CEOC) of the University of Aveiro, cofinanced by the European Community fund FEDER/POCTI. The first author was also supported by the postdoctoral fellowship SFRH/BPD/20934/2004.     

A Weak Maximum Principle for Optimal Control Problems with Nonsmooth Mixed Constraints

We derive a weak Maximum Principle for nonsmooth optimal control problem involving mixed constraints under some convexity assumptions. Notably we consider problems with possibly nonsmooth mixed constraints. A nonsmooth version of the positive linear independence of the gradients with respect to the control of the mixed constraints plays a key role in validation of our main result. The first author was support by FEDER and FCT-Portugal, grants POSC/EEA/SRI/61831/2004 and SFRH/BSAB/781/2008. G.N. Silva thanks the financial support of CNPq grant 200875/06-0 and FAPESP grant 07-5226-6.     

Isometries of Weighted Spaces of Harmonic Functions

We investigate the isometries between weighted spaces of harmonic functions. We show that, under some mild conditions, every isometry is a composition operator. Our research shows that the structure of isometries of weighted spaces of harmonic functions is, in general, simpler than that observed for weighted spaces of holomorphic functions. Supported by MEC and FEDER Project MTM 2005-08210.     

Dynamics of the differentiation operator on weighted spaces of entire functions

The continuity of the differentiation operator on weighted Banach spaces of entire functions with sup-norm has been characterized recently by Harutyunyan and Lusky. We give necessary and sufficient conditions to ensure that the differentiation operator on these weighted Banach spaces of entire functions is hypercyclic or chaotic, when it is continuous. This research was partially supported by MEC and FEDER Project MTM2007-62643.     

Credibility of Incentive Equilibrium Strategies in Linear-State Differential Games

We characterize in this paper the credibility of incentive equilibrium strategies for the class of linear-state differential games. We derive a general condition for credibility and illustrate its use on two differential games taken from the literature of environmental economics and knowledge accumulation. We show that the proposed linear incentive strategies are not always credible. Further, we provide alternative nonlinear credible strategies which suggest that we should not stick only to linear incentive strategies, even in a simple class of differential games such as the linear-state one.This research was completed when the first author was visiting professor at GERAD, HEC, Montréal. The first author’s research was partially supported by MCYT under project BEC2002-02361 and by JCYL under project VA051/03, confinanced by FEDER funds. The second author’s research was supported by NSERC, Canada.     

A maschke type theorem for weak Hopf algebras

In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. Also, from these results, we deduce a version of Maschke＇s Theorem for （H, B）-Hopf modules associated with a weak Hopf algebra H and a right H-comodule algebra B.     

A non-cooperative approach to the cost spanning tree problem

We associate to each cost spanning tree problem a non-cooperative game, which is inspired by a real-life problem. We study the Nash equilibria and subgame perfect Nash equilibria of this game. We prove that these equilibria are closely related with situations where agents connect sequentially to the source.Finicial support from the Ministerio de Ciencia y Tecnologia and FEDER, and Xunta de Galicia through grants BEC2002-04102-C02-01 and PGIDIT03PXIC30002PN is gratefully acknowledged.     

A polynomial of degree four not satisfying Rolle’s theorem in the unit ball of <Emphasis Type="Italic">l</Emphasis><Subscript>2</Subscript>

We give an example of a fourth degree polynomial which does not satisfy Rolles Theorem in the unit ball of l ₂.The author has been partially supported by MCyT and FEDER Project BFM2002-01423.     

Ideals of Multilinear Forms – a Limit Order Approach

A general theory of limit orders for ideals of multilinear forms is developed. We relate the limit order of an ideal to those of its maximal hull and its adjoint ideal. We study the limit orders of the ideals of dominated and multiple summing multilinear forms. Finally, estimates of the diagonal of a (non-necessarily diagonal) multilinear form are presented, in terms of the limit order of the ideals to which it belongs. The third author was supported by the MCYT and FEDER Project BFM2002-01423 and grant GV-GRUPOS04/45. The first and second authors were supported by CONICET-PIP 5272. The first author was also supported by UBACyT-X108 and ANPCyT-PICT 0315033.