Ordinary differential inequalities and quasimonotonicity in ordered topological vector spaces

A well known comparison theorem on ordinary differential inequalities with quasimonotone right-hand side was carried over by
Volkmann (1972) to (pre)ordered topological vector spaces. We prove that the quasimonotonicity of is a necessary condition here if is continuous. Then it is shown that quasimonotonicity can be verified by considering only a few positive continuous linear functionals in the definition (for instance in by taking coordinate functionals).

     

On ordered spaces of polynomials

We study quasimonotone increasing linear mappings in finite-dimensional spaces of real polynomials, ordered by the cone of nonnegative polynomials on . In particular, we prove a representation of the space of quasimonotone increasing and decreasing operators, which turns out to have dimension 3 or 4.     

On the null-controllability of linear discrete-time systems with restrained controls

This paper is devoted to the null-controllability problem for linear, autonomous, discrete-time systems with restrained controls, i.e., systems described by the equationx _k+1=Ax _k+u _k,u _k , where 0 , but is not necessarily an interior point of . Some criteria for local and global null controllability are given with proofs. These results can be considered as extensions to discrete-time systems of the known corresponding criteria for linear, continuous-time systems.     

椭圆方程组径向非负解的存在性

应用锥不动点定理证明了圆环中半线性椭圆方程径向非负解的存在性.     

Perturbation Approach to Sensitivity Analysis in Mathematical Programming

This paper presents a perturbation approach for performing sensitivity analysis of mathematical programming problems. Contrary to standard methods, the active constraints are not assumed to remain active if the problem data are perturbed, nor the partial derivatives are assumed to exist. In other words, all the elements, variables, parameters, Karush–Kuhn–Tucker multipliers, and objective function values may vary provided that optimality is maintained and the general structure of a feasible perturbation (which is a polyhedral cone) is obtained. This allows determining: (a) the local sensitivities, (b) whether or not partial derivatives exist, and (c) if the directional derivative for a given direction exists. A method for the simultaneous obtention of the sensitivities of the objective function optimal value and the primal and dual variable values with respect to data is given. Three examples illustrate the concepts presented and the proposed methodology. Finally, some relevant conclusions are drawn. The authors are indebted to the Ministry of Science and Education of Spain, Projects CICYT DPI2002-04172-C04-02 and CICYT DPI2003-01362, and to the Fulbright Commission for partial support. The authors are grateful to the referees for comments improving the quality of the paper.     

Positive periodic solutions and eigenvalue intervals for systems of second order differential equations

In this paper, we employ a well‐known fixed point theorem for cones to study the existence of positive periodic solutions to the n ‐dimensional system x ″ + A (t)x = H (t)G (x). Moreover, the eigenvalue intervals for x ″ + A (t)x = λH (t)G (x) are easily characterized. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the equivalence of differential systems

In this article. First, we construct some nonlinear differential systems which are equivalent to some known systems. Second, we discuss, in a different method, the equivalence between some linear differential systems. And then we apply the obtained results to the study of the qualitative properties of these systems simultaneously.     

On Solidness of Polar Cones

We investigate the properties of cones whose polars are solid in different polar topologies. By a standard duality argument, we obtain a number of necessary and sufficient conditions for closed convex cones to be solid in various locally convex spaces. From this, we can deduce easily the extensions of previous related results. Furthermore, we construct a class of closed convex cones in some Banach spaces, which are not solid but whose polars satisfy the angle property. This solves the Han conjecture in the negative.     

Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity

This paper is concerned with the existence of traveling wave fronts for delayed non-local diffusion systems without quasimonotonicity, which can not be answered by the known results. By using exponential order, upper-lower solutions and Schauder's fixed point theorem, we reduce the existence of monotone traveling wave fronts to the existence of upper-lower solutions without the requirement of monotonicity. To illustrate our results, we establish the existence of traveling wave fronts for two examples which are the delayed non-local diffusion version of the Nicholson's blowflies equation and the Belousov-Zhabotinskii model. These results imply that the traveling wave fronts of the delayed non-local diffusion systems without quasimonotonicity are persistent if the delay is small.     

On the planar integrable differential systems

In this paper, we prove that the C ¹ planar differential systems that are integrable and non-Hamiltonian roughly speaking are C ¹ equivalent to the linear differential systems ${\dot u= u}$ , ${\dot v= v}$ . Additionally, we show that these systems have always a Lie symmetry. These results are improved for the class of polynomial differential systems defined in ${\mathbb{R}^2}$ or ${\mathbb{C}^2}$ .     

On strong pseudomonotonicity and (semi)strict quasimonotonicity

New concepts of strong pseudomonotonicity, strict quasimonotonicity, and semistrict quasimonotonicity of a map are introduced and their properties are studied. In the case of a differentiable gradient map, we show that strong pseudomonotonicity of the gradient is equivalent to strong pseudoconvexity of the underlying function. This does not hold for a different concept of strong pseudomonotonicity in Ref. 1. Analogous results are shown for strict quasimonotonicity and semistrict quasimonotonicity.     

On the boundedness of outer polyhedral estimates for reachable sets of linear systems

New properties of outer polyhedral (parallelepipedal) estimates for reachable sets of linear differential systems are studied. For systems with a stable matrix, it is determined what the orientation matrices are for which the estimates possessing the generalized semigroup property are bounded/unbounded on an infinite time interval. In particular, criteria are found (formulated in terms of the eigenvalues of the system’s matrix and the properties of bounding sets) that guarantee for previously mentioned tangent estimates and estimates with a constant orientation matrix that either there are initial orientation matrices for which the corresponding estimate tubes are bounded or all these tubes are unbounded. For linear stationary systems, a system of ordinary differential equations and algebraic relations is derived that determines estimates with constant orientation matrices for reachable sets that have no generalized semigroup property but are tangent and also bounded if the matrix of the system is stable.     

Generalized linear complementarity problems

It has been shown by Lemke that if a matrix is copositive plus on ⁿ , then feasibility of the corresponding linear complementarity problem implies solvability. In this article we show, under suitable conditions, that feasibility of ageneralized linear complementarity problem (i.e., defined over a more general closed convex cone in a real Hilbert space) implies solvability whenever the operator is copositive plus on that cone. We show that among all closed convex cones in a finite dimensional real Hilbert Space, polyhedral cones are theonly ones with the property that every copositive plus, feasible GLCP is solvable. We also prove a perturbation result for generalized linear complementarity problems.This research has been partially supported by the Air Force Office of Scientific Research under grants #AFOSR-82-0271 and #AFOSR-87-0350.     

On strong pseudomonotonicity and (semi)strict quasimonotonicity

An example in Ref. 1 is corrected to show that indeed a strongly pseudoconvex function, which is only once but not twice differentiable, does not necessarily have a strongly pseudoconvex gradient.     

On stochastically perturbed differential systems

We give a result which refers to the global existence and pathwise uniqueness of solutions of Itô stochastic differential systems. The sufficient conditions we propose improve some results of Taniguchi [5] and Constantin [1].     

On the oscillation of nonlinear two-dimensional differential systems

An oscillation criterion is given for a certain form of nonlinear two-dimensional differential systems. This criterion originated in a well-known oscillation result due to Coles (as extended and improved by Wong) concerning second order nonlinear differential equations with alternating coefficients.