An infinite-dimensional linking theorem and applications

In this paper, we establish a new infinite-dimensional linking theorem without (PS)-type assumptions. The new theorem needs a weaker linking geometry and produces bounded (PS) sequences. The abstract result will be applied to the study of the existence of solutions of the strongly indefinite partial differential systems. For the first application, we consider the system

     

A lower closure theorem for abstract control problems withL p-bounded controls

A lower closure theorem for an abstract control problem is proved. The functional isJ(,u)= _G f ⁰(t, (M)(t),u(t))dt and the state equations areN(t)=f(t, (M)(t),u(t)). It is shown that, if {( _k ,u _k)} is a sequence of admissible controlsu _k and corre-sponding trajectories _k such that lim infJ( _k ,u _k)<+ and such that _k weakly,M _k M strongly,N _k N weakly, and {u _k} is bounded in someL _p norm, then there is a controlu such that (,u) is admissible and lim infJ( _k ,u _k)J(,u).Dedicated to Professor M. R. HestenesThis research was supported by the National Science Foundation, Grant No. GP-33551X.     

An averaging theorem for two-point boundary value problems with applications to optimal control

An asymptotic result is obtained for a two-point boundary value problem for a vector system of nonlinear ordinary differential equations involving “fast” and “slow” inputs. The asymptotically limiting system is obtained by an averaging procedure. Using this result, an approximate analysis of the original system may be carried out by considering two lower-order systems each involving only one time scale. It is shown that some optimal control problems for systems with multiple time scales may be analyzed by this method.     

An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications

A generalization of the classical Leray-Schauder fixed-point theorem based on the infinite-dimensional Borsuk-Ulam-type antipode construction is proposed. A new nonstandard proof of the classical Leray-Schauder fixed-point theorem and a study of the solution manifold of a nonlinear Hamilton-Jacobi-type equation are presented. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 100–106, January, 2008.     

An extension of gap functions for a system of vector equilibrium problems with applications to optimization problems

In this paper, the notion of gap functions is extended from scalar case to vector one. Then, gap functions and generalized functions for several kinds of vector equilibrium problems are shown. As an application, the dual problem of a class of optimization problems with a system of vector equilibrium constraints (in short, OP) is established, the concavity of the dual function, the weak duality of (OP) and the saddle point sufficient condition are derived by using generalized gap functions. This work was supported by the National Natural Science Foundation of China (10671135) and the Applied Research Project of Sichuan Province (05JY029-009-1).     

An infinite-dimensional generalization of the Jung theorem

A complete characterization of the extremal subsets of Hilbert spaces, which is an infinite-dimensional generalization of the classical Jung theorem, is given. The behavior of the set of points near the Chebyshev sphere of such a subset with respect to the Kuratowski and Hausdorff measures of noncompactness is investigated.     

An infinite-dimensional version of the Borsuk-Ulam theorem

We study the solvability of the equation a(x) = f(x) on a sphere in a Banach space, where a is a closed surjective linear operator and f is an odd a-compact map. We also estimate the topological dimension of the solution set and give applications of the corresponding theorem to some problems in differential equations and other fields of mathematics.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 1–5, 2004Original Russian Text Copyright © by B. D. GelmanSupported by the Russian Foundation for Basic Research, Grant No. 02-01-00189.     

Ergodic theorem for generalized long-range exclusion processes with positive recurrent transition probabilities

This paper is devoted to the ergodicity of generalized long-range exclusion processes with positive recurrent transition probabilities. The set of invariant probability measures and the corresponding domain of attraction for each invariant probability measure are described. This paper is partially supported the National Natural Science Foundation of China.     

An extension theorem to rough paths

We show that any continuous path of finite p-variation can be lifted to a geometric q  -rough path, where q>p$q>p$.     

Some applications of Ball's extension theorem

We present two applications of Ball's extension theorem. First we observe that Ball's extension theorem, together with the recent solution of Ball's Markov type problem due to Naor, Peres, Schramm and Sheffield, imply a generalization, and an alternative proof of, the Johnson-Lindenstrauss extension theorem. Second, we prove that the distortion required to embed the integer lattice , equipped with the metric, in any -uniformly convex Banach space is of order .

     

An extension of Burnside’s Theorem to infinite-dimensional spaces

The classical Burnside’s Theorem guarantees in a finite dimensional space the existence of invariant subspaces for a proper subalgebra of the matrix algebra. In this paper we give an extension of Burnside’s Theorem for a general Banach space, which also gives new results on invariant subspaces. Partially supported by a grant from the National Science Foundation.     

An extension to a theorem of Himmelberg

This paper extends a widely used theorem of Himmelberg to topological vector spaces whose completion have a separating dual.     

A finite dimensional extension of Lyusternik theorem with applications to multiobjective optimization

We consider a multiobjective program with inequality and equality constraints and a set constraint. The equality constraints are Fréchet differentiable and the objective function and the inequality constraints are locally Lipschitz. Within this context, a Lyusternik type theorem is extended, establishing afterwards both Fritz–John and Kuhn–Tucker necessary conditions for Pareto optimality.     

Abstract convergence theorem for quasi-convex optimization problems with applications

Abstract

Quasi-convex optimization is fundamental to the modelling of many practical problems in various fields such as economics, finance and industrial organization. Subgradient methods are practical iterative algorithms for solving large-scale quasi-convex optimization problems. In the present paper, focusing on quasi-convex optimization, we develop an abstract convergence theorem for a class of sequences, which satisfy a general basic inequality, under some suitable assumptions on parameters. The convergence properties in both function values and distances of iterates from the optimal solution set are discussed. The abstract convergence theorem covers relevant results of many types of subgradient methods studied in the literature, for either convex or quasi-convex optimization. Furthermore, we propose a new subgradient method, in which a perturbation of the successive direction is employed at each iteration. As an application of the abstract convergence theorem, we obtain the convergence results of the proposed subgradient method under the assumption of the Hölder condition of order p and by using the constant, diminishing or dynamic stepsize rules, respectively. A preliminary numerical study shows that the proposed method outperforms the standard, stochastic and primal-dual subgradient methods in solving the Cobb–Douglas production efficiency problem.     

An extension of Tarski's fixed point theorem and its application to isotone complementarity problems

Tarski's fixed point theorem is extended to the case of set-valued mappings, and is applied to a class of complementarity problems defined by isotone set-valued operators in a complete vector lattice.     

An extension of Lucas' theorem

Let be a prime. A famous theorem of Lucas states that if are nonnegative integers with . In this paper we aim to prove a similar result for generalized binomial coefficients defined in terms of second order recurrent sequences with initial values and .