Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities

The variational inequality problem with set-valued mappings is very useful in economics and nonsmooth optimization. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational inequalities (VVI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVI. It is shown that the optimization problem formulated by using gap functions can be transformed into a semi-infinite programming problem. We investigate also the existence of a solution for the generalized VVI with a set-valued mapping by virtue of the existence of a solution of the VVI with a single-valued function and a continuous selection theorem.     

An elementary sufficiency proof of an absolute minimum for a nonparametric variational problem

For a fixed endpoint, nonparametric simple integral variational problem, there is presented an expansion method proof of a sufficiency theorem for an absolute minimum. In particular, this sufficiency theorem yields readily the proof of a result of the type recently presented by Nehari (Ref. 1), but with an error in formulation and an incorrect proof. The present discussion is in a setting which permits considered arcs to be on the boundary of the set of admissible arcs; thus it contains as particular instances certain types of unilateral variational problems of a control nature.This research was supported by the National Science Foundation under Grant No. GP-36120.     

Robust Decentralized Stabilization of Uncertain Large-Scale Discrete-Time Systems with Delays

This paper describes the synthesis of robust decentralized controllers for uncertain large-scale discrete-time systems with time delays in the subsystem interconnections. Based on the Lyapunov method, a sufficient condition for robust stability is derived in terms of a linear matrix inequality (LMI). The solutions of the LMIs can be obtained easily using efficient convex optimization techniques. A numerical example is given to illustrate the proposed method.     

Self Improving Sobolev-Poincaré Inequalities,Truncation and Symmetrization

In Martín et al. (J Funct Anal 252:677–695, 2007) we developed a new method to obtain symmetrization inequalities of Sobolev type for functions in . In this paper we extend our method to Sobolev functions that do not vanish at the boundary. This paper is in final form and no version of it will be submitted for publication elsewhere.     

一个新的分数阶比较定理

研究了Caputo和Riemann-Liouville两型分数阶微分方程的比较定理.首先,讨论了一类线性分数阶微分不等式解得非负性.其次,引入单边Lipschitz条件,将微分方程解的比较问题化为线性微分不等式非负解问题,通过线性分数阶微分方程的求解,得到分数阶比较定理.最后,为进一步说明结论,给出了两个数值仿真例子.     

Weighted Lp‐theory for vector potential operators in three‐dimensional exterior domains

In the present paper, we study the vector potential problem in exterior domains of . Our approach is based on the use of weighted spaces in order to describe the behavior of functions at infinity. As a first step of the investigation, we prove important results on the Laplace equation in exterior domains with Dirichlet or Neumann boundary conditions. As a consequence of the obtained results on the vector potential problem, we establish useful results on weighted Sobolev inequalities and Helmholtz decompositions of weighted spaces. Copyright © 2015 John Wiley & Sons, Ltd.     

On Korn's first inequality for tangential or normal boundary conditions with explicit constants

We will prove that for piecewise C²‐concave domains in Korn's first inequality holds for vector fields satisfying homogeneous normal or tangential boundary conditions with explicit Korn constant . Copyright © 2016 John Wiley & Sons, Ltd.     

Solvability and optimal controls for semilinear fractional evolution hemivariational inequalities

This paper is concerned with the control systems of semilinear fractional evolution hemivariational inequalities and their optimal controls in Banach space. Firstly, the existence of mild solutions is obtained and proved mainly by using a well‐known fixed point theorem of multivalued maps and the properties of generalized Clarke subdifferential. Then, by applying generally mild conditions of cost functionals, we investigate the existence results of the optimal controls for fractional differential evolution hemivariational inequalities. Finally, an example is given to demonstrate the applicability of the main results. Copyright © 2016 John Wiley & Sons, Ltd.