带有单模糊映射的一般变分不等式解的存在性

众所周知，在各种变分不等式问题中，解的存在性是最基础也是最重要的问题之一．本文的目的是在Hilbert空间中研究带有单模糊映射的一般变分不等式解的存在性．此外，还讨论了几种特殊情况．     

模糊映射的完全广义混合型强变分不等式

研究关于模糊映射的一类新的变分不等式－模糊映射的完全广义混合型强变分不等式,得到此类变分不等式解的存在定理分解的一个逼近算法,推广了文[4]和文[8]的主要结果。     

A Fuzzy Extension of Generalized Implicit Vector Variational-like Inequalities

In this paper, we study the existence of solutions to the generalized implicit vector variational like inequality problems with fuzzy mappings in Hausdorff topological vector spaces using KKM-Fan Theorem.     

A class of differential fuzzy variational inequalities in finite-dimensional spaces

In this paper, a new class of differential fuzzy variational inequalities in finite dimensional spaces are introduced and studied. An existence theorem of the Carathéodory weak solution of differential fuzzy variational inequality is established under some suitable conditions. The convergence analysis for time-stepping method for finding the weak solution of the differential fuzzy variational inequality is given. Finally, an example is reported to verify the validity of the proposed algorithm.     

模糊映射的广义混合型强变分不等式

研究了新的一类模糊映射的广义混合型强变分不等式问题。证明了这类问题解的存在定理和收敛定理,给出解的带误差的Ishikawa型迭代算法。     

Iterative Algorithm for Solving Triple-Hierarchical Constrained Optimization Problem

Many practical problems such as signal processing and network resource allocation are formulated as the monotone variational inequality over the fixed point set of a nonexpansive mapping, and iterative algorithms to solve these problems have been proposed. This paper discusses a monotone variational inequality with variational inequality constraint over the fixed point set of a nonexpansive mapping, which is called the triple-hierarchical constrained optimization problem, and presents an iterative algorithm for solving it. Strong convergence of the algorithm to the unique solution of the problem is guaranteed under certain assumptions.     

A symmetry problem from probability

We examine solutions of two related boundary value problems for smooth domains in Euclidean space which arise from variational problems in probability. We show that the existence of solutions to each problem implies that the domain is a sphere.

     

模糊映象的完全广义变分包含

研究了一类新的关于模糊映射的完全广义变分包含问题 ,给出了解的逼近算法 ,证明了这类问题解的存在定理和序列收敛定理 ,推广了 N. J. Huang在文 [1 ]的主要结果 .     

A nonoverlapping domain decomposition method for variational inequalities derived from free boundary problems

The article proposes a nonoverlapping domain decomposition method for variational inequalities derived from free boundary problems. The free boundary value problem is broken up into two problems on nonoverlapping regions. In one region the problem is treated as a partial differential equation, while in the second region that contains the free boundary part, a variational inequality is considered. By solving these two related problems successively, we have shown that the successive solutions converge to the solution of the original problem. Application to a free surface seepage problem is given. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Free boundary problems with nonlinear source terms

Summary The method of lines is used to semi-discretize the non-linear Poisson equation over a domain with a free boundary. The resulting multipoint free boundary problem is solved with a line Gauss-Seidel method which is shown to converge monotonically. The method of lines solution is then shown to converge to the continuous solution of the variational inequality form of the obstacle problem. Some numerical results for the diffusion-reaction equation indicate that the method is applicable to more general free boundary problems for nonlinear elliptic equations.This research was supported by the U.S. Army Research Office under Contract DAAG-79-0145     

On a variational degenerate elliptic problem

In this paper we study a class of variational degenerate elliptic problems of the form in on , where is a bounded or unbounded domain in Received January 1999     

Hybrid boundary element formulation for acoustic wave propagation

The so‐called hybrid stress boundary element method (HSBEM) is introduced in a frequency domain formulation for the computation of acoustic radiation and scattering in closed and in finite domains. Different from other boundary element formulations, the HSBEM is based on an extended Hellinger‐Reissner variational principle and leads to a Hermitian, frequency‐dependent stiffness equation. Due to this, the method is very well suited for treating fluid structure interaction problems since the effort for the coupling the structure, discretized by a finite elements, and the fluid, discretized by the HSBEM is strongly reduced. To arrive at a boundary integral formulation, the field variables are separated into boundary variables, which are approximated by piecewise polynomial functions, and domain variables, which are approximated by a superposition of singular fundamental solutions weighed by source strength. This approximation cancels the domain integral over the equation of motion in the hybrid principle and leads to a boundary integral formulation, incorporating singular integrals. Comparing to previous results published by the authors, new considerations concerning the interpretation of singular contributions in the stiffness matrix for exterior domain problems are communicated here.     

Prototypical knowledge for interpreting fuzzy concepts and quantifiers

The ability to understand truly natural language expressions which involve fuzzy concepts and quantifiers (like many, few, most, etc.) presents many problems, some of which are worth mentioning: cardinality of a fuzzy set, extensions of the classical syllogisms to fuzzy syllogisms, dispositions, etc. Apart from these problems, which have been discussed in the literature, the main difficulty in evaluating such expressions is the strong interaction between the definition of the fuzzy concept and the domain knowledge.In this paper we will try to make such a claim apparent and describe some initial solutions, which provide an intelligent system with the capability of representing and understanding fuzzy concepts and quantifiers by taking into account domain knowledge.     

Weighted Variational Inequalities

In this paper, we introduce weighted variational inequalities over product of sets and system of weighted variational inequalities. It is noted that the weighted variational inequality problem over product of sets and the problem of system of weighted variational inequalities are equivalent. We give a relationship between system of weighted variational inequalities and systems of vector variational inequalities. We define several kinds of weighted monotonicities and establish several existence results for the solution of the above-mentioned problems under these weighted monotonicities. We introduce also the weighted generalized variational inequalities over product of sets, that is, weighted variational inequalities for multivalued maps and systems of weighted generalized variational inequalities. Extensions of weighted monotonicities for multivalued maps are also considered. The existence of a solution of weighted generalized variational inequalities over product of sets is also studied. The existence results for a solution of weighted generalized variational inequality problem give also the existence of solutions of systems of generalized vector variational inequalities. The first and third author express their thanks to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The authors are also grateful to the referees for comments and suggestions improving the final draft of this paper.     

Convergence Theorem for Variational Inequality in Hilbert Spaces with Applications

In this paper, we study variational inequality over the set of the common fixed points of a countable family of quasi-nonexpansive mappings. To tackle this problem, we propose an algorithm and use it to prove a strong convergence theorem under suitable conditions. As applications, we study variational inequality over the solution set of different nonlinear or linear problems, like minimization problems, split feasibility problems, convexly pseudoinverse problems, convex linear inverse problems, etc.     

Some homogenization problems for the system of elasticity with nonlinear boundary conditions in perforated domains

The system of linear elasticity is considered in a perforated domain with an ε-periodic structure. External forces nonlinearly depending on the displacements are applied to the surface of the cavities (or channels), while the body is fixed along the outer portion of its boundary. We investigate the asymptotic behavior of solutions to such boundary value problems asε→0 and construct the limit problem, according to the external surface forces and their dependence on the parameter ε. In some cases, this dependence results in the homogenized problem having the form of a variational inequality over a certain closed convex cone in a Sobolev space. This cone is described in terms of the functions involved in the nonlinear boundary conditions on the perforated boundary. A homogenization theorem is also proved for some unilateral problems with boundary conditions of Signorini type for the system of elasticity in a perforated domain. We discuss some cases when the homogenized tensor may depend on the functions specifying the boundary conditions.     

Global stress regularity of convex and some nonconvex variational problems

We derive a global regularity theorem for stress fields which correspond to minimizers of convex and some special nonconvex variational problems with mixed boundary conditions on admissible domains. These are Lipschitz domains satisfying additional geometric conditions near those points, where the type of the boundary conditions changes. In the first part it is assumed that the energy densities defining the variational problem are convex but not necessarily strictly convex and satisfy a convexity inequality. The regularity result for this case is derived with a difference quotient technique. In the second part the regularity results are carried over from the convex case to special nonconvex variational problems taking advantage of the relation between nonconvex variational problems and the corresponding (quasi-) convexified problems. The results are applied amongst others to the variational problems for linear elasticity, the p-Laplace operator, Hencky elasto-plasticity with linear hardening and for scalar and vectorial two-well potentials (compatible case).      

Signorini接触问题的边界变分不等式

本以Signorini接触问题为背景，讨论了变分不等式与边值问题的等价性，利用Green公式，基本解和基本解法向导数的性质，将区域型变分不等式化成等价的边界型变分不等式，并证明了边界变分不等式解的存在唯一性，为使用边界元方法数值求解提供理论依据。     

The use of possibility theory in the definition of fuzzy Pareto-optimality

Pareto-optimality conditions are crucial when dealing with classic multi-objective optimization problems. Extensions of these conditions to the fuzzy domain have been discussed and addressed in recent literature. This work presents a novel approach based on the definition of a fuzzily ordered set with a view to generating the necessary conditions for the Pareto-optimality of candidate solutions in the fuzzy domain. Making use of the conditions generated, one can characterize fuzzy efficient solutions by means of carefully chosen mono-objective problems and Karush-Kuhn-Tucker conditions to fuzzy non-dominated solutions. The uncertainties are inserted into the formulation of the studied fuzzy multi-objective optimization problem by means of fuzzy coefficients in the objective function. Some numerical examples are analytically solved to illustrate the efficiency of the proposed approach.     

Characterizations of an approximate minimum in optimal control

By using the classical variational methods based on geodesic coverings of a domain and on Hilbert's independent integral, further characterizations of an approximate solution in problems of control are described. The starting point is the Ekeland-type characterization, the variational principle. As consequences, sufficient conditions for optimality are obtained in a form similar to the Weierstrass conditions from the calculus of variations.The author is grateful to the referee for the valuable counterexamples to the first version of the paper.