On the Boundedness of Solutions of Degenerate Anisotropic Elliptic Variational Inequalities

In this article, for a class of degenerate anisotropic elliptic second-order variational inequalities we give conditions on the right-hand side and the set of constraints under which solutions of the variational inequalities are bounded. Our conditions on the set of constraints admit the consideration of a sufficiently large class of problems with pointwise constraints, and in particular, unilateral and bilateral problems. They also admit the consideration of the Dirichlet problem for the corresponding equations. We provide a series of examples which demonstrate the essentiality of the imposed conditions. In particular, we show that the condition assumed for the right-hand side of the variational inequalities in general is unimprovable in the scale of Lebesgue spaces.     

A Linearization Method for Nondegenerate Variational Conditions

We employ recent results about constraint nondegeneracy in variational conditions to design and justify a linearization algorithm for solving such problems. The algorithm solves a sequence of affine variational inequalities, but the variational condition itself need not be a variational inequality: that is, its underlying set need not be convex. However, that set must be given by systems of differentiable nonlinear equations with additional polyhedral constraints. We show that if the variational condition has a solution satisfying nondegeneracy and a standard regularity condition, and if the linearization algorithm is started sufficiently close to that solution, the algorithm will produce a well defined sequence that converges Q-superlinearly to the solution.     

On the Convergence of the Rayleigh-Ritz Method in Plane Subsonic Flow

The boundary value problem for plane compressible flow pastan aerofoil can be formulated as a variational problem. It isshown that for subsonic flow approximate solutions obtainedby the Rayleigh-Ritz method converge uniformly to the exactsolution provided that the aerofoil is sufficiently smooth andthe coordinate functions are properly chosen. The analysis appliesalso to certain other two-dimensional variational problems.     

Weak Sharp Solutions for Nonsmooth Variational Inequalities

In this paper, we study the weak sharp solutions for nonsmooth variational inequalities and give a characterization in terms of error bound. Some characterizations of solution set of nonsmooth variational inequalities are presented. Under certain conditions, we prove that the sequence generated by an algorithm for finding a solution of nonsmooth variational inequalities terminates after a finite number of iterates provided that the solutions set of a nonsmooth variational inequality is weakly sharp. We also study the finite termination property of the gradient projection method for solving nonsmooth variational inequalities under weak sharpness of the solution set.     

Some Methods Based on the D-Gap Function for Solving Monotone Variational Inequalities

The D-gap function has been useful in developing unconstrained descent methods for solving strongly monotone variational inequality problems. We show that the D-gap function has certain properties that are useful also for monotone variational inequality problems with bounded feasible set. Accordingly, we develop two unconstrained methods based on them that are similar in spirit to a feasible method of Zhu and Marcotte based on the regularized-gap function. We further discuss a third method based on applying the D-gap function to a regularized problem. Preliminary numerical experience is also reported.     

Some iterative schemes for general mixed variational inequalities

In this paper, we consider a class of variational inequalities which is called the general mixed variational inequality. It is known that the general mixed variational inequalities are equivalent to the fixed point problems. This equivalent formulation is used to suggest and analyze some three-step iterative schemes for finding the common element of the set of fixed points of a nonexpansive mappings and the set of solutions of the mixed variational inequalities. We also study the convergence criteria of three-step iterative method under some mild conditions. Our results include the previous results as special cases and may be considered as an improvement and refinement of the previously known results.     

Multiple solutions of a coupled nonlinear Schrödinger system

We will consider the relation between the number of positive standing waves solutions for a class of coupled nonlinear Schrödinger system in RN and the topology of the set of minimum points of potential V(x). The main characteristics of the system are that its functional is strongly indefinite at zero and there is a lack of compactness in RN. Combining the dual variational method with the Nehari technique and using the Concentration-Compactness Lemma, we obtain the existence of multiple solutions associated to the set of global minimum points of the potential V(x) for ? sufficiently small. In addition, our result gives a partial answer to a problem raised by Sirakov about existence of solutions of the perturbed system.     

Existence of infinite non-Birkoff periodic orbits for area-preserving monotone twist maps of cylinders

The exact monotone twist map of infinite cylinders in the Birkhoff region of instability is studied. A variational method based on Aubry-Mather theory is used to discover infinitely many non-Birkhoff periodic orbits of fixed rotation number sufficiently close to some irrational number for which the angular invariant circle does not exist.     

Steklov特征值问题的一种有效的Legendre-Galerkin谱逼近

本文给出Steklov特征值问题基于Legendre-Galerkin逼近的一种有效的谱方法.首先利用Legendre多项式构造了一组适当的基函数使得离散变分形式中的矩阵是稀疏的,然后推导了2维及3维情形下离散变分形式基于张量积的矩阵形式,由此可以快速地计算出离散的特征值和特征向量.文章还给出了误差分析和数值试验,数值结果表明本文提出的方法是稳定和有效的.     

On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space

Abstract

In this paper, new numerical algorithms are introduced for finding the solution of a variational inequality problem whose constraint set is the common elements of the set of fixed points of a demicontractive mapping and the set of solutions of an equilibrium problem for a monotone mapping in a real Hilbert space. The strong convergence of the iterates generated by these algorithms is obtained by combining a viscosity approximation method with an extragradient method. First, this is done when the basic iteration comes directly from the extragradient method, under a Lipschitz-type condition on the equilibrium function. Then, it is shown that this rather strong condition can be omitted when an Armijo-backtracking linesearch is incorporated into the extragradient iteration. The particular case of variational inequality problems is also examined.     

A Logarithmic-Quadratic Proximal Method for Variational Inequalities

We present a new method for solving variational inequalities on polyhedra. The method is proximal based, but uses a very special logarithmic-quadratic proximal term which replaces the usual quadratic, and leads to an interior proximal type algorithm. We allow for computing the iterates approximately and prove that the resulting method is globally convergent under the sole assumption that the optimal set of the variational inequality is nonempty.     

Estimates for normal solutions in problems with irregular zonal controls for the wave equation

For the wave equation with variable coefficients and boundary conditions of the first kind, we consider mutually dual problems with irregular zonal controls and regular zonal observations. Constructive estimates of well-posed solvability are obtained for the observation problem with strong generalized solutions on sufficiently large time intervals. These estimates contain information necessary for the construction of stable approximations to solutions of both problems with the use of the earlier suggested variational method.     

Structure of extremals of autonomous convex variational problems

In this paper we study the structure of approximate solutions of autonomous variational problems with convex integrands. We are interested in a turnpike property of the extremals which is independent of the length of the interval, for all sufficiently large intervals. To have this property means, roughly speaking, that the approximate solutions of the variational problems are determined mainly by the integrand, and are essentially independent of the choice of interval and endpoint conditions.     

On characterizing the set of possible effective tensors of composites: The variational method and the translation method

A general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarization-problem simplifies the derivation of the Hashin-Shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials. A general connection is established between two methods for bounding effective tensors of composites. The first method is based on the variational principles of Hashin and Shtrikman. The second method, due to Tartar, Murat, Lurie, and Cherkaev, uses translation operators or, equivalently, quadratic quasiconvex functions. A correspondence is established between these translation operators and bounding operators on the relevant non-local projection operator, T₁. An important class of bounds, namely trace bounds on the effective tensors of two-component media, are given a geometrical interpretation: after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors. A wide class of translation operators that generate these bounds is found. The extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors. These extremal translations generate attainable trace bounds even when the tensors of the two-components are not well ordered. In particular, they generate the bounds of Walpole on the effective bulk modulus. The variational principles of Gibiansky and Cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by Golden and Papanicolaou. An isomorphism is shown to underlie their variational principles. This isomorphism is used to obtain Dirichlet-type variational principles and bounds for the effective tensors of general non-selfadjoint problems. It is anticipated that these variational principles, which stem from the work of Gibiansky and Cherkaev, will have applications in many fields of science.     

Variationally derived scaling and variable metric updates from the preconvex part of the Broyden family

The paper is devoted to a general variational approach for scaling the variable metric updates. A new method from the preconvex part of the Broyden family is given which minimizes not only the generalized Frobenius norm but also the condition number of the update. The computational efficiency of this method, with both preliminary and controlled scaling strategy, is demonstrated on sufficiently difficult test problems.     

Cantor families of periodic solutions for wave equations via a variational principle

We prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a Cantor set of asymptotically full measure, via a variational principle. A Lyapunov-Schmidt decomposition reduces the problem to a finite dimensional bifurcation equation—variational in nature—defined on a Cantor set of non-resonant parameters. The Cantor gaps are due to “small divisors” phenomena. To solve the bifurcation equation we develop a suitable variational method. In particular, we do not require the typical “Arnold non-degeneracy condition” of the known theory on the nonlinear terms. As a consequence our existence results hold for new generic sets of nonlinearities.     

A Modified Alternating Direction Method for Variational Inequality Problems

The alternating direction method is an attractive method for solving large-scale variational inequality problems whenever the subproblems can be solved efficiently. However, the subproblems are still variational inequality problems, which are as structurally difficult to solve as the original one. To overcome this disadvantage, in this paper we propose a new alternating direction method for solving a class of nonlinear monotone variational inequality problems. In each iteration the method just makes an orthogonal projection to a simple set and some function evaluations. We report some preliminary computational results to illustrate the efficiency of the method. Accepted 4 May 2001. Online publication 19 October, 2001.     

Weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space

We know that variational inequality problem is very important in the nonlinear analysis. For a variational inequality problem defined over a nonempty fixed point set of a nonexpansive mapping in Hilbert space, the strong convergence theorem has been proposed by I. Yamada. The algorithm in this theorem is named the hybrid steepest descent method. Based on this method, we propose a new weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space. Using this result, we obtain some new weak convergence theorems which are useful in nonlinear analysis and optimization problem.     

Homogenization of Optimal Control Problems for Functional Differential Equations

The paper deals with the variational convergence of a sequence of optimal control problems for functional differential state equations with deviating argument. Variational limit problems are found under various conditions of convergence of the input data. It is shown that, upon sufficiently weak assumptions on convergence of the argument deviations, the limit problem can assume a form different from that of the whole sequence. In particular, it can be either an optimal control problem for an integro-differential equation or a purely variational problem. Conditions are found under which the limit problem preserves the form of the original sequence.