A Unifying Approach to Variational Relation Problems

The purpose of this paper is to present a unified approach to study the existence of solutions for two types of variational relation problems, which encompass several generalized equilibrium problems, variational inequalities and variational inclusions investigated in the recent literature. By using two well-known fixed point theorems, we establish several existence criteria for the solutions of these problems.     

An Optimum Solution for One-Dimensional Slitting Problems: A Dynamic Programming Approach

One-dimensional slitting problems are often encountered in textile, paper or sheet-metal industries where a roll of this product needs to be divided into various narrower-width rolls. The rolls are slit into pieces such that the total value of the sale is maximized depending on the width and the number of defects in the sheet. As the width of a piece increases, the value of the piece increases also, but on the other hand, the number of defects in the piece increases as well, causing reduction in its value. Based on these constraints, a dynamic programming formulation is framed so as to maximize the total value of the sale. Computational complexity with a numerical example is discussed to demonstrate how the procedure works.     

A Variational Approach to Nonlinear Singularly Perturbed Boundary-Value Problems

Carrier and Pearson introduced a nonlinear singularly perturbed boundary value problem that has served as a paradigm for problems where the method of matched asymptotic expansions (MAE) apparently fails. The “failure” of MAE is its inability to select the location of possible internal layers, though their structure is determined. Thus, a straightforward application of MAE leaves the positions of any internal layers arbitrary, though the asymptotic expansion of the exact solution to the problem exhibits internal layers only at specific locations. For this reason the solutions produced by MAE have been referred to as spurious solutions. We resolve the question of finding the positions of the interior layers by employing the variational approach of Grasman and Matkowsky. In addition, we show that this method tells how solutions bifurcate as the boundary values are varied, and give an alternative motivation for the variational approach via Newton”s method.     

Variational Problems with Symmetries: A Pfaffian System Approach

Reduction methods for completely integrable Pfaffian Systems with symmetry are applied to variational problems, providing analogues of the Arnold?CLiouville Theorem and Marsden?CWeinstein reduction in the Lagrangian setting. A?generalization of the Mishchenko?CFomenko Theorem for non Abelian integrability is given in terms of solvable structures.     

A Variational Approach Via Bipotentials for a Class of Frictional Contact Problems

In the reference (Cui and Yin, Pacific J. Math. 233:257–289, 2007), under the assumptions that the supersonic incoming flow is isothermal and symmetrically perturbed with respect to a uniform supersonic constant state, the authors have shown the global existence and stability of a symmetric supersonic conic shock for such a supersonic flow past a circular cone. In this paper, we will remove all the symmetric assumptions in the previous paper and study the global existence problem on a really multidimensional shock wave. More concretely, we establish the global existence and stability of a three-dimensional supersonic conic shock wave for a perturbed steady supersonic isothermal flow past an infinitely long conic body.     

A Variational Approach to Discontinuous Problems with Critical Sobolev Exponents

We employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form − Δu = λh(x)H(u − a)u^q + u^2* − 1 in R^N, where λ, a > 0 are parameters, h(x) is both nonnegative and integrable on R^N, H is the Heaviside function, 2* is the critical Sobolev exponent, and 0 ≤ q < 2* − 1. We obtain existence, multiplicity and regularity of solutions by distinguishing the cases 0 ≤ q ≤ 1 and 1 < q < 2* − 1.     

A Dual Variational Approach to a Class of Nonlocal Semilinear Tricomi Problems

The existence of at least one nontrivial solution to a class of semilinear Tricomi problems is established via an application of the dual variational method which captures the solution as the preimage of a minimum of a suitable dual action functional. The boundary conditions are homogeneous Dirichlet conditions on a suitable part of the boundary, as dictated by uniqueness theorems for the linear problem. While there are good compactness properties for the inverse operator for the linear problem, there is a manifest asymmetry in the linear part due to the form of the boundary conditions. The linear part is symmetrized by introducing suitable re ection operators on symmetric domains, which then results in a nonlocal character of the nonlinearity. Received July 1997; Revised February 1998     

分层变分包含问题中经由分层不动点途径的黏性方法

基于一原创技术—指标选取法,以黏性法分层逼近可数族非线性映射之公共不动点.并于适当条件下得一强收敛定理,用以解决Hilbert空间背景下之分层变分包含问题.     

New Numerical Approach for Fractional Variational Problems Using Shifted Legendre Orthonormal Polynomials

This paper reports a new numerical approach for numerically solving types of fractional variational problems. In our approach, we use the fractional integrals operational matrix, described in the sense of Riemann–Liouville, with the help of the Lagrange multiplier technique for converting the fractional variational problem into an easier problem that consisting of solving an algebraic equations system in the unknown coefficients. Several numerical examples are introduced, combined with their approximate solutions and comparisons with other numerical approaches, for confirming the accuracy and applicability of the proposed approach.     

A Variational Method for Inclusion and Indentation Problems

This paper considers a variational principle for Fredholm integralequations in which the range of integration is fixed but unknown.Equations of this type occur in the study of inclusion and indentationproblems in the theory of elasticity. It is shown that a functionalassociated with the integral equation is stationary with respectto variations of the range of integration around its exact value. The method is applied to indentation problems and to cylindricaland spherical inclusion problems. Where a comparison is possible,the present results agree closely with those obtained by othermethods.     

A Scalarization Approach for Vector Variational Inequalities with Applications

We consider an approach to convert vector variational inequalities into an equivalent scalar variational inequality problem with a set-valued cost mapping. Being based on this property, we give an equivalence result between weak and strong solutions of set-valued vector variational inequalities and suggest a new gap function for vector variational inequalities. Additional examples of applications in vector optimization, vector network equilibrium and vector migration equilibrium problems are also given Mathematics Subject Classification(2000). 49J40, 65K10, 90C29     

计算Pascal矩阵加数量矩阵之逆的一种新方法(英文)

In this paper,using the Jordan canonical form of the Pascal matrix Pn,we present a new approach for inverting the Pascal matrix plus a scalar Pn＋aIn for arbitrary real number a≠1.     

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.     

A Variational Approach to Perturbation Feedback Control for Optimal Control Problems with Terminal Constraints and Free Terminal Time

Set-Valued and Variational Analysis - Using the method of characteristics, for an optimal control problem with terminal constraints and free terminal time, we construct a family of extremals along...     

A Note on Branch-and-Price Algorithms for the One-Dimensional Cutting Stock Problems

One algorithm proposed in Branch-and-price algorithms for the one-dimensional cutting stock problems (COAP, vol. 9, pp. 211–228, 1998) may not find an optimal solution to the cutting stock problem, as it is stated.     

Duality for Variational Problems with Pseudo-Invexity

In this paper the duality results of Mond, Chandra and Husain for variational problems are extended to generalized invex (pseudo-invex) functions