On Implicit Multifunction Theorems

We give implicit multifunction results generalizing to multifunctions the Robinson’s implicit function theorem (Robinson, Math Oper Res 16(2):292–309, 1991). To this end, we use parametric error bounds estimates for a suitable function refining the one given in Azé and Corvellec (ESAIM Control Optim Calc Var 10:409–425, 2004). Sharp approximations of the implicit multifunctions are given extending the results of Nachi and Penot (Control Cybernet 35:871–901, 2005). Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

Implicit multifunction theorems with positively homogeneous maps

We extend Robinson’s and Ledyaev and Zhu’s implicit multifunction theorems using the language of generalized derivatives with positively homogeneous maps, allowing us to obtain results that more closely resemble the classical (single-valued) implicit function theorem. We highlight that using linear openness instead of metric regularity gives simpler proofs and stronger results. As part of our analysis, we study perturbations of generalized linear openness and metric regularity. Finally, we discuss how our methods may also be adapted to study generalized calmness.     

广义多值拟变分包含解的两步迭代

本文引入并研究了Hilbert空间中的一类广义多值拟变分包含问题.借助预解算子技巧构造了一个新的两步迭代算法来逼近广义多值拟变分包含的解,并且证明了其解的存在性以及迭代算法生成的迭代序列的收敛性.     

Optimal Control of Differential–Algebraic Equations of Higher Index, Part 1: First-Order Approximations

This paper deals with optimal control problems described by higher index DAEs. We introduce a class of these problems which can be transformed to index one control problems. For this class of higher index DAEs, we derive first-order approximations and adjoint equations for the functionals defining the problem. These adjoint equations are then used to state, in the accompanying paper, the necessary optimality conditions in the form of a weak maximum principle. The constructive way used to prove these optimality conditions leads to globally convergent algorithms for control problems with state constraints and defined by higher index DAEs.     

隐式Euler法关于Volterra延迟积分方程的数值稳定性

本文涉及隐式Euler法应用于非线性Volterra型延迟积分方程的稳定性,其探讨了基于非经典Lipschitz条件,其方法的整体与渐近稳定性结果被获得。     

Complementarity Problems and Variational Inequalities. A Unified Approach of Solvability by an Implicit Leray-Schauder Type Alternative

In several recent papers we obtained existence theorems for complementarity problems and variational inequalities using for each of them a particular notion of exceptional family of elements. Now, in this paper we introduce a new notion of exceptional family of elements. This notion is based on an Implicit Leray-Schauder Alternative. By this new notion we obtain a unification of the study of solvability of complementarity problems and of variational inequalities. The paper is finished with a section dedicated to variational inequalities with δ-pseudomonotone operators.     

Nonlinear models of suspension bridges

Nonlinear variational equations describing one type of suspension bridges are proposed and studied. The variational equations describe the behaviour of road bed, main cables and cable stays. The road bed is described by two functions connected with vertical and horizontal deformation of any cross section. The main cable is considered to be perfectly flexible and inextensible. The cable stays only resist tensile forces. The variational equations are derived from the principle of minimum potential energy. The existence of solution is based on the Brouwer Fixed Point Theorem. The local uniqueness and continuous dependence on the data represented by gravitational forces acting on the road bed are studied. The local results are based on the Implicit Function Theorem for Banach spaces. A certain stability criterion for suspension bridges is formulated and this criterion indicates how to influence the stability of suspension bridges.     

Fixed-Point Technique for Implicit Complementarity Problem in Hilbert Lattice

In this paper, we consider and study the implicit complementarity problem in the setting of a Hilbert lattice. It has been shown that this problem can be formulated as a fixed-point problem by using a suitable change of variables. Moreover, this formulation allows us to prove the existence and uniqueness of solutions of the implicit complementarity problem.     

A fast semi-implicit method for anisotropic diffusion

Simple finite differencing of the anisotropic diffusion equation, where diffusion is only along a given direction, does not ensure that the numerically calculated heat fluxes are in the correct direction. This can lead to negative temperatures for the anisotropic thermal diffusion equation. In a previous paper we proposed a monotonicity-preserving explicit method which uses limiters (analogous to those used in the solution of hyperbolic equations) to interpolate the temperature gradients at cell faces. However, being explicit, this method was limited by a restrictive Courant–Friedrichs–Lewy (CFL) stability timestep. Here we propose a fast, conservative, directionally-split, semi-implicit method which is second order accurate in space, is stable for large timesteps, and is easy to implement in parallel. Although not strictly monotonicity-preserving, our method gives only small amplitude temperature oscillations at large temperature gradients, and the oscillations are damped in time. With numerical experiments we show that our semi-implicit method can achieve large speed-ups compared to the explicit method, without seriously violating the monotonicity constraint. This method can also be applied to isotropic diffusion, both on regular and distorted meshes.     

A new simple method of implicit time integration for dynamic problems of engineering structures

This paper presents a new simple method of implicit time integration with two control parameters for solving initial-value problems of dynamics such that its accuracy is at least of order two along with the conditional and unconditional stability regions of the parameters. When the control parameters in the method are optimally taken in their regions, the accuracy may be improved to reach of order three. It is found that the new scheme can achieve lower numerical amplitude dissipation and period dispersion than some of the existing methods, e.g. the Newmark method and Zhai’s approach, when the same time step size is used. The region of time step dependent on the parameters in the new scheme is explicitly obtained. Finally, some examples of dynamic problems are given to show the accuracy and efficiency of the proposed scheme applied in dynamic systems. The project supported by the National Key Basic Research and Development Foundation of the Ministry of Science and Technology of China (G2000048702, 2003CB716707), the National Science Fund for Distinguished Young Scholars (10025208), the National Natural Science Foundation of China (Key Program) (10532040), the Research Fund for Oversea Chinese (10228028). The English text was polished by Keren Wang.