Fixed Points on Two Complete and Compact Metric Spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｆｏｌｌｏｗｉｎｇｒｅｌａｔｅｄｆｉｘｅｄｐｏｉｎｔｔｈｅｏｒｅｍｗａｓｐｒｏｖｅｄｉｎ [1 ] .Ｔｈｅｏｒｅｍ 1　Ｌｅｔ(Ｘ ,ｄ)ａｎｄ (Ｙ ,ρ)ｂｅｃｏｍｐｌｅｔｅｍｅｔｒｉｃｓｐａｃｅｓ,ｌｅｔＴｂｅａｍａｐｐｉｎｇｏｆＸｉｎｔｏＹａｎｄｌｅｔＳｂｅａｍａｐｐｉｎｇｏｆＹｉｎｔｏＸｓａｔｉｓｆｙｉｎｇｔｈｅｉｎｅｑｕａｌｉｔｉｅｓρ(Ｔｘ,ＴＳｙ) ≤ｃｍａｘ ｄ(ｘ ,Ｓｙ) ,ρ(ｙ,Ｔｘ) ,ρ(ｙ,ＴＳｙ) ,ｄ(Ｓｙ ,ＳＴｘ) ≤ｃｍａｘ ρ(ｙ ,Ｔｘ) ,ｄ(ｘ,Ｓｙ) ,ｄ(ｘ ,ＳＴｘ)…     

Fixed Points of Multi-valued Maps and Static Coulomb Friction Problems

The existence of solutions to the Signorini problem with Coulomb friction is a long standing open question. We prove the existence of generalized solutions that satisfy the pointwise Coulomb friction conditions on the entire interface and the normal nonpenetration condition on the complement of a subset with arbitrarily small but possibly positive measure. Furthermore, the penetration itself can also be made arbitrarily small. Although “measure zero” instead of “arbitrarily small measure” would be needed to fully resolve the issue, these generalized solutions seem to be the closest answer available to date. Their existence is proved by a suitable application of Ky Fan's fixed point theorem for multi-valued maps. The same method can be used with a number of variants involving contact of two or more elastic bodies and possible debonding phenomena. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Dynamical System Associated with the Fixed Points Set of a Nonexpansive Operator

We study the existence and uniqueness of (locally) absolutely continuous trajectories of a dynamical system governed by a nonexpansive operator. The weak convergence of the orbits to a fixed point of the operator is investigated by relying on Lyapunov analysis. We show also an order of convergence of \(o\left( \frac{1}{\sqrt{t}}\right) \) for the fixed point residual of the trajectory of the dynamical system. We apply the results to dynamical systems associated with the problem of finding the zeros of the sum of a maximally monotone operator and a cocoercive one. Several dynamical systems from the literature turn out to be particular instances of this general approach.     

ON THE ITERATIVE APPROXIMATION PROBLEM OF FIXED POINTS FOR ASYMPTOTICALLY NONEXPANSIVE TYPE MAPPINGS IN BANACH SPACES

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｒｏｕｇｈｏｕｔｔｈｉｓｐａｐｅｒｗｅａｓｓｕｍｅｔｈａｔＥｉｓａｒｅａｌＢａｎａｃｈｓｐａｃｅ ,Ｅ ｉｓｔｈｅｄｕａｌｓｐａｃｅｏｆＥ ,ＤｉｓａｎｏｎｅｍｐｔｙｓｕｂｓｅｔｏｆＥａｎｄＪ:Ｅ → 2 Ｅ ｉｓｔｈｅｎｏｒｍａｌｉｚｅｄｄｕａｌｉｔｙｍａｐｐｉｎｇｄｅｆｉｎｅｄｂｙＪ(ｘ) =ｆ∈Ｅ :〈ｘ ,ｆ〉 =‖ｘ‖·‖ｆ‖ ,　‖ｆ‖ =‖ｘ‖ (ｘ∈Ｅ) .　　Ｄｄｆｉｎｉｔｉｏｎ 1 1　ＬｅｔＴ :Ｄ →Ｄｂｅａｍａｐｐｉｎｇ1 )Ｔｉｓｓａｉｄｔｏｂｅａｓｙｍｐｔｏｔｉｃａｌｌｙｎｏ…     

三维摩擦接触问题不动点算法

三维摩擦接触问题具有多重非线性性质,使求解变得比较困难,为了解决此问题,建立了三维摩擦接触问题的模型,介绍了一种非光滑混合不动点算法.该算法克服了在接触面上由于可能的滑动状态有无穷多个而难以确定的难点,算法未引入人工变量,计算量较小,计算结果精确满足接触状态条件,收敛性得到保证.根据此算法编制程序,将叠合悬臂梁算例数值计算结果与商用有限元软件进行比较,也表明了不动点算法的有效性.     

Theory for Atomic Diffusion on Fixed and Deformable Crystal Lattices

We develop a theoretical framework, for the diffusion of a single unconstrained species of atoms on a crystal lattice, that provides a generalization of the classical theories of atomic diffusion and diffusion-induced phase separation to account for constitutive nonlinearities, external forces, and the deformation of the lattice. In this framework, we regard atomic diffusion as a microscopic process described by two independent kinematic variables: (i) the atomic flux, which reckons the local motion of atoms relative to the motion of the underlying lattice, and (ii) the time-rate of the atomic density, which encompasses nonlocal interactions between migrating atoms and characterizes the kinematics of phase separation. We introduce generalized forces power-conjugate to each of these rates and require that these forces satisfy ancillary microbalances distinct from the conventional balance involving the forces that expend power over the rate at which the lattice deforms. A mechanical version of the second law, which takes the form of an energy imbalance accounting for all power expenditures (including those due to atomic diffusion and phase separation), is used to derive restrictions on constitutive equations. With these restrictions, the microbalance involving the forces conjugate to the atomic flux provides a generalization of the usual constitutive relation between the atomic flux and the gradient of the diffusion potential, a relation that in conjunction with the atomic balance yields a generalized Cahn–Hilliard equation.     

Fixed point theorems of local contraction mappings on menger spaces

In this paper,we introduce the concept ofε-chainable PM-space,and give severalfixed point theorems of one-valued and multivalued local contraction mapping on the kindof spaces.     

用Dirac符号表述的非自由质点系动力学方程

研究Dirac符号在经典力学中的应用,给出一种用Dirac符号表述的非自由质点系统动力学方程,并举例说明该方程的广泛应用.     

非连续边界元积分的精确表达式及相关问题

以二维位势问题边界元分析为例,给出了利用线性非连续边界元离散边界积分方程时系数矩阵积分计算的精确表达式,通过和利用Gauss积分方法计算系数矩阵所得数值结果的比较表明：配位点选择不同对数值计算结果精度影响的主要原因是积分计算的精度,尤其当配位因子选择较大时,存在的准奇异积分（Nearly Singular Integrals)很难利用常规Gauss积分方法准确求得。     

Stability of Fixed Points and Associated Relative Equilibria of the 3-Body Problem on $${\mathbb {S}}^1$$ and $${\mathbb {S}}^2$$S2

We prove that the fixed points of the curved 3-body problem and their associated relative equilibria are Lyapunov stable if the solutions are restricted to \({\mathbb {S}}^1\), but unstable if the bodies are considered in \({\mathbb {S}}^2\).     

Mathematical and Physical Constraints on Large-Eddy Simulations

We examine the mathematical constraints on the filtering operation in large-eddy simulation (LES), and the physical constraints on the formulation of subgrid scale (SGS) models. The mathematical constraints on filtering arise from the assumption of commutativity between the filtering operation and the differential operators: this breaks down for finite domains and for non-uniform filter kernels. The physical constraints on the modelling are twofold; invariance principles require any model to be invariant under a group of transformations: the most stringent of these being the principle of frame indifference. In addition, realizability requires that the SGS model should have the same mathematical properties as the term it replaces, which leads to further constraints on the modelling. Finally, we investigate the effects of commutation and realizability for two simulated cases: forced homogeneous isotropic turbulence in a box and fully developed channel flow. In particular, we look for correlations between the commutation errors and the non-realizable regions, and the large-scale resolved flow structures in the simulations. Received: 17 June 1996 and accepted 3 April 1997     

Fixed points of nonexpansive mappings on star-shaped subsets of a convex metric space

In this paper, we give some characteristic properties of star-shaped sets which include a subset of a convex metric space. Using the characteristic properties, we discuss the existence problems of fixed points of nonexpansive type mappings on star-shaped subsets of convex metric spaces, which generalize the recent results obtained by Ding Xie-ping, Beg and Azam. Finally, we give an example which shows that our generalizations are essential.     

The application of Dirac matrices and Pauli matrices for the theory of plasticity

We are primarily concerned in this paper with the problem of plasticity. The solution of the problem of stress-increment for plasticity can be put into extremely compact form by famous Dirac matrices and Pauli matrices of quantum electrodynamics.     

Fixed point theorems in probabilistic inner product spaces

The purpose of this paper is to introduce a new maditying detinition for probabilistie inner product space, and to establish seveeral new fixed point theorems for mappings on such kind of spaces, As an example of applications. we utilize the results of this paper to study the existence and uniqueness of solution of Uryson’s integral equation in IV.     

连拱式大跨悬挑屋盖抗风气动措施研究

本文对连拱式大跨度悬挑屋盖进行了数值风洞试验,分别探讨了在屋盖悬挑前缘增设通风孔、导流板以及同时布设导流板与通风孔这三种气动措施对悬挑屋盖表面风荷载的影响及其作用机理．研究结果表明,同时布设导流板与通风孔的综合气动措施,能显著影响屋盖结构前缘气流的分离,从而减小屋盖的表面风压,并可减弱屋盖的风致振动．在此类屋盖结构设计中,运用综合气动措施可有效降低屋盖整体的升力系数与弯矩系数,对该类屋盖抗风设计较为有利．     

Physical and Numerical Experiments on an Under-Expanded Jet

The axial characteristics of axisymmetric under-expanded sonic jets are dealt with, both experimentally and numerically. Experiments were carried out in a test rig where a sonic nozzle discharged into an ambient which could be kept at different pressures. The density measurements were taken by means of a differential interferometer. The numerical simulation was obtained by adapting a code by Bird (1994) which is based on the MonteCarlo method. The comparison between experimental data and numerical results was carried out by considering the density distribution along the axis of the jet. The direct simulation was then applied to explore the jet characteristics in a rather wide range of the dimensionless products upon which the phenomenology depends. New expressions are presented for the density distribution along the jet axis and for the location of the Mach disk. Sommario. Si trattano sia sperimentalmente che numericamente le caratteristiche assiali di getti assialsimmetrici sonici sottoespansi. Gli esperimenti sono stati eseguiti con ugelli sonici in ambienti a pressione variabile. La densità è stata misurata mediante un interferometro differenziale e le simulazioni sono state condotte mediante un codice basato sul metodo MonteCarlo. Il confronto tra dati sperimentali e numerici è basato sulla distribuzione di densità lungo l'asse del getto. La simulazione diretta è stata poi applicata allo studio del getto in un ampio campo di valori dei numeri caratteristici. Si discutono la distribuzione della densità lungo l'asse e la posizione del disco di Mach, ottenute sia dai dati sperimentali che dai risultati numerici e si propongono per esse nuove espressioni.     

Fixed Final Time Optimal Control via Simple Cell Mapping

A cell mapping strategy based on Bellman'sPrinciple of optimality (BP) is proposed to solve the fixed final time optimalcontrol of strongly nonlinear systems with state and control bounds. Thefixed final time problem is transformed into a fixed final time-fixed finalstate optimal control problem by reversing the time. This transformationallows to solve the problem in the framework of the BP. Backward searchingalgorithms within the cell mapping context are used to obtain the solution ofthe new problem. This approach reduces considerably the computational effortrequired for the original problem when it is solved by a forward searchingapproach. The vibration control problem of damped and undamped variablestiffness oscillators with bounded stiffness tuning range is studied todemonstrate the effectiveness of the approach. A numerical study on theconvergence of the method is also presented.     

FIXED POINT THEOREMS FOR MAPPINGS SATISFYING IMPLICIT RELATION ON TWO COMPLETE AND COMPACT METRIC SPACES

First, the implicit relations were given. A common fixed point theorem was proved for two mappings satisfying implicit relation functions. A further fixed point theorem was proved for mappings satisfying implicit relation functions on two compact metric spaces.