三维空间中的有理样条插值

对三维空间某个多面体区域的四面体剖分，通过在每个四面体胞腔的棱和顶点设置适当的插值结点．本文给出了（１，１）型Ｃ⁰及Ｃ¹光滑的非奇异有理样条存在的充分必要条件．     

Optimality conditions for maximizations of set-valued functions

The maximization with respect to a cone of a set-valued function into possibly infinite dimensions is defined, and necessary and sufficient optimality conditions are established. In particular, an analogue of the Fritz John necessary optimality conditions is proved using a notion of derivative defined in terms of tangent cones.     

Convergence of the control parameterization Ritz method for nonlinear optimal control problems

This paper studies the local convergence properties of the control parameterization Ritz method in which the control variable is approximated over a finite-dimensional subspace. The nonlinear free-endpoint optimal control problem is considered, and error bounds are derived for both the cost functional and state-control convergence. Explicit error bounds are obtained for the particular case of approximations over spline spaces. On specializing the general results to the linear-quadratic regulator problem, global convergence results are obtained. Computational results supporting the theoretically derived error bounds are presented.This research was supported by the University Grants Committee of New Zealand.     

Approximations and Well-Posedness in Multicriteria Games

First, sufficient conditions of minimal character are given which guarantee the sequential closedness of the set-valued function defined by the parametric weak-multicriteria Nash equilibria of a parametric multicriteria game, that is to say: a convergent sequence of parametric weak-multicriteria Nash equilibria, corresponding to an approximate value of the parameter x_n, converges to a weak-multicriteria Nash equilibrium corresponding to the limit value x of the sequence (x_n)_n. Then, approximating sequences and parametrically well-posedness for a multicriteria game are introduced and investigated.     

On the Shadowing Property and Shadowable Point of Set-valued Dynamical Systems

In this article, the authors introduce the concept of shadowable points for set-valued dynamical systems, the pointwise version of the shadowing property, and prove that a set-valued dynamical system has the shadowing property iff every point in the phase space is shadowable; every chain transitive set-valued dynamical system has either the shadowing property or no shadowable points; and for a set-valued dynamical system there exists a shadowable point iff there exists a minimal shadowable point. In the end, it is proved that a set-valued dynamical system with the shadowing property is totally transitive iff it is mixing and iff it has the specification property.     

Stability of a Cubically Convergent Method for Generalized Equations

In Geoffroy et al, Acceleration of convergence in Dontchev's iterative method for solving variational inclusions Serdica Math. J. 29 (2003), pp. 45–54] we showed the convergence of a cubic method for solving generalized equations of the form 0 ∈ f(x) +- G(x) where f is a function and G stands for a set-valued map. We investigate here the stability of such a method with respect to some perturbations. More precisely, we consider the perturbed equation y ∈ f(x) +- G(x) and we show that the pseudo-Lipschitzness of the map (f +- G)⁻¹ is closely tied to the uniformity of our method in the sense that the attraction region does not depend on small perturbations of the parameter y. Finally, we provide an enhanced version of the convergence theorem established by Geoffroy, et al.     

Detection of C2‐singularities using uniform spline approximations

For the detection of C²‐singularities, we present lower estimates for the error in Schoenberg variation‐diminishing spline approximation with equidistant knots in terms of the classical second‐order modulus of smoothness. To this end, we investigate the behaviour of the iterates of the Schoenberg operator. In addition, we show an upper bound of the second‐order derivative of these iterative approximations. Finally, we provide an example of how to detect singularities based on the decay rate of the approximation error.     

An efficient nonpolynomial spline method for distributed order fractional subdiffusion equations

In this paper, we propose an efficient numerical method for a distributed order fractional subdiffusion problem using nonpolynomial spline approach. The solvability, stability, and convergence of the scheme are established rigorously, and it is shown that the spatial convergence order improves some previous work done. Simulation is then conducted to verify the accuracy of the proposed scheme as well as to compare with earlier work.     

An alternative theorem for set-valued maps via set relations and its application to robustness of feasible sets

Abstract

This paper contains a generalized Gordan-type alternative theorem for set-valued maps which characterizes set relations without any convexity assumptions using certain evaluation functions. As a direct consequence and as a good example, we discuss robustness (or stability) of linear programming problems for modelling error. Moreover, this theorem can be utilized for that of general vector optimization problems in special cases due to reformation of the evaluation functions.     

High order convergent modified nodal bi-cubic spline collocation method for elliptic partial differential equation

A high order modified nodal bi-cubic spline collocation method is proposed for numerical solution of second-order elliptic partial differential equation subject to Dirichlet boundary conditions. The approximation is defined on a square mesh stencil using nine grid points. The solution of the method exists and is unique. Convergence analysis has been presented. Moreover, the superconvergent phenomena can be seen in proposed one step method. The numerical results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency.