时标上的推广的Pachpatte型不等式及其在边值问题中的应用

该文利用单调化技巧研究了时标上的推广的Pachpatte型不等式, 该不等式右端有一个非常数项和三个包含未知函数与没有假设单调性的非线性函数的复合函数的积分项, 不等式左端是未知函数与非线性函数的复合函数. 所得不等式不仅把Pachpatte型不等式的离散形式和连续形式统一起来, 而且推广了已有的时标上的相应不等式. 最后, 用得到的结果研究时标上边值问题解的估计.     

非线性分数阶微分方程三点边值问题解的存在性

讨论了一类非线性分数阶微分方程三点边值问题解的存在性.微分算子是Riemann-Liouville导算子并且非线性项依赖于低阶分数阶导数.通过将所考虑的问题转化为等价的Fredholm型积分方程,利用Schauder不动点定理获得该三点边值问题至少存在一个解.     

非线性加权V时标动态方程共振问题的可解性与多解性

基于对应线性问题的谱结构,对一类非线性加权V时标动态方程共振问题,在允许非线性项无界,但是次线性增长的前提下,获得了符号条件下一个解和多个解的存在性,所用的研究方法是含参紧向量场的解集连通理论.     

带幂非线性项变系数非线性波动方程的李对称分类与等价性变换

从微分方程群理论分析角度,研究了一类含有3个任意函数和2个幂非线性项的变系数非线性波动方程.由于方程具有很强的任意性和非线性项,可通过等价性变换寻找方程的不变对称分类.首先给出了等价性变换的一般结果,其中包括一些包含任意元的非局部变换.然后对所研究的方程,利用广义扩展等价群和条件等价群给出了方程的完全对称分类.最后获得并分析了方程的特殊类相似解.     

非线性V时标动态系统解的存在性与唯一性

引入V时标的概念,并对一类非线性V时标动态系统非共振边值问题,在非线性项不跨相应线性问题的特征值时,运用临界点理论,获得了问题解的存在性和唯一性定理.     

时标上交互集值微分系统的单调迭代技术

2009年,Hong S讨论了时标上集值动力微分方程.因研究问题需要,首次引入Hukuhare-Hilger导数,简记为△H.基于Hong S提出的相关理论,将讨论常微分方程解的存在性的单调迭代方法推广到时标上的集值交互微分系统,并得到了时标上交互集值微分系统解的存在性结果.     

Banach空间非线性二阶混合型脉冲积分-微分方程的解

利用Mnch不动点定理以及一个脉冲积分不等式,研究二阶混合型脉冲积分-微分方程初值问题解的存在性.结果涉及右端项既包含导数又包含积分算子Sx的情形.最后给出了一个应用例子.     

一类具有Riemann-Liouville分数阶积分边值条件的奇异分数阶微分方程解的存在性

研究一类具有Riemann-Liouville分数阶积分边值条件的奇异分数阶微分方程解的存在性,其非线性项包含Caputo型分数阶导数,且在t=0具有奇异性.应用Schauder不动点定理获得了解的存在性定理,并给出了应用实例.     

左端固定右端被滑动夹子夹住的非线性弹性梁方程的一个存在定理

通过运用积分方程并考察非线性项的“高度”对一类含有各阶导数的四阶两点边值问题建立了一个解的存在定理.在力学上,这类问题描述了一个左端固定右端被滑动夹子夹住的非线性弹性的形态.     

带导数项的奇摄动非线性 Schrodinger方程孤波解的存在性及其集中性质

利用Lyapunov-Schmidt方法证明了带有一阶导数项和(V)α势函数的非线性Schrodinger方程半经典孤波解的存在性及其集中性质. 具体地讲,当相当于Planck常数的奇摄动参数趋于零时,证明了该非线性Schrodinger方程的孤波解存在并且这些解在其势函数的非退化临界点处集中. 研究的是椭圆型方程的奇摄动问题,方程带有一阶导数项是本文特征之一.     

Approximating crossed symmetric solutions of nonlinear dynamic equations via quasilinearization

Crossed symmetric solutions of nonlinear boundary value dynamic problems play an important role in many applications, in particular in adaptive algorithm designs. This article is devoted to the continuation of our investigation on second-order nonlinear companion dynamic boundary value problems on time scales. Monotonically convergent upper and lower solutions of the problems and their quasilinear approximations are investigated. It is shown that, under proper smoothness constraints, the iterative sequences constructed not only converge to the analytic solutions of the desired companion problems monotonically, but also preserve important crossed symmetry properties. The quasilinearization offers an efficient way in the solution approximation. Computational examples are given to illustrate our results.     

The Generic Stability and Existence of Essentially Connected Components of Solutions for Nonlinear Complementarity Problems

The aim of this paper is to develop the general generic stability theory for nonlinear complementarity problems in the setting of infinite dimensional Banach spaces. We first show that each nonlinear complementarity problem can be approximated arbitrarily by a nonlinear complementarity problem which is stable in the sense that the small change of the objective function results in the small change of its solution set; and thus we say that almost all complementarity problems are stable from viewpoint of Baire category. Secondly, we show that each nonlinear complementarity problem has, at least, one connected component of its solutions which is stable, though in general its solution set may not have good behaviour (i.e., not stable). Our results show that if a complementarity problem has only one connected solution set, it is then always stable without the assumption that the functions are either Lipschitz or differentiable.     

Symmetric positive solutions for second order boundary value problems with integral boundary conditions on time scales

This paper investigates the existence of symmetric positive solutions for a class of nonlinear boundary value problem of second order dynamic equations with integral boundary conditions on time scales. Under suitable conditions, the existence of symmetric positive solution is established by using monotone iterative technique     

一类非线性矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在Schur插值问题中遇到的含未知矩阵二次项之逆的非线性矩阵方程转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求非线性矩阵方程的对称解的双迭代算法.双迭代算法仅要求非线性矩阵方程有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定.数值算例表明,双迭代算法是有效的.     

Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains

We consider solutions of some nonlinear parabolic boundary value problems in radial bounded domains whose initial profile satisfies a reflection inequality with respect to a hyperplane containing the origin. We show that, under rather general assumptions, these solutions are asymptotically (in time) foliated Schwarz symmetric, that is, all elements in the associated omega limit set are axially symmetric with respect to a common axis passing through the origin and nonincreasing in the polar angle from this axis. In this form, the result is new even for equilibria (i.e., solutions of the corresponding elliptic problem) and time periodic solutions.     

Approximate solutions to one-dimensional nonlinear heat conduction problems with a given flux

One-dimensional (planar, cylindrically symmetric, and spherically symmetric) nonlinear heat conduction problems with the heat flux at the origin specified in the form of a power time dependence are considered. The initial temperature of the medium is assumed to be zero. Approximate solutions to the problems are obtained. The convergence of the resulting solutions is discussed.     

AN EFFECTIVE CONTINUOUS ALGORITHM FOR APPROXIMATE SOLUTIONS OF LARGE SCALE MAX-CUT PROBLEMS

An effective continuous algorithm is proposed to find approximate solutions of NP-hardmax-cut problems.The algorithm relaxes the max-cut problem into a continuous nonlinearprogramming problem by replacing n discrete constraints in the original problem with onesingle continuous constraint.A feasible direction method is designed to solve the resultingnonlinear programming problem.The method employs only the gradient evaluations ofthe objective function,and no any matrix calculations and no line searches are required.This greatly reduces the calculation cost of the method,and is suitable for the solutionof large size max-cut problems.The convergence properties of the proposed method toKKT points of the nonlinear programming are analyzed.If the solution obtained by theproposed method is a global solution of the nonlinear programming problem,the solutionwill provide an upper bound on the max-cut value.Then an approximate solution to themax-cut problem is generated from the solution of the nonlinear programming and providesa lower bound on the max-cut value.Numerical experiments and comparisons on somemax-cut test problems(small and large size)show that the proposed algorithm is efficientto get the exact solutions for all small test problems and well satisfied solutions for mostof the large size test problems with less calculation costs.     

Mathematical programming via the least-squares method

The least-squares method is used to obtain a stable algorithm for a system of linear inequalities as well as linear and nonlinear programming. For these problems the solution with minimal norm for a system of linear inequalities is found by solving the non-negative least-squares (NNLS) problem. Approximate and exact solutions of these problems are discussed. Attention is mainly paid to finding the initial solution to an LP problem. For this purpose an NNLS problem is formulated, enabling finding the initial solution to the primal or dual problem, which may turn out to be optimal. The presented methods are primarily suitable for ill-conditioned and degenerate problems, as well as for LP problems for which the initial solution is not known. The algorithms are illustrated using some test problems.     

一般Lidstone边值问题的n个正解的存在性

考察了含所有偶数阶导数的一般Lidstone边值问题的正解和对称正解.通过 选择合适的Banach空间和锥,对该问题建立了n个正解或者对称正解的存在性,其 中n是一个任意的自然数.基本工具是等价范数和锥拉伸与锥压缩型的Krasnosel'skii 不动点定理. 结论的主要条件是局部的.换言之,如果非线性项f在某些有界集上的 "高度"是适当的,则该问题可以具有n个正解.     

Complexity and algorithms for convex network optimization and other nonlinear problems

Nonlinear optimization algorithms are rarely discussed from a complexity point of view. Even the concept of solving nonlinear problems on digital computers is not well defined. The focus here is on a complexity approach for designing and analyzing algorithms for nonlinear optimization problems providing optimal solutions with prespecified accuracy in the solution space. We delineate the complexity status of convex problems over network constraints, dual of flow constraints, dual of multi-commodity, constraints defined by a submodular rank function (a generalized allocation problem), tree networks, diagonal dominant matrices, and nonlinear Knapsack problem's constraint. All these problems, except for the latter in integers, have polynomial time algorithms which may be viewed within a unifying framework of a proximity-scaling technique or a threshold technique. The complexity of many of these algorithms is furthermore best possible in that it matches lower bounds on the complexity of the respective problems. In general nonseparable optimization problems are shown to be considerably more difficult than separable problems. We compare the complexity of continuous versus discrete nonlinear problems and list some major open problems in the area of nonlinear optimization. MSC classification: 90C30, 68Q25