关于运输问题最优解的进一步讨论

首先给出了运输问题最优解的相关概念,将最优解扩展到广义范畴,提出狭义多重最优解和广义多重最优解的概念及其区别.然后给出了惟一最优解、多重最优解、广义有限多重最优解、广义无限多重最优解的判定定理及其证明过程.最后推导出了狭义有限多重最优解个数下限和广义有限多重最优解个数上限的计算公式,并举例验证了结论的正确性.     

On the existence of multiple steady-state solutions in the theory of electrodiffusion. Part I: the nonelectroneutral case. Part II: a constructive method for the electroneutral case

We give a constructive method for giving examples of doping functions and geometry of the device for which the nonelectroneutral voltage driven equations have multiple solutions. We show in particular, by performing a singular perturbation analysis of the current driven equations that if the electroneutral voltage driven equations have multiple solutions then the nonelectroneutral voltage driven equations have multiple solutions for sufficiently small normed Debye length. We then give a constructive method for giving examples of data for which the electroneutral voltage driven equations have multiple solutions.

     

具有多重解的非线性奇摄动问题

利用边界层法,研究了一类具有多重解的非线性奇摄动问题．在适当的假设下,通过给出外部解展开式系数及其对应边界条件的一般表达式,根据退化问题的边值作为某方程的根的重数,得到了此问题不同形式的渐近解．特别地,当这种根的重数为偶数时,问题具有二重解．另外,将相关结果应用于化学反应器理论,并通过对具有多重解的例子的渐近解和精确解的数值模拟说明如此构造的渐近解具有较高的精度．     

A Study of Multiple Solutions for the Navier-Stokes Equations by a Finite Element Method

In this paper, a finite element method is proposed to investigate multiple solutions of the Navier-Stokes equations for an unsteady, laminar, incompressible flow in a porous expanding channel. Dual or triple solutions for the fixed values of the wall suction Reynolds number $R$ and the expansion ratio $α$ are obtained numerically. The computed multiple solutions for the symmetric flow are validated by comparing them with approximate analytic solutions obtained by the similarity transformation and homotopy analysis method. Unlike previous works, our method deals with the Navier-Stokes equations directly and thus has no similarity and other restrictions as in previous works. Finally we use the method to study multiple solutions for three cases of the asymmetric flow (which has not been studied before using the similarity-type techniques).     

MULTIPLE POSITIVE SOLUTIONS OF NONRESONANT SINGULAR BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

This paper studies the existence of multiple positive solutions of nonresonant singular boundary value problem of second order ordinary differential equations. A sufficient condition for the existence of C[0,1] multiple positive solutions as well as C1[0, 1] multiple positive solutions is given by means of the fixed point theorems on cones.     

Banach空间中含脉冲边值问题的多重正解的存在性

In this paper, we first obtain some new results about the existence of multiple positive solutions for singular impulsive boundary value problems, and then to illustrate our main results we studied the existence of multiple positive solutions for an infinite system of scalar equations.     

EXISTENCE OF MULTIPLE POSITIVE PERIODIC SOLUTIONS TO A CLASS OF INTEGRO-DIFFERENTIAL EQUATION

In this paper,by the Avery-Henderson fixed point theorem,we investigate the existence of multiple positive periodic solutions to a class of integro-differential equation. Some suficient conditions are obtained for the existence of multiple positive periodic solutions.     

A variety of distinct kinds of multiple soliton solutions for a ( 3 + 1)‐dimensional nonlinear evolution equation

In this work, a variety of distinct kinds of multiple soliton solutions is derived for a ( 3 + 1)‐dimensional nonlinear evolution equation. The simplified form of the Hirota's method is used to derive this set of distinct kinds of multiple soliton solutions. The coefficients of the spatial variables play a major role in the existence of this variety of multiple soliton solutions for the same equation. The resonance phenomenon is investigated as well. Copyright © 2012 John Wiley & Sons, Ltd.     

Approximating closed form solutions to converging branch dynamic programming problems

This paper concerns a method for obtaining closed form approximate solutions to converging branch dynamic programming problems with quadratic returns and linear transitions. For quadratic returns, not necessarily convex, the optimum solutions for each stage generally contain multiple segments. A procedure for dealing with these multiple segments is presented. Problems with nonlinear returns at the converging branch have not been effectively dealt with in the literature. Utilizing a piecewise linear approximation to these nonlinear returns and the multiple segment procedure, closed form solutions to the resulting problems can be obtained. This method facilitates the computerization of this class of converging branch problems.     

Pareto memetic algorithm with path relinking for bi-objective traveling salesperson problem

The paper presents an effective version of the Pareto memetic algorithm with path relinking and efficient local search for multiple objective traveling salesperson problem. In multiple objective Traveling salesperson problem (TSP), multiple costs are associated with each arc (link). The multiple costs may for example correspond to the financial cost of travel along a link, time of travel, or risk in the case of hazardous materials. The algorithm searches for new good solutions along paths in the decision space linking two other good solutions selected for recombination. Instead of a simple local search it uses short runs of tabu search based on the steepest version of the Lin–Kernighan algorithm. The efficiency of local search is further improved by the techniques of candidate moves and locked arcs. In the final step of the algorithm the neighborhood of each potentially Pareto-optimal solution is searched for new solutions that could be added to this set. The algorithm is compared experimentally to the state-of-the-art algorithms for multiple objective TSP.     

非共振多参数脉冲奇异边值问题的多解性

该文讨论了非共振奇异脉冲多参数二阶微分方程边值问题的多个正解的存在性,利用上下解方法和不动点指数给出了多个正解存在的一个充分条件．     

Completely integrable coupled KdV and coupled KP systems

In this work we study two completely integrable coupled KdV and coupled KP systems. The Hirota’s bilinear method is employed to formally derive multiple soliton solutions and multiple singular soliton solutions for each system. The resonance phenomenon will be examined.     

Some further results on a semilinear equation with concave–convex nonlinearity

Following our previous work [J. Math. Anal. Appl. 295 (2004) 341], we give an exact growth order near zero for positive solutions of a class of elliptic equation and use it to give the existence of multiple solutions with negative energy, multiple positive solutions and sign changing solutions of the considered problem. Even in the particular case, our results extend a previous work by Ambrosetti–Garcia–Peral [J. Funct. Anal. 137 (1996) 219].     

The measurement of returns to scale under a simultaneous occurrence of multiple solutions in a reference set and a supporting hyperplane

This research theoretically explores the measurement of RTS (Returns to Scale) under a possible occurrence of multiple solutions in DEA (Data Envelopment Analysis). In this study, the occurrence of multiple solutions is classified into Type I and Type II. Type I is an occurrence of multiple solutions in a reference set. Type II is an occurrence of multiple solutions on a supporting hyperplane passing on the reference set. Both Types I and II are very well known among DEA researchers, but previous research has not sufficiently explored a simultaneous occurrence of Type I and Type II in the RTS measurement. The two types of multiple solutions influence a degree of RTS in the DEA measurement. Such a quantitative issue on RTS is examined from the perspective of the Type I and Type II problems. To deal with such difficulties, a new linear programming approach is proposed to identify all efficient DMUs (Decision Making Units) that consist of a reference set, even if multiple solutions occur on the reference set. Based upon the research result, we can identify when multiple solutions of Type I and/or Type II occur on the RTS measurement and how to deal with such difficulties. Our research result makes it possible to measure a degree of scale economies (RTS) under the simultaneous occurrence of Type I and Type II.     

Multiple lump solutions of the (2+1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, multiple lump solutions of the (2+1)-dimensional Konopelchenko–Dubrovsky equation are obtained by means of the Hirota bilinear method. With the aid of positive quartic-quadratic-functions, we can get the 1-lump solutions, 3-lump solutions, and 6-lump solutions. Via the density plots and three-dimensional plots, the dynamic properties of multiple lump solutions are discussed by choosing the appropriate parameters. It is expected that our results are valuable for revealing the high-dimensional dynamic phenomenon of the nonlinear evolution equations.     

Unique and multiple PHAM series solutions of a class of nonlinear reactive transport model

The purpose of this paper is to visit a class of nonlinear reactive transport model in the case including advective and diffusive transport with the Michaelis-Menten reaction term. We apply the method so-called predictor homotopy analysis method (PHAM) which has been recently proposed to predict multiplicity of solutions of nonlinear BVPs. Consequently two consequential matters are indicated which confirms the authority of PHAM to identify multiple solutions: (i) The Talylor series solutions are improved by the so-called convergence-controller parameter (ii) The possibility of existence of multiple solutions is discovered in some cases for the model.     

Multiplicity for Self-Dual Condensate Solutions in the Maxwell-Chern-Simons O(3) Sigma Model

In this paper, we study multiple existence of solutions of an elliptic system on a flat torus arising from the self-dual Maxwell-Chern-Simons O(3) sigma model. Using the Leray-Schauder degree, we find an explicit range of κ and q for which we have at least two vortex-antivortex solutions. For vortex solutions, we provide a complete picture of the range of κ and q for multiple existence of solutions.     

Some further results for a class of weighted nonlinear elliptic equations

Following our previous work [M. Lin, Multiple positive solutions for a class of weighted nonlinear elliptic equations, preprint, 2006], we give an exact growth order near zero for positive solutions of a class of weighted elliptic equations and use it to give the existence of multiple positive solutions and sign changing solutions of the considered problem.     

Multiplicity of positive periodic solutions to a generalized delayed predator–prey system with stocking

In this paper, by using the continuation theorem of coincidence degree theory, the existence of multiple positive periodic solutions for a generalized delayed predator–prey system with stocking is established. When our result is applied to a delayed predator–prey system with nonmonotonic functional response and stocking, we establish the sufficient condition for the existence of multiple positive periodic solutions for the system.     

Predictor homotopy analysis method and its application to some nonlinear problems

The purpose of this paper is to present a kind of analytical method so-called Predictor homotopy analysis method (PHAM) to predict the multiplicity of the solutions of nonlinear differential equations with boundary conditions. This method is very useful especially for those boundary value problems which admit multiple solutions and furthermore is capable to calculate all branches of the solutions simultaneously. As illustrative examples, the method is checked by the model of mixed convection flows in a vertical channel and a nonlinear model arising in heat transfer which both admit multiple (dual) solutions.