Some results on linear delay integral equations

Recently Borwein has proposed a definition for extending Geoffrion's concept of proper efficiency to the vector maximization problem in which the domination cone S is any nontrivial, closed convex cone. However, when S is the non-negative orthant, solutions may exist which are proper according to Borwein's definition but improper by Geoffrion's definition. As a result, when S is the non-negative orthant, certain properties of proper efficiency as defined by Geoffrion do not hold under Borwein's definition. To rectify this situation, we propose a definition of proper efficiency for the case when S is a nontrivial, closed convex cone which coincides with Geoffrion's definition when S is the non-negative orthant. The proposed definition seems preferable to Borwein's for developing a theory of proper efficiency for the case when S is a nontrivial, closed convex cone.     

Some results on linear nabla Riemann-Liouville fractional difference equations

In this paper, we establish some criteria for boundedness, stability properties, and separation of solutions of autonomous nonlinear nabla Riemann-Liouville scalar fractional difference equations. To derive these results, we prove the variation of constants formula for nabla Riemann-Liouville fractional difference equations.     

Stability results for the approximation of weakly coupled wave equations

In this Note, we consider the approximation of two coupled wave equations with internal damping. Our goal is to damp the spurious high frequency modes by introducing numerical viscosity terms in the approximation scheme. With these viscosity terms, we show the exponential or polynomial decay of the discrete scheme when the continuous problem has such a decay (since the spectrum of the spatial operator associated with the undamped system satisfies the generalized gap condition).     

Some results related to complex linear and nonlinear difference equations

In this paper, we investigate the complex oscillation problems of meromorphic solutions to some linear difference equations with meromorphic coefficients, and obtain some results about the relationships between the exponent of convergence of zeros, poles and the order of growth of meromorphic solutions to complex linear difference equations. We also study the existence of solution of certain types of nonlinear differential-difference equations, and partially answer a conjecture concerning the above problem posed by Yang and Laine (C.C. Yang and I. Laine, On analogies between nonlinear difference and differential equations, Proc. Japan Acad. Ser. A Math. Sci. 86(1) (2010), pp. 10–14).     

Further results on linear interval equations

As a continuation of my paper “New techniques for the analysis of linear interval equations” [Linear Algebra Appl. 58:273–325 (1984)], the interrelation between interval Gauss elimination and interval iteration is investigated. Main results are a new existence theorem for interval Gauss elimination (in the guise of a perturbation theorem), a convergence and comparison theorem for a general family of interval iteration schemes, and a new method for the calculation of the hull of the solution set of linear interval equations with inverse positive coefficient matrix.     

Some results on linear arboricity

The linear arboricity of the graph G is the minimum number of linear forests whose union is G. In the paper exact values and bounds of linear arboricity for some additional classes of graphs are determined.     

Some results on difference Riccati equations

Some properties of solutions for the difference Riccati equations are obtained. The existence and forms of rational solutions, and the Borel exceptional value, zeros, poles and fixed points of transcendental solutions are researched.     

Some linear fractional stochastic equations

Using the multiple stochastic integrals, we prove an existence and uniqueness result for a linear stochastic equation driven by the fractional Brownian motion with any Hurst parameter. We study both the one- and two-parameter cases. When the drift is zero, we show that in the one-parameter case the solution is an exponential—thus positive—function while in the two-parameter setting the solution is negative on a non-negligible set.     

On the standing wave in coupled non‐linear Klein–Gordon equations

This paper is concerned with the standing wave in coupled non‐linear Klein–Gordon equations. By an intricate variational argument we establish the existence of standing wave with the ground state. Then we derive out the sharp criterion for blowing up and global existence by applying the potential well argument and the concavity method. We also show the instability of the standing wave. Copyright © 2003 John Wiley & Sons, Ltd.     

Some comments on the solution of linear equations

Homogeneous, ill-conditioned and singular linear equations are considered and some methods for their solution are described.This research was supported in part by the National Aeronautics and Space Administration Grant No. NGR-33-015-013.     

Traveling wave solutions of the n-dimensional coupled Yukawa equations

We discuss traveling wave solutions to the Yukawa equations, a system of nonlinear partial differential equations which has applications to meson–nucleon interactions. The Yukawa equations are converted to a six-dimensional dynamical system, which is then studied for various values of the wave speed and mass parameter. The stability of the solutions is discussed, and the methods of competitive modes is used to describe parameter regimes for which chaotic behaviors may appear. Numerical solutions are employed to better demonstrate the dependence of traveling wave solutions on the physical parameters in the Yukawa model. We find a variety of interesting behaviors in the system, a few of which we demonstrate graphically, which depend upon the relative strength of the mass parameter to the wave speed as well as the initial data.     

On exact boundary controllability for linearly coupled wave equations

In this paper we study exact boundary controllability for a system of two linear wave equations coupled by lower order terms. We obtain square integrable control of Neuman type for initial state with finite energy, in nonsmooth domains of the plane.     

Global smooth solution for a coupled non-linear wave equations

In this paper, the global existence and uniqueness of smooth solution to the initial-value problem for coupled non-linear wave equations are studied using the method of a priori estimates.     

Variational principles for a class of coupled linear equations

This paper presents variational and extremum principles for pairs of coupled linear equations. The results are illustrated by several examples arising in mathematical physics.     

Some basic principles for linear coupled dynamic thermopiezoelectricity

According to the basic idea of classical yin-yang complementarity and modern dualcomplementarity, in a simple and unified way some basic principles for linear coupled dynamic thermopiezoelectricity can be established systematically. An important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in linear coupled dynamic thermopiezoelectricity, but also to derive systematically the complementary functionals for eleven-field, nine-field, six-field and three-field simplified Gurtin-type variational principles. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly. Project supported by the National Natural Science Foundation of China (Grant No. 19672074) and Research Grand Council of Hong Kong, No.RGC97/98, HKUST 6055/97E.     

Superconvergence results for linear second-kind Volterra integral equations

In this paper, Galerkin method is applied to approximate the solution of Volterra integral equations of second kind with a smooth kernel, using piecewise polynomial bases. We prove that the approximate solutions of the Galerkin method converge to the exact solution with the order \({\mathcal {O}}(h^{r}),\) whereas the iterated Galerkin solutions converge with the order \({\mathcal {O}}(h^{2r})\) in infinity norm, where h is the norm of the partition and r is the smoothness of the kernel. We also consider the multi-Galerkin method and its iterated version, and we prove that the iterated multi-Galerkin solution converges with the order \({\mathcal {O}}(h^{3r})\) in infinity norm. Numerical examples are given to illustrate the theoretical results.