Frequency Tests for the Existence and Stability of Bounded Solutions to Differential Equations of Higher Order

To study a vector-matrix differential equation of order n, the method of integral equations is used. When the Lipschitz condition holds, an existence and uniqueness theorem for a bounded solution and its estimates are obtained. This solution is almost periodic if the nonlinearity is almost periodic, and it is asymptotically Lyapunov stable if the matrix characteristic polynomial is a Hurwitz polynomial. Under a Lipschitztype condition, a theorem on the existence of at least one bounded solution is proved; among the bounded solutions, there is at least one recurrent solution if the nonlinearity is almost periodic. The equation is S-dissipative if the matrix characteristic polynomial is a Hurwitz polynomial.     

半正泛函微分方程边值问题的正解

本文证明了一个关于一类含参数半正奇异泛函微分边值问题正解的存在性结果。推广了已有的某些结果。作为应用,讨论了一个含参数多项式型泛函微分方程正解的存在性问题。     

Hamiltonians associated with the sixth Painlevé equation

We obtain the formula determining the general form of polynomial Hamiltonians associated with the sixth Painlevé equation and prove its uniqueness. We prove the existence of nonpolynomial Hamiltonians associated with this equation. We identify the Hamiltonian class for which the defining differential equation coincides with the equation (h-equation) for the simplest polynomial Hamiltonian (the Okamoto Hamiltonian). __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 54–65, April, 2007.     

完全域上矩阵多项式方程的可解性

本文利用一般域上的λ-矩阵理论，研究了矩阵多项式方程的可解性，证明了完全域上矩阵多项式方程有解的充要条件，这些条件同时提供了解此类矩阵方程的方法。     

Chebyshev rational matrix approximation with a priori error bounds for linear and Riccati matrix equations

This paper deals with initial value problems for Lipschitz continuous coefficient matrix Riccati equations. Using Chebyshev polynomial matrix approximations the coefficients of the Riccati equation are approximated by matrix polynomials in a constructive way. Then using the Fröbenius method developed in [1], given an admissible error ϵ > 0 and the previously guaranteed existence domain, a rational matrix polynomial approximation is constructed so that the error is less than ϵ in all the existence domain. The approach is also considered for the construction of matrix polynomial approximations of nonhomogeneous linear differential systems avoiding the integration of the transition matrix of the associated homogeneous problem.     

The Abhyankar-Jung theorem revisited

In this paper we use a technique introduced by P.M. Cohn in order to prove the so-called Abhyankar–Jung theorem, i.e. the existence of parametric equations for a quasiordinary hypersurface. The basic tool of the proof is the equivalence between the existence of solutions for a polynomial equation and the existence of an eigenvalue of its companion matrix.     

MULTIPLICITY OF SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION

We study a quasilinear elliptic equation with polynomial growth coefficients. The existence of infinitely many solutions is obtained by a dual method and a nonsmooth critical point theory.     

Eigenfunctions of Two-Scale Difference Equations with Dilation Parameter and Infinite Products

This paper deals with two-scale difference equations having an arbitrary dilation parameter and a formal power series as symbol. We investigate the equation concerning the existence of nonzero compactly supported distributional solutions. In order to include also continuous solutions it is advantageous to consider the two-scale difference equation as eigenvalue problem where the solutions are either compactly supported or integrals of compactly supported distributions. Such solutions are called eigenfunctions. As main result we determine the necessary and sufficient condition for the existence of eigenfunctions that the symbol must be a rational function with a special structure depending on the dilation parameter. We show that the eigenfunctions can be expressed by means of a finite sum of shifted eigenfunctions belonging to the case with a polynomial symbol (characteristic polynomial), which is well investigated.     

On a Solution of the Quaternion Matrix Equation X - A X B = C and Its Application

This paper first studies the solution of a complex matrix equation X - AXB = C, obtains an explicit solution of the equation by means of characteristic polynomial, and then studies the quaternion matrix equation X - A X B = C, characterizes the existence of a solution to the matrix equation, and derives closed-form solutions of the matrix equation in explicit forms by means of real representations of quaternion matrices. This paper also gives an application to the complex matrix equation X - AXB =C.     

带外力的Enskog方程

本文在初值对时间变量和空间变量都具有多项式衰减时,证明了带外力Ens-kog方程解的存在性和唯一性.     

Existence and Multiplicity of Solutions for a Class of Quasilinear Elliptic Equations with Exponential Growth北大核心CSCD

本文研究一类具有次临界多项式增长或次临界指数型(临界指数型)增长的(p,2)-拉普拉斯方程一个正解及无穷多非平凡解的存在性,运用山路定理及喷泉定理,得到了拉普拉斯方程非平凡解的一些存在性结果.     

A discrete and q asymptotic iteration method

ABSTRACT

We introduce a finite difference and q-difference analogues of the Asymptotic Iteration Method of Ciftci, Hall, and Saad. We give necessary, and sufficient condition for the existence of a polynomial solution to a general linear second-order difference or q-difference equation subject to a ‘terminating condition’, which is precisely defined. When a difference or q-difference equation has a polynomial solution, we show how to find the second solution.     

Solving a Problem of Robin Type for Biharmonic Equation

We investigate existence and uniqueness conditions for solution to one Robin type problem for inhomogeneous biharmonic equation in the unit ball. We construct polynomial solution to the problem when the boundary functions of the problems are polynomials.     

Computer algebra systems and theorems on real roots of polynomials

A computer algebra system is used to derive a theorem on the existence of roots of a quadratic equation on any bounded real interval. This is extended to a cubic polynomial. We discuss how students could be led to derive and prove these theorems.     

Polynomial solutions of certain differential equations arising in physics

Conditions for the existence of polynomial solutions of certain second‐order differential equations have recently been investigated by several authors. In this paper, a new algorithmic procedure is given to determine necessary and sufficient conditions for a differential equation with polynomial coefficients containing parameters to admit polynomial solutions and to compute these solutions. The effectiveness of this approach is illustrated by applying it to determine new solutions of several differential equations of current interest. A comparative analysis is given to demonstrate the advantage of this algorithmic procedure over existing software. Copyright © 2012 John Wiley & Sons, Ltd.     

一类拟线性椭圆方程的非平凡解

运用环绕理论和对称型山路理论对一类具有次临界多项式增长和次临界指数增长的$p$-Laplacian方程建立一个非平凡解(无穷多个非平凡解)的存在性结果.     

Ground state solution for a fourth-order nonlinear elliptic problem with logarithmic nonlinearity

In this paper, we study the existence of ground state solutions of nonlinear elliptic equation with logarithmic nonlinearity by the Linking theorem and logarithmic Sobolev inequality. Our results are quite different from those in the case of polynomial nonlinearity.     

The existence theorems of the random solutions for random Hammerstein equation with random kernel (II)

In this paper, we use the random fixed theorems of cone expansion and compression of random operator to obtain the existence theorems of random solutions for random Hammerstein integral equation of polynomial type with random kernel.     

On the properties of solutions of equations in the <Emphasis Type="Italic">K</Emphasis><Subscript>2</Subscript> hierarchy

For the sixth-order equation of the K ₂ hierarchy, we establish conditions for the existence of families of polar solutions, construct polynomial Hamiltonians and Bäcklund transformations, represent meromorphic solutions via a ratio of special entire functions, and give a criterion for the existence of rational solutions.     

The existence of random $\mathcal{D}$-pullback attractors for random dynamical system and its applications

In this paper, we establish a result on the existence of random $\mathcal{D}$-pullback attractors for norm-to-weak continuous non-autonomous random dynamical system. Then we give a method to prove the existence of random $\mathcal{D}$-pullback attractors. As an application, we prove that the non-autonomous stochastic reaction diffusion equation possesses a random $\mathcal{D}$-pullback attractor in $H_0^1$ with polynomial growth of the nonlinear term.