Markovian extensions of symmetric second order elliptic differential operators

Let be bounded with a smooth boundary Γ and let S be the symmetric operator in given by the minimal realization of a second order elliptic differential operator. We give a complete classification of the Markovian self‐adjoint extensions of S by providing an explicit one‐to‐one correspondence between such extensions and the class of Dirichlet forms in which are additively decomposable by the bilinear form of the Dirichlet‐to‐Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of an additive decomposition of the bilinear forms associated to the extensions, the second one uses the additive decomposition of the resolvents provided by Kre?n's formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell‐type boundary conditions. Some properties of the extensions, and of the corresponding Dirichlet forms, semigroups and heat kernels, like locality, regularity, irreducibility, recurrence, transience, ultracontractivity and Gaussian bounds are also discussed.     

Zeros of eigenfunctions of a class of generalized second order differential operators on the Cantor set

Generalized second order differential operators of the form when μ is a selfsimilar measure whose support is the classical Cantor set are considered. The asymptotic distribution of the zeros of the eigenfunctions is determined. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the existence of positive solutions of second order differential equations

Using the fixed point method we prove an existence result for positive solutions of nonlinear second order ordinary differential equations. An application to semilinear Schrödinger equations in exterior domains is also presented. Mathematics Subject Classification (2000) 34C10     

Comparison theorems for the third order trinomial differential equations with delay argument

In this paper we study asymptotic properties of the third order trinomial delay differential equation (*) y‴(t) − p(t)y′(t) + g(t)y(τ(t)) = 0 by transforming this equation to the binomial canonical equation. The results obtained essentially improve known results in the literature. On the other hand, the set of comparison principles obtained permits to extend immediately asymptotic criteria from ordinary to delay equations. Research was supported by S.G.A. No.1/003/09.     

Periodic solutions of the third order functional differential equations

We obtain sufficient conditions for the existence of periodic solutions of the following third order nonlinear functional differential equations

     

一类三阶泛函微分方程周期解的存在性

利用拓扑度理论中的Mawhin延拓定理,研究了三阶泛函微分方程的周期解的存在性,并改进推广了已有文献中的相应结论。     

On subnormal solutions of second order linear periodic differential equations

In this paper, we investigate the existence and the form of subnormal solution for a class of second order periodic linear differential equations, estimate the growth properties of all solutions, and answer the question raised by Gundersen and Steinbart.     

Periodic solutions of a second order forced sublinear differential equation with delay

In this paper, existence criteria are established for the existence of periodic solutions of a second order sublinear differential equation with delay and forcing function. Our method is based on careful a priori estimation and continuation theorems, and our sublinear condition is an improvement of the boundedness condition in some recent results.     

On self-adjointness of the product of two second-order differential operators

In this paper, the problem of self-adjointness of the product of two differential operators is considered. A number of results concerning self-adjointness of the productL ₂ L ₁ of two second-order self-adjoint differential operators are obtained by using the general construction theory of self-adjoint extensions of ordinary differential operators. Supported by the Royal Society and the National Natural Science Foundation of China and the Regional Science Foundation of Inner Mongolia     

高阶整函数系数线性微分方程解的增长性

主要研究了高阶整函数系数线性微分方程f(n) An-1f(n-1) … A1f′ A0f=0的解的增长性,我们证明了如果σ(Aj)>1,σ(Aj),j=1,…,n-1都不是整数,且0＜σ(A0)≤(1)(2)和每个Aj的所有零点都位于与它的亏格有关的角域内,那么方程的每个解f(≠)0具有无穷增长级,并得到其超级的一些估计.     

On oscillatory solutions of fourth order ordinary differential equations

The paper deals with the oscillation of a differential equation L ₄ y + P(t)L ₂ y + Q(t)y 0 as well as with the structure of its fundamental system of solutions.     

Liapunov-type inequality for delay-differential equations of third order

A Liapunov-type inequality for a class of third order delay-differential equations is derived.     

New canonical forms of self-adjoint boundary conditions for regular differential operators of order four

In this paper, we find new canonical forms of self-adjoint boundary conditions for regular differential operators of order two and four. In the second order case the new canonical form unifies the coupled and separated canonical forms which were known before. Our fourth order forms are similar to the new second order ones and also unify the coupled and separated forms. Canonical forms of self-adjoint boundary conditions are instrumental in the study of the dependence of eigenvalues on the boundary conditions and for their numerical computation. In the second order case this dependence is now well understood due to some surprisingly recent results given the long history and voluminous literature of Sturm-Liouville problems. And there is a robust code for their computation: SLEIGN2.     

Nonlinear boundary value problems for second order differential equations with causal operators

In this paper we deal with second order differential equations with causal operators. To obtain sufficient conditions for existence of solutions we use a monotone iterative method. We investigate both differential equations and differential inequalities. An example illustrates the results obtained.     

Global monotonicity and oscillation for second order differential equation

Oscillatory properties of the second order nonlinear equation

A note on differential operators of infinite order

A sufficient condition for entire functions f and g to be such that the series ∑_m=0^∞f^(m)(0)g^(m)/m! represents an entire function is established; and in that case, the growth of the resulting function is described.     

On Oscillation Criteria of Fourth Order Linear Differential Equations

The paper deals with oscillation criteria of fourth order linear differential equations with quasi-derivatives.     

Some examples of orthogonal matrix polynomials satisfying odd order differential equations

It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form

W(t)=t^αe^-te^Att^Bt^B*e^A*t,