Unicity of meromorphic solutions of partial differential equations

In this survey, results on the existence, growth, uniqueness, and value distribution of meromorphic (or entire) solutions of linear partial differential equations of the second order with polynomial coefficients that are similar or different from that of meromorphic solutions of linear ordinary differential equations have been obtained. We have characterized those entire solutions of a special partial differential equation that relate to Jacobian polynomials. We prove a uniqueness theorem of meromorphic functions of several complex variables sharing three values taking into account multiplicity such that one of the meromorphic functions satisfies a nonlinear partial differential equations of the first order with meromorphic coefficients, which extends the Brosch??s uniqueness theorem related to meromorphic solutions of nonlinear ordinary differential equations of the first order.     

A class of stochastic differential equations with pathwise unique solutions

We propose a new method viz., using stochastic partial differential equations to study the pathwise uniqueness of stochastic (ordinary) differential equations. We prove the existence and pathwise uniqueness of a class of stochastic differential equations with coefficients in suitable Hermite-Sobolev class using our approach.     

Ordinary differential equations,transport theory and Sobolev spaces

Summary We obtain some new existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces. These results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.     

Mean periodic solutions of a linear inhomogeneous first-order differential equation with random coefficients

We obtain deterministic first-order linear differential equations with ordinary and variational derivatives and deterministic initial conditions for the expectation and the second moment function of the solution of an ordinary scalar first-order linear inhomogeneous differential equation whose coefficients are random processes. We derive existence conditions for mean periodic solutions. In particular, we consider Gaussian and uniformly distributed random coefficients.     

Anticipating Stochastic Differential Equations of Stratonovich Type

We prove the existence and uniqueness of Stratonovich stochastic differential equations where the coefficients and the initial condition may depend on the whole path of the driving Wiener process. Our main hypothesis is that the diffusion coefficient satisfies the Frobenius condition. The solution is given in terms of solutions of ordinary differential equations and the Wiener process. We use this representation to study properties of the solution. Accepted 3 April 1996     

Inverse problem for differential pencils on a hedgehog graph

We study boundary value problems on a hedgehog graph for second-order ordinary differential equations with a nonlinear dependence on the spectral parameter. We establish properties of spectral characteristics and consider the inverse spectral problem of reconstructing the coefficients of a differential pencil on the basis of spectral data. For this inverse problem, we prove a uniqueness theorem and obtain a procedure for constructing its solution.     

UNIQUENESS OF VISCOSITY SOLUTIONS OF STOCHASTIC HAMILTON-JACOBI EQUATIONS

This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi(HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.     

Inverse spectral problem for differential operator pencils on noncompact spatial networks

We study boundary value problems on noncompact cycle-free graphs (i.e., trees) for second-order ordinary differential equations with a nonlinear dependence on the spectral parameter. We establish properties of the spectrum and analyze the inverse problem of reconstructing the coefficients of a differential equation on the basis of the so-called Weyl functions. For this inverse problem, we prove a uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mapping.     

Some remarks on stochastic differential equations in the plane with local lipschitz coefficients

We prove the existence and uniqueness of a local solution for stochastic differential equations in the plane with local Lipschitz coefficients, and state the existence of random points where the solution explodes. A sufficient condition to obtain a global solution is given.     

Recovering differential pencils on compact graphs

We study boundary value problems on compact graphs without circles (i.e. on trees) for second-order ordinary differential equations with nonlinear dependence on the spectral parameter. We establish properties of the spectral characteristics and investigate the inverse spectral problem of recovering the coefficients of the differential equation from the so-called Weyl vector which is a generalization of the Weyl function (m-function) for the classical Sturm-Liouville operator. For this inverse problem we prove the uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mappings.     

Multi-dimensional backward stochastic differential equations with one reflecting lower barrier of Itô diffusion type

This paper investigates a class of multi-dimensional stochastic differential equations with one reflecting lower barrier (RBSDEs in short), where the random obstacle is described as an Itô diffusion type of stochastic differential equation. The existence and uniqueness results for adapted solutions to such RBSDEs are established based on a penalization scheme and some higher moment estimates for solutions to penalized BSDEs under the Lipschitz condition and a higher moment condition on the coefficients. Finally, two examples are given to illustrate our theory and their applications.     

On existence,uniqueness and convergence of multi-valued stochastic differential equations driven by continuous semimartingales

In this paper we study the existence and uniqueness of solutions of multi-valued stochastic differential equations driven by continuous semimartingales when the coefficients are stochastically Lipschitz continuous. We also show the convergence results when the random coefficients or the differentials converge.     

一类常微分方程解的存在唯一性及其应用

提出并证明了一类常微分方程解的存在唯一性成立的一个充要条件,并给出了多项式形式增长函数的一列上界.最终将此结果应用到证明一类倒向随机微分方程的唯一解问题.     

Tensor approach to the problem of averaging differential equations with δ-correlated random coefficients

Linear ordinary differential equations with δ-correlated random coefficients are considered. We introduce the notion of linearizing tensor and use this notion to construct an algorithm for deriving differential equations for higher-order statistical moments of the solution of arbitrary positive integer orders.     

Nondegeneracy and uniqueness of positive solutions for Robin problem of second order ordinary differential equations and its applications

This article studies positive solutions of Robin problem for semi-linear second order ordinary differential equations. Nondegeneracy and uniqueness results are proven for homogeneous differential equations. Necessary and sufficient conditions for the existence of one or two positive solutions for inhomogeneous differential equations or differential equations with concave-convex nonlinearities are obtained by making use of the nondegeneracy and uniqueness results for positive solutions of homogeneous differential equations.     

具阶段结构害虫防治模型的脉冲效应

对于用微分方程描述的种群生态动力系统，其研究结果已十分丰富，但自然界中的许多变化规律都呈现出脉冲效应，因此用脉冲微分方程描述某些运动状态在固定或不固定时刻的快速变化或跳跃更切合实际，尤其在刻画种群生长和流行病动力学行为方面，脉冲微分方程的描述显得更科学更真实，具有脉冲效应的种群动力学模型的研究目前还处于刚刚起步阶段，本对符合实际的有脉冲效应的具阶段结构的常系数害早防治模型进行了研究，得到了系统存在周期解的充分条件，系统存在唯一周期解的充分条件，系统周期解轨道渐近稳定的充分条件。     

Galerkin Methods for Nonlinear Ordinary Differential Equation with Impulses

Methods based on fixed mesh variational formulations for ordinary differential equations in presence of a possibly infinite number of impulses on the righthand side are presented. A simple transformation allows us to show that the problem can be treated as an ordinary differential equation. Existence and uniqueness results for the solution and approximation schemes with their error estimates are obtained.     

Applications of maximum principles to dynamic equations on time scales

We apply the strong maximum principle to obtain a priori bounds and uniqueness of solutions for some initial value and boundary value problems as well as to establish oscillation results for second-order dynamic equations on time scales. Our comparison, uniqueness, and oscillation results are new and are extensions of results for ordinary differential equations to the times scale setting.     

Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise

Existence and uniqueness of the mild solutions for stochastic differential equations for Hilbert valued stochastic processes are discussed, with the multiplicative noise term given by an integral with respect to a general compensated Poisson random measure. Parts of the results allow for coefficients which can depend on the entire past path of the solution process. In the Markov case Yosida approximations are also discussed, as well as continuous dependence on initial data, and coefficients. The case of coefficients that besides the dependence on the solution process have also an additional random dependence is also included in our treatment. All results are proven for processes with values in separable Hilbert spaces. Differentiable dependence on the initial condition is proven by adapting a method of S. Cerrai.     

Pseudo almost automorphic solutions for dissipative differential equations in Banach spaces

This work aims to study the existence and uniqueness of pseudo compact almost automorphic solution for some dissipative ordinary and functional differential equations. We prove the existence and uniqueness of pseudo compact almost automorphic solution for dissipative differential equations in Banach spaces and then we apply this result to show the existence of pseudo compact almost automorphic solutions for some functional differential equations.