YoshiZawa型周期解定理和Massera型周期解定理研究进展简介

微分方程解的有界性和周期解的存在性是檄分方程理论研究中的两个重要课题，二者之间有着紧密的联系．在解的有界性与周期解的存在性的研究中，Yoshizawa周期解定理和Massera周期解定理是非常重要的结果，具有重要的理论意义和应用价值．本文以Yoshizawa型周期解定理和Massera型周期解定理的研究为主，简要介绍泛函微分方程周期解理论研究方面的一些新进展。     

Positive solutions for multi‐point boundary value problems of fractional differential equations with p‐Laplacian

This paper investigates the existence of solutions for multi‐point boundary value problems of higher‐order nonlinear Caputo fractional differential equations with p‐Laplacian. Using the five functionals fixed‐point theorem, the existence of multiple positive solutions is proved. An example is also given to illustrate the effectiveness of ourmain result. Copyright © 2015 John Wiley & Sons, Ltd.     

A Liouville theorem for the compressible Navier‐Stokes equations

This paper is concerned with 3‐dimensional steady compressible Navier‐Stokes equations. A Liouville‐type theorem is proved when some suitable conditions are satisfied.     

A note on the existence of stochastic integro‐differential equations with memory

In this study, we investigate the existence of mild solutions for a class of impulsive neutral stochastic integro‐differential equations with infinite delays, using the Krasnoselskii–Schaefer type fixed point theorem combined with theories of resolvent operators. As an application, an example is provided to illustrate the obtained result. Copyright © 2014 John Wiley & Sons, Ltd.     

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg‐de Vries‐type (KdV‐type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.     

连续系数倒向随机微分方程最小解的Levi定理

本文研究了倒向随机微分方程解的连续依赖性问题.利用文献[4]中使用的方法,提出并证明了连续系数的一维倒向随机微分方程最小解的Levi定理,推广了文献[10]中的相应结果.     

Anticipated backward stochastic differential equations with non-Lipschitz coefficients

This paper deals with a class of anticipated backward stochastic differential equations. We extend results of Peng and Yang (2009) to the case in which the generator satisfies non-Lipschitz condition. The existence and uniqueness of solutions for anticipated backward stochastic differential equations as well as a comparison theorem are obtained. The existence and uniqueness of L^p(p>2) solutions for anticipated backward stochastic differential equations are also studied.     

Sturm‐Picone comparison theorems for nonlinear impulsive differential equations with discontinuous solutions

The nonlinear versions of Sturm‐Picone comparison theorem as well as Leighton's variational lemma and Leighton's theorem for regular and singular nonlinear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions are established. Although discontinuity of the solutions causes some difficulties, these new comparison theorems cover the old ones where impulse effects are dropped.     

Characterization of Stochastic Viability of Any Nonsmooth Set Involving Its Generalized Contingent Curvature

Abstract

Thanks to the Stroock and Varadhan “Support Theorem” and under convenient regularity assumptions, stochastic viability problems are equivalent to invariance problems for control systems (also called tychastic viability), as it has been singled out by Doss in 1977 for instance. By the way, it is in this framework of invariance under control systems that problems of stochastic viability in mathematical finance are studied. The Invariance Theorem for control systems characterizes invariance through first‐order tangential and/or normal conditions whereas the stochastic invariance theorem characterizes invariance under second‐order tangential conditions. Doss's Theorem states that these first‐order normal conditions are equivalent to second‐order normal conditions that we expect for invariance under stochastic differential equations for smooth subsets. We extend this result to any subset by defining in an adequate way the concept of contingent curvature of a set and contingent epi‐Hessian of a function, related to the contingent curvature of its epigraph. This allows us to go one step further by characterizing functions the epigraphs of which are invariant under systems of stochastic differential equations. We shall show that they are (generalized) solutions to either a system of first‐order Hamilton‐Jacobi equations or to an equivalent system of second‐order Hamilton‐Jacobi equations.     

Malmquist型复差分方程组

近来一些论文里,Ablowitz,Halburd以及Herbst等人应用Nevanlinna理论证明了类似于复微分方程Malmquist定理的复差分方程一些结果.本文主要研究一类复差分方程组的Malmquist定理,推广和改进了他们的一些结论.     

Asymptotic behavior and oscillation of functional differential equations

Asymptotic relations between the solutions of a linear autonomous functional differential equation and the solutions of the corresponding perturbed equation are established. In the scalar case, it is shown that the existence of a nonoscillatory solution of the perturbed equation often implies the existence of a real eigenvalue of the limiting equation. The proofs are based on a recent Perron type theorem for functional differential equations.     

Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform

Dynamical behavior of many nonlinear systems can be described by fractional‐order equations. This study is devoted to fractional differential–difference equations of rational type. Our focus is on the construction of exact solutions by means of the (G'/G)‐expansion method coupled with the so‐called fractional complex transform. The solution procedure is elucidated through two generalized time‐fractional differential–difference equations of rational type. As a result, three types of discrete solutions emerged: hyperbolic, trigonometric, and rational. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of positive periodic solution for variable–coefficient third‐order differential equation with singularity

In this paper, we investigate a class of singular third‐order differential equations with variable coefficients. By application of Green's functions and Schauder's fixed point theorem, sufficient conditions for the existence of positive periodic solutions are established. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiple positive solutions of a class of Schrödinger–Poisson equation involving indefinite nonlinearity in R3

In this paper, we study the existence of multiple positive solutions of Schrödinger–Poisson type equations with indefinite nonlinearity. Our main tool is the mountain pass theorem.     

Reflected backward doubly stochastic differential equations with discontinuous coefficients

In this paper, we study one-dimensional reflected backward doubly stochastic differential equations (RBDSDEs) with one continuous barrier and discontinuous (left or right continuous) generator. We obtain an existence theorem and a comparison theorem for solutions of the class of RBDSDEs.     

Anticipated backward stochastic differential equations driven by the Teugels martingales

In this paper, a class of anticipated backward stochastic differential equations driven by Teugels martingales associated with Lévy process is investigated. We obtain the existence and uniqueness of solutions to these equations by means of the fixed-point theorem. We show that a comparison theorem for this type of ABSDEs also holds under some slight stronger conditions.     

On Solutions of Forward‐Backward Stochastic Differential Equations with Poisson Jumps

Abstract

In this paper, we use a purely probabilistic approach to study forward‐backward differential equations with Poisson jumps with stopping time as termination. Under some weak monotonicity conditions and Lipschitz conditions, the existence and uniqueness results of solutions are obtained, it may be served as the generalized results contrast to FBDE with Brownian motion. We also derive the convergence theorem of the solutions.     

中立型泛函微分方程的周期解

对于中立型泛函微分方程,证明了解的毕竟有界性蕴含周期解的存在性,把常微分方程中著名的Yoshizawa周期解存在定理推广到中立型泛函微分方程,然后利用所得结果给出一类产生于电力系统的中立型时滞泛函微分方程周期解存在惟一与吸引的条件。     

偶数阶非线性微分方程的周期解

This paper deals with even order nonlinear differential equations. Applying the Schauder fixed point theorem and the Schwartz inequality technique, we give a criterion to ensure the existence and uniqueness of periodic solutions. This criterion is a complement of the well known Lazer type results.     

Comparison Method of Stability Analysis of Semilinear Time-Delay Stochastic Evolution Equation

Abstract

In this paper, we set up the comparison theorem between the mild solution of semilinear time-delay stochastic evolution equation with general time-delay variable and the solution of a class (1-dimension) deterministic functional differential equation, by using the Razumikhin–Lyapunov type functional and the theory of functional differential inequalities. By applying this comparison theorem, we give various types of the stability comparison criteria for the semilinear time-delay stochastic evolution equations. With the aid of these comparison criteria, one can reduce the stability analysis of semilinear time-delay stochastic evolution equations in Hilbert space to that of a class (1-dimension) deterministic functional differential equations. Furthermore, these comparison criteria in special case have been applied to derive sufficient conditions for various stability of the mild solution of semilinear time-delay stochastic evolution equations. Finally, the theories are illustrated with some examples.