一类n维中立型泛函微分方程的周期解

通过对NFDE周期系统；周期解的讨论，给出了其周期解界的估计式，结合不动点原理研究了下列系统：周期解的存在性，唯一性等问题，得到一些新的结果．     

高维系统周期解的共振分支

本文利用Liapunov－Schmidt方法获得了高维自治系统在共振情况下决定周期解个数的分支函数，并通过计算分支函数的主项，分析分支函数的零点，研究了具两对共轭特征根的四维系统多个周期解的共振分支问题．     

竞争扩散系统周期解的唯一性

本文讨论了一类周期竞争扩散系统，给出了诺依曼边界条件下周期竞争扩散系统的周期解唯一的充分条件，并讨论了对应周期竞争扩散系统初边值问题解的渐近性态．     

具有Beddington-DeAnglis功能反应和脉冲输入的恒化器模型

本文研究具有周期脉冲输入营养基和Beddington-DeAnglis功能反应捕食一食饵系统．通过分析营养基和食饵的子系统，获得系统的边界周期解．对边界周期解稳定性的分析，得到了捕食者侵入的阈值．     

一类积分微分方程系统正周期解的存在性

本文利用重合度理论讨论了一类积分微分方程所描述的多物种生态竞争系统正周期解的存在性,得到了保证周期解存在的充分条件．     

反应扩散方程解的渐近性态

本文使用锥映象不动点指数的计算方法，讨论一类反应扩散方程正静态解的存在性，并给出方程的静态解渐适性态．然后，利用上，下解的方法讨论相应周期系统周期解的存在性及其渐近性态．     

中立型Lotka-Volterra系统正周期解的存在性

利用迭合度理论的连续定理，讨论了一类中立型系统的正周期解的存在性．得到了正周期解存在的一些充分条件．     

一类非齐次时滞微分方程概周期解的存在性

该文研究一类非齐次时滞微分方程的概周期解，结合运用指数二分法，给出了系统存在概周期解的一组充分条件．     

一类非线性时滞系统概周期解的存在唯一性

应用Liapunov，泛函，研究了一类时滞系统概周期解的存在唯一性，得到了保证系统存在唯一概周期解的一组充分条件．     

一类具有无穷多个周期解的二阶偏微分方程

本文首先给出了一类具有无穷多个周期解的无阻尼二阶线性偏微分方程所描述的系统。同时讨论了一类无阻尼非线性二阶偏微分方程存在多个周期解的情况，最后给出了一个判断有阻尼二阶偏微分系统存在周期解的方法。     

Exact solutions of one-dimensional nonlinear shallow water equations over even and sloping bottoms

We establish an equivalence of two systems of equations of one-dimensional shallow water models describing the propagation of surface waves over even and sloping bottoms. For each of these systems, we obtain formulas for the general form of their nondegenerate solutions, which are expressible in terms of solutions of the Darboux equation. The invariant solutions of the Darboux equation that we find are simplest representatives of its essentially different exact solutions (those not related by invertible point transformations). They depend on 21 arbitrary real constants; after “proliferation” formulas derived by methods of group theory analysis are applied, they generate a 27-parameter family of essentially different exact solutions. Subsequently using the derived infinitesimal “proliferation” formulas for the solutions in this family generates a denumerable set of exact solutions, whose linear span constitutes an infinite-dimensional vector space of solutions of the Darboux equation. This vector space of solutions of the Darboux equation and the general formulas for nondegenerate solutions of systems of shallow water equations with even and sloping bottoms give an infinite set of their solutions. The “proliferation” formulas for these systems determine their additional nondegenerate solutions. We also find all degenerate solutions of these systems and thus construct a database of an infinite set of exact solutions of systems of equations of the one-dimensional nonlinear shallow water model with even and sloping bottoms.     

Existence of periodic solutions of sublinear Hamiltonian systems

The variational method is used to obtain some existence theorems of periodic solutions of sublinear systems with or not with impacts under suitable growth conditions. Compared with normal systems, impact systems need additional conditions to ensure the existence of periodic bouncing solutions.     

EXISTENCE OF NONNEGATIVE SOLUTIONS FOR DEGENERATE ECOLOGICAL MODELS

In this paper,nonnegative solutions for the degenerate elliptic systems are considered.First,nonnegative solutions for scalar equation with spatial discontinuities are studied. Then themethod developed for scalar equation is applied to study elliptic systems. At last,the existence criteria of nonnegative solutions of elliptic systems are given.     

Equations of Advanced–Retarded Type and Solutions of Traveling-Wave Type for Infinite-Dimensional Dynamic Systems

In the paper we study infinite-dimensional dynamic systems with the Frenkel–Kontorova potentials. For such systems we describe their traveling-wave-type solutions, which are solutions for the corresponding boundary-value problem with nonlocal conditions. Describing the mentioned solutions is equivalent to describing the space of solutions for a functional differential equation that can be canonically derived from the original dynamic system. The stability of traveling-wave-type solutions is also investigated.     

Hodograph Transformations and Cauchy Problem to Systems of Nonlinear Parabolic Equations

In this paper, we provide a method to solve the Cauchy problem of systems of quasi‐linear parabolic equations, such systems can be transformed to the systems of linear parabolic equations with variable coefficients via the hodograph transformations. Our approach to solve the linear systems with variable coefficients is to use their fundamental solutions, which are constructed by using the Lie's symmetry method. In consequence, we can derive explicit solutions to the Cauchy problem of the quasi‐linear systems in terms of the solutions of the linear systems and the hodograph transformations relating to the quasi‐linear and the linear systems.     

Blow-up solutions of cubic differential systems

We investigate the existence problem for blow-up solutions of cubic differential systems. We find sets of initial values of the blow-up solutions. We also discuss a method of finding upper estimates for the blow-up time of these solutions. Our approach can be applied to systems of partial differential equations. We apply this approach to the Cauchy-Dirichlet problem for systems of semilinear heat equations with cubic nonlinearities.     

System of vector quasi-variational inclusions with some applications

In this paper, we consider systems of vector quasi-variational inclusions which include systems of vector quasi-equilibrium problems for multivalued maps, systems of vector optimization problems and several other systems as special cases. We establish existence results for solutions of these systems. As applications of our results, we derive the existence results for solutions of system vector optimization problems, mathematical programs with systems of vector variational inclusion constraints and bilevel problems. Another application of our results provides the common fixed point theorem for a family of lower semicontinuous multivalued maps. Further applications of our results for existence of solutions of systems of vector quasi-variational inclusions are given to prove the existence of solutions of systems of Minty type and Stampacchia type generalized implicit quasi-variational inequalities. The results of this paper can be seen as extensions and generalizations of several known results in the literature.     

On a Class of Systems of Linear and Nonlinear Fredholm Integral Equations of the Third Kind with Multipoint Singularities

Based on a new approach, we show that finding solutions for a class of systems of linear (respectively, nonlinear) Fredholm integral equations of the third kind with multipoint singularities is equivalent to finding solutions of systems of linear (respectively, nonlinear) Fredholm integral equations of the second kind with additional conditions. We study the existence, nonexistence, uniqueness, and nonuniqueness of solutions for this class of systems of Fredholm integral equations of the third kind with multipoint singularities.     

On Asymptotic Properties of Solutions to Weakly Nonlinear Systems in the Neighborhood of a Singular Point

In this paper, we consider the class of so-called weakly hyperbolic linear systems, which includes correct and hyperbolic (exponentially-dichotomous) systems. We prove new results generalizing related classical theorems on the conditional stability of solutions. We establish the existence of stable manifolds consisting of points corresponding to the solutions of such systems with negative Lyapunov exponents. We study the behavior of solutions starting outside the stable manifold. The technique used in our paper is similar to that in the theory of hyperbolic systems.     

Invariant subspaces and exact solutions of a class of dispersive evolution equations

The invariant subspace method is used to classify a class of systems of nonlinear dispersive evolution equations and determine their invariant subspaces and exact solutions. A crucial step is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that systems of evolution equations admit. A few examples of presenting exact solutions with generalized separated variables illustrate the effectiveness of the invariant subspace method in solving systems of nonlinear evolution equations.