由Lévy过程驱动的倒向随机微分方程在局部Bihari条件下解的存在唯一性

本文利用推广的Bihari不等式和截断函数,证明了由Lévy过程驱动的倒向随机微分方程在局部Bihari条件下解的存在唯一性。我们先给出在某种较弱的条件下,方程在局部区间[T0,T]上解的存在唯一性,然后加强条件,得到解的全局存在唯一性,从而推广了周和秦的结论。     

具有非Lipschitz和非增长条件的带跳倒向随机微分方程

本文研究一类带Poisson跳的倒向随机微分方程。在方程的系数满足非增长条件和非Lipschitz条件下,讨论方程适应解的存在唯一性和稳定性。为了证明解的存在性,首先通过函数变换,构造出一逼近序列,然后运用推广的Bihari不等式和Lebesgue控制收敛定理证明该逼近序列是收敛的,得到逼近序列的极限就是方程的适应解。解的唯一性和稳定性主要运用了Bihari不等式和推广的Bihari不等式来进行证明。     

一类反射型非线性倒向随机微分方程的适应解

本文研究了反射型非线性倒向随机微分方程yt=ξ ∫Ttf(s,ys,zs)ds-∫Ttg(s,ys,zs)dws KT-Kt,t∈[0,T],在非Lipschitz条件下,给出了其解的存在唯一性定理.文中所使用的主要方法是罚则函数法,主要工具是Bihari不等式的一个推广形式及凸函数次微分算子的Yosida逼近.     

一类非Lipschitz条件的Backward SDE适应解的存在唯一性

本文中，我们在非Lipschitz条件下证明了倒向随机微分方程的局部与整体适应解的存在唯一性，推广了PaLrdoux-Peng定理．     

等熵可压Navier-Stokes-Poisson方程局部强解适定性

该文得到了三维情形等熵可压Navier-Stokes-Poisson方程局部强解的存在性、唯一性及稳定性. 重要的是,该文允许初始密度真空的存在. 首先用推广形式的Gronwall不等式得到了强解的局部存在性,然后得到了较弱条件下的唯一性,在证明唯一性的同时得到了稳定性.     

带Poisson跳随机微分方程终值与边值问题的适应解

Poisson跳的拟线性倒向随机微分方程x(t) ∫tf(s,x(s),,x(s)) y(s)]dMs =ξ,t∈[0,1],这里M = (W,Q)T,其中W为Wiener过程,Q为补偿Poisson过程.利用区间延拓和 Bihari 不等式证明了在某种弱于Lipschitz条件下方程存在唯一适应解,并给出了解的估计,从而将文章[1]的结论推广到带 Poission 跳的情形.另外,本文还讨论了以下形式的边值问题:dx(t) = f(t,x(t),y(t))dt y(t)dMt,Ax(0) Bx(1) =ξ*,t∈[0,1],并证明了在Lipschitz条件下适应解的存在唯一性.     

非混合单调二元算子方程(组)解的存在唯一性

利用锥理论及Banach压缩映射原理,在不要求上、下解条件及算子紧性与连续性的条件下,建立了一类满足更一般序关系条件的非混合单调二元算子方程组(?)解的存在唯一性定理,以及非单调二元算子方程T(x,x)=x和非单调一元算子方程Lx=x解的存在唯一性定理,推广了最近相关文献的研究结果.     

更弱条件下随机微分方程解的存在性

（１）中的Ｌｉｐｓｃｈｚ条件下，证明了形如下列方程Ｘ＝Ф（Ｘ）＋Ｆ（Ｘ）．Ｍ的解的存在性和唯一性；（２）在局部Ｌｉｐｓｃｈｔｚ条件下，证明了上述方程解的存在性和唯一性；然而在实际应用中，有许多随机微分方程不满足Ｌｉｐｓｃｈｔｚ条件，但解存在却不唯一（见（５）中例子）本文利用非紧致度在更弱条件下证明（＊）至少存在一个解，从面推广了（１），（２），（３）中的存在性定理。     

一个非局部方程解的存在性

本文旨在讨论一个线性非局部方程和相应非线性方程解存在的条件，对非局部积分核作一定限制后，得到在积分核变号条件下解的存在唯一性结果。     

局部Lipschitz条件下的布朗运动和泊松过程混合驱动的正倒向随机微分方程

本文得到在局部Lipschiz条件下的布朗运动和泊松过程混合驱动的倒向随机微分方程的存在唯一性；同时也证明了布朗运动和泊松过程混合驱动的完全藕合的正倒向随机微分方程在局部Lipschitz条件下的解的存在唯一性。     

具有无限时滞的随机泛函微分方程的解的唯一存在性

本文主要证明了在相空间（B）中具有无限时滞随机泛函微分方程解的唯一存在性.推广了文献[2]中的相空间,并且给出了一些相空间存在的例子.另外,本文建立了一个Banach空间（M）^2t0（（-∞,T],Rd）依范数‖·‖,并在这个空间上讨论了具有无限时滞随机泛函微分方程的解的唯一存在性.     

Global exponential stability and periodicity of reaction–diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions

In this paper, global exponential stability and periodicity of a class of reaction–diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions are studied by constructing suitable Lyapunov functionals and utilizing some inequality techniques. We first prove global exponential convergence to 0 of the difference between any two solutions of the original neural networks, the existence and uniqueness of equilibrium is the direct results of this procedure. This approach is different from the usually used one where the existence, uniqueness of equilibrium and stability are proved in two separate steps. Secondly, we prove periodicity. Sufficient conditions ensuring the existence, uniqueness, and global exponential stability of the equilibrium and periodic solution are given. These conditions are easy to verify and our results play an important role in the design and application of globally exponentially stable neural circuits and periodic oscillatory neural circuits.     

On the Maxwell system under impedance boundary conditions with memory

We consider an initial boundary value problem for the system of the Maxwell equations in a bounded domain with smooth boundary on a finite time interval with new boundary conditions with memory. In appropriate function spaces, we define and study the nonselfadjoint operator that is generated by the Maxwell operator under a boundary condition with memory. Using the operator method, we prove an existence and uniqueness theorem for a solution to the initial boundary value problem.     

Stability and monotonicity of a conservative difference scheme for a multidimensional nonlinear scalar conservation law

We prove the existence, uniqueness, and monotonicity of the solution of an upwind conservative explicit difference scheme approximating an initial-boundary value problem for a many-dimensional nonlinear scalar conservation law with a quadratic nonlinearity under some specific conditions imposed only on the input data of the problem. We show that the resulting solution is not necessarily stable. Under some additional conditions on the input data, which provide the absence of shock waves, we prove the stability of the unique solution of the difference scheme for any finite time.     

Recovery of the unknown coefficients in a two-dimensional hyperbolic equation

In this paper, an inverse boundary value problem for a two-dimensional hyperbolic equation with overdetermination conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence to the original problem in a certain sense. We then use the Fourier method to reduce such an equivalent problem to a system of integral equations. Furthermore, we prove the existence and uniqueness theorem for the auxiliary problem by the contraction mappings principle. Based on the equivalency of these problems, the existence and uniqueness theorem for the classical solution of the original inverse problem is proved. Some discussions on the numerical solutions for this inverse problem are presented including some numerical examples.     

On nonlocal solutions of semilinear equations of the Sobolev type

We prove the existence and uniqueness of a classical solution of the Cauchy problem and the Showalter problem for some semilinear equations of the Sobolev type on a given interval. The abstract results are used to study initial-boundary value problems for some systems of equations unsolvable for the time derivative.     

A Functional Equation Whose Unknown is \mathcal{P}([0,1]) Valued

We study a functional equation whose unknown maps a Euclidean space into the space of probability distributions on [0,1]. We prove existence and uniqueness of its solution under suitable regularity and boundary conditions, we show that it depends continuously on the boundary datum, and we characterize solutions that are diffuse on [0,1]. A canonical solution is obtained by means of a Randomly Reinforced Urn with different reinforcement distributions having equal means. The general solution to the functional equation defines a new parametric collection of distributions on [0,1] generalizing the Beta family.     

Local Solutions to the Navier–Stokes Equations with Mixed Boundary Conditions

We study the existence and uniqueness of solutions to the system of three-dimensionalNavier-Stokes equations and the continuity equation for incompressible fluid with mixedboundary conditions. It is known that if the Dirichlet boundary conditions are prescribed on thewhole boundary and the total influx equals zero, then weak solutions exist globally in time andthey are even unique and smooth in the case of two-dimensional domains. The methods that havebeen used to prove these results fail if non-Dirichlet conditions are applied on a part of theboundary, since there is then no control over the energy flux on this part of the boundary. In thispaper, we prove the existence and the uniqueness of solutions on a (short) time interval. Theproof is performed for Lipschitz domains and a wide class of initial data. The length of the timeinterval on which the solution exists depends only on certain norms of the data.     

Initial-boundary value problem for the nonlinear string-beam system

This paper deals with a nonlinear string-beam system describing the torsional-vertical oscillations of a suspension bridge. We consider the initial-boundary value problem and study the existence and uniqueness question. We assume time independent right hand sides, but allow quite general nonlinear terms. Using the Faedo-Galerkin method we prove the existence of a unique solution on an arbitrary large time interval.     

On the Cauchy problem for the wave equation on time-dependent domains

We introduce a notion of solution to the wave equation on a suitable class of time-dependent domains and compare it with a previous definition. We prove an existence result for the solution of the Cauchy problem and present some additional conditions which imply uniqueness.