A note on the Cauchy problem of the generalized Davey–Stewartson equations

In this note, we establish some local and global existence results for the Cauchy problem of a class of nonlinear dispersive equations which generalize the nonlinear Schrödinger equations and the Davey–Stewartson equations. These results improve some previously obtained results by some other authors when they are restricted to certain special equations.     

On entire solutions of two types of systems of complex differential-difference equations

In this paper, we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations, and obtain some interesting results. It extends some results concerning complex differential (difference) equations to the systems of differential-difference equations.     

Hardy–Sobolev inequalities in half-space and some semilinear elliptic equations with singular coefficients

We study some semilinear elliptic equations with singular coefficients which relate to some Hardy–Sobolev inequalities. We obtain some existence results for these equations and give a theorem for prescribing the Palais–Smale sequence for these equations. Moreover, we find some interesting connections between these equations and some semilinear elliptic equations in hyperbolic space. Using these connections, we obtain many new results for these equations.     

一类复差分方程组的亚纯解

许多作者研究了复差分方程解的存在性及增长性问题,得到了较多理想的结果.本文利用亚纯函数Nevanlinna值分布理论,研究了一类复高阶非线性差分方程解的表达式问题,将复差分方程的一结果推广至复差分方程组中.     

高阶变系数泛函方程解的振动性

讨论了高阶变系数泛函微分方程解的振动性,并给出了这类高阶变系数函数方程解的若干新振动准则.结果推广了目前已有的某些结果.并且给出了在差分方程中的应用.     

一类非单调算子方程解的存在性定理

运用锥理论与非对称迭代方法,得到了Banach空间不具有单调性、连续性和紧性条件的一类算子方程解的存在唯一性,并给出了迭代序列收敛于解的误差估计.所得结果改进和推广了增(减)算子方程的某些已知结果.     

Asymptotic behavior of solutions of functional difference equations

For linear functional difference equations, we obtain some results on the asymptotic behavior of solutions, which correspond to a Perron-type theorem for linear ordinary difference equations. We also apply our results to Volterra difference equations with infinite delay.     

Oscillation of Third-order Delay Dynamic Equations on Time Scales

This paper is concerned with the oscillatory behavior of a class of third-order noonlinear variable delay neutral functional dynamic equations on time scale. By using the generalized Riccati transformation and inequality technique, we establish some new oscilla- tion criteria for the equations. Our results extend and improve some known results, but also unify the oscillation of third-order nonlinear variable delay functional differential equations and functional difference equations with a nonlinear neutral term. Some examples are given to illustrate the importance of our results.     

On the Equivalence of Differential Equations

In this article we use the reflecting function of Mironenko to study some complicated differential equations which are equivalent to the Riccati equation and some polynomial differential equations. The results are applied to discussion of the qualitative behavior of periodic solutions of these complicated differential equations.     

Banach空间中一类非单调算子方程解的存在性定理

运用锥理论与非对称迭代方法,得到了Banach空间不具有单调性、连续性和紧性条件的一类算子的不动点的存在唯一性,并给出了迭代序列收敛于解的误差估计,所得结果改进和推广了增(减)算子方程的某些已知结果.     

Generalized equations and the generalized Newton method

We give some convergence results on the generalized Newton method (referred to by some authors as Newton's method) and the chord method when applied to generalized equations. The main results of the paper extend the classical Kantorovich results on Newton's method to (nonsmooth) generalized equations. Our results also extend earlier results on nonsmooth equations due to Eaves, Robinson, Josephy, Pang and Chan. We also propose inner-iterative schemes for the computation of the generalized Newton iterates. These schemes generalize popular iterative methods (Richardson's method, Jacobi's method and the Gauss-Seidel method) for the solution of linear equations and linear complementarity problems and are shown to be convergent under natural generalizations of classical convergence criteria. Our results are applicable to equations involving single-valued functions and also to a class of generalized equations which includes variational inequalities, nonlinear complementarity problems and some nonsmooth convex minimization problems.     

Positive almost automorphic solutions for a class of non-linear delay integral equations

This article is concerned with a class of non-linear delay integral equations, which unify some extensively studied delay integral equations. We establish a new existence and uniqueness theorem about positive almost automorphic solutions of the delay integral equations. Our theorem can deal with some cases to which many known results are not applicable. Two examples are given to illustrate our results.     

ON MEROMORPHIC SOLUTIONS OF A TYPE OF SYSTEM OF COMPOSITE FUNCTIONAL EQUATIONS

In this article,we mainly investigate the growth and existence of meromorphic solutions of a type of systems of composite functional equations,and obtain some interesting results.It extends some result...     

Interpolation of nonlinear partial differential operators and generation of differentiale evolutions

By interpolating between Sobolev spaces we find that many partial differential operators become continuous when restricted to a sufficiently small domain. Hence some techniques from the theory of ordinary differential equations can be applied to some p.d.e.'s. Using these ideas, we study a class of nonlinear evolutions in a Banach space. We obtain some very simple existence and continuous dependence results. The theory is applicable to reaction-diffusion equations, dispersion equations, and hyperbolic equations before shocks develop.     

复微分-差分方程组的超越整函数解

利用亚纯函数的Nevanlinna值分布理论, 我们主要研究了一类复微分-差分方程和一类复微分-差分方程组的有限级超越整函数解的存在形式, 得到两个有趣的结论. 将复微分(差分)方程的一些结论推广到复微分-差分方程(组)中.     

长水波近似方程组的新精确解

依据齐次平衡法的思想 ,首先提出了求非线性发展方程精确解的新思路 ,这种方法通过改变待定函数的次序 ,优势是使求解的复杂计算得到简化 .应用本文的思路 ,可得到某些非线性偏微分方程的新解 .其次我们给出了长水波近似方程组的一些新精确解 ,其中包括椭圆周期解 ,我们推广了有关长波近似方程的已有结果 .     

A class of discrete time generalized Riccati equations

In this paper, a class of discrete-time backward non-linear equations defined on some ordered Hilbert spaces of symmetric matrices is considered. The problem of the existence of some global solutions is investigated. The class of considered discrete-time non-linear equations contains, as special cases, a great number of difference Riccati equations both from the deterministic and the stochastic framework. The results proved in the paper provide the sets of necessary and sufficient conditions that guarantee the existence of some special solutions of the considered equations as: the maximal solution, the stabilizing solution and the minimal positive semi-definite solution. These conditions are expressed in terms of the feasibility of some suitable systems of linear matrix inequalities (LMI). One shows that in the case of the equations with periodic coefficients to verify the conditions that guarantee the existence of the maximal or the stabilizing solution, we have to check the solvability of some systems of LMI with a finite number of inequations. The proofs are based on some suitable properties of discrete-time linear equations defined by the positive operators on some ordered Hilbert spaces chosen adequately. The results derived in this paper provide useful conditions that guarantee the existence of the maximal solution or the stabilizing solution for different classes of difference matrix Riccati equations involved in many problems of robust control both in the deterministic and the stochastic framework. The proofs are deterministic and are accessible to the readers less familiarized with the stochastic reasonings.     

一类非单调二元算子方程解的Mann迭代序列的收敛性

利用Mann迭代技巧,讨论了不具有连续性和紧性条件的非单调二元算子方程解的存在唯一性,并给出了迭代序列收敛于解的误差估计,所得结果是某些已知结果本质改进和推广.     

两类复微分-差分方程组的整函数解

利用Nevanlinna值分布理论以及复差分和复微分理论,讨论了两类复微分-差分方程组的有限级超越整函数解问题,得到了两个结果,并将涉及微分或差分方程的某些结果推广至复微分-差分方程组中.     

Almost automorphic solutions to some classes of partial evolution equations

We give in this work some sufficient conditions for the existence and uniqueness of almost automorphic (mild) solutions to some classes of partial evolution equations. Then we use our abstract results to discuss the existence and uniqueness of almost automorphic solutions to some partial differential equations.