一类拟线性椭圆方程径向整体正解的分类性质 献给余家荣教授100华诞

本文对如下拟线性方程整体径向正解进行分类研究:{r~(-γ)(r~α|u′|~βu′)′+|u|~(p-1)u=0, 0 r ∞,u(0)=ρ 0, u′(0)=0.这类方程中的微分算子包含了径向函数空间中通常的Laplace算子、m-Laplace算子和k-Hessian算子.本文研究该类方程的任意两个解(包括奇异解)之间的相交和分离的性质,完整地给出各种情形下它们之间的相交数,解决了Miyamoto (2016)未解的一种情形.     

从Hardy空间到Zygmund-型空间的加权微分复合算子

设φ为单位圆盘D上的解析自映射,H(D)表示D上的所有解析函数的集合,u∈H(D).研究了从Hardy空间到Zygmund-型空间及小Zygmund-型空间的加权微分复合算子D_(φ,u)~n,的有界性和紧性,其中n∈N_0.     

某一卷积函数类在Orlicz空间内宽度的精确估计

该文讨论了由实系数线性微分算子定义的2π周期函数类■在Orlicz空间内的宽度问题.得到该函数类在Orlicz空间内的n-K宽度,n-G宽度,n-L宽度,n-B宽度的精确值和相应的极子空间.     

某一周期函数类在Orlicz空间内的n宽度

研究了由实系数线性微分算子定义的周期函数类?rM在Orlicz空间内的宽度估计问题,利用求解变分问题的方法,得到了该函数类在Orlicz空间内的Kolmogorov宽度, Linear宽度, Gelfand宽度以及Bernstein宽度的精确估计,并给出相应的极子空间与最佳线性算子.     

从混合模空间到 Bloch -型空间微分算子与乘子的积

设H(D) 表示单位圆盘D上的解析函数空间,u ∈ H(D). 该文研究了从混合模空间到Bloch -型空间微分算子与乘子的积DM_u 的有界性与紧性.     

关于某一重要函数类的n-K宽度的极子空间

由微分算子和积分算子确定的函数类在分析学中往往被认为是重要函数类,本文研究了由高阶微分算子和N函数M(u)确定的r阶广义函数类Ω_M~r在L_1空间内n-K宽度的极子空间的问题,所得结果比起前人的同类结果具有一定的拓展意义.     

退化拟线性椭圆方程的均匀化

该文获得了下列退化椭圆方程的均匀化结果-div a(x/ε,u,▽u)+g(x/ε,u)=f(x),其中a(y,α,λ)和g(y,α)是变量y的周期函数.     

Sobolev临界增长椭圆方程注

利用空间H10(Ω)的正交分解和极小值原理给出了具临界指数2*的椭圆方程-Δu=λ1u-|u|2*-2u+g(x,u)+h(x)1解的存在性定理,这里次临界项g(x,u)关于u是非线性的,λ1为算子-Δ在H10(Ω)中最小特征值.特别当h≡0时,本文还获得了非零解的存在性结论.     

Zygmund空间上的微分复合算子

讨论Zygmund空间E={f∈H(D):sup_(z∈D)(l-|z|~2)|f″(z)|∞}上的微分复合算子DC_φ,这里C_φ是复合算子,D是微分算子.得到了DC_φ在Zygmund空间E和小Zygmund空间E_0上是有界算子与紧算子的充分必要条件.     

一类非线性椭圆边值问题解的存在性

目前 ,对 s——拉普拉斯算子△s的研究是较为活跃的数学课题 .原因在于算子 -△s与许多物理现象有关 .比如 :反射扩散问题 ,石油提取问题等等 .基于此因 ,在文 [3]的基础上 ,我们将继续研究以下非线性边值问题在 Ls(Ω) ,( 1 2 nn+1 )中解的存在条件 .-△su +g( x,u) =f几乎处处在Ω中-〈 ,| u|s- 2 u〉 =0几乎处处在Γ上其中 f∈Ls( Ω)给定 ,Ω Rn( n 1 ) ,△su=div( | u|s- 2 u) ,g∶Ω× R→ R满足 Caratheodory条件 .本文把文 [3]关于非线性边值问题 @在 Lp( Ω) ( 2 p<+∞ )空间中解的存在性的研究推广到 Ls( Ω) ( 1 2 nn+1 )空间中 .     

Existence of generalized homoclinic solutions of a coupled KdV-type Boussinesq system under a small perturbation

This paper considers the coupled KdV-type Boussinesq system with a small perturbation $u_{xx}=6cv-6u-6uv+\varepsilon f(\varepsilon,u,u_{x},v,v_{x}),$ $ v_{xx}=6cu-6v-3u^{2}+\varepsilon g(\varepsilon,u,u_{x},v,v_{x}),$ where $c=1+\mu$, $\mu>0$ and $\varepsilon$ are small parameters. The linear operator has a pair of real eigenvalues and a pair of purely imaginary eigenvalues. We first change this system into an equivalent system with dimension 4, and then show that its dominant system has a homoclinic solution and the whole system has a periodic solution if the perturbation functions $g$ and $h$ satisfy some conditions. By using the contraction mapping theorem, the perturbation theorem, and the reversibility, we theoretically prove that this homoclinic solution, when higher order terms are added, will persist and exponentially approach to the obtained periodic solution (called generalized homoclinic solution) for small $\varepsilon$ and $\mu>0$.     

有限图上高阶Yamabe型方程的非平凡解

该文研究了以下高阶Yamabe型方程Lm,pu?g|u|^p?2u=λf|u|α?2u在有限图上的非平凡正解的存在性,其中Lm,p是一个2m阶差分算子,它是一种p次(?Δ)^m算子更一般化,α≥p≥2,g>0和f>0是定义在G的所有顶点上的实函数,m≥1是一个整数.     

Approximation and existence of periodic solutions for controlled diffusion equations

This paper concerns the existence of control functions such that a system of controlled stochastic differential equations (SDE) with periodic coefficients has a solution which is periodic in distribution. We show that bounded periodic controls acting in the same direction as the Driving Wiener process will achieve this under some nondegeneracy condition on the diffusion part. The main tool is an approximation theorem for solutions of SDEs which enables one to check certain stability conditions on a more suitable differential equation     

On the existence of more positive solutions of periodic bvps with singularity

We consider Periodic boundary value problems for ordinary second order differential equations of the form u′′=f(t,u,u′), Where f satisfies the (local) Carathéodory conditions and can have a singularity in the second variable.Writing our problem in an operator can be computed on. These sets are not convex, in general. Using the degree theory we get at least one fixed point of the operator at each such set which leads to the existence and localization of more solutions of the related Periodic boundary value problem. Our results are based on the generalized lower and upper functions method from Rach?nková and Tvrdý[15].     

Estimates for the embedding operator of a sobolev space of periodic functions and for the solutions of differential equations with periodic coefficients

We obtain estimates for the embedding operator of a Sobolev space in the space of continuous periodic functions and use them to estimate the solutions of differential equations with periodic coefficients. We prove a theorem on a necessary and sufficient condition for the invertibility of a differential operator with unbounded operator coefficients.     

抽象半线性发展方程正周期解的存在唯一性

讨论有序Banach空间E中半线性发展方程 u′(t)+Au(t)=f(t,u(t)),t∈R, ω-周期解的存在性,其中A为E中正C0-半群的生成元,f:R×E→E连续,关于t 以ω为周期．我们对相应的线性发展方程建立了周期解的存在唯一性,并对周期解算子的谱半径作了精确估计．借助于这个估计,我们用单调迭代方法获得了半线性发展方程正ω-周期解的存在唯一性．     

非线性中立型微分差分方程非振动解的渐近性

考虑非线性中立型微分差分方程［ｙ（ｔ）＋Ｐ（ｔ）ｇ（ｙ（ｔ－τ）］′＋Ｑ（ｔ）ｈ（ｙ（ｔ－σ））＝０（１）的非振动解的渐近性．若无特别申明，本文总假设Ａ函数Ｐ（ｔ），Ｑ（ｔ），ｇ（ｕ），ｈ（ｕ）皆为连续函数；ＢＱ（ｔ）＞０；ｕｇ（ｕ）＞０，ｕｈ（ｕ）＞０（ｕ≠０）；Ｃｇ（ｕ）＝ｈ（ｕ）＝０当且仅当ｕ＝０．     

含反馈控制和参数的非线性泛函微分系统的正周期解的分析

我们研究下面具有反馈控制和参数的非线性微分系统的正周期解的存在性与不存在性：{dx/dt=-r（t）x（t）＋λF（t,xt,u（t-δ（t）））,du/dt=-h（t）u（t）＋g（t）x（t-σ（t））.在一定条件下通过应用Leggett．Williams不动点定理,证明该系统至少有三个正周期解;在另外的条件下,通过用反证法证明了该系统的正周期解不存在．     

Characteristic functions of almost solvable extensions of Hermitian operators

The characteristic operator-functions W() are studied of the almost solvable extensions of an Hermitian operator. The inverse problem is solved, a multiplication theorem is proved, and a formula is derived expressing W() in terms of the Weyl function and the boundary operator. Characteristic functions are computed of various differential and difference operators, with the help of which are proved theorems of the completeness of the systems of proper and adjoint vectors.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 435–459, April, 1992.     

一类三维微分系统正值解的结构(英文)

研究三维微分系统:u′1=a1(t)|u2|λ1sgn u2,u′2=a2(t)|u3|λ2sgn u3,u′3=-a3(t)|u1|λ3sgn u1.假设λi(i=1,2,3)是正的常数,ai(t)(i=1,2,3)在区间[0,∞)上是正的连续函数,根据u的分量ui的特殊渐近条件定义了正值解的几种类型。系统满足条件∫0∞ai(t)dt=∞,i=1,2.