单位圆上一类高阶线性微分方程解的性质

结合微分方程理论和函数空间理论,研究了单位圆上一类特殊高阶线性微分方程解的性质,得到当方程系数满足某些条件时,其解属于某类函数空间的充分条件.     

多圆盘的Bergman空间上Hankel乘积

对多圆盘上的平方可积函数f和g,研究了Bergman空间上稠密定义的Hankel乘积H_fH_g~*的有界性和紧性.给出了这些算子有界和紧的一些必要条件和充分条件.当f是解析函数时,对混合Haplitz乘积H_gT_(f~-)和T_fH_g~*得到了相似的结果.     

Lipschitz空间上的加权复合算子

设D是复空间C中的单位圆盘,ψ是D到自身的一个全纯映射,ψ(z)是D上的全纯函数,0＜α＜1.本文给出了单位圆盘中Lipschitz空间Lipa(D)上由ψ和ψ诱导的加权复合算子Wψ,ψ的有界性及紧性的充要条件.     

多圆盘的Bergman空间上Hankel乘积

对多圆盘上的平方可积函数f和g,研究了Bergman空间上稠密定义的Hankel乘积HfH*g的有界性和紧性.给出了这些算子有界和紧的一些必要条件和充分条件.当f是解析函数时,对混合Haplitz乘积HgT-f和TfH*g得到了相似的结果.     

算子方程X+A~*X~(-t)A=Q的正算子解的研究

研究了算子方程X+A*X~(-t)A=Q(t>1)的正算子解问题,分别给出了算子方程X+A*X~(-t)A=Q有正算子解的一些充分条件和必要条件,并确定了解的范围,用迭代的方法得到方程的正算子解.     

Bergman空间上Cowen-Douglas算子的换位

令H~∞(D)表示单位圆盘D上的有界解析函数全体构成的代数,对于■∈H~∞(D),在某种条件下,证明T=M■是一个指标为n的Cowen-Douglaus算子,并且给出了■(T)/rad■(T)可交换的一个充分条件.当n=1时,刻画了T的换位.     

Bergman空间上一类解析Toeplitz算子的强不可约性

令D为单位圆盘D={z∈C:|z|<1},L_a~2(D)为L~2(D)中解析函数构成的Bergman空间.设f(z)=a_0+a_1z+a_2z~2+…,用算子理论的技巧给出解析Toeplitz算子T_f为强不可约算子的一个充分条件.     

一类不动点定理及其在微分方程中的应用

在完备的距离空间中,建立了一类含有广义变距离函数的压缩映射不动点定理.将这一结果应用于一阶常微分方程周期边值问题中,得到了方程存在唯一解的充分条件.我们的结果比已有结果的条件更弱且完全去掉了下解存在的条件.     

一类非线性算子方程的正解问题

研究算子方程X^s+A^*X^-tA=Q的正算子解的存在性问题,通过构造有效的迭代序列,给出了算子方程X^s+A^*X^-tA=Q有正算子解的一些充分条件和必要条件,同时给出了该方程有极大解和唯一解的条件.     

多圆盘上的本质交换对偶Toeplitz算子

在单位多圆盘的Bergman空间上,本文分别刻画了以有界可测函数和有界多重调和函数为符号的本质交换对偶Toeplitz算子.     

The Topoligical Sphere Theorem for Complete Submanifolds

A topological sphere theorem is obtained from the point of view of submanifold geometry. An important scalar is defined by the mean curvature and the squared norm of the second fundamental form of an oriented complete submanifold Mn in a space form of nonnegative sectional curvature. If the infimum of this scalar is negative, we then prove that the Ricci curvature of Mn has a positive lower bound. Making use of the Lawson–Simons formula for the nonexistence of stable k-currents, we eliminate Hk (Mn, Z) for all 1 ` k ` n – 1.We then observe that the fundamental group of Mn is trivial. It should be emphasized that our result is optimal.     

Some results on the occurrence of compact minimal submanifolds

Varying the situation considered in Myers theorem, we show, via standard index form techniques, that a complete Riemannian manifold which admits a compact minimal submanifold is necessarily compact, provided a suitable curvature object is positive on the average along the geodesies issuing orthogonally from the minimal submanifold. By slightly recasting this result, one establishes the nonexistence of compact minimal submanifolds (in particular, closed geodesies) in complete noncompact manifolds which obey an appropriate curvature condition. A generalization of a result of Tipler concerning the occurrence of zeros of solutions to the scalar Jacobi equation is also obtained.     

Entire radial solutions of elliptic systems and inequalities of the mean curvature type

In this paper we study first nonexistence of radial entire solutions of elliptic systems of the mean curvature type with a singular or degenerate diffusion depending on the solution u. In particular we extend a previous result given in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389]. Moreover, in the scalar case we obtain nonexistence of all entire solutions, radial or not, of differential inequalities involving again operators of the mean curvature type and a diffusion term. We prove that in the scalar case, nonexistence of entire solutions is due to the explosion of the derivative of every nonglobal radial solution in the right extremum of the maximal interval of existence, while in that point the solution is bounded. This behavior is qualitatively different with respect to what happens for the m-Laplacian operator, studied in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389], where nonexistence of entire solutions is due, even in the vectorial case, to the explosion in norm of the solution at a finite point. Our nonexistence theorems for inequalities extend previous results given by Naito and Usami in [Y. Naito, H. Usami, Entire solutions of the inequality div(A(|Du|)Du)?f(u), Math. Z. 225 (1997) 167-175] and Ghergu and Radulescu in [M. Ghergu, V. Radulescu, Existence and nonexistence of entire solutions to the logistic differential equation, Abstr. Appl. Anal. 17 (2003) 995-1003].     

Global nonexistence of positive initial energy solution for coupled quasilinear system

In this paper, we consider global nonexistence of a solution for coupled quasilinear system with damping and source under Dirichlet boundary condition. We obtain a global nonexistence result of solution by using the perturbed energy method, where the initial energy is assumed to be positive. Copyright © 2017 John Wiley & Sons, Ltd.     

非线性微分方程正解的不存在性

本文研究了非线性微分方程正解不存在性     

An optimal parabolic estimate and its applications in prescribing scalar curvature on some open manifolds with Ricci \ge 0

By establishing an optimal comparison result on the heat kernel of the conformal Laplacian on open manifolds with nonnegative Ricci curvature, (a) we show that many manifolds with positive scalar curvature do not possess conformal metrics with scalar curvature bounded below by a positive constant; (b) we identify a class of functions with the following property: If the manifold has a scalar curvature in this class, then there exists a complete conformal metric whose scalar curvature is any given function in this class. This class is optimal in some sense; (c) we have identified all manifolds with nonnegative Ricci curvature, which are “uniformly” conformal to manifolds with zero scalar curvature. Even in the Euclidean case, we obtain a necessary and sufficient condition under which the main existence results in [Ni1] and [KN] on prescribing nonnegative scalar curvature will hold. This condition had been sought in several papers in the last two decades. Received: 11 November 1998 / Revised: 7 April 1999     

非强制混合向量变分不等式解的存在性研究

本文利用例外簇方法研究非强制混合向量变分不等式的弱有效解的存在性:首先证明若混合向量变分不等式问题不存在例外簇,则混合向量变分不等式问题的弱有效解集为非空集合:利用向量值映射的渐近映射给出自反Banach空间中非强制混合向量变分不等式的弱有效解集不存在例外簇的充分条件,从而得到混合向量变分不等式问题的弱有效解的存在性结果;我们研究了当算子为余正仿射算子时,给出混合仿射向量变分不等式不存在例外簇的充分条件,得到混合仿射向量变分不等式弱有效解的存在性,给出了混合仿射向量变分不等式的弱有效解集为非空紧致集的充分条件.将Iusem等人(2019)在有限维空间中标量混合变分不等式解的存在性结果推广到自反Banach空间中混合向量变分不等式.     

Existence results for solutions of mixed tensor variational inequalities

By employing the notion of exceptional family of elements, we establish existence results for the mixed tensor variational inequalities. We show that the nonexistence of an exceptional family of elements is a sufficient condition for the solvability of mixed tensor variational inequality. For positive semidefinite mixed tensor variational inequalities, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the nonemptiness of the solution sets. We derive several sufficient conditions of the nonemptiness and compactness of the solution sets for the mixed tensor variational inequalities with some special structured tensors. Finally, we show that the mixed tensor variational inequalities can be defined as a class of convex optimization problems.

     

加权Dirichlet空间上的复合算子

本文研究复平面中单位圆盘的加权Ｄｉｒｉｃｈｌｅｔ空间上的复合算子Ｃψ。利用Ｃａｒｌｅｓｏｎ测度的概念，给出了Ｃｐ为有界算子，紧算子的充要条件。同时，对复合算子Ｃｐ为Ｓｃｈａｔｔｅｎ Ｐ－类算子时，函数ψ应满足的条件作了讨论，给出了积分形式的刻划。     

A necessary and sufficient condition for the nirenberg problem

We seek metrics conformal to the standard ones on Sⁿ having prescribed Gaussian curvature in case n = 2 (the Nirenberg Problem), or prescribed scalar curvature for n ≧ 3 (the Kazdan-Warner problem). There are well-known Kazdan-Warner and Bourguignon-Ezin necessary conditions for a function R(x) to be the scalar curvature of some conformally related metric. Are those necessary conditions also sufficient? This problem has been open for many years. In a previous paper, we answered the question negatively by providing a family of counter examples. In this paper, we obtain much stronger results. We show that, in all dimensions, if R(x) is rotationally symmetric and monotone in the region where it is positive, then the problem has no solution at all. It follows that, on S², for a non-degenerate, rotationally symmetric function R(θ), a necessary and sufficient condition for the problem to have a solution is that Rθ changes signs in the region where it is positive. This condition, however, is still not sufficient to guarantee the existence of a rotationally symmetric solution, as will be shown in this paper. We also consider similar necessary conditions for non-symmetric functions. ©1995 John Wiley & Sons, Inc.