RIESZ IDEMPOTENT AND BROWDER＇S THEOREM FOR ABSOLUTE-（p,r ）-PARANORMAL OPERATORS

An operator T is said to be paranormal if ||T 2x|| ≥ ||T x||2 holds for every unit vector x.Several extensions of paranormal operators are considered until now,for example absolute-k-paranormal and p-paranormal introduced in [10],[14],respectively.Yamazaki and Yanagida [38] introduced the class of absolute-(p,r)-paranormal operators as a further generalization of the classes of both absolute-k-paranormal and p-paranormal operators.An operator T ∈ B(H) is called absolute-(p,r)-paranormal operator if |||T |p|T |r x||r ≥ |||T |rx||p+r for every unit vector x ∈ H and for positive real numbers p > 0 and r > 0.The famous result of Browder,that self adjoint operators satisfy Browder’s theorem,is extended to several classes of operators.In this paper we show that for any absolute-(p,r)paranormal operator T,T satisfies Browder’s theorem and a-Browder’s theorem.It is also shown that if E is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a absolute-(p,r)-paranormal operator T,then E is self-adjoint if and only if the null space of T μ,N(T μ) N(T μ).     

Approximation of the Fractional Schrödinger Propagator on Compact Manifolds

Let L be a second order positive, elliptic differential operator that is self-adjoint with respect to some C~∞ density dx on a compact connected manifold M. We proved that if 0 α 1,α/2 s α and f ∈ H~s(M) then the fractional Schr?dinger propagator e~(it Lα/2) on M satisfies e~(it Lα/2) f(x)-f(x) = o(t~(s/α-ε)) almost everywhere as t → 0~+, for any ε 0.     

POWERS OF AN INVERTIBLE ω-HYPONORMAL OPERATOR

§1　IntroductionIn whatfollows,H means a complex Hilbertspace.A bounded linear operator T on His said to be positive(in symbol:T≥0 ) if(Tx,x)≥0 for any x∈H.Also an operator T isstrictly positive(in symbol:T>0 ) if T is positive and invertible.If A and B are invertiblepositive operators,itis well known that A≥B implies log A≥log B.However[1 ] ,log A≥log B does notnecessarily imply A≥B.Let T be a bounded linear operator and p≥0 .T is said to be a p-hyponormal operatorif(T* T)…     

Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω R n be a bounded domain, H = L 2 (Ω), L : D(L) H → H be an unbounded linear operator, f ∈ C(■× R, R) and λ∈ R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = λf (x, u), u ∈ D(L), which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.     

EXISTENCE AND ASYMPTOTIC ESTIMATE OF SOLUTION FOR THIRD ORDER BOUNDARY VALUE PROBLEM

This article shows the existence and asymptotic estimates of solutions of singularly perturbed boundary value problems for a class of third order nonlinear differential equations εx'" = f(t,x,x',ε), x(0) = A, x'(0) = x'(1), x"(0) = x"(1).     

STATIONARY STRUCTURES FOR A WEAKLY COUPLED ELLIPTIC SYSTEM ARISING IN TWO-PREDATOR, TWO-PREY MODELS

1 IntroductionIn tl1is paper we coIlsider positive steady-statc solutiolls to the fOllowi1lg weakly-c()uI)ledparabolic systenlwllere fl is a bouuded don1ain in Rn with sufficiently smooth bouudary, t1T = n x [0, T) and0f1T = 0fl x [0, T) l' is the olltward ul1it 11orIIlaJ vector. The col1stants ai, bi, (i = 1. 2, 3, 4, j =l, 2, 3, 4) are positive.The systenl (1.1) describes tlle LotkaVOlterra two-predator, two-prey nlodcl. u1 a11d u2represeIlt tl1e del1sitie8 of twDeprey wllile u3, '('4 …     

STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES

A pair of Banach spaces(X, Y) is said to be stable if for every ε-isometry f :X → Y, there exist γ 0 and a bounded linear operator T : L(f) → X with ||T|| ≤α such that ||T f(x)-x|| ≤γε for all x ∈ X, where L(f) is the closed linear span of f(X). In this article, we study the stability of a pair of Banach spaces(X, Y) when X is a C(K) space.This gives a new positive answer to Qian's problem. Finally, we also obtain a nonlinear version for Qian's problem.     

OSCILLATION CRITERIA FOR SECOND ORDER QUASILINEAR DIFFERENTIAL EQUATIONS

1 IntroductionThis paper is concerned with the problem of oscillation of second-orderquasilinear differential equationsFOr equation (1), the following conditions, which are referred to (FO), arealways assumed to be valid(a) a > 0 is a constant;(b) F: [to, co) x R x R x R x R - R is a continuous function;(c) sgn F(t, x) u? v3 w) = sgn x for each t 2 to and x, u? v? w E R,(d) r(t) E C([to, co); (0, co)) and jco ddt = co.' (e) T: [to, co) - R is continuous, T(t) 5 t for t 2 to and ill7T(t)…     

一个正反馈时滞微分方程的Hopf分支

Under some minor technical hypotheses, for each T larger than a certain rS > 0, Krisztin, Walther and Wu showed the existence of a periodic orbit for the positive feedback delay differential equation x(t) = -rμx(t) + rf(x(t-1)), where r and μ are positive constants and f : R → R satisfies f(0) = 0 and f' > 0. Combining this with a unique result of Krisztin and Walther, we know that this periodic orbit is the one branched out from 0 through Hopf bifurcation. Using the normal form theory for delay differential equations, we show the same result under the condition that f ∈ C3(R,R) is such that f'(0) = 0 and f'(0) < 0, which is weaker than those of Krisztin and Walther.     

Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraint

In this paper, we study the well-posedness in the generalized sense for variational inclusion problems and variational disclusion problems, the well-posedness for optimization problems with variational inclusion problems, variational disclusion problems and scalar equilibrium problems as constraint.     

Quasi-Newton Methods for Nonlinear Least Squares Focusing on Curvatures

A new quasi-Newton method for nonlinear least squares problems is proposed. Two advantages of the method are accomplished by utilizing special geometrical properties in the problem class. First, fast convergence is established for well-conditioned problems by interpolating both the current and the previous step in each iteration. Second, high accuracy is achieved for certain difficult problems, such as ill-conditioned problems and problems with large curvatures in the tangent space. Numerical results for artificial problems and standard test problems are presented and discussed.     

Well-posedness for parametric optimization problems with variational inclusion constraint

In this paper, we study the well-posedness for the parametric optimization problems with variational inclusion problems as constraint (or the perturbed problem of optimization problems with constraint). Furthermore, we consider the relation between the well-posedness for the parametric optimization problems with variational inclusion problems as constraint and the well-posedness in the generalized sense for variational inclusion problems.     

Omni-optimizer: A generic evolutionary algorithm for single and multi-objective optimization

Due to the vagaries of optimization problems encountered in practice, users resort to different algorithms for solving different optimization problems. In this paper, we suggest and evaluate an optimization procedure which specializes in solving a wide variety of optimization problems. The proposed algorithm is designed as a generic multi-objective, multi-optima optimizer. Care has been taken while designing the algorithm such that it automatically degenerates to efficient algorithms for solving other simpler optimization problems, such as single-objective uni-optimal problems, single-objective multi-optima problems and multi-objective uni-optimal problems. The efficacy of the proposed algorithm in solving various problems is demonstrated on a number of test problems chosen from the literature. Because of its efficiency in handling different types of problems with equal ease, this algorithm should find increasing use in real-world optimization problems.     

On System of Generalized Vector Quasi-equilibrium Problems with Set-valued Maps

In this paper, we introduce four new types of the system of generalized vector quasi-equilibrium problems with set-valued maps which include system of vector quasi-equilibrium problems, system of vector equilibrium problems, system of variational inequality problems, and vector equilibrium problems in the literature as special cases. We prove the existence of solutions for such kinds of system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of vector quasi-equilibrium problems and the generalized Debreu type equilibrium problem for vector-valued functions.     

New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems

In this article we present a new fixed point theorem for a class of general mixed monotone operators, which extends the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators. Based on them the local existence-uniqueness of positive solutions for nonlinear boundary value problems which include Neumann boundary value problems, three-point boundary value problems and elliptic boundary value problems for Lane-Emden-Fowler equations is proved. The theorems for nonlinear boundary value problems obtained here are very general.     

变分不等式与互补问题、双层规划与平衡约束数学规划问题的若干进展

考虑有限维变分不等式与互补问题、双层规划以及均衡约束的数学规划问题. 在简单介绍这些问题之后,重点介绍近年来这些领域中发展迅速的几个研究方向,包括对称锥互补问题的理论与算法、变分不等式的投影收缩算法、随机变分不等式与随机互补问题的模型与方法、双层规划以及均衡约束数学规划问题的新方法. 最后提出几个进一步研究的方向.     

Well-posedness for generalized quasi-variational inclusion problems and for optimization problems with constraints

In this paper, well-posedness of generalized quasi-variational inclusion problems and of optimization problems with generalized quasi-variational inclusion problems as constraints is introduced and studied. Some metric characterizations of well-posedness for generalized quasi-variational inclusion problems and for optimization problems with generalized quasi-variational inclusion problems as constraints are given. The equivalence between the well-posedness of generalized quasi-variational inclusion problems and the existence of solutions of generalized quasi-variational inclusion problems is given under suitable conditions.     

非线性特征值问题的二次近似方法

本文研究求解非线性特征值问题的数值方法.基于矩阵值函数的二次近似,将非线性特征值问题转化为二次特征值问题,提出了求解非线性特征值问题的逐次二次近似方法,分析了该方法的收敛性.结合求解二次特征值问题的Arnoldi方法和Jacobi-Davidson方法,给出求解非线性特征值问题的一些二次近似方法.数值结果表明本文所给算法是有效的.     

伪单调型向量F-互补问题可行集的最小元问题

本文引入了几类向量F-互补问题并给出了向量F-互补问题与广义向量变分不等式之间的关系.通过定义向量F-互补问题的可行集,研究了伪单调型向量F-互补问题的可行集的最小问题,推广了已有的一些结果.