New approach to differential equations with countable impulses

This paper provides a new approach to study the solutions of a class of generalized Jacobi equations associated with the linearization of certain singular flows on Riemannian manifolds with dimension n + 1.A new class of generalized differential operators is defined.We investigate the kernel of the corresponding maximal operators by applying operator theory.It is shown that all nontrivial solutions to the generalized Jacobi equation are hyperbolic,in which there are n dimension solutions with exponential...     

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.     

The monotone method for controllability of the nonlinear evolution systems

In this paper,we study the controllability of the nonlinear evolution systems.We establish the controllability results by using the monotone operator theory.No compactness assumptions are imposed in the main results.We present an example to illustrate our results.     

Solutions to the polynomial Dirac equations on unbounded domains in Clifford analysis

In this paper we study polynomial Dirac equation p(??)f = 0 including (?? ? λ)f = 0 with complex parameter λ and ??^kf = 0(k?1) as special cases over unbounded subdomains of ?^{n + 1}. Using the Clifford calculus, we obtain the integral representation theorems for solutions to the equations satisfying certain decay conditions at infinity over unbounded subdomains of ?^{n + 1}. Copyright © 2010 John Wiley & Sons, Ltd.     

带P-Laplacian算子的四点四阶奇异边值问题的对称正解

本文主要利用上下解方法和Schauder不动点定理,在更广泛的条件下研究了一类带PLaplacian算子的四点四阶奇异边值问题的对称正解的存在性.克服了对非线性微分算子[φp(u″)]″Fredholm抉择定理和极大值原理不能使用的困难,改进并推广了最近的一些已知结果.     

Completeness of the System of Root Vectors of 2 × 2 Upper Triangular Infinite-Dimensional Hamiltonian Operators in Symplectic Spaces and Applications

The authors investigate the completeness of the system of eigen or root vectors of the 2 × 2 upper triangular infinite-dimensional Hamiltonian operator H ₀. First, the geometrical multiplicity and the algebraic index of the eigenvalue of H ₀ are considered. Next, some necessary and sufficient conditions for the completeness of the system of eigen or root vectors of H ₀ are obtained. Finally, the obtained results are tested in several examples.     

UNIVERSAL INTERPOLATING SEQUENCES ON SPACES OF ANALYTIC FUNCTIONS

This article derives the relation between universal interpolating sequences and some spectral properties of the multiplication operator by the independent variable z in case the underlying space is a Hilbert space of functions analytic on the open unit disk.     

APPLICATION OF TWO OPERATORS TRANSFORM FROM CLASS A OPERATOR TO THE CLASS OF HYPONORMAL OPERATOR

In this article, we give an operator transform T (*) from class A operator to the class of hyponormal operators. It is different from the operator transform T defined by M. Ch and T. Yamazaki. Then, we show that σ(T ) = σ( T (*)) and σa(T )\{0} = σa( T (*))\{0}, in case T belongs to class A. Next, we obtain some relations between T and T (9).     

L~p CONTINUITY OF HRMANDER SYMBOL OPERATORS OpS_（0,0） ~m AND NUMERICAL ALGORITHM

If we use Littlewood-Paley decomposition, there is no pseudo-orthogonality for Ho¨rmander symbol operators OpS m 0 , 0 , which is different to the case S m ρ,δ (0 ≤δ < ρ≤ 1). In this paper, we use a special numerical algorithm based on wavelets to study the L p continuity of non infinite smooth operators OpS m 0 , 0 ; in fact, we apply first special wavelets to symbol to get special basic operators, then we regroup all the special basic operators at given scale and prove that such scale operator’s continuity decreases very fast, we sum such scale operators and a symbol operator can be approached by very good compact operators. By correlation of basic operators, we get very exact pseudo-orthogonality and also L 2 → L 2 continuity for scale operators. By considering the influence region of scale operator, we get H 1 (= F 0 , 2 1 ) → L 1 continuity and L ∞→ BMO continuity. By interpolation theorem, we get also L p (= F 0 , 2 p ) → L p continuity for 1 < p < ∞ . Our results are sharp for F 0 , 2 p → L p continuity when 1 ≤ p ≤ 2, that is to say, we find out the exact order of derivations for which the symbols can ensure the resulting operators to be bounded on these spaces.