p-adic 域上的拟微分算子

本文研究p-adic 域Q_p 上的一类拟微分算子{T^α : α∈R}, 其中算子T^α 在Q_p 的检验函数空 间S(Q_p) 以及相应的分布空间S′(Q_p) 中作为一个运算是封闭的. 本文构造了算子T^α 的卷积核, 指 出算子与其卷积核在解决p-adic 域上的相关问题中所起的重要作用. 最后研究算子T^α 的谱性质, 构 造了算子T^α 的全体固有值与固有函数, 并证明全体固有函数所成的集合组成L²(Q_p) 空间中的一组 完整正交基.     

Hahn Difference Operator and Associated Jackson–Nörlund Integrals

This paper is devoted for a rigorous investigation of Hahn’s difference operator and the associated calculus. Hahn’s difference operator generalizes both the difference operator and Jackson’s q-difference operator. Unlike these two operators, the calculus associated with Hahn’s difference operator receives no attention. In particular, its right inverse has not been constructed before. We aim to establish a calculus of differences based on Hahn’s difference operator. We construct a right inverse of Hahn’s operator and study some of its properties. This inverse also generalizes both Nörlund sums and the Jackson q-integrals. We also define families of corresponding exponential and trigonometric functions which satisfy first and second order difference equations, respectively.     

一类非线性偏微分方程组的解析解

首先,利用共轭算子的性质,将张鸿庆等提出的求伴随算子对的方法推广到了求一类非线性(即部分非线性的)算子矩阵的伴随算子向量．其次,利用机械化的构造方法给出了求解一类非线性(即,部分非线性的,且以所有线性的为其特例)非齐次微分方程组的统一理论,即通过齐次化和三角化求得恰当的变换,从而将原方程组化为较简单的形式,一般为对角化的．最后利用该方法求得了一些弹性力学方程组的解析解．     

关于一类有限格蕴涵代数方程的解法

研究一类含有有限个未知量且含有蕴涵算子的格蕴涵代数方程,给出方程有解的充分必要条件。在方程的解集非空时,讨论解集的一些结构性质。并刻画出方程在具有某种限制条件下的整个解集。最后,相关结论被应用到一个数值例子。     

Two-dimensional Toda field equations related to the exceptional algebra \mathfrak{g}_2 : Spectral properties of the lax operators

We analyze spectral properties of the Lax operator corresponding to the two-dimensional Toda field equations related to the algebra $\mathfrak{g}_2 $ . We construct two minimal sets of scattering data $\mathcal{T}_s $ , s = 1, 2, understanding the map between the potential and each of the sets $\mathcal{T}_s $ as a generalized Fourier transformation. We construct explicit recursion operators with special factorization properties.     

Properties of minimal mathematical expectations

The aim of this paper is to discuss how to compute a class of minimal mathematical expectations by using backward stochastic differential equations. We also prove that the minimal mathematical expectation operator still preserves some properties of the mathematical expectation operator.     

The Conditions for Some Linear Partial Differential Equations to Be Solvable in G

Using Bargmann's transformation and some basic results of theory of analytic functions with several complex variables, we have disscussed two classes of LPDOs in this paper. We prove that each operator of one class of them is surjective both from G to G and from L² to L², but not injective, and each operator of another class is injective from G to G but not surjective. And in the letter case, the necessary and suffcient conditions for the corresponding equations to be solvable in G are given.     

Elements of the Theory of Linear Volterra Operators in Banach Spaces

The new definition of Volterra operator introduced in [5] allows specification of the classical theory of linear equations in Banach spaces to equations with such operators. Here we specially address relations between properties of the given linear equation with Volterra operator and properties of its conjugate. As well we treat the theory of Noetherian and Fredholm equations.     

反向混合单调算子方程解的存在唯一性定理

运用锥与半序理论和非对称迭代方法,讨论半序Banach空间一类反向混合单调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,作为其应用着重讨论了非反向混合单调算子方程解的存在唯一性,所得结果改进和推广了混合单调算子方程某些已知相应结果.     

On One Class of Matrix Differential Operators

We consider one class of matrix differential operators in the whole space. For this class of operators we establish the isomorphic properties in some special scales of weighted Sobolev spaces and study the regularity properties for solutions to the system of differential equations defined by these operators. The class of operators under consideration contains the stationary Navier–Stokes operator.     

On certain properties of linear iterative equations

An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is given both for equations in standard form and for equations in reduced normal form. The operator that generates an iterative equation of a general order in reduced normal form is also obtained and some other properties of iterative equations are established. An expression for the parameters of the source equation of the transformed equation under equivalence transformations is obtained, and this gives rise to the derivation of important symmetry properties for iterative equations. The transformation mapping a given iterative equation to the canonical form is obtained in terms of the simplest determining equation, and several examples of application are discussed.     

Positive solutions for nonlinear operator equations and several classes of applications

In this paper, we study a class of nonlinear operator equations x = Ax + x ₀ on ordered Banach spaces, where A is a monotone generalized concave operator. Using the properties of cones and monotone iterative technique, we establish the existence and uniqueness of solutions for such equations. In particular, we do not demand the existence of upper-lower solutions and compactness and continuity conditions. As applications, we study first-order initial value problems and two-point boundary value problems with the nonlinear term is required to be monotone in its second argument. In the end, applications to nonlinear systems of equations and to nonlinear matrix equations are also considered.     

Solvability conditions of a boundary value problem with operator coefficients and related estimates of the norms of intermediate derivative operators

Sufficient conditions for the proper and unique solvability in the Sobolev space of vector functions of the boundary value problem for a certain class of second-order elliptic operator differential equations on a semiaxis are obtained. The boundary condition at zero involves an abstract linear operator. The solvability conditions are established by using properties of operator coefficients. The norms of intermediate derivative operators, which are closely related to the solvability conditions, are estimated.     

Trace formula for a perturbed operator of Jacobi type

There is a broad class of problems of mathematical physics that lead to the solution of second-order differential equations of some special form. In particular, systems of solutions of such equations are given by classical polynomials (Jacobi, Laguerre, and Hermite polynomials). Such equations are naturally related to second-order differential operators in appropriate Hilbert spaces and the corresponding spectral problems. We consider a Jacobi operator and its perturbation by the operator of multiplication by a function. We derive a trace formula for the perturbed operator and a closed-form expression for the first correction.     

The Existence and Uniqueness of Solutions of a Class of Evolution Equations

In this paper, a class of evolution equations which have special boundary conditions is discussed. The spectrum properties of an operator which is associated with this class of equations arc provided, and the existence and uniqueness of solutions of this class of equations is proved by the theory of pertu rbation of C_0-semigroups. In the last part of this paper, a condition which ensures the solutions to be non-negative is given. Some results of the time-independent population systems in [1-5] are our examples.     

On the spectrum of well-defined restrictions and extensions for the Laplace operator

The study of the spectral properties of operators generated by differential equations of hyperbolic or parabolic type with Cauchy initial data involve, as a rule, Volterra boundary-value problems that are well posed. But Hadamard’s example shows that the Cauchy problem for the Laplace equation is ill posed. At present, not a single Volterra well-defined restriction or extension for elliptic-type equations is known. Thus, the following question arises: Does there exist a Volterra well-defined restriction of a maximal operator $\hat L$ or a Volterra well-defined extension of a minimal operator L ₀ generated by the Laplace operator? In the present paper, for a wide class of well-defined restrictions of the maximal operator $\hat L$ and of well-defined extensions of the minimal operator L ₀ generated by the Laplace operator, we prove a theorem stating that they cannot be Volterra.     

On shape preserving semigroups

Motivated by positivity-, monotonicity-, and convexity preserving differential equations, we introduce a definition of shape preserving operator semigroups and analyze their fundamental properties. In particular, we prove that the class of shape preserving semigroups is preserved by perturbations and taking limits. These results are applied to partial delay differential equations.     

Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response

A class of reaction-diffusion equations with time delay and nonlocal response is considered. Assuming that the corresponding reaction equations have heteroclinic orbits connecting an equilibrium point and a periodic solution, we show the existence of traveling wave solutions of large wave speed joining an equilibrium point and a periodic solution for reaction-diffusion equations. Our approach is based on a transformation of the differential equations to integral equations in a Banach space and the rigorous analysis of the property for a corresponding linear operator. Our approach eventually reduces a singular perturbation problem to a regular perturbation problem. The existence of traveling wave solution therefore is obtained by the application of Liapunov-Schmidt method and the Implicit Function Theorem.     

Implicit function theorem via the DSM

Sufficient conditions are given for an implicit function theorem to hold. The result is established by an application of the Dynamical Systems Method (DSM). It allows one to solve a class of nonlinear operator equations in the case when the Fréchet derivative of the nonlinear operator is a smoothing operator, so that its inverse is an unbounded operator.     

An indicator for Wiener-Hopf integral equations with invertible analytic symbol

For Wiener-Hopf integral equations with an operator or matrix valued kernel and with an invertible symbol which is analytic on the real line and at infinity an indicator is introduced. In general this indicator is a bounded linear operator, but when the kernel is matrix valued and the symbol is rational it is a (possibly non-square) matrix. From the indicator the invertibility properties and Fredholm characteristics of the integral equation can be read off. The class of Wiener-Hopf equations studied here is also described in terms of growth conditions on the derivatives of the kernel.