非线性Lipschitz算子半群的渐近性质及其应用

本文对一类非线性算子半群————Lipschitz算子半群的渐近性质进行研究,刻划了非线性Lipschitz算子半群所具有的基本渐近性质（这些性质与线性算子半群所具有的基本渐近性质相一致）,证明了作为线性算子对数范数的非线性推广,Dahlquist数能用于刻划非线性Lipschitz算子半群的渐近性质.为克服Dahlquist数只对Lips-chitz算子有定义的缺点,本文引入一个全新的特征数:广义 Dahlquist数,并证明广义Dahlquist数比Dahlquist数能更为精确地刻划Lipschitz算子半群的渐近性质.作为应用,得到关于 Hopfield型神经网络全局指数稳定性的一个新结果.     

A particle transport problem with non-homogeneous reflection boundary conditions

We consider particle transport in a three-dimensional convex region V, bounded by the regular surface ∂V. We assume that particles are specularly reflected by ∂V and that a source q is assigned on ∂V; more general non-homogeneous boundary conditions are also discussed. The problem is non-linear because the boundary condition is not homogeneous. We prove existence of a unique strict solution and by using the theory of semigroups we derive the explicit expression of such a solution in terms of the boundary source q. In the appendix, we indicate how some properties of affine operators can be used to derive the solution. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Bounded approximation properties in non‐archimedean Banach spaces

Some non‐archimedean bounded approximation properties are introduced and studied in this paper. As an application, an affirmative answer is given, for non‐spherically complete base fields, to the following problem, posed in 13 , p. 95: Does there exist an absolutely convex edged set B in a non‐archimedean locally convex space such that its closure $\overline{B}Some non‐archimedean bounded approximation properties are introduced and studied in this paper. As an application, an affirmative answer is given, for non‐spherically complete base fields, to the following problem, posed in 13 , p. 95: Does there exist an absolutely convex edged set B in a non‐archimedean locally convex space such that its closure $\overline{B}$ is not edged?     

A Factorization Property of R‐Bounded Sets of Operators on Lp‐Spaces

Let T be a bounded operator on L^p‐space, with 1 ≤ p < ∞. A theorem of W. B. Johnson and L. Jones asserts that after an appropriate change of density, T actually extends to a bounded operator on L². We show that if 𝒯 ⊂ B (L^p) is an R‐bounded set of operators, then the latter result holds for any T ∈ 𝒯 with a common change of density. Then we give applications including results on R‐sectorial operators.     

Fractional Schrödinger equation; solvability and connection with classical Schrödinger equation

We consider the Dirichlet boundary problem for semilinear fractional Schrödinger equation with subcritical nonlinear term. Local and global in time solvability and regularity properties of solutions are discussed. But our main task is to describe the connections of the fractional equation with the classical nonlinear Schrödinger equation, including convergence of the linear semigroups and continuity of the nonlinear semigroups when the fractional exponent α approaches 1.     

m次积分半群逼近及在抽象Cauchy问题中的应用

研究了指数有界的m次积分半群的离散逼近问题,利用可积的离散参数半群,获得了相关离散逼近结果.另外,给出了该逼近理论在非齐次抽象Cauchy问题中的应用.     

B-Bounded semigroups and applications

Summary We study a new one-parameter family of linear bounded operators Y(t), tO which is uniquely defined by a linear operator A, not necessarily bounded, and by a linear bounded operator B. The definition of Y(t), t 0 given in Section 2 was suggested by a particle transport problem with multiplying boundary conditions.     

On singular mono‐energetic transport equations in slab geometry

In this paper we establish the well posedness of the Cauchy problem associated to transport equations with singular cross‐sections (i.e. unbounded collisions frequencies and unbounded collision operators) in L₁ spaces for specular reflecting boundary conditions. In addition, we discuss the weak compactness of the second‐order remainder term of the Dyson–Phillips expansion. This allows us to estimate the essential type of the transport semigroup from which the asymptotic behaviour of the solution is derived. The case of singular transport equations with periodic boundary conditions is also discussed. The proofs make use of the Miyadera perturbation theory of positive semigroups on AL‐spaces. Copyright © 2002 John Wiley & Sons, Ltd.     

Coagulation and fragmentation with discrete mass loss

A nonlinear integro-differential equation that models a coagulation and multiple fragmentation process in which discrete fragmentation mass loss can occur is examined using the theory of strongly continuous semigroups of operators. Under the assumptions that the coagulation kernel K is bounded and the fragmentation rate function a satisfies a linear growth condition, global existence and uniqueness of solutions that lose mass in accordance with the model are established. In the case when no coagulation is present and the fragmentation process is governed by power-law kernels, an explicit formula is given for the substochastic semigroup associated with the resulting mass-loss fragmentation equation.     

Continuous state branching semigroups

Summary This paper is concerned with Markov processes with continuous creation where the phase space is a general separable compact metric space. The transition probabilities for such a process determine a semigroup of operators acting on a function space over the collection of bounded Borel measures on the phase space. Such a semigroup is characterized by a particular convolution condition and is called a continuous state branching semigroup. A connection is established between continuous state branching semigroups and certain semigroups of nonlinear operators and then this connection is exploited to establish an existence theorem for the former.Research associated with a project in probability at Princeton University supported by the Office of Army Research.     

Local Controllability of Functional Integrodifferential Equations in Banach Space

In this paper, we establish a set of sufficient conditions for the local controllability of functional integrodifferential equations in Banach space. The results are obtained by using the methods of analytic semigroups, fractional powers of operators, and a fixed-point argument. These results generalize previous results on bounded linear operators to unbounded linear operators in which the equation involves a nonlinear delay term. An application to a partial integrodifferential equation is given.     

On pointwise convergence of the family of Urysohn‐type integral operators

In this paper, we consider a family of nonlinear integral operators of Urysohn‐type and study the pointwise convergence of the family at characteristic points of L₁?function. The kernel K_λ(x,t,u(t)) depends on the positive parameter λ changing on the set of numbers with the accumulation point at infinity and K_λ(x,t,u(t)) is an entire analytic function of variable u, which is a bounded function belonging to L₁(R).     

Fredholm Operators, Essential Spectra and Application to Transport Equations

In this paper the essential spectra of closed, densely defined linear operators is characterized on a Banach spaces under perturbations of n-strictly power compact operators. Further we apply the obtained results to investigate the essential spectra of one-dimensional transport equation with general boundary conditions and the essential spectra of singular neutron transport equations in bounded geometries.     

Semilinear evolution equations with nonlinear constraints and applications

This paper deals with a general class of evolution problems for semilinear equations coupled with nonlinear constraints. Those constraints may contain compositions of nonlinear operators and unbounded linear operators, and hence the associated operators are not necessarily formulated in the form of continuous perturbations of linear operators. Accordingly, a family of equivalent norms is introduced to discuss 'quasidissipativity' in a local sense of the operators and a generation theory for nonlinear semigroups is employed to construct solution operators on bounded sets. It is a feature of our treatment that the resultant solution operators are obtained as nonlinear semigroups on the whole space which are not 'quasicontractive' but locally equi-Lipschitz continuous.     

The Existence of Positive Almost Periodic Type Solutions for Some Nonlinear Delay Integral Equations

This paper presents the conditions of existence of positive almost periodic type solutions for some nonlinear delay integral equations, by using a fixed point theorem in the mixed monotone operators （Ma, Y.： On a class of mixed monotone operators and a kind of two-point bounded value problem. Indian J. Math., 41（2）, 211-220 （1999]]. Some known results are operators.     

On a Dirichlet Problem Involving an Ornstein–Uhlenbeck Operator

We consider an elliptic Dirichlet problem which involves Ornstein–Uhlenbeck operators of special form in a half space of R ⁿ . We obtain necessary and sufficient conditions under which global Schauder estimates in spaces of Hölder continuous and bounded functions hold. For this purpose we use analytical tools, in particular semigroups and interpolation theory. Moreover we extend a theorem on the analiticity of subordinated semigroups (see Carasso and Kato; Trans. Amer. Math. Soc. 327 (1990, 867–877)) to a class of Markov type semigroups. We also provide explicit formulas for the Poisson kernels.     

Nonlinear semigroup approach to transport equations with delayed neutrons

This paper deal with a nonlinear transport equation with delayed neutron and general boundary conditions. We establish, via the nonlinear semigroups approach, the existence and uniqueness of the mild solution, weak solution, strong solution and local solution on L~p-spaces(1 ≤ p +∞). Local and non local evolution problems are discussed.     

Unbounded functional calculus for bounded groups with applications

In this paper, we develop the unbounded extension of the Hille–Phillips functional calculus for generators of bounded groups. Mathematical applications include the generalised Lévy–Khintchine formula for subordinate semigroups, the analyticity of semigroups generated by fractional powers of group generators, where the power is not an odd integer, and a shifted abstract Grünwald formula. We also give an application of the theory to subsurface hydrology, modeling solute transport on a regional scale using fractional dispersion along flow lines. M. Kovács is partially supported by postdoctoral grant No. 623-2005-5078 of the Swedish Research Council (VR).     

Matrix multiplication operators on Banach function spaces

In this paper, we study the matrix multiplication operators on Banach function spaces and discuss their applications in semigroups for solving the abstract Cauchy problem.     

C_0半群在非线性Lipschitz扰动下的范数连续性保持

研究有界线性算子强连续半群在非线性Lipschitz扰动下的正则性质保持问题.具体地,我们证明:如果强连续半群是直接范数连续的,则非线性扰动半群是直接Lipschitz范数连续的.结论推广了线性算子半群的范数连续性质保持,丰富和完善了非线性算子半群的理论.