Global solvability for abstract semilinear evolution equations

The Cauchy problem for the abstract semilinear evolution equation u^′(t) = Au (t) + B (u (t)) + C (u (t)) is discussed in a general Banach space X. Here A is the so‐called Hille‐Yosida operator in X, B is a differentiable operator from D (A) into X, and C is a locally Lipschitz continuous operator from D (A) into itself. A vectorvalued functional defined only on X is used and appropriate conditions on the nonlinear operators B and C are imposed so that a vector‐valued functional defined on the domain of the operator A may be constructed in order to specify the growth of a global solution. The advantage of our formulation lies in the fact that it is possible to obtain a global solution by checking some energy inequalities concerning only low order derivatives (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Criteria for the single-valued metric generalized inverses of multi-valued linear operators in banach spaces

Let X, Y be Banach spaces and M be a linear subspace in X × Y = {{x, y}|x ∈ X, y ∈ Y }. We may view M as a multi-valued linear operator from X to Y by taking M (x) = {y|{x, y} ∈ M }. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M . The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces.     

On the nonlinear matrix equation

Based on fixed point theorems for monotone and mixed monotone operators in a normal cone, we prove that the nonlinear matrix equation always has a unique positive definite solution. A conjecture which is proposed in [X.G. Liu, H. Gao, On the positive definite solutions of the matrix equation X^s±A^TX^-tA=I_n, Linear Algebra Appl. 368 (2003) 83–97] is solved. Multi-step stationary iterative method is proposed to compute the unique positive definite solution. Numerical examples show that this iterative method is feasible and effective.     

On extensions of some Flugede–Putnam type theorems involving (p,k)‐quasihyponormal,spectral, and dominant operators

A Hilbert space operator S is called (p, k)‐quasihyponormal if S *^k ((S *S)^p – (SS *)^p )S^k ≥ 0 for an integer k ≥ 1 and 0 < p ≤ 1. In the present note, we consider (p, k)‐quasihyponormal operator S ∈ B (H) such that SX = XT for some X ∈ B (K,H) and prove the Fuglede–Putnam type theorems when the adjoint of T ∈ B (K) is either (p, k)‐quasihyponormal or dominant or a spectral operator (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The alternative Daugavet property of C *‐algebras and JB *‐triples

A Banach space X is said to have the alternative Daugavet property if for every (bounded and linear) rank‐one operator T: X → X there exists a modulus one scalar ω such that ∥Id+ωT ∥ = 1 + ∥T ∥. We give geometric characterizations of this property in the setting of C *‐algebras, JB *‐triples, and of their isometric preduals. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Hermitian positive defnite solution of the nonlinear matrix equation X + A*X −1 A + B*X −1 B = I

In this paper, we study the matrix equation X + A*X ⁻¹ A + B*X ⁻¹ B = I, where A, B are square matrices, and obtain some conditions for the existence of the positive definite solution of this equation. Two iterative algorithms to find the positive definite solution are given. Some numerical results are reported to illustrate the effectiveness of the algorithms. This research supported by the National Natural Science Foundation of China 10571047 and Doctorate Foundation of the Ministry of Education of China 20060532014.     

PRV property and the <Emphasis Type="Italic">ϕ</Emphasis>-asymptotic behavior of solutions of stochastic differential equations

In this paper, we investigate the a.s. asymptotic behavior of the solution of the stochastic differential equation dX(t) = g(X(t)) dt + σ(X(t))dW(t), X(0) ≢ 1, where g(·) and σ(·) are positive continuous functions, and W(·) is a standard Wiener process. By means of the theory of PRV functions we find conditions on g(·), σ(·), and ϕ(·) under which ϕ(X(·)) may be approximated a.s. by ϕ(μ(·)) on {X(t) → ∞}, where μ(·) is the solution of the ordinary differential equation dμ(t) = g(μ(t)) dt with μ(0) = 1. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 445–465, October–December, 2007.     

Small Data Scattering for the One-Dimensional Nonlinear Dirac Equation with Power Nonlinearity

We study scattering problems for the one-dimensional nonlinear Dirac equation (?_t + α?_x + iβ)Φ = λ|Φ|^p?1Φ. We prove that if p > 3 (resp. p > 3 + 1/6), then the wave operator (resp. the scattering operator) is well-defined on some 0-neighborhood of a weighted Sobolev space. In order to prove these results, we use linear operators D(t)xD(?t) and t?_x + x?_t ? α/2, where {D(t)}_t∈? is the free Dirac evolution group. For the reader's convenience, in an appendix we list and prove fundamental properties of D(t)xD(?t) and t?_x + x?_t ? α/2.     

Approximate solution of u′=–A2u using a relation between exp(–tA2) and exp(itA)

Let A be a positive definite operator in a Hilbert space and consider the initial value problem for u _t =–A² u. Using a representation of the semi group exp(–A² t) in terms of the group exp(iAt) we express u in terms of the solution of the standard heat equation w _t = w _yy , with initial values v solving the initial value problem for v _y = iAv. This representation is used to construct a method for approximating u in terms of approximations of v. In the case that A is a 2^nd order elliptic operator the method is combined with finite elements in the spatial variable and then reduces the solution of the 4th order equation for u to that of the 2^nd order equation for v, followed by the solution of the heat equation in one space variable.     

Approximation of solutions to evolution equations

The convergence of the Galerkin approximations to solutions of abstract evolution equations of the form u′(t)= ? Au(t) + M(u(t)) is shown. Here A is a closed, positive definite, self-adjoint linear operator with domain D(A) dense in a Hilbert space H and M is a non-linear map defined on D(A^½) which satisfies a Lipschitz condition on balls in D(A^½).     

On positive definite solution of a nonlinear matrix equation

In this paper, some necessary and sufficient conditions for the existence of the positive definite solutions for the matrix equation X + A^*X^?αA = Q with α ∈ (0, ∞) are given. Iterative methods to obtain the positive definite solutions are established and the rates of convergence of the considered methods are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

More on extremal ranks of the matrix expressions A − BX ± X*B* with statistical applications

Through a Hermitian‐type (skew‐Hermitian‐type) singular value decomposition for pair of matrices (A, B) introduced by Zha (Linear Algebra Appl. 1996; 240 :199–205), where A is Hermitian (skew‐Hermitian), we show how to find a Hermitian (skew‐Hermitian) matrix X such that the matrix expressions A ? BX ± X^*B^* achieve their maximal and minimal possible ranks, respectively. For the consistent matrix equations BX ± X^*B^* = A, we give general solutions through the two kinds of generalized singular value decompositions. As applications to the general linear model {y, Xβ, σ²V}, we discuss the existence of a symmetric matrix G such that Gy is the weighted least‐squares estimator and the best linear unbiased estimator of Xβ, respectively. Copyright © 2007 John Wiley & Sons, Ltd.     

An inner product lemma

Given the operator product BA in which both A and B are symmetric positive‐definite operators, for which symmetric positive‐definite operators C is BA symmetric positive‐definite in the C inner product 〈x, y〉C? This question arises naturally in preconditioned iterative solution methods, and will be answered completely here. Copyright © 2004 John Wiley & Sons, Ltd.     

Measurability of inverses of random operators and existence theorems

Let (Ω,B,μ) be ameasure space andX a separable Hubert space. LetT be a random operator from Ω ×X intoX. In this paper we investigate the measurability ofT ^-1. In our main theorems we show that ifT is a separable random operator withT(w) almost sure invertible and monotone and demicontinuous thenT ^-1is also a random operator. As an application of this we give an existence theorem for random Hammerstein operator equation.     

Weighted inequalities for iterated maximal functions in Orlicz spaces

Let M be the classical Hardy‐Littlewood maximal operator. The object of our investigation in this paper is the iterated maximal function M^kf(x) = M(M^k?1f) (x) (k ≥ 2). Let Φ be a φ‐function which is not necessarily convex and Ψ be a Young function. Suppose that w is an A′_∞ weight and that k is a positive integer. If there exist positive constants C₁ and C₂ such that ((I)) then there exist positive constants C₃ and C₄ such that ((II)) where the functions a(t) and b(t) are the right derivatives of Φ(t) and Ψ(t), respectively. Conversely, if w is an A₁ weight, then (II) implies (I). Another necessary and sufficient condition will be given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solutions and perturbation estimates for the matrix equation

In this paper, the nonlinear matrix equation X^s+A^*X^-tA=Q is investigated. Necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions are derived. An effective iterative method to obtain the special solution X_L (We proved that if there is a maximal Hermitian positive definite solution, then it must be X_L) is established. Moreover, some new perturbation estimates for X_L are obtained. Several numerical examples are given to illustrate the effectiveness of the algorithm and the perturbation estimates.     

Global continuation for first order systems over the half‐line involving parameters

Let X be one of the functional spaces W^1,p ((0, ∞), ?^N ) or C₀¹ ([0, ∞), ?^N ), we study the global continuation in λ for solutions (λ, u, ξ) ∈ ? × X × ?^k of the following system of ordinary differential equations: where ?^N = X₁ ⊕ X₂ is a given decomposition, with associated projection P: ?^N → X₁. Under appropriate conditions upon the given functions F and φ, this problem gives rise to a nonlinear Fredholm operator which is proper on the closed bounded subsets of ? × X × ?^k and whose zeros correspond to the solutions of the original problem. Using a new abstract continuation result, based on a recent degree theory for proper Fredholm mappings of index zero, we reduce the continuation problem to that of finding a priori estimates for the possible solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Combination of standard Galerkin and subspace methods for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

We consider a combination of the standard Galerkin method and the subspace decomposition methods for the numerical solution of the two‐dimensional time‐dependent incompressible Navier‐Stokes equations with nonsmooth initial data. Because of the poor smoothness of the solution near t = 0, we use the standard Galerkin method for time interval [0, 1] and the subspace decomposition method time interval [1, ∞). The subspace decomposition method is based on the solution into the sum of a low frequency component integrated using a small time step Δt and a high frequency integrated using a larger time step pΔt with p > 1. From the H¹‐stability and L²‐error analysis, we show that the subspace decomposition method can yield a significant gain in computing time. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

On the supercyclicity and hypercyclicity of the operator algebra

Let B（X） be the operator algebra for a separable infinite dimensional Hilbert space H, endowed with the strong operator topology or ＊-strong topology. We give sufficient conditions for a continuous linear mapping L ： B（X） →B（X） to be supercyclic or ,-supercyclic. In particular our condition implies the existence of an infinite dimensional subspace of supercyclic vectors for a mapping T on H. Hypercyclicity of the operator algebra with strong operator topology was studied＇ by Chan and here we obtain an analogous result in the case of ＊-strong operator topology.     

Vanishing theorem for sheaves of microfunctions at the boundary on cr–manifolds

Let X be a complex analytic manifold. Consider S?M?Xreal analytic submonifolds with codium ^R _MS=1,and let ω be a connected component of M\S. Let p∈S X_MT_M ^*X where T^* _Xdenotes the conormal bundle to M in X, and denote by ν(p) the complex radial Euler field at p. Denote by μ*(Ox) (for * = M, ω) the microlocalization of the sheaf of holomorphic functions along *.

Under the assumption dim^R(T_pT_M ^*X? ν(p)) = 1, a theorem of vanishing for the cohomology groups H^jμM(Ox)p is proved in [K-S 1, Prop. 11.3.1], j being related to the number of positive and negative eigenvalue for the Levi form of M.

Under the hypothesis dim^R(T_pT_S ^*X∩ν(p))=1, a similar result is proved here for the cohomology groups of the complex of microfunctions at the boundary μω(Ox).Stating this result in terms of regularity at the boundary for CR–hyperfunctions a local Bochner–type theorem is then obtained.