Corrigendum to “Discrepancy principle for the dynamical systems method” [Communications in Nonlinear Science and Numerical Simulation 10 (2005) 95–101]

Consider an operator equation B(u) − f = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator B′(u) is not boundedly invertible, and well-posed otherwise. The dynamical systems method (DSM) for solving this equation consists of a construction of a Cauchy problem, which has the following properties: (1) it has a global solution for an arbitrary initial data, (2) this solution tends to a limit as time tends to infinity, (3) the limit is the minimal-norm solution to the equation B(u) = f. A global convergence theorem is proved for DSM for equation B(u) − f = 0 with monotone operators B.     

Optimal bounds for bilinear forms associated with linear equations

The construction of upper and lower bounds to the bilinear quantity g₀, f, where f is the solution of an operator equation Tf = f₀, requires either an approximation for f or one for T^?1. In this paper the question of “best” approximation of T^?1 by an operator of the form B = βI, where β is a real constant, is investigated for linear operators that are either self-adjoint or can be related by suitable manipulations to others that are. Particular attention is paid to a special operator, previously studied by Robinson, of importance in predicting the dynamic polarisabilities of quantum-mechanical systems.     

Solution of the spectral problem for the curl and Stokes operators with periodic boundary conditions

In this paper, we establish relations between eigenvalues and eigenfunctions of the curl operator and Stokes operator (with periodic boundary conditions). These relations show that the curl operator is the square root of the Stokes operator with ν = 1. The multiplicity of the zero eigenvalue of the curl operator is infinite. The space L ₂(Q, 2π) is decomposed into a direct sum of eigenspaces of the operator curl. For any complex number λ, the equation rot u + λu = f and the Stokes equation −ν(Δv + λ ²v) + ∇p = f, div v = 0, are solved. Bibliography: 15 titles. Dedicated to the memory of Olga Aleksandrovna Ladyzhenskaya __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 246–276.     

Explicit solutions of the equation –div z = f under the constraint ∥z ∥∞ = 1

We consider the eigenvalue problem for the 1-Laplace operator and the corresponding variational problem. Since the underlying functionals are not differentiable, the derivation of the Euler-Lagrange equation is highly nontrivial and involves methods of nonsmooth analysis. A remarkable fact is, that there are infinitely many functions f for which the equation –div z = f has to be satisfied. The purpose of the present work is the investigation of this equation. We show that in a square 2-dimensional domain for each right hand side f there are infinitely many vector fields z satisfying the Euler-Lagrange equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The mathieu-hill operator equation with dissipation and estimates of its instability index

The behavior ast→∞ of solutions of the equation

Least squares and bounded variation regularization with nondifferentiable functionals

Let A be an operator from a real Banach space into a real Hilbert space. In this paper we study least squares regularization methods for the ill-posed operator equation A(u) = f using nonlinear nondifferentiable penalty functionals. We introduce a notion of distributional approximation, and use constructs of distributional approximations to establish convergence and stability of approximations of bounded variation solutions of the operator equation. We also show that the results provide a framework for a rigorous analysis of numerical methods based on Euler-Lagrange equations to solve the minimization problem. This justifies many of the numerical implementation schemes of bounded variation minimization that have been recently proposed.     

The Point Spectrum and Spectral Geometry for Riemannian Manifolds with Cusps

The bifurcation of a solution of the equation f(x, λ) = 0 at the point (x₀, λ₀) is investigated. In the case that B: = –f_x(x₀, λ₀) is a FREDHOLM operator by the method of LJAPUNOV/SCHMIDT the original equation is equivalent to a system consisting of a locally uniquely solvable equation and an equation in a finit dimensional subspace, the so-called bifurcation equation. For analytical/recursion formulas are deduced to determine the locally unique solution. In the case of FREDHOLM operators B with index zero practicable criteria are given for the applicability of a theorem of IZE being a generalization of a well known theorem of KRASNOSELSKIJ.     

Sur un théorème variationnel de Langenbach

Using some imbedding theorems, the Langenbach variational theorem for operator equation Pu=f* is obtained as a natural consequence of the Kerner-Vainberg potentiality theorem coupled with standard results of the variational calculus; in our approach operator P is defined not over all the space X but only on a dense subspace (compare with [1], pp. 33–34, propositions 3.2 and 3.3).     

Convergence rates in regularization for the case of monotone perturbations

Convergence rates are justified for regularized solutions of a Hammerstein operator equation of the form x + F ₂ F ₁(x) = f in the Banach space with monotone perturbations f ₂ ^h and f ₁ ^h .     

Estimate of error of an approximated solution by the method of moments of an operator equation

For an equationAu = f whereA is a closed densely defined operator in a Hilbert spaceH, f εH, we estimate the deviation of its approximated solution obtained by the moment method from the exact solution. All presented theorems are of direct and inverse character. The paper refers to direct methods of mathematical physics, the development of which was promoted by Yu. D. Sokolov, the well-known Ukrainian mathematician and mechanic, a great humanitarian and righteous man. We dedicate this paper to his blessed memory.     

On the shape of the ground state eigenvalue density of a random Hill's equation

Consider the Hill's operator Q = ?d²/dx² + q(x) in which q(x), 0 ≤ x ≤ 1, is a white noise. Denote by f(μ) the probability density function of ?λ₀(q), the negative of the ground state eigenvalue, at μ. We prove the detailed asymptotics as μ → + ∞. This result is based on a precise Laplace analysis of a functional integral representation for f(μ) established by S. Cambronero and H. P. McKean in 5 . © 2005 Wiley Periodicals, Inc.     

On various types of homogeneous Riemannian spaces with an isotropy group which decomposes

Homogeneous Riemannian spaces are considered whose isotropy group H decomposes into the direct product of irreducible subgroups and the identity operator acting in mutually orthogonal planes in the tangent space of a point M. We exclude the special cases when an irreducible subgroup in the decomposition of H is semisimple and acts on a plane whose dimension is a multiple of four. These spaces admit a rigid tensor structuref satisfying the conditionf ³ +f = 0.Translated from Matematicheskie Zametki, Vol. 5, No. 3, pp. 361–372, March, 1969.     

Stabilization of solutions of linear differential equations in Hilbert space

Conditions, less stringent than those known at present, are found for the stabilization of a solution of a linear differential equation of the form (du/dt) + A(t) u =f(t) in Hilbert space to a solution of the operational equation Ax =f, where A is a positive self-adjoint operator. Some regularization algorithms (in A. N. Tikhonov's sense) for this equation are investigated.Translated from Matematicheskie Zametki, Vol. 9, No. 4, pp. 415–420, April, 1971.I wish to thank Ya. I. Al'ber, O. A. Liskovts, and A. M. Il'in for their advice and useful comments.     

The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations

We study the asymptotic behavior of linear evolution equations of the type t_∂g=Dg+Lg−λg, where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator Dg+Lg. In the case Dg=−x_∂g, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg=−x_∂(xg), it is known that λ=1 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation t_∂f=Lf.By means of entropy–entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In other words, the linear operator has a spectral gap in the natural L² space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part.     

Hyers–Ulam stability of linear differential operator with constant coefficients

Let P(z) be a polynomial of degree n with complex coefficients and consider the n–th order linear differential operator P(D). We show that the equation P(D)f = 0 has the Hyers–Ulam stability, if and only if the equation P(z) = 0 has no pure imaginary solution. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On holomorphic solutions of some inhomogeneous linear differential equations in a banach space over a non-archimedean field

Let A be a closed linear operator on a Banach space $ \mathfrak{B} $ \mathfrak{B} over the field Ω of complex p-adic numbers having an inverse operator defined on the whole $ \mathfrak{B} $ \mathfrak{B} , and f be a locally holomorphic at 0 $ \mathfrak{B} $ \mathfrak{B} -valued vector function. The problem of existence and uniqueness of a locally holomorphic at 0 solution of the differential equation y ^(m) − Ay = f is considered in this paper. In particular, it is shown that this problem is solvable under the condition $ \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left\| {A^{ - n} } \right\|}} $ \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left\| {A^{ - n} } \right\|}} = 0. It is proved also that if the vector-function f is entire, then there exists a unique entire solution of this equation. Moreover, the necessary and sufficient conditions for the Cauchy problem for such an equation to be correctly posed in the class of locally holomorphic functions are presented.     

Generalized equations,variational inequalities and a weak Kantorovich theorem

We prove a Kantorovich-type theorem on the existence and uniqueness of the solution of a generalized equation of the form f(u)+g(u) ' 0f(u)+g(u)\owns 0 where f is a Fréchet-differentiable function and g is a maximal monotone operator defined on a Hilbert space. The depth and scope of this theorem is such that when we specialize it to nonlinear operator equations, variational inequalities and nonlinear complementarity problems we obtain novel results for these problems as well. Our approach to the solution of a generalized equation is iterative, and the solution is obtained as the limit of the solutions of partially linearized generalized Newton subproblems of the type Az+g(z) ' bAz+g(z)\owns b where A is a linear operator.     

hp‐FEM for second moments of elliptic PDEs with stochastic data. II: Exponential convergence for stationary singular covariance functions

We prove exponential rates of convergence of a class of hp Galerkin Finite Element approximations of solutions to a model tensor nonhypoelliptic equation in the unit square □ = (0, 1)² which exhibit singularities on ?□ and on the diagonal Δ = {(x, y) ∈ □ : x = y}, but are otherwise analytic in □. As we explained in the first part (Pentenrieder and Schwab, Research Report, Seminar for Applied Mathematics, 2010) of this work, such problems arise as deterministic second moment equations of linear, second order elliptic operator equations Au = f with Gaussian random field data f. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Hyponormality of Toeplitz Operators with Polynomial Symbols: An Extremal Case

If T_φ is a hyponormal Toeplitz operator with polynomial symbol φ = ḡ + f (f, g ∈ H^∞ (𝕋 )) such that g divides f, and if ψ := then where μ is the leading coefficient of ψ and 𝒵(ψ) denotes the set of zeros of ψ. In this paper we present a necessary and sufficient condition for T_φ to be hyponormal when φ enjoys an extremal case in the above inequality, that is, equality holds in the above inequality.