Higher order differentiability properties of the composition and of the inversion operator

This paper contains theorems of r-th order Fréchet differentiability, with r≥1, for the autonomous composition operator and for the inversion operator in Schauder spaces. The optimality of the differentiability theorems for the composition is indicated by means of an ‘inverse result’. A main point of this paper is that (higher order) ‘sharp’ differentiability theorems for the composition operator can be proved by approximating the operator by composition operators whose superposing function is a polynomial, an idea which may be employed in other function space settings.     

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

Summary This paper considers the optimal quadratic cost problem (regulator problem) for a class of abstract differential equations with unbounded operators which, under the same unified framework, model in particular «concrete» boundary control problems for partial differential equations defined on a bounded open domain of any dimension, including: second order hyperbolic scalar equations with control in the Dirichlet or in the Neumann boundary conditions; first order hyperbolic systems with boundary control; and Euler-Bernoulli (plate) equations with (for instance) control(s) in the Dirichlet and/or Neumann boundary conditions. The observation operator in the quadratic cost functional is assumed to be non-smoothing (in particular, it may be the identity operator), a case which introduces technical difficulties due to the low regularity of the solutions. The paper studies existence and uniqueness of the resulting algebraic (operator) Riccati equation, as well as the relationship between exact controllability and the property that the Riccati operator be an isomorphism, a distinctive feature of the dynamics in question (emphatically not true for, say, parabolic boundary control problems). This isomorphism allows one to introduce a «dual» Riccati equation, corresponding to a «dual» optimal control problem. Properties between the original and the «dual» problem are also investigated.Research partially supported by the National Science Foundation under Grant NSF-DMS-8301668 and by the Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

Cauchy-type problem for an abstract differential equation with fractional derivatives

The uniform well-posedness of a Cauchy-type problem with two fractional derivatives and bounded operator A is proved. For an unbounded operator A we present a test for the uniform well-posedness of the problem under consideration consistent with the test for the uniform well-posedness of the Cauchy problem for an equation of second order.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 28–41.Original Russian Text Copyright © 2005 by A. V. Glushak.This revised version was published online in April 2005 with a corrected issue number.     

Asymptotics of the eigenvalues and the formula for the trace of perturbations of the Laplace operator on the sphere \mathbb{S}^2

In this paper, we study the asymptotics of the eigenvalues of the Laplace operator perturbed by an arbitrary bounded operator on the sphere . For the first time, for the partial differential operator of second order, the leading term of the second correction of perturbation theory is obtained. A connection between the coefficient of the second term of the asymptotics of the eigenvalues and the formula for the traces of the operator under consideration is established.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 434–448Original Russian Text Copyright © 2005 by V. A. Sadovnichii, Z. Yu. Fazullin.This revised version was published online in April 2005 with a corrected issue number.     

Iterationsverfahren höherer Ordnung in Banach-Räumen

The Newton process for operator equations in say a linear normed complete space converges under certain hypothesis about the Fréchet-derivatives of the operator with at least the order two. There are different ways to improve this Newton process. For instance you obtain a process of order three if you add a correction element containing the second Fréchet-derivative of the operator [1]. In the following note we will generalize this idea. In a recursive manner — by adding higher derivatives — we will construct iterative processes of any orderk (k > 1). A general theorem due toCollatz provides us error estimates for this processes. Last we will illustrate the processes by several examples.     

The fundamental solution of an operator-differential equation with singular boundary condition

The fundamental solution for a differential equation of any order, both even and odd, with operator coefficients with a boundary condition at the singular end is constructed. This solution is a function analytic in λ in a neighborhood of the set, being a supplement of the limiting spectrum of a self-adjoint extension of the minimal operator, on the real axis in the complex plane.     

Dirichlet空间上一类乘法算子的酉等价类

本文研究了Dirichlet空间D上由二阶Blaschke积φ定义的乘法算子M_φ的酉等价类.主要结果表明对于D的一个乘子ψ,乘法算子M_ψ酉等价于M_φ当且仅当存在常数θ,|θ|θ=1,使得ψ(z)=φ(θz).这一结果是完全不同于Hardy空间与Bergman空间上的相应结果.     

The linear factorization of polynomial operator pencils

Sufficient conditions are found for the linear factorization of polynomial operator pencils of arbitrary order in a Banach space. This factorization is generated by the solution of an appropriate operator equation.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 551–559, April, 1973.The author wishes to express his thanks to A. G. Kostyuchenko for the formulation of the problem.     

Implication in fuzzy logic

Intermediate truth values and the order relation “as true as” are interpreted. The material implication A → B quantifies the degree by which “B is at least as true as A.” Axioms for the → operator lead to a representation of → by the pseudo-Lukasiewicz model. A canonical scale for the truth value of a fuzzy proposition is selected such that the → operator is the Lukasiewicz operator and the negation is the classical 1−. operator. The mathematical structure of some conjunction and disjunction operators related to → are derived.     

Analysis of the spectrum of a Cartesian Perfectly Matched Layer (PML) approximation to acoustic scattering problems

In this paper, we study the spectrum of the operator which results when the Perfectly Matched Layer (PML) is applied in Cartesian geometry to the Laplacian on an unbounded domain. This is often thought of as a complex change of variables or “complex stretching.” The reason that such an operator is of interest is that it can be used to provide a very effective domain truncation approach for approximating acoustic scattering problems posed on unbounded domains. Stretching associated with polar or spherical geometry lead to constant coefficient operators outside of a bounded transition layer and so even though they are on unbounded domains, they (and their numerical approximations) can be analyzed by more standard compact perturbation arguments. In contrast, operators associated with Cartesian stretching are non-constant in unbounded regions and hence cannot be analyzed via a compact perturbation approach. Alternatively, to show that the scattering problem PML operator associated with Cartesian geometry is stable for real nonzero wave numbers, we show that the essential spectrum of the higher order part only intersects the real axis at the origin. This enables us to conclude stability of the PML scattering problem from a uniqueness result given in a subsequent publication.     

Extremal Solutions and Strong Relaxation for Second Order Multivalued Boundary Value Problems

In this paper we study semilinear second order differential inclusions involving a multivalued maximal monotone operator. Using notions and techniques from the nonlinear operator theory and from multivalued analysis, we obtain “extremal” solutions and we prove a strong relaxation theorem. This paper has been partially supported by the State Committee for Scientific Research of Poland (KBN) under research grants No. 2 P03A 003 25 and No. 4 T07A 027 26.     

On the spectrum of a differential operator of high-order

In this paper, sufficient conditions for the spectrum of the operator of high order to be discrete and unbounded below are obtained.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 188–193.Original Russian Text Copyright © 2005 by M. G. Gimadislamov.This revised version was published online in April 2005 with a corrected issue number.     

Numerical analysis of a non‐singular boundary integral method: Part I. The circular case

In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we present here a method which consists in inverting, on a finite element space, a non‐singular integral operator. This operator is a geometrical perturbation of the Steklov operator, and we precisely define the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme. Furthermore, this numerical scheme does not give rise to any singular integral. The scheme can also be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single layer potential, the special quadrature formula being defined by the introduction of a neighbouring curve. In the present paper, we prove stability and we give error estimates of our numerical scheme when the Laplace problem is set on a disk. We will extend our results to any domains by using compact perturbation arguments, in a second paper. Copyright © 2001 John Wiley & Sons, Ltd.     

Eigenvalue estimates for Dirac operators with parallel characteristic torsion

Assume that the compact Riemannian spin manifold (Mn,g) admits a G-structure with characteristic connection ∇ and parallel characteristic torsion (∇T=0), and consider the Dirac operator D1/3 corresponding to the torsion T/3. This operator plays an eminent role in the investigation of such manifolds and includes as special cases Kostant's “cubic Dirac operator” and the Dolbeault operator. In this article, we describe a general method of computation for lower bounds of the eigenvalues of D1/3 by a clever deformation of the spinorial connection. In order to get explicit bounds, each geometric structure needs to be investigated separately; we do this in full generality in dimension 4 and for Sasaki manifolds in dimension 5.     

p-Quasihyponormal operators have scalar extensions of order 6

In this paper we show that every p-quasihyponormal operator has a scalar extension of order 6, i.e., is similar to the restriction to a closed invariant subspace of a scalar operator of order 6, where 0<p<1. As a corollary, we get that every p-quasihyponormal operator with rich spectra has a nontrivial invariant subspace. Also we show that Aluthge transforms preserve an analogue of the single-valued extension property for W²(D,H) and an operator T.     

Direct and converse approximation theorems for the Shepard operator

Direct and converse approximation theorems for the Shepard operator (1) are given in uniform metric. The main result is Theorem 3 which completes the characterization of Lipschitz classes by the order of approximation by the Shepard operator for λ>2. Research supported by National Science Foundation of the Hungarian Academy of Sciences, Grant No. 1801     

Convergence of the iteration of Halley’s family and Smale operator class in Banach space

Smale operator classes of any order for nonlinear operators in Banach space are introduced. For an operatorf in Smale operator class of orderk, a proper condition for the convergence and the exact estimations error for the iteration of Halley’s family {H _j,k ⁿ } _n=0 ^∞ (1≤j≤k) are given. This Halley’s family is a higher order explicit generalization of Newton iteration. Project supported by China State Major Key Project for Basic Research and Zhejiang Provincial Natrural Science Foundation.     

Existence of fractional differential chains and factorizations based on transformations

In this work, we deal with the existence of the fractional integrable equations involving two generalized symmetries compatible with nonlinear systems. The method used is based on the Bä cklund transformation or B‐transformation. Furthermore, we shall factorize the fractional heat operator in order to yield the fractional Riccati equation. This is done by utilizing matrix transform Miura type and matrix operators, that is, matrices whose entries are differential operators of fractional order. The fractional differential operator is taken in the sense of Riemann–Liouville calculus. Copyright © 2014 John Wiley & Sons, Ltd.     

Maximum principle for quasilinear SPDE’s on a bounded domain without regularity assumptions

We prove a maximum principle for local solutions of quasi-linear parabolic stochastic PDEs, with non-homogeneous second order operator on a bounded domain and driven by a space–time white noise. Our method based on an approximation of the domain and the coefficients of the operator, does not require regularity assumptions. As in previous works by Denis et al. (2005, 2009)  and , the results are consequences of Itô’s formula and estimates for the positive part of local solutions which are non-positive on the lateral boundary.     

One-parameter orthogonality relations for basic hypergeometric series

The second order hypergeometric q-difference operator is studied for the value c = −q. For certain parameter regimes the corresponding recurrence relation can be related to a symmetric operator on the Hilbert space ℓ²( ). The operator has deficiency indices (1, 1) and we describe as explicitly as possible the spectral resolutions of the self-adjoint extensions. This gives rise to one-parameter orthogonality relations for sums of two ₂₁-series. In particular, we find that the Ismail-Zhang q-analogue of the exponential function satisfies certain orthogonality relations.