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.     

On the concept of generalized solution of operator equations in banach spaces

We introduce a new concept of generalized solution of operator equations with closed linear operator in a Banach space as an element of the completion of the space in certain locally convex topology. We prove a theorem on the existence and uniqueness of a generalized solution and give examples of finding the generalized solution for infinite systems of the linear algebraic equations. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9. pp. 1286–1290, September, 1996.     

A Generalized Analytic Operator-Valued Function Space Integral and a Related Integral Equation

We introduce a generalized Wiener measure associated with a Gaussian Markov process and define a generalized analytic operator-valued function space integral as a bounded linear operator from L _p into L _p^\prime (1<p ≤ 2) by the analytic continuation of the generalized Wiener integral. We prove the existence of the integral for certain functionals which involve some Borel measures. Also we show that the generalized analytic operator-valued function space integral satisfies an integral equation related to the generalized Schr?dinger equation. The resulting theorems extend the theory of operator-valued function space integrals substantially and previous theorems about these integrals are generalized by our results.     

A Generalized Analytic Operator-Valued Function Space Integral and a Related Integral Equation

We introduce a generalized Wiener measure associated with a Gaussian Markov process and define a generalized analytic operator-valued function space integral as a bounded linear operator from L _p into L _p^\prime (1<p ≤ 2) by the analytic continuation of the generalized Wiener integral. We prove the existence of the integral for certain functionals which involve some Borel measures. Also we show that the generalized analytic operator-valued function space integral satisfies an integral equation related to the generalized Schr?dinger equation. The resulting theorems extend the theory of operator-valued function space integrals substantially and previous theorems about these integrals are generalized by our results.     

On a time nonlocal problem for inhomogeneous evolution equations

Under consideration is some problem for inhomogeneous differential evolution equation in Banach space with an operator that generates a C ₀-continuous semigroup and a nonlocal integral condition in the sense of Stieltjes. In case the operator has continuous inhomogeneity in the graph norm. We give the necessary and sufficient conditions for existence of a generalized solution for the problem of whether the nonlocal data belong to the generator domain. Estimates on solution stability are given, and some conditions are obtained for existence of the classical solution of the nonlocal problem. All results are extended to a Sobolev-type linear equation, the equation in Banach space with a degenerate operator at the derivative. The time nonlocal problem for the partial differential equation, modeling a filtrating liquid free surface, illustrates the general statements.     

Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators

In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator P consisting of finitely or countably many distributional operators P _n , which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green function G with respect to L := P ^*T P now becomes a conditionally positive function. In order to support this claim we ensure that the distributional adjoint operator P ^* of P is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function G can be embedded into or even be equivalent to a generalized Sobolev space. As an application, we take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant s _f,X to data values sampled from an unknown generalized Sobolev function f at data sites located in some set X ì \mathbbR^d{X \subset \mathbb{R}^d}. We provide several examples, such as Matérn kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of well-known reproducing kernels are equivalent to a generalized Sobolev space. These examples further illustrate how we can rescale the Sobolev spaces by the vector distributional operator P. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the “best” kernel function for kernel-based approximation methods.     

Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. I. Abstract theory

N. Dunford and J.T. Schwartz (1963) striking Hilbert space theory about completeness of a system of root vectors (generalized eigenvectors) of an unbounded operator has been generalized by J. Burgoyne (1995) to the Banach spaces framework. We use the Burgoyne's theorem and prove n-fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. The theory will allow to consider, in application, boundary value problems for ODEs and elliptic PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions.     

Invariant distributions on a non-isotropic pseudo-Riemannian symmetric space of rank one

We investigate the structure of invariant distributions on a non-isotropic non-Riemannian symmetric space of rank one. Especially, the J-criterion related to the generalized Gelfand pair is shown for this space without imposing the condition on the eigenfuction of the Laplace-Bertrami operator.     

Study of generalized eigenfunctions of a perturbed isotropic elastic half‐space

We consider the self‐adjoint operator governing the propagation of elastic waves in a perturbed isotropic half‐space (perturbation with compact support of a homogeneous isotropic half‐space) with a free boundary condition. We propose a method to obtain, numerical values included, a complete set of generalized eigenfunctions that diagonalize this operator. The first step gives an explicit representation of these functions using a perturbative method. The unbounded boundary is a new difficulty compared with the method used by Wilcox [25], who set the problem in the complement of bounded open set. The second step is based on a boundary integral equations method which allows us to compute these functions. For this, we need to determine explicitly the Green's function of (A₀ – ω²), where A₀ is the self‐adjoint operator describing elastic waves in a homogeneous isotropic half‐space. Copyright © 2000 John Wiley & Sons, Ltd.     

On the solvability of an evolution variational inequality with a nonlocal space operator

By using semidiscretization and penalty methods, we prove the existence of a generalized solution of a variational inequality of parabolic type with a space operator monotone in the gradient and depending on an integral characteristic of the solution with respect to the space variables.     

An existence principle for a periodic solution of a differential equation in Banach space

The equation d²x/dt²=Ax +f(t, x) is considered in a Banach space E, where A is a fixed unbounded linear operator, andf(t, x) is a nonlinear operator which is periodic in t and satisfies a Lipschitz condition with respect to x E. Existence conditions have been obtained for a well defined generalized periodic solution of this equation, and also when this solution coincides with the true solution. Similar results have been obtained for the first order equation.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 105–112, July, 1968.     

Gaussian white noise with trajectories in the space S′(H)

In this paper we construct a Gaussian white noise with trajectories in the space of generalized functions over S with values in a separable Hilbert space H. We obtain a solution to the Cauchy problem for a linear differential-operator equation with additive white noise as a generalized random process with trajectories in the space of exponential distributions. We prove the existence of a solution in the case when the operator coefficient A generates a C ₀-semigroup and in the case when A generates an integrated semigroup.     

Non‐compactness of the solution operator to $ \bar \partial $ on the Fock‐space in several dimensions

We consider the canonical solution operator to restricted to (0, 1)‐forms with coefficients in the generalized Fock‐spaces (1) We will show that the canonical solution operator restricted to (0, 1)‐forms with ‐coefficients can be interpreted as a Hankel‐operator. Furthermore we will show that the canonical solution operator is not compact for m ≥ 2. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bilateral inequalities and implicit unilateral systems of the non-variational type

We prove that ageneralized solution exists for a bilateral problem relative to second order linear elliptic operator with principal part not in divergence form. We also prove that this solution isregular, when the obstacles are sufficiently regular, and that it provides an appropriate substitute for the solution when the hypotheses of regularity are not satisfied. A probabilistic characterization of this solution is also given.     

Linear boundary value problems for normally solvable operator equations in a Banach space

We consider linear boundary value problems for operator equations with generalized-invertible operator in a Banach or Hilbert space. We obtain solvability conditions for such problems and indicate the structure of their solutions. We construct a generalized Green operator and analyze its properties and the relationship with a generalized inverse operator of the linear boundary value problem. The suggested approach is illustrated in detail by an example.     

Generalized resolvents of closed dissipative operators in a Hilbert space with aF metric

Generalized resolvents of a closedF-dissipative operator in a Hilbert space with aF metric are defined and studied; in particular, a formula of generalized resolvents of such an operator is obtained.Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 261–272, August, 1976.In conclusion the author expresses his gratitude to A. V. Shtraus for his scientific guidance.     

On a Class of Analytic Operator Functions and Their Linearizations

We consider an operator function T in a Krein space which can formally be written as (0.1)but the last term on the right of (0.1) is replaced by a relatively form‐compact perturbation of a similar form. We study relations between the operator function T, a selfadjoint operator M in some Krein space, associated with T, and an operator which can be constructed with the help of the operator function –T^–1. The results are applied to a Sturm‐Liouville problem with a coefficient depending rationally on the eigenvalue parameter.     

On the number of eigenvalues of a matrix operator

We consider a matrix operator H in the Fock space. We prove the finiteness of the number of negative eigenvalues of H if the corresponding generalized Friedrichs model has the zero eigenvalue (0 = min σ _ess(H)). We also prove that H has infinitely many negative eigenvalues accumulating near zero (the Efimov effect) if the generalized Friedrichs model has zero energy resonance. We obtain asymptotics for the number of negative eigenvalues of H below z as z → −0.     

On generalized resolvents and spectral functions of differential operators of even order in a space of vector-valued functions

We consider a self-adjoint differential expression of even order in a space of vector-valued functions. We prove that an arbitrary generalized resolvent of the minimal operator L₀ induced by this differential expression is an integral operator and we derive a formula for all of the spectral functions (orthogonal and nonorthogonal) of the operator L₀.