增算子的不动点和耦合不动点

Under some general continuous and compact conditions, the existence problems of fiked points andd coupled fixed points for increasing operators are studied. an application, we utilize the results obtained to study the existence of solutions for differential inclusions in Banach spaces.     

The existence of Hall polynomials for x2-bounded invariant subspaces of nilpotent linear operators

We prove the existence of Hall polynomials for $x^2$-bounded invariant subspaces of nilpotent linear operators.     

A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.     

<Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>-calculus for Products of Non-Commuting operators

It is shown that the product of two sectorial operators A and B admits a bounded H^∞-calculus on a Banach space X provided suitable commutator estimates and Kalton-Weis type assumptions on A and B are satisfied.     

Double families of integrated evolution operators

A theory of double families of evolution operators, which involves mappings between different spaces, is presented. Causality is in a form analogous to that in integrated semi-groups, but involves an integrated relationship between members of both families. The approach is from a dynamic systems point of view, and focuses very much on the trajectories defined by the evolution operators and their relationship to the solutions of implicit evolution equations. Illustrative examples concerned with dynamic boundary conditions are provided.     

The spectra of closed interpolated operators

Let (E ₀,E ₁) be a compatible couple of Banach spaces, and letE : 0Re1 be the complex interpolation spaces ofE ₀,E ₁. LetT be a closed linear operator onE ₀+E ₁, then the restrictionT ofT to eachE is closed. If we denote by the extended spectrum ofT inE , then, under appropriate conditions, it is shown that the map is an analytic multifunction in the strip {C0<Re<1}. We use these results to give some applications to the spectral theory of semigroups.     

Strongly continuous M,N-families of bounded operators

Corresponding to the strong wellpssedness of the Cauchy problems for complete second order linear differential equations in Banach spaces, we introduce a notion of strongly continuous M, N-families and establish some basic results for these operator families.The project supported by National Science Fundation of China.     

Local asymptotic attractivity for nonlinear quadratic functional integral equations

In this paper, an existence result for local asymptotic attractivity of the solutions is proved for a nonlinear quadratic functional integral equation under certain growth conditions which in turn gives the existence as well as asymptotic stability of solutions. A couple of examples are provided for indicating the natural realizations of abstract theory presented in the paper.     

Compact linear operators on functional spaces with two norms

Some principles of the operator theory in a linear space with two norms are established in this paper. The well-known Hilbert-Schmidt theorem on the eigenfunction expansion of sourcewise represented functions, Mercer's theorem and other results can be consider as special cases of the statements presented. The general approach proposed is used to construct the theory of symmetrizable operators and to investigate the asymptotic behaviour of eigenvalues of compact operators.This paper was translated by M. Gorbuchuk and V. GorbachukThis paper was translated by M. Gorbuchuk and V. Gorbachuk.     

On certain classes of functional inclusions with causal operators in Banach spaces

We introduce the notion of a multivalued causal operator and consider an abstract Cauchy problem in a Banach space for various classes of functional inclusions with causal operators. The methods of the topological degree theory for condensing maps are applied to obtain local and global existence results for this problem and to study the continuous dependence of a solution set on initial data. As application we generalize some existence results for semilinear functional differential inclusions and Volterra integro-differential inclusions with delay.     

Extremal feedback perturbations for families of closed operators

Let T- S, be a family of not necessarily bounded semi-Fredholm operators, where T and S are operators acting between Banach spaces X and Y, and where S is bounded with D(S) D(T). For compact sets , as well as for certain open sets , we investigate existence and minimal rank of bounded feedback perturbations of the form F=BE such that min.ind (T-S+F)=0 for all . Here B is a given operator from a linear space Z to Y and E is some operator from X to Z.We give a simple characterization of that situation, when such regularizing feedback perturbations exist and show that for compact sets the minimal rank never exceeds max { min.ind (T-S) }+1. Moreover, an example shows that the minimal rank, in fact, may increase from max {...} to max {...}+1, if the given B enforces a certain structure of the feedbachk perturbation F.However, the minimal rank is equal to max { min.ind (T-S) }, if is an open set such that min.ind (T-S) already vanishes for all but finitely many points . We illustrate this result by applying it to the stabilization of certain infinite-dimensional dynamical systems in Hilbert space.     

Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional operators

In this paper, by using the Leray-Schauder alternative, we have investigated the existence of mild solutions to first-order impulsive partial functional integrodifferential equations with nonlocal conditions in an α-norm. We assume that the linear part generates an analytic compact bounded semigroup, and that the nonlinear part is a Lipschitz continuous function with respect to the fractional power norm of the linear part. An example is also given to illustrate our main results.     

Generalization of the determinants for trace-potent linear operators

SupposeA is a bounded linear operator on a separable Hilbert space withA ^m of trace class for some positive integerm. A generalized determinant for the operatorI–A is defined, its properties studied and this determinant is then used to exhibit an inversion formula forI–A.     

Existence for bounded positive solutions of quasilinear elliptic equations in two-dimensional exterior domains

In this paper, by using the fixed point theory, we obtain a new existence result for bounded positive solutions of the quasilinear elliptic equations in two-dimensional exterior domains.     

Unimodular eigenvalues and invariant measures for linear operators

For a bounded linear operator in a complex separable Hilbert space we show that it accepts an invariant Borel probability measure of square integrable norm whose essential support spans the space, iff its eigenvectors with unimodular eigenvalues span the space.This research has been supported by the Economic Research Center (KOE) of the Athens University of Economics and Business.     

Invariant subspaces for polynomially bounded operators

Let T be a polynomially bounded operator on a Banach space X whose spectrum contains the unit circle. Then T∗ has a nontrivial invariant subspace. In particular, if X is reflexive, then T itself has a nontrivial invariant subspace. This generalizes the well-known result of Brown, Chevreau, and Pearcy for Hilbert space contractions.     

Maximal dissipativity for degenerate Kolmogorov operators

In this paper we are interested in studying the properties of an elliptic degenerate operator N₀ in the space L^p of with respect to an invariant measure μ. The existence of μ is proven under suitable conditions on coefficients of the operator. We prove that the closure of N₀ is m-dissipative in      

Trace class norm inequalities for localization operators

We give sharp estimates on the norms in the trace class of localization operators in terms of their symbols.