Eventually norm continuous semigroups on Hilbert space and perturbations

The eventually norm continuous semigroups on Hilbert space and perturbation are studied in this paper. By resolvent of infinitesimal generator, the sufficient and necessary conditions for eventually norm continuous semigroups are given. Using the result obtained, it is proved that if is infinitesimal generator of an eventually norm continuous semigroup T(t), then there is a subspace Ξ_A of such that, for any , the semigroup S(t) generated by preserves the property of T(t).     

An iff fixed point criterion for continuous self-mappings on a complete metric space

Summary Letf be a self-map on a metric space (X, d). We give necessary and sufficient conditions for the sequences {f ⁿ x} (x X) to be equivalent Cauchy. As a typical application we get the following result. Letf be continuous and (X, d) be complete. If, for anyx, y X d(f ⁿ x, f ⁿ y) 0 and for somec > 0, this convergence is uniform for allx, y inX withd(x, y) c thenf has a unique fixed pointp, andf ⁿ x p, for eachx inX. This theorem includes among others results of Angelov, Browder, Edelstein, Hicks and Matkowski.     

A noncooperativen-person semi-Markov game with a separable metric state space

A noncooperativen-person semi-Markov game with a separable metric state space is studied in this paper. We define the equilibrium condition of the noncooperative game, and find a necessary and sufficient condition for which the equilibrium condition holds, and prove that in our game system there possesses an equilibrium point so that each player has his own stationary equilibrium strategy.Partly supported by NSC Taiwan, R.O.C.     

Comparison inequalities for heat semigroups and heat kernels on metric measure spaces

We prove a certain inequality for a subsolution of the heat equation associated with a regular Dirichlet form. As a consequence of this inequality, we obtain various interesting comparison inequalities for heat semigroups and heat kernels, which can be used for obtaining pointwise estimates of heat kernels. As an example of application, we present a new method of deducing sub-Gaussian upper bounds of the heat kernel from on-diagonal bounds and tail estimates.     

Nuclearity of a nonnegative definite integral kernel on a separable metric space

The aim of this paper is to characterize the nuclearity of an integral operator, defined by a continuous non-negative definite square integrable kernel on a separable metric space, in terms of the integrability of the trace of the kernel function. Nuclearity here plays a role forU-statistics.     

A unified method for a class of convex separable nonlinear knapsack problems

In this paper, a unified algorithm is proposed for solving a class of convex separable nonlinear knapsack problems, which are characterized by positive marginal cost (PMC) and increasing marginal loss–cost ratio (IMLCR). By taking advantage of these two characteristics, the proposed algorithm is applicable to the problem with equality or inequality constraints. In contrast to the methods based on Karush–Kuhn–Tucker (KKT) conditions, our approach has linear computation complexity. Numerical results are reported to demonstrate the efficacy of the proposed algorithm for different problems.     

A library for continuous convex separable quadratic knapsack problems

The Continuous Convex Separable Quadratic Knapsack problem (CQKnP) is an easy but useful model that has very many different applications. Although the problem can be solved quickly, it must typically be solved very many times within approaches to (much) more difficult models; hence an efficient solution approach is required. We present and discuss a small open-source library for its solution that we have recently developed and distributed.     

Continuity and generators of dynamical semigroups for infinite Bose systems

For a class of quasifree quantum dynamical semigroups on the algebra of the canonical commutation relations (CCR) we give sufficient conditions for these semigroups to extend to ultraweakly continuous semigroups of normal operators on the von Neumann algebra associated with a representation of the CCR. Then the explicit form of the generators of the extended semigroups is calculated.     

A novel dual approach to nonlinear semigroups of Lipschitz operators

Lipschitzian semigroup refers to a one-parameter semigroup of Lipschitz operators that is strongly continuous in the parameter. It contains -semigroup, nonlinear semigroup of contractions and uniformly -Lipschitzian semigroup as special cases. In this paper, through developing a series of Lipschitz dual notions, we establish an analysis approach to Lipschitzian semigroup. It is mainly proved that a (nonlinear) Lipschitzian semigroup can be isometrically embedded into a certain -semigroup. As application results, two representation formulas of Lipschitzian semigroup are established, and many asymptotic properties of -semigroup are generalized to Lipschitzian semigroup.

     

New results on general observers for nonlinear systems

In this paper, we establish that detectability is a necessary condition for the existence ofgeneral observers (asymptotic or exponential) for nonlinear systems. Using this necessary condition, we show that there does not exist any general observer (asymptotic or exponential), for nonlinear systems with real parametric uncertainty, if the state equilibrium does not change with the parameter values and if the plant output function is purely a function of the state. Next, using center manifold theory, we derive necessary and sufficient conditions for the existence of general exponential observers for Lyapunov stable nonlinear systems. As an application of this result, we show that for the existence of general exponential observers for Lyapunov stable nonlinear systems, the dimension of the state of the general exponential observer should not be less than the number of critical eigenvalues of the linearization matrix of the state dynamics of the plant.     

A characterization of well-posed minimum problems in a complete metric space

Given a functionalI:X R, defined on a complete metric space (X,d), we give a necessary and sufficient condition for the minimum problem forI onX to be well posed or well posed in the generalized sense.This work was performed under the auspices of the Italian Research Council, Grant No. 115-3083-0-5179.     

Restricted algebras on inverse semigroups I,representation theory

The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. This is the first in a series of papers in which we have investigated a similar relation on inverse semigroups. We use a new concept of “restricted” representations and study the restricted semigroup algebras and corresponding C *‐algebras. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A relation for general and inverse semigroups

The relation in the title is S defined by

Well-posedness for a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions

We consider a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions. Well-posedness in the Hadamard sense for strong and weak solutions is proved by using the theory of nonlinear semigroups.