Exponential Expansiveness and Complete Admissibility For Evolution Families

Connections between uniform exponential expansiveness and complete admissibility of the pair are studied. A discrete version for a theorem due to Van Minh, Räbiger and Schnaubelt is presented. Equivalent characterizations of Perron type for uniform exponential expansiveness of evolution families in terms of complete admissibility are given.     

On the Stability of Functional Equations and a Problem of Ulam

In this paper, we study the stability of functional equations that has its origins with S. M. Ulam, who posed the fundamental problem 60 years ago and with D. H. Hyers, who gave the first significant partial solution in 1941. In particular, during the last two decades, the notion of stability of functional equations has evolved into an area of continuing research from both pure and applied viewpoints. Both classical results and current research are presented in a unified and self-contained fashion. In addition, related problems are investigated. Some of the applications deal with nonlinear equations in Banach spaces and complementarity theory.     

Stability problem of Ulam for generalized forms of Cauchy functional equation

Let G₁ be a vector space and G₂ a Banach space. In this paper, we solve the generalized Hyers-Ulam-Rassias stability problem for a generalized form

     

Abstract Cauchy Problems for Quasi-Linear Evolution Equations in the Sense of Hadamard

This paper is devoted to the well-posedness of abstract Cauchyproblems for quasi-linear evolution equations. The notion ofHadamard well-posedness is considered, and a new type of stabilitycondition is introduced from the viewpoint of the theory offinite difference approximations. The result obtained here generalizesnot only some results on abstract Cauchy problems closely relatedwith the theory of integrated semigroups or regularized semigroupsbut also the Kato theorem on quasi-linear evolution equations.An application to some quasi-linear partial differential equationof weakly hyperbolic type is also given. 2000 Mathematics SubjectClassification 34G20, 47J25 (primary), 47D60, 47D62 (secondary).     

Monodromy and Stability of a Class of Degenerate Planar Critical Points

Necessary and also sufficient monodromy conditions for a large class of degenerate singular points of planar differential systems are given. For these systems, we also find a computable expression of the principal term of the asymptotic expansion of the return map, which gives the stability of the point.     

Stability and continuity of polling systems

The stability of a polling system with exhaustive service and a finite number of users, each with infinite buffers is considered. The arrival process is more general than a Poisson process and the system is not slotted. Stochastic continuity of the stationary distributions, rates of convergence and functional limit theorems for the queue length and waiting time processes have also been proved. The results extend to the gated service discipline.     

Existence and Nonlinear Stability of Stationary Solutions to the Viscous Two-Phase Flow Model in a Half Line

The outflow problem for the viscous two-phase flow model in a halfline is investigated in the present paper. The existence and uniqueness of thestationary solution is shown for both supersonic state and sonic state at spatialfar field, and the nonlinear time stability of the stationary solution is also established in the weighted Sobolev space with either the exponential time decayrate for supersonic flow or the algebraic time decay rate for sonic flow.     

R-Boundedness of Smooth Operator-Valued Functions

In this paper we study R-boundedness of operator families , where X and Y are Banach spaces. Under cotype and type assumptions on X and Y we give sufficient conditions for R-boundedness. In the first part we show that certain integral operator are R-bounded. This will be used to obtain R-boundedness in the case that is the range of an operator-valued function which is in a certain Besov space . The results will be applied to obtain R-boundedness of semigroups and evolution families, and to obtain sufficient conditions for existence of solutions for stochastic Cauchy problems. Mark Veraar is supported by the Alexander von Humboldt foundation. His visit to Helsinki, which started this project, was funded by the Finnish Centre of Excellence in Analysis and Dynamics Research.