On the quantitative asymptotic behavior of strongly nonexpansive mappings in banach and geodesic spaces

We give explicit rates of asymptotic regularity for iterations of strongly nonexpansive mappings T in general Banach spaces as well as rates of metastability (in the sense of Tao) in the context of uniformly convex Banach spaces when T is odd. This, in particular, applies to linear norm-one projections as well as to sunny nonexpansive retractions. The asymptotic regularity results even hold for strongly quasi-nonexpansive mappings (in the sense of Bruck), the addition of error terms and very general metric settings. In particular, we get the first quantitative results on iterations (with errors) of compositions of metric projections in CAT(?)-spaces (? > 0). Under an additional compactness assumption we obtain, moreover, a rate of metastability for the strong convergence of such iterations.     

Inclusion mappings between lp spaces

It is shown that the inclusion mapping from $\text{l}{}^{\mathrm{p}}\text{into}\text{l}{}^{\mathrm{q}}\text{is}\text{(}\text{2pq}\text{(pq ? 2p + 2q)}\text{, 1)}$-absolutely summing if 1 ? p ? q ? 2 and (p, 1)-absolutely summing if 1 ? p ? 2 ? q ? ∞ or 2 ? p ? q ? ∞. As a corollary, it follows that a (p, 1)-absolutely summing operator, p > 1, need not have the Dunford-Pettis property; this observation answering a question raised by Pelczynski.     

On the asymptotic equilibrium and asymptotic equivalence of differential equations in Banach spaces

We present some conditions for the asymptotic equilibrium of nonlinear differential equations and, in particular, a linear inhomogeneous equation in Banach spaces. We also discuss analogous problems for a linear equation with unbounded operator. Some obtained results are applied to problems of asymptotic equivalence. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 626–635, May, 2008.     

On asymptotic pointwise contractions in modular function spaces

We discuss the existence of fixed points of asymptotic pointwise contractions in modular function spaces.     

On strongly stable normal operators in Krein spaces

For bounded normal operators in Krein spaces we give a necessary and sufficient condition for strong stability. The same result for unitary operators was obtained by M.G.Krein [1] (see also [2]). For selfadjoint operators we refer to the papers of P.Jonas, H.Langer [3] and H.Langer [4].     

On Riesz spaces with b-property and strongly order bounded operators

A Riesz space E is said to have the b-property if each subset that is order bounded in the bidual remains to be order bounded in E. Properties of a Riesz space with the b-property, the relationship between the b-property and various classes of operators are studied.     

On asymptotic pointwise nonexpansive mappings in modular function spaces

We prove the existence of fixed points of asymptotic pointwise nonexpansive mappings in modular function spaces.     

Schatten properties,nuclearity and minimality of phase shift invariant spaces

We extend Feichtinger's minimality property on the smallest non-trivial time-frequency shift invariant Banach space, to the quasi-Banach case. Analogous properties are deduced for certain matrix spaces.We use these results to prove that the pseudo-differential operator $\mathrm{Op}(a)$ is a Schatten-q operator from $M^{\infty }$ to $M^p$ and r-nuclear operator from $M^{\infty }$ to $M^r$ when $a\in M^r$ for suitable p, q and r in $(0,\infty ]$.     

Universal spaces for asymptotic dimension

We construct a universal space for the class of proper metric spaces of bounded geometry and of given asymptotic dimension. As a consequence of this result, we establish coincidence of asymptotic dimension with asymptotic inductive dimension.     

On the asymptotic normality of frequency polygons for strongly mixing spatial processes

This paper establishes the asymptotic normality of frequency polygons in the context of stationary strongly mixing random fields indexed by $\mathbb {Z}^d$ . Our method allows us to consider only minimal conditions on the width bins and provides a simple criterion on the mixing coefficients. In particular, we improve in several directions a previous result by Carbon, Francq and Tran 2010.     

Corank and asymptotic filling-invariants for symmetric spaces

Let X be a Riemannian symmetric space of noncompact type. We prove that there exists an embedded submanifold which is quasi-isometric to a manifold with strictly negative sectional curvature, which intersects a given flat F in a geodesic line and which satisfies dim(Y) — 1 = dim(X) — rank(X). This yields an estimate of the hyperbolic corank of X. As another application we show that certain asymptotic filling invariants of X are exponential. Submitted: February 1999, Revised version: November 1999.     

Tree-graded spaces and asymptotic cones of groups

We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic groups, and to construct the first example of a finitely generated group with a continuum of non-π₁-equivalent asymptotic cones. Note that by a result of Kramer, Shelah, Tent and Thomas, continuum is the maximal possible number of different asymptotic cones of a finitely generated group, provided that the Continuum Hypothesis is true.     

On two generalizations of Pareto minimality

In this paper we investigate two generalizations of the Pareto minimality concept: infimality and approximate minimality. It is shown that existence conditions for these optimality notions are much weaker and that they allow a more complete characterization via linear and nonlinear scalarization than Pareto minimality. We further study some relations between those optimality structures and apply the results to the image of a vector-valued mapping.The author is indebted to Professor J. Jahn for his encouragement and for helpful discussions, and to the anonymous referees for their suggestions.