Christian Bargetz

Convergence properties of dynamic string-averaging projection methods in the presence of perturbations

Assuming that the absence of perturbations guarantees weak or strong convergence to a common fixed point, we study the behavior of perturbed products of an infinite family of nonexpansive operators. Our main result indicates that the convergence rate of unperturbed products is essentially preserved in the presence of perturbations. This, in particular, applies to the linear convergence rate of dynamic string-averaging projection methods, which we establish here as well. Moreover, we show how this result can be applied to the superiorization methodology.

σ-porosity of the set of strict contractions in a space of non-expansive mappings

We consider the space of non-expansive mappings on a bounded, closed and convex subset of a Banach space equipped with the metric of uniform convergence. We show that the set of strict contractions is a σ-porous subset. If the underlying Banach space is separable, we exhibit a σ-porous subset of the space of non-expansive mappings outside of which all mappings attain the maximal Lipschitz constant one at typical points of their domain.

Solution to the Cauchy problem for quasi-hyperbolic systems by vector-valued convolution

We solve the initial value problem for quasi-hyperbolic systems of partial differential equations with distributional initial data by vector-valued convolution. We show that every initial value problem of a quasi-hyperbolic system of partial differential equations with temperate initial data has a distributional solution in the sense of L. Hörmander which can be represented as a convolution of the fundamental matrix with the initial data. Moreover, we show that for systems that are correct in the sense of Petrovsky, the solution of the initial value problem is a differentiable distribution-valued function and the solution converges to the given initial data if converges to zero. The special case of the distributional Cauchy problem for the heat equation is considered by R. Dautray and J. L. Lions and by Z. Szmydt and, more recently, by M. A. Chaudhry and M. H. Kazi and by W. Kierat and K. Skornik. The use of the theory of vector-valued distributions allows for a generalization to systems of partial differential equations which are correct in the sense of Petrovsky or, yet more general, quasi-hyperbolic systems.

Linear convergence rates for extrapolated fixed point algorithms

ABSTRACT We establish linear convergence rates for a certain class of extrapolated fixed point algorithms which are based on dynamic string-averaging methods in a real Hilbert space. This applies, in particular, to the extrapolated simultaneous and cyclic cutter methods. Our analysis covers the cases of both metric and subgradient projections.

Pivot duality of universal interpolation and extrapolation spaces

Abstract It is a widely used method, for instance in perturbation theory, to associate with a given C 0 -semigroup its so-called interpolation and extrapolation spaces. In the model case of the shift semigroup acting on L 2 ( R ) , the resulting chain of spaces recovers the classical Sobolev scale. In 2014, the second named author defined the universal interpolation space as the projective limit of the interpolation spaces and the universal extrapolation space as the completion of the inductive limit of the extrapolation spaces, provided that the latter is Hausdorff. In this note we use the notion of the dual with respect to a pivot space in order to show that the aforementioned inductive limit is Hausdorff and already complete if we consider a C 0 -semigroup acting on a reflexive Banach space. If the space is Hilbert, then the inductive limit can be represented as the dual of the projective limit whenever a power of the generator of the initial semigroup is a self-adjoint operator. In the case of the classical Sobolev scale we show that the latter duality holds, and that the two universal spaces were already studied by Laurent Schwartz in the 1950s. Our results and examples complement the approach of Haase, who in 2006 gave a different definition of universal extrapolation spaces in the context of functional calculi. Haase avoids the inductive limit topology precisely for the reason that it a priori cannot be guaranteed that the latter is always Hausdorff. We show that this is indeed the case provided that we start with a semigroup defined on a reflexive Banach space.

Porosity Results for Sets of Strict Contractions on Geodesic Metric Spaces

We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic

A separable Fréchet space of almost universal disposition

\mathrm{CAT}(\kappa)

Convolution of vector-valued distributions: A survey and comparison

-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in these settings. We prove that the strict contractions form a negligible subset of this space in the sense that they form a

Convolvability and regularization of distributions

\sigma

Completing the Valdivia–Vogt tables of sequence-space representations of spaces of smooth functions and distributions

-porous subset. For certain separable and complete metric spaces we show that a generic nonexpansive mapping has Lipschitz constant one at typical points of its domain. These results contain the case of nonexpansive self-mappings and the case of nonexpansive set-valued mappings as particular cases.