1,300 research outputs found
Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees
We study the computational difficulty of the problem of finding fixed points
of nonexpansive mappings in uniformly convex Banach spaces. We show that the
fixed point sets of computable nonexpansive self-maps of a nonempty, computably
weakly closed, convex and bounded subset of a computable real Hilbert space are
precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A
uniform version of this result allows us to determine the Weihrauch degree of
the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is
equivalent to a closed choice principle, which receives as input a closed,
convex and bounded set via negative information in the weak topology and
outputs a point in the set, represented in the strong topology. While in finite
dimensional uniformly convex Banach spaces, computable nonexpansive mappings
always have computable fixed points, on the unit ball in infinite-dimensional
separable Hilbert space the Browder-Goehde-Kirk theorem becomes
Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is
equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive
mappings may not have any computable fixed points in infinite dimension. We
also study the computational difficulty of the problem of finding rates of
convergence for a large class of fixed point iterations, which generalise both
Halpern- and Mann-iterations, and prove that the problem of finding rates of
convergence already on the unit interval is equivalent to the limit operator.Comment: 44 page
Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov-Galerkin, and Monotone Mixed Methods
This work presents a comprehensive discretization theory for abstract linear
operator equations in Banach spaces. The fundamental starting point of the
theory is the idea of residual minimization in dual norms, and its inexact
version using discrete dual norms. It is shown that this development, in the
case of strictly-convex reflexive Banach spaces with strictly-convex dual,
gives rise to a class of nonlinear Petrov-Galerkin methods and, equivalently,
abstract mixed methods with monotone nonlinearity. Crucial in the formulation
of these methods is the (nonlinear) bijective duality map.
Under the Fortin condition, we prove discrete stability of the abstract
inexact method, and subsequently carry out a complete error analysis. As part
of our analysis, we prove new bounds for best-approximation projectors, which
involve constants depending on the geometry of the underlying Banach space. The
theory generalizes and extends the classical Petrov-Galerkin method as well as
existing residual-minimization approaches, such as the discontinuous
Petrov-Galerkin method.Comment: 43 pages, 2 figure
Continuous selections of multivalued mappings
This survey covers in our opinion the most important results in the theory of
continuous selections of multivalued mappings (approximately) from 2002 through
2012. It extends and continues our previous such survey which appeared in
Recent Progress in General Topology, II, which was published in 2002. In
comparison, our present survey considers more restricted and specific areas of
mathematics. Note that we do not consider the theory of selectors (i.e.
continuous choices of elements from subsets of topological spaces) since this
topics is covered by another survey in this volume
Spectra of weighted algebras of holomorphic functions
We consider weighted algebras of holomorphic functions on a Banach space. We
determine conditions on a family of weights that assure that the corresponding
weighted space is an algebra or has polynomial Schauder decompositions. We
study the spectra of weighted algebras and endow them with an analytic
structure. We also deal with composition operators and algebra homomorphisms,
in particular to investigate how their induced mappings act on the analytic
structure of the spectrum. Moreover, a Banach-Stone type question is addressed.Comment: 25 pages Corrected typo
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