77 research outputs found
Weak dentability index of spaces
We compute the weak-dentability index of the spaces where is a
countable compact space. Namely , whenever . More generally,
if is a scattered compact whose height
satisfies with an
countable
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
Hyper-extensions in metric fixed point theory
We apply a modern axiomatic system of nonstandard analysis in metric fixed
point theory. In particular, we formulate a nonstandard iteration scheme for
nonexpansive mappings and present a nonstandard approach to fixed-point
problems in direct sums of Banach spaces.Comment: 12 page
Necessary Conditions for Nonsmooth Optimization Problems with Operator Constraints in Metric Spaces
This paper concerns nonsmooth optimization problems involving operator constraints given by mappings on complete metric spaces with values in nonconvcx subsets of Banach spaces. We derive general first-order necessary optimality conditions for such problems expressed via certain constructions of generalized derivatives for mappings on metric spaces and axiomatically defined subdifferentials for the distance function to nonconvex sets in Banach spaces. Our proofs arc based on variational principles and perturbation/approximation techniques of modern variational analysis. The general necessary conditions obtained are specified in the case of optimization problems with operator constraints dDScribcd by mappings taking values in approximately convex subsets of Banach spaces, which admit uniformly Gateaux differentiable renorms (in particular, in any separable spaces)
Algorithm for solutions of nonlinear equations of strongly monotone type and applications to convex minimization and variational inequality problems
Real-life problems are governed by equations which are nonlinear in nature.
Nonlinear equations occur in modeling problems, such as minimizing costs in
industries and minimizing risks in businesses. A technique which does not
involve the assumption of existence of a real constant whose calculation is
unclear is used to obtain a strong convergence result for nonlinear equations
of (p, {\eta})-strongly monotone type, where {\eta} > 0, p > 1. An example is
presented for the nonlinear equations of (p, {\eta})-strongly monotone type. As
a consequence of the main result, the solutions of convex minimization and
variational inequality problems are obtained. This solution has applications in
other fields such as engineering, physics, biology, chemistry, economics, and
game theory.Comment: 11 page
Kuelbs-Steadman spaces on Separable Banach spaces
The purpose of this paper is to construct a new class of separable Banach
spaces \K^p[\mathbb{B}], \; 1\leq p \leq \infty. Each of these spaces contain
the \mcL^p[\mathbb{B}] spaces, as well as the space \mfM[\R^\iy], of
finitely additive measures as dense continuous compact embeddings. These spaces
are of interest because they also contain the Henstock-Kurzweil integrable
functions on . Finally, we offer a interesting approach to the
Fourier transform on \K^p[\mathbb{B}].Comment: pages 17. arXiv admin note: text overlap with arXiv:2001.0000
Dual Connections in Nonparametric Classical Information Geometry
We construct an infinite-dimensional information manifold based on
exponential Orlicz spaces without using the notion of exponential convergence.
We then show that convex mixtures of probability densities lie on the same
connected component of this manifold, and characterize the class of densities
for which this mixture can be extended to an open segment containing the
extreme points. For this class, we define an infinite-dimensional analogue of
the mixture parallel transport and prove that it is dual to the exponential
parallel transport with respect to the Fisher information. We also define
{\alpha}-derivatives and prove that they are convex mixtures of the extremal
(\pm 1)-derivatives
Discrete Approximations of Differential Inclusions in Infinite-Dimensional Spaces
In this paper we study discrete approximations of continuous-time evolution systems governed by differential inclusions with nonconvex compact values in infinite-dimensional spaces. Our crucial result ensures the possibility of a strong Sobolev space approximation of every feasible solution to the continuous-time inclusion by its discrete-time counterparts extended as Euler\u27s broken lines. This result allows us to establish the value and strong solution convergences of discrete approximations of the Bolza problem for constrained infinite-dimensional differential/evolution inclusions under natural assumptions on the initial data
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