Realistic error bounds for a simple eigenvalue and its associated eigenvector

Summary An algorithm is described which, given an approximate simple eigenvalue and a corresponding approximate eigenvector, provides rigorous error bounds for improved versions of them. No information is required on the rest of the eigenvalues, which may indeed correspond to non-linear elementary divisors. A second algorithm is described which gives more accurate improved versions than the first but provides only error estimates rather than rigorous bounds. Both algorithms extend immediately to the generalized eigenvalue problem.Dedicated to A.S. Householder on his 75th birthday     

A resolvent operator technique for approximate solving of generalized system mixed variational inequality and fixed point problems

The main aim of this work is to use the resolvent operator technique to find the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point problems for Lipschitz mappings in Hilbert spaces. An example of applying the main result is also given. The results obtained in this work may be viewed as an extension, refinement and improvement of the previously known results.     

The approximate fixed point property in product spaces

In this paper we generalize to unbounded   convex subsets C$C$ of hyperbolic   spaces results obtained by W.A. Kirk and R. Espínola on approximate fixed points of nonexpansive mappings in product spaces _(C×M)∞${(C\times M)}_{\infty }$, where M$M$ is a metric space and C$C$ is a nonempty, convex, closed and bounded subset of a normed or a CAT(0)-space. We extend the results further, to families _(Cu)u∈M${\left(C_u\right)}_{u\in M}$ of unbounded convex subsets of a hyperbolic space. The key ingredient in obtaining these generalizations is a uniform quantitative version of a theorem due to Borwein, Reich and Shafrir, obtained by the authors in a previous paper using techniques from mathematical logic. Inspired by that, we introduce in the last section the notion of uniform approximate fixed point property   for sets C$C$ and classes of self-mappings of C$C$. The paper ends with an open problem.     

A subspace preconditioning algorithm for eigenvector/eigenvalue computation

We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigenspaces of a symmetric positive definite operatorA defined on a finite dimensional real Hilbert spaceV. In our applications, the dimension ofV is large and the cost of invertingA is prohibitive. In this paper, we shall develop an effective parallelizable technique for computing these eigenvalues and eigenvectors utilizing subspace iteration and preconditioning forA. Estimates will be provided which show that the preconditioned method converges linearly when used with a uniform preconditioner under the assumption that the approximating subspace is close enough to the span of desired eigenvectors.     

The approximate fixed point property in Banach and hyperbolic spaces

A geometrical characterization is given for those convex subsets of a Banach space (more generally a hyperbolic space) which possess the approximate fixed point property for nonexpansive mappings.     

An extension and simple proof of a constrained lattice fixed point theorem

We extend a result of Roth dealing with fixed points of lattice mappings which satisfy certain constraints. Presented by J. Mycielski.     

The approximation to a fixed point

In a spherically complete ultrametric space, a strictly contracting mapping has a fixed point. We indicate in this paper how this fixed point can either be reached or approximated.     

Fixed point, approximate fixed point and Kantorovich-Rubinstein maximum principle in convex metric spaces

We obtain necessary conditions for the existence of fixed point and approximate fixed point of nonexpansive and asymptotically nonexpansive maps defined on a closed bounded convex subset of a uniformly convex complete metric space and study the structure of the set of fixed points. We construct Mann type iterative sequences in convex metric space and study its convergence. As a consequence of fixed point results, we prove best approximation results. We also prove Kantorovich-Rubinstein maximum principle in convex metric spaces.     

A fixed point approach to stability of a quadratic equation

Using the fixed point alternative theorem we establish the orthogonal stability of the quadratic functional equation of Pexider type f (x+y)+g(x−y) = h(x)+k(y), where f, g, h, k are mappings from a symmetric orthogonality space to a Banach space, by orthogonal additive mappings under a necessary and sufficient condition on f.     

A fixed point theorem

We establish a fixed point theorem for a Lie group of isometries acting on a Riemannian manifold with nonnegative curvature.     

On fixed point sets and Lefschetz modules for sporadic simple groups

We consider 2-local geometries and other subgroup complexes for sporadic simple groups. For six groups, the fixed point set of a noncentral involution is shown to be equivariantly homotopy equivalent to a standard geometry for the component of the centralizer. For odd primes, fixed point sets are computed for sporadic groups having an extraspecial Sylow p-subgroup of order p³, acting on the complex of those p-radical subgroups containing a p-central element in their centers. Vertices for summands of the associated reduced Lefschetz modules are described.     

A combinatorial proof of a fixed point property

A class of finite simplicial complexes, called pseudo cones, is developed that has a number of useful combinatorial properties. A partially ordered set is a pseudo cone if its order complex is a pseudo cone. Pseudo cones can be constructed from other pseudo cones in a number of ways. Pseudo cone ordered sets include finite dismantlable ordered sets and finite truncated noncomplemented lattices. The main result of the paper is a combinatorial proof of the fixed simplex property for finite pseudo cones in which a combinatorial structure is constructed that relates fixed simplices to one another. This gives combinatorial proofs of some well known non-constructive results in the fixed point theory of finite partially ordered sets.     

A Schauder-type fixed point theorem

In this brief note we study Schauder's second fixed point theorem in the space $(\mathit{BC},6\cdot 6)$ of bounded continuous functions $\varphi :[0,\infty )\to {\Re }^n$ with a view to reducing the requirement that there is a compact map to the requirement that the map is locally equicontinuous. Several examples are given, both motivating and applying the theory.