We exhibit a class of nonlinear operators with the property that their iterates converge to their unique fixed points even when computational errors are present. We also show that most (in the sense of the Baire category) elements in an appropriate complete metric space of operators do, in fact, possess this property.

We consider continuous descent methods for the minimization of convex functions defined on a general Banach space. In our previous work we showed that most of them (in the sense of Baire category) converged. In the present paper we show that convergent continuous descent methods are stable under small perturbations.

In this work we show that, in the class of L ? ((0, T ); 2 (T 3 )) distributional solutions incompressible Navier-Stokes system, ones which are smooth some open interval times meagre sense Baire category, and Leray a nowhere dense set.

We introduce a covering conjecture and show that it holds below ADR + “Θ is regular”. We then use it to show that in the presence of mild large cardinal axioms, PFA implies that there is a transitive model containing the reals and ordinals and satisfying ADR + “Θ is regular”. The method used to prove the Main Theorem of this paper is the core model induction. The paper contains the first applic...

We study a class of equilibrium problems which is identified with a complete metric space of functions. For most elements of this space of functions in the sense of Baire category, we establish that the corresponding equilibrium problem possesses a unique solution and is well-posed.

For a subset A of an -group B, r(A,B) denotes the relative uniform closure of A in B. RX denotes the -group of all real-valued functions on the set X, and when X is a topological space, C∗(X) is the -group of all bounded continuous real-valued functions, and B(X) is the -group of all Baire functions. We show that B(X) = r (C∗(X), B(X)) = r ¡ C∗(X), R ¢ . This would appear to be a purely order-t...

Abstract We show that for a given initial point the typical, in sense of Baire category, nonexpansive compact valued mapping F has following properties: there is unique sequence successive approximations and this converges to fixed . In case separable Banach spaces we typical residual set points have trajectory.

One of the main goals of computable analysis is that of formalizing the complexity of theorems from real analysis. In this setting Weihrauch reductions play the role that Turing reductions do in standard computability theory. Via coding, we can transfer computability and topological results from the Baire space ω to any space of cardinality 2א0 , so that e.g. functions over R can be coded as fu...

We investigate the computable content of certain theorems which are sometimes called the “principles” of the theory of Banach spaces. Among these the main theorems are the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also study closely related theorems, as Banach’s Inverse Mapping Theorem, the Theorem on Condensation of Singularities and the BanachStein...