We continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL0 over RCA0. We show that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for proving these geometrical Hahn–Banach...

The aim of the present paper is to introduce the asymmetric locally convex spaces and to prove some basic properties. Among these I do mention the analogs of the EidelheitTuckey separation theorems, of the Alaoglu-Bourbaki theorem on the weak compactness of the polar of a neighborhood of 0, and a Krein-Milman-type theorem. These results extend those obtained by Garcı́a-Raffi et al. (2003) and Co...

For a one-dimensional nonlocal nonconvex singular perturbation problem with a noncoercive periodic well potential, we prove a Γ-convergence theorem and show compactness up to translation in all L and certain Orlicz spaces for sequences of bounded energy. This generalizes work of Alberti, Bouchitté and Seppecher [1] for the coercive two-well case. The theorem has applications to a certain thin-f...

This paper proves a general Uhlenbeck compactness theorem for sequences of solutions Yang–Mills flow on Riemannian manifolds dimension $$n \ge 4,$$ including rectifiability the singular set at finite or infinite time.

Existence and multiplicity results are obtained for superlinear p-Laplacian equations without the Ambrosetti and Rabinowitz condition. To overcome the difficulty that the Palais-Smale sequences of the EulerLagrange functional may be unbounded, we consider the Cerami sequences. Our results extend the recent results of Miyagaki and Souto [ J. Differential Equations 245 (2008), 3628–3638].

1. Properties of J-holomorphic curves 1 1.1. Basic definitions 1 1.2. Unique continuation and critical points 5 1.3. Simple curves 8 1.4. Adjunction inequality 9 2. Gromov compactness 12 2.1. Gromov compactness theorem 12 2.2. Energy estimate and bubbling 15 2.3. The isoperimetric inequality 19 2.4. Bubbles connect 22 3. Moduli spaces of J-holomorphic curves 25 3.1. The Fredholm setup 25 3.2. T...

The concept of quasi-continuity and the new concept of almost compactness for a function are the basis for the extension of the contraction principle in large devi­ ations presented here. Important equivalences for quasi-continuity are proved in the case of metric spaces. The relation between the exponential tightness of a sequence of stochastic processes and the exponential tightness of its tr...

This paper regards initial value problem for second order impulsive integro-differential equations as some nonlinear vector system. By means of the Mönch′s fixed point theorem, some existence theorems of solutions of the initial value problem are established. The results are newer than all of the previous ones because of the more general form compactness-type condition and the weaker restrictio...

A standard example of an impredicative definition is that of the way-below relation [11], and thus of local compactness in (point-free) topology [6]. Local compactness can be dealt with predicatively as in [8], [5]; the particularly convenient re-formulation to be presented is based on a recent result by Peter Aczel [2], and on the observation that, in set-presented topologies, the way-below re...

A strong compactness result in the spirit of the Lions-Aubin-Simon lemma is proven for piecewise constant functions in time (uτ ) with values in a Banach space. The main feature of our result is that it is sufficient to verify one uniform estimate for the time shifts uτ − uτ (· − τ) instead of all time shifts uτ − uτ (· − h) for h > 0, as required in Simon’s compactness theorem. This simplifies...