Cleon S. Barroso

Krasnoselskii's fixed point theorem for weakly continuous maps

Abstract We consider the sum A+B : M→X , where M is a weakly compact and convex subset of a Banach space X, A : M→X is weakly continuous, and B∈ L (X) with ||Bp||⩽1, p⩾1. An alternative condition is given in order to guarantee the existence of fixed points in M for A+B. Some illustrative applications are given.

Semilinear elliptic equations and fixed points

In this paper, we deal with a class of semilinear elliptic equations in a bounded domain Q ⊂ R N , N ≥ 3, with C 1,1 boundary. Using a new fixed point result of the Krasnoselskii type for the sum of two operators, an existence principle of strong solutions is proved. We give two examples where the nonlinearity can be critical.

Optimal approximate fixed point results in locally convex spaces

Abstract Let C be a convex subset of a locally convex space. We provide optimal approximate fixed point results for sequentially continuous maps f : C → C ¯ . First, we prove that, if f ( C ) is totally bounded, then it has an approximate fixed point net. Next, it is shown that, if C is bounded but not totally bounded, then there is a uniformly continuous map f : C → C without approximate fixed point nets. We also exhibit an example of a sequentially continuous map defined on a compact convex set with no approximate fixed point sequence. In contrast, it is observed that every affine (not-necessarily continuous) self-mapping of a bounded convex subset of a topological vector space has an approximate fixed point sequence. Moreover, we construct an affine sequentially continuous map from a compact convex set into itself without fixed points.

Monotonicity inequalities for ther-area and a degeneracy theorem forr-minimal graphs

We establish monotonicity inequalities for the r-area of a complete oriented properly immersed r-minimal hypersurface in Euclidean space under appropriate quasi-positivity assumptions on certain invariants of the immersion. The proofs are based on the corresponding first variational formula. As an application, we derive a degeneracy theorem for an entire r-minimal graph whose defining function ƒ has first and second derivatives decaying fast enough at infinity: Its Hessian operator D2 ƒ has at least n − r null eigenvalues everywhere.

On topological groups with an approximate fixed point property

A topological group G has the Approximate Fixed Point (AFP) property on a bounded convex subset C of a locally convex space if every continuous affine action of G on C admits a net ( x i ) , x i ∈ C , such that x i - g ⁢ x i ⟶ 0 for all g ∈ G . In this work, we study the relationship between this property and amenability.