Some extremely amenable groupsQuelques groupes extrêmement moyennables

A topological group G is extremely amenable if every continuous action of G on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology but not with the uniform one. Strengthening a de la Harpe's result, we show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group. To cite this article: T. Giordano, V. Pestov, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 273–278.     

On Boundary Conditions for Wedge Operators on Radial Sets

We present a theorem about calculation of fixed point index for k-ψ-contractive operators with 0 ≤ k < 1 defined on a radial set of a wedge of an infinite-dimensional Banach space. Then, results on the existence of eigenvectors and nonzero fixed points are obtained.     

Best proximity points: global optimal approximate solutions

Let A and B be non-empty subsets of a metric space. As a non-self mapping \({T:A\longrightarrow B}\) does not necessarily have a fixed point, it is of considerable interest to find an element x in A that is as close to Tx in B as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x in A such that the error d(x, Tx) is minimum, where d is the distance function. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, to the fixed point equation Tx = x when there is no exact solution. As the distance between any element x in A and its image Tx in B is at least the distance between the sets A and B, a best proximity pair theorem achieves global minimum of d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). The purpose of this article is to establish best proximity point theorems for contractive non-self mappings, yielding global optimal approximate solutions of certain fixed point equations. Besides establishing the existence of best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.     

Existence of solutions of sublinear elliptic systems with quadratic growth in the gradient

In this paper, we deal with the following sublinear elliptic system: $${{\left\{\begin{array}{ll}-\Delta u + u + |\nabla u|^2 = a(x)|v|^p+ f,\quad x \in \mathbb{R}^N,\\ -\Delta v + v + |\nabla v|^2 = b(x)|u|^q+ g,\quad x \in \mathbb{R}^N,\\\end{array}\right.}}$$ where 0 <  p <  1 and 0 <  q <  1. Under suitable assumptions on the terms a, b, f and g and by using the Schauder fixed point theorem, we obtain a solution for an approximated system. The limit of the approximated solutions is a nonnegative solution.     

On Olaleru’s open problem on Gregus fixed point theorem

Let (X, d) be a complete metric space and ${TX \longrightarrow X }$ be a mapping with the property d(Tx, Ty) ≤ ad(x, y) + bd(x, Tx) + cd(y, Ty) + ed(y, Tx) + fd(x, Ty) for all ${x, y \in X}$ , where 0 < a < 1, b, c, e, f ≥ 0, a + b + c + e + f = 1 and b + c > 0. We show that if e + f > 0 then T has a unique fixed point and also if e + f ≥ 0 and X is a closed convex subset of a complete metrizable topological vector space (Y, d), then T has a unique fixed point. These results extend the corresponding results which recently obtained in this field. Finally by using our main results we give an answer to the Olaleru’s open problem.     

On bounds for total absolute curvature of surfaces in hyperbolic 3-space

We construct examples of surfaces in hyperbolic space which do not satisfy the Chern–Lashof inequality (which holds for immersed surfaces in Euclidean space). To cite this article: R. Langevin, G. Solanes, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Best proximity points: approximation and optimization

A best proximity point theorem explores the existence of an optimal approximate solution, known as a best proximity point, to the equations of the form Tx = x where T is a non-self mapping. The purpose of this article is to establish some best proximity point theorems for non-self non-expansive mappings, non-self Kannan- type mappings and non-self Chatterjea-type mappings, thereby producing optimal approximate solutions to some fixed point equations. Also, algorithms for determining such optimal approximate solutions are furnished in some cases.     

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets ${K\subset H}$ of a Hilbert space H by the metric entropy of the set K where the covering numbers ${N(K, \varepsilon)}$ of K by ${\varepsilon}$ -balls of H satisfy the Lorentz condition $$ \int\limits_{0}^{\infty} \left(\log N(K,\varepsilon) \right)^{r/s}\, d\varepsilon^{s} < \infty $$ for some fixed ${0 < r, s \le \infty}$ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 < r < ∞, s = ∞. The integral here is an improper Stieltjes integral. Moreover, we obtain optimal estimates of Gelfand numbers of absolutely convex hulls if the metric entropy satisfies the entropy condition $$\sup_{\varepsilon >0 }\varepsilon \left(\log N(K,\varepsilon) \right)^{1/r}\left(\log(2+\log N(K,\varepsilon))\right)^\beta < \infty$$ for some fixed 0 < r < ∞, ?∞ < β < ∞. Using inequalities between Gelfand and entropy numbers we also get optimal estimates of the metric entropy of the absolutely convex hull cov(K). As an interesting feature of the estimates, a sudden jump of the asymptotic behavior of Gelfand numbers as well as of the metric entropy of absolutely convex hulls occurs for fixed s if the parameter r crosses the point r = 2 and, if r = 2 is fixed, if the parameter β crosses the point β = 1. The results established in Hilbert spaces extend and recover corresponding results of several authors.     

An extension of Brouwer's fixed point theorem allowing discontinuities

In this article, we extend Brouwer's fixed point theorem – which states that every continuous mapping $\text{f}\text{:B→B}$ (a closed ball of $\text{R}{}^{\mathrm{n}}$) must have a fixed point – by allowing discontinuities of f, and we apply this extension to equilibrium theory in Economics. To cite this article: P. Bich, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Positive-fraction intersection results and variations of weak epsilon-nets

A connected Finsler space (M, F) is said to be homogeneous if it admits a transitive connected Lie group G of isometries. A geodesic in a homogeneous Finsler space (G / H, F) is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of G. In this paper, we study the problem of the existence of homogeneous geodesics on a homogeneous Finsler space, and prove that any homogeneous Finsler space of odd dimension admits at least one homogeneous geodesic through each point.     

A Krasnosel’skii theorem for orthogonal polygons starshaped via staircase (n + 1)-paths

Let S be a simply connected orthogonal polygon in the plane, and let n be fixed, n ≥ 1. If every two points of S are visible via staircase n-paths from a common point of S, then S is starshaped via staircase (n + 1)-paths. Moreover, the associated staircase (n + 1)-kernel is staircase (n + 1)-convex. The number two is best possible, and the number n + 1 is best possible for n ≥ 2.     

Hyers–Ulam Stability of General Jensen-Type Mappings in Banach Algebras

Using the fixed point method, we prove the Hyers–Ulam stability of homomorphisms in complex Banach algebras and complex Banach Lie algebras and also of derivations on complex Banach algebras and complex Banach Lie algebras for the general Jensen-type functional equation f(α x + β y) + f(α x ? β y) = 2α f(x) for any \({\alpha, \beta \in \mathbb{R}}\) with \({\alpha, \beta \neq 0}\) . Furthermore, we prove the hyperstability of homomorphisms in complex Banach algebras for the above functional equation with α = β = 1.     

On the Laplace equation with a supercritical nonlinear Robin boundary condition in the half-space

We study the Laplace equation in the half-space ${\mathbb{R}_{+}^{n}}$ with a nonlinear supercritical Robin boundary condition ${\frac{\partial u}{\partial\eta }+\lambda u=u\left\vert u\right\vert^{\rho-1}+f(x)}$ on ${\partial \mathbb{R}_{+}^{n}=\mathbb{R}^{n-1}}$ , where n ≥ 3 and λ ≥ 0. Existence of solutions ${u \in E_{pq}= \mathcal{D}^{1, p}(\mathbb{R}_{+}^{n}) \cap L^{q}(\mathbb{R}_{+}^{n})}$ is obtained by means of a fixed point argument for a small data $f \in {L^{d}(\mathbb{R}^{n-1})}$ . The indexes p, q are chosen for the norm ${\Vert\cdot\Vert_{E_{pq}}}$ to be invariant by scaling of the boundary problem. The solution u is positive whether f(x) > 0 a.e. ${x\in\mathbb{R}^{n-1}}$ . When f is radially symmetric, u is invariant under rotations around the axis {x _n  = 0}. Moreover, in a certain L ^q -norm, we show that solutions depend continuously on the parameter λ ≥ 0.     

Distortion and stability of the fixed point property for non-expansive mappings

Let X be a Banach space. We say that X satisfies the fixed point property (weak fixed point property) if every non-expansive mapping defined from a convex closed bounded (convex weakly compact) subset of X into itself has a fixed point. We say that X satisfies the stable fixed point property (stable weak fixed point property) if the same is true for every equivalent norm which is close enough to the original one. Denote by P(X) the set formed by all equivalent norms with the topology of the uniform convergence on the unit ball of X. We prove that the subset of P(X) formed by the norms failing the fixed point property is dense in P(X) when X is a non-distortable space which fails the fixed point property. In particular, no renorming of ?₁ can satisfy the stable fixed point property. Furthermore, we show some examples of distortable spaces failing the weak fixed point property, which can be renormed to satisfy the stable weak fixed point property. As a consequence we prove that every separable Banach space can be renormed to satisfy the stable weak fixed point property.     

Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes,I

We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of Abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic contribution. We show how counting curves over finite fields provides us with detailed information about Siegel modular forms. To cite this article: C. Faber, G. van der Geer, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Jacobians of Sobolev homeomorphisms

Let ${\Omega\subset\mathbb{R}^{n}}$ be a domain. We show that each homeomorphism f in the Sobolev space ${W^{1,1}_{\rm loc}(\Omega,\mathbb{R}^{n})}$ satisfies either J _f  ≥ 0 a.e or J _f  ≤ 0 a.e. if n = 2 or n = 3. For n > 3 we prove the same conclusion under the stronger assumption that ${f\in W^{1,s}_{\rm loc}(\Omega,\mathbb{R}^{n})}$ for some s > [n/2] (or in the setting of Lorentz spaces).     

Extensions of Banach's Contraction Principle

Let A and B be nonempty subsets of a metric space. As a non-self mapping T: A → B does not necessarily have a fixed point, it is of considerable interest to find an element x that is as close to Tx as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x such that the error d(x, Tx) is minimum. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, of the fixed point equation Tx = x when there is no exact solution. As d(x, Tx) is at least d(A, B), a best proximity point theorem achieves an absolute minimum of the error d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). This article furnishes extensions of Banach's contraction principle to the case of non-self mappings. On account of the preceding argument, the proposed generalizations are formulated as best proximity point theorems for non-self contractions.     

Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes,II

We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of Abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic contribution. We show how counting curves over finite fields provides us with detailed information about Siegel modular forms. To cite this article: C. Faber, G. van der Geer, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Weak convergence results for inhomogeneous rotating fluid equations

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vector B(x): this is a generalization of the usual rotating fluid model (where B is constant). We prove the weak convergence of Leray-type solutions towards a vector field which satisfies the usual 2D Navier–Stokes equation in the regions of space where B is constant, with Dirichlet boundary conditions, and a heat–type equation elsewhere. The method of proof uses weak compactness arguments. To cite this article: I. Gallagher, L. Saint-Raymond, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Orthogonal sets: The axiom of choice and proof of a fixed point theorem

In this paper, we prove some fixed point theorem on orthogonal spaces. Our result improve the main result of the paper by Eshaghi Gordji et al. [On orthogonal sets and Banach fixed point theorem, to appear in Fixed Point Theory]. Also we prove a statement which is equivalent to the axiom of choice. In the last section, as an application, we consider the existence and uniqueness of a solution for a Volterra-type integral equation in L ^p space.