The higher fixed point theorem for foliations I. Holonomy invariant currents

In this paper, we prove a higher Lefschetz formula for foliations in the presence of a closed Haefliger current. To this end, we associate with such a current an equivariant cyclic cohomology class of Connes' C^∗-algebra of the foliation, and compute its pairing with the localized equivariant K-theory in terms of local contributions near the fixed points.     

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 a fixed point theorem of Kirk

W.A. Kirk [J. Math. Anal. Appl. 277 (2003) 645-650] first introduced the notion of asymptotic contractions and proved the fixed point theorem for this class of mappings. In this note we present a new short and simple proof of Kirk's theorem.     

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.     

Four functionals fixed point theorem

The Four Functionals Fixed Point Theorem is a generalization of the original, as well as the functional generalizations, of the Leggett–Williams Fixed Point Theorem. In the Four Functionals Fixed Point Theorem, neither the upper nor the lower boundary of the underlying set is required to map below or above the boundary in the functional sense. As an application, the existence of a positive solution to a second-order right focal boundary value problem is considered by applying both standard and nonstandard choices of functionals. An extension to multivalued maps is provided for completeness.     

The Kakutani fixed point theorem for Roberts spaces

Roberts spaces were the first examples of compact convex subsets of Hausdorff topological vector spaces (HTVS) where the Krein–Milman theorem fails. Because of this exotic quality they were candidates for a counterexample to Schauder's conjecture: any compact convex subset of a HTVS has the fixed point property. However, extending the notion of admissible subsets in HTVS of Klee [Math. Ann. 141 (1960) 286–296], Ngu [Topology Appl. 68 (1996) 1–12] showed the fixed point property for a class of spaces, including the Roberts spaces, he called weakly admissible spaces. We prove the Kakutani fixed point theorem for this class and apply it to show the non-linear alternative for weakly admissible spaces.     

The Schauder fixed point theorem in geodesic spaces

We give a direct proof of Schauder's fixed point theorem in the setting of geodesic metric spaces, generalizing the classical Schauder's theorem and improving a recent version of this theorem in CAT(κ)$\mathrm{CAT}(\kappa )$ spaces. As an application we prove an existence result for a variational inequality in the setting of CAT(κ)$\mathrm{CAT}(\kappa )$ spaces.     

A surjection theorem and a fixed point theorem for a class of positive operators

This paper is concerned with α-convex operators on ordered Banach spaces. A surjection theorem for 1-convex operators in order intervals is established by means of the properties of cone and monotone iterative technique. It is assumed that 1-convex operator A is increasing and satisfies Ay−Ax?M(y−x) for θ?x?y?v₀, where θ denotes the zero element and v₀ is a constant. Moreover, we prove a fixed point theorem for -convex operators by using fixed point theorem of cone expansion. In the end, we apply the fixed point theorem to certain integral equations.     

A fixed point theorem equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz theorem

In this note we prove a fixed point theorem and show that this fixed point theorem is equivalent to a recent generalization of the Knaster-Kuratowski-Mazurkiewicz theorem by Ky Fan.     

A fixed point theorem involving a hybrid inwardness-contraction condition

A fixed point theorem is proved which involves a hybrid inwardness-contraction condition and simultaneously extends three earlier results of NADLER, CLARKE and REICH, respectively. The aim of this paper is to prove a fixed point theorem which will simultaneously extends three earlier results of NADLER [5], CLARKE [2], and REICH [7], respectively.     

Euler’s fixed point theorem: The axis of a rotation

We give an elementary proof of what is perhaps the earliest fixed point theorem; namely Leonhard Euler’s theorem of 1775 on the existence of an axis v for any three-dimensional rotation R. The proof is constructive and shows that no multiplications are required to compute v. Dedicated to the memory of Leonhard Euler, “The Master of us all”, on the occasion of the 300th anniversary of his birth