Some notes on fixed points of quasi-contraction maps

In this paper, we shall give some results about fixed points of quasi-contraction maps on cone metric spaces. These results generalize some recent results.     

On Monch type multi-valued maps and fixed points

In this work, some fixed point theorems for a new class of Chandrabhan maps are proved which in turn include the fixed point theorems of Monch, Sadovskii, Darbo, Krasnoselskii, Dhage, and Covitz and Nadler as special cases.     

Surjectivity and fixed points of relaxed dissipative multifunctions. Differential inclusions approach

One from the most important properties of accretive and monotone operators is the existence of zeros and surjectivity. In the paper we introduce relaxed variants of dissipative, accretive and monotone operators. Using essentially the properties of the solution set of appropriate differential inclusions we study the existence of zeros of such operators. As corollaries the existence of fixed points of relaxed contractive and relaxed nonexpansive multifunctions are obtained.     

Existence and comparison results for fixed points of multifunctions with applications to normal-form games

In this paper we apply generalized iteration methods to prove comparison results which show how fixed points of a multifunction can be bounded by least and greatest fixed points of single-valued functions. As an application we prove existence and comparison results for fixed points of multifunctions. These results are applied to normal-form games, by proving existence and comparison results for pure and mixed Nash equilibria and their utilities.     

On fixed points of permutations

The number of fixed points of a random permutation of {1,2,…,n} has a limiting Poisson distribution. We seek a generalization, looking at other actions of the symmetric group. Restricting attention to primitive actions, a complete classification of the limiting distributions is given. For most examples, they are trivial – almost every permutation has no fixed points. For the usual action of the symmetric group on k-sets of {1,2,…,n}, the limit is a polynomial in independent Poisson variables. This exhausts all cases. We obtain asymptotic estimates in some examples, and give a survey of related results. This paper is dedicated to the life and work of our colleague Manfred Schocker. We thank Peter Cameron for his help. Diaconis was supported by NSF grant DMS-0505673. Fulman received funding from NSA grant H98230-05-1-0031 and NSF grant DMS-0503901. Guralnick was supported by NSF grant DMS-0653873. This research was facilitated by a Chaire d’Excellence grant to the University of Nice Sophia-Antipolis.     

Efficacy of cones on topological vector spaces and application to common fixed points of multifunctions

Let (E, τ) be a topological vector space and P a cone in E. We shall define a topology τ _P on E so that (E, τ _P ) is a normable topological vector space and P is a normal cone with normal constant M = 1. Then by using the norm, we shall give some results about common fixed points of two multifunctions on cone metric spaces.     

On multifunctions

In this paper, we introduce and study γ-continuous multifunctions as a generalization of quasi-continuous multifunctions due to Popa in 1985 and precontinuous multifunctions due to Popa in 1988. Some characterizations and several properties concerning upper (lower) γ-continuous multifunctions are obtained. The relationships between upper (lower) γ-continuous multifunctions and some known concepts are also discussed.     

A random fixed point theorem for multifunctions

A Class of multifunctions is introduced and a random fixed point theorem for pairs of measurable multifunctions belonging to this class is proved. The result is then used to study the existence of solutions for a class of random operator equations     

On cyclic fixed points of spectra

For a finite $p$ -group $G$ and a bounded below $G$ -spectrum $X$ of finite type mod  $p$ , the $G$ -equivariant Segal conjecture for $X$ asserts that the canonical map $X^G \rightarrow X^{hG}$ , from $G$ -fixed points to $G$ -homotopy fixed points, is a $p$ -adic equivalence. Let $C_{p^n}$ be the cyclic group of order  $p^n$ . We show that if the $C_p$ -equivariant Segal conjecture holds for a $C_{p^n}$ -spectrum $X$ , as well as for each of its geometric fixed point spectra $\varPhi ^{C_{p^e}}(X)$ for $0 < e < n$ , then the $C_{p^n}$ -equivariant Segal conjecture holds for  $X$ . Similar results also hold for weaker forms of the Segal conjecture, asking only that the canonical map induces an equivalence in sufficiently high degrees, on homotopy groups with suitable finite coefficients.     

On fixed points of conformal pseudogroups

We show in this paper Theorem 2 that if (H, H ₁) is a pseudogroup generated by a finite numberH ₁ of germs of conformal diffeomorphisms of defined on a sufficiently small discD, which is not linearizable and such that the linear group (L,H ₁)={g(0)/g(H,H ₁)}* is dense in *, then the set of fixed points of the pseudogroup (H, H ₁) is dense inD. This implies the abundance of distinct homotopy classes of loops in leaves of foliations defined in ² by generic polynomial vector fields as well as for germs of holomorphic vector fields in ² beginning with generic jets, both of degree at least 2. These homotopy classes may be realized arbitrarily close to the line at infinity or to 0, respectively. This shows the genericity of polynomial vector fields with infinite Petrovsky-Landis genus ([5]).The idea of the proof is very simple. Ifg is a non-linear conformal diffeomorphism with multiplier =g'(0), then the map obtained by the composition ofg and the linear map with multiplier ^–1 will have at 0 a fixed point of multiplicity at least 2. Since we may approximate ^–1 by elementsh in the pseudogroup and the multiplicity of fixed points satisfy a law of conservation of number, we obtain thath o g has fixed points close to 0. These fixed points appear as a by product of the relative nonlinearity of the generators of the pseudogroup, since linearizable pseudogroups have 0 as an isolated fixed point. The fixed points obtained are not conjugate since they have distinct multipliers.The main technical tool is the angular derivative introduced in [8]. It allows one to split the search for fixed points into two parts: One is to obtain a contraction and the other is to return arbitrarily close to the starting point without modifying the property of contraction. This is carried out since the angular derivative is multiplicative for compositions and is identically 1 for linear maps.Supported by CONACYT-CNRS and CONACYT 3398-E9307.     

On fixed points of holomorphic automorphisms

Summary Banach M-lattices are studied from the view point whether all the biholomorphic automorphisms of their unit balls admit fixed points when continuously extended to the closure of the unit ball. A characterization of compact topologieal F-spaces is found in terms of the fixed points of the elements of AutB(C()) which enables to establish some particular properties also of the topological automorphisms of compact F-spaces. Finally it is shown that if the M- lattice E admits a predual then each member of Aut B(E) has fixed point if and only if E is isometrically isomorphic with some l-space.     

On fixed points of Knaster continua

Aarts and Fokkink [Proc. Amer. Math. Soc. 126 (1998) 881] have shown that any homeomorphism of the bucket handle has at least two fixed points. Using their methods, we determine the minimum number of fixed points homeomorphisms on generalized one-dimensional Knaster continua can have. We show that there is a class of these continua that admit homeomorphisms with a single fixed point. Among the examples is one that shows that Theorem 15 in [Proc. Amer. Math. Soc. 126 (1998) 881] is incorrect. We also show that there are generalized Knaster continua on which every homeomorphism has either uncountably many fixed points or uncountably many points of period two.     

On fixed points of stratified maps

Nielsen fixed point theory deals with the fixed point sets of self maps on compact polyhedra. In this note, we shall extend it to stratified maps, to consider fixed points on (noncompact) strata. The extension was motivated by our recent work on the braid forcing problem in which the deleted symmetric products are indispensable. The stratified viewpoint is theoretically as natural as the equivariant Nielsen fixed point theory, while it can be more tractable computationally and more flexible in applications. This work was partially supported by an NSFC grant and a BMEC grant.