Existence of fixed points on compact epilipschitz sets without invariance conditions
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We provide a new result of existence of equilibria of a single-valued Lipschitz function f on a compact set K of Rn which is locally the epigraph of a Lipschitz functions (such a set is called epilipschitz set). Equivalently this provides existence of fixed points of the map x → x − f (x). The main point of our result lies in the fact that we do not impose that f (x) is an “inward vector” for all point x of the boundary of K. Some extensions of the existence of equilibria result are also discussed for continuous functions and set-valued maps. 1. Introduction This paper is devoted to the following result. Theorem 1.1. Let K be an epilipschitz compact subset of Rn ; f : Rn → Rn be a (locally) Lipschitz function. Assume that Ks is closed and that the Euler characteristic χ(Ks ) is well defined. If χ(K) = χ(Ks ) then there exists an equilibria in K that is a point x ∈ K such that f (x) = 0. In the above Theorem 1.1, the set Ks (or Ks ( f )) is the set of elements x of the boundary of K such that the solution to the Cauchy problem
x (t) = f x(t) ,
t ≥ 0, x(0) = x,
(1.1)
leaves K immediately (that is there exists σ > 0 such that (x((0,σ)) ∩ K = ∅)). Epilipschitz sets are sets which are locally the epigraph of a Lipschitz function (an equivalent formulation is given in [25]). It is worth pointing out that when f (x) is “inward” for any x ∈ ∂K, we have that K is invariant by the differential equation
x (t) = f x(t) ,
Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 267–279 DOI: 10.1155/FPTA.2005.267
t ≥ 0,
(1.2)
268
Fixed point without invariance
and consequently Ks = ∅. So our theorem, contains for example the famous fixed point Brouwer theorem, viewed as an existence result for equilibria of the map x → x − g(x) for convex compact closed sets. It contains also several results of existence of equilibria which impose inwardness conditions of the type ∀x ∈ ∂K,
f (x) ∈ CK (x)
(1.3)
where CK (x) denotes Clarke’s tangent cone. Since pioneering results of Fan and Browder [5, 15], several theorems have been obtained in this direction [10, 12, 13, 19, 18, 23, 22], among them we wish to quote one of the most recent result (in a version adapted for single valued map). Proposition 1.2 [11, Corollary 4.1]. If f continuous, K is a compact epilipschitz subset of Rn with χ(K) = 0 and if (1.3) holds true, then there is an equilibria of f in K.
We also wish to underline that more general results with condition (1.3) have been obtained for set-valued maps and for normed spaces more general than Rn (cf. for instance for L retracts in normed spaces). We are mainly interested to weaken the condition (1.3) for a class of epilischitz sets of Rn which is large enough because it contains for instance convex sets with nonempty interiors, C 1 submanifolds with boundary. Our approach is mainly based on properties of trajectories of the differential equation associated with f . Indeed the set Ks appears in the so called topological Wa˙zewski principle, which gives sufficien
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