A fixed point theorem for commuting families of relational homomorphisms. Applications to metric spaces, ordered sets and oriented graphs

Abstract

We extend to binary relational systems the notion of compact and normal structure, introduced by J.P.Penot for metric spaces, and we prove that for the involutive and reflexive ones, every commuting family of relational homomorphisms has a common fixed point. The proof is based upon the clever argument that J.B.Baillon discovered in order to show that a similar conclusion holds for bounded hyperconvex metric spaces and then refined by the first author to metric spaces with a compact and normal structure. Since the nonexpansive mappings are relational homomorphisms, our result includes those of T.C.Lim, J.B.Baillon and the first author. We show that it extends the Tarski's fixed point theorem to graphs which are retracts of reflexive oriented zigzags of bounded length. Doing so, we illustrate the fact that the consideration of binary relational systems or of generalized metric spaces are equivalent.

Generalized metric spaces. Relations with graphs, ordered sets and automata : A survey

“…In [61], Khamsi and Pouzet proved that: Theorem 5.3. If a generalized metric space E ∶= (E,d) has a compact normal structure then every commuting family F of nonexpansive self maps has a common fixed point.…”

“…This notion, defined for ordinary metric space, extends to metric spaces over a Heyting algebra. In fact, it extends to metric spaces over an ordered monoid equipped with an involution and more generally to binary structures which are reflexive and involutive in the sense of [61]. It plays a crucial role in the fixed point theorem presented in the next section.…”

“…The possible extension to commuting set of nonexpansive maps was left unresolved (only the case of a countable set was settled). In [61] the notion of compact normal structure for metric spaces over Heyting algebra (and more generally for systems of binary relations) was introduced and Khamsi's theorem extended to families of nonexpansive maps which commute on a space endowed with a compact and normal structure.…”

“…The genesis of this topic is to be found in two theses [46], [66] and a paper [47]. The theme was subsequently developped in [52], [75], [53], [85], [54], [55], [56], [10], [11] and [61].…”

“…Khamsi [60] extended the conclusion to metric spaces with a compact normal structure. Quite recently, Khamsi and the second author ( [61]) extended it to generalized metric spaces endowed with a compact normal structure. As a consequence, every set of commuting order-preserving maps on a retract of a power of a finite fence, has a fixed point (the case of one map followed from a result due to I.…”