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Register a new userA fixed point theorem for commuting families of relational homomorphisms. Application to metric spaces, oriented graphs and ordered sets
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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 non-expansive 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 Tarskis 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.
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In this survey we present a generalization of the notion of metric space and some applications to discrete structures as graphs, ordered sets and transition systems. Results in that direction started in the middle eighties based on the impulse given by Quilliot (1983). Graphs and ordered sets were considered as kind of metric spaces, where - instead of real numbers - the values of the distance functions $d$ belong to an ordered semigroup equipped with an involution. In this frame, maps preserving graphs or posets are exactly the nonexpansive mappings (that is the maps $f$ such that $d(f(x),f(y))leq d(x,y)$, for all $x,y$). It was observed that many known results on retractions and fixed point property for classical metric spaces (whose morphisms are the nonexpansive mappings) are also valid for these spaces. For example, the characterization of absolute retracts, by Aronszajn and Panitchpakdi (1956), the construction of the injective envelope by Isbell (1965) and the fixed point theorem of Sine and Soardi (1979) translate into the Banaschewski-Bruns theorem (1967), the MacNeille completion of a poset (1933) and the famous Tarski fixed point theorem (1955). This prompted an analysis of several classes of discrete structures from a metric point of view. In this paper, we report the results obtained over the years with a particular emphasis on the fixed point property.
The aim of this paper is to provide characterizations of a Meir-Keeler type mapping and a fixed point theorem for the mapping in a metric space endowed with a transitive relation.
We obtain an extended Reich fixed point theorem for the setting of generalized cone rectangular metric spaces without assuming the normality of the underlying cone. Our work is a generalization of the main result in cite{AAB} and cite{JS}.
We establish two fixed point theorems for certain mappings of contractive type. The first result is concerned with the case where such mappings take a nonempty, closed subset of a complete metric space $X$ into $X$, and the second with an application of the continuation method to the case where they satisfy the Leray-Schauder boundary condition in Banach spaces.
Lins theorem states that for all $epsilon > 0$, there is a $delta > 0$ such that for all $n geq 1$ if self-adjoint contractions $A,B in M_n(mathbb{C})$ satisfy $|[A,B]|leq delta$ then there are self-adjoint contractions $A,B in M_n(mathbb{C})$ with $[A,B]=0$ and $|A-A|,|B-B|<epsilon$. We present full details of the approach in arXiv:0808.2474, which seemingly is the closest result to a general constructive proof of Lins theorem. Constructive results for some special cases are presented along with applications to the problem of almost commuting matrices where $B$ is assumed to be normal and also to macroscopic observables.
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