Unique Tarski Fixed Points

Abstract

We establish su¢ cient conditions that ensure the uniqueness of Tarski-type …xed points of monotone operators. Several applications are presented. IntroductionIn this paper we establish su¢ cient conditions that ensure in ordered vector spaces the uniqueness of …xed points a la Tarski [36], often a highly desirable property in the many applications in economics and operations research in which such …xed points appear (cf. Topkis [38]).More speci…cally, our results establish the existence and uniqueness of …xed points of monotone operators that are either order concave or subhomogenous. Their common feature is to require that no …xed points belong to the lower perimeter of the domain. This novel notion, which we introduce in Section 3, is thus a keystone of our analysis.We establish our main results in Sections 4 and 5. The results of the latter section rely on a close relation between the subhomogeneous case and the contractive property according to a metric introduced by Thompson [37]. This novel connection, elaborated in the Appendix, permits to prove the uniqueness and global attractiveness of …xed points of subhomogeneous operators. Besides the role of lower perimeters, this connection is the other main contribution of this paper.We illustrate our uniqueness results with some applications on recursive utilities, integral equations, complementary problems, variational inequalities, and operator equations in Section 6. We conclude by discussing the related literature in Section 7.Spaces Throughout the paper V is a (partially) ordered vector space with order relation and K will always denote its positive cone. If V is Dedekind -complete, then it is Archimedean. 2 When V is a lattice, it is called Riesz space. In this case, to be Dedekind complete amounts to say that the order intervals [a; b] V are complete lattices. Fixed pointsA …xed point theorem due to Tarski [36] p. 286 says that the set of …xed points of a monotone self-map de…ned on a complete lattice is a nonempty complete lattice. A generalized version of this result says that set of …xed points of a monotone self-map de…ned on a chain complete poset is a nonempty chain complete poset. 3 A self-map T : A ! A is order continuous if, given any countable chain fa n g A for which sup a n exists, we have T (sup a n ) = sup T (a n ). Clearly, order continuous self-maps are monotone. A …xed point theorem, essentially due to Kantorovich [17] p. 68, says that the set of …xed points of a order continuous self-map de…ned on a chain -complete poset has a least …xed point.Concavity A subset A of an V is order convex if a c b and a; b 2 A imply c 2 A. This amounts to say that A contains all order intervals (and so all segments) determined by its elements.A self-map T : A ! A de…ned on an order convex subset is order concave iffor all t 2 [0; 1] and all a; b 2 A with a b. Order concave and order convex operators are studied in Amann [3, Chapter V], along with their di¤erential characterizations.Subhomogeneity The study of subhomogeneity for operators was pioneered by ...

Recursive Utility and Thompson Aggregators, Ii: Uniqueness of the Recursive Utility Representation

“…We must exclude null consumption sequences as well as all other consumption sequences in the positive cone's boundary. These restrictions are the same as the ones imposed in Martins-da-Rocha and Vailakis [32], Bloise and Vailakis [12] and Marinacci and Montrucchio [31]. They are formally more restrictive than constraints imposed by Marinacci and Montrucchio [30] who exploit properties of the aggregator in de…ning their commodity space restrictions.…”

“…27 However, this does not mean we cannot say something useful about the subset of consumption sequences where the extremal …xed points deliver the same utility value. The papers by Marinacci and Montrucchio ([30], [31]), Martins-da Rocha and Vailakis [32] and Bloise and Vailakis [12], prove their uniqueness theorems by restricting the commodity space's domain for the utility functions. We do so here as well.…”

“…This approach di¤ers from the methods used in the existing literature. The concave operator techique is an alternative for demonstrating uniqueness results compared to the contraction operator theorems based on the Thompson metric employed by Marinacci and Montrucchio ([30], [31]) or the 0-local contractions introduced by Rincón-Zapatero and Rodriguez-Palmero ( [36], [37]) and Martins-da-Rocha and Vailakis ([32], [33]). It is worth noting the uniqueness contribution by Marinacci and Montrucchio [30] accommodates some endogenous growth models, whereas we are not able to do so.…”

“…Marinacci and Montrucchio [31] present a uniqueness theorem for Thompson aggregators that is valid on the interior of the commodity space's positive cone. That work is independent of ours; it applies to their original axiom list de…ning the Thompson class.…”

“…Marinacci and Montrucchio [31] base their results on a careful order theoretic description of the boundary of both the utility function and commodity spaces. 6 We emphasize the topological features of the positive cone's boundary relative to the positive cone as well as the conditions characterizing interior points in the positive cone.…”