Suprema of chains of operators

Herzog, Gerd; Schmoeger, Christoph

doi:10.1007/s11117-010-0079-3

Cited by 1 publication

(4 citation statements)

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“…R + , and # k 2 (0; 1). The resolution of the existence of …xed points of these operators is closely related to what studied in [16].…”

Section: Operator Equationsmentioning

confidence: 79%

“…and that (16) is equivalent to the condition T c v v by using the auxiliary operator (12). Proposition 21 implies that T c has a unique …xed point w in l 1 + .…”

Section: Proposition 23mentioning

confidence: 98%

“…Rather than solving directly the Koopmans equation (11), it is often useful to analyze an auxiliary "parametric" problem. 16 Speci…cally, for a given consumption plan c = (c t ) t 0 2 l 1 + , we introduce the operator T c : l 1…”

Section: Recursive Utilitiesmentioning

confidence: 99%

“…(X + A k ) # k are monotone and concave in L + s (H). Moreover, for > 0 large enough, T maps the interval [0; I] into itself (see [16] for more details). By the chain completeness of [0; I], it follows the existence of …xed points in [0; I] :…”

Section: Operator Equationsmentioning

confidence: 99%

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Unique Tarski Fixed Points

Marinacci

Montrucchio

2019

Mathematics of OR

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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 ...

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“…R + , and # k 2 (0; 1). The resolution of the existence of …xed points of these operators is closely related to what studied in [16].…”

Section: Operator Equationsmentioning

confidence: 79%

“…and that (16) is equivalent to the condition T c v v by using the auxiliary operator (12). Proposition 21 implies that T c has a unique …xed point w in l 1 + .…”

Section: Proposition 23mentioning

confidence: 98%

Section: Recursive Utilitiesmentioning

confidence: 99%

Section: Operator Equationsmentioning

confidence: 99%

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