A New Minimal Point Theorem in Product Spaces

Abstract

We derive a minimal point theorem for a subset A in a cone in product spaces under a weak assumption concerning the boundedness of the considered set A. Using this result we improve two vectorial variants of Ekeland's variational principle. Finally, a new characterization of well-based cones is given.

Minimal elements for product orders

“…a Banach space V with a convex cone K V generating the quasiorder K , the question arises if it is possible to obtain minimal element theorems on product spaces X Â V where the set in question cannot necessarily be interpreted as the epigraph of a function f : X ! V. Gö pfert, Tammer and Za˘linescu established results in this direction in a series of papers [7][8][9]. Compare also the book [10], Section 3.10.…”

Nuclear and Full Nuclear Cones in Product Spaces: Pareto Efficiency and an Ekeland Type Variational Principle

“…Many authors have been considered Ekeland's Principle from several points of view, on metric spaces [1, 8-10, 15-20, 22, 29, 44], in locally convex topological vector spaces [10,35] and also in general topological vector spaces [21]. For a long period of time, this principle has considered for real-valued functions but recently it has also been considered for vector-valued functions, as for example in [9,13,14,25,28,34,41,42,53]. In 1983, it was introduced the notion of nuclear cone as useful mathematical tool for Pareto optimization [31].…”