A New Maximal Point Theorem

Abstract

The aim of our paper is to generalize the maximal point theorem of Bishop and Phelps and to apply this result to derive a new multicriteria Ekeland's principle in a direct way by induction without making use of Ekeland's original scalar result.

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

“…This supplementary assumption is not necessary if the cone K(ε) is replaced by the cone K ε used in [25]. In this case the sequence converging to (x ε , y ε ) will be defined by the method described in [25]. (4) Our method presented in this paper and based on the cone K(ε) can be generalized to Hausdorff locally convex spaces.…”

“…We note that in this case the element x ε satisfies conclusion (1) and (2) as in Theorem 9, but to have that conclusion (3) is also satisfied we must have that y ε = f (x ε ). This supplementary assumption is not necessary if the cone K(ε) is replaced by the cone K ε used in [25]. In this case the sequence converging to (x ε , y ε ) will be defined by the method described in [25].…”

“…Göpfert and Chr. Tammer proved recently in [25] a maximal point theorem using the following convex cone in a product of two Banach spaces.…”

“…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].…”

Phelps’ lemma, Danes’ drop theorem and Ekeland’s principle in locally convex spaces

“…Very general versions of the maximal point lemma (in spaces X ×Z, X a complete metric space, Z a locally convex Hausdorff space) can be found for example in [11], [12]. Note that it is also possible to derive Theorem 3 from Fang's Theorem 3.2 in [9].…”