Variational Analysis in Cone Pseudo-Quasimetric Spaces and Applications to Group Dynamics

Abstract

In this paper, we generalize Ekeland's variational principle in the new context of cone pseudo-quasimetric spaces. We propose this extension for applications to group dynamics in behavioral sciences. In this setting, a cone pseudo-quasimetric helps to model, in a crude way, multidimensional aspects of resistance to change for a group, where each component represents resistance to change of one agent in the group. At the behavioral level, our new version of Ekeland's variational principle shows how a group, forming and breaking routines each period by balancing between motivations and resistances to change of all members, can improve step by step their payoffs to end in a trap worthwhile to approach and reach, but not worthwhile to leave.

Variational principles in set optimization with domination structures and application to changing jobs

“…When (X, q) is a metric, the forward-limit set − − → {x n } is singleton and thus it extends the socalled strictly decreasing forward-lower-semicontinuity for the class of functions ϕ : X → R ∪ {±∞} in [4, Definition 3.18]; known also as lower-semicontinuity from above in [36]. It also extend the concept of decreasing forward-lower-semicontinuity studied in [6,10] for vectorvalued functions.…”

“…In [18], the authors studied convex-cone-valued domination structures. In a majority of publications in the literature, X = Y and domination sets are convex cones; see, for example, [6,9,10,19,56] and the references therein. Definition 2.6.…”

“…; see the references above. Recently, set-valued and set optimization with variable ordering structure have been extensively studied in [5,6,7,8,9,10,18,19,33,34] and references therein.…”

“…In set-valued optimization with ordering structure, many versions of Ekeland's variational principle have been formulated; for example, [6,7,10] in quasimetric spaces and [4] in pseudo quasimetric spaces with applications in behavioral sciences. There have been several versions of Ekeland's variational principle in set optimization with fixed ordering cone.…”

“…We use the idea in the proof of extended versions of Dancs-Hegedüs-Medvegyev's fixedpoint theorem in [11]; for more details, see [6,7,10] that were used to establish versions of Ekeland's variational principles in set-valued optimization with ordering structure. In addition, we use marginal functions used in [30,54] associated with Gerstewitz' (nonlinear separation) scalarization function introduced in [21,22] (see also Krasnoselskii [27] for assertions in the context of operator theory and compare the scalar optimization problem by Pascoletti, Serafini [52]) is an important tool in nonconvex vector optimization.…”