Minimal element theorems revisited

Abstract

Starting with the Brezis-Browder principle, we give stronger versions of many variational principles and minimal element theorems which appeared in the recent literature. Relationships among the elements of different sets of assumptions are discussed and clarified, i.e., assumptions to the metric structure of the underlying space and boundedness assumptions. New results involving set-valued maps and the increasingly popular set relations are obtained along the way. applications consist in a mere check if the order relation in question satisfies the required assumptions which links it to the metric structure. Therefore, the mentioned results can be proven without involving the Brezis-Browder principle as shown in the alternative proof of Theorem 3.7 below.The second line of research is concerned with order relations on arbitrary sets (without a metric structure, for example). Of course, alternative requirements have to be added. A prominent result of this type is the Brezis-Browder principle [1] (BB-principle) where the existence of a real-valued function is assumed which is bounded from below and increasing wrt the order relation. Further examples for such results can be found in [25,27] and also in [31]. Of course, BB-type theorems can also be used to obtain the results on metric spaces discussed in the previous paragraph with a suitable monotone function; the proof of the BB-principle also involves a countable induction argument (see the first proof of Theorem 3.2 below).A third goal is to lift the results from order relations on a set X to such relations on a product set X × Z. Results in this direction have been obtained first by Göpfert and Tammer (see [5, Section 3.10] and the references 140-144 therein) and are usually called minimal element theorems.In this note, all three of the above lines are followed providing very general results and discussing the relationships between different sets of assumptions in detail. Consequently, most relevant results in the literature are obtained as special cases. In particular, it is shown that variational principles and minimal element theorems involving (very general) set relations can be obtained (see, for example, Corollary 3.14 below). Such results can be found in [8] for complete metric spaces (manuscript version [7] from 2002) and more general ones in [12] (preprint version [11] from 2002), the latter reference already giving a proof based on the BB-principle and nonlinear scalarization.

Robust Ekeland variational principles. Application to the formation and stability of partnerships

“…and it has several applications in mathematics and behavioural sciences, many authors have extended EVP to vector-valued functions, and, more recently, to set-valued maps and bimaps. We refer the reader to books [8,11,13] and several recent articles [2][3][4][5][6][7]10,12,[14][15][16][18][19][20][21][22][23][24][25][26][27][28] for a good reporting of the results and various approaches. Moreover, inspired by set-valued optimization, there are two approaches to extend EVP to set-valued maps, the vector approach, see, for example, [2,[5][6][7]19,20,24] and the set approach, see, for example, [3,17,18,[21][22][23]26,28].…”

“…We refer the reader to books [8,11,13] and several recent articles [2][3][4][5][6][7]10,12,[14][15][16][18][19][20][21][22][23][24][25][26][27][28] for a good reporting of the results and various approaches. Moreover, inspired by set-valued optimization, there are two approaches to extend EVP to set-valued maps, the vector approach, see, for example, [2,[5][6][7]19,20,24] and the set approach, see, for example, [3,17,18,[21][22][23]26,28]. The vector criterion provides the existence of an approximate minimal (efficient) point of the range set of a set-valued map.…”

Set-valued variational principles. When migration improves quality of life

“…More recently, some researchers obtained several extensions of the EVP with applications in behavioral sciences; see, e.g., [7,9,10,11,15,16,24,27] and the references therein. However, there are "two types of EVP in the case of set-valued mappings": the vector criterion (see, e.g., [2,7,8,9,13,14,19,23]) and the set criterion (see, e.g., [3,4,20,21,22,28]). The first criterion is to find an approximate minimal (efficient) point of the range set of a set-valued mapping.…”

On Ekeland's variational principle for interval-valued functions with applications

“…It is well known that Ekeland's variational principle becomes a powerful tool in the study of many problems arising in nonlinear analysis, dynamical system, critical points theory, economics and finance, optimization and control theory (see, for instance, [2,4,12,16,17,19,35]). Thereafter, extensive efforts have been devoted into many different versions of Ekeland's variational principle with vector-valued functions or set-valued mappings and bifunctions (see, for instance, [1,8,9,13,20,21,25,28,40,47]).…”