Critical Point Theorems and Ekeland Type Variational Principle with Applications

Abstract

We introduce the notion of λ-spaces which is much weaker than cone metric spaces defined by Huang and X. Zhang 2007. We establish some critical point theorems in the setting of λ-spaces and, in particular, in the setting of complete cone metric spaces. Our results generalize the critical point theorem proposed by Dancs et al. 1983 and the results given by Khanh and Quy 2010 to λ-spaces and cone metric spaces. As applications of our results, we characterize the completeness of λ-space cone metric spaces and quasimetric spaces are special cases of λ-space and studying the Ekeland type variational principle for single variable vector-valued functions as well as for multivalued bifunctions in the setting of cone metric spaces.

Variational principles with generalized distances and the modelization of organizational change

“…-In [22], the authors introduced the notion of λ-spaces which is much weaker than cone metric spaces and then established some critical point theorems and Ekeland-type variational theorems in the setting of λ-spaces. In [44], a modification of the notion of a w-distance was presented to further extend some fixed-point results for generalized contractive set-valued maps on complete preordered quasimetric spaces.…”

“…Note also that a left-convergent sequence in a quasimetric space is not necessarily left-Cauchy and that a quasimetric space might not be left-Hausdorff; see [4,22], and the references therein. It was proved in [22,Example 3.16] that the bifunction…”

On extended versions of Dancs-Hegedüs-Medvegyev’s fixed-point theorem

“…This feature allows us to obtain new applications to Ekeland's variational principle and Caristi's fixed point theorem by using our alternative result (Theorem 4.5). Following this way, we plan to extend this research to the setting of λ-spaces and to the setting of complete cone metric spaces introduced by Lin et al in [27].…”

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

“…It is important to emphasize that a majority of generalizations of EVP have been established in a less general context of cone metric spaces, where the symmetric axiom is preserved and no more real justification is presented (see, among others, [8][9][10][11][12]), and that the case of cone metric spaces does not offer a true generalization of metric spaces when the cone enjoys some good properties [13][14][15][16][17][18]. The new version of EVP in cone pseudo-quasimetric established in Theorem 3.1 and its Corollary 3.1 is far-going extensions of EVP presented in [12,13] in cone metric spaces.…”