Ekeland's variational principle in premetric spaces

In Part 1, we show that the 1993 Cârjă–Ursescu Ordering Principle [An. Şt. Univ. “A. I. Cuza” Iasi (Mat.) 39, 367–396 (1993)] is equivalent with the Dependent Choice Principle (DC); and as such, equivalent with Ekeland’s Variational Principle (EVP) [J. Math. Anal. Appl. 47, 324–353 (1974)]. This conclusion is valid for all intermediary principles; hence, in particular, for the Brezis–Browder’s (BB) [Adv. Math. 21, 355–364 (1976)]. In Part 2, it is established that the vectorial Zhu–Li Variational Principle in Fang spaces is in the logical segment between (BB) and (EVP); hence, it is equivalent with both (BB) and (EVP). In particular, the conclusion is applicable to Hamel’s Variational Principle (HVP); moreover, a proof of (HVP) ⇔ (EVP) is provided, by means of a direct approach that avoids (DC). Finally, in Part 3, we show that the gauge Brezis–Browder Principle in Turinici [Bull. Acad. Pol. Sci. (Math.) 30, 161–166 (1982)] is obtainable from (DC) and implies (EVP); hence, it is equivalent with both (DC) and (EVP). This is also true for the gauge variational principle deductible from it, including the one in Bae, Cho, and Kim [Bull. Korean Math. Soc. 48, 1023–1032 (2011)].

A Study of Ekeland’s Variational Principle and Related Theorems and Applications by Jessica Robinson ???David Costa ???, Examination Committee Chair Professor of Mathematical Sciences University of Nevada, Las Vegas Ekeland’s Variational Principle has been a key result used in various areas of analysis such as fixed point analysis, optimization, and optimal control theory. In this paper, the application of Ekeland’s Variational Principle to Caristi’s Fixed Point Theorem, Clarke’s Fixed Point Theorem, and Takahashi’s Minimization theorem is the focus. In addition, Ekeland produced a version of the classical Pontryagin Minimum Principle where his variational principle can be applied. A further look at this proof and discussion of his approach will be contrasted with the classical method of Pontryagin. With an understanding of how Ekeland’s Variational Princple is used in these settings, I am motivated to explore a multi-valued version of the principle and investigate its equivalence with a multi-valued version of Caristi’s Fixed Point Theorem and Takahashi’s Minimization theorem. iii

Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces

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

In this article, following the idea used by Göpfert et al . [A. Göpfert, Chr. Tammer and C. Zalinescu (2000). On the vectorial Ekeland's variational principle and minimal points in product spaces. Nonlinear Analysis, Theory, Methods & Applications , 39 , 909-922] to derive an Ekeland's variational principle for vector-valued functions, we derive a new variant of Ekeland's variational principle for set-valued maps. Finally, we apply this variational principle to obtain an approximate necessary optimality condition for a class of set-valued optimization problems.

Vectorial Ekeland's Variational Principle with a W-Distance and its Equivalent Theorems

Set-valued Ekeland variational principles in fuzzy metric spaces

An extension of Ekeland's variational principle in fuzzy metric space and its applications

Local completeness, drop theorem and Ekeland's variational principle

The density of extremal points in Ekeland's variational principle

This paper deals with Ekeland's variational principle for vector optimization problems. By using a set-valued metric, a set-valued perturbed map, and a cone-boundedness concept based on scalarization, we introduce an original approach to extending the well-known scalar Ekeland's principle to vector-valued maps. As a consequence of this approach, we obtain an Ekeland's variational principle that does not depend on any approximate efficiency notion. This result is related to other Ekeland's principles proved in the literature, and the finite-dimensional case is developed via an $\varepsilon$-efficiency notion that we introduced in [Math. Methods Oper. Res., 64 (2006), pp. 165–185; SIAM J. Optim., 17 (2006), pp. 688–710].

Ekeland's variational principle (EVP) has many equivalent formulations and generalizations. In this article, we present new minimal point theorems in product spaces and the corresponding vector variational principles for set-valued functions. As special cases we derive many of the existing variational principles of Ekeland's type. Moreover, we use our new approach to get extensions of EVPs of Isac–Tammer and Ha types, as well as extensions of EVPs for bi-functions.

Chapter 6 - Variational Principles and Variational Analysis for the Equilibrium Problems

We establish a vectorial Ekeland’s variational principle where the objective function is from bornological vector spaces into real vector spaces, and the ordering cone in real vector spaces is not necessarily solid. Meanwhile, a vectorial Caristi’s fixed point theorem and a vectorial Takahashi’s nonconvex minimization theorem are obtained and the equivalences between the three theorems are shown.

In this paper, we introduce cancellation laws for weighted set relations, which are a mixture of lower and upper set relations. Applying nonlinear scalarizing techniques for sets, we propose a new type of minimal element theorem and generalized Ekeland's variational principles for weighted set relation with set perturbation. An application of generalized Ekeland's variational principles is given to vector-valued game theory under uncertainty.