Versions Of Ekeland Research Articles

Ekeland, Takahashi and Caristi principles in preordered quasi-metric spaces

We prove versions of Ekeland, Takahashi and Caristi principles in pre-ordered quasi-metric spaces, the equivalence between these principles, as well as their equivalence to some completeness results for the underlying quasi-metric space. These extend the results proved in S. Cobzaş, Topology Appl. 265 (2019), 106831, 22, for quasi-metric spaces. The key tools are Picard sequences for some special set-valued mappings on a pre-ordered quasi-metric space X, defined in terms of the preorder and of a function φ on X.

A New Equilibrium Version of Ekeland’s Variational Principle and Its Applications

In this note, a new equilibrium version of Ekeland’s variational principle is presented. It is a modification and promotion of previous results. Subsequently, the principle is applied to discuss the equilibrium points for binary functions and the fixed points for nonlinear mappings.

On multiple solutions to a nonlocal fractional p(cdot )-Laplacian problem with concave\u2013convex nonlinearities

The aim of this paper is to examine the existence of at least two distinct nontrivial solutions to a Schrödinger-type problem involving the nonlocal fractional p(cdot )-Laplacian with concave–convex nonlinearities when, in general, the nonlinear term does not satisfy the Ambrosetti–Rabinowitz condition. The main tools for obtaining this result are the mountain pass theorem and a modified version of Ekeland’s variational principle for an energy functional with the compactness condition of the Palais–Smale type, namely the Cerami condition. Also we discuss several existence results of a sequence of infinitely many solutions to our problem. To achieve these results, we employ the fountain theorem and the dual fountain theorem as main tools.

A concave–convex Kirchhoff type elliptic equation involving the fractional p-Laplacian and steep well potential

In this paper, we study the following Schrödinger–Kirchhoff type equation: , where , , N>ps, and , are three real parameters. By using the mountain pass theorem and a variant version of Ekeland's variational principle in Zhong [A generalization of Ekeland's variational principle and application to the study of relation between the weak P.S. condition and coercivity. Nonlinear Anal. 1997;29:1421–1431], we get some results. Firstly, we obtain a positive energy solution by a truncated functional and discuss their asymptotical behavior for when p<r<2p and b lies in suitable range. Secondly, for all b>0, we obtain a positive energy solution and discuss their asymptotical behavior for when , where . Moreover, we obtain other asymptotical behavior of positive energy solution when . Finally, we get a negative energy solution and the non-existence of the non-trivial solutions.

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

In this paper, we obtain a version of Ekeland's variational principle for interval-valued functions by means of the Dancs-Hegedüs-Medvegyev theorem (Dancs et al. (1983) [14]). We also derive two versions of Ekeland's variational principle involving the generalized Hukuhara Gâteaux differentiability of interval-valued functions as well as a version of Ekeland's variational principle for interval-valued bifunctions. Finally, we apply these new versions of Ekeland's variational principle to fixed point theorems, to interval-valued optimization problems, to the interval-valued Mountain Pass Theorem, to noncooperative interval-valued games, and to interval-valued optimal control problems described by interval-valued differential equations.

Ekeland\u2019s variational principle in weak and strong systems of arithmetic

We analyze Ekeland’s variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to Pi ^1_1text{- }mathsf {CA}_0, a strong theory of second-order arithmetic, while natural restrictions (e.g. to compact spaces or to continuous functions) yield statements equivalent to weak König’s lemma (mathsf {WKL}_0) and to arithmetical comprehension (mathsf {ACA}_0). We also find that the localized version of Ekeland’s variational principle is equivalent to Pi ^1_1text{- }mathsf {CA}_0, even when restricted to continuous functions. This is a rare example of a statement about continuous functions having great logical strength.

Ekeland’s variational principle with weighted set order relations

The main results of the paper are a minimal element theorem and an Ekeland-type variational principle for set-valued maps whose values are compared by means of a weighted set order relation. This relation is a mixture of a lower and an upper set relation which form the building block for modern approaches to set-valued optimization. The proofs rely on nonlinear scalarization functions which admit to apply the extended Brezis–Browder theorem. Moreover, Caristi’s fixed point theorem and Takahashi’s minimization theorem for set-valued maps based on the weighted set order relation are obtained and the equivalences among all these results is verified. An application to generalized intervals is given which leads to a clear interpretation of the weighted set order relation and versions of Ekeland’s principle which might be useful in (computational) interval mathematics.

Ekeland’s variational principle and fixed point theorems in partial b-metric spaces

In this paper, a new version of Ekeland’s variational principle is presented in partial b-metric spaces via Cantor’s intersection theorem. As a consequence, some Caristi type fixed point theorems are proved in partial b-metric spaces. Furthermore, the construction and number of fixed points are discussed.

Cyclically antimonotone vector equilibrium problems

ABSTRACTIn this paper, we extend the notion of cyclic antimonotonicity (known for scalar bifunctions) to the vector case, in order to obtain a vectorial equilibrium version of Ekeland's variational principle. We characterize the cyclic antimonotonicity in terms of a suitable approximation from below of the vector bifunction, which allows us to avoid the demanding triangle inequality property, usually required in the literature, when dealing with Ekeland's principle for bifunctions. Furthermore, a result for weak vector equilibria in the absence of convexity assumptions is given, without passing through the existence of approximate solutions.

Fixed point theorem for generalized set-valued contractions on partial metric space

In this paper, we give a new version of Ekeland’s variationalprinciple on partial metric space. Using this variational principle,we establich a general result on the existence of a fixed point for aclass of genralized set-valued contractions on partiel metric space.

A generalization of Ekeland\u2019s variational principle by using the \u03c4-distance with its applications

In this paper, a new version of Ekeland’s variational principle by using the concept of τ-distance is proved and, by applying it, an approximate minimization theorem is stated. Moreover, by using it, two versions of existence results of a solution for the equilibrium problem in the setting of complete metric spaces are investigated. Finally some examples in order to illustrate the results of this note are given.

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

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.

Versions of Ekeland’s variational principle involving set perturbations

We consider Ekeland’s variational principle for multivalued maps. Instead of dealing with directional perturbations in a direction of the positive cone of the image space, we perturb the map under question by a convex subset of the positive cone to get stronger and more general versions. Many example are provided to highlight relations of our results to existing ones, including their advantages.

On Ha’s version of set-valued Ekeland’s variational principle

By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124, 187–206 (2005)] established a new version of Ekeland’s variational principle for set-valued maps, which is expressed by the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement of Ha’s version of set-valued Ekeland’s variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev theorem. From the improved Ha’s version, we deduce a Caristi-Kirk’s fixed point theorem and a Takahashi’s nonconvex minimization theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other.

P-distances, q-distances and a generalized Ekeland’s variational principle in uniform spaces

In this paper, we attempt to give a unified approach to the existing several versions of Ekeland’s variational principle. In the framework of uniform spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance (particularly, a p-distance), which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized Takahashi’s nonconvex minimization theorem, a generalized Ekeland’s variational principle and a generalized Caristi’s fixed point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland’s variational principle, we deduce a number of particular versions of Ekeland’s principle, which include many known versions of the principle and their improvements.

Ekeland's variational principle in locally convex spaces and the density of extremal points

In this paper, we prove a general version of Ekeland's variational principle in locally convex spaces, where perturbations contain subadditive functions of topology generating seminorms and nonincreasing functions of the objective function. From this, we obtain a number of special versions of Ekeland's principle, which include all the known extensions of the principle in locally convex spaces. Moreover, we give a general criterion for judging the density of extremal points in the general Ekeland's principle, which extends and improves the related known results.

Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems

In this paper, Levitin-Polyak well-posedness for two classes ofgeneralized vector quasi-equilibrium problems is introduced.Criteria and characterizations of the Levitin-Polyak well-posednessare investigated. By virtue of gap functions for the generalizedvector quasi-equilibrium problems, some equivalent relations areobtained between the Levitin-Polyak well-posedness for optimizationproblems and the Levitin-Polyak well-posedness for generalizedvector quasi-equilibrium problems. Finally, a set-valued version ofEkeland's variational principle is derived and applied to establisha sufficient condition for Levitin-Polyak well-posedness of a classof generalized vector quasi-equilibrium problems.

On some nonlinear problems induced by an abstract maximal element principle

In this paper, by applying an abstract maximal element principle on quasi-ordered sets established by Lin and Du, we obtain a generalized Brézis–Browder principle, system (vectorial) versions of Ekeland's variational principle and maximal element principle and a vectorial version of Takahashi's nonconvex minimization theorem. Moreover, we investigate the equivalence between scalar versions and vectorial versions of these results.

Ekeland's variational principle in locally p-convex spaces and related results

In the framework of locally $p$-convex spaces, two versions of Ekeland's variational principle and two versions of Caristi's fixed point theorem are given. It is shown that the four results are mutually equivalent. Moreover, by using the local completenes

Ekeland's variational principle in Fréchet spaces and the density of extremal points

By modifying the method of Phelps, we obtain a new version of Ekeland's variational principle in the framework of Fréchet spaces, which admits a very general form of perturbations. Moreover we give a density result concerning extremal points of lower semi