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We first give a pre-order principle whose form is very general. Combining the pre-order principle and generalized Gerstewitz functions, we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective function is a set-valued bimap defined on the product of quasi-metric spaces and taking values in a quasi-ordered linear space, and the perturbation consists of a subset of the ordering cone multiplied by the quasi-metric. From this, we obtain a number of new results which essentially improve the related results. Particularly, the earlier lower boundedness condition has been weakened. Finally, we apply the new EVPs to Psychology. [ABSTRACT FROM AUTHOR]

By using a pre-order principle in [Qiu JH. A pre-order principle and set-valued Ekeland variational principle. J Math Anal Appl. 2014;419:904–937], we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective function is a set-valued bimap defined on the product of left-complete quasi-metric spaces and taking values in a quasi-ordered linear space, and the perturbation consists of a cone-convex subset of the ordering cone multiplied by the quasi-metric. Moreover, we obtain an equilibrium EVP, where the perturbation contains a σ-convex subset and the quasi-metric. From the above two general EVPs, we deduce several interesting corollaries, which extend and improve the related known results. Several examples show that the obtained set-valued EVPs are new. Finally, applying the above EVPs to organizational behavior sciences, we obtain some interesting results on organizational change and development with leadership. In particular, we show that the existence of robust organizational traps. [ABSTRACT FROM AUTHOR]

By using generalized nonconvex separation functionals and a pre-order principle in Qiu [J Math Anal Appl. 2014;419:904–937], we establish a general set-valued Ekeland variational principle (briefly, denoted by EVP), where the objective function is a set-valued map taking values in a real vector space quasi-ordered by a convex cone K and the perturbation consists of a cone-convex subset H of K multiplied by the distance function. Here, the assumption on lower semi-continuity of the objective function is replaced by a weaker one: sequentially lower monotony. And the assumption on lower boundedness of the objective function is taken to be the weakest of several different kinds. From the general set-valued EVP, we deduce a number of particular versions of set-valued EVP, which extend and improve the related results in the literature. In particular, we give several EVPs for approximately efficient solutions in set-valued optimization, which not only extend the related results from vector-valued objective functions into set-valued objective functions, but also improve the related results by removing a usual assumption for K-boundedness (by scalarization) of the objective function's range. [ABSTRACT FROM AUTHOR]

In this paper, we first generalize Gerstewitz's functions from a single positive vector to a subset of the positive cone. Then, we establish a partial order principle, which is indeed a variant of the pre-order principle [Qiu, J. H.: A pre-order principle and set-valued Ekeland variational principle. J. Math. Anal. Appl., 419, 904-937 (2014)]. By using the generalized Gerstewitz's functions and the partial order principle, we obtain a vector EVP for ε-efficient solutions in the sense of Németh, which essentially improves the earlier results by completely removing a usual assumption for boundedness of the objective function. From this, we also deduce several special vector EVPs, which improve and generalize the related known results. [ABSTRACT FROM AUTHOR]

In my former paper 'A pre-order principle and set-valued Ekeland variational principle' (see [ J. Math. Anal. Appl., 419, 904-937 (2014)]), we established a general pre-order principle. From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles (denoted by EVPs) in set containing forms and their improvements. But the pre-order principle could not imply Khanh and Quy's EVP in [On generalized Ekeland's variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49, 381-396 (2011)], where the perturbation contains a weak τ-function, a certain type of generalized distances. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle, but also can be applied to obtain the above Khanh and Quy's EVP. In particular, we give several new set-valued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances. [ABSTRACT FROM AUTHOR]

By using Gerstewitz functions, we establish a new equilibrium version of Ekeland variational principle, which improves the related results by weakening both the lower boundedness and the lower semi-continuity of the objective bimaps. Applying the new version of Ekeland principle, we obtain some existence theorems on solutions for set-valued vector equilibrium problems, where the most used assumption on compactness of domains is weakened. In the setting of complete metric spaces ( Z, d), we present an existence result of solutions for set-valued vector equilibrium problems, which only requires that the domain X ⊂ Z is countably compact in any Hausdorff topology weaker than that induced by d. When ( Z, d) is a Féchet space (i.e., a complete metrizable locally convex space), our existence result only requires that the domain X ⊂ Z is weakly compact. Furthermore, in the setting of non-compact domains, we deduce several existence theorems on solutions for set-valued vector equilibrium problems, which extend and improve the related known results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a set-valued pseudo-metric family on a fuzzy metric space and the notion of compatibility between the set-valued pseudo-metric family and the original fuzzy metric. By means of this notion, we prove a general set-valued EVP, where the perturbation involves a set-valued pseudo-metric family compatible with the original fuzzy metric. From the general EVP, we deduce several particular EVPs, which extend the EVPs in Qiu (2013) [36] and in Gutiérrez et al. (2008) [20] to fuzzy metric spaces. By using set-valued pseudo-metric families and using the unified approach for approximate solutions introduced by Gutiérrez, Jiménez and Novo, we deduce a general version of set-valued EVP based on ( C , ϵ ) -efficient solutions in fuzzy metric spaces, where C is a coradiant set contained in the order cone. By choosing two specific versions of the coradiant set C in the general version of EVP, we obtain several particular set-valued EVPs for ϵ -efficient solutions in the sense of Németh and of Dentcheva and Helbig, respectively. These EVPs improve and generalize the related known results. [ABSTRACT FROM AUTHOR]

We present an equilibrium version of vectorial Ekeland variational principle, where the objective bimap F is defined on the product of sequentially lower complete spaces (see Zhu et al., 2013) and taking values in a quasi-ordered locally convex space. Besides, the perturbation consists of a subset of the ordering cone and a non-negative function p which only needs to satisfy p ( x , y ) = 0 iff x = y . Applying the equilibrium version of Ekeland principle to equilibrium problems, we obtain a general existence theorem on solutions of vectorial equilibrium problems in setting of sequentially lower complete spaces, which implies several improvements of known results. Particularly, under the framework of complete metric spaces ( Z , d ) , we obtain an existence result of solutions for equilibrium problems which only requires that the domain X ⊂ Z is sequentially compact in any Hausdorff topology weaker than that induced by d . By using the theory of angelic spaces, we extend some results in Alleche and Raˇdulescu (2015) from reflexive Banach spaces to the strong duals of weakly compact generated spaces. [ABSTRACT FROM AUTHOR]

By using sequentially lower complete spaces (see [Zhu, J., Wei, L., Zhu, C. C.: Caristi type coincidence point theorem in topological spaces. J. Applied Math., 2013, ID 902692 (2013)]), we give a new version of vectorial Ekeland's variational principle. In the new version, the objective function is defined on a sequentially lower complete space and taking values in a quasi-ordered locally convex space, and the perturbation consists of a weakly countably compact set and a non-negative function p which only needs to satisfy p( x, y) = 0 iff x = y. Here, the function p need not satisfy the subadditivity. From the new Ekeland's principle, we deduce a vectorial Caristi's fixed point theorem and a vectorial Takahashi's non-convex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. By considering some particular cases, we obtain a number of corollaries, which include some interesting versions of fixed point theorem. [ABSTRACT FROM AUTHOR]

We give a general vectorial Ekeland's variational principle, where the objective function is defined on an F-type topological space and taking values in a pre-ordered real linear space. Being quite different from the previous versions of vectorial Ekeland's variational principle, the perturbation in our version is no longer only dependent on a fixed positive vector or a fixed family of positive vectors. It contains a family of set-valued functions taking values in the positive cone and a family of subadditive functions of topology generating quasi-metrics. Hence, the direction of the perturbation in the new version is a family of variable subsets which are dependent on the objective function values. The general version includes and improves a number of known versions of vectorial Ekeland's variational principle. From the general Ekeland's principle, we deduce the corresponding versions of Caristi-Kirk's fixed point theorem and Takahashi's nonconvex minimization theorem. Finally, we prove that all the three theorems are equivalent to each other. [ABSTRACT FROM AUTHOR]

We establish a pre-order principle. From the principle, we obtain a very general set-valued Ekeland variational principle, where the objective function is a set-valued map taking values in a quasi-ordered linear space and the perturbation contains a family of set-valued maps satisfying certain property. From this general set-valued Ekeland variational principle, we deduce a number of particular versions of set-valued Ekeland variational principle, which include many known Ekeland variational principles, their improvements and some new results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we establish a general set-valued Ekeland's variational principle in fuzzy metric spaces, where the objective function is a set-valued map defined on a fuzzy metric space and taking values in a pre-ordered locally convex space, and the perturbation involves a quasi-metric family generating the fuzzy topology of the domain space. Moreover, the direction of the perturbation is a convex subset of the positive cone instead of a single positive vector. In our general version, the assumption that the objective function is lower semi-continuous and one that the range of the function is lower bounded are both weakened. From the general Ekeland's variational principle, we obtain several particular set-valued Ekeland's variational principles in fuzzy metric spaces, which generalize and improve some related known results. From these, we deduce the corresponding Caristi's fixed point theorems for set-valued maps and the corresponding Takahashi's non-convex minimization theorems in set-valued optimization. Finally, we extend the obtained results to F-type topological spaces. [Copyright &y& Elsevier]

In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle (in short EVP), where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space (which is a uniform space) has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi's fixed point theorem and a general vectorial Takahashi's nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: In this paper, we give a generalized Ekeland vector variational principle. By using the principle, we extend and improve the related results in sharp efficiency. In the framework of locally convex spaces, we introduce two kinds of generalized sharp efficiencies and prove that they are equivalent. In particular, we show that a sharp efficient solution with respect to an interior point of the ordering cone is also one with respect to every interior point. Moreover, we introduce the generalized Takahashi’s condition and the generalized Hamel’s condition for vector-valued functions. From the generalized Ekeland principle we deduce that the two conditions are equivalent. From this, we discuss the relationship between the ‘distance’ of from and the distance of from , where denotes the efficient point set of and denotes the efficient solution set. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the density of extremal points appeared in Ekeland''s variational principle. By introducing radial intersections of sets, we give a very general result on the density of extremal points in the framework of locally convex spaces. This solves a problem proposed by G. Isac in 1997. From the general result we deduce several convenient criterions for judging the density of extremal points, which extend and improve a result of F. Cammaroto and A. Chinni. Using the equivalence between Ekeland''s variational principle and Caristi''s fixed point theorem, we obtain some density results on Caristi''s fixed points. [Copyright &y& Elsevier]

Abstract: By using a very general drop theorem in locally convex spaces we obtain some extended versions of Ekeland''s variational principle, which only need assume local completeness of some related sets and improve Hamel''s recent results. From this, we derive some new versions of Caristi''s fixed points theorems. In the framework of locally convex spaces, we prove that Daneš'' drop theorem, Ekeland''s variational principle, Caristi''s fixed points theorem and Phelps lemma are equivalent to each other. [Copyright &y& Elsevier]