A generalized distance and enhanced Ekeland’s variational principle for vector functions

We propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation. This result is shown to contain probably a large part of existing versions of Ekeland’s variational principle (EVP). We introduce the notion of a weak τ-function p as a generalized distance and use it together with the above result on minimal elements to establish enhanced EVP for various settings, under relaxed lower semicontinuity assumptions. These principles conclude the existence not only of p-strict minimizers of p-perturbations of the considered vector function, but also p-sharp and p-strong minimizers. Our results are proved to be stronger than the classical EVP and many generalizations in the literature, even in the usual one-dimensional case, by numerous corollaries and examples. We include equivalent formulations of our enhanced EVP as well.     

The KKM principle in abstract convex spaces: Equivalent formulations and applications

The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. A KKM space is an abstract convex space satisfying the partial KKM principle and its “open” version. In this paper, we clearly derive a sequence of a dozen statements which characterize the KKM spaces and equivalent formulations of the partial KKM principle. As their applications, we add more than a dozen statements including generalized formulations of von Neumann minimax theorem, von Neumann intersection lemma, the Nash equilibrium theorem, and the Fan type minimax inequalities for any KKM spaces. Consequently, this paper unifies and enlarges previously known several proper examples of such statements for particular types of KKM spaces.     

Fixed points,coincidence points and maximal elements with applications to generalized equilibrium problems and minimax theory

In this paper, using a generalization of the Fan–Browder fixed point theorem, we obtain a new fixed point theorem for multivalued maps in generalized convex spaces from which we derive several coincidence theorems and existence theorems for maximal elements. Applications of these results to generalized equilibrium problems and minimax theory will be given in the last sections of the paper.     

General existence theorems, alternative theorems and applications to minimax problems

We establish general theorems on maximal elements, coincidence points and nonempty intersections for set-valued mappings on GFC-spaces and show their equivalence. Applying them we derive equivalent forms of alternative theorems. As applications, we develop in detail general types of minimax theorems. The results obtained improve or include as special cases several recent ones in the literature.     

New generalizations of basic theorems in the KKM theory

In the present paper, we obtain a new KKM type theorem for intersectionally closed-valued KKM maps and some useful new basic consequences. Typical examples of them are abstract forms of Fan’s matching theorem, Fan’s geometric lemma, the Fan-Browder fixed point theorem, maximal element theorems, Fan’s minimax inequality, variational inequalities, and others.     

Some existence theorems in nonlinear analysis for mappings on GFC-spaces and applications

We establish a maximal element theorem, an intersection theorem and a coincidence-point theorem in product GFC-spaces. As examples of wide ranges of applications, we first deduce sufficient conditions for the solution existence of a mixed system of inclusions. Then using this we obtain existence results for systems of vector quasi-optimization problems and for multiobjective mathematical programs constrained by systems of inclusions. Our results are shown to improve and include recent ones in the literature.     

Bead spaces and fixed point theorems

The notion of a bead metric space defined here (see Definition 6) is a nice generalization of that of the uniformly convex normed space. In turn, the idea of a central point for a mapping when combined with the “single central point” property of the bead spaces enables us to obtain strong and elegant extensions of the Browder-Göhde-Kirk fixed point theorem for nonexpansive mappings (see Theorems 14-17). Their proofs are based on a very simple reasoning. We also prove two theorems on continuous selections for metric and Hilbert spaces. They are followed by fixed point theorems of Schauder type. In the final part we obtain a result on nonempty intersection.     

On mappings covering at a nonlinear rate and their perturbation stability

The present paper contains a study of covering (alias, openness) properties at a nonlinear rate for set-valued mappings between metric spaces. Such study is focussed on the stability of these properties in the presence of perturbations. A crucial result valid for linear openness, known as Milyutin’s theorem, is extended to set-valued mappings covering at a nonlinear rate under possibly non-Lipschitz perturbations. Consequently, a Lyusternik type theorem is derived from such extension and a general penalization principle for constrained optimization problems, which exploits nonlinear covering properties, is presented.     

A coercive James’s weak compactness theorem and nonlinear variational problems

We discuss a perturbed version of James’s sup theorem for weak compactness that not only properly generalizes that classical statement, but also some recent extensions of this central result: the sublevel sets of an extended real valued and coercive function whose subdifferential is surjective are relatively weakly compact. Furthermore, we apply it to generalize and unify some facts in mathematical finance and to prove that the unique possible framework in the development of an existence theory for a wide class of nonlinear variational problems is the reflexive one.     

Convergence theorems for continuous descent methods

We examine continuous descent methods for the minimization of Lipschitzian functions defined on a general Banach space. We establish several convergence theorems for those methods which are generated by regular vector fields. Since the complement of the set of regular vector fields is -porous, we conclude that our results apply to most vector fields in the sense of Baires categories.     

Remarks on some basic concepts in the KKM theory

In the KKM theory, some authors adopt the concepts of the compact closure (ccl), compact interior (cint), transfer compactly closed-valued multimap, transfer compactly l.s.c. multimap, and transfer compactly local intersection property, respectively, instead of the closure, interior, closed-valued multimap, l.s.c. multimap, and possession of a finite open cover property. In this paper, we show that such adoption is inappropriate and artificial. In fact, any theorem with a term with “transfer” attached is equivalent to the corresponding one without “transfer”. Moreover, we can invalidate terms with “compactly” attached by giving a finer topology on the underlying space. In such ways, we obtain simpler formulations of KKM type theorems, Fan-Browder type fixed point theorems, and other results in the KKM theory on abstract convex spaces.     

The optimal value and optimal solutions of the proximal average of convex functions

The proximal average of a finite collection of convex functions is a parameterized convex function that provides a continuous transformation between the convex functions in the collection. This paper analyzes the dependence of the optimal value and the minimizers of the proximal average on the weighting parameter. Concavity of the optimal value is established and implies further regularity properties of the optimal value. Boundedness, outer semicontinuity, single-valuedness, continuity, and Lipschitz continuity of the minimizer mapping are concluded under various assumptions. Sharp minimizers are given further attention. Several examples are given to illustrate our results.     

Applications of proximal calculus to fixed point theory on Riemannian manifolds

We prove a general form of a fixed point theorem for mappings from a Riemannian manifold into itself which are obtained as perturbations of a given mapping by means of general operations which in particular include the cases of sum (when a Lie group structure is given on the manifold) and composition. In order to prove our main result we develop a theory of proximal calculus in the setting of Riemannian manifolds.     

Proximal Calculus on Riemannian Manifolds

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold M. We give some applications of this theory, concerning, for instance, a Borwein-Preiss type variational principle on a Riemannian manifold M, as well as differentiability and geometrical properties of the distance function to a closed subset C of M. The first-named author was supported by a Marie Curie Intra-European Fellowship of the European Community, Human Resources and Mobility Programme under contract number MEIF CT2003-500927. The second-named author was supported by BFM2003-06420.     

Strong convergence theorem by a new hybrid method for equilibrium problems and variational inequality problems

In this paper, we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of the solutions of the variational inequality problem by using a new hybrid method. We obtain a new result for finding a solution of an equilibrium problem and the solutions of the variational inequality problem.     

Equilibrium existence theorems in KKM spaces

An abstract convex space satisfying the KKM principle is called a KKM space. This class of spaces contains G$G$-convex spaces properly. In this work, we show that a large number of results in KKM theory on G$G$-convex spaces also hold on KKM spaces. Examples of such results are theorems of Sperner and Alexandroff–Pasynkoff, Fan–Browder type fixed point theorems, Horvath type fixed point theorems, Ky Fan type minimax inequalities, variational inequalities, von Neumann type minimax theorems, Nash type equilibrium theorems, and Himmelberg type fixed point theorems.     

Compositions and averages of two resolvents: Relative geometry of fixed points sets and a partial answer to a question by C. Byrne

We show that the set of fixed points of the average of two resolvents can be found from the set of fixed points for compositions of two resolvents associated with scaled monotone operators. Recently, the proximal average has attracted considerable attention in convex analysis. Our results imply that the minimizers of proximal-average functions can be found from the set of fixed points for compositions of two proximal mappings associated with scaled convex functions. When both convex functions in the proximal average are indicator functions of convex sets, least squares solutions can be completely recovered from the limiting cycles given by compositions of two projection mappings. This provides a partial answer to a question posed by C. Byrne. A novelty of our approach is to use the notion of resolvent average and proximal average.     

A variational inequality index for condensing maps in Hilbert spaces and applications to semilinear elliptic inequalities

A variational inequality index for γ-condensing maps is established in Hilbert spaces. New results on existence of nonzero positive solutions of variational inequalities for such maps are proved by using the theory of variational inequality index. Applications of such a theory are given to existence of nonzero positive weak solutions for semilinear second order elliptic inequalities, where previous results of variational inequalities for S-contractive maps cannot be applied.     

Minimax theorems for limits of parametrized functions having at most one local minimum lying in a certain set

In this paper, we establish some minimax theorems, of purely topological nature, that, through the variational methods, can be usefully applied to nonlinear differential equations. Here is a (simplified) sample: Let X be a Hausdorff topological space, I⊆R an interval and . Assume that the function Ψ(x,⋅) is lower semicontinuous and quasi-concave in I for all x∈X, while the function Ψ(⋅,q) has compact sublevel sets and one local minimum at most for each q in a dense subset of I. Then, one has