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In this paper we investigate quasi equilibrium problems in a real Banach space under the assumption of Brezis pseudomonotonicity of the function involved. To establish existence results under weak coercivity conditions we replace the quasi equilibrium problem with a sequence of penalized usual equilibrium problems. To deal with the non compact framework, we apply a regularized version of the penalty method. The particular case of set-valued quasi variational inequalities is also considered.

Exploiting recent results obtained in the theory of equilibrium problems, we focus on the study of the existence of antiperiodic solutions for a first order hemivariational inequality problem. The problem involved is associated to time dependent pseudomonotone and quasimonotone operators. Our results are new and lead to introduce a new and efficient approach for the study of hemivariational inequality problems.

We first derive an Ekeland type variational principle for vector-valued bifunctions without assuming triangular inequality for bifunctions. Then, by using such variational principle, we derive some...

In this paper we prove the existence of solutions of regularized set-valued variational inequalities involving Brezis pseudomonotone operators in reflexive and locally uniformly convex Banach spaces. By taking advantage of this result, we approximate a general set-valued variational inequality with suitable regularized set-valued variational inequalities, and we show that their solutions weakly converge to a solution of the original one. Furthermore, by strengthening the coercivity conditions, we can prove the strong convergence of the approximate solutions.

In this paper, we focus on vector equilibrium problems whose final space is a real linear space not necessarily endowed with a topology. In this framework, some exact and approximate Ekeland variational principles are obtained. The main mathematical tool is a kind of strict fixed point theorem for set-valued mappings, from which the Ekeland variational principles are derived by means of algebraic notions and a concept of approximate solution for vector equilibrium problems based on free-disposal sets. The obtained results improve other recent ones of the literature, and clarify the role of some assumptions usually required on the vector bifunction to state Ekeland variational principles in vector equilibrium problems.

In this paper, we establish Takahashi’s minimization theorem in the setting of quasi-metric spaces and provide its equivalence with Ekeland’s variational principle given in Cobzas (Topol Appl 158:1073–1084, 2011). We present an equilibrium version of Ekeland’s variational principle and extended Takahashi’s minimization theorem in the setting of quasi-metric spaces but without using the triangle inequality of the involved bifunction. We establish an equivalent chain of theorems containing Takahashi’s minimization theorem, Ekeland’s variational principle, the equilibrium version of Ekeland’s variational principle and Caristi–Kirk’s fixed point theorem for set-valued maps in the setting of quasi-metric spaces. As applications, we give an error bound for the solution set of the equilibrium problems and provide sufficient conditions for the existence of weak sharp solutions of equilibrium problems.

In this paper, we study vector quasi-equilibrium problems for the sum of two multivalued bifunctions. The assumptions are required separately on each of these bifunctions. Sufficient conditions for the existence of solutions of such problems are shown in the setting of topological vector spaces. The results in this paper unify, improve and extend some well-known existence theorems from the literature.

In this paper we study the linear openness of the composition of set-valued maps carried out thanks to applications of Nadler’s fixed point theorem and Lim’s lemma. As a byproduct, we obtain the Lipschitz property of the solution map of a generalized parametric equation and of parametric approximate variational inequalities, as well.

In this paper, we provide sufficient conditions entailing the existence of weak sharp efficient points of a multiobjective optimization problem. The approach uses variational analysis techniques, like regularity and subregularity of the diagonal subdifferential map related to a suitable scalar equilibrium problem naturally associated to the multiobjective optimization problem.

We obtain existence results for the weak vector equilibrium problem where the function involved is a sum of two functions, and the assumptions are required separately on each of these functions. We show that some earlier results of this type contain too demanding assumptions. We relax several of these assumption without loosing the results. The special case of reflexive Banach spaces is also studied, where we make use of the fact that closed balls are weakly compact.

In this paper we investigate the Aubin property of the solution map of a parametric equilibrium problem, by providing a connection with a suitable behaviour of the diagonal subdifferential operator associated to the equilibrium bifunction. In particular, we shed some light on the relationship between metric regularity and subregularity of the diagonal subdifferential, on one side, and some properties of the bifunction, on the other side.

We present existence results for variational inequalities given by generalized monotone operators. As a consequence, we deduce the existence of zeros, or even more, the surjectivity of some classes of set-valued operators. We show that by strengthening the continuity assumptions, similar surjectivity results can be obtained without any monotonicity assumption. In the framework of reflexive Banach spaces, we extend a related result due to Inoan and Kolumban (Nonlinear Anal. 68:47–53, 2008).

In this paper we show that the local monotonicity in the sense of Minty and Browder on some residual sets assure the global monotonicity and, according to an earlier result, the convexity of the inverse images. We pay some special attention to the residual sets arising as complements of some special first Baire category sets, namely the \(\sigma \)-affine sets, the \(\sigma \)-compact sets and the \(\sigma \)-algebraic varieties. We achieve this goal gradually by showing, at first, that the continuous real valued functions of one real variable, which are locally nondecreasing on sets whose complements have no nonempty perfect subsets, are globally nondecreasing. The convexity of the inverse images combined with their discreteness, in the case of local injective operators, ensure the global injectivity. Note that the global monotonicity and the local injectivity of regular enough operators is guaranteed by the positive definiteness of the symmetric part of their Gâteaux differentials on the involved residual sets. We close this work with a short subsection on the global convexity which is obtained out of its local counterpart on some residual sets.

New existence results for the strong vector equilibrium problem are presented, relying on a well-known separation theorem in infinite-dimensional spaces. The main results are applied to strong cone saddle-points and strong vector variational inequalities providing new existence results, and furthermore they allow recovery of an earlier result from the literature.

We prove strong convergence theorems for three iterative algorithms which approximate solutions to systems of variational inequalities for mappings of monotone type. All the theorems are set in reflexive Banach spaces and take into account possible computational errors.

In this article we prove that the local increasing monotonicity of an operator on the complement of a certain type of closed set implies its global increasing monotonicity. As applications we obtain some global injectivity/convexity results based on local injectivity/convexity properties and some extra analytic requirements.

In this paper we first provide a geometric interpretation of the Minty–Browder monotonicity which allows us to extend this concept to the so called h -monotonicity, still formulated in an analytic way. A topological concept of monotonicity is also known in the literature: it requires the connectedness of all preimages of the operator involved. This fact is important since combined with the local injectivity, it ensures global injectivity. When a linear structure is present on the source space, one can ask for the preimages to even be convex. In an earlier paper, the authors have shown that Minty–Browder monotone operators defined on convex open sets do have convex preimages, obtaining as a by-product global injectivity theorems. In this paper we study the preimages of h -monotone operators, by showing that they are not divisible by closed connected hypersurfaces, and investigate them from the dimensional point of view. As a consequence we deduce that h -monotone local homeomorphisms are actually global homeomorphisms, as the proved properties of their preimages combined with local injectivity still produce global injectivity.

In this paper we introduce some notions of well-posedness for scalar equilibrium problems in complete metric spaces or in Banach spaces. As equilibrium problem is a common extension of optimization, saddle point and variational inequality problems, our definitions originates from the well-posedness concepts already introduced for these problems.We give sufficient conditions for two different kinds of well-posedness and show by means of counterexamples that these have no relationship in the general case. However, together with some additional assumptions, we show via Ekeland's principle for bifunctions a link between them.Finally we discuss a parametric form of the equilibrium problem and introduce a well-posedness concept for it, which unifies the two different notions of well-posedness introduced in the first part. (C) 2009 Elsevier Ltd. All rights reserved.

We introduce and study two notions of well-posedness for vector equilibrium problems in topological vector spaces; they arise from the well-posedness concepts previously introduced by the same authors in the scalar case, and provide an extension of similar definitions for vector optimization problems. The first notion is linked to the behaviour of suitable maximizing sequences, while the second one is defined in terms of Hausdorff convergence of the map of approximate solutions. In this paper we compare them, and, in a concave setting, we give sufficient conditions on the data in order to guarantee well-posedness. Our results extend similar results established for vector optimization problems known in the literature. © 2008 Springer-Verlag.

The aim of this paper is to introduce and study a dual problem associated to a generalized equilibrium problem (GEP). We show that the solutions of (GEP) and its dual are strictly related to the saddle points of an associated Lagrangian function, and, under some suitable conditions, to the solutions of a family of parametric optimization problems and their dual problems. Our results allow us to show that well-known concepts and results from duality theory of some important particular cases of (GEP) like variational inequalities and optimization problems can be recovered.

We deal with duality for almost convex finite dimensional optimization problems by means of the classical perturbation approach. To this aim some standard results from the convex analysis are extended to the case of almost convex sets and functions. The duality for some classes of primal-dual problems is derived as a special case of the general approach. The sufficient regularity conditions we need for guaranteeing strong duality are proved to be similar to the ones in the convex case.

In this paper we study the convergence of an iterative algorithm for finding zeros with constraints for not necessarily monotone set-valued operators in a reflexive Banach space. This algorithm, which we call the proximal-projection method is, essentially, a fixed point procedure and our convergence results are based on new generalizations of Lemma Opial. We show how the proximal-projection method can be applied for solving ill-posed variational inequalities and convex optimization problems with data given or computable by approximations only. The convergence properties of the proximal-projection method we establish also allow us to prove that the proximal point method (with Bregman distances), whose convergence was known to happen for maximal monotone operators, still converges when the operator involved in it is monotone with sequentially weakly closed graph., 38 pages

The main purpose of this paper is the study of sufficient and/or necessary conditions for existence of solutions of equilibrium problems. We discuss some of the assumptions of the problem, under which the introduced conditions are sufficient and/or necessary, and also analyze the effect of these assumptions on the connection between the solution sets of the equilibrium problem and of a related convex feasibility problem.

In this paper, the authors deal with bifunctions defined on complete metric spaces and with values in locally convex spaces ordered by closed convex cones. The aim is to provide a vector version of Ekeland's theorem related to equilibrium problems. To prove this principle, a weak notion of continuity of a vector-valued function is considered, and some of its properties are presented. Via the vector Ekeland's principle, existence results for vector equilibria are proved in both compact and noncompact domains. (c) 2006 Elsevier Ltd. All rights reserved.

In this paper, we consider first the most important classes of cone convexlike vector-valued functions and give a dual characterization for some of these classes. It turns out that these characterizations are strongly related to the closely convexlike and Ky Fan convex bifunctions occurring within minimax problems. Applying the Lagrangian perturbation approach, we show that some of these classes of cone convexlike vector-valued functions show up naturally in verifying strong Lagrangian duality for finite-dimensional optimization problems. This is achieved by extending classical convexity results for biconjugate functions to the class of so-called almost convex functions. In particular, for a general class of finite-dimensional optimization problems, strong Lagrangian duality holds if some vector-valued function related to this optimization problem is closely K-convexlike and satisfies some additional regularity assumptions. For K a full-dimensional convex cone, it turns out that the conditions for strong Lagrangian duality simplify. Finally, we compare the results obtained by the Lagrangian perturbation approach worked out in this paper with the results achieved by the so-called image space approach initiated by Giannessi.

In this paper, we present the notions of openness and metric regularity for a set-valued map with respect to two fixed sets, proving their equivalence. By using different approaches, we show the stability, with respect to the sum of maps, of the openness property, both in the setting of Banach spaces and of metric spaces. Finally, we infer the regularity of the map solving a generalized parametric equation defined via a parametric map that is, in its turn, perturbed by the sum with another map.

In this paper we discuss the level set method of Joo and how to use it to give an elementary proof of the well-known minimax theorem of Sion. Although this proof technique was initiated by Joo and based on the intersection of upper level sets and a clever use of the topological notion of connectedness, it is not very well known and accessible for researchers in optimization. At the same time we simplify the original proof of Joo and give a more elementary proof of the celebrated minimax theorem of Sion.

In this paper, strong duality for nearly-convex optimization problems is established. Three kinds of conjugate dual problems are associated to the primal optimization problem: the Lagrange dual, Fenchel dual, and Fenchel-Lagrange dual problems. The main result shows that, under suitable conditions, the optimal objective values of these four problems coincide.

In the literature, when dealing with equilibrium problems and the existence of their solutions, the most used assumptions are the convexity of the domain and the generalized convexity and monotonicity, together with some weak continuity assumptions, of the function. In this paper, we focus on conditions that do not involve any convexity concept, neither for the domain nor for the function involved. Starting from the well-known Ekeland's theorem for minimization problems, we find a suitable set of conditions on the function f that lead to an Ekeland's variational principle for equilibrium problems. Via the existence of c-solutions, we are able to show existence of equilibria on general closed sets for equilibrium problems and systems of equilibrium problems. (c) 2004 Elsevier Inc. All rights reserved.

Stampacchia and Minty type generalized implicit variational inequality problems are considered. We extended the notion of dense pseudomonotonicity for multivalued maps and established several existence results for solutions of these problems in the setting of segment-dense sets. We also studied the existence of solutions of Minty type generalized implicit quasi-variational inequality problems. Some particular cases are also studied. It is shown that our results contain several existing results in the literature as special cases.

In this article, the well-known minimax theorems of Wald, Ville and von Neumann are generalized under weaker topological conditions on the payoff function f and/or extended to the larger set of the Borel probability measures, instead of the set of mixed strategies.

In this paper we review known minimax theorems with applications in game theory and show that these theorems can be proved using the first minimax theorem for a two-person zero-sum game with finite strategy sets published by von Neumann in 1928. Among these results are the well known minimax theorems of Wald, Ville and Kneser and their generalizations due to Kakutani, Ky Fan, Konig, Neumann and Gwinner–Oettli. Actually, it is shown that these results form an equivalent chain and this chain includes the strong separation result in finite dimensional spaces between two disjoint closed convex sets of which one is compact. To show the implications the authors only use simple properties of compact sets and the well-known Weierstrass–Lebesgue lemma.

In this paper sufficient regularity and coercivity conditions for Minty and Stampacchia type variational inequality systems are offered. The typical results state that if the independent inequalities are solvable, and the functions involved are lower semicontinuous, then the system has also a solution, that is, a factorization principle holds. As an application, a variational inequality system consisting of two partial differential inequalities is considered. This result is analogous to that of Chen [Journal of Mathematical Analysis and Application 231 (1999) 177] obtained for one variational inequality.

In this paper, we apply a new fixed-point theorem and use various monotonicity and some coercivity conditions to establish equilibrium theorems for multimaps. As a simple consequence, we give a unified approach to vectorial equilibria for multimaps. We show that, from our results, some well-known classical results, such as the Ky Fan minimax inequality theorem and the Browder and Hartman-Stampacchia theorems concerning the existence for variational inequalities, can be derived easily.

In this paper, we review and unify some classes of generalized convex functions introduced by different authors to prove minimax results in infinite-dimensional spaces and show the relations between these classes. We list also for the most general class already introduced by Jeyakumar (Ref. 1) an elementary proof of a minimax result. The proof of this result uses only a finite-dimensional separa- tion theorem; although this minimax result was already presented by Neumann (Ref. 2) and independently by Jeyakumar (Ref. 1), we believe that the present proof is shorter and more transparent.

In this paper, we consider variational inequalities with pseudomonotone maps which depend on a parameter and study the behavior of their solutions. The main result gives sufficient conditions for the stability of the initial variational inequality problem under small perturbations of the parameter. As an application, we obtain a stability result for a class of parametric optimization problems.

In this paper, we introduce several classes of generalized convex functions already discussed in the literature and show the relation between these classes. Moreover, a Gordan–Farkas type theorem is proved for all these classes and it is shown how these theorems can be used to verify strong Lagrangian duality results in finite-dimensional optimization.

A general alternative theorem for convexlike functions is given. This permits the establishment of optimality conditions for convexlike programming problems in which both inequality and equality constraints are considered. It is shown that the main results of the paper contain, in particular, those of Craven, Giannessi, Jeyakumar, Hayashi, and Komiya, Simons, Zalinescu, and a recent result of Tamminen.

This work deals with non-global inverse map results for the sum of two maps. We prove two theorems which shed some new light on this aspect. Some implications in terms of sensitivity of parametric generalized equations are investigated. Finally, a class of well-conditioned operators is identified.

In this paper, necessary and sufficient conditions for solvability of nonlinear inequality systems are given using certain generalized convexity concepts. Our results imply some theorems of Kirszbraun, Fan, Minty, Simons, Sebestyen, and Gwinner-Oettli.

We introduce a regularized equilibrium problem in Banach spaces, involving generalized Bregman functions. For this regularized problem, we establish the existence and uniqueness of solutions. These regularizations yield a proximal-like method for solving equilibrium problems in Banach spaces. We prove that the proximal sequence is an asymptotically solving sequence when the dual space is uniformly convex. Moreover, we prove that all weak accumulation points are solutions if the equilibrium function is lower semicontinuous in its first variable. We prove, under additional assumptions, that the proximal sequence converges weakly to a solution. Refereed/Peer-reviewed

The main purpose of this paper is the study of sufficient conditions for the existence of solutions of vector equilibrium problems. Our main results permit to obtain some existence results already established for the scalar case and, to provide some applications in the area of vector optimization and vector saddle point problems.

This chapter deals with an extended form of the scalar equilibrium problem called the weak multifunction equilibrium problem. New existence results are obtained in the general setting using a well-known separation theorem in infinite dimensional spaces. These results are applied for the particular case of real-valued multifunctions. Furthermore, two gap functions associated with the studied problems are constructed, where, for one of them, the Fenchel duality theory of optimization is used.

In this chapter we give an overview of the theory of scalar equilibrium problems. To emphasize the importance of this problem in nonlinear analysis and in several applied fields we first present its most important particular cases as optimization, Kirszbraun’s problem, saddlepoint (minimax) problems, and variational inequalities. Then, some classical and new results together with their proofs concerning existence of solutions of equilibrium problems are exposed. The existence of approximate solutions via Ekeland’s variational principle – extended to equilibrium problems – is treated within the last part of the chapter.

textabstractIn this chapter we give an overview on the theory of noncooperative games. In the first part we consider in detail for zero-sum (and constant-sum) noncooperative games under which necessary and sufficient conditions on the payoff function and different (extended) strategy sets for both players an equilibrium saddlepoint exists. This is done by using the most elementary proofs. One proof uses the separation result for disjoint convex sets, while the other proof uses linear programming duality and some elementary properties of compact sets. Also, for the most famous saddlepoint result given by Sion's minmax theorem an elementary proof using only the definition of connectedness is given. In the final part we consider n-person nonzero-sum noncooperative games and show by a simple application of the KKM lemma that a so-called Nash equilibrium point exists for compact strategy sets and concavity conditions on the payoff functions.