Densely defined equilibrium problems

In the present work we deal with set-valued equilibrium problems for which we provide sufficient conditions for the existence of a solution. The conditions that we consider are imposed not on the whole domain, but rather on a self segment-dense subset of it, a special type of dense subset. As an application, we obtain a generalized Debreu-Gale-Nikaido-type theorem, with a considerably weakened Walras law in its hypothesis. Further, we consider a non-cooperative n-person game and prove the existence of a Nash equilibrium, under assumptions that are less restrictive than the classical ones.Comment: 21 page

Golden ratio algorithms for solving equilibrium problems in Hilbert spaces

In this paper, we design a new iterative algorithm for solving pseudomonotone equilibrium problems in real Hilbert spaces. The advantage of our algorithm is that it requires only one strongly convex programming problem at each iteration. Under suitable conditions we establish the strong and weak convergence of the proposed algorithm. The results presented in the paper extend and improve some recent results in the literature. The performances and comparisons with some existing methods are presented through numerical examples.Comment: 25 pages, 5 figure

Vector Equilibrium Problems on Dense Sets

In this paper we provide sufficient conditions that ensure the existence of the solution of some vector equilibrium problems in Hausdorff topological vector spaces ordered by a cone. The conditions that we consider are imposed not on the whole domain of the operators involved, but rather on a self segment-dense subset of it, a special type of dense subset. We apply the results obtained to vector optimization and vector variational inequalities.Comment: arXiv admin note: substantial text overlap with arXiv:1405.232

Some Minimax Results on Dense Sets and Dense Families of Functionals in C(K) and B(K)

In this paper, by an example we show that the general minimax results of Fan and Sion cannot be extended on usual dense sets. Nevertheless, we obtain some new minimax results on a special type of dense set that we call self-segment-dense. We apply our results in order to obtain denseness of some family of functionals in function spaces.Comment: 27 page

Densely defined perturbed vector equilibrium problems

In this work, we considered perurbed vector equilibrium problems involving set-valued monotone mapping and prove some existence results with and without compactness assump- tions by employing KKM Fan lemma on self segment dense set, a special type of dense set, instead of whole domain