Generalized Proximal Distances for Bilevel Equilibrium Problems

Bento, G. C.; Neto, J. X. Cruz; Lopes, J. O.; Soares, P. A.; Soubeyran, Antoine

doi:10.1137/140975589

Cited by 22 publications

(11 citation statements)

References 29 publications

(48 reference statements)

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“…Other extensions, with applications to various areas of social sciences and psychology, were given in [3], [4], [5], [6], [7], [9], [32].…”

Section: Introductionmentioning

confidence: 99%

Ekeland, Takahashi and Caristi principles in quasi-pseudometric spaces

Cobzaş

2019

Topology and its Applications

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We prove versions of Ekeland, Takahashi and Caristi principles in sequentially right K-complete quasi-pseudometric spaces (meaning asymmetric pseudometric spaces), the equivalence between these principles, as well as their equivalence to the completeness of the underlying quasi-pseudometric space.The key tools are Picard sequences for some special set-valued mappings corresponding to a function ϕ on a quasi-pseudometric space, allowing a unitary treatment of all these principles.Classification MSC 2010: 58E30 47H10 54E25 54E35 54E50

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“…Other extensions, with applications to various areas of social sciences and psychology, were given in [3], [4], [5], [6], [7], [9], [32].…”

Section: Introductionmentioning

confidence: 99%

Ekeland, Takahashi and Caristi principles in quasi-pseudometric spaces

Cobzaş

2019

Topology and its Applications

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“…In 2014, Quy [23] introduced the algorithm by combining the proximal method with the Halpern method for solving bilevel monotone equilibrium and fixed point problem. For more details and most recent works on the methods for solving bilevel equilibrium problems, we refer the reader to [2, 5, 24]. The authors considered the method for monotone and pseudoparamonotone equilibrium problem.…”

Section: Introductionmentioning

confidence: 99%

Extragradient subgradient methods for solving bilevel equilibrium problems

et al. 2018

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In this paper, we propose two algorithms for finding the solution of a bilevel equilibrium problem in a real Hilbert space. Under some sufficient assumptions on the bifunctions involving pseudomonotone and Lipschitz-type conditions, we obtain the strong convergence of the iterative sequence generated by the first algorithm. Furthermore, the strong convergence of the sequence generated by the second algorithm is obtained without a Lipschitz-type condition.

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“…We can mention, here, vector variational inequalities, vector optimization problems, or Debreu‐type equilibrium problems. For further relevant information, the reader is referred to the following recent publications available in our bibliography …”

Section: Introductionmentioning

confidence: 99%

Existence of solutions for systems of generalized quasi‐equilibrium problems

Patriche

2017

Math Methods in App Sciences

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In this paper, first of all, we consider a generalized game in choice form with 2 constraints and its corresponding equilibrium in choice. We assert new conditions under which the equilibrium in choice exists. As a consequence, we establish the existence of the equilibrium for generalized abstract economies.Then, we apply the obtained theorems to prove the existence of solutions for systems of quasi-equilibrium problems. We do this by considering new hypotheses for the properties of the involved correspondences. This approach leads us to results which differ a lot from the ones existing in literature. KEYWORDS equilibrium in choice, generalized game in choice form, quasi-equilibrium problem Math Meth Appl Sci. 2018;41:803-817.wileyonlinelibrary.com/journal/mma 804 PATRICHE the approached topic. New techniques of proof are developed by defining new correspondences which are requested to verify the local intersection property. Further, a selection lemma and the Brouwer fixed-point theorem are appealed. Two versions of equilibrium, slightly different, are discussed: weak equilibrium and strong equilibrium. As direct applications, we prove that the equilibrium exists for generalized abstract economies in which the preference correspondences do not have convex values, and their upper or lower sections do not verify topological properties. This study is a continuation of the author's research work, concerning the existence of equilibrium, as it can be seen, for instance, in Hervés-Beloso and Patriche. [10][11][12] Secondly, we study the existence of the solutions for some systems of vector quasi-equilibrium problems. We emphasize the importance of research studies on this topic, justified by the fact that it is, in fact, a unified model of other several problems. We can mention, here, vector variational inequalities, vector optimization problems, or Debreu-type equilibrium problems. For further relevant information, the reader is referred to the following recent publications available in our bibliography. [13][14][15][16][17][18][19][20][21][22][23][24][25][26] We define the notion of "weak solution," and we show its existence by considering an associate generalized abstract economy model which admits weak equilibrium. In this way, we obtain results which are totally new in literature, and here, we refer to the hypotheses, as well as to the tools of demonstration. We show that very slight continuity properties of the involved correspondences can also lead to the existence of the solutions for various classes of equilibrium problems.The rest of the paper is organized as follows: In Section 2, the systems of vector quasi-equilibrium problems are presented, and some new definitions for correspondences are introduced. Section 3 contains results concerning the existence of equilibrium. In Section 4, the existence of solutions for systems of vector quasi-equilibrium problems is proved. The concluding remarks are presented in Section 5.

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Generalized Proximal Distances for Bilevel Equilibrium Problems

Cited by 22 publications

References 29 publications

Ekeland, Takahashi and Caristi principles in quasi-pseudometric spaces

Ekeland, Takahashi and Caristi principles in quasi-pseudometric spaces

Extragradient subgradient methods for solving bilevel equilibrium problems

Existence of solutions for systems of generalized quasi‐equilibrium problems

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