Coercivity Conditions for Equilibrium Problems

Bianchi, Monica; Pini, Rita

doi:10.1007/s10957-004-6466-9

Cited by 108 publications

(76 citation statements)

References 10 publications

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“…Furthermore, if condition (vi) is realized uniformly in λ, then the coercivity assumption (vi) is satisfied. We point out that similar conditions to (vi) have been considered in [6] and [37].…”

Section: Theorem 23 Suppose That the Conditions (I) (Ii) (Iii) (Imentioning

confidence: 79%

“…(b) In comparison with related results in [6] on the existence of solutions for mixed equilibrium problems, we can see that Theorem 2.3 improves Theorem 1A in [6] in the setting where X is a reflexive Banach space endowed with the weak topology. More precisely, in the coercivity assumption (vi) , the weak compact set W is not needed to be convex and hence we present a more general coercivity assumption than the condition (vi) in [6, Theorem 1A] and some related coercivity conditions used in [37]. Furthermore, if G is weakly upper semicontinuous with respect to the first argument, then it is pseudomonotone.…”

Section: Theorem 23 Suppose That the Conditions (I) (Ii) (Iii) (Imentioning

confidence: 99%

See 1 more Smart Citation

Mixed Equilibrium Problems and Anti-periodic Solutions for Nonlinear Evolution Equations

Chadli

Ansari

Yao

2015

J Optim Theory Appl

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By using some new developments in the theory of equilibrium problems, we study the existence of anti-periodic solutions for nonlinear evolution equations associated with time-dependent pseudomonotone and quasimonotone operators in the topological sense. More precisely, we establish new existence results for mixed equilibrium problems associated with pseudomonotone and quasimonotone bifunctions in the topological sense. The results obtained are therefore applied to study the existence of anti-periodic solutions for nonlinear evolution equations in the setting of reflexive 123 J Optim Theory Appl Banach spaces. This new approach leads us to improve and unify most of the recent results obtained in this direction.

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Section: Theorem 23 Suppose That the Conditions (I) (Ii) (Iii) (Imentioning

confidence: 79%

Section: Theorem 23 Suppose That the Conditions (I) (Ii) (Iii) (Imentioning

confidence: 99%

Mixed Equilibrium Problems and Anti-periodic Solutions for Nonlinear Evolution Equations

Chadli

Ansari

Yao

2015

J Optim Theory Appl

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“…Meanwhile, Bianchi and Pini [6] established similar results by using the coercivity condition for equilibrium problem with pseudomonotone functions, and further extended to quasimonotone functions. Analogous to the scalar case, necessary and/or sufficient conditions on the nonemptiness and boundedness of the solution set for C-psedomonotone (generalized) (VEP) were established in [14] for finite-dimensional spaces and in [3,11,25] for reflexive Banach spaces.…”

Section: Introductionmentioning

confidence: 77%

On the characterization of the solution set for vector equilibrium problem

Wang¹,

Gao²

2017

J. Nonlinear Sci. Appl.

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In this article, we investigate the nonemptiness and compactness of the solution set for vector equilibrium problem defined in finite-dimensional spaces. We show that vector equilibrium problem has nonempty and compact solution set if and only if linearly scalarized equilibrium problem has nonempty and compact solution set provided that R 1 = {0} holds. Furthermore, we obtain that vector equilibrium problem has nonempty and compact solution set if and only if linearly scalarized equilibrium problem has nonempty and compact solution set when coercivity condition holds. As applications, we employ the obtained results to derive Levitin-Polyak well-posedness, stability analysis and connectedness of the solution set of the vector equilibrium problem.

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“…The concept of the C-hemicontinuity was introduced by Bianchi and Pini [1] and then by Hadjisavvas [6] for single-valued mapping in the framework of variational inequality problems. (2) F is upper (respectively, lower) C-continuous atx ∈ D if for any neighborhood V of the origin in Y there is a neighborhood U ofx such that:…”

Section: Variational Inclusion Problemsmentioning

confidence: 99%

“…− f (y, αx1 + (1 − α)x 2 ) ∈ − f (y, x 2 ) + V.This yieldsG(y, αx 1 + (1 − α)x 2 , t) = f (y, t) − f (y, αx 1 + (1 − α)x 2 ) ∈ f (y, t) − f (y, x 2 ) + V.…”

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On the existence of solutions to generalized quasi-equilibrium problems

Duong

Tan

2011

J Glob Optim

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In this paper, we apply new results on variational relation problems obtained by D. T. Luc (J Optim Theory Appl 138:65-76, 2008) to generalized quasi-equilibrium problems. Some sufficient conditions on the existence of its solutions of generalized quasiequilibrium problems are shown. As special cases, we obtain several results on the existence of solutions of generalized Pareto and weak quasi-equilibrium problems concerning C-pseudomonotone multivalued mappings. We deduce also some results on the existence of solutions to generalized vector Pareto and weakly quasivariational inequality and vector Pareto quasi-optimization problems with multivalued mappings.Keywords Generalized quasi-equilibrium problems · Upper and lower quasivariational inclusions · Quasi-optimization problems · Upper and lower C-quasiconvex · Upper and lower-quasiconvex-like multivalued mappings · Upper and lower C-continuous multivalued mappings · C-pseudomonotone · C-strong pseudomonotone multivalued mappings Mathematics Subject Classification (2000) 49J27 · 49J53 · 91B50 · 90C48

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Coercivity Conditions for Equilibrium Problems

Cited by 108 publications

References 10 publications

Mixed Equilibrium Problems and Anti-periodic Solutions for Nonlinear Evolution Equations

Mixed Equilibrium Problems and Anti-periodic Solutions for Nonlinear Evolution Equations

On the characterization of the solution set for vector equilibrium problem

On the existence of solutions to generalized quasi-equilibrium problems

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