Parametric Equilibrium Problems Governed by Topologically Pseudomonotone Bifunctions

2015 ◽  
Vol 65 (5) ◽  
Author(s):  
Marcel Bogdan ◽  
Eduardo Pascali
Keyword(s):  
Equilibrium Problems ◽  
Sufficient Conditions ◽  
Limit Problem ◽  
Solution Map ◽  
Pseudomonotone Bifunction ◽  
Parametric Equilibrium Problems

AbstractClosedness of the solution map is investigated for a sequence of parametric inequality problems related to a “limit” problem governed by a pseudomonotone bifunction. The main result gives sufficient conditions for closedness of the solution map defined on the set of parameters.

scholarly journals Generalized Knaster-Kuratowski-Mazurkiewicz type theorems and applications to minimax inequalities

2017 ◽  
Vol 20 (K2) ◽  
pp. 131-140
Author(s):  
Linh Manh Ha
Keyword(s):  
Variational Inequalities ◽  
Equilibrium Problems ◽  
Applied Mathematics ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Minimax Theorems ◽  
Minimax Inequalities ◽  
Special Cases ◽  
Kkm Mappings ◽  
Definition Of

Knaster-Kuratowski-Mazurkiewicz type theorems play an important role in nonlinear analysis, optimization, and applied mathematics. Since the first well-known result, many international efforts have been made to develop sufficient conditions for the existence of points intersection (and their applications) in increasingly general settings: Gconvex spaces [21, 23], L-convex spaces [12], and FCspaces [8, 9]. Applications of Knaster-Kuratowski-Mazurkiewicz type theorems, especially in existence studies for variational inequalities, equilibrium problems and more general settings have been obtained by many authors, see e.g. recent papers [1, 2, 3, 8, 18, 24, 26] and the references therein. In this paper we propose a definition of generalized KnasterKuratowski-Mazurkiewicz mappings to encompass R-KKM mappings [5], L-KKM mappings [11], T-KKM mappings [18, 19], and many recent existing mappings. Knaster-KuratowskiMazurkiewicz type theorems are established in general topological spaces to generalize known results. As applications, we develop in detail general types of minimax theorems. Our results are shown to improve or include as special cases several recent ones in the literature.

scholarly journals ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF STOCHASTIC, PARAMETRIC ELLIPTIC MULTISCALE PDEs

2013 ◽  
Vol 11 (01) ◽  
pp. 1350001 ◽  
Author(s):  
V. H. HOANG ◽  
CH. SCHWAB
Keyword(s):  
Polynomial Chaos ◽  
Sufficient Conditions ◽  
Limit Problem ◽  
Elliptic Pdes ◽  
Random Solution ◽  
Anisotropic Conductivity ◽  
Scale Parameters ◽  
Regularity Results ◽  
Analytic Regularity ◽  
Scale Limit

A class of second order, elliptic PDEs in divergence form with stochastic and anisotropic conductivity coefficients and n known, separated microscopic length scales εi, i = 1, …, n in a bounded domain D ⊂ ℝd is considered. Neither stationarity nor ergodicity of these coefficients is assumed. Sufficient conditions are given for the random solution to converge ℙ-a.s, as εi → 0, to a stochastic, elliptic one-scale limit problem in a tensorized domain of dimension (n + 1)d. It is shown that this stochastic limit problem admits best N-term "polynomial chaos" type approximations which converge at a rate σ > 0 that is determined by the summability of the random inputs' Karhúnen–Loève expansion. The convergence of the polynomial chaos expansion is shown to hold ℙ-a.s. and uniformly with respect to the scale parameters εi. Regularity results for the stochastic, one-scale limiting problem are established. An error bound for the approximation of the random solution at finite, positive values of the scale parameters εi is established in the case of two scales, and in the case of n > 2, scales convergence is shown, albeit without giving a convergence rate in this case.

scholarly journals Approximation Results for Equilibrium Problems Involving Strongly Pseudomonotone Bifunction in Real Hilbert Spaces

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 137
Author(s):  
Wiyada Kumam ◽  
Kanikar Muangchoo
Keyword(s):  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Minimax Problems ◽  
Strong Convergence Theorem ◽  
Test Problems ◽  
Subgradient Extragradient Method ◽  
Step Size ◽  
Pseudomonotone Bifunction ◽  
Equilibrium Test

A plethora of applications in non-linear analysis, including minimax problems, mathematical programming, the fixed-point problems, saddle-point problems, penalization and complementary problems, may be framed as a problem of equilibrium. Most of the methods used to solve equilibrium problems involve iterative methods, which is why the aim of this article is to establish a new iterative method by incorporating an inertial term with a subgradient extragradient method to solve the problem of equilibrium, which includes a bifunction that is strongly pseudomonotone and meets the Lipschitz-type condition in a real Hilbert space. Under certain mild conditions, a strong convergence theorem is proved, and a required sequence is generated without the information of the Lipschitz-type cost bifunction constants. Thus, the method operates with the help of a slow-converging step size sequence. In numerical analysis, we consider various equilibrium test problems to validate our proposed results.

scholarly journals LOWER SEMICONTINUITY OF PARAMETRIC GENERALIZED WEAK VECTOR EQUILIBRIUM PROBLEMS

2009 ◽  
Vol 81 (1) ◽  
pp. 85-95 ◽  
Author(s):  
SHENG-JIE LI ◽  
HUI-MIN LIU ◽  
CHUN-RONG CHEN
Keyword(s):  
Equilibrium Problem ◽  
Lower Semicontinuity ◽  
Equilibrium Problems ◽  
Sufficient Conditions ◽  
Vector Equilibrium Problems ◽  
Solution Mapping ◽  
Scalarization Method ◽  
Set Valued Mappings ◽  
Vector Equilibrium ◽  
Weak Vector

AbstractIn this paper, using a scalarization method, we obtain sufficient conditions for the lower semicontinuity and continuity of the solution mapping to a parametric generalized weak vector equilibrium problem with set-valued mappings.

Existence of semiclassical ground-state solutions for semilinear elliptic systems

2012 ◽  
Vol 142 (4) ◽  
pp. 867-895 ◽  
Author(s):  
Jun Wang ◽  
Junxiang Xu ◽  
Fubao Zhang
Keyword(s):  
Ground State ◽  
Sufficient Conditions ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Limit Problem ◽  
State Solution ◽  
Ground State Solutions ◽  
Semilinear Elliptic ◽  
Small Positive Parameter ◽  
Image Position

This paper is concerned with the following semilinear elliptic equations of the formwhere ε is a small positive parameter, and where f and g denote superlinear and subcritical nonlinearity. Suppose that b(x) has at least one maximum. We prove that the system has a ground-state solution (ψε, φε) for all sufficiently small ε > 0. Moreover, we show that (ψε, φε) converges to the ground-state solution of the associated limit problem and concentrates to a maxima point of b(x) in certain sense, as ε → 0. Furthermore, we obtain sufficient conditions for nonexistence of ground-state solutions.

Second-order sensitivity analysis for parametric equilibrium problems in set-valued optimization

2019 ◽  
Vol 53 (4) ◽  
pp. 1245-1260
Author(s):  
Nguyen Le Hoang Anh
Keyword(s):  
Sensitivity Analysis ◽  
Equilibrium Problems ◽  
Second Order ◽  
Weak Perturbation ◽  
Parametric Equilibrium Problems ◽  
Perturbation Map ◽  
Derivatives Of

In the paper, we first establish relationships between second-order contingent derivatives of a given set-valued map and that of the weak perturbation map. Then, these results are applied to sensitivity analysis for parametric equilibrium problems in set-valued optimization.

scholarly journals Existence and Well-Posedness for Symmetric Vector Quasi-Equilibrium Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Kaihong Wang ◽  
Wenyan Zhang ◽  
Min Fang
Keyword(s):  
Existence Result ◽  
Equilibrium Problems ◽  
Sufficient Conditions ◽  
Solution Set ◽  
Quasi Equilibrium ◽  
Well Posedness

An existence result for the solution set of symmetric vector quasi-equilibrium problems that allows for discontinuities is obtained. Moreover, sufficient conditions for the generalized Levitin-Polyak well-posedness of symmetric vector quasi-equilibrium problems are established.

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