Generalized Equilibrium Problems

If X is a convex subset of a topological vector space and f is a real bifunction defined on X×X, the problem of finding a point x0∈X such that f(x0,y)≥0 for all y∈X, is called an equilibrium problem. When the bifunction f is defined on the cartesian product of two distinct sets X and Y we will call...

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Published inMathematics (Basel) Vol. 11; no. 9; p. 2146
Main Authors Balaj, Mircea, Serac, Dan Florin
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.05.2023
Subjects
Equilibrium
Equilibrium conditions
equilibrium problem
fixed point
generalized equilibrium problem
Mathematical research
quasi-equilibrium problem
Theorems
variational inequality
Vector space
Vector spaces
United States
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Summary:If X is a convex subset of a topological vector space and f is a real bifunction defined on X×X, the problem of finding a point x0∈X such that f(x0,y)≥0 for all y∈X, is called an equilibrium problem. When the bifunction f is defined on the cartesian product of two distinct sets X and Y we will call it a generalized equilibrium problem. In this paper, we study the existence of the solutions, first for generalized equilibrium problems and then for equilibrium problems. In the obtained results, apart from the bifunction f, another bifunction is introduced, the two being linked by a certain compatibility condition. The particularity of the equilibrium theorems established in the last section consists of the fact that the classical equilibrium condition (f(x,x)=0, for all x∈X) is missing. The given applications refer to the Minty variational inequality problem and quasi-equilibrium problems.
ISSN:2227-7390
2227-7390
DOI:10.3390/math11092146