Abstract
Purpose The purpose of this paper is to study the Hölder calmness of solutions to equilibrium problems and apply it to economics. Design/methodology/approach The authors obtain the Hölder calmness by using an effective approach. More precisely, under the key assumption of strong convexity, sufficient conditions for the Hölder continuity of solution maps to equilibrium problems are established. Findings A new result in stability analysis for equilibrium problems and applications in economics is archived. Originality/value The authors confirm that the paper has not been published previously, is not under consideration for publication elsewhere and is not being simultaneously submitted elsewhere.
Highlights
Many important problems such as optimization problems, variational inequality problems, complementarity problems, Nash equilibrium problems, minimax problems, fixed-point and coincidence-point problems and traffic network problems are considered as special cases of an equilibrium problem (Blum and Oettli, 1994)
The paper aims at investigating the stability analysis in the sense of Hölder calmness of the solution maps to equilibrium problems
G is called h.b -strongly convex-like in B (B not necessarily convex) if and only if, for all x1; x2 2 B and t 2 ð0; 1Þ, there is z 2 B such that: gðzÞ # ð1 À tÞgðx1Þ þ tgðx2Þ À htð1 À tÞdb ðx1; x2Þ: Recall that a function g : X Â X ! R is called monotone on B X if: gðx; yÞ þ gðy; xÞ # 0; 8 x; y 2 B: In what follows, we use the following assumptions that play an important role in investigating conditions for the Hölder calmness of solution maps to the equilibrium problems: H1
Summary
Many important problems such as optimization problems, variational inequality problems, complementarity problems, Nash equilibrium problems, minimax problems, fixed-point and coincidence-point problems and traffic network problems are considered as special cases of an equilibrium problem (Blum and Oettli, 1994). The paper aims at investigating the stability analysis in the sense of Hölder calmness of the solution maps to equilibrium problems. The Hölder calmness of solution maps to mean-variance portfolio and Nash equilibrium problems is derived. R is called monotone on B X if: gðx; yÞ þ gðy; xÞ # 0; 8 x; y 2 B: In what follows, we use the following assumptions that play an important role in investigating conditions for the Hölder calmness of solution maps to the equilibrium problems: H1. The map m 7!w ðx; y; m Þ is n.g -Hölder calm on M, u -uniformly over K(K)
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