Quasi-Variational Inequalities without Concavity Assumptions

Tian, Guoqiang and Zhou, Jianxin (1991): Quasi-Variational Inequalities without Concavity Assumptions. Published in: Journal of Mathematical Analysis and Applications , Vol. 172, (1993): pp. 289-299.

Abstract

This paper generalizes a foundational quasi-variationalinequality by relaxing the (0-diagonal) concavity condition. The approach adopted in this paper is based on continuous selection-type arguments and hence it is quite different from the approach used in the literature. Thus it enables us to prose the existence of equilibrium of the constrained noncooperative games without assuming the (quasi) convexity of loss functions.

• Item Type: MPRA Paper
• Language: English
• Keywords: Quasi-Variational; Inequalities; Concavity
• Subjects: D - Microeconomics > D6 - Welfare Economics > D63 - Equity, Justice, Inequality, and Other Normative Criteria and Measurement

References

G. Allen, Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. Appl. 58 (1977), 1-10.

K. Arrow and G. Debreu, Existence of equilibrium for a competitive economy, Econometrica 22 (1954), 265-290.

J.P. Aubin, “Mathematical Methods of Game and Economic Theory,” North-Holland, Amsterdam, 1979.

J.P. Aubin and I. Ekeland, “Applied Nonlinear Analysis,” Wiley, New York, 1984.

T. C. Bergstrom. R. P. Parks and T. Pader. Pereference which have open graphs, j. Math. Econom. 3 (1976), 265-268

G. Debreu, A social equilibrium existence theorem, Proc. Natl. Acad. Sci. U.S.A. 38 (1952).

K. Fan, A minimax inequality and applications, in “inequalities” (O. Shisha, Ed.), Vol.3, pp. 103-113, Academic Press, New York, 1972.

K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519-537.

U. Mosco, Implicit variational problems and quasi-variational inequalities, in “Lecture Notes in Math.,” Vol. 543, pp. 83-156, Springer-Verlag, New York/Berlin, 1976.

J. Nash, Equilibrium points in N-person games, Proc. Natl. Acad. Sci. U.S.A. 36 (1950), 48-49.

W. Shafer and H. Sonnenschein, Equilibrium in abstract economies without ordered preferences, J. Math. Econom. 2 (1975), 345-348.

M. H. Shin and K. K. Tan, Generalized quasi-variational inequalities in locally convex topological vector spaces, J. Math. Anal. Appl. 108 (1985), 333-343.

W. Takahshi, Non-linear variational inequalities and fixed point theorems, J. Math. Soc. Japan 28 (1976), 477-481.

E. Tarafdar, A fix point theorem equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz theorem, J. Math. Anal. Appl. 128 (1987), 475-479.

G. Tian, Minimax inequalities equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz theorems, to appear.

G. Tian, Equilibrium in abstract economies with a non-compact infinite dimensional strategy space, an infinite number of agents and without ordered preferences, Econom. Lett., in press.

G. Tian, Fixed points theorems for mappings with non-compact and non-convex domains, J. Math. Anal. Appl., in press.

J.X. Zhou and G. Chen, Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl. 132 (1988), 213-225.