Exceptional Families and Existence Theorems for Variational Inequality Problems

This paper introduces the concept of exceptional family for nonlinear variational inequality problems. Among other things, we show that the nonexistence of an exceptional family is a sufficient condition for the existence of a solution to variational inequalities. This sufficient condition is weaker than many known solution conditions and it is also necessary for pseudomonotone variational inequalities. From the results in this paper, we believe that the concept of exceptional families of variational inequalities provides a new powerful tool for the study of the existence theory for variational inequalities.

[1]  S. Karamardian.  An existence theorem for the complementarity problem , 1976 .

[2]  Vyacheslav Kalashnikov,et al.  Exceptional Families, Topological Degree and Complementarity Problems , 1997, J. Glob. Optim..

[3]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[4]  J. J. Moré,et al.  On P- and S-functions and related classes of n-dimensional nonlinear mappings , 1973 .

[5]  Richaard W. Cottle.  Nonlinear Programs with Positively Bounded Jacobians , 1966 .

[6]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[7]  P. Harker.  A note on the existence of traffic equilibria , 1986 .

[8]  Tony E. Smith,et al.  A solution condition for complementarity problems: with an application to spatial price equilibrium , 1984 .

[9]  S. Karamardian,et al.  The complementarity problem , 1972, Math. Program..

[10]  S. Karamardian.  Complementarity problems over cones with monotone and pseudomonotone maps , 1976 .

[11]  Nimrod Megiddo.  A monotone complementarity problem with feasible solutions but no complementary solutions , 1977, Math. Program..

[12]  Masakazu Kojima,et al.  A unification of the existence theorems of the nonlinear complementarity problem , 1975, Math. Program..

[13]  Siegfried Schaible,et al.  Quasimonotone variational inequalities in Banach spaces , 1996 .

[14]  S. Karamardian.  Generalized complementarity problem , 1970 .

[15]  Jen-Chih Yao,et al.  Pseudo-monotone complementarity problems in Hilbert space , 1992 .

[16]  J. J. Moré.  Coercivity Conditions in Nonlinear Complementarity Problems , 1974 .

[17]  G. Isac.  Complementarity Problems , 1992 .

[18]  B. Curtis Eaves,et al.  On the basic theorem of complementarity , 1971, Math. Program..

[19]  Jorge J. Moré,et al.  Classes of functions and feasibility conditions in nonlinear complementarity problems , 1974, Math. Program..

[20]  S. Karamardian,et al.  Seven kinds of monotone maps , 1990 .

[21]  G. Habetler,et al.  Existence theory for generalized nonlinear complementarity problems , 1971 .

[22]  G. Stampacchia,et al.  On some non-linear elliptic differential-functional equations , 1966 .