Generalized Variational and Quasi-Variational Inequalities with Discontinuous Operators

Abstract We study the existence of solutions to generalized variational and generalized quasi-variational inequalities with discontinuous operators. We obtain results which generalize and extend previously known theorems. We also compare our new continuity condition on the operator in the variational inequality to previously used continuity conditions. We then apply our results to generalized variational inequalities which involve pseudo-monotone operators.

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

[2]  Paolo Cubiotti.  An existence theorem for generalized quasi-variational inequalities , 1993 .

[3]  Mau-Hsiang Shih,et al.  Browder-Hartman-Stampacchia variational inequalities for multi-valued monotone operators☆ , 1988 .

[4]  K. Fan.  A generalization of Tychonoff's fixed point theorem , 1961 .

[5]  Won Kyu Kim,et al.  Remark on a generalized quasivariational inequality , 1988 .

[6]  Jen-Chih Yao,et al.  The generalized quasi-variational inequality problem with applications , 1991 .

[7]  Sangho Kum,et al.  A generalization of generalized quasi-variational inequalities , 1994 .

[8]  E. Michael.  Continuous Selections. I , 1956 .

[9]  Felix E. Browder,et al.  Existence theorems for nonlinear partial di erential equations , 1970 .

[10]  Jen-Chih Yao,et al.  Generalized quasi-variational inequality and implicit complementarity problems , 1989 .

[11]  Jen-Chih Yao,et al.  Variational and Generalized Variational Inequalities with Discontinuous Mappings , 1994 .

[12]  Bronisław Knaster,et al.  Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe , 1929 .

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

[14]  Felix E. Browder,et al.  Pseudo-monotone operators and the direct method of the calculus of variations , 1970 .

[15]  Jen-Chih Yao,et al.  On the generalized complementarity problem , 1994, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[16]  Mau-Hsiang Shih,et al.  Generalized quasi-variational inequalities in locally convex topological vector spaces , 1985 .

[17]  F. Browder.  Nonlinear operators and nonlinear equations of evolution in Banach spaces , 1976 .