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Existence of generalized variational inequalities. (English) Zbl 0874.49012

Summary: We investigate certain generalized variational inequalities in Hilbert spaces which contain general mildly nonlinear variational inequalities and the classical variational inequalities as special cases. We employ the fixed point technique used by Glowinski, Lions and Stampacchia to obtain some existence results for these nonlinear inequalities. In particular, we obtain some existence results for both general mildly nonlinear variational inequalities and implicit complementarity problems.

MSC:

49J40 Variational inequalities
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
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[1] Allen, G., Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. Appl., 58, 1-10 (1977) · Zbl 0383.49005
[2] Browder, F. E., Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc., 71, 781-785 (1965) · Zbl 0138.39902
[3] Browder, F. E., On the unification of the calculus of variations and the theory of monotone operators in Banach spaces, (Proc. National Academy of Sciences of the United States of America, 56 (1966)), 419-425 · Zbl 0143.36902
[4] Glowinski, R.; Lions, J.; Tremolieres, R., Numerical Analysis of Variational Inequalities (1981), North-Holland: North-Holland Amsterdam · Zbl 0508.65029
[5] Isac, G., On the implicit complementarity problem in Hilbert spaces, Bull. Austral. Math. Soc., 32, 251-260 (1985)
[6] Isac, G., (Complementarity Problems, Lecture Notes in Mathematics (1992), Springer: Springer Berlin), 1528
[7] Harker, P. T.; Pang, J. S., Finite-dimensional variational inequality and nonlinear complementary problems: A survey of theory, algorithms and applications, Math. Programming, Ser. B, 48, 161-221 (1990)
[8] Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and Their Applications (1980), Academic Press: Academic Press New York · Zbl 0457.35001
[9] Lions, J.; Stampacchia, G., Variational inequalities, Comm. Pure Appl. Math., 20, 493-519 (1967) · Zbl 0152.34601
[10] Nagurney, A., Network Economics: A Variational Inequality Approach (1993), Kluwer Academic Publishers: Kluwer Academic Publishers Boston, MA · Zbl 0873.90015
[11] Noor, M. A., An iterative algorithm for variational inequalities, J. Math. Anal. Appl., 158, 448-455 (1991) · Zbl 0733.65047
[12] Noor, M. A., Mixed variational inequalities, Appl. Math. Lett., 3, 73-75 (1990) · Zbl 0714.49014
[13] Noor, M. A., On Variational Inequalities, (Ph.D. Thesis (1975), Brunel University) · Zbl 0859.49009
[14] Noor, M. A., General variational inequalities, Appl. Math. Lett., 1, 119-122 (1988) · Zbl 0655.49005
[15] Yao, J. C., Nonlinear inequalities in Banach spaces, Comput. Math. Appl., 23, 12, 95-98 (1992) · Zbl 0805.47064
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