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On the generalized monotonicity of variational inequalities. (English) Zbl 1122.49006

Summary: Three new classes of generalized monotone operators are introduced: the relaxed \(\mu \) pseudomonotone operator, relaxed \(\mu \) quasimonotone operator and densely relaxed pseudomonotone operator. The relations between these three classes and the relations between them and some other well known classes of generalized monotone operators are discussed in detail. Various existence results for variational inequalities in normed spaces are derived. The results generalize some known results.

MSC:

49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
65K10 Numerical optimization and variational techniques
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[1] Chen, Y.Q., On the semimonotone operator theory and applications, J. math. anal. appl., 231, 177-192, (1999)
[2] Siddqi, A.H.; Ansari, Q.H.; Kazmi, K.R., On nonlinear variational inequalities, Indian J. pure appl. math., 25, 969-973, (1994) · Zbl 0817.49012
[3] Naniewicz, Z.; Panagiotopoulos, P.D., ()
[4] Hartman, G.J.; Stampacchia, G., On some nonlinear elliptic differential functional equations, Acta math., 115, 271-310, (1966) · Zbl 0142.38102
[5] Verma, R.U., On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators, J. math. anal. appl., 213, 387-392, (1997) · Zbl 0902.49009
[6] Fang, Y.P.; Huang, N.J., Variational-like inequalities with generalized monotone mappings in Banach spaces, J. optim. theory appl., 118, 327-338, (2003) · Zbl 1041.49006
[7] Harker, P.T.; Pang, J.S., Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms, and applications, Math. program., 48, 161-220, (1990) · Zbl 0734.90098
[8] Cottle, R.W.; Yao, J.C., Pseudomonotone complementarity problems in Hilbert spaces, J. optim. theory appl., 78, 281-295, (1992) · Zbl 0795.90071
[9] Hadjisavvas, N.; Schaible, S., Quasimonotone variational inequalities in Banach spaces, J. optim. theory appl., 90, 95-111, (1996) · Zbl 0904.49005
[10] Karamardian, S.; Schaible, S., Seven kinds of monotone maps, J. optim. theory appl., 66, 37-46, (1990) · Zbl 0679.90055
[11] Karamardian, S., Complementarity over cones with monotone and pseudomonotone maps, J. optim. theory appl., 18, 445-454, (1976) · Zbl 0304.49026
[12] Karamardian, S.; Schaible, S.; Crouzeix, J.P., Characterizations of generalized monotone maps, J. optim. theory appl., 76, 399-413, (1993) · Zbl 0792.90070
[13] Konnov, I.V.; Schaible, S., Duality for equalibrium problems under generalized monotonicity, J. optim. theory appl., 104, 395-408, (2000) · Zbl 1016.90066
[14] Bai, M.R.; Zhou, S.Z.; Ni, G.Y., Variational-like inequalities with relaxed \(\eta - \alpha\) pseudomonotone mappings in Banach spaces, Appl. math. lett., 19, 547-554, (2006) · Zbl 1144.47047
[15] Luc, D.T., Existence results for densely pseudomonotone variational inequalities, J. math. anal. appl., 254, 309-320, (2001)
[16] Fan, K., A generalization of tychonoff’s fixed point theorem, Math. ann., 142, 305-310, (1961) · Zbl 0093.36701
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