zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:
49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
65K10Optimization techniques (numerical methods)
WorldCat.org
Full Text: DOI
References:
[1] Chen, Y. Q.: On the semimonotone operator theory and applications. J. math. Anal. appl. 231, 177-192 (1999) · Zbl 0934.47031
[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.: Mathematical theory of hemivariational inequalities and applications. Monographs and textbooks in pure and applied mathematics 118 (1995)
[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) · Zbl 0974.49006
[16] Fan, K.: A generalization of tychonoff’s fixed point theorem. Math. ann. 142, 305-310 (1961) · Zbl 0093.36701