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On generalized variational-like inequalities with generalized monotone multivalued mappings. (English) Zbl 1158.49010
Summary: Let $E$ be a reflexive Banach space with the dual space $E^{*}$ and $K$ be a nonempty closed convex subset of $E$. Let us have $\varPsi :K\times K\times E^{*}\rightarrow R$ and $A:E^{*}\rightarrow E^{*}$. We introduce the class of generalized $\alpha $-monotone multifunctions $T:K\rightarrow 2^{E^{*}}$ with respect to $\varPsi $ and $A$ where $\alpha :E\times E\rightarrow R$. By using the KKM technique and the concept of the Hausdorff metric, we establish some existence results for generalized variational-like inequalities with generalized monotone multivalued mappings in $E$.

MSC:
49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
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References:
[1] Ansari, Q. H.; Siddiqi, A. H.; Yao, J. C.: Generalized vector variational-like inequalities and their scalarizations, Vector variational inequalities and vector equilibria, 17-37 (2002) · Zbl 0992.49013
[2] Fang, Y. P.; Huang, N. J.: Variational-like inequalities with generalized monotone mappings in Banach spaces, J. optim. Theory appl. 118, No. 3, 327-337 (2003) · Zbl 1041.49006 · doi:10.1023/A:1025499305742
[3] V. Preda, M. Beldiman, A. Batatorescu, On variational-like inequalities with generalized monotone mappings, Preprint
[4] Konnov, I. V.; Yao, J. C.: On the generalized vector variational inequality problems, J. math. Anal. appl. 206, 42-58 (1997) · Zbl 0878.49006 · doi:10.1006/jmaa.1997.5192
[5] Schaible, S.: Generalized monotonicity: concepts and uses, Variational inequalities and network equilibrium problems, 289-299 (1995) · Zbl 0847.49013
[6] Konnov, I. V.; Schaible, S.: Duality for equilibrium problems under generalized monotonicity, J. optim. Theory appl. 104, No. 2, 395-408 (2000) · Zbl 1016.90066 · doi:10.1023/A:1004665830923
[7] Ansari, Q. H.; Konnov, I. V.; Yao, J. C.: Existence of a solution and variational principles for vector equilibrium problems, J. optim. Theory appl. 110, No. 3, 481-492 (2001) · Zbl 0988.49004 · doi:10.1023/A:1017581009670
[8] Zeng, L. C.; Schaible, S.; Yao, J. C.: Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, J. optim. Theory appl. 124, 725-738 (2005) · Zbl 1067.49007 · doi:10.1007/s10957-004-1182-z
[9] Yao, J. C.: Variational inequalities and generalized monotone operators, Math. oper. Res. 19, 691-705 (1994) · Zbl 0813.49010 · doi:10.1287/moor.19.3.691
[10] Fan, K.: A generalization of tychonoff’s fixed-point theorem, Math. ann. 142, 305-310 (1961) · Zbl 0093.36701 · doi:10.1007/BF01353421
[11] Chen, Y. Q.: On the semimonotone operator theory and applications, J. math. Anal. appl. 231, 177-192 (1999) · Zbl 0934.47031 · doi:10.1006/jmaa.1998.6245
[12] Jr., S. B. Nadler: Multi-valued contraction mappings, Pacific J. Math. 30, 475-488 (1969) · Zbl 0187.45002