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Existence of solutions of generalized variational inequalities in reflexive Banach spaces. (English) Zbl 1163.47307
Summary: In this work, we study the following Generalized Variational Inequality Problem (for short, GVIP): Given a closed convex set $K$ in a reflexive Banach space $E$ with the dual $E^{*}$, a multifunction $T:K\rightarrow 2^{E^{*}}$, and a vector $b\in E^{*}$, find $\bar x \in K$ such that there exists $\bar u \in T(\bar x)$ satisfying $$\langle \bar u - b, y-\bar x\rangle \ge 0,\qquad \text{for  all}\,\, y \in K.$$ By using generalized projection and Ky Fan’s well-known KKM theorem, we prove existence results for solutions of GVIP. Our results extend some recent results from the literature.

47J20Inequalities involving nonlinear operators
46N10Applications of functional analysis in optimization and programming
47N10Applications of operator theory in optimization, convex analysis, programming, economics
49J40Variational methods including variational inequalities
Full Text: DOI
[1] Alber, Ya.: Metric and generalized projection operators in Banach spaces: properties and applications, Theory and applications of nonlinear operators of accretive and monotone type, 15-50 (1996) · Zbl 0883.47083
[2] Ansari, Q. H.; Lin, Y. C.; Yao, J. C.: General KKM theorem with applications to minimax and variational inequalities, J. optim. Theory appl. 104, No. 1, 41-57 (2000) · Zbl 0956.49004 · doi:10.1023/A:1004620620928
[3] Cioranescu, I.: Geometry of Banach spaces duality mappings and nonlinear problems, (1990) · Zbl 0712.47043
[4] Deimling, K.: Nonlinear functional analysis, (1985) · Zbl 0559.47040
[5] Li, J. L.: On the existence of solutions of variational inequalities in Banach spaces, J. math. Anal. appl. 295, 115-126 (2004) · Zbl 1045.49008 · doi:10.1016/j.jmaa.2004.03.010
[6] Li, J. L.: The generalized projection operator on reflexive Banach spaces and its applications, J. math. Anal. appl. 306, 55-71 (2005) · Zbl 1129.47043 · doi:10.1016/j.jmaa.2004.11.007
[7] Shih, M. -H.; Tan, K. -K.: Browder--hartman--stampacchia variational inequalities for multi-valued monotone operators, J. math. Anal. appl. 134, 431-440 (1988) · Zbl 0671.47043 · doi:10.1016/0022-247X(88)90033-9
[8] Yao, J. C.: Multi-valued variational inequalities with K-pseudomonotone operators, J. optim. Theory appl. 83, No. 2, 391-403 (1994) · Zbl 0812.47055 · doi:10.1007/BF02190064
[9] Zeidler, E.: Nonlinear functional analysis and its application, II/B: nonlinear monotone operators, (1990) · Zbl 0684.47029
[10] Zeidler, E.: Nonlinear functional analysis and its application, I fixed-point theorems, (1993) · Zbl 0794.47033
[11] Zeng, L. C.; Yao, J. C.: Existence theorems for variational inequalities in Banach spaces, J. optim. Theory appl. 132, No. 2, 321-337 (2007) · Zbl 1149.49015 · doi:10.1007/s10957-006-9139-z