A VARIANT OF THE GENERALIZED VECTOR VARIATIONAL INEQUALITY WITH OPERATOR SOLUTIONS

Title & Authors
A VARIANT OF THE GENERALIZED VECTOR VARIATIONAL INEQUALITY WITH OPERATOR SOLUTIONS
Kum, Sang-Ho;

Abstract
In a recent paper, Domokos and $\small{Kolumb\`{a}}n}$ [2] gave an interesting interpretation of variational inequalities (VI) and vector variational inequalities (VVI) in Banach space settings in terms of variational inequalities with operator solutions (in short, OVVI). Inspired by their work, in a former paper [4], we proposed the scheme of generalized vector variational inequality with operator solutions (in short, GOVVI) which extends (OVVI) into a multivalued case. In this note, we further develop the previous work [4]. A more general pseudomonotone operator is treated. We present a result on the existence of solutions of (GVVI) under the weak pseudomonotonicity introduced in Yu and Yao [8] within the framework of (GOVVI) by exploiting some techniques on (GOVVI) or (GVVI) in [4].
Keywords
vector variational inequality;weakly C-pseudemonotone operator;generalized hemicontinuity;Fan-Browder fixed point theorem;
Language
English
Cited by
1.
On generalized operator quasi-equilibrium problems, Journal of Mathematical Analysis and Applications, 2008, 345, 1, 559
2.
Semicontinuity of the solution multifunctions of the parametric generalized operator equilibrium problems, Nonlinear Analysis: Theory, Methods & Applications, 2009, 71, 12, e2182
References
1.
F. E. Browder, The fixed point theory of multivalued mappings in topological vector space, Math. Ann. 177 (1968), 283-30l

2.
A. Domokos and J. Kolumban, Variational inequalities with operator solutions, J. Global Optim. 23 (2002), 99-110

3.
I. V. Konnov and J. C. Yao, On the generalized vector variational inequality problem, J. Math. Anal. Appl. 206 (1997), 42-58

4.
S. H. Kum and W. K. Kim, Generalized vector variational and quasi-variational inequalities with operator solutions, J. Global Optim. 32 (2005), 581-595

5.
G. M. Lee and S. H. Kum, On implicit vector variational inequalities, J. Optim. Theory Appl. 104 (2000), 409-425

6.
S. Park, Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps, J. Korean Math. Soc. 31 (1994), 493-519

7.
W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1974

8.
S. J. Yu and J. C. Yao, On vector variational inequalities, J. Optim. Theory Appl. 89 (1996), 749-769