Existence of Solutions of Systems of Generalized Implicit Vector Variational Inequalities
- 98 Downloads
We consider five different types of systems of generalized vector variational inequalities and derive relationships among them. We introduce the concept of pseudomonotonicity for a family of multivalued maps and prove the existence of weak solutions of these problems under these pseudomonotonicity assumptions in the setting of Hausdorff topological vector spaces as well as real Banach spaces. We also establish the existence of a strong solution of our problems under lower semicontinuity for a family of multivalued maps involved in the formulation of the problems. By using a nonlinear scalar function, we introduce gap functions for our problems by which we can solve systems of generalized vector variational inequalities using optimization techniques.
KeywordsSystems of generalized implicit vector variational inequalities Pseudomonotonicity Existence results for a solution Gap functions Lower semicontinuity Upper hemicontinuity ℋ-Hemicontinuity
Unable to display preview. Download preview PDF.
- 1.Ansari, Q.H., Siddiqi, A.H.: A generalized vector variational-like inequality and optimization over an efficient set. In: Brokate, M., Siddiqi, A.H. (eds.) Functional Analysis with Current Applications in Science, Technology and Industry. Pitman Research Notes in Mathematics, vol. 377, pp. 177–191. Addison-Wesley, Essex (1998) Google Scholar
- 4.Lee, G.M.: On relations between vector variational inequality and vector optimization problem. In: Yang, X.Q. et al. (eds.) Progress in Optimization, pp. 167–179. Kluwer Academic, Dordrecht (2000) Google Scholar
- 14.Huang, N.-J., Li, J., Yao, J.C.: Gap function and existence of solutions for a system of vector equilibrium problems. Preprint, National Sun Yat-sen University, Kaohsiung, Taiwan (2006) Google Scholar
- 21.Zeng, L.C., Yao, J.C.: Existence of solutions of generalized vector variational inequalities in reflexive Banach spaces. Preprint, National Sun Yat-sen University, Kaohsiung, Taiwan (2006) Google Scholar