Projection methods for nonconvex variational inequalities
- First Online:
- 231 Downloads
In this paper, we introduce and consider a new class of variational inequalities, which is called the nonconvex variational inequalities. We establish the equivalence between the nonconvex variational inequalities and the fixed-point problems using the projection technique. This equivalent formulation is used to discuss the existence of a solution of the nonconvex variational inequalities. We also use this equivalent alternative formulation to suggest and analyze a new iterative method for solving the nonconvex variational inequalities. We also discuss the convergence of the iterative method under suitable conditions. Our method of proof is very simple as compared with other techniques.
KeywordsMonotone operators Iterative method Resolvent operator Convergence
Unable to display preview. Download preview PDF.
- 1.Brezis, H.: Operateurs maximaux monotone.Mathematical Studies, vol. 5.North-Holland, Amsterdam (1973)Google Scholar
- 2.Bounkhel M., Tadji L., Hamdi A.: Iterative schemes to solve nonconvex variational problems. J. Inequal. Pure Appl. Math. 4, 1–14 (2003)Google Scholar
- 6.Aslam Noor, M.: On Variational Inequalities, Ph.D. Thesis. Brunel University, London (1975)Google Scholar
- 18.Aslam Noor, M.: Some iterative methods for general nonconvex variational inequalities. Comput. Math. Model. 21 (2010)Google Scholar
- 19.Aslam Noor M.: On a class of general variational inequalities. J. Adv. Math. Stud. 1, 75–86 (2008)Google Scholar
- 21.Aslam Noor, M.: Variational Inequalities and Applications. Lecture Notes, Mathematics Department. COMSATS Institute of Information Technology, Islamabad, 2007–2009Google Scholar
- 24.Aslam Noor, M., Inayat Noor, K., Yaqoob, H.: On general mixed variational inequalities. Acta Appl. Math. (2008). doi:10.1007/s10440-008-9402.4