We investigate the contraction and nonexpansiveness properties of the marginal mappings for gap functions in generalized variational inequalities dealing with strongly monotone and co-coercive operators in a real Hilbert space. We show that one can choose regularization operators such that the solution of a strongly monotone variational inequality can be obtained as the fixed point of a certain contractive mapping. Moreover a solution of a co-coercive variational inequality can be computed by finding a fixed point of a certain nonexpansive mapping. The results give a further analysis for some methods based on the auxiliary problem principle. They also lead to new algorithms for solving generalized variational inequalities involving co-coercive operators. By the Banach contraction mapping principle the convergence rate can be easily established.
- Generalized variational inequality
- contractive and nonexpansive mapping
- Banach iterative method
This work was completed during the visit of the second author at the Department of Mathematics, University of Namur (FUNDP), Namur, Belgium
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Anh, P.N., Muu, L.D., Nguyen, V.H., Strodiot, JJ. (2005). On the Contraction and Nonexpansiveness Properties of the Marginal Mappings in Generalized Variational Inequalities Involving Co-Coercive Operators. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds) Generalized Convexity, Generalized Monotonicity and Applications. Nonconvex Optimization and Its Applications, vol 77. Springer, Boston, MA. https://doi.org/10.1007/0-387-23639-2_5
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