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Generalized Convexity, Generalized Monotonicity and Applications pp 89–111Cite as

On the Contraction and Nonexpansiveness Properties of the Marginal Mappings in Generalized Variational Inequalities Involving Co-Coercive Operators

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Part of the Nonconvex Optimization and Its Applications book series (NOIA,volume 77)

Abstract

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.

Keywords

  • Generalized variational inequality
  • co-coercivity
  • 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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Author information

Authors and Affiliations

  1. Posts and Telecommunications, Institute of Technology, Vietnam

    Pham Ngoc Anh

  2. Hanoi Institute of Mathematics, Vietnam

    Le Dung Muu

  3. Department of Mathematics, University of Namur (FUNDP), Belgium

    Van Hien Nguyen & Jean-Jacques Strodiot

Authors
  1. Pham Ngoc Anh
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  2. Le Dung Muu
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  3. Van Hien Nguyen
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  4. Jean-Jacques Strodiot
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Editor information

Editors and Affiliations

  1. RMIT University, Australia

    Andrew Eberhard

  2. University of the Aegean, Greece

    Nicolas Hadjisavvas

  3. University of Avignon, France

    Dinh The Luc

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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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