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Generalized System for Relaxed Cocoercive Variational Inequalities and Projection Methods

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  • R. U. Verma

    (University of Toledo)

Abstract

Let K be a nonempty closed convex subset of a real Hilbert space H. The approximate solvability of a system of nonlinear variational inequality problems, based on the convergence of projection methods, is discussed as follows: find an element (x*, y*)∈K×K such that $$\begin{gathered} \left\langle {\rho {\rm T}(y^* ,x^* ) + x^* - y^* ,x - x^* } \right\rangle \geqslant 0,{\text{ }}\forall x \in K{\text{ and }}\rho > 0, \hfill \\ \left\langle {\eta {\rm T}(x^* ,y^* ) + y^* - x^* ,x - x^* } \right\rangle \geqslant 0,{\text{ }}\forall x \in K{\text{ and }}\eta > 0, \hfill \\ \end{gathered}$$ where T: K×K→H is a nonlinear mapping on K×K.

Suggested Citation

  • R. U. Verma, 2004. "Generalized System for Relaxed Cocoercive Variational Inequalities and Projection Methods," Journal of Optimization Theory and Applications, Springer, vol. 121(1), pages 203-210, April.
    • Handle: RePEc:spr:joptap:v:121:y:2004:i:1:d:10.1023_b:jota.0000026271.19947.05
    DOI: 10.1023/B:JOTA.0000026271.19947.05
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    References listed on IDEAS

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    1. N.H. Xiu & J.Z. Zhang, 2002. "Local Convergence Analysis of Projection-Type Algorithms: Unified Approach," Journal of Optimization Theory and Applications, Springer, vol. 115(1), pages 211-230, October.
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    Cited by:

    1. Wu, Zeng-bao & Zou, Yun-zhi & Huang, Nan-jing, 2016. "A class of global fractional-order projective dynamical systems involving set-valued perturbations," Applied Mathematics and Computation, Elsevier, vol. 277(C), pages 23-33.
    2. R. P. Agarwal & R. U. Verma, 2010. "Inexact A-Proximal Point Algorithm and Applications to Nonlinear Variational Inclusion Problems," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 431-444, March.
    3. R. U. Verma, 2006. "General System of A-Monotone Nonlinear Variational Inclusion Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 131(1), pages 151-157, October.
    4. Changqun Wu & Meijuan Shang & Xiaolong Qin, 2007. "A General Projection Method for the System of Relaxed Cocoercive Variational Inequalities in Hilbert Spaces," Modern Applied Science, Canadian Center of Science and Education, vol. 1(3), pages 1-24, September.
    5. P. E. Maingé, 2008. "New Approach to Solving a System of Variational Inequalities and Hierarchical Problems," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 459-477, September.
    6. Q. Z. Yang, 2006. "On a Generalized System for Relaxed Cocoercive Variational Inequalities and Projection Methods," Journal of Optimization Theory and Applications, Springer, vol. 130(3), pages 547-549, September.
    7. Kyung Soo Kim, 2018. "System of Extended General Variational Inequalities for Relaxed Cocoercive Mappings in Hilbert Space," Mathematics, MDPI, vol. 6(10), pages 1-12, October.
    8. R. U. Verma, 2006. "A-Monotonicity and Its Role in Nonlinear Variational Inclusions," Journal of Optimization Theory and Applications, Springer, vol. 129(3), pages 457-467, June.

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