Generalized System for Relaxed Cocoercive Variational Inequalities and Projection Methods

  • R. U. Verma


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×KH is a nonlinear mapping on K×K.
Relaxed cocoercive nonlinear variational inequalities projection methods relaxed cocoercive mappings cocoercive mappings convergence of projection methods 


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  1. 1.
    VERMA, R. U., Projection Methods, Algorithms, and a New System of Nonlinear Variational Inequalities, Computers and Mathematics with Applications, Vol.41, 1025–1031, 2001.Google Scholar
  2. 2.
    NIE, H., LIU, Z., KIM, K. H., and KANG, S. M., A System of Nonlinear Variational Inequalities Involving Strongly Monotone and Pseudocontractive Mappings, Advances in Nonlinear Variational Inequalities, Vol.6, 91–99, 2003.Google Scholar
  3. 3.
    VERMA, R. U., Projection Methods and a New System of Cocoercive Variational Inequality Problems, International Journal of Differential Equations and Applications, Vol.6, 359–367, 2002.Google Scholar
  4. 4.
    DUNN, J. C., Convexity, Monotonicity, and Gradient Processes in Hilbert Spaces, Journal of Mathematical Analysis and Applications, Vol.53, 145–158, 1976.Google Scholar
  5. 5.
    HE, B. S., A New Method for a Class of Linear Variational Inequalities, Mathematical Programming, Vol.66, 137–144, 1994.Google Scholar
  6. 6.
    KINDERLEHRER, D., and STAMPACCHIA, G., An Introduction to Variational Inequalities, Academic Press, New York, NY, 1980.Google Scholar
  7. 7.
    VERMA, R. U., A Class of Quasivariational Inequalities Involving Cocoercive Mappings, Advances in Nonlinear Variational Inequalities, Vol.2, 1–12, 1999.Google Scholar
  8. 8.
    VERMA, R. U., An Extension of a Class of Nonlinear Quasivariational Inequality Problems based on a Projection Method, Mathematical Sciences Research Hotline, Vol.3, 1–10, 1999.Google Scholar
  9. 9.
    VERMA, R. U., Variational Inequalities in Locally Convex Hausdorff Topological Vector Spaces, Archiv der Mathematik, Vol.71, 246–248, 1998.Google Scholar
  10. 10.
    VERMA, R. U., Nonlinear Variational and Constrained Hemivariational Inequalities Involving Relaxed Operators, Zeitschrift fur Angewandte Mathematik und Mechanik, Vol.77, 387–391, 1997.Google Scholar
  11. 11.
    WITTMANN, R., Approximation of Fixed Points of Nonexpansive Mappings, Archiv der Mathematik, Vol.58, 486–491, 1992.Google Scholar
  12. 12.
    ZEIDLER, E., Nonlinear Functional Analysis and Its Applications, Vol. II/B, Springer Verlag, New York, NY, 1990.Google Scholar
  13. 13.
    XIU, N. H., and ZHANG, J. Z., Local Convergence Analysis of Projection Type Algorithms: Unified Approach, Journal of Optimization Theory and Applications, Vol.115, 211–230, 2002.Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • R. U. Verma
    • 1
  1. 1.Department of MathematicsUniversity of ToledoToledo

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