Projection iterative methods for solving some systems of general nonconvex variational inequalities

Abstract

In this article, we introduce and consider a new system of general nonconvex variational inequalities involving four different operators. We use the projection operator technique to establish the equivalence between the system of general nonconvex variational inequalities and the fixed points problem. This alternative equivalent formulation is used to suggest and analyse some new explicit iterative methods for this system of nonconvex variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of nonconvex variational inequalities, variational inequalities and related optimization problems as special cases, results obtained in this article continue to hold for these problems. Our results can be viewed as a refinement and an improvement of the previously known results for variational inequalities.

Keywords:

• iterative algorithms
• system of nonconvex variational inequalities with different mappings
• relaxed cocoercive mappings
• Lipschitz continuity
• convergence

AMS Subject Classifications::

• 49J40
• 90C33

Acknowledgements

The author would like to thank Dr S.M. Junaid Zaidi, Rector, CIIT, for providing excellent research facilities. The author is grateful to the referees for their constructive and valuable suggestions.