International Journal of Nonlinear Analysis and Applications

Comments on relaxed $(\gamma, r)$-cocoercive mappings

Document Type : Research Paper

Author

Department of Mathematics, University of Kurdistan, Sanandaj 416, Kurdistan, Iran.

Abstract

We show that the variational inequality $VI(C,A)$ has a unique solution for a relaxed $(\gamma , r)$-cocoercive, $\mu$-Lipschitzian mapping $A: Cto H$ with $r>\gamma \mu^2$, where $C$ is a nonempty closed convex subset of a Hilbert space $H$. From this result, it can be derived that, for example, the recent algorithms given in the references of this paper, despite their becoming more complicated, are not general as they should be.

Keywords

  1. M. Aslam Noor, General variational inequalities and nonexpansive mappings, J. Math. Anal. Appl., 331(2007), 810-822.
  2. M. Aslam Noor and Z. Huang, Some resolvent iterative methods for variational inclusions and nonexpansive mappings, Appl. Math. Comput., 194(2007), 267-275.
  3. M. Aslam Noor and Z. Huang, Three-step methods for nonexpansive mappings and variational inequalities, Appl. Math. Comput., 187(2007), 680-685.
  4. M. Aslam Noor and Z. Huang, WienerHopf equation technique for variational inequalities and nonexpansive mappings, Appl. Math. Comput., 191(2007), 504-510.
  5. X. Gao and Y. Guo, Strong Convergence of a Modified Iterative Algorithm for MixedEquilibrium Problems in Hilbert Spaces, J. Inequal. Appl., Vol. 2008, Article ID 454181.
  6. Z. Huang and M. Aslam Noor, Some new unified iteration schemes with errors for nonexpansive mappings and variational inequalities, Appl. Math. Comput., 194(2007), 135-142.
  7. X. Qin, M. Shang and Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Anal., 69(2008), 3897-3909.
  8. X. Qin, M. Shang and Y. Su, Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems, Math. Comput. Model., 48(2008), 1033-1046.
  9. X. Qin, M. Shang and H. Zhou, Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces, Appl. Math. Comput., 200(2008), 242-253.
  10. R. U. Verma, Generalized system for relaxed cocoercive variational inequalities and projection methods, J. Optim. Theory Appl., 121(2004), 203-210.
  11. R. U. Verma, General convergence analysis for two-step projection methods and applications to variational problems, Appl. Math. Lett., 18(2005) 1286-1292.
International Journal of Nonlinear Analysis and Applications
Volume 1, Issue 1 - Serial Number 1
January 2010
Pages 54-57
  • Receive Date: 20 April 2009
  • Revise Date: 05 November 2009
  • Accept Date: 15 November 2009