Regularization and iterative method for general variational inequality problem in hilbert spaces

Without the strong monotonicity assumption of the mapping, we provide a regularization method for general variational inequality problem, when its solution set is related to a solution set of an inverse strongly monotone mapping. Consequently, an iterative algorithm for finding such a solution is constructed, and convergent theorem of the such algorithm is proved. It is worth pointing out that, since we do not assume strong monotonicity of general variational inequality problem, our results improve and extend some well-known results in the literature.

[1]  M. Noor.  General variational inequalities , 1988 .

[2]  W. Oettli,et al.  From optimization and variational inequalities to equilibrium problems , 1994 .

[3]  M. O. Osilike,et al.  Demiclosedness Principle and Convergence Theorems for Strictly Pseudocontractive Mappings of Browder–Petryshyn Type☆ , 2001 .

[4]  Hong-Kun Xu.  Iterative Algorithms for Nonlinear Operators , 2002 .

[5]  Wataru Takahashi,et al.  Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings , 2003 .

[6]  Muhammad Aslam Noor,et al.  Some developments in general variational inequalities , 2004, Appl. Math. Comput..

[7]  Wataru Takahashi,et al.  Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings , 2005 .

[8]  Muhammad Aslam Noor,et al.  GENERAL VARIATIONAL INEQUALITIES AND NONEXPANSIVE MAPPINGS , 2007 .

[9]  Muhammad Aslam Noor,et al.  Wiener-Hopf equation technique for variational inequalities and nonexpansive mappings , 2007, Appl. Math. Comput..

[10]  Muhammad Aslam Noor,et al.  General Wiener-Hopf equation technique for nonexpansive mappings and general variational inequalities in Hilbert spaces , 2008, Appl. Math. Comput..

[11]  N. Petrot.  Existence and algorithm of solutions for general set-valued Noor variational inequalities with relaxed (μ,ν)-cocoercive operators in Hilbert spaces , 2010 .

[12]  J. Kim,et al.  Regularization Inertial Proximal Point Algorithm for Monotone Hemicontinuous Mapping and Inverse Strongly Monotone Mappings in Hilbert Spaces , 2010 .