A Simple proof for the algorithms of relaxed $(u, v)$-cocoercive mappings and $alpha$-inverse strongly monotone mappings

In this paper, a simple proof is presented for the Convergence of the algorithms for the class of relaxed $(u, v)$-cocoercive mappings and $alpha$-inverse strongly monotone mappings. Based on $alpha$-expansive maps, for example, a simple proof of the convergence of the recent iterative algorithms by relaxed $(u, v)$-cocoercive mappings due to Kumam-Jaiboon is provided. Also a simple proof for the convergence of the iterative algorithms by inverse-strongly monotone mappings due to Iiduka-Takahashi in a special case is provided. These results are an improvement as well as a refinement of previously known results.

[1]  J. Jung.  A general composite iterative method for generalized mixed equilibrium problems, variational inequality problems and optimization problems , 2011 .

[2]  Meijuan Shang,et al.  A General Iterative Method for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces , 2008 .

[3]  Kaleem Raza Kazmi,et al.  A hybrid extragradient method for approximating the common solutions of a variational inequality, a system of variational inequalities, a mixed equilibrium problem and a fixed point problem , 2012, Appl. Math. Comput..

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

[5]  Meijuan Shang,et al.  Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces , 2008, Appl. Math. Comput..

[6]  W. Takahashi.  Strong Convergence Theorems for Maximal and Inverse-Strongly Monotone Mappings in Hilbert Spaces and Applications , 2013, J. Optim. Theory Appl..

[7]  Hybrid methods for common solutions in Hilbert spaces with applications , 2014 .

[8]  A. Kangtunyakarn.  Iterative Methods for Finding Common Solution of Generalized Equilibrium Problems and Variational Inequality Problems and Fixed Point Problems of a Finite Family of Nonexpansive Mappings , 2010 .

[9]  P. Kumam,et al.  The Shrinking Projection Method for Common Solutions of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Strictly Pseudocontractive Mappings , 2011 .

[11]  Poom Kumam,et al.  A New Hybrid Algorithm for a System of Mixed Equilibrium Problems, Fixed Point Problems for Nonexpansive Semigroup, and Variational Inclusion Problem , 2011 .

[12]  Shin Min Kang,et al.  Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method , 2010, Comput. Math. Appl..

[13]  Poom Kumam,et al.  The shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-ϕ-nonexpansive mappings , 2011 .

[14]  Tie-Gang Fan,et al.  Viscosity approximation methods for nonexpansive mappings and monotone mappings , 2007 .

[15]  Gang Cai,et al.  Strong convergence theorems based on a new modified extragradient method for variational inequality problems and fixed point problems in Banach spaces , 2011, Comput. Math. Appl..

[16]  Hybrid Algorithms for Minimization Problems over the Solutions of Generalized Mixed Equilibrium and Variational Inclusion Problems , 2011 .

[17]  P. Kumam,et al.  A Viscosity Hybrid Steepest Descent Method for Generalized Mixed Equilibrium Problems and Variational Inequalities for Relaxed Cocoercive Mapping in Hilbert Spaces , 2010 .

[18]  P. Kumam,et al.  Strong Convergence for Generalized Equilibrium Problems, Fixed Point Problems and Relaxed Cocoercive Variational Inequalities , 2010 .

[19]  K. R. Kazmi,et al.  Hybrid iterative method for split monotone variational inclusion problem and hierarchical fixed point problem for a finite family of nonexpansive mappings , 2017, Numerical Algorithms.

[20]  W. Takahashi,et al.  Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space , 2008 .

[21]  Poom Kumam,et al.  Convergence of iterative sequences for fixed points of an infinite family of nonexpansive mappings based on a hybrid steepest descent methods , 2012 .

[22]  P. Kumam,et al.  A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems , 2009 .

[23]  Xiaolong Qin,et al.  Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems , 2008, Math. Comput. Model..

[24]  Mingliang Zhang.  Iterative algorithms for common elements in fixed point sets and zero point sets with applications , 2012 .

[25]  A Hybrid Method for Generalized Equilibrium, Variational Inequality and Fixed Point Problems of Finite Family of Nonexpansive Mappings , 2012 .

[26]  F. Browder,et al.  Construction of fixed points of nonlinear mappings in Hilbert space , 1967 .

[27]  Poom Kumam,et al.  A new hybrid algorithm for a system of equilibrium problems and variational inclusion , 2011 .

[28]  Strong convergence of a general implicit algorithm for variational inequality problems and equilibrium problems and a continuous representation of nonexpansive mappings , 2014 .

[29]  Hossein Piri,et al.  A general iterative method for finding common solutions of system of equilibrium problems, system of variational inequalities and fixed point problems , 2012, Math. Comput. Model..