An Iteration to a Common Point of Solution of Variational Inequality and Fixed Point-Problems in Banach Spaces

We introduce an iterative process which converges strongly to a common point of solution of variational inequality problem for a monotone mapping and fixed point of uniformly Lipschitzian relatively asymptotically nonexpansive mapping in Banach spaces. As a consequence, we provide a scheme that converges strongly to a common zero of finite family of monotone mappings under suitable conditions. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

[1]  Habtu Zegeye,et al.  A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems , 2011 .

[2]  R. Rockafellar.  Characterization of the subdifferentials of convex functions , 1966 .

[3]  W. Takahashi.  Nonlinear Functional Analysis , 2000 .

[4]  Habtu Zegeye,et al.  A hybrid approximation method for equilibrium, variational inequality and fixed point problems , 2010 .

[5]  Wataru Takahashi,et al.  STRONG AND WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS AND RELATIVELY NONEXPANSIVE MAPPINGS IN BANACH SPACES , 2009 .

[6]  Wataru Takahashi,et al.  Strong Convergence of Modified Iteration Processes for Relatively Asymptotically Nonexpansive Mappings , 2010 .

[7]  Hong-Ku N Xu.  Another control condition in an iterative method for nonexpansive mappings , 2002, Bulletin of the Australian Mathematical Society.

[8]  W. Takahashi,et al.  Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups , 2003 .

[9]  Wataru Takahashi,et al.  Strong convergence studied by a hybrid type method for monotone operators in a Banach space , 2008 .

[10]  Habtu Zegeye,et al.  Strong convergence theorems for variational inequality problems and quasi-$${\phi}$$-asymptotically nonexpansive mappings , 2012, J. Glob. Optim..

[11]  G. Minty.  Monotone (nonlinear) operators in Hilbert space , 1962 .

[12]  Habtu Zegeye,et al.  Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings , 2010, Appl. Math. Comput..

[13]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[14]  P. Maingé.  Strong Convergence of Projected Subgradient Methods for Nonsmooth and Nonstrictly Convex Minimization , 2008 .

[15]  Wataru Takahashi,et al.  A strong convergence theorem for relatively nonexpansive mappings in a Banach space , 2005, J. Approx. Theory.

[16]  Wataru Takahashi,et al.  Weak and Strong Convergence Theorems for Maximal Monotone Operators in a Banach Space , 2004 .

[17]  Habtu Zegeye,et al.  Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces , 2011, Optim. Lett..

[18]  Wataru Takahashi,et al.  APPROXIMATION OF SOLUTIONS OF VARIATIONAL INEQUALITIES FOR MONOTONE MAPPINGS , 2004 .

[19]  Wataru Takahashi,et al.  Weak and Strong Convergence Theorems for a Nonexpansive Mapping and an Equilibrium Problem , 2007 .

[20]  Habtu Zegeye,et al.  Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings , 2009 .

[21]  Y. Alber.  Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications , 1993, funct-an/9311001.

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

[23]  Wataru Takahashi,et al.  Weak convergence of a projection algorithm for variational inequalities in a Banach space , 2008 .

[24]  Wataru Takahashi,et al.  Strong Convergence of a Proximal-Type Algorithm in a Banach Space , 2002, SIAM J. Optim..