Algorithms for solutions of extended general mixed variational inequalities and fixed points

In this paper, we introduce and study a new class of extended general nonlinear mixed variational inequalities and a new class of extended general resolvent equations and establish the equivalence between the extended general nonlinear mixed variational inequalities and implicit fixed point problems as well as the extended general resolvent equations. Then by using this equivalent formulation, we discuss the existence and uniqueness of solution of the problem of extended general nonlinear mixed variational inequalities. Applying the aforesaid equivalent alternative formulation and a nearly uniformly Lipschitzian mapping S, we construct some new resolvent iterative algorithms for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping S which is the unique solution of the problem of extended general nonlinear mixed variational inequalities. We study convergence analysis of the suggested iterative schemes under some suitable conditions. We also suggest and analyze a class of extended general resolvent dynamical systems associated with the extended general nonlinear mixed variational inequalities and show that the trajectory of the solution of the extended general resolvent dynamical system converges globally exponentially to the unique solution of the extended general nonlinear mixed variational inequalities. The results presented in this paper extend and improve some known results in the literature.

[1]  Muhammad Aslam Noor,et al.  EXISTENCE RESULTS FOR QUASI VARIATIONAL INEQUALITIES , 2007 .

[2]  Jun Wang,et al.  A recurrent neural network for solving linear projection equations , 2000, Neural Networks.

[3]  Muhammad Aslam Noor,et al.  Three-step methods for nonexpansive mappings and variational inequalities , 2007, Appl. Math. Comput..

[4]  A. Nagurney,et al.  On the stability of projected dynamical systems , 1995 .

[5]  Muhammad Aslam Noor.  Projection iterative methods for extended general variational inequalities , 2010 .

[6]  Yeol Je Cho,et al.  An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems , 2009 .

[7]  Muhammad Aslam Noor,et al.  An Iterative Scheme for a Class of Quasi Variational Inequalities , 1985 .

[8]  Paul Tseng,et al.  A Modified Forward-backward Splitting Method for Maximal Monotone Mappings 1 , 1998 .

[9]  Yeol Je Cho,et al.  Generalized nonlinear mixed quasi-variational inequalities , 2000 .

[10]  Ravi P. Agarwal,et al.  Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces , 2011 .

[11]  Muhammad Aslam Noor.  Sensitivity Analysis Of Extended General Variational Inequalities , 2009 .

[12]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[13]  Themistocles M. Rassias,et al.  General auxiliary problem principle and solvability of a class of nonlinear mixed variational inequalities involving partially relaxed monotone mappings , 2002 .

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

[15]  Yeol Je Cho,et al.  A new class of generalized nonlinear mixed quasi-variational inequalities in Banach spaces , 2003 .

[16]  Yeong-Cheng Liou,et al.  Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality problems , 2011 .

[17]  Daya Ram Sahu,et al.  Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces , 2005 .

[18]  Yeol Je Cho,et al.  Regularization and iterative method for general variational inequality problem in hilbert spaces , 2011 .

[19]  Yeol Je Cho,et al.  On systems of generalized nonlinear variational inequalities in Banach spaces , 2008, Appl. Math. Comput..

[20]  Muhammad Aslam Noor.  ON MERIT FUNCTIONS FOR QUASIVARIATIONAL INEQUALITIES , 2007 .

[21]  P. Lions,et al.  Splitting Algorithms for the Sum of Two Nonlinear Operators , 1979 .

[22]  Yeol Je Cho,et al.  Systems of generalized nonlinear variational inequalities and its projection methods , 2008 .

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

[24]  A. Nagurney,et al.  Projected Dynamical Systems and Variational Inequalities with Applications , 1995 .

[25]  Gang Feng,et al.  On Convergence Conditions of an Extended Projection Neural Network , 2005, Neural Computation.

[26]  P. Hartman.  Ordinary Differential Equations , 1965 .

[27]  M Aslam Noor.  Auxiliary Principle Technique for Extended General Variational Inequalities , 2008 .

[28]  Humaira Yaqoob,et al.  On General Mixed Variational Inequalities , 2010 .

[29]  Muhammad Aslam Noor,et al.  Differentiable non-convex functions and general variational inequalities , 2008, Appl. Math. Comput..

[30]  Ralph Tyrell Rockafellar,et al.  Part I. Basic Concepts , 1970 .

[31]  W. Han,et al.  Contact problems in elasticity , 2002 .

[32]  Muhammad Aslam Noor,et al.  Sensitivity Analysis for Quasi-Variational Inequalities , 1997 .

[33]  William A. Kirk,et al.  Asymptotic pointwise contractions , 2008 .

[34]  Yeol Je Cho,et al.  Iterative methods for nonlinear operator equations in banach spaces , 2002 .

[35]  William A. Kirk,et al.  A FIXED POINT THEOREM FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS , 1972 .

[36]  Yeol Je Cho,et al.  ON THE TWO-STEP PROJECTION METHODS AND APPLICATIONS TO VARIATIONAL INEQUALITIES , 2007 .

[37]  Muhammad Aslam Noor,et al.  Iterative methods for solving general quasi-variational inequalities , 2010, Optim. Lett..

[38]  Jun Wang,et al.  On the Stability of Globally Projected Dynamical Systems , 2000 .

[39]  Xinlong Weng,et al.  Fixed point iteration for local strictly pseudo-contractive mapping , 1991 .

[40]  Yeol Je Cho,et al.  ALGORITHMS FOR SYSTEMS OF NONLINEAR VARIATIONAL INEQUALITIES , 2004 .

[41]  Anna Nagurney,et al.  A projected dynamical systems model of general financial equilibrium with stability analysis , 1996 .

[42]  Yeol Je Cho,et al.  Generalized systems for relaxed cocoercive variational inequalities and projection methods in Hilbert spaces , 2009 .

[43]  Yeol Je Cho,et al.  Iterative algorithms for variational inclusions, mixed equilibrium and fixed point problems with application to optimization problems , 2011 .

[44]  Muhammad Aslam Noor.  Resolvent dynamical systems for mixed variational inequalities , 2002 .

[45]  Peter Shi,et al.  Equivalence of variational inequalities with Wiener-Hopf equations , 1991 .

[46]  Ioannis K. Argyros,et al.  Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems , 2010, Comput. Math. Appl..

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

[48]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[49]  Terry L. Friesz,et al.  Dynamic Systems, Variational Inequalities and Control Theoretic Models for Predicting Time-Varying Urban Network Flows , 1996, Transp. Sci..

[50]  M. Noor.  Some algorithms for general monotone mixed variational inequalities , 1999 .

[51]  M. Noor,et al.  Some aspects of variational inequalities , 1993 .

[52]  Anna Nagurney,et al.  Dynamical systems and variational inequalities , 1993, Ann. Oper. Res..

[53]  Lei Wang,et al.  Nonzero solutions for a system of variational inequalities in reflexive Banach spaces , 2009, Math. Comput. Model..

[54]  M. Noor.  New approximation schemes for general variational inequalities , 2000 .