Some New Classes of Quasi Split Feasibility Problems

In thispaper, we introduce and consider a new problem of finding u2 K(u) such that Au2 C, where K : u! K(u) is a closed convex-valued set in the real Hilbert space H1, C is closed convex set in the real Hilbert space H2 respectively and A is linear bounded self-adjoint operator from H1 and H2. This problem is called the quasi split feasibility problem. We show that the qu asi feasibility problem is equivalent to the fixed point problem and quasi variational ine quality. These s alternative equivalent formulations are used to consider the existence of a solution of the quasi split feasibility problem. So me special cases are also considered. Problems considered in this paper may open further research opportunities in these fields.

[1]  Muhammad Aslam Noor,et al.  Extended general variational inequalities , 2009, Appl. Math. Lett..

[2]  Christoph Schnörr,et al.  A class of quasi-variational inequalities for adaptive image denoising and decomposition , 2012, Computational Optimization and Applications.

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

[4]  F. Giannessi,et al.  Variational inequalities and network equilibrium problems , 1995 .

[5]  Panos M. Pardalos,et al.  Encyclopedia of Optimization , 2006 .

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

[7]  G. M. Korpelevich.  The extragradient method for finding saddle points and other problems , 1976 .

[8]  Qingzhi Yang,et al.  Several solution methods for the split feasibility problem , 2005 .

[9]  Yair Censor,et al.  A multiprojection algorithm using Bregman projections in a product space , 1994, Numerical Algorithms.

[10]  Qingzhi Yang.  The relaxed CQ algorithm solving the split feasibility problem , 2004 .

[11]  Y. Censor,et al.  Perturbed projections and subgradient projections for the multiple-sets split feasibility problem , 2007 .

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

[13]  M. Noor,et al.  Iterative Methods for Solving Nonconvex Equilibrium Variational Inequalities , 2012 .

[14]  Muhammad Aslam Noor,et al.  Auxiliary Principle Technique for Solving Split Feasibility Problems , 2013 .

[15]  Muhammad Aslam Noor,et al.  Quasi variational inequalities , 1988 .

[16]  Muhammad Aslam Noor,et al.  SENSITIVITY ANALYSIS OF SOME QUASI VARIATIONAL INEQUALITIES , 2013 .

[17]  Akhtar A. Khan,et al.  Generalized solutions of quasi variational inequalities , 2012, Optim. Lett..

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

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

[20]  Y. Censor,et al.  On some optimization techniques in image reconstruction from projections , 1987 .

[21]  Hong-Kun Xu.  Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces , 2010 .

[22]  Muhammad Aslam Noor,et al.  On descent-projection method for solving the split feasibility problems , 2012, J. Glob. Optim..

[23]  Otmar Scherzer,et al.  Variational Methods in Imaging , 2008, Applied mathematical sciences.

[24]  C. Byrne,et al.  A unified treatment of some iterative algorithms in signal processing and image reconstruction , 2003 .

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