A generalized f-projection method for countable families of weak relatively nonexpansive mappings and the system of generalized Ky Fan inequalities

AbstractThe purpose of this paper is to present new hybrid Ishikawa iteration process by the generalized f-projection operator for finding a common element of the fixed point set for two countable families of weak relatively nonexpansive mappings and the set of solutions of the system of generalized Ky Fan inequalities in a uniformly convex and uniformly smooth Banach space. Furthermore, we show that our new iterative scheme converges strongly to a common element of the afore mentioned sets. As applications, we apply our results to obtain some new results for finding a solution of a common fixed point of two countable in finite families, a system of generalized Ky Fan inequalities and a common zero-point problem for general B-monotone and maximal monotone operators in Banach spaces. The results presented in this paper improve and extend important recent results.

[1]  Poom Kumam,et al.  A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping , 2009 .

[2]  R. Rockafellar.  On the maximality of sums of nonlinear monotone operators , 1970 .

[3]  Dan Butnariu,et al.  Weak Convergence of Orbits of Nonlinear Operators in Reflexive Banach Spaces , 2003 .

[4]  S. Ishikawa.  Fixed points by a new iteration method , 1974 .

[5]  P. L. Combettes,et al.  Equilibrium programming in Hilbert spaces , 2005 .

[6]  Hong-Kun Xu,et al.  Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications , 2010 .

[7]  Xu Su-mei.  A Strong Convergence Theorem for Nonexpansive Mapping , 2010 .

[8]  Dennis F. Cudia.  The geometry of Banach spaces , 1964 .

[9]  Y. Censor,et al.  Iterations of paracontractions and firmaly nonexpansive operators with applications to feasibility and optimization , 1996 .

[10]  Poom Kumam,et al.  Strong Convergence Theorems for Countable Families of Uniformly Quasi--Asymptotically Nonexpansive Mappings and a System of Generalized Mixed Equilibrium Problems , 2011 .

[11]  W. R. Mann,et al.  Mean value methods in iteration , 1953 .

[12]  Poom Kumam,et al.  A modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in Banach spaces , 2011, Comput. Math. Appl..

[13]  Jinlu Li.  The generalized projection operator on reflexive Banach spaces and its applications , 2005 .

[14]  S. Reich,et al.  Asymptotic Behavior of Relatively Nonexpansive Operators in Banach Spaces , 2001 .

[15]  Satit Saejung,et al.  Strong Convergence to Common Fixed Points of Countable Relatively Quasi-Nonexpansive Mappings , 2008 .

[16]  Nan-Jing Huang,et al.  The generalised f -projection operator with an application , 2006 .

[17]  Nawel Gharbi,et al.  Numerical investigation of finite-source multiserver systems with different vacation policies , 2010, J. Comput. Appl. Math..

[18]  Lai-Jiu Lin,et al.  On a Generalized Ky Fan Inequality and Asymptotically Strict Pseudocontractions in the Intermediate Sense , 2011, J. Optim. Theory Appl..

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

[20]  Xi Li,et al.  Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications , 2010, Comput. Math. Appl..

[21]  Shin Min Kang,et al.  Strong convergence of shrinking projection methods for quasi-ϕ-nonexpansive mappings and equilibrium problems , 2010, J. Comput. Appl. Math..

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

[23]  Shih-sen Chang,et al.  On a Hybrid Method for Generalized Mixed Equilibrium Problem and Fixed Point Problem of a Family of Quasi--Asymptotically Nonexpansive Mappings in Banach Spaces , 2010 .

[24]  I. Ciorǎnescu.  Geometry of banach spaces, duality mappings, and nonlinear problems , 1990 .

[25]  Jiang-hua Fan,et al.  Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces , 2009 .

[26]  Poom Kumam,et al.  A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space , 2010 .

[27]  Y. Cho,et al.  A strong convergence theorem for common fixed points of two relatively nonexpansive mappings and its applications , 2009 .

[28]  Nan-Jing Huang,et al.  Variational inclusions with a general H-monotone operator in Banach spaces , 2007, Comput. Math. Appl..

[29]  Yekini Shehu,et al.  Convergence analysis for finite family of relatively quasi nonexpansive mappings and systems of equilibrium problems , 2011, Appl. Math. Comput..

[30]  Poom Kumam,et al.  Strong Convergence Theorems of Modified Ishikawa Iterations for Countable Hemi-Relatively Nonexpansive Mappings in a Banach Space , 2009 .

[31]  Yeol Je Cho,et al.  Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces , 2009 .

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

[33]  Nenad Ujević,et al.  An iterative method for solving nonlinear equations , 2007 .

[34]  Shi-sheng Zhang,et al.  Generalized mixed equilibrium problem in Banach spaces , 2009 .

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

[36]  S. Reich.  Book Review: Geometry of Banach spaces, duality mappings and nonlinear problems , 1992 .

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

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

[39]  A. Iusem,et al.  Iterative Algorithms for Equilibrium Problems , 2003 .

[40]  Poom Kumam,et al.  A general iterative method for solving equilibrium problems, variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings , 2010 .

[41]  Chi Kin Chan,et al.  A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications☆ , 2010 .

[42]  K. Deimling.  Nonlinear functional analysis , 1985 .

[43]  B. Pettis,et al.  A proof that every uniformly convex space is reflexive , 1939 .

[44]  Poom Kumam,et al.  Generalized Mixed Equilibrium Problems for Maximal Monotone Operators and Two Relatively Quasi-Nonexpansive Mappings , 2012 .

[45]  S. Yau.  Mathematics and its applications , 2002 .

[46]  Yongfu Su,et al.  Strong convergence theorems for relatively nonexpansive mappings in a Banach space , 2007 .

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

[48]  A. Kartsatos.  Theory and applications of nonlinear operators of accretive and monotone type , 1996 .

[49]  Wataru Takahashi,et al.  Strong Convergence Theorem by a New Hybrid Method for Equilibrium Problems and Relatively Nonexpansive Mappings , 2008 .

[50]  Ya. I. Alber.  Generalized Projection Operators in Banach Spaces: Properties and Applications , 1993 .

[51]  Poom Kumam,et al.  Convergence theorems for mixed equilibrium problems, variational inequality problem and uniformly quasi-ϕ-asymptotically nonexpansive mappings , 2011, Appl. Math. Comput..

[52]  Suthep Suantai,et al.  Convergence Analysis for a System of Equilibrium Problems and a Countable Family of Relatively Quasi-Nonexpansive Mappings in Banach Spaces , 2010 .

[53]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[54]  A. Moudafi,et al.  SECOND-ORDER DIFFERENTIAL PROXIMAL METHODS FOR EQUILIBRIUM PROBLEMS , 2003 .