A New Multi-Step Iterative Algorithm for Approximating Common Fixed Points of a Finite Family of Multi-Valued Bregman Relatively Nonexpansive Mappings

In this article, we introduce a new multi-step iteration for approximating a common fixed point of a finite class of multi-valued Bregman relatively nonexpansive mappings in the setting of reflexive Banach spaces. We prove a strong convergence theorem for the proposed iterative algorithm under certain hypotheses. Additionally, we also use our results for the solution of variational inequality problems and to find the zero points of maximal monotone operators. The theorems furnished in this work are new and well-established and generalize many well-known recent research works in this field.

[1]  S. Reich,et al.  Two Strong Convergence Theorems for a Proximal Method in Reflexive Banach Spaces , 2010 .

[2]  Benar Fux Svaiter,et al.  Forcing strong convergence of proximal point iterations in a Hilbert space , 2000, Math. Program..

[3]  Habtu Zegeye,et al.  Convergence theorem for common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings , 2014 .

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

[5]  Shih-sen Chang,et al.  Halpern-Mann’s iterations for Bregman strongly nonexpansive mappings in reflexive Banach spaces with applications , 2013 .

[6]  A. Abkar,et al.  STRONG CONVERGENCE OF A MULTI-STEP ITERATIVE PROCESS FOR RELATIVELY QUASI-NONEXPANSIVE MULTIVALUED MAPPINGS AND EQUILIBRIUM PROBLEM IN BANACH SPACES , 2012 .

[7]  L. Wang,et al.  Strong convergence theorems for Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces , 2014, Appl. Math. Comput..

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

[9]  Dan Butnariu,et al.  Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces , 2006 .

[10]  Jen-Chih Yao,et al.  Bregman weak relatively nonexpansive mappings in Banach spaces , 2013 .

[11]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[12]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

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

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

[15]  Jen-Chih Yao,et al.  The modified Ishikawa iterative algorithm with errors for a countable family of Bregman totally quasi-D-asymptotically nonexpansive mappings in reflexive Banach spaces , 2015 .

[16]  S. Reich,et al.  Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces , 2010 .

[17]  R. Phelps.  Convex Functions, Monotone Operators and Differentiability , 1989 .

[18]  Shoham Sabach,et al.  Existence and Approximation of Fixed Points of Bregman Firmly Nonexpansive Mappings in Reflexive Banach Spaces , 2011, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[19]  Eskandar Naraghirad,et al.  Strong convergence theorems for Bregman W-mappings with applications to convex feasibility problems in Banach spaces , 2015 .

[20]  R. T. ROCKAFELLARi.  LEVEL SETS AND CONTINUITY OF CONJUGATE CONVEX FUNCTIONS , 2010 .

[21]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[22]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[23]  Shoham Sabach,et al.  A projection method for solving nonlinear problems in reflexive Banach spaces , 2011 .

[24]  Wei Li,et al.  A new iterative algorithm with errors for maximal monotone operators and its applications , 2005, 2005 International Conference on Machine Learning and Cybernetics.

[25]  Y. Censor,et al.  An iterative row-action method for interval convex programming , 1981 .

[26]  Yekini Shehu,et al.  Convergence theorems for maximal monotone operators and fixed point problems in Banach spaces , 2014, Appl. Math. Comput..

[27]  Yeol Je Cho,et al.  Halpern's iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces , 2012, Comput. Math. Appl..

[28]  Yi Li,et al.  Strong convergence results of two-steps modifying Halpern’s iteration for Bregman strongly nonexpansive multi-valued mappings in reflexive Banach spaces with application , 2014 .

[29]  Dan Butnariu,et al.  A Proximal-Projection Method for Finding Zeros of Set-Valued Operators , 2007, SIAM J. Control. Optim..

[30]  Habtu Zegeye,et al.  Convergence results for a common solution of a finite family of variational inequality problems for monotone mappings with Bregman distance function , 2013 .

[31]  Heinz H. Bauschke,et al.  ESSENTIAL SMOOTHNESS, ESSENTIAL STRICT CONVEXITY, AND LEGENDRE FUNCTIONS IN BANACH SPACES , 2001 .

[32]  P. L. Combettes,et al.  The Convex Feasibility Problem in Image Recovery , 1996 .

[33]  P. Gács,et al.  Algorithms , 1992 .

[34]  L. Bregman.  The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[35]  Osman Güer.  On the convergence of the proximal point algorithm for convex minimization , 1991 .

[36]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[37]  Yi Li,et al.  Halpern’s iteration for Bregman strongly nonexpansive multi-valued mappings in reflexive Banach spaces with application , 2013 .

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

[39]  R. T. Rockafellar,et al.  Gradients of convex functions , 1969 .

[40]  Heinz H. Bauschke,et al.  Bregman Monotone Optimization Algorithms , 2003, SIAM J. Control. Optim..

[41]  Chung-Chien Hong,et al.  Fixed point and weak convergence theorems for point-dependent λ-hybrid mappings in Banach spaces , 2011 .

[42]  Heinz H. Bauschke,et al.  Legendre functions and the method of random Bregman projections , 1997 .

[43]  G. Stampacchia,et al.  On some non-linear elliptic differential-functional equations , 1966 .

[44]  Gregory B. Passty.  Ergodic convergence to a zero of the sum of monotone operators in Hilbert space , 1979 .