A Modified Iterative Algorithm for Split Feasibility Problems of Right Bregman Strongly Quasi-Nonexpansive Mappings in Banach Spaces with Applications

In this paper, we present a new iterative scheme for finding a common element of the solution set F of the split feasibility problem and the fixed point set F ( T ) of a right Bregman strongly quasi-nonexpansive mapping T in p-uniformly convex Banach spaces which are also uniformly smooth. We prove strong convergence theorem of the sequences generated by our scheme under some appropriate conditions in real p-uniformly convex and uniformly smooth Banach spaces. Furthermore, we give some examples and applications to illustrate our main results in this paper. Our results extend and improve the recent ones of some others in the literature.

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

[2]  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 .

[3]  C. Byrne,et al.  Iterative oblique projection onto convex sets and the split feasibility problem , 2002 .

[4]  Hong-Kun Xu.  Iterative Algorithms for Nonlinear Operators , 2002 .

[5]  Habtu Zegeye,et al.  Convergence Theorems for Right Bregman Strongly Nonexpansive Mappings in Reflexive Banach Spaces , 2014 .

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

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

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

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

[10]  Poom Kumam,et al.  Convergence theorem for equilibrium problem and Bregman strongly nonexpansive mappings in Banach spaces , 2016 .

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

[12]  F. Smithies.  Linear Operators , 2019, Nature.

[13]  S. Reich,et al.  Right Bregman Nonexpansive Operators in Banach Spaces , 2012 .

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

[15]  S. Reich,et al.  Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings , 1984 .

[16]  Shoham Sabach,et al.  Existence and approximation of fixed points of right Bregman nonexpansive operators , 2013 .

[17]  Yekini Shehu,et al.  Further investigation into approximation of a common solution of fixed point problems and split feasibility problems , 2016 .

[18]  Poom Kumam,et al.  Iterative Scheme for System of Equilibrium Problems and Bregman Asymptotically Quasi-Nonexpansive Mappings in Banach Spaces , 2016 .

[19]  Poom Kumam,et al.  A New Multi-Step Iterative Algorithm for Approximating Common Fixed Points of a Finite Family of Multi-Valued Bregman Relatively Nonexpansive Mappings , 2016, Algorithms.

[20]  F. Schöpfer,et al.  An iterative regularization method for the solution of the split feasibility problem in Banach spaces , 2008 .

[21]  Yekini Shehu,et al.  An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces , 2015, Numerical Algorithms.

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

[23]  Fenghui Wang.  A New Algorithm for Solving The Multiple-Sets Split Feasibility Problem in Banach Spaces , 2014 .

[24]  Yekini Shehu,et al.  Convergence analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces , 2016 .