A self-adaptive hybrid inertial algorithm for split feasibility problems in Banach spaces

In this paper, we introduce a new self-adaptive hybrid algorithm of inertial form for solving Split Feasibility Problem (SFP) which also solve a Monotone Inclusion Problem (MIP) and a Fixed Point Problem (FPP) in $p$-uniformly convex and uniformly smooth Banach spaces. Motivated by the self-adaptive technique, we incorporate the inertial technique to accelerate the convergence of the proposed method. Under standard and mild assumption of monotonicity of the SFP associated mapping, we establish the strong convergence of the sequence generated by our algorithm which does not require a prior knowledge of the norm of the bounded linear operator. Some numerical examples are presented to illustrate the performance of our method as well as comparing it with some related methods in the literature.

[1]  T. O. Alakoya,et al.  A parallel combination extragradient method with Armijo line searching for finding common solutions of finite families of equilibrium and fixed point problems , 2019, Rendiconti del Circolo Matematico di Palermo Series 2.

[2]  Qamrul Hasan Ansari,et al.  Split Feasibility and Fixed Point Problems , 2014 .

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

[4]  Yekini Shehu,et al.  A cyclic iterative method for solving Multiple Sets Split Feasibility Problems in Banach Spaces , 2016 .

[5]  Qingzhi Yang,et al.  Generalized KM theorems and their applications , 2006 .

[6]  O. T. Mewomo,et al.  Inertial-Type Algorithm for Solving Split Common Fixed Point Problems in Banach Spaces , 2021, J. Sci. Comput..

[7]  O. T. Mewomo,et al.  Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity , 2021, Numerical Algorithms.

[8]  Oluwatosin Temitope Mewomo,et al.  Iterative algorithm with self-adaptive step size for approximating the common solution of variational inequality and fixed point problems , 2021, Optimization.

[9]  Y. Shehu,et al.  Iterative methods for the split feasibility problem and the fixed point problem in Banach spaces , 2019, Optimization.

[10]  Paul-Emile Maingé,et al.  The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces , 2010, Comput. Math. Appl..

[11]  S. Khan,et al.  An extragradient algorithm for split generalized equilibrium problem and the set of fixed points of quasi-φ-nonexpansive mappings in Banach spaces , 2020 .

[12]  T. O. Alakoya,et al.  A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications , 2020, Journal of Industrial & Management Optimization.

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

[14]  Y. Shehu,et al.  Further investigation into split common fixed point problem for demicontractive operators , 2016 .

[15]  Lateef Olakunle Jolaoso,et al.  On split equality variation inclusion problems in Banach spaces without operator norms , 2021 .

[16]  T. O. Alakoya,et al.  Inertial algorithm for solving equilibrium, variational inclusion and fixed point problems for an infinite family of strict pseudocontractive mappings , 2021, Journal of Nonlinear Functional Analysis.

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

[18]  C. Izuchukwu,et al.  Iterative algorithm for a family of monotone inclusion problems in cat(0) spaces , 2020, Quaestiones Mathematicae.

[19]  Yekini Shehu,et al.  Nonlinear iterative methods for solving the split common null point problem in Banach spaces , 2019, Optim. Methods Softw..

[20]  T. O. Alakoya,et al.  Inertial extragradient method via viscosity approximation approach for solving equilibrium problem in Hilbert space , 2020 .

[21]  A. Gibali,et al.  A new approximation scheme for solving various split inverse problems , 2020, Afrika Matematika.

[22]  C. Izuchukwu,et al.  An inertial extrapolation method for solving generalized split feasibility problems in real hilbert spaces , 2021 .

[23]  A. A. Mebawondu,et al.  A new method for solving split variational inequality problems without co-coerciveness , 2020, Journal of Fixed Point Theory and Applications.

[24]  Chinedu Izuchukwu,et al.  A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem , 2021, Numerical Algebra, Control & Optimization.

[25]  C. C. Okeke,et al.  Inertial approximation method for split variational inclusion problem in Banach spaces , 2020 .

[26]  T. O. Alakoya,et al.  Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces , 2020, Numerical Algorithms.

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

[28]  Rudong Chen,et al.  A Relaxed CQ Algorithm for Solving Split Feasibility Problem , 2011, 2011 International Conference on Control, Automation and Systems Engineering (CASE).

[29]  T. O. Alakoya,et al.  Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings , 2021, Demonstratio Mathematica.

[30]  G. N. Ogwo,et al.  An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions , 2020, Optimization.

[31]  C. E. Chidume,et al.  Geometric Properties of Banach Spaces and Nonlinear Iterations , 2009 .

[32]  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.

[33]  O. T. Mewomo,et al.  Strong convergence theorems for finite families of pseudomonotone equilibrium and fixed point problems in Banach spaces , 2021, Afrika Matematika.

[34]  N. Xiu,et al.  A note on the CQ algorithm for the split feasibility problem , 2005 .

[35]  C. C. Okeke,et al.  ON SPLIT EQUILIBRIUM PROBLEM , VARIATIONAL INEQUALITY PROBLEM AND FIXED POINT PROBLEM FOR MULTI-VALUED MAPPINGS ∗ , 2017 .

[36]  Y. Censor,et al.  The multiple-sets split feasibility problem and its applications for inverse problems , 2005 .

[37]  D. R. Sahu,et al.  Bregman Distance and Strong Convergence of Proximal-Type Algorithms , 2013, Abstract and Applied Analysis.

[38]  Boris Polyak.  Some methods of speeding up the convergence of iteration methods , 1964 .

[39]  Chinedu Izuchukwu,et al.  Approximating common fixed points of mean nonexpansive mappings in hyperbolic spaces , 2021 .

[40]  L. Jolaoso,et al.  Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces , 2021, Journal of Industrial & Management Optimization.

[41]  T. O. Alakoya,et al.  An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings , 2021 .

[42]  T. O. Alakoya,et al.  A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem , 2019, Computational and Applied Mathematics.

[43]  Y. Censor,et al.  A unified approach for inversion problems in intensity-modulated radiation therapy , 2006, Physics in medicine and biology.

[44]  O. Mewomo,et al.  A strong convergence algorithm for a fixed point constrained split null point problem , 2020 .

[45]  Y. Liou,et al.  Regularized Methods for the Split Feasibility Problem , 2012 .

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

[47]  Oluwatosin Temitope Mewomo,et al.  A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings , 2021, Numerical Algebra, Control & Optimization.

[48]  O. T. Mewomo,et al.  Inertial algorithm with self-adaptive step size for split common null point and common fixed point problems for multivalued mappings in Banach spaces , 2021, Optimization.