Applied Numerical Mathematics 161 (2021) 437–451 Contents lists available at ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum Generalized split feasibility problem for multi-valued Bregman quasi-nonexpansive mappings in Banach spaces ✩ Suliman Al-Homidan a,∗ , Bashir Ali b , Yusuf I. Suleiman c a Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia b Department of Mathematical Sciences, Bayero University, Kano-Nigeria c Department of Mathematics, Kano University of Science and Technology, Wudil-Nigeria a r t i c l e i n f o a b s t r a c t Article history: Received 19 October 2019 Received in revised form 16 November 2020 Accepted 1 December 2020 Available online xxxx Keywords: Generalized split feasibility problems Uniformly convex Banach spaces Multi-valued Bregman quasi-nonexpansive mappings In this paper, the notion of generalized split feasibility problem (GSFP) is studied in p- uniformly convex Banach spaces. Some special cases of the GSFP are highlighted. A self- adaptive step-size iterative algorithm which converges strongly to solution of the GSFP is proved. The implementation of the method is demonstrated with two numerical examples. Our method does not require prior information of operator norms. Our results extend, improve and enrich recently announced related results in the literature.  2020 Published by Elsevier B.V. on behalf of IMACS. 1. Introduction Let E be a real Banach space E and f : E → R a lower semicontinuous convex function. In this paper, we are concerned with the generalized split feasibility problem GSFP find z ∗ ∈ N  i =1 Fix (T i ) such that f (z ∗ ) = 0, (1.1) where T i : K → CB ( K ), for each i ∈{1, 2, ..., N} is a finite family of multi-valued Bregman quasi-nonexpansive mappings such that  N i=1 Fix(T i )  =∅, CB ( K ) is the family of nonempty closed bounded subsets of K and Fix (T ) ={x ∈ D(T ) : x ∈ T (x)} denotes the fixed point set of T . The GSFP is interesting on the grounds that the algorithmic approach for solving the GSFP is characterized by self-adaptive step-size that do not depend on prior information about any operator norm. Recall that a mapping T : K → CB ( K ) is multi-valued Bregman quasi-nonexpansive [17] if Fix (T )  =∅ and  p ( w, x ∗ ) ≤  p (x, x ∗ ), for all x ∗ ∈ Fix (T ), w ∈ T (x), x ∈ K . T : K → K is Bregman quasi-nonexpansive [16] if Fix (T )  =∅ and  p (Tx, x ∗ ) ≤  p (x, x ∗ ), for all x ∈ K , x ∗ ∈ Fix (T ). ✩ The authors are grateful to the reviewers for suggestions to improve the presentation of this paper. The first author grateful to King Fahd University of Petroleum & Minerals for providing excellent research facilities to carry out this research work. * Corresponding author. E-mail addresses: homidan@kfupm.edu.sa (S. Al-Homidan), bashiralik@yahoo.com (B. Ali), yubram@yahoo.com (Y.I. Suleiman). https://doi.org/10.1016/j.apnum.2020.12.001 0168-9274/ 2020 Published by Elsevier B.V. on behalf of IMACS.