Computational and Applied Mathematics (2020) 39:223 https://doi.org/10.1007/s40314-020-01246-z Convergence results of forward–backward method for a zero of the sum of maximally monotone mappings in Banach spaces Getahun Bekele Wega 1 · Habtu Zegeye 1 Received: 31 January 2020 / Revised: 31 January 2020 / Accepted: 3 July 2020 / Published online: 17 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020 Abstract The purpose of this paper is to study a forward–backward algorithm for approximating a zero of the sum of maximally monotone mappings in the setting of Banach spaces. Under some mild conditions, we prove a new strong convergence theorem for the algorithm produced by the method in real reflexive Banach spaces. In addition, we give some applications to the minimization problems. Finally, we provide a numerical example, which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings. Keywords Banach spaces · Forward–backward algorithm · Monotone mapping · Maximally monotone mapping · Strong convergence · Zero points Mathematics Subject Classification 47H05 · 47J25 · 49M27 · 90C25 1 Introduction Let E be a real Banach space with its dual E ∗ . A mapping F : E → 2 E ∗ is called monotone if: 〈x − y , x ∗ − y ∗ 〉≥ 0, ∀(x , x ∗ ), ( y , y ∗ ) ∈ Gph( F ), where Gph( F ) ={(x , x ∗ ) ∈ E × E ∗ : x ∗ ∈ Fx } is graph of F . It is also called maximally monotone if its graph is not properly contained in the graph of any other monotone mapping. Communicated by Carlos Conca. B Habtu Zegeye habtuzh@yahoo.com Getahun Bekele Wega getahunbekele2012@gmail.com 1 Department of Mathematics and Statistical Sciences, Botswana International University of Science and Technology, Pvt., Bag 0016, Palapye, Botswana 123