Received: 5 July 2020 Revised: 18 July 2021 Accepted: 23 December 2021 DOI: 10.1002/mma.8095 RESEARCH ARTICLE A novel iterative scheme for solving delay differential equations and nonlinear integral equations in Banach spaces Godwin Amechi Okeke 1,2 Austine Efut Ofem 3 1 Functional Analysis and Optimization Research Group Laboratory (FANORG), Department of Mathematics, School of Physical sciences, Federal University of Technology Owerri, P.M.B. 1526 Owerri, Imo State, Nigeria 2 Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan 3 Department of Mathematics, University of Uyo, Nigeria Correspondence Godwin Amechi Okeke, Functional Analysis and Optimization Research Group Laboratory (FANORG), Department of Mathematics, School of Physical sciences, Federal University of Technology Owerri, P.M.B. 1526 Owerri, Imo State, Nigeria. Email: gaokeke1@yahoo.co.uk Communicated by: Q. Wang We propose a three-step iteration process for finding the common fixed points of nonexpansive mapping and strongly pseudocontractive mapping in a real Banach space. We weaken the necessity of condition (C) imposed by a previous study on the mappings by using a quite simple and different method to obtain strong convergence of our proposed iterative scheme to the common fixed point of nonexpansive mapping and strongly pseudocontractive mapping. Numeri- cally, we also show that our proposed iterative scheme converges faster than some existing iterative schemes. Furthermore, we apply our proposed iterative process in solving mixed type Volterra-Fredholm functional nonlinear integral equations and delay differential equations. KEYWORDS Banach spaces, delay differential equations, fixed point, iterative scheme, mixed-type Volterra-Fredholm functional nonlinear integral equations, nonexpasive mapping, strong convergence, strongly pseudocontractive mapping MSC CLASSIFICATION 47H09; 47H10 1 INTRODUCTION Let E be a real Banach space and J ∶ E → 2 E ∗ denotes the normalized duality mapping defined by J ()={ ∗ ∈ E ∗ ∶ ⟨, ∗ ⟩ = |||| 2 = || ∗ || 2 }, ∀ ∈ E, (1.1) where E ∗ denotes the dual space of E and ⟨· , ·⟩ denotes the generalized duality pairing of E and E ∗ . In the sequel, we shall use j to denote the single-valued duality mapping and F(R) denotes the set of fixed points of mapping R, that is, F(R)={ ∈ E ∶ R = }. Definition 1.1. Let C be a nonempty subset of real Banach space E. A mapping R : C → C is said to be: • nonexpansive if, ||R - R|| ≤ || - ||, ∀, ∈ C; (1.2) Math Meth Appl Sci. 2022;45:5111–5134. wileyonlinelibrary.com/journal/mma © 2022 John Wiley & Sons, Ltd. 5111