A novel iterative scheme for solving delay differential equations and nonlinear integral equations in Banach spaces

Abstract

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. Numerically, 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.

A novel Picard–Ishikawa–Green's iterative scheme for solving third‐order boundary value problems

“…Over the years, researchers have been successful in developing and using fixed point iterative schemes in approximating the solution of a variety of problems beginning from when Banach [6] in 1922 first used Picard iterative scheme [7], followed chronologically by Mann's [8], Krasnosel'skiı̆'s [9], Ishikawa's [10], Noor's [11], and other numerous schemes that have evolved with time and have been used to approximate the solutions of initial value problems (IVPs) and other related problems in application (see, e.g., [12–15]). For instance, Khan in 2013 introduced a Picard–Mann hybrid iterative [16] defined as $$$$ \left\{\begin{array}{l}{p}_0\in X,\\ {}{p}_{n+1}=J{q}_n,\\ {}{q}_n=\left(1-{\xi}_n\right){p}_n+{\xi}_nJ{p}_n,n\in \mathrm{\mathbb{N}},\end{array}\right.$$…”

“…Over the years, researchers have been successful in developing and using fixed point iterative schemes in approximating the solution of a variety of problems beginning from when Banach [6] in 1922 first used Picard iterative scheme [7], followed chronologically by Mann's [8], Krasnosel'skiı's [9], Ishikawa's [10], Noor's [11], and other numerous schemes that have evolved with time and have been used to approximate the solutions of initial value problems (IVPs) and other related problems in application (see, e.g., [12][13][14][15]). For instance, Khan in 2013 introduced a Picard-Mann hybrid iterative [16] defined as…”

On a Faster Iterative Method for Solving Fractional Delay Differential Equations in Banach Spaces

“…Many iterative methods have been introduced in the past few years. Some well-known iterative methods in the literature are given in [9][10][11][12][13].…”