A solution of delay differential equations via Picard\u2013Krasnoselskii hybrid iterative process

Godwin Amechi Okeke,Mujahid Abbas

doi:10.1007/s40065-017-0162-8

Abstract

The purpose of this paper is to introduce Picard–Krasnoselskii hybrid iterative process which is a hybrid of Picard and Krasnoselskii iterative processes. In case of contractive nonlinear operators, our iterative scheme converges faster than all of Picard, Mann, Krasnoselskii and Ishikawa iterative processes in the sense of Berinde (Iterative approximation of fixed points, 2002). We support our analytic proofs with a numerical example. Using this iterative process, we also find the solution of delay differential equation.

Highlights

Introduction and preliminariesThroughout this paper, N denotes the set of all positive integers
[11] established some convergence theorems for multivalued nonexpansive mappings for a Krasnoselskii-type sequence which is known to be superior to the Mann-type and Ishikawa-type iterations
Motivated by the facts above, we introduce the Picard–Krasnoselskii hybrid iterative process defined by the sequence {xn} :

Summary

Introduction and preliminaries

Throughout this paper, N denotes the set of all positive integers. Let C be a nonempty convex subset of a normed space E and T : C → C a mapping. Chidume and Olaleru [13] established some interesting fixed points results using the Picard iteration process. [11] established some convergence theorems for multivalued nonexpansive mappings for a Krasnoselskii-type sequence which is known to be superior to the Mann-type and Ishikawa-type iterations (see [11]). Definition 1.2 [6] Let {un} and {vn} be two fixed point iteration processes that converge to a certain fixed point p of a given operator T. The aim of this paper is to introduce the Picard–Krasnoselskii hybrid iterative process and to show that this new iterative process is faster than all of Picard, Mann, Krasnoselskii and Ishikawa iterative processes in the sense of Berinde [6]. We show that our iterative process can be used to find the solution of delay differential equations

Rate of convergence

Application to delay differential equations