Approximation of the Solution of Delay Fractional Differential Equation Using AA-Iterative Scheme

Mujahid Abbas,Muhammad Waseem Asghar,Manuel De la Sen

doi:10.3390/math10020273

Mujahid Abbas, Muhammad Waseem Asghar + Show 1 more

Open Access

https://doi.org/10.3390/math10020273

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Abstract

The aim of this paper is to propose a new faster iterative scheme (called AA-iteration) to approximate the fixed point of (b,η)-enriched contraction mapping in the framework of Banach spaces. It is also proved that our iteration is stable and converges faster than many iterations existing in the literature. For validity of our proposed scheme, we presented some numerical examples. Further, we proved some strong and weak convergence results for b-enriched nonexpansive mapping in the uniformly convex Banach space. Finally, we approximate the solution of delay fractional differential equations using AA-iterative scheme.

Highlights

Introduction and PreliminariesThe proof of the Banach contraction principle (BCP) [1] is based on convergence of the most simplest iterative process named as the sequence of successive approximations or Picard iterative process
The aim of this paper is to show that an AA-iterative scheme has a faster rate of convergence than (3)–(11)
It is proved that the proposed iterative scheme is stable and converges faster than Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur, M-iteration and F-iteration

Summary

Introduction and Preliminaries

The proof of the Banach contraction principle (BCP) [1] is based on convergence of the most simplest iterative process named as the sequence of successive approximations or Picard iterative process. It was shown that (b, η )-enriched contraction mapping on Ω has a unique fixed point, which can be approximated by means of the Krasnoselskii’s iterative scheme [11]. An iterative scheme { pn : n ∈ Z+ } introduced in [21] has a faster rate of convergence than S− iteration for approximating the fixed points of contraction mappings. This scheme is given as: p(5) = (1 − k ) Tq(5) + k Tr (5). We approximate the solution of delay fractional differential equations by using our proposed scheme

Convergence and Stability Results

Convergence Results for b-Enriched Nonexpansive Mappings

Application

Conclusions