Abstract
Purpose
In this paper, Picard–S hybrid iterative process is defined, which is a hybrid of Picard and S-iterative process. This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid and Picard–Ishikawa hybrid iterative processes for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard–S hybrid iterative process for nonexpansive mappings.
Design/methodology/approach
This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings.
Findings
Showed the fastest convergence of this new iteration and then other iteration defined in this paper. The author finds the solution of delay differential equation using this hybrid iteration. For new iteration, the author also proved a theorem for nonexpansive mapping.
Originality/value
This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard–S hybrid iterative process for nonexpansive mappings.
Keywords
Citation
Srivastava, J. (2022), "Introduction of new Picard–S hybrid iteration with application and some results for nonexpansive mappings", Arab Journal of Mathematical Sciences, Vol. 28 No. 1, pp. 61-76. https://doi.org/10.1108/AJMS-08-2020-0044
Publisher
:Emerald Publishing Limited
Copyright © 2020, Julee Srivastava
License
Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
Let E be a normed linear space and C be a non-empty convex subset of E. A mapping
Let C be a non-empty subset of a normed linear space E and
A sequence
In this paper, N denotes the set of all positive integers.
The Picard iterative process [1] is defined by the sequence
The Krasnoselskii iterative process [2] is defined by the sequence
The Mann iteration [3] is defined by the sequence
The Ishikawa iterative process [4] is defined by the sequence
S-iterative process [5] is defined by the sequence
Many important non-linear problems of applied mathematics are usually constructed in the form of fixed point equation. These problems are related with physical problem of applied sciences and engineering.
The Picard iteration is the simple iteration for approximate solution of fixed point equation for non-linear contraction mapping. Some results based on Picard iteration are introduced by Chidume and Olaleru [6].
Khan [7] introduced the Picard–Mann hybrid iterative process defined by the sequence
Okeke and Abbas [8] introduced the Picard–Krasnoselskii hybrid iterative process defined by the sequence
Okeke [9] introduced the Picard–Ishikawa hybrid iterative process defined by the sequence
Let
2. Preliminaries
Let
Let
Let X be a Banach space. Then a function
It is easy to see that
Let
The aim of this paper is to introduce the Picard–S hybrid iterative process and to show that this new iterative process is faster than all of Picard, Krasnoselskii, Mann, Ishikawa in sense of Berinde [20], S-iteration in sense of Agarwal [5], Picard–Mann hybrid in sense of Khan [7], Picard–Krasnoselskii hybrid in sense of Okeke [8] and Picard–Ishikawa hybrid in the sense of Okeke [9].
Okeke already proved that Picard–Krasnoselskii hybrid iterative process converges faster than Picard, Krasnoselskii, Mann and Ishikawa. Khan [7] proved that Picard–Mann hybrid iterative process converges faster than Picard, Mann, Ishikawa iterative processes. Therefore, I show that my new Picard–S hybrid iterative process converges faster than S-iteration, Picard–Mann hybrid iteration, Picard–Krasnoselskii hybrid iteration and Picard–Ishikawa hybrid iterative process in the topic Rate of Convergence. In 2020, Zhao [21] proved existence and uniqueness of pseudo almost periodic solution for a class of iterative functional differential equations with delays depending on state. In next section, I find the solution of delay differential equation using Picard–S hybrid iterative process. Aynur Sahin [22] proved some strong convergence results of Picard–Krasnoselskii hybrid iterative process for a general class of contractive-like operator in hyperbolic space. In next section, I prove some results of Picard–S hybrid iterative process for nonexpansive mappings in uniformly convex Banach space.
3. Rate of convergence
Let C be a non-empty closed convex subset of a normed space E and let T be a contraction of C into itself. Suppose that each of the iterative process 1.6, 1.7, 1.8, 1.9 and 1.10 converges to the same fixed point p of T where
Proof: Suppose that p is the fixed point of the operator T. Using (1.1) and S-iterative process (1.6), we have
Using (1.1) and Picard–Krasnoselskii hybrid iteration (1.8)
Using (1.1) and Picard–Ishikawa hybrid iterative process (1.9), we have
Now compute the rate of convergence of Picard–S iterative process (1.10) as follows:
Thus,
Thus,
Thus,
Thus,
In [8], Okeke proved that the rate of convergence of Picard–Krasnoselskii hybrid iterative process is faster than Picard, Krasnoselskii, Mann and Ishikawa iterations. Agarwal et al. [5] proved that S-iteration converges faster than Picard, Krasnoselskii, Mann and Ishikawa iterative processes, and Okeke [9] proved that rate of convergence of Picard–Ishikawa hybrid iterative process is faster than Picard–Mann hibrid and Picard–Krasnoselskii iterations. Therefore, I give an example to show that rate of convergence of Picard–S hybrid iterative process is faster than Picard–Mann hybrid, Picard–Krasnoselskii hybrid and S-iteration. This will show that Picard–S hybrid defined by (1.10) converges to fixed point faster than all other iterations defined in this paper.
Let
4. Application to delay differential equation
Here, I use this new Picard–S hybrid iterative process to find the solution of delay differential equations.
Let
Space
By the solution of above problem, we mean a function
Now we can reformulate problems (4.1) and (4.2) by the following integral equation:
Coman [23] et al. established the following result.
Assume that the conditions
Using Picard–S hybrid iterative process, I prove the following result.
Assume that
Proof: Let
Let p be a fixed point of T, now I prove that
Now,
Now,
Using (4.6) in (4.7), we get
From (4.5) we get
Using (4.5) and (4.9) in (4.4), we get
Note that
5. Picard–S hybrid iterative process for nonexpansive mappings
Let E be a normed space, C a non-empty convex subset of E and
Proof: Set an = xn − Txn for all
Now,
From inequality (5.2) and (5.3), we have
Now,
Now,
Using (5.6) and (5.8) in (5.5), we get
From (5.1), (5.4) and (5.10), we get
So that
Let C be a non-empty closed convex (not necessary bounded) subset of a uniformly convex Banach space X and
Then, for arbitrary initial value
Proof: Lemma (5.1) implies that
Observe that
Now
Using (5.12) in (5.13), we have
This shows that
Let
Let
Using theorem (5.2) and (5.16), we obtain that
It follows from (5.14) that
This gives us
Or
Using restriction
Therefore,
We have
Observe that
Thus,
It follows that
Figures
A comparison of Picard–S hybrid with other iterative processes
Step | Picard–S hybrid | Picard–Ishikawa hybrid | Picard–Mann hybrid | Picard–Krasnoselskii hybrid | S-iteration |
---|---|---|---|---|---|
0 | 5.0000000000000 | 5.0000000000000 | 5.000000000000 | 5.000000000000 | 5.000000000000 |
1 | 2.053665985829 | 2.053665985829 | 2.251284354073 | 2.251284354073 | 2.330713309124 |
2 | 2.001174310362 | 2.001174310362 | 2.024068969098 | 2.024068969098 | 2.042698929425 |
3 | 2.000024999687 | 2.000024999687 | 2.002336639386 | 2.002336639386 | 2.005602850705 |
4 | 2.000000433332 | 2.000000549760 | 2.000227141158 | 2.000227141158 | 2.000738954904 |
5 | 2.000000009450 | 2.000000012089 | 2.000022082864 | 2.000022082864 | 2.000097495596 |
6 | 2.000000000207 | 2.000000000265 | 2.000002146942 | 2.000002146942 | 2.000012863908 |
7 | 2.000000000000 | 2.000000000005 | 2.000000208730 | 2.000000208730 | 2.000001697319 |
8 | 2.000000000000 | 2.000000020293 | 2.000000020293 | 2.000000223951 | |
9 | 2.000000001973 | 2.000000001973 | 2.000000029549 | ||
10 | 2.000000000191 | 2.000000000191 | 2.000000003898 | ||
11 | 2.000000000018 | 2.000000000018 | 2.000000000514 | ||
12 | 2.000000000002 | 2.000000000002 | 2.000000000067 | ||
13 | 2.000000000000 | 2.000000000000 | 2.000000000008 | ||
14 | 2.000000000000 |
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Further reading
[24]Nelson PW, Murray JD and Perelson AS. A model of HIV-1 pathogenesis that includes an intracellular delay, Math Biosci. 2000; 163: 201-15.
[25]Soltuz SM and Otrocol D. Classical results via Mann-Ishikawa iteration. Revue d'Analyse Numerique et de Theorie de I'Approximation. 2007; 36(2): 195-9.