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Article

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

by
Mujahid Abbas
1,2,
Muhammad Waseem Asghar
1 and
Manuel De la Sen
3,*
1
Department of Mathematics, Government College University, Lahore 54000, Pakistan
2
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
3
Institute of Research and Development of Processes, University of the Basque Country, 48940 Leioa, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2022, 10(2), 273; https://doi.org/10.3390/math10020273
Submission received: 6 December 2021 / Revised: 11 January 2022 / Accepted: 12 January 2022 / Published: 16 January 2022
(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

Abstract

:
The aim of this paper is to propose a new faster iterative scheme (called A A -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 A A -iterative scheme.

1. 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. This principle solves a fixed point problem for contraction mapping defined on a complete metric space and has become an important tool to prove the existence and approximation of solutions of nonlinear functional equations such as differential equations, integral equations and partial differential equations. In certain cases, the existence of solution of fixed point problem is guaranteed, but finding the exact solution is not possible. In such a situation, an approximation of the solution of the given problem is much desired, which gave rise to development of the different iterative processes [2,3,4,5,6,7].
Many authors have proposed and applied different fixed point iterative schemes for approximation of the solution of linear and nonlinear equations and inclusion. It is always preferred to develop an iterative scheme, which is better than others in the sense that the solution is approximated in a fewer number of steps. In this paper, we shall develop an iterative process and compare it with some well-known iterative processes existing in the literature. Throughout this paper, the set { 0 , 1 , 2 , } is denoted by Z + . Let U be a normed space, Ω a nonempty closed convex, Ω a nonempty bounded closed convex subsets of U and T a self mapping on Ω . The set { p * Ω : p * = T p * } of all fixed points of T is denoted by F ( T ) .
In 1991, Sahu [8] proved the following:
Lemma 1
([8]). Suppose that U is a uniformly convex Banach space and 0 < a k n b < 1 for all n N . Let { p n } and { q n } be two sequences in U such that
lim sup n p n l , lim sup n q n l a n d lim sup n ( 1 k n ) p n + k n q n = l
hold for some l 0 . Then lim n p n q n = 0 .
Recall that a mapping T is called Lipschitzian if there exists a constant L > 0 such that
T p T q L p q
holds for all p , q Ω . If we take L ( 0 , 1 ) in the above inequality, then T is called a contraction. The mapping T is called nonexpansive if we set L = 1 in the above inequality.
Berinde [9] introduced the concept of enriched nonexpansive mapping on a normed space as follows:
A self mapping T on Ω is said to be an enriched nonexpansive if for all p , q Ω , there exists b [ 0 , ) such that
| | b ( p q ) + T p T q | | ( b + 1 ) | | p q | | .
To highlight the parameter b in ( 1 ) , T is termed as b-enriched nonexpansive mapping.
A mapping T : Ω Ω is said to be an enriched contraction [10] if for all p , q Ω , there exists b [ 0 , ) and η ( 0 , b + 1 ) such that
| | b ( p q ) + T p T q | | η | | p q | | .
Again to highlight the parameter b in ( 2 ) , T is called a ( b , η ) -enriched contraction. 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].
Definition 1
([12]). A mapping T : Ω U is said to be demiclosed at q * , if whenever a sequence p n p * in Ω and T p n q * in U, it follows that T p * = q * .
Lemma 2
([13]). Let Ω be nonempty closed convex subset of a uniformly convex Banach space U and T a nonexpansive mapping on Ω. Then I T is demiclosed at zero.
Remark 1
([14]). Let T be a self mapping on Ω. For any λ [ 0 , 1 ] , the averaged mapping T λ on Ω given by
T λ p = ( 1 λ ) p + λ T p
has the property that F ( T ) = F ( T λ ) . Clearly, T 0 = I and T 1 = T are the trivial cases.
There are several iterative processes which are used to approximate fixed point of a certain nonlinear operator. One of the most important factors to decide the preference of one iterative process over the other is the rate of convergence. In order to compare convergence rates between two iteration processes, we use the following useful definition of Berinde [15].
Definition 2.
Suppose that sequences { α n } and { β n } converge to the same point l * with the following error estimates
α n l * p n , β n l * q n .
If lim n p n q n = 0 , then { α n } converges faster than { β n } .
Let us recall that for the given p 0 ( 0 ) in Ω , the sequence { p n ( 0 ) : n Z + } defined by
p n + 1 ( 0 ) = T p n ( 0 ) , n Z +
is known as the sequence of successive approximations or Picard iteration [16]. The well-known Banach contraction principle states that { p n ( 0 ) } converges to a unique fixed point of T for any choice of a starting point p 0 ( 0 ) in Ω provided that T is a contraction mapping. However, the Picard iteration does not need to converge to fixed point of nonexpansive mapping. For instance, the self mapping T x = 1 x on [ 0 , 1 ] does not converge to its fixed point 1 2 , for any choice of x [ 0 , 1 ] other than 1 2 . On the other hand, averaged operator for any λ ( 0 , 1 ) converges to fixed point of T for any choice of x and hence in certain cases, it is useful to consider an averaged operator in an iterative scheme than a mapping T itself.
Let us choose an initial guess p 0 ( 1 ) in Ω . The sequence { p n ( 1 ) : n Z + } defined by
p n + 1 ( 1 ) = ( 1 k n ) p n ( 1 ) + k n T p n ( 1 ) , n Z +
is called Mann iteration sequence [17], where the sequence { k n } of parameters in ( 0 , 1 ) satisfies certain conditions.
The sequence { p n ( 2 ) : n Z + } defined by
p 0 ( 2 ) Ω p n + 1 ( 2 ) = ( 1 k n ) p n ( 2 ) + k n T q n ( 2 ) q n ( 2 ) = ( 1 o n ) p n ( 2 ) + o n T p n ( 2 ) , n Z +
is known as Ishikawa iteration scheme [18], where { o n } and { k n } are some appropriate sequences in ( 0 , 1 ) .
Noor [19] proposed a three step iteration scheme to construct a sequence { p n ( 3 ) : n Z + } as follows:
p 0 ( 3 ) p n + 1 ( 3 ) = Ω ( 1 k n ) p n ( 3 ) + k n T q n ( 3 ) q n ( 3 ) = ( 1 o n ) p n ( 3 ) + o n T r n ( 3 ) r n ( 3 ) = ( 1 w n ) p n ( 3 ) + w n T p n ( 3 ) n Z +
where { w n } , { o n } , { k n } in ( 0 , 1 ) satisfy certain conditions.
In 2007, Agarwal et al. [20] defined a sequence { p n ( 4 ) : n Z + } known as S-iteration scheme given by
p 0 ( 4 ) p n + 1 ( 4 ) = Ω ( 1 k n ) T p n ( 4 ) + k n T q n ( 4 ) q n ( 4 ) = ( 1 o n ) p n ( 4 ) + o n T p n ( 4 ) , n Z +
where { o n } , { k n } are appropriate sequences in ( 0 , 1 ) .
It was proved that the rate of convergence of iteration scheme (7) is same as the Picard iteration scheme but faster than Mann iteration scheme for the class of contraction mappings [20].
An iterative scheme { p n ( 5 ) : 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 0 ( 5 ) Ω p n + 1 ( 5 ) = ( 1 k n ) T q n ( 5 ) + k n T r n ( 5 ) q n ( 5 ) = ( 1 o n ) T p n ( 5 ) + o n T r n ( 5 ) r n ( 5 ) = ( 1 w n ) p n ( 5 ) + w n T p n ( 5 ) , n Z +
where { w n } , { o n } and { k n } in ( 0 , 1 ) satisfy certain appropriate conditions.
An iterative sequence { p n ( 6 ) : n Z + }
p 0 ( 6 ) Ω p n + 1 ( 6 ) = ( 1 k n ) T r n ( 6 ) + k n T q n ( 6 ) q n ( 6 ) = ( 1 o n ) r n ( 6 ) + o n T r n ( 6 ) r n ( 6 ) = ( 1 w n ) p n ( 6 ) + w n T p n ( 6 ) , n Z +
proposed by Thakur et al. [6] has a better rate of convergence than iterative sequence in [21], where the sequences { w n } , { o n } and { k n } are given sequences in ( 0 , 1 ) .
In 2018, Ullah et al. [22] introduced M-iteration sequence { p n ( 7 ) : n Z + } as follows
p 0 ( 7 ) Ω p n + 1 ( 7 ) = T q n ( 7 ) q n ( 7 ) = T r n ( 7 ) r n ( 7 ) = ( 1 w n ) p n ( 7 ) + w n T p n ( 7 ) , n Z +
for approximating the fixed points of Suzuki’s generalized nonexpansive mappings, where { w n } ( 0 , 1 ) .
Ali et al. [2] modified an M-iteration sequence by introducing F-iteration sequence { p n ( 8 ) : n Z + } given by
p 0 ( 8 ) Ω p n + 1 ( 8 ) = T q n ( 8 ) q n ( 8 ) = T r n ( 8 ) r n ( 8 ) = T ( ( 1 w n ) p n ( 8 ) + w n T p n ( 8 ) ) , n Z +
where { w n } ( 0 , 1 ) . They showed that F-iteration sequence has a better rate of convergence than M-iteration and all other iterative schemes presented in [2].
For practical purposes, we deal with an approximate sequence { a n } ; we obtain it because of numerical approximation of operator and round off errors, instead of a theoretical sequence { p n } obtained through an iterative process p n + 1 = f ( T , p n ) for some given function f.
The approximate sequence { a n } converges to fixed point of T if and only if the given fixed point iterative scheme is stable. The notion of the stability for a fixed point iterative scheme was introduced by Ostrowski [23].
Definition 3.
Let { a n } be an approximate sequence of { p n } in a subset Ω of a Banach space U. Then a given iterative process p n + 1 = f ( T λ , p n ) for some function f, converging to a fixed point p * of self mapping T on Ω, is said to be T-stable or stable with respect to T provided that lim n e n = 0 if and only if lim n a n = p * , where { e n } is given by
e n = a n + 1 f ( T , a n ) , n Z + .
Following results are needed in the sequel.
Lemma 3
([15]). Let { u n } and { e n } be sequences of positive real numbers satisfying the following inequality:
u n + 1 ( 1 υ n ) u n + e n
where υ n ( 0 , 1 ) for all n Z + with Σ n = 0 υ n = . If lim n e n υ n = 0 , then lim n u n = 0 .
Question: Continuing in this direction, we pose the following question:
Is it possible to construct an iterative scheme which is faster than iterative schemes (3)–(11).
To answer the above question in affirmative, we introduce an A A -iterative scheme { p n : n Z + } for an averaged mapping to approximate the fixed points of enriched contraction mappings as follows:
p 0 Ω p n + 1 = T λ q n q n = T λ ( ( 1 k n ) T λ s n + k n T λ r n ) r n = T λ ( ( 1 o n ) s n + o n T λ s n ) s n = ( 1 w n ) p n + w n T λ p n , n Z +
where { w n } , { o n } and { k n } are sequences of parameters in ( 0 , 1 ) .
The aim of this paper is to show that an A A -iterative scheme has a faster rate of convergence than ( 3 ) ( 11 ) . Strong and week convergence results are also established for a b-enriched nonexpansive mapping. Numerical examples are presented to compare the rate of convergence with Ishikawa, Noor, Agarwal et al., Abbas et al. and Thakur et al., M-iteration and F-iteration for classes of contraction and ( b , η ) -enriched contraction mappings. As an application, we approximate the solution of delay fractional differential equations by using our proposed scheme.

2. Convergence and Stability Results

In this section, we establish convergence and stability of AA-iterative scheme ( 12 ) constructed with ( b , η ) -enriched contractions mapping in arbitrary Banach space.
Theorem 1.
Let Ω be a nonempty closed and convex subset of a Banach space U and T : Ω Ω a ( b , η ) -enriched contraction mapping with F ( T ) . Then, the sequence { p n } defined by ( 12 ) converges to a fixed point of T .
Proof. 
Take b = 1 λ 1 , it follows that λ ( 0 , 1 ) . Then ( 2 ) becomes
( 1 λ 1 ) ( p q ) + T p T q η p q ,
which can be written in an equivalent form as follows:
T λ p T λ q ε p q ,
where ε = λ η . As η ( 0 , b + 1 ) , ε ( 0 , 1 ) . Thus an averaged operator T λ is a contraction with contractive constant ε . Let p * F ( T ) . Then, we have
s n p * = ( 1 w n ) p n + w n T λ p n p * ( 1 w n ) p n p * + w n ε p n p * = ( 1 ( 1 ε ) w n ) p n p * .
Further,
r n p * = T λ ( ( 1 o n ) s n + o n T λ s n ) p * ε [ ( 1 o n ) s n p * + o n ε s n p * = ε [ 1 o n + o n ε ] s n p * ε [ 1 ( 1 ε ) o n ) ] [ 1 ( 1 ε ) w n ) ] p n p * .
Moreover,
q n p * = T λ ( ( 1 k n ) T λ s n + k n T λ r n ) p * ε ( 1 k n ) T λ s n + k n T λ r n p * ε [ ( 1 k n ) ε s n p * + k n ε r n p * ] ε 2 [ ( 1 k n ) s n p * + k n r n p * ] .
Note that
( 1 k n ) s n p * ( 1 k n ) ( 1 ( 1 ε ) w n ) p n p * = [ ( 1 ( k n + w n ) + ( w n k n w n ) ε ] p n p * ,
and
k n r n p * k n [ ε [ 1 ( 1 ε ) o n ) ] [ 1 ( 1 ε ) w n ) ] p n p * ] = ε [ k n k n o n k n w n + ε ( k n o n + k n w n ) + k n o n w n 2 ε k n o n w n + ε 2 k n o n w n ] p n p * .
Now by ( 14 ) , we have
q n p * ε 2 ( 1 k n w n + k n w n + ε [ k n + w n 2 k n w n k n o n + k n o n w n ] + ε 2 [ k n o n + k n w n 2 k n o n w n ] + ε 3 [ k n o n w n ] ) p n p * ε 2 ( 1 k n w n + k n w n + ε [ k n + w n 2 k n w n k n o n + k n o n w n ] + ε [ k n o n + k n w n 2 k n o n w n ] + ε [ k n o n w n ] ) p n p * = ε 2 [ 1 k n w n + k n w n + ε k n + ε w n ε k n w n ] p n p * = ε 2 [ 1 ( 1 ε ) ( k n + w n k n w n ) ] p n p * .
Thus,
p n + 1 p * = T λ q n p * ε q n p * ε ε 2 [ 1 ( 1 ε ) ( k n + w n k n w n ) ] p n p * ε 3 [ 1 ( 1 ε ) ( k n + w n k n w n ) ] p n p * .
Inductively, we can obtain that
p n + 1 p * ε 3 n [ 1 ( 1 ε ) ( k n + w n k n w n ) ] p 0 p * .
As 0 < ε 3 n 1 ( 1 ε ) ( k n + w n k n w n ) < 1 , { p n } converges to p * . □
Theorem 2.
Let Ω be a closed convex subset of a uniformly convex Banach space U and T a ( b , η ) -enriched contraction with F ( T ) . Suppose that the sequences { p n ( 1 ) } , { p n ( 2 ) } , { p n ( 3 ) } , { p n ( 4 ) } , { p n ( 5 ) } , { p n ( 6 ) } , { p n ( 7 ) } , { p n ( 8 ) } , { p n } given by iterative schemes ( 4 ) , ( 5 ) , ( 6 ) , ( 7 ) , ( 8 ) , ( 9 ) , ( 10 ) , ( 11 ) , ( 12 ) , respectively, converge to p * F ( T ) . Then, { p n } converges at a rate faster than the other schemes.
Proof. 
Let p * F ( T ) . As proved in Theorem 3 of [21], we have
p n + 1 ( 4 ) p * ε n 1 ( 1 ε ) k n o n w n n p 1 ( 4 ) p * n N . p n + 1 ( 4 ) p * ε n 1 ( 1 ε ) k n o n w n n p 1 ( 4 ) p * = a n ( say ) .
Now, by ( 16 ) in above theorem, we have
p n + 1 p * = ε 3 n 1 ( 1 ε ) ( k n + w n k n w n ) n p 1 p * = b n ( say ) .
Then,
b n a n = ε 3 n 1 ( 1 ε ) ( k n + w n k n w n ) n p 1 p * ε n 1 ( 1 ε ) k n o n w n n p 1 ( 4 ) p * = ε 2 n 1 ( 1 ε ) ( k n + w n k n w n ) 1 ( 1 ε ) k n o n w n n p 1 p * p 1 ( 4 ) p * .
So, we have b n a n 0 as n . Definition (2) implies that { p n } converges faster than { p n ( 4 ) } to the fixed point p * . Now, the inequality proved in Theorem 3.1 of Thakur et al. [6]
p n + 1 ( 5 ) p * ε n ( 1 ( 1 ε ) w n ) n p 1 ( 5 ) p * .
Let
p n + 1 ( 5 ) p * = ε n ( 1 ( 1 ε ) w n ) n p 1 ( 5 ) p * = c n .
So,
b n c n = ε 3 n 1 ( 1 ε ) ( k n + w n k n w n ) n p 1 p * ε n ( 1 ( 1 ε ) w n ) n p 1 ( 5 ) p * = ε 2 n 1 ( 1 ε ) ( k n + w n k n w n ) 1 ( 1 ε ) w n n p 1 p * p 1 ( 5 ) p * .
So, we have b n c n 0 as n . It implies that { p n } converges faster than { p n ( 5 ) } to the fixed point p * .
Similarly, we can show that the sequence { p n } has better rate of convergence than all other sequences define above. □
Theorem 3.
Let Ω be a nonempty closed and convex subset of a Banach space U and T : Ω Ω a ( b , η ) -enriched contraction mapping. Then, the iterative scheme defined in ( 12 ) is T λ -stable for λ = 1 b + 1 .
Proof. 
Let { a n } be an approximate sequence of { p n } in Ω . The sequence defined by iteration ( 12 ) is:
p n + 1 = f ( T λ , p n ) and e n = a n + 1 f ( T λ , a n ) , n N .
We need to show that lim n e n = 0 if and only if lim n a n = p * .
If lim n e n = 0 , it follows from ( 12 ) that
a n + 1 p * a n + 1 f ( T λ , a n ) + f ( T λ , a n ) p * = e n + p n + 1 p * .
By Theorem 1, we have
a n + 1 p * e n + ε 3 [ 1 ( 1 ε ) ( k n + w n k n w n ) ] a n p * .
Let
α n = a n p * and β n = ( 1 ε ) ( k n + w n k n w n ) .
Then,
α n ε 3 ( 1 β n ) α n + e n .
Since, lim n e n = 0 , e n β n 0 as n .
Now, by Lemma 3, we have lim n α n = 0 and hence lim n a n = p * .
Conversely, if lim n a n = p * , then we have
e n = a n + 1 f ( T λ , a n ) a n + 1 p * + f ( T λ , a n ) p * a n + 1 p * + ε 3 [ 1 ( 1 ε ) ( k n + w n k n w n ) ] a n p * .
This implies that lim n e n = 0 . Hence, the iterative scheme ( 12 ) is T λ -stable. □
We now present an example to support our assertion that our iterative process ( 12 ) converges faster than all other iterative schemes considered herein for the class of ( b , η ) -enriched contraction mappings.
Example 1.
Let U = R and Ω = [ 0 , 10 ] . Let T : Ω Ω be a mapping given by T ( p ) = 10 p for all p Ω . Choose k n = 10 11 , o n = 10 13 and w n = 5 6 , with the initial value of p 1 = 8 .
Note that T is ( 3 5 , 5 8 ) -enriched contraction with fixed point 5. So T 5 8 ( p ) = 25 p 4 .
Our iteration ( 12 ) , F-iteration ( 11 ) , M-iteration ( 10 ) , Thakur et al. ( 9 ) , Abbas and Nazir ( 8 ) , Agarwal et al. ( 7 ) and Noor ( 6 ) iterative processes are given in Table 1.
Note that all sequences converge to p * = 5 . The comparison shows that our iteration scheme ( 12 ) converges faster than all the other schemes.
Here we present another numerical example to support our claim.
Example 2.
Let U = R , and Ω = [ 1 , 50 ] . Let T : Ω Ω be defined as
T ( p ) = p 2 6 p + 48
for all p Ω . Choose k n = 3 4 , o n = 1 2 and w n = 1 4 , with the initial value of p 1 = 32 .
Our iteration scheme ( 12 ) along with other iterative schemes for λ = 1 are given in Table 2.
All sequences converge to p * = 8 . The comparison shows that our iteration scheme ( 12 ) has a better rate of convergence than other schemes.
In Table 1 we have the following observations regarding the convergence of fixed point of ( b , η ) -enriched contraction mapping mapping by taking initial point x 1 = 8 . Observe that our iterative scheme converges to the fixed point p * = 5 in eight iterations while other described iterative schemes took more than 10 iterations for convergence.
In Table 2 we have the following observations regarding the convergence of fixed point of contraction mapping mapping by taking initial point x 1 = 32 . Observe that our iterative scheme converges to the fixed point p * = 8 in 12 iterations, F Iteration in 14 iterations, M-iteration in 17 iterations and other iterative schemes took more than 30 iterations for convergence.
In Figure 1 and Figure 2, we test the convergence of different iteration processes for ( b , η ) -enriched contraction and contraction, respectively. Observe that in both cases our iterative scheme is more stable and converges faster to their fixed points than other iterative scheme.

3. Convergence Results for b-Enriched Nonexpansive Mappings

This section deals with some convergence results of an iterative process ( 12 ) .
Theorem 4.
Let Ω be a nonempty closed bounded convex subset of a uniformly convex Banach space U and T : Ω Ω a b-enriched nonexpansive mapping. Then F ( T ) .
Proof. 
Since T is a b-enriched nonexpansive mapping, take b = 1 λ 1 .It follows that λ ( 0 , 1 ) . Then by ( 1 ) we have
( 1 λ 1 ) ( p q ) + T p T q ( b + 1 ) p q ,
which can be written in equivalent form as
T λ p T λ q p q .
That is the averaged operator T λ is nonexpansive. By means of the Browder’s fixed point theorem, it follows that T λ has at least one fixed point. By remark 1 ,   F ( T λ ) = F ( T ) . □
Lemma 4.
Let Ω be a nonempty bounded closed convex subset of a uniformly convex Banach space U and T : Ω Ω a b-enriched nonexpansive mapping. If { p n } is a sequence defined in ( 12 ) and F ( T ) , then lim n p n p * exists for all p * F ( T ) .
Proof. 
By Theorem 4, an averaged operator T λ is nonexpansive. Let p * F ( T λ ) . Then,
s n p * = ( 1 w n ) p n + w n T λ p n p * ( 1 w n ) p n p * + w n T λ p n p * ( 1 w n ) p n p * + w n p n p * p n p * .
Thus,
r n p * = T λ ( ( 1 o n ) s n + o n T λ s n ) p * ( 1 o n ) s n + o n T λ s n p * ( 1 o n ) s n p * + o n T λ s n p * ( 1 o n ) s n p * + o n s n p * s n p * p n p * ,
and
q n p * = T λ ( ( 1 k n ) T λ s n + k n T λ r n ) p * ( 1 k n ) T λ s n + k n T λ r n p * ( 1 k n ) T λ s n p * + k n T λ r n p * ( 1 k n ) s n p * + k n r n p * ( 1 k n ) s n p * + k n s n p * s n p * p n p * .
Now,
p n + 1 p * = T λ q n p * q n p * p n p * .
So, { p n p * } is a bounded monotone decreasing sequence. Therefore, lim n p n p * exists for all p * F ( T λ ) = F ( T ) .
Lemma 5.
Let T, Ω and U be as given in Lemma 4 and { p n } a sequence defined by ( 12 ) and F ( T ) . Then lim n p n T λ p n = 0 , where λ = 1 b + 1 .
Proof. 
Let p * F ( T ) . Then, by Lemma 4, lim n p n p * exists. Suppose that
lim n p n p * = l .
By taking limit supremum as n on both sides of ( 23 ) ( 25 ) , we have
lim sup n s n p * l , lim sup n r n p * l , and lim sup n q n p * l .
Since, T λ is nonexpansive mapping, we have
T λ p n p * p n p * , T λ r n p * r n p * , and T λ q n p * q n p * .
By taking limit supremum on both sides, we obtain
lim sup n T λ p n p * l lim sup n T λ s n p * l lim sup n T λ r n p * l lim sup n T λ q n p * l .
Since,
l = lim inf n p n + 1 p * = lim inf n T q n p * lim inf n q n p * .
So, from ( 28 ) and ( 30 ) , we have
lim n q n p * = l
and
l = lim n q n p * = lim n q n p * .
So,
q n p * = T λ ( ( 1 k n ) T λ s n + k n T λ r n ) p * ( 1 k n ) T λ s n + k n T λ r n p * ( 1 k n ) s n p * + k n r n p * s n p * .
Taking lim inf as n on both sides of the above, we obtain
l lim inf n s n p * .
This implies that
l = lim n s n p * = lim n ( 1 w n ) p n + w n T λ p n p * = lim n ( 1 w n ) ( p n p * ) + w n ( T λ p n p * ) .
from ( 27 ) , ( 29 ) and ( 32 ) and by Lemma 1, we have
lim n p n T λ p n = 0 .
Theorem 5.
Let Ω be a nonempty closed bounded convex subset of a real uniformly convex Banach space U which satisfies the Opial’s condition [24], and T : Ω Ω a b-enriched nonexpansive mapping with F ( T ) . If { p n } is a sequence defined by ( 12 ) , then { p n } converges weakly to a fixed point of T.
Proof. 
Let p * F ( T ) . Then lim n p n p * exists. We prove that { p n } has a unique subsequential limit in F ( T ) . Let a and b be two weak limits of the subsequences { p n i } and { p n j } of { p n } , respectively. As, lim n p n T λ p n k = 0 and I T λ is demiclosed with respect to zero, where λ = 1 b + 1 , by Lemma 2 we obtain that T λ a = a . Similarly, we can show that b F ( T λ ) . From Lemma 4, lim n p n b exists. Next, we prove the uniqueness. If a b , then by the Opial’s condition, we have
lim n p n a lim n i p n i a < lim n i p n i b = lim n p n b = lim n j p n j b < lim n j p n j a = lim n p n a ,
a contradiction, so a = b and hence { p n } converges weakly to a fixed point of T λ = F ( T ) . □
Theorem 6.
Let Ω be a nonempty closed bounded convex subset of a uniformly convex Banach space U and T : Ω Ω a b-enriched nonexpansive mapping. If { p n } is a sequence defined by ( 12 ) and F ( T ) , then { p n } converges to a point in F ( T ) if and only if lim inf n d ( p n , F ( T ) ) = 0 or lim sup n d ( p n , F ( T ) ) = 0 , where d ( p n , F ( T ) ) = inf { p n p * : p * F ( T ) } .
Proof. 
Necessity is obvious.
Conversely, Since p * F ( T λ ) , so d ( p n , F ( T ) ) = d ( p n , F ( T λ ) ) .
Suppose that lim inf n d ( p n , F ( T λ ) ) = 0 . As proved in Lemma 5, lim n p n p * exists for all p * F ( T λ ) , by the given assumption we have lim inf n d ( p n , F ( T λ ) ) = 0 , therefore lim n d ( p n , F ( T λ ) ) = 0 . We now show that { p n } is a Cauchy sequence in Ω . As lim n d ( p n , F ( T λ ) ) = 0 , for given ϵ > 0 there is m 0 N such that for all n m 0 , we have
d ( p n , F ( T λ ) ) < ϵ 2 ,
which implies that
inf { p n p * : p * F ( T λ ) } < ϵ 2 .
In particular, inf { p m 0 p * : p * F ( T λ ) } < ϵ 2 . Hence there exists p * F ( T λ ) such that
p m 0 p * < ϵ 2 .
Now, for m , n m 0
p m + n p n p m + n p * + p n p * p m 0 p * + p m 0 p * 2 p m 0 p * ϵ .
This shows that { p n } is a Cauchy sequence in Ω . As Ω is closed and bounded subset of the Banach space U, there exists a point q * Ω such that lim n p n = q * . Now, lim n d ( p n , F ( T λ ) ) = 0 gives that
d ( p n , F ( T λ ) ) = 0 q * F ( T λ ) = F ( T ) .
Sentor and Dotson [25] introduced the notion of mapping satisfying Condition (I) which is given as follows:
Definition 4.
A mapping T : Ω Ω is said to satisfy Condition (I), if there is a nondecreasing function h : [ 0 , ) [ 0 , ) with h ( 0 ) = 0 and h ( r ) > 0 , r > 0 such that
d ( p , T p ) h ( d ( p , F ( T ) ) ) ,
for all p Ω , where d ( p , F ( T ) ) = inf { d ( p , p * ) : p * F ( T ) } .
We now prove a strong convergence result by using the Condition (I).
Theorem 7.
Let Ω be a nonempty closed bounded convex subset of a uniformly convex Banach space U and T : Ω Ω a b-enriched nonexpansive mapping. Let { p n } be a sequence defined by ( 12 ) and F ( T ) . If T λ satisfies Condition (I) for the value of λ = 1 b + 1 , then { p n } converges strongly to a fixed point of T.
Proof. 
We proved in Lemma 5 that
lim n p n T λ p n = 0 .
From condition (I), we get
lim n h ( d ( p n , F ( T λ ) ) ) lim n p n T λ p n = 0 .
That is, lim n h ( d ( p n , F ( T λ ) ) ) = 0 . Since, h : [ 0 , ) [ 0 , ) is a nondecreasing function with h ( 0 ) = 0 and h ( r ) > 0 , r ( 0 , ) , we have lim n d ( p n , F ( T λ ) ) = 0 . By Theorem 6, the sequence { p n } converges strongly to a point in F ( T λ ) = F ( T ) . □

4. Application: Solution of Delay Fractional Differential Equations

In 1967, Caputo proposed a new form of fractional differentiation called Caputo’s fractional derivatives, which is defined as:
C D t α f ( t ) = 1 Γ ( α n ) a t f ( n ) ( τ ) ( t τ ) n α 1 d τ , ( n 1 < α < n ) .
Under natural conditions on the function f ( t ) , for α n the Caputo derivative becomes a conventional n-th derivative of the function f ( t ) . The main advantage of Caputo’s approach is that the initial conditions for fractional differential equations with Caputo derivatives take on the same form as for integer-order differential equations, i.e., contain the limit value of integer-order derivatives of unknown functions at the lower terminal t = a .
In 2017, Cong and Tuan [26] obtained the existence and uniqueness of global solution of delay fractional differential equations by using properties of Mittag–Leffler functions and BCP. Many authors have solved the delay differential equations of fractional order using different approaches. For more details, we refer to [27,28,29,30,31].
Here, we estimate the solution of a delay fractional differential equation [26] by using an iterative scheme ( 12 ) with α ( 0 , 1 ) . Let h > 0 be any constant and ρ C ( [ j h , j ] : R n ) be a continuous mapping.
Consider the following delay Caputo fractional differential equation
c D α p ( t ) = f ( t , p ( t ) , p ( t h ) ) , t [ j , M ]
with initial condition
p ( t ) = ρ ( t ) , t [ j τ , j ] ,
where p R n , f : [ j , M ] × R n × R n R n is continuous, τ > 0 and M > 0 . Suppose that the following conditions are fulfilled.
( c 1 )
f satisfies the Lipschitz condition with respect to 2nd and 3rd variables: That is, there exists a positive constant L f (depending on f) such that
f ( t , p , q ) f ( t , p ^ , q ^ ) L f ( p p ^ + q q ^ )
for all t R + and p , p ^ , q , q ^ R n .
( c 2 )
There exists a positive constant β L depending upon L such that β L > 2 L , that is, 2 L β L < 1 .
A function p * C ( [ j h , M ] : R n ) C 1 ( [ j , M ] : R n ) is called solution of the initial value problem if it satisfies ( 35 ) and ( 36 ) .
It is known that [32] finding the solution of ( 35 ) and ( 36 ) is equivalent to finding the solution of the following integral equation
p ( t ) = ρ ( t ) + 1 Γ ( α ) j t ( t τ ) ( α 1 ) f ( τ , p ( τ ) , p ( τ h ) ) d τ t [ j , M ] ,
with p ( t ) = ρ ( t ) , t [ j h , M ] . Define a norm . β L on C ( [ j h , j ] : R n ) by
ρ β L = sup ρ ( t ) E α ( β L t α ) , for any ρ C ( [ j h , j ] : R n )
where E α : R R is a Mittag–Leffler function defined as:
E α ( t ) = Σ n = 0 t n Γ ( α n + 1 ) , t R .
Note that C ( [ j h , j ] : R n , . β L ) is a Banach space.
Wang et al. [27] proved the existence and uniqueness of solution of delay differential Equations ( 35 ) and ( 36 ) provided that the condition ( c 1 ) holds. In the following theorem, we obtain an approximation of the solution using an iterative scheme ( 12 ) for λ = 1 .
Theorem 8.
Let ρ and f be functions as given above. If the conditions ( c 1 ) and ( c 2 ) are satisfied, then the problem ( 35 ) and ( 36 ) has a unique solution p * C ( [ j h , M ] : R n ) C 1 ( [ j , M ] : R n ) and the sequence { p n } defined by ( 12 ) converges to p * .
Proof. 
The existence of unique solution p * is followed from [26]. Let { p n } be a sequence of defined by ( 12 ) . Define an operator T on C ( [ j h , M ] : R n ) C 1 ( [ j , M ] : R n ) by:
T p ( t ) = ρ ( j ) + 1 Γ ( α ) j t ( t τ ) ( α 1 ) f τ , p ( τ ) , p ( τ h ) d τ , t [ j , M ] , ρ ( t ) t [ j τ , j ] .
We now prove that p n p * as n .
For t [ j τ , j ] , it is easy to see that p n p * as n .
Now, if t [ j , M ] , then using the proposed iterative process and conditions ( c 1 ) and ( c 2 ) , we obtain
s n p * = ( 1 w n ) p n + w n T 1 p n p * ( 1 w n ) p n p * + w n T 1 p n p * .
Taking supremum over [ j h , M ] on both sides of the above inequality, we have
sup t [ j h , M ] s n p * = ( 1 w n ) sup t [ j h , M ] p n p * + w n sup t [ j h , M ] T 1 p n T 1 p * = ( 1 w n ) sup t [ j h , M ] p n p * + w n sup t [ j h , M ] ρ ( j ) + 1 Γ ( α ) j t ( t τ ) ( α 1 ) f ( τ , p n ( τ ) , p n ( τ h ) ) d τ ρ ( j ) 1 Γ ( α ) j t ( t τ ) ( α 1 ) f ( τ , p * ( τ ) , p * ( τ h ) ) d τ = ( 1 w n ) sup t [ j h , M ] p n p * + w n sup t [ j h , M ] 1 Γ ( α ) j t ( t τ ) ( α 1 ) f ( τ , p n ( τ ) , p n ( τ h ) ) d τ 1 Γ ( α ) j t ( t τ ) ( α 1 ) f ( τ , p * ( τ ) , p * ( τ h ) ) d τ ( 1 w n ) sup t [ j h , M ] p n p * + w n sup t [ j h , M ] 1 Γ ( α ) j t ( t τ ) ( α 1 ) L f ( p n ( τ ) p * ( τ ) + p n ( τ h ) ) p * ( τ h ) ) d τ ( 1 w n ) sup t [ j h , M ] p n p * + w n L f Γ ( α ) j t ( t τ ) ( α 1 ) d τ sup t [ j h , M ] ( p n ( τ ) p * ( τ ) + sup t [ j h , M ] p n ( τ h ) ) p * ( τ h ) .
Dividing by E α ( β L t α ) on both sides of the above equality, we obtain
sup t [ j h , M ] s n p * E α ( β L t α ) = ( 1 w n ) sup t [ j h , M ] p n p * E α ( β L t α ) + w n L f Γ ( α ) j t ( t τ ) ( α 1 ) d τ ( sup t [ j h , M ] p n ( τ ) p * ( τ ) E α ( β L t α ) + sup t [ j h , M ] p n ( τ h ) ) p * ( τ h ) E α ( β L t α ) ) s n p * β L = ( 1 w n ) p n p * β L + w n Γ ( α ) j t ( t τ ) ( α 1 ) d τ L f ( p n ( τ ) p * ( τ ) β L + p n ( τ h ) p * ( τ h ) β L )
= ( 1 w n ) p n p * β L + w n ( 2 L f ) p n p * β L 1 Γ ( α ) j t ( t τ ) ( α 1 ) d τ multiply and divide the right end term by E α ( β L t α ) , then we have = ( 1 w n ) p n p * β L + w n ( 2 L f ) E α ( β L t α ) p n p * β L 1 Γ ( α ) j t ( t τ ) ( α 1 ) E α ( β L t α ) d τ = ( 1 w n ) p n p * β L + w n ( 2 L f ) E α ( β L t α ) p n p * β L . c I α c D α ( E α ( β L t α ) β L ) = ( 1 w n ) p n p * β L + w n ( 2 L f ) E α ( β L t α ) . E α ( β L t α ) β L p n p * β L = ( 1 w n ) p n p * β L + w n ( 2 L f ) β L p n p * β L .
As 2 L f β L < 1 , we obtain
s n p * β L p n p * β L .
Let u n = ( 1 o n ) s n + o n T λ s n . Then by following similar arguments to those given above, we have
u n p * β L s n p * β L p n p * β L .
Thus,
r n p * = T u n T p * = ρ ( j ) + 1 Γ ( α ) j t ( t τ ) ( α 1 ) f ( τ , u n ( τ ) , u n ( τ h ) ) d τ ρ ( j ) 1 Γ ( α ) j t ( t τ ) ( α 1 ) f ( τ , p * ( τ ) , p * ( τ h ) ) d τ 1 Γ ( α ) j t ( t τ ) ( α 1 ) L f u ( τ ) p * ( τ ) + u ( τ h ) p * ( τ h ) d τ .
Taking supremum over [ j h , M ] and dividing by E α ( β L t α ) on both sides of the above inequality, we obtain
sup t [ j h , M ] r n p * E α ( β L t α ) = 1 Γ ( α ) j t ( t τ ) ( α 1 ) L f [ sup t [ j h , M ] U ( τ ) p * ( τ ) E α ( β L t α ) + sup t [ j h , M ] U ( t h ) p * ( t h ) E α ( β L t α ) ] d t r n p * β L 2 L f E α ( β L t α ) u n p * β L 1 Γ ( α ) j t ( t τ ) ( Γ 1 ) E α ( β L t α ) d τ = 2 L f E α ( β L t α ) u n p * β L . c I α c D α ( E α ( β L t α ) β L ) = 2 L f E α ( β L t α ) . E α ( β L t α ) β L u n p * β L = 2 L f β L u n p * β L .
Again by 2 L f β L < 1 , we obtain
r n p * β L u n p * β L .
So,
r n p * β L p n p * β L .
Similarly,
q n p * β L p n p * β L .
p n + 1 p * β L p n p * β L .
If we set, p n p * β L = ν n , then we have
ν n + 1 ν n n N .
Thus { ν n } is a monotone decreasing sequence of positive real numbers. Further, it is bounded sequence, we obtain
lim n ν n = inf { ν n } = 0 .
So,
lim n p n p * β L = 0 .

5. Conclusions

In this paper we approximate the fixed point of ( b , η ) -enriched contraction mapping by using a new iterative scheme (define by Abbas and Asghar) in the frame work of Banach spaces. It is also proved that the proposed iterative scheme is stable and converges faster than Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur, M-iteration and F-iteration. We presented some numerical examples to support our claim. Further, we proved some strong and weak convergence results for b-enriched nonexpansive mapping in uniformly convex Banach space. In the end, using our proposed iterative scheme we approximated the solution of delay fractional differential equations.

Author Contributions

Conceptualization, M.A. and M.W.A.; methodology, M.W.A.; validation, M.A., M.W.A. and M.D.l.S.; formal analysis, M.W.A.; investigation, M.A. and M.W.A.; writing—original draft preparation, M.W.A.; writing—review and editing, M.A. and M.W.A.; supervision, M.A. and M.D.l.S.; project administration, M.D.l.S.; funding acquisition, M.D.l.S. All authors have read and agreed to the published version of the manuscript.

Funding

The authors are very grateful to the Basque Government for their support through Grant no. IT1207-19.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

Authors are thankful to the reviewers and editor for their constructive comments which helped us a lot in improving the presentation of the paper.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Banach, S. Surles operations dans les ensembles abstraites et leurs applications. Fund. Math. 1922, 2, 133–187. [Google Scholar] [CrossRef]
  2. Ali, J.; Ali, F. A new iterative scheme to approximating fixed points and the solution of a delay differential equation. J. Nonlinear Convex Anal. 2020, 21, 2151–2163. [Google Scholar]
  3. Jajarmi, A.; Baleanu, D. A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems. Front. Phys. 2020, 8, 2020. [Google Scholar] [CrossRef]
  4. Okeke, G.A.; Abbas, M. A solution of delay differential equations via Picard–Krasnoselskii hybrid iterative process. Arab. J. Math. 2017, 6, 21–29. [Google Scholar] [CrossRef] [Green Version]
  5. Okeke, G.A.; Abbas, M.; de la Sen, M. Approximation of the fixed point of multivalued quasi-nonexpansive mappings via a faster iterative process with applications. Discret. Dyn. Nat. Soc. 2020, 2020, 8634050. [Google Scholar] [CrossRef]
  6. Thakur, B.S.; Thakur, D.; Postolache, M. A new iteration scheme for approximating fixed points of nonexpansive mappings. Filomat 2016, 30, 2711–2720. [Google Scholar] [CrossRef] [Green Version]
  7. Zhou, H.Y.; Cho, Y.J.; Kang, S.M. A new iterative algorithm for approximating common fixed points for asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2007, 2007, 1–10. [Google Scholar] [CrossRef] [Green Version]
  8. Schu, J. Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43, 153–159. [Google Scholar] [CrossRef] [Green Version]
  9. Berinde, V.; Păcurar, M. Approximating fixed points of enriched contractions in Banach spaces. J. Fixed Point Theory Appl. 2020, 22, 38. [Google Scholar] [CrossRef] [Green Version]
  10. Berinde, V. Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces. Carpathian J. Math. 2019, 35, 293–304. [Google Scholar] [CrossRef]
  11. Krasnosel’skii, M.A. Two comments on the method of successive approximations. Usp. Math. Nauk. 1955, 10, 123–127. [Google Scholar]
  12. Gallagher, T.M. The demiclosedness principle for mean nonexpansive mappings. J. Math. Anal. Appl. 2016, 439, 832–842. [Google Scholar] [CrossRef]
  13. Suzuki, T. Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 2008, 340, 1088–1095. [Google Scholar] [CrossRef] [Green Version]
  14. Górnicki, J.; Bisht, R.K. Around averaged mappings. J. Fixed Point Theory Appl. 2021, 23, 48. [Google Scholar] [CrossRef]
  15. Berinde, V. Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. Fixed Point Theory Appl. 2004, 2004, 716359. [Google Scholar] [CrossRef] [Green Version]
  16. Picard, E. Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives. J. Math. Pures Appl. 1890, 6, 145–210. [Google Scholar]
  17. Mann, W.R. Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4, 506–510. [Google Scholar] [CrossRef]
  18. Ishikawa, S. Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44, 147–150. [Google Scholar] [CrossRef]
  19. Noor, M.A. New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 2000, 251, 217–229. [Google Scholar] [CrossRef] [Green Version]
  20. Agarwal, R.P.; Regan, D.O.; Sahu, D.R. Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 2007, 61, 2007. [Google Scholar]
  21. Abbas, M.; Nazir, T. Some new faster iteration process applied to constrained minimization and feasibility problems. Mat. Vesn. 2014, 2014, 223–234. [Google Scholar]
  22. Ullah, K.; Arshad, M. Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process. Filomat 2018, 32, 187–196. [Google Scholar] [CrossRef] [Green Version]
  23. Ostrowski, A.M. The Round-off Stability of Iterations. ZAMM-J. Appl. Math. Mech. Angew. Math. Mech. 1967, 47, 77–81. [Google Scholar] [CrossRef]
  24. Opial, Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73, 591–597. [Google Scholar] [CrossRef] [Green Version]
  25. Senter, H.F.; Dotson, W.G. Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 1974, 44, 375–380. [Google Scholar] [CrossRef]
  26. Cong, N.; Tuan, H. Existence, uniqueness, and exponential boundedness of global solutions to delay fractional differential equations. Mediterr. J. Math. 2017, 10, 1–12. [Google Scholar] [CrossRef] [Green Version]
  27. Wang, F.-F.; Chen, D.-Y.; Zhang, X.-G.; Wu, Y. The existence and uniqueness theorem of the solution to a class of nonlinear fractional order system with time delay. Appl. Math. Lett. 2016, 53, 45–51. [Google Scholar] [CrossRef]
  28. Boutiara, A.; Matar, M.M.; Kaabar, M.K.A.; Martínez, F.; Sina, E.; Rezapour, S. Some Qualitative Analyses of Neutral Functional Delay Differential Equation with Generalized Caputo Operator. J. Funct. Spaces 2021, 2021, 9993177. [Google Scholar] [CrossRef]
  29. Daftardar-Gejji, V.; Sukale, Y.; Bhalekar, S. Solving fractional delay differential equations: A new approach. Fract. Calc. Appl. Anal. 2015, 18, 400–418. [Google Scholar] [CrossRef]
  30. Garrappa, R.; Kaslik, E. On initial conditions for fractional delay differential equations. Commun. Nonlinear Sci. Numer. Simul. 1955, 90, 105359. [Google Scholar] [CrossRef]
  31. Jhinga, A.; Daftardar-Gejji, V. A new numerical method for solving fractional delay differential equations. Comput. Appl. Math. 2019, 38, 166. [Google Scholar] [CrossRef]
  32. Kilbas, A.; Marzan, S. Cauchy problem for differential equation with Caputo derivative. Fract. Calc. Appl. Anal. 2004, 7, 297–321. [Google Scholar]
Figure 1. Convergence behavior of our scheme, F-Iteration, M-Iteration, Thakur, Abbas, Agarwal and Noor iterations for ( b , η ) -enriched contraction mapping given in Example 1.
Figure 1. Convergence behavior of our scheme, F-Iteration, M-Iteration, Thakur, Abbas, Agarwal and Noor iterations for ( b , η ) -enriched contraction mapping given in Example 1.
Mathematics 10 00273 g001
Figure 2. Convergence behavior of our scheme, F-Iteration, M-Iteration, Thakur, Abbas, Agarwal, Noor and Ishikawa iterations for contraction mapping given in Example 2.
Figure 2. Convergence behavior of our scheme, F-Iteration, M-Iteration, Thakur, Abbas, Agarwal, Noor and Ishikawa iterations for contraction mapping given in Example 2.
Mathematics 10 00273 g002
Table 1. Convergence comparison of iterative schemes for ( b , η ) -enriched contraction mapping.
Table 1. Convergence comparison of iterative schemes for ( b , η ) -enriched contraction mapping.
StepsOur SchemeF-IterationM-IterationThakurAbbasAgarwal (S)Noor
18.00000000008.00000000008.00000000008.00000000008.00000000008.00000000008.0000000000
25.06763782415.37696754444.40762243014.89981768793.25905649575.27550135814.5632864505
35.00152495845.04736817655.11697039505.00334549856.01029476165.02530033275.0635729080
45.00003438165.00595208834.97690312054.99988828004.41371129905.00232342534.9907456165
55.00000077515.00074791475.00456069115.00000373075.34023183515.00021336895.0013471715
65.00000001745.00009397984.99909944964.99999987544.80255853225.00001959444.9998038906
75.00000000035.00001180915.00017782195.00000000415.11457814695.00000179945.0000285478
85.00000000005.00000148384.99996488744.99999999984.93350863975.00000016524.9999958442
95.00000000005.00000018645.00000693335.00000000005.03858590055.00000001515.0000006049
105.00000000005.00000002344.99999863094.99999999994.97760804235.00000000134.9999999119
115.00000000005.00000000295.00000027035.00000000005.01299437775.00000000015.0000000128
125.00000000005.00000000034.99999994665.00000000004.99245917415.00000000004.9999999981
135.00000000005.00000000005.00000001055.00000000005.00437605055.00000000005.0000000002
145.00000000005.00000000004.99999999795.00000000004.99746051435.00000000004.9999999999
155.00000000005.00000000005.00000000045.00000000005.00147370035.00000000005.0000000000
Table 2. Convergence comparison of iterative schemes for contraction mapping.
Table 2. Convergence comparison of iterative schemes for contraction mapping.
StepsOur SchemeF-IterationM-IterationThakurAbbasAgarwal (S)NoorIshikawa
132.00000032.00000032.00000032.00000032.00000032.00000032.00000032.000000
222.27820523.56709325.72460928.24476428.59792528.80951129.18463229.393312
314.41330416.36070120.01043024.65594225.32852625.73335226.45974326.862041
49.77359311.24911815.14463421.27794122.22323422.79660223.84242024.419226
58.3065408.83083211.51439618.16903119.32243720.03107821.35360922.080669
68.0444448.1017739.37366015.40246916.67670517.47642419.01860819.865335
78.0062388.0289798.44514313.06100814.34548615.17993216.86703517.795540
88.0008718.0051048.13118311.21680212.38958113.19297114.93154215.896548
98.0001228.0008968.0373689.89475610.85375011.56118613.24429214.195061
108.0000178.0001578.0105359.0425669.74250310.30770111.83049512.715924
118.0000028.0000278.0029618.5450439.0065519.41651110.70001411.476899
128.0000008.0000048.0008328.2757458.5572628.8306429.84093510.482609
138.0000008.0000018.0002348.1369168.2998598.4708719.2199509.720547
148.0000008.0000008.0000658.0673128.1585918.2609968.7904979.162136
158.0000008.0000008.0000188.0329268.0830608.1426938.5038618.768990
168.0000008.0000008.0000058.0160668.0432718.0773968.3175128.501076
178.0000008.0000008.0000018.0078298.0224798.0417938.1985698.322954
188.0000008.0000008.0000008.0038138.0116608.0225128.1235718.206601
198.0000008.0000008.0000008.0018578.0060448.0121118.0766598.131513
208.0000008.0000008.0000008.0009048.0031318.0065108.0474628.083445
218.0000008.0000008.0000008.0004408.0016228.0034988.0293498.052835
228.0000008.0000008.0000008.0002148.0008408.0018798.0181358.033409
238.0000008.0000008.0000008.0001048.0004358.0010108.0112008.021107
248.0000008.0000008.0000008.0000518.0002258.0005428.0069158.013328
258.0000008.0000008.0000008.0000258.0001178.0002918.0042698.008413
268.0000008.0000008.0000008.0000128.0000608.0001568.0026358.005309
278.0000008.0000008.0000008.0000068.0000318.0000848.0016268.003350
288.0000008.0000008.0000008.0000038.0000168.0000458.0010048.002114
298.0000008.0000008.0000008.0000018.0000088.0000248.0006198.001334
308.0000008.0000008.0000008.0000008.0000048.0000138.0003828.000841
318.0000008.0000008.0000008.0000008.0000028.0000078.0002368.000531
328.0000008.0000008.0000008.0000008.0000018.0000038.0001468.000335
338.0000008.0000008.0000008.0000008.0000008.0000028.0000908.000211
348.0000008.0000008.0000008.0000008.0000008.0000018.0000558.000133
358.0000008.0000008.0000008.0000008.0000008.0000008.0000348.000084
368.0000008.0000008.0000008.0000008.0000008.0000008.0000218.000053
378.0000008.0000008.0000008.0000008.0000008.0000008.0000138.000033
388.0000008.0000008.0000008.0000008.0000008.0000008.0000088.000021
398.0000008.0000008.0000008.0000008.0000008.0000008.0000048.000013
408.0000008.0000008.0000008.0000008.0000008.0000008.0000038.000008
418.0000008.0000008.0000008.0000008.0000008.0000008.0000018.000005
428.0000008.0000008.0000008.0000008.0000008.0000008.0000008.000003
438.0000008.0000008.0000008.0000008.0000008.0000008.0000008.000002
448.0000008.0000008.0000008.0000008.0000008.0000008.0000008.000001
458.0000008.0000008.0000008.0000008.0000008.0000008.0000008.000000
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Abbas, M.; Asghar, M.W.; De la Sen, M. Approximation of the Solution of Delay Fractional Differential Equation Using AA-Iterative Scheme. Mathematics 2022, 10, 273. https://doi.org/10.3390/math10020273

AMA Style

Abbas M, Asghar MW, De la Sen M. Approximation of the Solution of Delay Fractional Differential Equation Using AA-Iterative Scheme. Mathematics. 2022; 10(2):273. https://doi.org/10.3390/math10020273

Chicago/Turabian Style

Abbas, Mujahid, Muhammad Waseem Asghar, and Manuel De la Sen. 2022. "Approximation of the Solution of Delay Fractional Differential Equation Using AA-Iterative Scheme" Mathematics 10, no. 2: 273. https://doi.org/10.3390/math10020273

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