# Convergence and Stability of a NovelM‐Iterative Algorithm with an Application

Authors: Manjinder Kaur; Sumit Chandok
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this manuscript, we propose a novel three - step iteration scheme calledM - iteration to approximate the invariant points for the class of weak contractions in the sense of Berinde and obtain that M - iteration strongly converges to one and only one fixed point for Berinde mappings. Proving ‘almost - stability’ of M - iteration, we compare the M - iterative scheme with other chief iterative algorithms, namely. Picard, Mann, Ishikawa, Noor, S, normal-S, Abbas, Thakur, Varat, Ullah and F∗ and claim that the framed iterative procedure converges to the invariant points of weak contractions at faster rate than other vital algorithms. Some numerical illustrations are adduced to strengthen our claim. We further ascertain data dependency through M - iteration. Finally, we establish the solution of Caputo type fractional differential equation as an application and exhibit that M - fixed point procedure converges to the solution of a fractional differential equation. The obtained results are not only new but, also, extend the scope of previous findings.


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## Body

## 1. Introduction

Several nonlinear problems can be mathematically modelled via real valued self-mapping(1)Mx=x.Endowed with specific properties like continuity, contraction, nonexpansiveness etc. Acknowledging the work of Banach, mappings with contraction condition have proven to be fascinating in the theory of fixed points. But an obvious question arises:What if the contraction condition on self - mapping is weaken?Answering to this, Berinde in 2003 (see [1]) came up with a notion called weak contraction, sometimes termed as almost - contractions. He established that the class of these weak contractions is wider than contraction mappings, Zamfirescu mappings and developed the existence and uniqueness theorem for fixed points in the ambient domain.Further, finding an analytical solution of (1) is not so straightforward that urges the researchers to numerical reckoning of fixed points. Owing to this limitation, researchers broached a number of approximation procedures to estimate fixed points of varying classes of nonlinear mappings, Picard [2], Mann [3], Ishikawa [4], S [5], K∗ [6] are to name a few. Motivated by these works, we propose a new three - step approximation procedure called M - iteration and prove that the suggested iterative scheme converges strongly to a unique fixed point for the class of Berinde’s weak contractions. Also, we establish almost M- stability of our proposed algorithm in Subsection 3.1. In Section 4, we present that our algorithm has faster rate of convergence than certain other significant approximation procedures. Subsection 4.1, hooks our assertion of Section 4 numerically via few illustrations. In Section 5, we exhibit data dependency by M- iteration. From application perspective, we obtain a solution of Caputo type fractional differential equation in the last section.


## 2. Brief Preliminaries

In this section, we present few requisite definitions, notations and results.Berinde in 2003 [7] came up with a broader class of contraction mappings called weak contractions, sometimes known as ‘almost contractions’, which can be defined as:Definition 1.
Letℬˇ be a Banach space. A self - mapping M on ℬˇ is called weak contraction if there exist a constant δ∈0,1 and a nonnegative constant L, such that(2)Mx−My≤δx−y+Ly−Mx,for allx,y∈ℬ⌣.
It is to be emphasized that weak contractions are different from contraction mappings (see Example 2.11 of [1]). He then manifests the subsequent theorem for the case of weak contraction mappings.Theorem 1.
The self - mappingM on a Banach space ℬ⌣ agreeing (2) together with(3)Mx−My≤δx−y+Lx−Mx,for allx,y∈ℬ⌣,ensues that the mappingM has a unique fixed point in ℬ⌣.Our study incorporates a comparison of the various iterative procedures see Table1 framed for the self - mapping M:G⟶G, where G is a nonempty subset of Banach space ℬ⌣, and ln,mn and en are sequences in 0,1.The following two definitions were accorded by Berinde [14] to size up the rate of convergence of two iterative procedures.Definition 2.
Letın and j be two positive sequences in reals, converging to ı and respectively. Suppose(4)ļ=limn⟶∞ın−ıjn−.(i)
Ifļ=0, then ın converges faster to 1 as compare to jn converging to.(ii)
If0<ļ<∞, then the rate of convergence of both ın and jn are coequal.Definition 3.
Assumekn and ρn are two iterative procedures such that each of them is converging to the common point k with error estimates:(5)kn−k≤ın,ρn−k≤.
Iflimn⟶∞ın/jn=0, then kn converges faster than ρn.
An iterative procedure, in general, is prone to errors that may cause harm to its aspect of convergence. To address this problem, the concept of stability of a fixed point procedure was introduced in 1967 by Ostrowski [15]. However, it was Harder et al. [16], who, using the idea of Ostrowski, clearly defined the concept for the first time in 1988 and established the stability of Picard iteration with respect to α - contractions and Zamfirescu maps (see Theorems 1 and 2 of [16]) in the setting of metric space. He further showed that certain Mann iterations are stable with respect to Zamfirescu maps in the framework of normed linear space.
The following definition is based on the idea of Theorem2 of Ostrowski [15].Definition 4.
Suppose, for a self - mappingM defined on Banach space ℬˇ, we have:(i)
Mg=g,(ii)
gn+1=θM,gn,n∈ℕ∪0, with g0∈ℬ⌣ where θ is certain function,(iii)
an approximate sequencern of gn in ℬˇ,(iv)
u˜n=rn+1−θM,rn,
Then the iterative proceduregn+1 may be regarded as M - stable on condition that:(6)limn⟶∞u˜n=0implieslimn⟶∞rn=g.
In 1990, Rhoades [17] extended the results of Harder et al. [16] and showed the stability for some fixed point iterations (Picard, Mann). Osilike [18] introduced the concept of weak stability called ‘almost-stability’ for iterative algorithms as follows:Definition 5.
Suppose, for a self - mappingM defined on Banach space ℬˇ, we have:(i)
Mg=g,(ii)
gn+1=θM,gn,n∈ℕ∪0, with g0∈ℬ⌣ where θ is certain function,(iii)
an approximate sequencern of gn in ℬ⌣,(iv)
u˜n=rn+1−θM,rn,
Then the iterative proceduregn+1 may be regarded as almost M - stable if:(7)∑n=0∞u˜n<∞implieslimn⟶∞rn=g.
Determination of fixed points is a challenging task, especially, in the case, when behaviour of an operator is not known. This propel the researchers to study about data dependency of fixed points. By data dependency for an iterative scheme, we mean, there exists an approximation by which one can find the fixed point of an unknown operator using the fixed point of a known operator. Markin [19] in 1973 was first to talk about continuous data dependency of fixed points. In 1998, Rus and Muresan [20] studied data dependency of weakly Picard operators. Then, again Rus et.al. in 2001 [21] and 2003 [22], discussed data dependence of fixed points of some multivalued weakly Picard operators. Şoltuz [23] in 2004 established data dependence result for Mann- Ishikawa iteration scheme. He, together with Grosan [24] in 2008, demonstrated the result for Ishikawa iterative procedure for contractive - like mappings. In 2009, Olatinwo [25] documented couple of results on continuous data dependency of fixed points.
The following results will be needed in the sequel.Definition 6 (see [1]).
SupposeM and Q be two self - operators defined on a nonvoid set G⊂ℬ⌣ then the operator Q may be called an approximate operator of M if a positive constant ε>0 exists such that(8)Mx−Qx≤εfor all x∈G.Lemma 1.
[26] Let u˜n and v˜n be two nonnegative sequences in ℝ and 0≤q<1, such that v˜n+1≤qv˜n+u˜n, for all n≥0. If limn⟶∞u˜n=0, then limn⟶∞v˜n=0.We need the following result to prove data dependency which is according to Şoltuz et al. [24], modified by Ali et al. [13].Lemma 2.
Let there exists a positive integerm for a sequence a`n of positive reals, such that(9)a`n+1≤1−b`na`n+b`nc`nfor alln≥m,where b`n∈0,1 for all n∈ℕ∪0 such that series b`n is divergent and c`n is nonnegative bounded sequence. This gives:(10)0≤limn⟶∞supa`n≤limn⟶∞supc`n.
Upcoming lemma is given by Şoltuz et al. [27] in 2007.Lemma 3.
For a sequencee˜n of positive reals satisfying following inequality,(11)e˜n+1≤1−d˜ne˜n,
Ifd˜n∈0,1 and ∑d˜n=∞ then limn⟶∞e˜n=0.


## 3. Main Results

We begin with the introduction of a new three - step iteration calledM - iteration along these lines:LetM:G⟶G be a self - mapping where G is nonvoid, closed and convex subset of Banach space ℬ⌣. We define sequence gn in the following fashion:M - iteration(12)g0,∈G,gn+1,=M1−enMhn+enhn,hn,=M1−lnfn+lnMfn,fn,=M1−mngn+mnMgn,n∈ℕ∪0,where ln,mn,en are sequences in 0,1.We prove thatM - iteration defined by (12) converges strongly to fixed point for the class of weak contractions.Theorem 2.
For a Banach spaceℬ⌣ and nonvoid, closed and convex set G⊆ℬ⌣, let M:G⟶G is Berinde weak contraction agreeing (3), then the sequence gn specified by M - iteration (12) converges to unique fixed point of M subject to the sequences ln,mn,en generated by (12) obeys one of the following:(13)CM1∑ln=∞,CM2∑mn=∞,CM3∑en=∞.Proof.
From Theorem1, M possess a unique fixed point say g and using (3), we have(14)Mgn−g=Mgn−Mg≤δgn−g+Lg−Mg=δ‖gn−g‖for alln∈ℕ∪0.
ByM- iterative algorithm (12), we have,(15)fn−g=M1−mngn+mnMgn−g≤δ1−mngn+mnMgn−g≤δ1−mngn−g+mnMgn−g≤δ1−mngn−g+δmngn−g=δ1−mn+δmngn−g=δ1−1−δmngn−g,
And(16)hn−g=M1−lnfn+lnMfn−g≤δ1−lnfn+lnMfn−g≤δ1−lnfn−g+lnMfn−g≤δ1−lnfn−g+δlnfn−g=δ1−ln+δlnfn−g=δ1−1−δlnfn−g.
Using (15) and (16), we get(17)gn+1−g=M1−enMhn+enhn−Mg≤δ1−enMhn+enhn−g≤δ1−enMhn−Mg+enhn−g≤δ1−enδhn−g+enhn−g≤δ1−1−δenhn−g≤δ31−1−δln1−1−δmn1−1−δengn−g.
Inductively, we get(18)‖gn+1−g‖≤δ3n+1∏t=1t=n1−1−δlt1−1−δmt1−1−δetg0−g.
Implementing one of the condition (CM 1)-(CM 3), we get(19)gn+1−g≤δ3n+1g0−g.
Because0<δ<1, consequently gn strongly converges to g. □


### 3.1. Stability

This subsection embraces almostM - stability of M - iteration (12) in the sense of Osilike.Theorem 3.
For a Banach spaceℬ⌣ and nonvoid, closed and convex set G⊆ℬ⌣, let M:G⟶G is Berinde weak contraction agreeing (3), then, iterative algorithm M - iteration (12) with real sequences ln,mn,en satisfying l≤ln,m≤mn,e≤en for some l,m,e∈0,1 and for all n∈ℕ∪0 is almost M - stable.Proof.
Consider an arbitrary sequencern⊆G. The sequence defined by M - iteration (12) is gn+1=θM,gn and u˜n=rn+1−θM,rn,n∈ℕ∪0.
Now, we have to prove that(20)∑n=0∞u˜n<∞implieslimn⟶∞rn=g.
Let∑n=0∞u˜n<∞ then by (18) and M - iteration (12), we get(21)rn+1−g≤rn+1−θM,rn+θM,rn−g≤u˜n+δ31−1−δln1−1−δmn1−1−δenrn−g≤u˜n+δ31−1−δl1−1−δm1−1−δern−g.
Definev˜n=rn−g and q=δ31−1−δl1−1−δm1−1−δe, then 0≤q<1. Therefore, we have(22)v˜n+1≤qv˜n+u˜n.
Thus from Lemma 1, we get the result.


## 3.1. Stability

This subsection embraces almostM - stability of M - iteration (12) in the sense of Osilike.Theorem 3.
For a Banach spaceℬ⌣ and nonvoid, closed and convex set G⊆ℬ⌣, let M:G⟶G is Berinde weak contraction agreeing (3), then, iterative algorithm M - iteration (12) with real sequences ln,mn,en satisfying l≤ln,m≤mn,e≤en for some l,m,e∈0,1 and for all n∈ℕ∪0 is almost M - stable.Proof.
Consider an arbitrary sequencern⊆G. The sequence defined by M - iteration (12) is gn+1=θM,gn and u˜n=rn+1−θM,rn,n∈ℕ∪0.
Now, we have to prove that(20)∑n=0∞u˜n<∞implieslimn⟶∞rn=g.
Let∑n=0∞u˜n<∞ then by (18) and M - iteration (12), we get(21)rn+1−g≤rn+1−θM,rn+θM,rn−g≤u˜n+δ31−1−δln1−1−δmn1−1−δenrn−g≤u˜n+δ31−1−δl1−1−δm1−1−δern−g.
Definev˜n=rn−g and q=δ31−1−δl1−1−δm1−1−δe, then 0≤q<1. Therefore, we have(22)v˜n+1≤qv˜n+u˜n.
Thus from Lemma 1, we get the result.


## 4. Comparison with Various Iterative Algorithms

In this section, we prove thatM - iteration (12) converges faster than other leading algorithms for the case of weak contraction mappings.Theorem 4.
For a Banach spaceℬ⌣ and nonvoid, closed and convex 1set G⊆ℬ⌣, let M:G⟶G is Berinde weak contraction agreeing (3). Let all the iterative algorithms listed in Table 1 together with M− iterative algorithm converges to common fixed point, g (say). Then, M - iteration (12) converges to g faster than those mentioned in Table 1.Proof.
Forn∈ℕ∪0, using Theorems 1 and 2 and inequality (18), we get(23)gn+1−g≤δ3n+1∏t=1t=n1−1−δlt1−1−δmt1−1−δetg0−g≤δ3n+11−1−δln1−1−δmn1−1−δeng0−g=μn.
Symbolising Noor iteration byg1,n, from [8],(24)f1,n−g=1−mng1,n+mnMg1,n−g≤1−mng1,n−g+δmng1,n−g,h1,n−g=1−lng1,n+lnMf1,n−g≤1−lng1,n−g+δlnf1,n−g,g1,n+1−g=1−eng1,n+enMh1,n−g≥1−eng1,n−g−enMh1,n−g≥1−eng1,n−g−δenh1,n−g≥1−eng1,n−g−δen1−lng1,n−g+δlnf1,n−g≥1−en−δen1−lng1,n−g−δ2lnen1−1−δmng1,n−g≥1−en1+δ1−ln1−δ1−mn1−δg1,n−g.≥1−1+δetg1,n−g⋯≥∏t=0t=n1−1+δetg1,0−g.
Therefore(25)gn+1−gg−g≤δ3n+11−l1−δn1−m1−δn1−e1−δn∏t=0t=n1−1+δet=W1,nsay.
Also, we get(26)W1,n+1W1,n=δ3n+1+11−l1−δn+11−m1−δn+11−e1−δn+1∏t=0t=n1−1+δet∏t=0t=n+11−1+δetδ3n+11−l1−δn1−m1−δn1−e1−δn=limn⟶∞δ31−l1−δ1−m1−δ1−e1−δ1−en+11+δ=δ31−l1−δ1−m1−δ1−e1−δ<1.
In accordance with the Ratio test, (26) implies ∑W1,n is convergent, which says limW1,n=0.
It validates thatgn converges faster than g1,n to g.
If normal-S - iterative algorithm [9] be denoted by g2,n, then(27)g2,n+1−g=M1−lng2,n+lnMg2,n−g≤δ1−lng2,n+lnMg2,n−g≤δ1−lng2,n−g+δlng2,n−g=δ1−1−δlng2,n−g≤δ1−1−δlg2,n−g.
Inductively, we get(28)g2,n+1−g≤δn+11−1−δlng2,0−g=μ2,n.
Therefore(29)limn⟶∞μnμ2,n=limn⟶∞δ3n+11−1−δln1−1−δmn1−1−δeng0−gδn+11−1−δlng2,0−g=limn⟶∞δ2n+11−1−δmn1−1−δen=0.
Hence,gn converges faster than g2,n to g.
Ifg3,n represents Abbas iterative scheme, then as shown in Theorem 3 of [10], we have(30)g3,n+1−g≤δn+11−1−δlmeng3,0−g=μ3,n.
Accordingly(31)μnμ4,n=δ3n+11−1−δln1−1−δmn1−1−δeng0−gδn+11−1−δlmeng3,0−g⟶0asn⟶∞.
This givesgn converges faster than g3,n to g.
Considering Theorem3 of [11] by Thakur et al. and using g4,n to mark the corresponding iterative scheme, we get(32)g4,n+1−g≤δn1−1−δmng4,0−g=μ4,n,μnμ4,n=δ3n+11−1−δln1−1−δmn1−1−δeng0−gδn1−1−δmng4,0−g=δ2n+31−1−δln1−1−δen⟶0asn⟶∞.
Which establishes thatgn converges faster than g4,n to g.
As proved by Sintunavarat et al. ([12], Theorem 2)(33)g5,n+1−g≤δn+11−1−δleng5,0−g=μ5,n,where g5,n denotes Varat’s iterative algorithm.
Therefore,(34)μnμ5,n=δ3n+11−1−δln1−1−δmn1−1−δeng0−gδn+11−1−δleng5,0−g⟶0asn⟶∞.
Hence,gn converges faster than g5,n to g.
Theorem4 of [6] by Ullah et al. elucidates that(35)g6,n−g≤δ2n+1g6,0−g=μ6,nwhere g6,n signifies AK iteration, which further gives(36)μnμ6,n=δ3n+11−1−δln1−1−δmn1−1−δeng0−gδ2n+1g6,0−g=δn+11−1−δln1−1−δmn1−1−δeng0−gg6,0−g⟶0asn⟶∞.
Consequently,gn converges faster than g6,n to g.
Letg7,n stands for F∗ iteration. Theorem 1 of [13] conveys,(37)g7,n+1−g≤δ2n+1g7,0−g=μ7,n.
Therefore(38)μnμ6,n=δ3n+11−1−δln1−1−δmn1−1−δeng0−gδ2n+1g7,0−g=δn+11−1−δln1−1−δmn1−1−δeng0−gg7,0−g⟶0asn⟶∞.
Hence,gn converges even faster than g7,n to g.
Also, in [12], it is shown that Varat iterative scheme has faster rate of convergence than Picard, Mann, Ishikawa and S - iteration. Henceforth Equations (26) - (38) confirms M - iteration converges faster than all the approximation procedures enlisted in Table 1.Table 1
Table listing different iterative algorithms.
Picard [2]g0∈Ggn+1=Mgn,n∈ℕ∪0Mann [3]g0∈Ggn+1=1−engn+enMgn,n∈ℕ∪0Ishikawa [4]g0∈Ggn+1=1−engn+enMhnhn=1−lngn+lnMgn,n∈ℕ∪0Noor [8]g0∈Ggn+1=1−engn+enMhnhn=1−lngn+lnMfnfn=1−mngn+mnMgn,n∈ℕ∪0S (see [5])g0∈Ggn+1=1−enMgn+enMhnhn=1−lngn+lnMgn,n∈ℕ∪0Normal - S [9]g0∈Ggn+1=M1−engn+enMgn,n∈ℕ∪0Abbas [10]g0∈Ggn+1=1−enMhn+enMfnhn=1−lnMgn+lnMfnfn=1−mngn+mnMgn,n∈ℕ∪0Thakur et al. [11]g0∈Ggn+1=1−enMfn+enMhnhn=1−lnfn+lnMfnfn=1−mngn+mnMgn,n∈ℕ∪0Varat [12]g0∈Ggn+1=1−enMhn+enMfnhn=1−lngn+lnfnfn=1−mngn+mnMgn,n∈ℕ∪0K∗ (Ullah et al.) [6]gn+1=Mhn , g0∈Ghn=M1−lnfn+lnMfnfn=1−mngn+mnMgn,n∈ℕ∪0F∗ [13]g0∈Ggn+1=Mhnhn=M1−lngn+lnMgn,n∈ℕ∪0


### 4.1. Comparison Results

In this section, we give some examples to strengthen the above claim thatM - iteration converges faster than various interesting algorithms existing in the literature.Example 1.
Let us take Banach spaceℬ⌣=ℝ equipped with usual norm. We define a self map M on a subset G=0,100 of Banach space ℬ⌣ by:(39)Mx=x−tan−1xfor allx∈G.
It can be eventually establish thatM is a weak contraction in sense of Berinde with unique fixed point g=0. Choosing control sequences ln=0.6021,mn=0.8654,en=0.8453, initial guess g0=45, using Matlab R 2019(b) software, we prove our claim (see Figure 1 and Table 2, Table 3 for your reference).Figure 1
Convergence behaviour of approximation procedures corresponding to example4.2.Table 2
Table depicting comparison of iterative algorithms in context of Example4.2.
Iter.No.MF∗UllahVaratThakurAbbas137.849440.972239.634042.194441.967941.9550230.721536.950534.279439.391638.939038.9134323.627732.936128.940236.591935.913935.8755416.591428.931123.622433.795932.893332.842259.672424.938018.336331.004229.877929.814163.120820.960813.102428.217526.868826.792575.7901e617.00597.969425.436823.867523.77878013.08443.109222.663620.876020.7749909.21810.021919.899517.897217.78401005.4594017.147314.935414.8105.....................4900000050000000Table 3
Table depicting comparison of iterative algorithms in context of Example4.2.
Iter.No.Normal-SSNoorIshikawaMannPicard142.143142.663742.221942.903243.691043.4514239.289040.329239.446540.808042.382541.9036336.438337.996836.674338.714441.074740.3567433.591335.666833.905836.622639.767538.8107530.748833.339531.141534.532938.460937.2656627.911731.015228.382432.445637.155135.7217725.080928.694425.629230.360835.850134.1789822.258026.377722.883628.279034.545832.6373919.445024.065820.147226.200633.242531.09721016.644921.759617.422724.126231.940129.5585.....................49005.5178e−271.2394e−221.0624e−11050008.5360e−281.9174e−231.6436e−120Example 2.
Letℬ⌣ and G be same as in Example 1: we define M:G⟶G by(40)Mx=x2−9x+45for allx∈G.
HereM is a weak contraction with unique fixed point g=5. Through Matlab 2019(b)software, it is shown that M - iteration converges faster than other iterative algorithms by selecting control sequences ln=0.7021,mn=0.8484,en=0.7558 starting with guess value gn=35 (see Figure 2 and Table 4, Table 5 for your reference).Figure 2
Convergence behaviour of approximation procedures corresponding to example4.3.Table 4
Table depicting comparison of iterative algorithms in context of Example4.3.
Iter No.MF∗UllahVaratThakurAbbas116.163524.087320.793927.727126.797426.753125.188913.99708.749820.708218.950718.867535.00006.54255.020314.158811.851211.745345.00005.01395.00008.64526.67666.603355.00005.00015.00005.54435.08285.0745655.00005.00005.02175.00175.0015755.00005.00005.00065.00005.0000855.000055.00005.00005.0000955.000055.00005.00005.0000105555.00005.00005.0000.....................4955555550555555Table 5
Table depicting comparison of iterative algorithms in context of Example4.3.
Iter No.Normal-SSNoorIshikawaMannPicard127.851528.760327.974929.760831.903530.9031220.945822.693821.216624.641028.841026.8677314.482316.911914.950919.700025.820222.914348.971911.65739.732115.047622.852019.074555.68297.50936.589810.904619.951515.400265.03155.42345.45017.703817.140811.981875.00105.02955.12235.919714.45308.984885.00005.00165.03295.266111.93936.698095.00005.00015.00885.07369.67965.4389105.00005.00005.00245.02017.79175.0628..................4955555550555555Remark 4.1.
It can be easily infer from the above two examples that in the category of weak contractions, Picard iterative scheme has good speed of convergence when compared to Mann and Ishikawa iterative schemes (see Table6).Table 6
Table enlisting count of iterations involved in Example4.2 and Example 4.3.
IterativeIteration CountIteration CountProcedureFor example4.2For example4.3Picard3425Mann43543Ishikawa42234Noor41632S2418Normal-S1916Abbas1814Thakur1814Varat2016Ullah108F∗1410M86


## 4.1. Comparison Results

In this section, we give some examples to strengthen the above claim thatM - iteration converges faster than various interesting algorithms existing in the literature.Example 1.
Let us take Banach spaceℬ⌣=ℝ equipped with usual norm. We define a self map M on a subset G=0,100 of Banach space ℬ⌣ by:(39)Mx=x−tan−1xfor allx∈G.
It can be eventually establish thatM is a weak contraction in sense of Berinde with unique fixed point g=0. Choosing control sequences ln=0.6021,mn=0.8654,en=0.8453, initial guess g0=45, using Matlab R 2019(b) software, we prove our claim (see Figure 1 and Table 2, Table 3 for your reference).Figure 1
Convergence behaviour of approximation procedures corresponding to example4.2.Table 2
Table depicting comparison of iterative algorithms in context of Example4.2.
Iter.No.MF∗UllahVaratThakurAbbas137.849440.972239.634042.194441.967941.9550230.721536.950534.279439.391638.939038.9134323.627732.936128.940236.591935.913935.8755416.591428.931123.622433.795932.893332.842259.672424.938018.336331.004229.877929.814163.120820.960813.102428.217526.868826.792575.7901e617.00597.969425.436823.867523.77878013.08443.109222.663620.876020.7749909.21810.021919.899517.897217.78401005.4594017.147314.935414.8105.....................4900000050000000Table 3
Table depicting comparison of iterative algorithms in context of Example4.2.
Iter.No.Normal-SSNoorIshikawaMannPicard142.143142.663742.221942.903243.691043.4514239.289040.329239.446540.808042.382541.9036336.438337.996836.674338.714441.074740.3567433.591335.666833.905836.622639.767538.8107530.748833.339531.141534.532938.460937.2656627.911731.015228.382432.445637.155135.7217725.080928.694425.629230.360835.850134.1789822.258026.377722.883628.279034.545832.6373919.445024.065820.147226.200633.242531.09721016.644921.759617.422724.126231.940129.5585.....................49005.5178e−271.2394e−221.0624e−11050008.5360e−281.9174e−231.6436e−120Example 2.
Letℬ⌣ and G be same as in Example 1: we define M:G⟶G by(40)Mx=x2−9x+45for allx∈G.
HereM is a weak contraction with unique fixed point g=5. Through Matlab 2019(b)software, it is shown that M - iteration converges faster than other iterative algorithms by selecting control sequences ln=0.7021,mn=0.8484,en=0.7558 starting with guess value gn=35 (see Figure 2 and Table 4, Table 5 for your reference).Figure 2
Convergence behaviour of approximation procedures corresponding to example4.3.Table 4
Table depicting comparison of iterative algorithms in context of Example4.3.
Iter No.MF∗UllahVaratThakurAbbas116.163524.087320.793927.727126.797426.753125.188913.99708.749820.708218.950718.867535.00006.54255.020314.158811.851211.745345.00005.01395.00008.64526.67666.603355.00005.00015.00005.54435.08285.0745655.00005.00005.02175.00175.0015755.00005.00005.00065.00005.0000855.000055.00005.00005.0000955.000055.00005.00005.0000105555.00005.00005.0000.....................4955555550555555Table 5
Table depicting comparison of iterative algorithms in context of Example4.3.
Iter No.Normal-SSNoorIshikawaMannPicard127.851528.760327.974929.760831.903530.9031220.945822.693821.216624.641028.841026.8677314.482316.911914.950919.700025.820222.914348.971911.65739.732115.047622.852019.074555.68297.50936.589810.904619.951515.400265.03155.42345.45017.703817.140811.981875.00105.02955.12235.919714.45308.984885.00005.00165.03295.266111.93936.698095.00005.00015.00885.07369.67965.4389105.00005.00005.00245.02017.79175.0628..................4955555550555555Remark 4.1.
It can be easily infer from the above two examples that in the category of weak contractions, Picard iterative scheme has good speed of convergence when compared to Mann and Ishikawa iterative schemes (see Table6).Table 6
Table enlisting count of iterations involved in Example4.2 and Example 4.3.
IterativeIteration CountIteration CountProcedureFor example4.2For example4.3Picard3425Mann43543Ishikawa42234Noor41632S2418Normal-S1916Abbas1814Thakur1814Varat2016Ullah108F∗1410M86


## 5. Data Dependency

The present section marks the data dependency result forM - iteration when applied to weak contractions. Here, instead of computing the fixed point of an operator, we approximate the operator with a given weak contraction operator for which it is possible to compute the fixed point and therefore, approximate the fixed point of an initial operator.Theorem 5.
LetM be a weak contraction with Q an approximate operator of it and gn be a sequence governed by M - iterative procedure (12) for M. Define sequence qn for Q as follows:(41)qn+1=Q1−enQsn+ensn,sn=Q1−lntn+lnQtn,tn=Q1−mnqn+mnQqn,n∈ℕ∪0,where ln,mn,en are sequences in 0,1 satisfying 1/2≤ln, 1/2≤mn, 1/2≤en for all n∈ℕ∪0 and ∑n=0∞mn=∞,∑n=0∞ln=∞,∑n=0∞en=∞. If Mg=g and Qq=q such that qn⟶q as n⟶∞, then(42)g−q≤14ε1−δ,where ε>0 is a fixed constant.Proof.
Consider(43)gn+1−qn+1=M1−enMhn+enhn−Q1−enQsn+ensn≤M1−enMhn+enhn−M1−enQsn+ensn+M1−enQsn+ensn−Q1−enQsn+ensn≤δ1−enMhn+enhn−1−enQsn+ensn+L1−enMhn+enhn−M1−enMhn+enhn+ε≤δ1−enMhn−Qsn+δenhn−sn+Lδ+1−δenhn−Mhn+ε≤δ1−enMhn−Msn+Msn−Qsn+δenhn−sn+Lδ+1−δenhn−Mhn+ε≤δ1−enδhn−sn+Lhn−Mhn+ε+δenhn−sn+Lδ+1−δenhn−Mhn+ε≤δ1−1−δ1−enhn−sn+L1−1−2δ1−enhn−Mhn+δ1−enε+ε.
Let us take(44)hn−sn=M1−lnfn+lnMfn−Q1−lntn+lnQtn≤M1−lnfn+lnMfn−M1−lntn+lnQtn+M1−lntn+lnQtn−Q1−lntn+lnQtn,hn−sn≤δ1−lnfn+lnMfn−1−lntn+lnQtn+L1−lnfn+lnMfn−M1−lnfn+lnMfn+ε≤δ1−lnfn−tn+δlnMfn−Qtn+L1−1−δln1+δ‖fn−Mfn‖+ε≤δ1−lnfn−tn+δlnMfn−Mtn+δlnMfn−Qtn+L1−1−δln1+δfn−Mfn+ε≤δ1−lnfn−tn+δlnδfn−tn+Lfn−Mfn+δlnε+L1−1−δln1+δfn−Mfn+ε≤δ1−1−δlnfn−tn+δlnLfn−Mfn+δlnε+L1−1−δln1+δfn−Mfn+ε,(45)fn−tn=M1−mngn+mnMgn−Q1−mnqn+mnQqn≤M1−mngn+mnMgn−M1−mnqn+mnQqn+ε≤δ1−mngn+mnMgn−1−mnqn+mnQqn+L1−mngn+mnMgn−M1−mngn+mnMgn+ε,fn−tn≤δ1−mngn−qn+δmnMgn−Qqn+L1−1−δmn1+δgn−g+ε≤δ1−1−δmngn−qn+δmnLgn−Mgn+L1−1−δmn1+δgn−g+δmnε+ε.
It follows from (43), (44) and (45),(46)gn+1−qn+1≤δ21−1−δln1−1−δ1−enfn−tn+δ2lnL1−1−δ1−enfn−Mfn+δ2ln1−1−δ1−enε+δL1−1−δln1−1−δ1−en1+δfn−Mfn+δ1−1−δ1−enε+L1−1−2δ1−enhn−Mhn+δ1−enε+ε≤δ31−1−δln1−1−δmn1−1−δ1−engn−qn+δ31−1−δln1−1−δ1−enmnLgn−Mgn+δ2L1−1−δln1−1−δmn1−1−δ1−en1+δgn−g+δ31−1−δln1−1−δ1−enmnε+δ21−1−δln1−1−δ1−enε+δ2lnL1−1−δ1−enfn−Mfn+δ2ln1−1−δ1−enε+δL1−1−δln1−1−δ1−en1+δfn−Mfn+δ1−1−δ1−enε+L1−1−2δ1−enhn−Mhn+δ1−enε+ε.
Sinceδ<1, therefore, δ2<1,δ3<1. Also, ln,mn,en<1, which gives, 1−1−δln<1,1−1−δmn<1 and 1−1−δ1−en<1. We get(47)gn+1−qn+1≤1−1−δlngn−qn+mnLgn−Mgn+2Lgn−g+mnε+ε+lnLfn−Mfn+lnε+2Lfn−Mfn+Lhn−Mhn+4ε≤1−1−δlngn−qn+2lnLgn−Mgn+4lnLgn−g+2lnLfn−Mfn+4lnLfn−Mfn+2lnLhn−Mhn+14lnε.
Define(48)a`n≔gn−qn;,b`n≔ln1−δ;,c`n:=2Lgn−Mgn+4Lgn−g+6Lfn−Mfn+2Lhn−Mhn+14ε1−δ.
Thus, equation (46) can be rewritten as(49)a`n+1≤1−b`na`n+b`nc`n.
This calls for the application of lemma (2.3), which gives(50)0≤limsupn⟶∞gn−qn≤limsupn⟶∞2Lgn−Mgn+4Lgn−g+6Lfn−Mfn+2Lhn−Mhn+14ε1−δ=14ε1−δ.
Using Theorem2 and given hypotheses, we get(51)g−q≤14ε1−δ.
The essence of Theorem5 can be realised from the following example which has been inspired from [13].Example 3.
Let us take Banach spaceℬ⌣=ℝ equipped with usual norm. We define a self map M on a subset G=−1,1 of ℬ⌣ by:(52)Mx=1329sin13x29,if−1≤x<0,−1329sin13x29,if0≤x≤1.
It can be observed thatM is a weak contraction in the sense of Berinde with δ∈169/841,1 and a unique fixed point g=0∈G.
We further define another self mapQ on G as follows:(53)Qx=x−0.095.01+x+0.2397.06−x−0.657601.16−x+0.87129996.03,if−1≤x<0,−x6.02−x−0.53108.95−x+0.257598.27+x−0.67230050.03,if0≤x≤1.
Hereq=−0.0224 is a unique fixed point lying in the domain of Q . Using MATLAB 2019(b) software, we get maxx∈GMx−Qx=0.0272. Therefore, if we choose ε=0.0272, then as per Berinde’s definition of an approximate operator (Definition 6), Q is an approximate operator of G. Moreover, distance between fixed points of M and Q is g−q=0.0224.
Forn∈ℕ∪0, choosing ln=mn=en=n+1/n+2 in (36) and using (48), we obtain approximated fixed point of the operator Q as visualized from Table 7.
However, using Theorem5, we compute(54)g−q≤140.02721−169/841=0.4765.
From here, we can conclude that without knowing the fixed point ofQ and in fact, without calculating it, we can approximate its fixed point by directly applying Theorem 5.Table 7
Approximated fixed point of operatorQ by applying M iterative procedure.
Iteration CountValues obtained using algorithm (5.1)10.94842−0.02233−0.0224


## 6. Application to Nonlinear Caputo Type Fractional Differential Equation

Fractional derivatives have been used through and through for mathematical modelling over recent years. For instance, numerous real world problems, signal processing, image restoration, determination of fluid level, biological algorithms, traffic flow, telecommunication etc., are being modelled in terms of nonlinear fractional order differential equations, majority of which do not possess exact solution and call for the approximate numerical solutions. Caputo et al. in [28] developed a novel fractional differential operator with a special feature of non singular kernel having exponential decay due to which it has attracted many researchers (see [29‐31]).(55)Dt♭u`t=ςt,u`t0<♭<1,u`0=u`0.where Dt♭ denotes Caputo type fractional differential equation, ς:0,1×ℝ⟶ℝ is continuous and u` is real valued with t∈0,1.The time dependent Caputo type fractional differential operator is defined by(56)Dt♭u`t=2−♭A♭21−♭∫0texp−♭t−ω1−♭U′`ζdζ,t≥0,0<♭<1,where A♭ is a normalization function satisfying A0=A1=1. The following lemma is due to [31].Lemma 4.
The initial value problem (IVP)(57)Dt♭u`t=ςt0<♭<1,u`0=u`0∈ℝ.
Possess following integeral solution(58)u`t=u`0+21−♭2−♭A♭ςt−ς0+2♭2−♭A♭∫0tςζdζ.
Letα=0,1 and Cα be the space of continuous functions defined on 0,1. Let the Banach space ℬ⌣=u`t|u`t∈Cα, with norm u`ℬ⌣=maxt∈αu`t.
By Lemma 4, IVP (55) can be expressed as(59)u`t=u`0+21−♭2−♭A♭ςt,u`t−ς0,u`0+2♭2−♭A♭∫0tςζ,u`ζdζ.
Let us define the operatorM:ℬ⌣⟶ℬ⌣ by(60)Mu`t=u`0+21−♭2−♭A♭ςt,u`t−ς0,u`0+2♭2−♭A♭∫0tςζ,u`ζdζ.
Then the solution of IVP (55) is equivalent to the fixed point operator M.
We prove the following theorem.Theorem 6.
For0≤α≤1, let ς:0,1×ℝ⟶ℝ be continuous function such that:(i
ςt,u`t−ςt,v`t≤δu`t−v`t+Lu`t−Mu`t(ii)
maxt∈α21−♭/2−♭A♭+2♭/2−♭A♭t<1.
Then the IVP (55) has unique solution which is fixed point of weak contraction M.Proof.
We denote fixed point operatorM by(61)Mu`t=K+21−♭2−♭A♭ςt,u`t+2♭2−♭A♭∫0tςζ,u`ζdζ,where K=u0`−21−♭/2−♭A♭ς0,u0`.
Foru`t,v`t∈ℬ⌣, we have(62)Mu`t−Mv`t=|K+21−♭2−♭A♭ςt,u`t+2♭2−♭A♭∫0tςζ,u`ζdζ−K+21−♭2−♭A♭ςt,v`t+2♭2−♭A♭∫0tςζ,v`ζdζ|≤21−♭2−♭A♭ςt,u`t−ςt,v`t+2♭2−♭A♭∫0tςζ,u`ζ−ςζ,v`ζdζ≤21−♭2−♭A♭δu`t−v`t+Lu`t−Mu`t+2♭2−♭A♭∫0tδu`ζ−v`ζ+Lu`ζ−Mu`ζdζ≤21−♭2−♭A♭δ+2♭2−♭A♭δ∫0tdζu`t−v`tℬ⌣+21−♭2−♭A♭Lu`t−Mu`t+2♭2−♭A♭L∫0tu`ζ−Mu`ζdζ,Mu`t−Mv`t≤21−♭2−♭A♭δ+2♭2−♭A♭δtu`t−v`tℬ⌣+21−♭2−♭A♭L+2♭2−♭)A(♭Ltu`t−Mu`tℬ⌣maxt∈αMu`t−Mv`t≤maxt∈α21−♭2−♭A♭+2♭2−♭A♭tδu`t−v`tℬ⌣+21−♭2−♭A♭+2♭2−♭A♭tLu`t−Mu`tℬ⌣Mu`t−Mv`tℬ⌣≤δu`t−v`tℬ⌣+Lu`t−Mu`tℬ⌣.
ThusM:ℬ⌣⟶ℬ⌣ is a weak contraction. In view of Theorem 1, we get M has a unique fixed point which is the solution of IVP (55).
Losada et al. [31] proved the following theorem in context of nonlinear Caputo type fractional differential equation.Theorem 7.
For0<♭<1,T>0, suppose the following conditions hold:
(CM1)ς:0,T×ℝ⟶ℝ is a continuous function such that there exists N∈0,1 satisfying,
ςt,τ1−ςt,τ2≤Nτ1−τ2 for all. τ1,τ2∈ℝ,
(CM2)21−♭/2−♭A♭+2♭/2−♭A♭T<1.
Then the IVP (55) has unique solution (59) on $ C0,T$.
We now prove that - iteration converges to the solution of IVP (55)Theorem 8.
If the conditionsCM1−CM2 are satisfied, then for ∑mn=∞, the sequence gn generated by M - iterative procedure converges to the solution say u# of IVP (55) on C0,T, for some g0∈C0,T.Proof.
Let us define the operatorM by:(63)Mu`t=u`0+21−♭2−♭A♭ςt,u`t−ς0,u`0+2♭2−♭A♭∫0tςζ,u`ζdζ.
Which can be rewritten as:(64)Mu`t=K+21−♭2−♭A♭ςt,u`t+2♭2−♭A♭∫0tςζ,u`ζdζ,where K=u0`−21−♭/2−♭Aγς0,u0`
In view of Theorem7, the solution u# of IVP (55) is fixed point of operator M.
Letgn be a sequence defined by M - iterative procedure on fixed point operator M. We shall establish that gn⟶u# as n⟶∞.
Definetn=1−mngn+Mgn,n∈ℕ∪0
Using conditionCM1, we have:(65)tn−u#=1−mngn+mnMgn−u#≤1−mngn−u#+mnMgn−Mu#=1−mngn−u#+mn|21−♭2−♭A♭ςt,gnt−ςt,u#t+2♭2−♭A♭∫0tςt,gnt−ςt,u#tdt|,tn−u#≤1−mngn−u#+mn21−♭2−♭A♭ςt,gnt−ςt,u#t+2♭2−♭A♭∫0tςt,gnt−ςt,u#tdt≤1−mngn−u#+mn21−♭2−♭A♭Ngn−u#+2♭2−♭A♭∫0tNgn−u#dttn−u#≤1−mngn−u#+21−♭2−♭A♭+2♭2−♭A♭TmnNgn−u#≤1−mn1−Ngn−u#.
Using (65) and condition (CM1), we get:(66)fn−u#=Mtn−Mu#=21−♭2−♭A♭ςt,tnt−ςt,u#t+2♭2−♭A♭∫0tςt,tnt−ςt,u#tdt≤21−♭2−♭A♭ςt,tnt−ςt,u#t+2♭2−♭A♭∫0tςt,tnt−ςt,u#tdt≤21−♭2−♭A♭Ntn−u#+2♭2−♭A♭∫0tNtn−u#dtfn−u#≤21−♭2−♭A♭+2♭2−♭A♭TNtn−u#.
Due to conditionCM2, we obtain that:(67)fn−u#≤N1−mn1−Ngn−u#
If we definekn=1−lnfn+lnMfn, then in similar fashion as above, we can obtain:(68)kn−u#≤1−ln1−Nfn−u#≤1−ln1−NN1−mn1−Ngn−u#
Therefore using conditionCM1−CM2,(69)hn−u#=Mkn−Mu#≤21−♭2−♭A♭+2♭2−♭A♭TNkn−u#≤N21−ln1−N1−mn1−Ngn−u#.
Likewise, we retrieve:(70)gn+1−u#≤N31−ln1−N1−mn1−N1−en1−Ngn−u#.
SinceN<1,ln<1,en<1, which gives N3<1,1−ln1−N<1 and 1−en1−N<1.
Therefore (70) becomes:(71)gn+1−u#≤1−mn1−Ngn−u#.
Define:(72)d˜n=mn1−N,e˜n=gn−u#.where d˜n∈0,1 such that ∑d˜n=∞.
Therefore, (71) can be viewed as:(73)e˜n+1=1−d˜ne˜n.
Executing Lemma 3, we getM - iteration converges to the solution u# of IVP (55).


## 7. Conclusion

This article is endowed with introduction of a new three - step iterative technique calledM - iteration which converges strongly to the invariant point of almost - contraction mappings in Banach space. Followed by almost M - stability of M approximation procedure, we presented that this novel technique has fast rate of convergence as compare to other vital procedures for the class of weak contractions which can be visualized from two numerical illustrations, namely. Example 1 and Example 2 presented here. We further established data dependency of our algorithm and demonstrated that Caputo type fractional differential equation possess a solution which, under certain circumstances, is a fixed point of Berinde contractions. Lastly, we showed that M - iterative procedure converges to the solution of fractional differential equation.Nevertheless, few obvious questions are engrossed in the field:(1)
Can one define an iterative technique which is even faster thanM iterative procedure for the class of weak contractions in the setting of Banach space?(2)
DoesM - iteration strongly converges to the fixed point of weak contractions in the framework of a space weaker than Banach space (quasi-Banach space or metric space, for instance)?(3)
DoesM - iterative algorithm converges for certain other class of mappings like quasi - nonexpansive or enriched contractions ?



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