# Estimating Fixed Points via New Iterative Scheme with an Application

Authors: Mohd Jubair; Javid Ali; Santosh Kumar
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we introduce a new scheme and prove convergence results for nonexpansive mappings as well as for weak contractions in the frame of Banach spaces. Moreover, we prove analytically and numerically that the proposed scheme converges to a fixed point of a weak contraction faster than some known and leading schemes. Further, we prove that the new scheme is almost stable with respect to weak contraction. For supporting the main results, we give a couple of nontrivial numerical examples, and the visualization is shown by using the Matlab program. Finally, the solution of a nonlinear fractional differential equation is approximated by operating the main result of the paper.


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## Body

## 1. Introduction

Throughout the paper,ℤ0+ denotes the set of nonnegative integers. Let E be a nonempty subset of a Banach space Z, K:E⟶E is a mapping and FK=w∈E:Kw=w. A mapping K:E⟶E is said to be nonexpansive if for each x,y∈E,
(1)Kx−Ky≤x−y.A self-mapK on E is said to be a weak contraction [1] if for all x,y∈E∃ a constant δ∈0,1 and some L≥0 such that
(2)Kx−Ky≤δx−y+Lx−Ky,Kx−Ky≤δx−y+Ly−Kx.Theorem 1 (see [1]).
LetK:Z⟶Z be a weak contraction with δ∈0,1 and some L≥0 such that
(3)Kx−Ky≤δx−y+Lx−Kx,∀x,y∈Z.
Then,K has a unique fixed point, and Picard sequence converges to the fixed point.But if we take an initial guess different from a fixed point in case of nonexpansive mapping, it is to be noted that the Picard iterative scheme fails to converge to the fixed points of such mappings; hence, we need some other iterative schemes. In the sequel, many authors gave the generalizations of nonexpansive mapping and proved existence and convergence results in linear space, e.g., see [2].However, to find the fixed points of numerous nonlinear mappings is not an easy task. So, to overcome this kind of problem, several researchers constructed iterative schemes to approximate fixed points of mappings. A few of them are Picard-S [3], Thakur-New [4], and others [5–10].Here, we consider some iterative schemes which are frequently used to approximate the fixed points of nonlinear mappings introduced by Picard [11], Mann [12], Ishikawa [13], Noor [14], and Agarwal et al. (S) [15], respectively, where the sequence τn is developed by an arbitrary point τ0∈E as follows:
(4)τn+1=Kτ0,n∈ℤ0+,(5)τn+1=1−θnτn+θnKτn,n∈ℤ0+,(6)τn+1=1−θnτn+θnKσn,σn=1−μnτn+μnKτn,n∈ℤ0+,(7)τn+1=1−θnτn+θnKσn,σn=1−μnτn+μnKξn,ξn=1−γnτn+γnKτn,n∈ℤ0+,(8)τn+1=1−θnKτn+θnKσn,σn=1−μnτn+μnKτn,n∈ℤ0+,where θn, μn, and γn are sequences in 0,1.Motivated by the previous work, we define a new iterative scheme for finding the fixed point of a weak contraction, where the sequenceτn is developed iteratively by τ0∈E and
(9)τn+1=K1−θnσn+θnKσn,σn=1−μnKτn+μnKξn,ξn=1−γnτn+γnKτn,n∈ℤ0+,where θn, μn, and γn are sequences in 0,1.Our main focus is to consider those iterative schemes which save time when we approximate fixed points of mappings. Berinde [16] gave the following definitions about the rate of convergence of iterative schemes which is defined as follows:Definition 2.
Letαn and βn be two sequences of positive numbers that converge to α and β, respectively. Assume that
(10)ℓ=limn⟶∞αn−αβn−β.(i)
Ifℓ=0, then αn converges to α faster than βn to β(ii)
If0<ℓ<∞, then αn and βn have the same rate of convergenceDefinition 3.
Considerτn and σn as two fixed point iterative schemes both converging to the same point t of a mapping with error estimates
(11)τn−t≤αn,σn−t≤βn.
Iflimn⟶∞αn/βn=0, then τn converges faster than σn.Now, we discuss another concept related to iterative schemes called stability. Letτn be a theoretical sequence and tn an approximate sequence which is due to rounding errors and numerical approximation of functions. We say that the approximate sequence tn converges to the fixed point of mapping K if and only if the given fixed point iterative scheme would be stable. Because of this fact, the concept of stability for a fixed point iterative scheme was coined by Ostrowski [17] which defined as follows.Definition 4 (see [17]).
Letτn+1=fK,τn be an iteration procedure, converging to a fixed point w, which is said to be K-stable or stable with respect to K, if for ϵn=tn+1−fK,tn, n∈ℤ0+, we have limn⟶∞ϵn=0⇔limn⟶∞tn=w, where tn is an approximate sequence in a subset E of a Banach space Z.
In 1998, a weaker concept of stability, called almost stability, was coined by Osilike [18] which is defined as follows.Definition 5 (see [18]).
Letτn+1=fK,τn be an iteration procedure, converging to fixed point w, which is said to be almost K-stable or almost stable with respect to K, if for ϵn=tn+1−fK,tn, n∈ℤ0+, we have ∑n=0∞ϵn<∞⇒limn⟶∞tn=w.Remark 6 (see [18]).
It can be easily seen that anyK-stable iteration procedure is almost K-stable, but reverse may fail.Lemma 7 (see [19]).
Let0≤δ<1 and ϵn and un be any two sequences of nonnegative numbers satisfying un+1≤δun+ϵn, n∈ℤ0+. If ∑n=0∞ϵn<∞, then ∑n=0∞un<∞.


## 2. Preliminaries

Definition 8.
A self-mappingK on a Banach space Z is said to be demiclosed at y, if for any sequence τn which converges weakly to x, and if the sequence Kτn converges strongly to y, then Kx=y.Definition 9.
A sequenceτn in a normed space Z is said to be weakly convergent (denoted by ⇀) if ∃ an element x∈Z such that
(12)limn⟶∞Kτn=Kx,∀K∈Z∗.Definition 10.
A Banach spaceZ is said to satisfy Opial’s property [20] if for any τn⇀f in Z⇒limn⟶∞infτn−f<limn⟶∞infτn−g for all g∈Z with g≠f.Lemma 11 (see [21]).
LetE be a nonempty closed and convex subset of a uniformly convex Banach space Z and K a nonexpansive mapping on E. Then, I−K is demiclosed at zero.Lemma 12 (see [22]).
LetZ be a uniformly convex Banach space and 0<a≤ωn≤b<1 for all n∈ℕ. Assume that τn and σn are two sequences in Z such that limn⟶∞supτn≤ω, limn⟶∞supσn≤ω, and limn⟶∞supωnτn+1−ωnσn=ω holds, for some ω≥0. Then, limn⟶∞τn−σn=0.Definition 13 (see [23]).
A mappingK:E⟶E is said to satisfy property A, if ∃ a nondecreasing mapping ψ:0,∞⟶0,∞ with ψ0=0 and ψz>0, ∀z>0, such that dx,Kx≥ψdx,FK, ∀x∈E.


## 3. Convergence Result for Weak Contractions

Throughout this section, we presume thatE is a nonempty closed and convex subset of a normed linear space Z and K:E⟶E a weak contraction satisfying (3) with FK≠∅.Theorem 14.
Letτn be a sequence developed by new iterative scheme (9), then τn converges to a fixed point of K.Proof.
From (9), for any w∈FK,
(13)ξn−w=1−γnτn+γnKτn−w≤1−γnτn−w+δγnτn−w=1−1−δγnτn−w,σn−w=1−μnKτn+μnKξn−w≤δ1−μnτn−w+δμnξn−w≤δ1−1−δμnγnτn−w,τn+1−w=K1−θnσn+θnKσn−w≤δ1−θnσn+θnKσn−w≤δ21−1−δθn1−1−δμnγnτn−w.
By using the fact that0<1−1−δθn≤1 and 0<1−1−δμnγn≤1, we have
(14)τn+1−w≤δ2τn−w.
Inductively, we get(15)τn+1−w≤δ2n+1τ0−w.
Since0<δ<1, τn converges to w.
Now, we prove almost stability of new iterative scheme (9) with respect to a weak contraction.Theorem 15.
Letτn be a sequence developed by iterative scheme (9), then τn is almost K-stable.Proof.
Considertn an approximate sequence of τn in E. Suppose sequence defined by (9) is τn+1=fK,τn converging to a fixed point w (by Theorem 14) and ϵn=tn+1−fK,tn, n∈ℤ0+. Now, we will prove that ∑n=0∞ϵn<∞⇒limn⟶∞tn=w.Let∑n=0∞ϵn<∞, then by iterative scheme (9), we have
(16)tn+1−w≤tn+1−fK,tn+fK,tn−w=ϵn+fK,tn−w≤ϵn+δ21−1−δθn1−1−δμnγntn−w.Since0<1−1−δθn≤1 and 0<1−1−δμnγn≤1 and using (16), we get
(17)tn+1−w≤ϵn+δ2tn−w.Defineun=tn−w, then
(18)un+1≤δ2un+ϵn.Since∑n=0∞ϵn<∞, by Lemma 7, we have ∑n=0∞un<∞. This implies limn⟶∞un=0, i.e., limn⟶∞tn=w. This shows that new iterative scheme (9) is almost K-stable.There is analytical comparison of the rate of convergence of iterative schemes with new iterative scheme (9) for weak contraction.Theorem 16.
Suppose that the sequenceτ1,n is introduced by Picard (4), τ2,n by Mann (5), τ3,n by Ishikawa (6), τ4,n by Noor (7), τ5,n by Agrawal (8), and τn by (9) iterative scheme which converges to the same point w. Then iterative scheme (9) converges faster than all the schemes (4)–(8) to a fixed point of K.Proof.
Using equation (15) of Theorem 14, we have
(19)τn+1−w≤δ2n+1τ0−w=αn,n∈ℤ0+.
From equation (7), we get
(20)ξn−w=1−γnτn+γnKτn−w≤1−1−δγnτn−w.
It can be easily seen that0<1−1−δγn≤1, so we get
(21)ξn−w≤τn−w.
Using (21), we obtain that
(22)σn−w=1−μnτn+μnKξn−w≤1−μnτn−w+δμnξn−w≤1−μnτn−w+δμnτn−w≤1−1−δμnτn−w.
Again, it can be easily seen that0<1−1−δμn≤1, so we get
(23)σn−w≤τn−w.
Using (23), we obtain that
(24)τn+1−w=1−θnτn+θnKσn−w≤1−θnτn−w+δθnσn−w≤1−θnτn−w+δθnτn−w≤1−1−δθnτn−w.
By using the fact that0<1−1−δθn≤1, we have
(25)τn+1−w≤τn−w.
Inductively, we get(26)τn+1−w≤τ0−w.
Let(27)τ4,n−w≤τ4,0−w=α4,n,n∈ℤ0+.
Then(28)αnα4,n=δ2n+1τ0−wτ4,0−w.
Thus,τn converges faster than τ4,n to w because 0<δ<1, then αn/α4,n⟶0 as n⟶∞.By applying a similar approach, we can also show that the rate of convergence of all the other leading iterative schemes tow is slower than iterative scheme (9).We embellish the following example to support our assertion.Example 17.
LetZ=ℝ2 be a Banach space with respect to the norm x=x1,x2=x1+x2 and E=x=x1,x2:x1,x2∈0,1×0,1 be a subset of Z. Let K:E⟶E be defined by
(29)Kx1,x2=12sinx1,14sinx2,ifx1,x2∈0,12×0,12,12x1,14x2,ifx1,x2∈12,1×12,1.
ThenK is a weak contraction satisfying (3) for δ=1/2=L, but K is not a contraction mapping.By Matlab 2015a, we exhibit that new iterative scheme (9) converges to a fixed point w=0,0 of the mapping K faster than the iterative schemes Picard, Mann, Ishikawa, Noor, and S with initial point τ0=0.20,0.40 and control sequences θn=0.35, μn=0.45, and γn=0.75, n∈ℤ0+, which can be easily seen in Tables 1 and 2 and Figure 1.Table 1
Comparison of speed of the convergence of different iterative schemes.
Iter.PicardMannIshikawa1(0.200000, 0.400000)(0.200000, 0.400000)(0.200000, 0.400000)2(0.099335, 0.097355)(0.164767, 0.294074)(0.156965, 0.282817)⋮19(0.000001, 0.000000)(0.006241, 0.001655)(0.002593, 0.000796)20(0.000000, 0.000000)(0.005149, 0.001221)(0.002037, 0.000564)⋮54(0.000000, 0.000000)(0.000007, 0.000000)(0.000001, 0.000000)55(0.000000, 0.000000)(0.000006, 0.000000)(0.000000, 0.000000)⋮67(0.000000, 0.000000)(0.000001, 0.000000)(0.000000, 0.000000)⋮68(0.000000, 0.000000)(0.000001, 0.000000)(0.000000, 0.000000)69(0.000000, 0.000000)(0.000000, 0.000000)(0.000000, 0.000000)70(0.000000, 0.000000)(0.000000, 0.000000)(0.000000, 0.000000)Table 2
Comparison of speed of the convergence of different iterative schemes.
Iter.NoorSNew scheme1(0.200000, 0.400000)(0.200000, 0.400000)(0.200000, 0.400000)2(0.154063, 0.280745)(0.091532, 0.086097)(0.034027, 0.013435)⋮8(0.032294, 0.033698)(0.000873, 0.000010)(0.000001, 0.000000)9(0.024894, 0.023670)(0.000402, 0.000002)(0.000000, 0.000000)⋮17(0.003104, 0.001403)(0.000001, 0.000000)(0.000000, 0.000000)18(0.002392, 0.000985)(0.000000, 0.000000)(0.000000, 0.000000)⋮50(0.000001, 0.000000)(0.000000, 0.000000)(0.000000, 0.000000)51(0.000000, 0.000000)(0.000000, 0.000000)(0.000000, 0.000000)⋮69(0.000000, 0.000000)(0.000000, 0.000000)(0.000000, 0.000000)70(0.000000, 0.000000)(0.000000, 0.000000)(0.000000, 0.000000)Figure 1
Graphical representation of iterative schemes.


## 4. Convergence Results for Nonexpansive Mapping

Throughout this section, we presume thatE is a nonempty, closed, and convex subset of a uniformly convex Banach space Z and K:E⟶E is a nonexpansive mapping. Now, we prove the following useful lemmas which are used to prove the next results of this section.Lemma 18.
Letτn be a sequence developed by new iterative scheme (9), then limn⟶∞τn−w exists for all w∈FK.Proof.
Supposew∈FK and τn∈E. From (9), we have
(30)ξn−w=1−γnτn+γnKτn−w≤τn−w,(31)σn−w=1−μnKτn+μnKξn−w≤1−μnτn−w+μnξn−w≤1−μnτn−w+μnτn−w≤τn−w.
Using (31), we get
(32)τn+1−w=K1−θnσn+θnKσn−w≤1−θnσn+θnKσn−w≤1−θnσn−w+θnσn−w=σn−w≤τn−w,
which exhibit thatτn−w is decreasing and bounded below. Therefore, limn⟶∞τn−w exists.Lemma 19.
LetFK≠∅ and τn be the iterative scheme developed by equation (9). Then limn⟶∞τn−Kτn=0.Proof.
Sincelimn⟶∞τn−w exists by Lemma 18 and it is given that FK≠∅ with w∈FK. Presume that limn⟶∞τn−w=c.
By the inequalities (30) and (31), we get
(33)limn⟶∞supξn−w≤c,(34)limn⟶∞supσn−w≤c,respectively. Since K is nonexpansive mapping, we have
(35)Kτn−w≤τn−w,Kσn−w≤σn−w,Kξn−w≤ξn−w.
Using (35), we get
(36)limn⟶∞supKτn−w≤c,(37)limn⟶∞supKσn−w≤c,(38)limn⟶∞supKξn−w≤c.
Since(39)τn+1−w=K1−θnσn+θnKσn−w≤1−θnσn+θnKσn−w≤1−θnσn−w+θnσn−w≤σn−w.
Now,(40)c=limn⟶∞infτn+1−w≤limn⟶∞infσn−w.
So that (33) and (40) give
(41)limn⟶∞σn−w=c,(42)c=limn⟶∞σn−w=limn⟶∞1−μnKτn+μnKξn−w=limn⟶∞1−μnKτn−w+μnKξn−wby using Lemma 12 and Inequality (36) and (38), we have
(43)limn⟶∞Kτn−Kξn=0.
Now,(44)σn−w=1−μnKτn+μnKξn−w≤Kτn−w+μnKξn−Kτn≤Kτn−w,which gives
(45)c≤limn⟶∞infKτn−w,using (36) and (45), we get
(46)limn⟶∞Kτn−w=c.
On the other hand, we have(47)Kτn−w≤Kτn−Kξn+Kξn−w≤Kτn−Kξn+ξn−w.
Applyingliminf on both sides, we get
(48)c≤limn⟶∞infξn−w,by using (34) and (48), we have
(49)limn⟶∞ξn−w=c.
So,(50)c=limn⟶∞ξn−w=limn⟶∞1−γnτn+γnKτn−w=limn⟶∞1−γnτn−w+γnKτn−w.
Using Lemma12 and Inequality (50), we get
(51)limn⟶∞τn−Kτn=0.
Now, we prove a weak convergence result for nonexpansive mapping.Theorem 20.
Presume thatZ enjoys Opial’s condition, then the sequence τn developed by iterative algorithm (9) converges weakly to a point of FK.Proof.
Letτn be a sequence with two subsequences τnj and τnk and l and m are two weak subsequential limits of τnj and τnk, respectively. From Lemmas 18 and 19, we get limn⟶∞τn−w exists and limn⟶∞τn−Kτṅ=0, respectively. Now, we have to show that τn cannot have different weak subsequential limits in FK. Also, from Lemma 11, I−K is demiclosed at 0. This implies that I−Kl=0, i.e., l=Kl, similarly m=Km. We have to show that l=m.
Let on contraryl≠m, by Opial’s condition, we have
(52)limn⟶∞τn−l=limnj⟶∞τnj−l<limnj⟶∞τnj−m=limn⟶∞τn−m=limnk⟶∞τnk−m<limnk⟶∞τnk−l=limn⟶∞τn−l,which is absurd, hence l=m. Consequently, τn⇀l∈FK.
There is a strong convergence result for nonexpansive mapping.Theorem 21.
Letτn be the sequence developed by equation (9). Then limn⟶∞infdτn,FK=0 if and only if τn converges to a point of FK, where dτn,FK=inf∥τn−w∥:w∈FK.Proof.
If the sequenceτn converges to a point w∈FK, then it is obvious that limn⟶∞infdτn,FK=0.
Now, for the first part takinglimn⟶∞infdτn,FK=0 for any fixed point w∈FK. From Lemma 18,
(53)limn⟶∞τn−wexists ∀w∈FK; therefore, limn⟶∞dτn,FK=0.
Now, our assertion is thatτn is a Cauchy sequence in E. Since limn⟶∞dτn,FK=0, and for a given α>0, there exists w0∈ℤ0+ such that for all n≥w0(54)dτn,FK<α2,infτn−w:w∈FK<α2.
Precisely,infτw0−w:w∈FK<α/2. Therefore, there exists w∈FK such that
(55)τw0−w<α2.
Now, form,n≥w0,
(56)τn+m−τn≤τn+m−w+τn−w≤τw0−w+τw0−w=2τw0−w<α.
Thus,τn is a Cauchy in E. Since E is closed, limn⟶∞τn=q for some q∈E. Now, limn⟶∞dτn,FK=0 implies dq,FK=0; hence, we get q∈FK.
We now prove a strong convergence result by applying propertyA.Theorem 22.
LetK:E⟶E be a nonexpansive mapping with property A. Then τn defined by (9) converges strongly to a fixed point of K.Proof.
From equation (51) of Lemma 19, we have
(57)limn⟶∞τn−Kτn=0.
Using (57) and property A, we get
(58)0≤limn⟶∞ψdτn,FK≤limn⟶∞τn−Kτn=0,limn⟶∞ψdτn,FK=0.
Sinceψ enjoy the conditions ψz>0 and ψ0=0, ∀z>0, then we obtain
(59)limn⟶∞dτn,FK=0.
So by Theorem21, we obtain the desired result.


## 5. An Illuminate Numerical Example

The purpose of this section is to present a numerical example to compare the rate of convergence for nonexpansive mapping.Example 23.
SupposeZ=ℝ3a Banach space with usual norm and letK:Z⟶Z be a mapping defined as
(60)Kx=Kx1,x2,x3=0,x1,x2,∀x=x1,x2,x3∈Z.
ThenK is nonexpansive, but not contraction.Proof.
Letx=x1,x2,x3 and y=y1,y2,y3∈Z. Then
(61)Kx−Ky=0,x1,x2−0,y1,y2=x1−y12+x2−y22≤x1−y12+x2−y22+x3−y32=x1,x2,x3−y1,y2,y3=x−y.
Hence,K is a nonexpansive mapping, but not contraction.Now, by taking control sequencesan=0.5, bn=0.4, and cn=0.3 with initial guess x1=0.5, x2=0.25, and x3=0.15, we can show that new iterative scheme (9) converges faster than all other leading iterative schemes which is shown in Tables 3 and 4 and Figure 2.Table 3
Numerical comparison of iterative schemes.
Iter.New schemeS1(0.500000, 0.250000, 0.150000)(0.500000, 0.250000, 0.150000)2(0.000000, 0.000000, 0.220000)(0.000000, 0.400000, 0.300000)3(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.320000)4(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)Table 4
Numerical comparison of iterative schemes.
Iter.IshikawaMann1(0.500000, 0.250000, 0.150000)(0.500000, 0.250000, 0.150000)2(0.250000, 0.275000, 0.250000)(0.250000, 0.375000, 0.200000)3(0.125000, 0.212500, 0.257500)(0.125000, 0.312500, 0.287500)⋮29(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000001)30(0.000000, 0.000000, 0.000000)(0.000000, 0.000000, 0.000000)Figure 2
Graphical representation of iterative schemes.


## 6. Application to Nonlinear Fractional Differential Equation

In recent years, many authors pointed out that derivatives and integrals of noninteger order are very suitable for the description of properties of various real materials, e.g., polymers. It has been shown that new fractional order models are more adequate than previously used integer order models. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional derivatives in comparison with classical integer order models, in which such effects are in fact neglected. The advantages of fractional derivatives become apparent in modelling mechanical and electrical properties of real materials, as well as in the description of rheological properties of rocks, and in many other fields. Most nonlinear fractional differential equations have no exact solution, so the approximate solution or numerical solution may be a good choice [24]. Related to this topic, we may refer the readers to [25–27] and the references therein.Consider the following fractional differential equation:(62)cDβ=ft,xt0,≤t≤1,1<β≤2,x0=0,x1=∫0ηxsds0<η<1,where cDβ denotes the Caputo fractional derivative of order β and f:0,1×ℝ⟶ℝ is a continuous function.In this section, we approximate the solution of problem (62) via new scheme (9) with Z=C0,1 which is a Banach space of continuous function from 0,1 into ℝ endowed with the maximum norm.C1 Assume that
(63)ft,a−ft,b≤Γβ+15a−bfor all t∈0,1 and a,b∈ℝ.Theorem 24.
LetZ=C0,1 and K:Z⟶Z be an operator defined by
(64)Kxt=1Γβ∫0tt−sβ−1fs,xsds−2t2−η2Γβ∫011−sβ−1fs,xsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,xmdmds,
t∈0,1, ∀x∈Z. Assume that the condition C1 is satisfied. Then the new scheme (9) converges to a solution of the problem (62), say x∗∈Z.Proof.
Observe thatx∗∈Z is a solution of (62) if and only if x∗ is a solution of the integral equation
(65)xt=1Γβ∫0tt−sβ−1fs,xsds−2t2−η2Γβ∫011−sβ−1fs,xsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,xmdmds.
Now, lett∈0,1 and x,y∈Z. Using C1, we get
(66)Kxt−Kyt=1Γβ∫0tt−sβ−1fs,xsds−2t2−η2Γβ∫011−sβ−1fs,xsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,xmdmds−1Γβ∫0tt−sβ−1fs,ysds+2t2−η2Γβ∫011−sβ−1fs,ysds−2t2−η2Γβ∫0η∫0ss−mβ−1fm,ymdmds≤Γβ+15x−ysupt∈0,11Γβ∫0tt−sβ−1ds+2t2−η2Γβ∫011−sβ−1ds+2t2−η2Γβ∫0η∫0ss−mβ−1dmds≤x−y.
Thus, fort∈0,1 and for each x,y∈Z, we get
(67)Kx−Ky≤x−y.
Thus,K is nonexpansive mapping. Hence, iterative scheme (9) converges to the solution of (62).Now, for the effectiveness of Theorem24, we present the following example.Example 25.
(68)cD1.25=sint0≤t≤1,x0=0,x1=∫00.5xsds.
The exact solution of problem (68) is given by
(69)xt=1Γβ∫0tt−sβ−1sinsds−2t2−η2Γβ∫011−sβ−1sinsds+2t2−η2Γβ∫0η∫0ss−mβ−1sinmdmds.
The operatorK:C0,1⟶C0,1 is defined by
(70)Kxt=1Γβ∫0tt−sβ−1sinsds−2t2−η2Γβ∫011−sβ−1sinsds+2t2−η2Γβ∫0η∫0ss−mβ−1sinmdmds.Taking initial hypothesisτ0t=t1−t,t∈0,1, β=1.25 and η=0.5, choose control sequences θn=0.95, μn=0.65 and ξn=0.50, n∈ℤ0+. It is shown in Table 5 and Figure 3 that iterative scheme (9) converges to the exact solution of problem (68) for the operator constructed in (70).Table 5
By using new iterative scheme observation between approximate solution and exact solution.
S.no.txtτ1τ3τ5τ7τ10100.000000000.000000000.000000000.000000000.000000000.0000000020.1-0.03804451-0.03804451-0.03804451-0.03804451-0.03804451-0.0380445130.2-0.07003487-0.07003487-0.07003487-0.07003487-0.07003487-0.0700348740.3-0.09478803-0.09478803-0.09478803-0.09478803-0.09478803-0.0947880350.4-0.11165674-0.11165674-0.11165674-0.11165674-0.11165674-0.1116567460.5-0.12026046-0.12026046-0.12026046-0.12026046-0.12026046-0.1202604670.6-0.12040960-0.12040960-0.12040960-0.12040960-0.12040960-0.1204096080.7-0.11207095-0.11207095-0.11207095-0.11207095-0.11207095-0.1120709590.8-0.09534778-0.09534778-0.09534778-0.09534778-0.09534778-0.09534778100.9-0.07046638-0.07046638-0.07046638-0.07046638-0.07046638-0.07046638111.0-0.03776548-0.03776548-0.03776548-0.03776548-0.03776548-0.03776548Figure 3
Exact solution and approximated solution of problem (68).


## 7. Conclusion

In this paper, convergence and stability results of a new three step iterative scheme has been studied. Further, the solution of a fractional differential equation is approximated by applying Theorem24. For nonlinear mappings, we compared the rate of convergence of remarkable iterative schemes analytically and numerically. To support the main result, we gave nontrivial examples.



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*Source: 3740809-2022-03-04.xml*