# Approximating Fixed Points and the Solution of a Nonlinear Fractional Difference Equation via an Iterative Method

Authors: Faizan Ahmad Khan
Journal: Journal of Mathematics
 (2022)

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## Abstract

The main intent of this article is to innovate a new iterative method to approximate fixed points of contraction and nonexpansive mappings. We prove that the new iterative method is stable for contraction and has a better rate of convergence than some distinctive iterative methods. Furthermore, some convergence results are proved for nonexpansive mappings. Finally, the solution of a nonlinear fractional difference equation is approximated via the proposed iterative method. Some numerical examples are constructed to support the analytical results and to illustrate the efficiency of the proposed iterative method.


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## Body

## 1. Introduction

The fixed point theory is an imperative arm of mathematics. It has become not only a field with significant advancement but also a vital tool for solving different kinds of problems in several fields of mathematics. The fixed point theory is a powerful tool because it has a variety of applications in different fields like differential and integral equations, variational inequalities, approximation theory, economics, biological sciences, medical sciences, engineering, optimization theory, fractal theory, game theory, and control theory. Indeed, the strength of fixed point theory lies in its wide range of applications inside and beyond mathematics.All over this paper, we presume thatℤ+ is the set of all non-negative integers, ℬ a Banach space, ∅≠D⊂ℬ, and FG=k∈D:G:D⟶DandGk=k. A mapping G:D⟶D is said to be a contraction mapping if ∃β∈0,1, such that ∀s,t∈D:(1)Gs−Gt≤βs−t.Ifβ=1, then G is a nonexpansive mapping on D.In 1965, an underlying existence result for fixed points of nonexpansive mappings was proved independently by Browder [1], Göhde [2], and Kirk [3]. After that, fixed point theory for nonexpansive and allied classes of mappings has been examined broadly and has provoked a parallel study in Banach space geometry.On the other hand, it is well known that the Picard iterative method failed to estimate the fixed points of nonexpansive mappings. So, in 1953, Mann [4] introduced a one-step iterative method to estimate the fixed points of nonexpansive mappings. But, it can be effortlessly seen that Mann’s iterative method was unsuccessful to estimate the fixed points of pseudo-contractive mappings. Therefore, in 1974, Ishikawa [5] defined a two-step Mann iterative method to estimate the fixed points of such type mappings. The speed of convergence of iterative sequence is important from the practical point of view. The faster iterative method saves time while approximating fixed points of nonlinear mappings. Recently, Ali et al. [6], and Garodia and Uddin [7] introduced new iterative methods to achieve a better rate of convergence. Quite recently, several Man-type and Ishikawa-type iterative methods have been studied by different researchers for the approximation of fixed points of single-valued nonlinear functions, e.g., see [8–10]. The following iterative methods are generated by an initial point τ0∈D, where G is a self-mapping on D.The S-iterative method (Agarwal et al. [11]) is as follows:(2)τn+1=1−μnGτn+μnGσn,σn=1−θnτn+θnGτn,n∈ℤ+.The Picard-S iterative method (Gursoy and Karakaya [12]) can be given as(3)τn+1=Gσn,σn=1−μnGτn+μnGξn,ξn=1−θnτn+θnGτn,n∈ℤ+.The Vatan iterative method (Karakaya et al. [13]) can be given as(4)τn+1=G1−μnσn+μnGσn,σn=G1−θnτn+θnGτn,n∈ℤ+.The Thakur-New iterative method (Thakur et al. [14]) can be given as(5)τn+1=Gσn,σn=G1−μnτn+μnξn,ξn=1−θnτn+θnGτn,n∈ℤ+.TheM∗ iterative method (Ullah and Arshad [15]) can be given as(6)τn+1=Gσn,σn=G1−μnτn+μnGξn,ξn=1−θnτn+θnGτn,n∈ℤ+.TheM iterative method (Ullah and Arshad [16]) can be given as(7)τn+1=Gσn,σn=Gξn,ξn=1−μnτn+μnGτn,n∈ℤ+.Strongly inspired by the above equations, we define the following a novel iterative method:(8)τ0∈D,τn+1=G1−μnσn+μnGσn,σn=G1−θnGτn+θnξn,ξn=GGτn,n∈ℤ+,where μn and θn are control sequences in (0, 1).Remark 1.
This iterative method (8) is different from all the iterative methods existing in the literature.
The rest of the manuscript is unified as follows: Section2 contains some definitions and lemmas that will be used in the main findings. In Section 3, the rate of convergence of the proposed iterative method is compared with some known and remarkable iterative methods by analytically and numerically. The stability of the new iterative method is also discussed with respect to contraction. In Section 4, weak and strong convergence theorems are proven for nonexpansive mappings via the newly defined method. In Section 5, the solution of a nonlinear fractional difference equation is estimated via the proposed iterative method. The conclusion of the paper is given in the last section.


## 2. Preliminaries

For the purpose of convenience, we recall the following concepts and results that will be used in the sequel.Lemma 1 (see [17]).
Letun and ϵn be sequences in ℝ+ that satisfy the following inequality:(9)un+1≤1−vnun+ϵn,where vn∈0,1, ∀n∈ℤ+ with ∑n=0∞vn=∞. If limn⟶∞ϵn/vn=0, then limn⟶∞un=0.Definition 1 (see [18]).
A Banach spaceℬ satisfies Opial’s condition if for each τn⇀p∈ℬ, limn⟶∞infτn−p<limn⟶∞infτn−q holds true, and ∀q∈ℬ with q≠p.Definition 2 (see [19]).
Letℬ be a Banach space, ∅≠D⊂ℬ and G:D⟶D be a function. G satisfies condition I if ∃ is an increasing mapping ξ:0,∞⟶0,∞ such that ξ0=0, ξw>0, ∀w>0, and dp,Gp≥ξdp,FG, ∀p∈D.Lemma 2 (see [20]).
Letτn and σn be sequences in a uniformly convex Banach space ℬ with limn⟶∞supτn≤c, limn⟶∞supσn≤c,m and limn⟶∞snτn+1−snσn=c, for some c≥0. Then, limn⟶∞τn−σn=0, where 0<a≤sn≤b<1 and ∀n≥1.Lemma 3 (see [21]).
IfG:D⟶ℬ is a nonexpansive function, where ℬ is a uniformly convex Banach space and D is a convex closed subset of ℬ. Then, I−G is semiclosed on ℬ, where I stands for identity map on D.Definition 3 (see [22]).
Letℬ,⋅ be a normed space and G be a self-mapping on ℬ and let two iterative methods τn and σn converge identical to the point k. Further, assume that the following error estimates are available:(10)τn−k≤dn,σn−k≤en,where dn, en⊂ℝ+ such that dn⟶0 and en⟶0 as n⟶∞.Definition 4 (see [22]).
Letdn, en⊂ℝ+ such that dn⟶d and en⟶e as n⟶∞. If(11)limn⟶∞dn−den−e=0,then dn converges to d faster than en to e.
And if(12)0<limn⟶∞dn−den−e<∞,then dn and en have the equal rate of convergence.Definition 5 (see [23]).
Letψn⊂D be an approximate sequence of τn. Then, an iterative method is defined as(13)τ0∈D,τn+1=ξG,τn,n∈ℤ+,for some relation ξ, such that τn⟶k as n⟶∞, is called G-stable; if ϵn=ψn+1−ξG,ψn, n∈ℤ+, we get limn⟶∞ϵn=0⇔limn⟶∞ψn=k.


## 3. Rate of Convergence and the Stability of the Proposed Method

The motive of the current section is to demonstrate convergence, rate of convergence, and stability results for contraction in an arbitrary Banach space by the proposed iterative method.Theorem 1.
Letℬ be a Banach space and D a nonempty, convex closed subset of ℬ and G:D⟶D be a contraction mapping. Then, the iterative sequence τn defined in the method (8) converges to one and only one point of FG. Moreover, the sequence τn is stable with respect to contraction mapping.Proof.
SinceG is a contraction mapping, β∈0,1 and ∀s∈D, and k∈FG:(14)Gs−k=Gs−Gk≤βs−k.
By iterative method (8), we get(15)ξn−k=GGτn−k≤β2τn−k.
Using equation (15), we get(16)σn−k=G1−θnGτn+θnξn−k≤β1−θnGτn+θnξn−k≤β1−θnGτn−k+θnξn−k≤β21−θnτn−k+θnβτn−k=β21−1−βθnτn−k.
Since0≤β<1 and θn∈0,1, so with the fact 0<1−1−βθn<1, we get(17)σn−k≤β2ξn−k.Using equation (17), we get(18)τn+1−k=G1−μnσn+μnGσn−k≤β1−μnσn+μnGσn−k≤β1−μnσn−k+μnGσn−k≤β1−μnσn−k+βμnσn−k≤β1−1−βμnσn−k≤β3τn−k.
Inductively, we get(19)τn+1−k≤β3n+1τ0−k.
Since0≤β<1, it concludes that τn converges to k.
Here, we prove the stability of the method (8). Let ψn be an estimate sequence of τn in D, then the sequence defined by the iterative method (8) is τn+1=ξG,τn and ϵn=ψn+1−ξG,ψn,n∈ℤ+. Now, we show that limn⟶∞ϵn=0⇔limn⟶∞ψn=k.
Letlimn⟶∞ϵn=0, then by the method (8), we obtain(20)ψn+1−k≤ψn+1−ξG,ψn+ξG,ψn−k=ϵn+ξG,ψn−k≤ϵn+β31−1−βμnψn−k.
Putun=ψn−k and vn=1−βμn∈0,1, then(21)un+1≤β31−vnun+εn.
Sincelimn⟶∞ϵn=0, ϵn/vn⟶0 as n⟶∞. Thus, by Lemma 1, limn⟶∞un=0, that is, limn⟶∞ψn=k.
Conversely, assume thatlimn⟶∞ψn=k, then we have(22)ϵn=ψn+1−ξG,ψn≤ψn+1−k+ξG,ψn−k≤ψn+1−k+β31−1−βμnψn−k.
This shows thatlimn⟶∞ϵn=0. Thus, the iterative method (8) is stable with respect to contraction.
The following theorem shows that the iterative method (8) has the better speed of convergence than the methods (2)–(7) for contraction.Theorem 2.
Letℬ be a Banach space and D a nonempty, convex closed subset of ℬ, and G:D⟶D be a contraction. Assume that the sequence τ1,n is defined by S (2), τ2,n by Picard-S (3), τ3,n by Vatan (4), τ4,n by Thakur-New (5), τ5,n by M∗ (6), τ6,n by M (7) and τn by (8) iterative method. Then, the iterative method (8) converges to a fixed point k of G faster than the M∗, S, Vatan, Picard-S, Thakur-New and M iterative methods.Proof.
In view of equation (19), we obtain(23)τn+1−k≤β3n+1τ0−k=αn,n∈ℤ+.
As proved by Sahu ([24], Theorem 3.6), we get(24)τ1,n−k≤βn+11−1−βμnθnn+1τ1,0−k=α1,n,n∈ℤ+.
From Gursoy and Karakaya’s study [12], we get(25)τ2,n−k≤β2n+11−1−βμnθnn+1τ2,0−k=α2,n,n∈ℤ+.
As proved by Ullah and Arshad ([25], Theorem 4), we get(26)τ3,n−k≤β2n+11−1−βμnn+1τ3,0−k=α3,n,n∈ℤ+.
Now, by iterative method (5), we obtain(27)ξn−k=1−θnτn+θnGτn−k≤1−θnτn−k+βθnτn−k=1−θn+βθnτn−k=1−1−βθnτn−k.
Sinceβ∈0,1 and θn∈0,1, so with the fact 0<1−1−βθn<1, we have(28)ξn−k≤τn−k.
By equation (28), we get(29)σn−k=G1−μnτn+μnξn−k≤β1−μnτn+μnξn−k≤β1−μnτn−k+μnξn−k≤β1−μnτn−k+μnτn−k=βτn−k.
Thus, using the equation (29), we obtain that(30)τn+1−k=Gσn−k≤βσn−k≤β2τn−k.
Inductively, we get(31)τn+1−k≤β2n+1τ0−k.
Let(32)τ4,n−k≤β2n+1τ4,0−k=α4,n,n∈ℤ+.
By iterative method (6) and using equation (28), we get(33)ξn−k≤τn−k.
Now, using equation (33), we get(34)σn−k=G1−μnτn+μnGξn−k≤β1−μnτn+μnGξn−k≤β1−μnτn−k+βμnξn−k≤β1−μnτn−k+βμnτn−k=β1−1−βμnτn−k≤βτn−k.
Thus, using equation (34), we obtain that(35)τn+1−k=Gσn−k≤βσn−k≤β2τn−k.
Inductively, we get(36)τn+1−k≤β2n+1τ0−k.
Let(37)τ5,n−k≤β2n+1τ5,0−k=α5,n,n∈ℤ+.
By iterative method (7) and using equation (28), we get(38)ξn−k≤τn−k.
Now, using equation (38), we have(39)σn−k=Gξn−k≤βξn−k≤βτn−k.By equation (39), we obtain(40)τn+1−k=Gσn−k≤βσn−k≤β2τn−k.
Inductively, we obtain(41)τn+1−k≤β2n+1τ0−k.
Let(42)τ6,n−k≤β2n+1τ6,0−k=α6,n,n∈ℤ+.
Now,(43)αnα1,n=β3n+1τ0−kβ1−1−βμnθnn+1τ1,0−k=βn+1β1−1−βμnθnn+1τ0−kτ1,0−k.
Sinceβ<1 and β/1−1−βμnθn<1, we have
αn/α1,n⟶0 as n⟶∞. Thus, τn has a better speed of convergence than τ1,n and(44)αnα2,n=β3n+1τ0−kβ2n+11−1−βμnθnn+1τ2,0−k=β1−1−βμnθnn+1τ0−kτ2,0−k,implies αn/α2,n⟶0 as n⟶∞. Thus, τn has better speed of convergence than τ2,n.
Next,(45)αnα3,n=β3n+1τ0−kβ2n+11−1−βμnn+1τ3,0−k=β1−1−βμnn+1τ0−kτ3,0−k.
Thus,αn/α3,n⟶0 as n⟶∞. Hence, τn has better speed of convergence than τ3,n.
Furthermore,(46)αnα4,n=β3n+1τ0−kβ2n+1τ4,0−k=βn+1τ0−kτ4,0−k⟶0,asn⟶∞.αnα5,n=β3n+1τ0−kβ2n+1τ5,0−k=βn+1τ0−kτ5,0−k⟶0,asn⟶∞.And(47)αnα6,n=β3n+1τ0−kβ2n+1τ6,0−k=βn+1τ0−tτ6,0−k⟶0,asn⟶∞.
Thus, the sequenceτn has better speed of convergence than the sequences τ4,n, τ5,n, and τ6,n.
The following example supports the above result.Example 1.
Takeℬ=ℝ and D=1,100⊂ℝ. Let G:D⟶D be given by Gs=s2−s+1, for all s∈D. Then, G is a contraction mapping and admits an unique fixed point k=1. Here, we choose the control sequences μn=0.25 and θn=0.35, for all n∈ℤ+ with the initial guess τ0=5.
The proposed method (8) has better speed of convergence than S, Picard-S, Vatan, Thakur-New, M∗, and M iterative methods with control sequences μn=0.25, θn=0.35, n∈ℤ+, and the initial point τ0=5 (Tables 1 and 2 and Figure 1). With the same inputs, we compare the CPU time for distinct iterative methods (Table 2).Table 1
A comparison table for the speed of convergence of iterative methods.
Iter.NewSPicard-SVatanThakur-NewM∗M15.0000005.0000005.0000005.0000005.0000005.0000005.00000023.4109954.5672084.1583874.0137004.1583844.0585414.07327232.0645734.1433973.3585213.0902383.3585143.1727453.199868⋮⋮⋮⋮⋮⋮⋮⋮101.0000001.4493461.0008661.0000981.0008661.0001961.000234⋮⋮⋮⋮⋮⋮⋮⋮151.0000001.0464731.0000031.0000001.0000031.0000001.000001161.0000001.0235801.0000011.0000001.0000011.0000001.000000171.0000001.0117711.0000001.0000001.0000001.0000001.000000181.0000001.0058261.0000001.0000001.0000001.0000001.000000⋮⋮⋮⋮⋮⋮⋮⋮321.0000001.0000001.0000001.0000001.0000001.0000001.000000Table 2
A comparison table for the CPU time for convergence of iterative methods.
Iterative methodsNew (s)S (s)Picard-S (s)Vatan (s)Thakur-New (s)M∗ (s)M (s)CPU time (in seconds)0.0150.0280.0180.0160.0180.0180.016Figure 1
Comparison by graph.


### 3.1. Observations

In the present article, we compare the speed of convergence of different iterative methods only in number of iterations. In Table1, we observe that proposed method (8) converges to a fixed point k=1 in 10 iterations and other methods, as shown in Figure 1, and converges more than 10 iterations. Thus, method (8) converges faster than methods (2)–(7).We also compare the convergence behavior of the proposed iterative method for different initial points. We noticed that the rate of convergence also depends on the initial points, and one can easily see it in Table3 and Figure 2.Table 3
A comparison table for the speed of convergence of iterative method (8) for different initial points.
Iter.τ0=0.5τ0=1.5τ0=2.0τ0=4.0τ0=15τ0=2510.5000001.5000002.0000004.00000015.00000025.00000020.9810671.0699721.2171592.53717013.13288623.08746330.9985081.0061441.0225861.47412811.28268921.180400⋮⋮⋮⋮⋮⋮⋮71.0000001.0000001.0000011.0000374.25454513.63314381.0000001.0000001.0000001.0000032.75286811.77791691.0000001.0000001.0000001.0000001.6031119.943726⋮⋮⋮⋮⋮⋮⋮151.0000001.0000001.0000001.0000001.0000001.166649⋮⋮⋮⋮⋮⋮⋮201.0000001.0000001.0000001.0000001.0000001.000001211.0000001.0000001.0000001.0000001.0000001.000000Figure 2
Comparison by the graph.


## 3.1. Observations

In the present article, we compare the speed of convergence of different iterative methods only in number of iterations. In Table1, we observe that proposed method (8) converges to a fixed point k=1 in 10 iterations and other methods, as shown in Figure 1, and converges more than 10 iterations. Thus, method (8) converges faster than methods (2)–(7).We also compare the convergence behavior of the proposed iterative method for different initial points. We noticed that the rate of convergence also depends on the initial points, and one can easily see it in Table3 and Figure 2.Table 3
A comparison table for the speed of convergence of iterative method (8) for different initial points.
Iter.τ0=0.5τ0=1.5τ0=2.0τ0=4.0τ0=15τ0=2510.5000001.5000002.0000004.00000015.00000025.00000020.9810671.0699721.2171592.53717013.13288623.08746330.9985081.0061441.0225861.47412811.28268921.180400⋮⋮⋮⋮⋮⋮⋮71.0000001.0000001.0000011.0000374.25454513.63314381.0000001.0000001.0000001.0000032.75286811.77791691.0000001.0000001.0000001.0000001.6031119.943726⋮⋮⋮⋮⋮⋮⋮151.0000001.0000001.0000001.0000001.0000001.166649⋮⋮⋮⋮⋮⋮⋮201.0000001.0000001.0000001.0000001.0000001.000001211.0000001.0000001.0000001.0000001.0000001.000000Figure 2
Comparison by the graph.


## 4. Convergence Theorems

First, we demonstrate the following fruitful lemmas that help us to obtain the sequel.Lemma 4.
Letℬ be a uniformly convex Banach space, D≠∅ be a convex closed subset of ℬ, and let G:D⟶D be a nonexpansive mapping. If τn is defined by (8), then limn⟶∞τn−k exists when ∀k∈FG.Proof.
AsG is a nonexpansive mapping, so we get(48)Gτn−k≤τn−k.
∀τn∈D and ∀k∈FG. By the iterative method (8), we obtain (49)ξn−k=GGτn−k≤τn−k.
Using equation (49), we have(50)σn−k=G1−θnGτn+θnξn−k≤1−θnGτn+θnξn−k≤1−θnGτn−k+θnξn−k≤1−θnτn−k+θnξn−k≤τn−k.Using equation (50), we have(51)τn+1−k=G1−μnσn+μnGσn−k≤1−μnσn+μnGσn−k≤1−μnσn−k+μnGσn−k≤1−μnσn−k+μnσn−k≤1−μnτn−k+μnτn−k=τn−k.
Hence,τn−k is nonincreasing ∀k∈FG. Hence, limn⟶∞τn−k exists.Lemma 5.
Let all the assumptions of Lemma4 be true. If τn is defined in the method (8), then FG≠∅ if and only if τn is bounded and limn⟶∞τn−Gτn=0.Proof.
Presume thatFG≠∅ and k∈FG. Then, limn⟶∞τn−k exists by Lemma 4 and τn is bounded. Presume that(52)limn⟶∞τn−k=α.
By equation (50) and (52), we obtain(53)limn⟶∞supσn−k≤limn⟶∞supτn−k≤α.
AsG is a nonexpansive mapping, we get(54)Gσn−k=Gσn−Gk≤σn−klimn⟶∞supGσn−k≤limn⟶∞supσn−k≤α.
Now,(55)α=limn⟶∞τn+1−k=limn⟶∞G1−μnσn+μnGσn−k≤limn⟶∞1−μnσn+μnGσn−k=limn⟶∞1−μnσn−k+μnGσn−k≤limn⟶∞1−μnσn−k+μnGσn−k≤limn⟶∞1−μnτn−k+μnτn−k≤α.
Thus,(56)limn⟶∞1−μnσn−k+μnGσn−k=α.
By (53)–(56) and applying Lemma 2, we get(57)limn⟶∞σn−Gσn=0.
Now, by using equation (57), we obtain(58)τn+1−Gτn+1=G1−μnσn+μnGσn−Gτn+1≤1−μnσn+μnGσn−τn+1=1−μnσn−1−μnGσn+Gσn−τn+1≤1−μnσn−Gσn+Gσn−τn+1=1−μnσn−Gσn+Gσn−G1−μnσn+μnGσn≤1−μnσn−Gσn+σn−1−μnσn−μnGσn=1−μnσn−Gσn+μnσn−Gσn=σn−Gσn.
Thus,(59)limn⟶∞τn−Gτn=0.
Conversely, presume thatτn is bounded and limn⟶∞τn−Gτn=0. Let k∈AD,τn, then we have(60)rGk,τn=limn⟶∞supτn−Gk≤limn⟶∞supτn−Gτn+Gτn−Gk≤limn⟶∞supτn−k=rk,τn=rD,τn.
This implies thatGk∈AD,τn. Thus, we know that AD,τn consists of only one element because ℬ is uniformly convex; hence, Gk=k.Theorem 3.
Let all the assumptions of Lemma4 be true. Presume that ℬ contents Opial’s property, then the sequence τn given in (8) converges to a point of FG weakly.Proof.
Assumeτnj and τnk are two subsequences of τn such that τnj⇀k and τnj⇀q as j⟶∞ (⇀, stands for weakly convergent), where k,q∈FG. In view of Lemma 4, limn⟶∞τn−k exists and by Lemma 5, limn⟶∞τn−Gτn=0. In view of Lemma 3, I−Gk=0, i.e., k=Gk, in the same way q=Gq.
Now, we prove thatk=q. If k≠q, then using Opial’s condition, we get(61)limn⟶∞τn−k=limnj⟶∞τnj−k<limnj⟶∞τnj−q=limn⟶∞τn−q=limnk⟶∞τnk−q<limnk⟶∞τnk−k=limn⟶∞τn−k.A contradiction implies that k=q. Thus, τn⇀k and ∀k∈FG.Theorem 4.
Let all the assumptions of Lemma4 be true. Then, τn defined by (8) goes to an fixed element of G if and only if limn⟶∞infdτn,FG=0, where dτn,FG=infτn−k:k∈FG.Proof.
One can prove the first part easily. For reverse, presumelimn⟶∞infdτn,FG=0. In view of Result 4.1, limn⟶∞τn−k exists, ∀k∈FG and limn⟶∞dτn,FG=0 is given.
We now prove thatτn converges to an element of D. Since limn⟶∞dτn,FG=0, for β>0, ∃P∈ℕ with ∀n≥P.(62)dτn,FG<β2⇒infτn−k:k∈FG<β2.
Specifically,infτP−k:k∈FG<β/2. So ∃k∈FG with τP−k〈β/2. Now, for m,n≥P, we have(63)τn+m−τn≤τn+m−k+τn−k≤τP−k+τP−k=2τP−k<β.
This shows that the sequenceτn is Cauchy in D, so that ∃an element r∈D and limn⟶∞τn=r. Now, limn⟶∞dτn,FG=0⇒dr,FG=0, and thus, r∈FG.Theorem 5.
Let all the assumptions of Lemma4 be true and G contents condition I. Then, τn defined by (8) converges strongly to a point of FG.Proof.
Using Lemma5 and condition I, we obtain(64)0≤limn⟶∞ξdτn,FG≤limn⟶∞τn−Gτn=0⇒limn⟶∞ξdτn,FG=0.
This implies that(65)limn⟶∞dτn,FG=0.
Thus, the result is followed by Theorem4.Example 2.
Letℬ=ℝ2 with the norm s,t=s+t, ∀s,t∈ℬ, and D=s,t:s,t∈0,1×0,1⊂ℬ. A function G:D⟶D is given as(66)Gs,t=coss,cost,∀s,t∈D.
Then,G is a nonexpansive mapping and has a fixed point k,q=0.739085,0.739085, whether G is not a contraction.
The proposed method (8) goes to the fixed point k,q=0.739085,0.739085 of the function G better than S, Picard-S, Vatan, Thakur-new, M∗, and M iterative methods with the control sequences μn=0.85, θn=0.35, n∈ℤ+, and initial point (0.5, 0.8) (Tables 4–6 and Figure 3). With the same inputs, we compare the CPU time for the conbergence of distinct iterative methods (Table 6).Table 4
A comparison table for the rate of convergence of iterative methods for Example2.
Iter.NewSPicard-S1(0.500000, 0.800000)(0.500000, 0.800000)(0.500000, 0.800000)2(0.728448, 0.742483)(0.817381, 0.718359)(0.684134, 0.752887)3(0.738520, 0.739267)(0.712404, 0.746080)(0.726516, 0.742226)⋮⋮⋮⋮6(0.739085, 0.739085)(0.740115,0.738814)(0.738936, 0.739122)⋮⋮⋮⋮10(0.739085, 0.739085)(0.739099, 0.739082)(0.739085, 0.739085)11(0.739085, 0.739085)(0.739081, 0.739086)(0.739085, 0.739085)12(0.739085, 0.739085)(0.739087, 0.739085)(0.739085, 0.739085)13(0.739085, 0.739085)(0.739085, 0.739085)(0.739085, 0.739085)Table 5
A comparison table for the rate of convergence of iterative methods for Example2.
Iter.VatanThakur-NewM∗M1(0.500000, 0.800000)(0.500000, 0.800000)(0.500000, 0.800000)(0.500000, 0.800000)2(0.758961, 0.734316)(0.683455, 0.752848)(0.758026, 0.734260)(0.691844, 0.750366)3(0.737514, 0.739464)(0.726326, 0.742215)(0.737582, 0.739468)(0.730107, 0.741200)4(0.739210, 0.739055)(0.736173, 0.739798)(0.739205, 0.739055)(0.737394, 0.739482)5(0.739075, 0.739088)(0.738422, 0.739248)(0.739076, 0.739088)(0.738767, 0.739160)6(0.739086, 0.739085)(0.738934, 0.739122)(0.739086, 0.739085)(0.739025, 0.739099)7(0.739085, 0.739085)(0.739051, 0.739094)(0.739085, 0.739085)(0.739074, 0.739088)8(0.739085, 0.739085)(0.739077, 0.739087)(0.739085, 0.739085)(0.739083, 0.739086)9(0.739085, 0.739085)(0.739083, 0.739086)(0.739085, 0.739085)(0.739085, 0.739085)10(0.739085, 0.739085)(0.739085, 0.739085)(0.739085, 0.739085)(0.739085, 0.739085)Table 6
A comparison table for the CPU time for convergence of iterative methods.
Iterative methodsNew (s)S (s)Picard-S (s)Vatan (s)Thakur-New (s)M∗ (s)M (s)CPU time (in seconds)0.0170.0170.0230.0180.0220.0220.018Figure 3
Comparison by the graph.


## 5. Solution of a Nonlinear Fractional Difference Equation

The main intent of the present section is to approximate the solution of a fractional difference equation via an iterative method (8). Consider the following:(67)Dδyz=Fz+δ−1,yz+δ−1,z∈I1−δ,0<δ≤1,y0=y0,where Dδ indicates the Caputo-like discrete fractional difference of order δ, F:0,∞×ℬ⟶ℬ is a continuous function, I1−δ=1−δ,2−δ,… and ℬ=CI1−δ is a real Banach space with the norm(68)y∞=maxz∈I1−δyz,∀y∈ℬ.It is shown in [26] that yz is a solution of the initial value problem (IVP) (67) if and only if yz is a solution of the following relation:(69)yz=y0+1Γδ∑b=1−δz−δz−b−1δ−1Fb+δ−1,yb+δ−1,0<δ≤1,y0=y0.The following lemma plays a key part to demonstrate the main finding of this segment.Lemma 6 (see [26]).
We have(70)∑b=1−δz−δz−b−1δ−1=Γz+δδΓz.
Now, setIk=1,2,3,…,k and ℬ=CIk is a real Banach space, where k∈ℕ. Presume that the following assumptions are true.
C1: Assume F:0,k×ℬ⟶ℬ is a locally Lipschitz continuous function with constant L, i.e.,(71)Fz,y−Fz,w∞≤Ly−w∞.
C2:(72)LΓk+δΓδ+1Γk<1.
The existence and uniqueness of a solution of the IVP (67) can be found in [26].
Presently, we are going to demonstrate the main result of the present section.Theorem 6.
Presume that the conditionsC1 and C2 are satisfied. Let an operator G:ℬ⟶ℬ be given by(73)Gyz=y0+1Γδ∑b=1−δz−δz−b−1δ−1Fb+δ−1,yb+δ−1,for z∈Ik. Then, the sequence τn developed by iterative method (8) converges to a unique solution of IVP (67).Proof.
It is enough to show the operatorG is a contraction mapping on ℬ. For any y,w∈ℬ, we obtain that(74)Gyz−Gwz∞≤1Γδ∑b=1−δz−δz−b−1δ−1Fb+δ−1,yb+δ−1−Fb+δ−1,wb+δ−1∞≤1Γδ∑b=1−δz−δz−b−1δ−1y−w∞≤LΓz+δδΓδΓzy−w∞≤LΓk+δΓδ+1Γky−w∞.
Thus,G is a contraction on ℬ. Hence, by applying Theorem 1, the sequence τn converges to a unique solution of IVP (67).


## 6. Conclusion

The main intent of this article was to propose a novel and effective iterative method for the estimation of fixed points of contraction and nonexpansive mappings in the frame work of Banach space. It is also shown that the new iterative method is stable with respect to contraction mapping. The rate of convergence of the distinct iterative methods has been discussed. Therefore, the newly introduced iterative method is more efficient and effective than the previously defined iterative methods. The researchers may apply the new iterative method to approximate the solution of nonlinear problems to achieve a better rate of convergence. Besides, the results of the present paper generalize and amplify the relevant results in the literature. Thus, it can be concluded that the work done in the paper is new and useful in the area of nonlinear analysis. The fixed point iterative methods are very useful to estimate the solution of nonlinear differential equations, nonlinear fractional differential equations, nonlinear integrodifferential equations, etc. The interested researchers may apply the proposed iterative method to estimate the solution of above discussed problems.



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