# Convergence Analysis of an Iteration Process for a Class of Generalized Nonexpansive Mappings with Application to Fractional Differential Equations

Authors: Kifayat Ullah; Sabri T. M. Thabet; Anwar Kamal; Junaid Ahmad; Fayyaz Ahmad
Journal: Discrete Dynamics in Nature and Society
 (2023)

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## Abstract

We consider the class of generalizedα-nonexpansive mappings in a setting of Banach spaces. We prove existence of fixed point and convergence results for these mappings under the K∗-iterative process. The weak convergence is obtained with the help of Opial’s property while strong convergence results are obtained under various assumptions. Finally, we construct two numerical examples and connect our K∗-iterative process with them. An application to solve a fractional differential equation (FDE) is also provided. It has been eventually shown that the K∗- iterative process of this example gives more accurate numerical results corresponding to some other iterative processes of the literature. The main outcome is new and improves some known results of the literature.


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## Body

## 1. Introduction and Preliminaries

In recent years, theory of fixed points gained the attention of many authors [1, 2]. Whenever the ordinary analytical techniques cannot yield a solution to a differential or an integral equation, we are interested in finding the approximate value of the requested solution (see, e.g., the recent results in [3–5] and others). Before employing the appropriate iterative processes on such problems, one needs to convert it into a form of equation of fixed point. In this way a sequence is generated by the algorithm. The intended fixed point value of the equation of fixed point and the given equation’s solution is the limit of the series. In case of contraction mappings, Banach fixed point theorem [6] signals the fundamental Picard iteration xn+1=Axn. However, when the Picard iterative process for a given mapping does not converge, we employ alternative iterative procedures with different steps. One of the other iterative processes that have been studied by authors are the Mann [7], Noor [8], Ishikawa [9], and SP iteration (Phuengrattana and Suantai) [10]; S-iteration (Agarwal et al.) [11]; S∗ iteration (Karahan and Ozdemie) [12]; Picard–Mann hybrid [13]; Normal-S [14]; Krasnoselskii–Mann [15]; Abbas [16]; Picard-S [17]; and Thakur [18].On the other hand, Ullah and Arshad [19] presented a new iterative process for generalized nonexpansive mappings and call it as a K∗ iterative process. The K∗-iterative process reads as follows:(1)x1∈C,zn=1−βnxn+βnAxn,yn=A1−αnzn+αnAzn,xn+1=Ayn,n≥0,where αn,βn∈0,1.They demonstrated that among the many other iterative processes, theK∗ iteration (1) gives very high accurate results in very less steps of iterations in the setting of Suzuki mappings. We improve here their results to the larger class of generalized α-nonexpansive mappings.Definition 1.
LetA:C⟶C. Then, A is referred to as(i)
Nonexpansive ifAa−Aa′≤a−a′, for every two a,a′∈C;(ii)
Satisfying Condition (C) (or Suzuki mapping) if1/2a−Aa≤a−a′ implies Aa−Aa′≤a−a′, for every two a,a′∈C;(iii)
Generalizedα-nonexpansive if 1/2a−Aa≤a−a′ implies Aa−Aa′≤αa−Aa′+αa′-Aa+1−2αa−a′, for every two a,a′∈C and α∈0,1;(iv)
Satisfying conditionI [20] if there is nondecreasing f with f0=0 and fr>0 at r>0 and a−Aa≥fda,DA for all a∈C.
First time, a fixed point existence result for nonexpansive mappings established by Gohde [21] and Browder [22] in a setting of uniform convex Banach space (UCBS), and in the same year, Kirk [23] obtained the same result in a setting of reflexive Banach space. In [24], Suzuki suggested a very interesting generalized of nonexpansive mappings and any mapping of this class if named as a mapping satisfying condition C (or Suzuki mapping). He established several convergence and existence results for these mappings in different Banach space settings. Since mappings with condition (C) are more general than the concept of nonexpansive mappings. Thus, Pant and Shukla [25] generalized the Suzuki mappings by introducing the class of generalized α-nonexpansive mappings. They proved that every Suzuki mapping is generalized α-nonexpansive but the converse is not valid in general; that is, they proved that the class of generalized α-nonexpansive mappings properly includes the class of Suzuki mappings. Moreover, they used the Agarwal iteration [11] for establishing the main convergence results. The purpose of this work is to obtain the strong and weak convergence for the K∗-iterative processes for generalized α-nonexpansive mappings. In this way, we extend some main results of Pant and Shukla [25], Ullah and Arshad, and many others.Definition 2 (see [26, 27]).
LetC be any nonvoid closed convex subset of a UCBS H. an⊆H is bounded. If h∈H is any fixed element then we set the following:(b1) For a bounded sequencean at point h, rh,an:=limsupn⟶∞h−an is termed as asymptotic radius;(b2) For a bounded sequencean with the connection of C, rC,an=infrh,an:h∈C is termed as asymptotic radius;(b3) For a bounded sequencean with the connection of C, AC,an=h∈C:rh,an=rC,an is termed as asymptotic center.Definition 3 (see [28]).
A Banach spaceH space is said to satisfy the Opial’s condition in the case of any weakly convergent sequence an⊆H whose weak limit is h∈H, one is able to obtain the following:(2)limsupn⟶∞an−h<limsupn⟶∞an−h′foreachh′∈C−h.
In [25], the authors obtained some characterizations for the class of generalized α-nonexpansive mappings. We write all these characterizations in the following proposition:Proposition 1.
Assume thatA is a self-map on any nonempty subset C of a Banach space. Then,(p1) IfA satisfy condition C, then A is essentially generalized α-nonexpansive.(p2) IfA is generalized α-nonexpansive having a nonempty fixed point set, then Aa−a∗≤a−a∗ for a∈C and a∗ in DA.(p3) IfA is generalized α-nonexpansive, then DAis closed. Moreover, if the given space H is strictly convex, C is convex, then the set DA is also convex.(p4) IfA is generalized α-nonexpansive, then for all pair of elements a,a′∈C, one has(3)a-Aa′≤3+α1−αa-Aa+a−a′;.(p5) If the spaceH is endowed with the Opial condition, A is generalized α-nonexpansive, sn is any weakly convergent sequence to r with the property limn⟶∞Asn−sn=0, then r∈DA.Following lemma is a well-known property of any UCBS that is needed for our main results.Lemma 1 (see [29]).
LetH be any UCBS. For 0<i≤αn≤j<1, an,bn⊆H so that limsupn⟶∞an≤u, limsupn⟶∞bn≤u and limn⟶∞αnan+1−αnbn=u for some u≥0. Subsequently, we have limn⟶∞an−bn=0.


## 2. Main Findings

To obtain our weak and strong convergence results, we need a key lemma as follows:Lemma 2.
IfA is generalized α-nonexpansive self-map on a closed convex subset C of a Banach space H with DA≠∅ and xn is a sequence of iterates obtained from the K∗-iterative process (1). Subsequently,limn⟶∞xn−a∗ exists for each a∗∈DA.Proof.
Let us take anya∗∈DA. Using Proposition 1p2, we see that(4)zn−a∗=1−βnxn+βnAxn−a∗≤1−βnxn+βnxn−a∗≤1−βnxn−a∗+βnxn−a∗=xn−a∗.
This implies that(5)xx+1−a∗=Ayn−a∗≤yn−a∗=A1−αnzn+αnAzn−a∗≤1−αnzn+αnAzn−a∗≤1−αnzn−a∗+αnAzn−a∗≤1−αnzn−a∗+αnzn−a∗≤zn−a∗≤xn−a∗.
Hence,xn+1−a∗≤xn−a∗, which shows that xn−a∗ is bounded and nonincreasing. This gives result limn⟶∞xn−a∗ exists for each a∗∈DA.Now we discuss about necessary and sufficient conditions that must be met in order for any generalized nonexpansive mapping in a Banach space to have fixed points.Theorem 1.
IfA is generalized α-nonexpansive self-map on a closed convex subset C of a UCBS H and xn is a sequence of iterates obtained from the K∗-iterative process (1). Subsequently, DA≠∅ if and only if xn is bounded in C and limn⟶∞Axn−xn=0.Proof.
LetDA≠∅ and a∗∈DA. Applying Lemma 2, we get existence of limn⟶∞xn−a∗ and xn is bounded. Let u be that limit; thus,(6)limn⟶∞xn−a∗=u.
As we have demonstrated in the proof of Lemma2 that(7)zn−a∗≤xn−a∗.
This together with (6) gives(8)limsupn⟶∞zn−a∗≤limsupn⟶∞xn−a∗=u.
Sincea∗ is in DA, so Proposition 1 (p2) can be applied to obtain the following:(9)Axn−a∗≤xn−a∗,⟹limsupn⟶∞Axn−a∗≤limsupn⟶∞xn−a∗=u.
Now by the proof of Lemma2, we have the following:(10)xn+1−a∗≤zn−a∗,⟹u=liminfn⟶∞xn+1−a∗≤liminfn⟶∞zn−a∗.
From (8) and (10), we have(11)u=limn⟶∞zn−a∗.
By (1) and (11), one has(12)u=limn⟶∞zn−a∗=limn⟶∞1−βnxn+βnAxn−a∗=limn⟶∞1−βnxn−a∗+βnAxn−a∗.
If and only if(13)u=limn⟶∞1−βnxn−a∗+βnAxn−a∗.
Lemma1, can be applied to obtain,(14)limn⟶∞Axn−xn=0.
On the other hand we aim to demonstrateDA≠∅ under the suppositions of a bounded xn in the sense that limn⟶∞Axn−xn=0. We may select a point a∗∈AA,xn. If Proposition 1p4 is applied, then we must have the following:(15)rAa∗,xn=limsupn⟶∞xn-Aa∗≤3+α1−αlimsupn⟶∞Axn−xn+limsupn⟶∞xn−a∗=limsupn⟶∞xn−a∗=ra∗,xn.
We observe thatAa∗∈AC,xn. As this set has only element in any UCBS setting, we deduce Aa∗=a∗, hence the set DA is nonempty.Among the convergence results, we first obtain our weak convergence result for theK∗-iterative process in the setting of generalized α-nonexpansive mappings as follows:Theorem 2.
IfA is generalized α-nonexpansive self-map on a closed convex subset C of a UCBS H and xn is a sequence of iterates obtained from the K∗-iterative process (1). Subsequently,xn is weakly convergent to a point of DA.Proof.
Using Theorem1, the provided sequence xn is bounded. As H is UCBS, H a is reflexive Banach space. That is why, we can built a weakly convergent sequence of xn so that xni be the subsequence with a1∈C as weak limit. Applying Theorem 1 on the subsequence, we may have limi⟶∞xni-Axni=0. Hence, by Proposition 1p5, we have a1∈DA. It is enough to prove that xn is weakly convergent to a1. Thus, if xn does not weakly converge to a1. Then, a subsequence xnj of xn and a2∈C with xnj converging weakly to a2 and a2≠a1 exists. Also by Proposition 1p5, a2∈DA, by Lemma 2 with Opial’s property, we get(16)limn⟶∞xn−a1=limi⟶∞xni−a1<limi⟶∞xni−a2=limn⟶∞xn−a2=limj⟶∞xnj−a2<limj⟶∞xnj−a1=limn⟶∞xn−a1.
This is a contradiction. Hence, we havea1=a2. So, xn is weakly convergent to a1∈DA.Now we discuss the strong convergence of theK∗-iterative process for generalized α-nonexpansive mappings on compact domains.Theorem 3.
IfA is generalized α-nonexpansive self-map on a compact convex subset C of a UCBS H and xn is a sequence of iterates obtained from theK∗-iterative process (1). Subsequently,xn is strongly convergent to a point of DA.Proof.
As the domainC is a compact subset of H and also xn⊆C due to the convexity of C. Thus, a subsequence xni of xn exists with limi⟶∞xni−a∗∗=0 for some a∗∗∈C. In the view of Theorem 1, limi⟶∞Axni−xni=0. Applying Proposition 1p4, one has(17)xni-Aa∗∗≤3+α1−αxni-Axni+xni−a∗∗.
Hence, ifi⟶∞, then Aa∗∗=a∗∗. In the view of Lemma 2, a∗∗ is the strong limit of xn. This finishes the proof.A strong convergence of theK∗-iterative process in the setting of generalized α-nonexpansive mappings on a noncompact domain is established as follows. It should be noted that this result holds in general Banach spaces.Theorem 4.
IfA is generalized α-nonexpansive self-map on a closed convex subset C of a Banach space H and xn is a sequence of iterates obtained from the K∗-iterative process (1). Subsequently, xn is strongly convergent to a point of DAifliminfn⟶∞dxn,DA=0.Proof.
Using Lemma2, we have existence of limn⟶∞xn−a∗, for each fixed point of A. This provides us that limn⟶∞dxn,DA exist. Accordingly(18)limn⟶∞dxn,DA=0.
Two subsequencexni and ai of xn and DA are, respectively, generated by the above limit. Hence(19)xni−ai≤12ifor eachi≥1.
From the proof of Lemma2, we get nonincreasing xn, that is why(20)xni+1−ai≤xni−ai≤12i.
It follows that(21)ai+1−ai≤ai+1−xni+1+xni+1−ai≤12i+1+12i≤12i−1⟶0,asi⟶∞.
Consequently,ai is Cauchy sequence in DA as limi⟶∞ai+1−ai=0; thus, ai converges to a∗∗. Using Proposition 1p3, DA is closed; thus, a∗∗∈DA. By Lemma 2, limn⟶∞xn−a∗∗ exists so, a∗∗ is the strong limit of xn.Theorem 5.
IfA is generalized α-nonexpansive self-map on a closed convex subset C of a UCBS H and xn is a sequence of iterates obtained from the K∗-iterative process (1). Subsequently,xn is strongly convergent to a point of DAifA satisfies condition (I).Proof.
Using Theorem1, we can have(22)liminfn⟶∞Axn−xn=0.
Due to condition (I) ofA, we have(23)xn-Axn≥fdxn,DA.
Applying (22) on (23), we get(24)liminfn⟶∞fdxn,DA=0.
It follows(25)liminfn⟶∞dxn,DA=0.
Now applying Theorem4, xn converges strongly to DA.


## 3. Application to Fractional Differential Equations

Fractional calculus is important and an active field of research on its own [30–32]. It is well-known that fractional calculus has a crucial role in fluid, electromagnetic theory, and, especially, in electrical networks. In recent years, many papers appeared on the existence and approximation of solutions for certain FDEs (see e.g., Karapinar et al. [33] and others). However, all these authors used the concept of nonexpansive mappings to achieve the main objective that are continuous on their domain of definitions.Our alternative in this paper is to solve a FDE in the setting of generalizedα-nonexpansive mappings that are in general discontinuous. Unlike, other iterative schemes, we suggest the K∗ iterative scheme (1) to find the solution for the following FDE.Now we consider the following FDE and also assume thatS is a solution set of it:(26)Dξhu+Υu,hu=0,h0=h1=0,,where 0≤u≤1, 1<ξ<2, and Dξ stands for the Caputo fractional derivative endowed with the order ξ and Υ:0,1×R⟶R.Now we considerC=C0,1, where C0,1 is the Banach space of continuous maps on 0,1 to R equiped with the maximum norm. The corresponding Green’s function with (26) is defined by(27)Gu,v=1Γξu1−vξ−1−u−vξ−1,if0≤v≤u≤1,u1−vξ−1Γξ,if0≤u≤v≤1.The main result is provided in the following way:Theorem 6.
IfC=C0,1, then set an operator A:C⟶C by the formula(28)Ahu=∫01Gu,vΥv,hvdv,foreachhu∈C.If(29)Υv,hv−Υv,gv≤αhv-Agv+αgv-Ahv+1−2αgv−hv,where α is some real number in [0, 1). Subsequently, theK∗ iterates (1) associated with the A (as defined above) essentially converges to some point of the solution set S of (26) provided that liminfn⟶∞dxn,S=0.Proof.
Notice that the elementh of C solves (26) if it solves(30)hu=∫01Gu,vΥv,hvdv.
Now for every choice ofh,g∈C and 0≤u≤1, it follows that(31)Ahu-Agu≤∫01Gu,vΥv,hvdv−∫01Gu,vΥv,gvdv=∫01Gu,vΥv,hv−Υv,gvdv≤∫01Gu,vΥv,hv−Υv,gvdv≤∫01Gu,vαhv-Agv+αgv-Ahv+1−2αgv−hvdv≤αhv-Agv+αgv-Ahv+1−2αgv−hv∫01Gu,vdv≤αhv-Agv+αgv-Ahv+1−2αgv−hv.
Consequently, we get(32)Ah-Ag≤αh-Ag+αg-Ah+1−2αg−h.
Hence,A is generalized α-nonexpansive mapping. In the view of Theorem 4, the sequence of K∗ iterates converges to a fixed point of A and hence to the solution of the given equation.


## 4. Numerical Example

First, we construct a novel example of generalizedα-nonexpansive mappings on closed convex subset of a UCBS. Using this example, we perform a comparative numerical experiment using our K∗ and other iterative processes of the literature.Example 1.
LetC=0,6 and a self-map on C by the following rule:(33)Aa=a+154,if0≤a≤5,4,if5<a≤6.
In this case, we prove thatA is generalized 1/3-nonexpansive but does not satisfy the condition C.Proof.
Leta,a′∈A, then following are the all possible cases:Case I.
If5<a,a′≤6, we have(34)13a-Aa′+13a′-Aa+1−213a−a′≥0=Aa-Aa′.Case II.
If0<a,a′≤5, we have(35)13a-Aa′+13a′-Aa+1−213a−a′=13a−a′+154+13a′−a+154+13a−a′≥14a−a′=Aa-Aa′.Case III.
If0<a≤5 and 5<a′≤6, we have(36)13a-Aa′+13a′-Aa+1−213a−a′=13a−4+13a′−a+154+13a−a′≥14a−1=Aa-Aa′.
Hence,Aa-Aa′≤1/3a-Aa′+1/3a′-Aa+1−21/3a−a′ for every two points a,a′∈C. Now let a=5 and a′=11/2, then 1/2a-Aa≤a−a′⇒Aa-Aa′>a−a′. Thus, A does not satisfy condition (C).To show the high accuracy of the proposedK∗ iteration, we compare it with the one-step Mann iteration [7], two-step Ishikawa [9], leading two-stepS-iteration of Agarwal [11] and a leading iterative scheme studied by Thakur [18]. We may take αn=0.85 and βn=0.65. Table 1 shows some values for the initial value of x1=4.9. Additionally, Figure 1 offers detail on the behavior of the different schemes. Moreover, if xn−0<10−4, then further comparison is given in Table 2. For generalized α-nonexpansive mapping it is evident that the K∗ iterative method performs better than the other methods.We finish this section with an example. This example uses a subset of a two-dimensional Euclidian space.Table 1
Comparison of various iterative processes.
nK∗ThakurSIshikawaMann14.94.94.94.94.924.998838874.996339844.985359384.974109384.9637500034.999986524.999866034.997856524.993296764.9868593844.999999844.999995104.999686184.998264494.99523652554.999999824.999954054.999550674.99827324654.999999994.999993274.999883664.999374057554.999999024.999969884.999773098554.999999864.99999224.999917759554.999999984.999997984.99997018105554.999999484.99998919115554.999999864.99999608125554.999999964.99999858135554.999999994.999999401455554.999999811555554.999999931655554.999999981755554.999999991855555Figure 1
Graphical comparison of various iterative processes.Table 2
Comparison of the iterates for different choice of parameters and initial points.
IterationsInitial points0.21.22.23.24.24.9Forαn=n/n+110/9,βn=1/n+32/3Mann1413131299Ishikawa13121212118S666659Thakur444555K∗334444Example 2.
LetC=0,1×0,1 and set a self-map on A by the following rule:(37)Aa,a′=a2,a′3,ifa,a′∈0,12×0,12,a4,a′5,ifa,a′∈12,1×12,1.Here A is a generalized α-nonexpansive mapping with fixed point (0, 0). The numerical results are shown in Table 3. In this case, it is also clear that our K∗ iterative scheme is moving fast to the fixed point (0, 0).Table 3
Sequences generated byK∗-, Thakur-, and S-iteration processes.
nK∗ThakurS1(0.1, 0.2)(0.1, 0.2)(0.1, 0.2)2(0.00970, 0.00272)(0.01962, 0.01585)(0.03925, 0.04755)3(0.00094, 0.00007)(0.00385, 0.00125)(0.01540, 0.01130)4(0.00009, 0)(0.00075, 0.00009)(0.00604, 0.00268)50,0(0.00014, 0)(0.00237, 0.00063)6(0, 0)(0.00002, 0)(0.00093, 0.00015)7(0, 0)0,0(0.00036, 0.00003)8(0, 0)(0, 0)(0.00014, 0)9(0, 0)(0, 0)(0.00005, 0)10(0, 0)(0, 0)0,0


## 5. Conclusions

In this research, we obtained the following new finding:(i)
We studied theK∗ iterative scheme of Ullah and Arshad for approximating fixed points of generalized α-nonexpansive mappings.(ii)
We successfully carried out some weak and strong convergence results under various mild conditions.(iii)
We carried out an application of our main outcome for solution of a FBVP in a Banach space setting.(iv)
A new example of generalizedα-nonexpansive mappings is constructed and proved that it exceeds properly the class of mappings with condition (C).(v)
Using our new example, we showed that theK∗ iterative scheme is more effective and suggests very high accurate numerical results in the setting of generalized α-nonexpansive mappings in the setting of generalized α-nonexpansive mappings.(vi)
Accordingly, our main outcome improved some recent results of Ullah and Arshad [19] form the case of mappings with condition (C) to the general case of mappings called generalized α-nonexpansive mappings. In a similar way, our results are the improvement and refinements of the results due to Agarwal [11], Abbas [16], Thakur [18], and many others from the setting of nonexpansive and Suzuki mappings to the general setting of generalized α-nonexpansive mappings.



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