# Iterative Approximation of Fixed Points by UsingF Iteration Process in Banach Spaces

Authors: Junaid Ahmad; Kifayat Ullah; Muhammad Arshad; Manuel de la Sen
Journal: Journal of Function Spaces
 (2021)

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## Abstract

We connect theF iteration process with the class of generalized α-nonexpansive mappings. Under some appropriate assumption, we establish some weak and strong convergence theorems in Banach spaces. To show the numerical efficiency of our established results, we provide a new example of generalized α-nonexpansive mappings and show that its F iteration process is more efficient than many other iterative schemes. Our results are new and extend the corresponding known results of the current literature.


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## Body

## 1. Introduction and Preliminaries

Once an existence of a solution for an operator equation is established then in many cases, such solution cannot be obtained by using ordinary analytical methods. To overcome such cases, one needs the approximate value of this solution. To do this, we first rearrange the operator equation in the form of fixed-point equation. We apply the most suitable iterative algorithm on the fixed point equation, and the limit of the sequence generated by this most suitable algorithm is in fact the value of the desired fixed point for the fixed point equation and the solution for the operator equation. The Banach Fixed Point Theorem [1] (BFPT, for short) suggests the elementry Picard iteration wt+1=Gwt in the case of contraction mappings. Since for the class of nonexpansive mappings, Picard iterates do not always converge to a fixed point of a certain nonexpansive mapping, we, therefore use some other iterative processes involving different steps and set of parameters. Among the other things, Mann [2], Ishikawa [3], Noor [4], S iteration of Agarwal et al. [5], SP iteration of Phuengrattana and Suantai [6], S∗ iteration of Karahan and Ozdemir [7], Normal-S [8], Picard-Mann hybrid [9], Krasnoselskii-Mann [10], Abbas [11], Thakur [12], and Picard-S [13] are the most studied iterative processes. In 2018, Ullah and Arshad introduced M [14] iteration process for Suzuki mappings and proved that it converges faster than all of these iteration processes.Very recently, Ali and Ali [15] introduced the novel iteration process, namely, F iterative scheme for generalized contractions as follows:
(1)w1∈P,ut=G1−αtwt+αtGwt,vt=Gut,wt+1=Gvt,t≥1,where αt∈0,1.They showed that theF iteration (1) is stable and has a better rate of convergence when compared with the other iterations in the setting of generalized contractions.Definition 1.
LetG:P→P. Then G is said to be
(i)
nonexpansive provided thatGp′−Gp″≤p′−p″, for every two p′,p″∈P(ii)
endowed with condition (C) provided that1/2p′−Gp′≤p′−p″ implies Gp′−Gp″≤p′−p″, for every two p′,p″∈P(iii)
generalizedα-nonexpansive provided that 1/2p′−Gp′≤p′−p″ implies Gp′−Gp″≤αp′−Gp″+αp″−Gp′+1−2αp′−p″, for every two p′,p″∈P and α∈0,1(iv)
endowed with conditionI [16] if one has a nondecreasing function f such that f0=0 and fa>0 at a>0 and p′−Gp′≥f(dp′,FG for all p′∈PIn 1965, Browder [17] and Gohde [18] are in a uniformly convex Banach space (UCBS), while Kirk [19] in a reflexive Banach space (RBS) established an existence of fixed point for nonexpansive maps. In 2008, Suzuki [20] showed that the class of maps endowed with condition C is weaker than the notion of nonexpansive maps and proved some related fixed point theorems in Banach spaces. In 2017, Pant and Shukla [21] proved that the notion of generalized α-nonexpansive maps is weaker than the notion of maps endowed with condition C. They proved some convergence theorems using Agarwal iteration [5] for these maps. Very recently, Ullah et al. [22] used M iteration for finding fixed points of generalized α-nonexpansive maps in Banach spaces. In this paper, we show under some conditions that F iteration converges better to a fixed point of generalized α-nonexpansive map as compared to the leading M iteration and hence many other iterative schemes.Definition 2.
Select a Banach spaceJ such that P⊆J is nonempty and wt⊆J is bounded. We set for fix j∈J the following.
a1 asymptotic radius of the bounded sequence wt at the point j by rj,wt≔limsupt→∞j−wt;
a2 asymptotic radius of the bounded sequence wt with the connection of P by rP,wt=infrj,wt:j∈P;
a3 asymptotic center of the bounded sequence wt with the connection of P by AP,wt=j∈P:rj,wt=rP,wt.It is worth mentioning thatAP,wt has a cardinality equal to one in the case of UCBS and nonempty convex in the case of weak compactness and convexity of P (see [23, 24]).Definition 3 (see [25]).
A Banach spaceJ is called with Opial’s condition in the case when every sequence wt⊆J which is weakly convergent to j∈J, then one has the following
(2)limsupt→∞wt−j<limsupt→∞wt−j′foreachj′∈P−j.
Pant and Shukla [21] observed the following facts about generalized α-nonexpansive operators.Proposition 4.
IfJ is a Banach space such that P⊆J is closed and nonempty, then for G:P→P and α∈0,1, the following hold
(i)
IfG is endowed with condition C, then G is generalized α-nonexpansive(ii)
IfG is generalized α-nonexpansive endowed with a nonempty fixed point, then Gp′−p∗≤p′−p∗ for p′∈P and p∗ is a fixed point of G(iii)
IfG is generalized α-nonexpansive, then FG is closed. Furthermore, when the underlying space J is strictly convex and the set P is convex, then the set FG is also convex(iv)
IfG is generalized α-nonexpansive, then for every choice of p′,p″∈P(3)p′−Gp″≤3+α1−αp′−Gp′+p′−p″.(v)
If the underlying spaceJ is with Opial condition, the operator G is generalized α-nonexpansive, wt is weakly convergent to l and limt→∞Gwt−wt=0, then l∈FG
We now state an interesting property of a UCBS from [26].Lemma 5.
SupposeJ is any UCBS. Choose 0<r≤αt≤s<1 and wt,xt⊆J such that limsupt→∞wt≤q, limsupt→∞xt≤q, and limt→∞αtwt+1−αtxt=q for some q≥0. Then, consequently, limt→∞wt−xt=0.


## 2. Main Results

We first provide a very basic lemma.Lemma 6.
SupposeJ is any UCBS and P⊆J is convex nonempty and closed. If G:P→P is generalized α-nonexpansive operator satisfying with FG≠∅ and wt is a sequence of F iterates (1), then, consequently, one has limt→∞wt−p∗ always exists for every taken p∗∈FG.Proof.
We may take anyp∗∈FG. Using Proposition 4(ii), we see that
(4)ut−p∗=G1−αtwt+αtGwt−p∗≤1−αtwt+αtGwt−p∗≤1−αtwt−p∗+αtGwt−p∗≤1−αtwt−p∗+αtwt−p∗≤wt−p∗.
This implies that(5)wt+1−p∗=Gvt−p∗≤vt−p∗=Gut−p∗≤ut−p∗≤wt−p∗.
Consequently,wt+1−p∗≤wt−p∗, that is, wt−p∗ is bounded as well as nonincreasing. This follows that limt→∞wt−p∗ exists for each p∗∈FG.
We now provide the necessary and sufficient requirements for the existence of fixed points for any given generalized nonexpansive mappings in a Banach space.Theorem 7.
SupposeJ is any UCBS and P⊆J is convex nonempty and closed. If G:P→P is generalized α-nonexpansive operator and wt is a sequence of F iterates (1). Then, FG≠∅ if and only if wt is bounded and limt→∞Gwt−wt=0.Proof.
Suppose thatFG≠∅ and p∗∈FG. Take any p∗∈FG, and so applying Lemma 6, we have limt→∞wt−p∗ exists and wt is bounded. Suppose that this limit is equal to some ε, that is,
(6)limt→∞wt−p∗=ε.
As we have established in the proof of Lemma6 that
(7)ut−p∗≤wt−p∗.
This together with (6) gives that
(8)limsupt→∞ut−p∗≤limsupt→∞wt−p∗=ε.
Sincep∗ is in the set FG, so we may apply Proposition 4(ii) to obtain the following
(9)Gwt−p∗≤wt−p∗,⇒limsupt→∞Gwt−p∗≤limsupt→∞wt−p∗=ε.
Now, if we look in the proof of Lemma6, we can see the following
(10)wt+1−p∗≤ut−p∗⇒ε=liminft→∞wt+1−p∗≤liminft→∞ut−p∗.
From (8) and (10), we have
(11)ε=limt→∞ut−p∗.
By (11) and (1), one has
(12)ε=limt→∞ut−p∗=limt→∞G1−αtwt+αtGwt−p∗≤limt→∞1−αtwt−p∗+αtGwt−p∗≤limt→∞1−αtwt−p∗+limt→∞αtGwt−p∗≤limt→∞1−αtwt−p∗+limt→∞αtwt−p∗=limt→∞wt−p∗≤ε.
If and only if(13)ε=limt→∞1−αtwt−p∗+αtGwt−p∗.
One can now apply the Lemma5, to obtain
(14)limt→∞Gwt−wt=0.
Conversely, we want to show that the setFG is nonempty under the assumptions that wt is bounded such that limt→∞Gwt−wt=0. We may choose a point p∗∈AP,wt. If we apply Proposition 4(iv), then one can observe the following
(15)rGp∗,wt=limsupt→∞wt−Gp∗≤3+α1−αlimsupt→∞Gwt−wt+limsupt→∞wt−p∗=limsupt→∞wt−p∗=rp∗,wt.
We observed thatGp∗∈A(P,wt. By using the facts that this set has only element in the case of UCBS J, one concludes Gp∗=p∗, accordingly the set FG is nonempty.The weak convergence ofF iteration is established as follows.Theorem 8.
SupposeJ is any UCBS with Opial condition and P⊆J is convex nonempty and closed. If G:P→P is generalized α-nonexpansive operator with FG≠∅ and wt is a sequence of F iterates (1). Then, consequently, wt converges weakly to a fixed point of G.Proof.
By Theorem7, the given sequence wt is bounded. Since J is UCBS, J is RBS. Therefore, some one construct a weakly convergent sequence of wt. We may assume that wtr be this subsequence having weak limit x1∈P. If we apply Theorem 7 on this subsequence, we obtain limr→∞wtr−Gwtr=0. Thus, by Proposition 4(v), one has x1∈FG. It is sufficient to show that wt converges weakly to x1. In fact, if wt does not converge weakly to x1. Then, there exists a subsequence wts of wt and x2∈P such that wts converges weakly to x2 and x2≠x1. Again by Proposition 4(v), x2∈FG. By Lemma 6 together with Opial property, we have
(16)limt→∞xn−l1=limr→∞wtr−x1<limr→∞wtr−x2=limt→∞wt−x2=lims→∞wts−x2<lims→∞wts−x1∣=limt→∞wt−x1.
This is a contradiction. So, we havex1=x2. Thus, wt converges weakly to x1∈FG.Now we provide some strong convergence results.Theorem 9.
SupposeJ is any UCBS and P⊆J is convex nonempty and compact. If G:P→P is generalized α-nonexpansive operator with FG≠∅ and wt is a sequence of F iterates (1). Then, consequently, wt converges strongly to a fixed point of G.Proof.
Since the domainP is a compact subset of J and wt⊆P. It follows that a subsequence wtr of wt exists such that limr→∞wt−p∗∗=0 for some p∗∗∈P. In the view of Theorem 7, limr→∞Pwtr−wtr=0. Applying Proposition 4(iv), one has
(17)wtr−Gp∗∗≤3+α1−αwtr−Gwtr+wtr−p∗∗.
Hence, if we letr→∞, then Gp∗∗=p∗∗. The fact that p∗∗ is the strong limit of wt now follows from the existence of limt→∞wt−p∗∗.Theorem 10.
SupposeJ is any UCBS and P⊆J is convex nonempty and closed. If G:P→P is generalized α-nonexpansive operator with FG≠∅ and wt is a sequence of F iterates (1) and liminft→∞dwt,FG=0. Then, consequently, wt converges strongly to a fixed point of G.Proof.
By using Lemma6, one has limt→∞wt−p∗ exists, for every fixed point of G. It follows that limt→∞dwt,FG exists. Accordingly
(18)limt→∞dwn,FG=0.
The above limit provides us two subsequencewtr and pr of wt and FG, respectively, in the following way
(19)wtr−pr≤12rforeachr≥1.
By looking into the proof of Lemma6, we see that wt is nonicreasing, therefore
(20)wtr+1−pr≤wtr−pr≤12r.
It follows that(21)pr+1−pr≤pr+1−wtr+1+wtr+1−pr≤12r+1+12r≤12r−1→0,asr→∞.
Consequently, we obtained thatlimr→∞pr+1−pr=0 which show that pr is Cauchy sequence in FG and so it converges to an element p∗∗. Applying Proposition 4(iii), FG is closed and so p∗∗∈FG. By Lemma 6, limt→∞wt−p∗∗ exists and hence p∗∗ is the strong limit of wt.Theorem 11.
SupposeJ is any UCBS and P⊆J is convex nonempty and closed. If G:P→P is generalized α-nonexpansive operator satisfying condition I with FG≠∅ and wt is a sequence of F iterates (1). Then, consequently, wt converges strongly to a fixed point of G.Proof.
Keeping Theorem7 in mind, one can write
(22)liminft→∞Gwt−wt=0.
From the definition of condition (I), we see that(23)wt−Gwt≥fdwt,FG.
Applying (22) on (23), we have
(24)liminft→∞fdwt,FG=0.
It follows that(25)liminft→∞dwt,FG=0.
Now applying Theorem10, wt is strongly convergent to a fixed point of G.


## 3. Example

To support the main results, we provide an example of generalizedα-nonexpansive mappings, which is not endowed with condition (C). Using this example, we compare F with other iterations in the setting of generalized α-nonexpansive mappings.Example 12.
We take a setP=7,13 and set a self map on G by the following rule:
(26)Gp′=p′+72ifp′<13,7ifz=13.
We show thatG is generalized α-nonexpansive having α=1/2, but not Suzuki mapping. This example thus exceeds the class of Suzuki mappings.Case I.
Whenp′=13=p″, we have
(27)12p′−Gp″+12p″−Gp′+1−212p′−p″≥0=Gp′−Gp″.Case II.
Choosep′,p″<13, we have
(28)12p′-Gp′′+12p′′-Gp′+1-212p′-p′′=12p′′-p′+72+12p′-p′′+72≥12p′′-p′+72-p′-p′′+72=122p′′-p′-7-2p′+p′′+72=123p′′-3p′2=34p′-p′′≥12p′-p′′=Gp′-Gp′′.Case III.
Whenp′=13 and p″<13, we have
(29)12p′-Gp′′+12p′′-Gp′+1-212p′-p′′=12p′-7+12p′′-p′+72≥12p′-7=p′-72=Gp′-Gp′′.
Consequently,Gp′−Gp″≤1/2∣p′−Gp″∣+1/2∣p″−Gp′∣+1−21/2∣p′−p″∣ for every two points p′,p″∈G. Now if one chooses p′=11.8 and p″=13, we must have ∣p′−p″∣=1.2,∣Gp′−Gp″∣=2.4 and 1/2∣p′−Gp′∣=1.2. It has been observed, 1/2∣p′−Gp′∣≤∣p′−p′′∣ and ∣Gp′−Gp″∣>∣p′−p″∣. Thus, G exceeded the class of Suzuki mappings.We now compare the effectiveness of the iterative schemeF [15] with the leading M [14] and Picard [13] and the elementry S [5], Ishikawa [3] and Mann [2] approximation scheme. We may take αt=0.85 and βt=0.65. For the strating w1=7.9, we can see some values in Table 1. Furthermore, Figure 1 provides information about the behavior of the leading schemes. Clearly, F iterative scheme is more effective than the other schemes in the general context of generalized α-nonexpansive maps.Remark 13.
In the view of the above discussion, we noted that the main theorems and outcome of this paper improved and extended the main results of Ullah and Arshad [14] from Suzuki mappings to generalized α-nonexpansive mappings and from the setting of M iteration to the more general setting of F iteration process. Moreover, the main results of this paper improved the results of Ali and Ali [15] from the setting of contractions to the general context of generalized α-nonexpansive mappings. We have also improved the results of Ullah et al. [22] in the sense of better rate of convergence.Table 1
Numerical data generated byF, M, Picard-S, S, Ishikawa, and Mann iterative approximation schemes for the self map given in Example 12.
FMPicard-SSIshikawaMann17.97.97.97.97.97.927.064687507.129375007.162843757.32568757.39318757.5175000037.004649417.018597667.029464547.117858167.171773797.2975625047.000334187.002673417.005331247.042649927.075043677.1710984457.000024027.000384307.000964627.015433947.032784717.0983816067.000001737.000055247.000174547.005585167.014322827.0565694277.000000127.000007947.000031587.002021137.006257287.0325274287.000000017.000001147.000005717.00073147.002733657.01870326977.000000167.000001037.000264677.001194267.010754381077.000000027.000000197.000095787.000521747.0061837711777.000000037.000034667.000227947.0035556712777.000000017.000012547.000099597.00204451137777.000004547.000043507.00117559147777.000001647.000019017.00067597157777.000000597.000008307.00038868167777.000000227.000003637.00022349177777.000000087.000001587.00012851187777.000000037.000000697.00007389197777.000000017.000000307.000042492077777.000000137.000024432177777.000000067.000014052277777.000000037.000008082377777.000000017.0000046424777777.00000260Figure 1
Convergence analysis view ofF (cyan), M (red), Picard-S (green), S (blue), Ishikawa (magenta), and Mann (yellow) iteration process for the mapping given in Example 12.



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