We connect the F 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.

Basque GovernmentIT1207-19
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  (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 , Ishikawa , Noor , S iteration of Agarwal et al. , SP iteration of Phuengrattana and Suantai , S iteration of Karahan and Ozdemir , Normal-S , Picard-Mann hybrid , Krasnoselskii-Mann , Abbas , Thakur , and Picard-S  are the most studied iterative processes. In 2018, Ullah and Arshad introduced M  iteration process for Suzuki mappings and proved that it converges faster than all of these iteration processes.

Very recently, Ali and Ali  introduced the novel iteration process, namely, F iterative scheme for generalized contractions as follows: (1)w1P,ut=G1αtwt+αtGwt,vt=Gut,wt+1=Gvt,t1,where αt0,1.

They showed that the F 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.

Let G:PP. Then G is said to be

nonexpansive provided that GpGppp, for every two p,pP

endowed with condition (C) provided that 1/2pGppp implies GpGppp, for every two p,pP

generalized α-nonexpansive provided that 1/2pGppp implies GpGpαpGp+αpGp+12αpp, for every two p,pP and α0,1

endowed with condition I  if one has a nondecreasing function f such that f0=0 and fa>0 at a>0 and pGpf(dp,FG for all pP

In 1965, Browder  and Gohde  are in a uniformly convex Banach space (UCBS), while Kirk  in a reflexive Banach space (RBS) established an existence of fixed point for nonexpansive maps. In 2008, Suzuki  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  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  for these maps. Very recently, Ullah et al.  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 space J such that PJ is nonempty and wtJ is bounded. We set for fix jJ the following.

a1 asymptotic radius of the bounded sequence wt at the point j by rj,wtlimsuptjwt;

a2 asymptotic radius of the bounded sequence wt with the connection of P by rP,wt=infrj,wt: jP;

a3 asymptotic center of the bounded sequence wt with the connection of P by AP,wt=jP:rj,wt=rP,wt.

It is worth mentioning that AP,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 [<xref ref-type="bibr" rid="B25">25</xref>]).

A Banach space J is called with Opial’s condition in the case when every sequence wtJ which is weakly convergent to jJ, then one has the following (2)limsuptwtj<limsuptwtjforeachjPj.

Pant and Shukla  observed the following facts about generalized α-nonexpansive operators.

Proposition 4.

If J is a Banach space such that PJ is closed and nonempty, then for G:PP and α0,1, the following hold

If G is endowed with condition C, then G is generalized α-nonexpansive

If G is generalized α-nonexpansive endowed with a nonempty fixed point, then Gpppp for pP and p is a fixed point of G

If G 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

If G is generalized α-nonexpansive, then for every choice of p,pP

(3)pGp3+α1αpGp+pp.

If the underlying space J is with Opial condition, the operator G is generalized α-nonexpansive, wt is weakly convergent to l and limtGwtwt=0, then lFG

We now state an interesting property of a UCBS from .

Lemma 5.

Suppose J is any UCBS. Choose 0<rαts<1 and wt,xtJ such that limsuptwtq, limsuptxtq, and limtαtwt+1αtxt=q for some q0. Then, consequently, limtwtxt=0.

2. Main Results

We first provide a very basic lemma.

Lemma 6.

Suppose J is any UCBS and PJ is convex nonempty and closed. If G:PP is generalized α-nonexpansive operator satisfying with FG and wt is a sequence of F iterates (1), then, consequently, one has limtwtp always exists for every taken pFG.

Proof.

We may take any pFG. Using Proposition 4(ii), we see that (4)utp=G1αtwt+αtGwtp1αtwt+αtGwtp1αtwtp+αtGwtp1αtwtp+αtwtpwtp.

This implies that (5)wt+1p=Gvtpvtp=Gutputpwtp.

Consequently, wt+1pwtp, that is, wtp is bounded as well as nonincreasing. This follows that limtwtp exists for each pFG.

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.

Suppose J is any UCBS and PJ is convex nonempty and closed. If G:PP is generalized α-nonexpansive operator and wt is a sequence of F iterates (1). Then, FG if and only if wt is bounded and limtGwtwt=0.

Proof.

Suppose that FG and pFG. Take any pFG, and so applying Lemma 6, we have limtwtp exists and wt is bounded. Suppose that this limit is equal to some ε, that is, (6)limtwtp=ε.

As we have established in the proof of Lemma 6 that (7)utpwtp.

This together with (6) gives that (8)limsuptutplimsuptwtp=ε.

Since p is in the set FG, so we may apply Proposition 4(ii) to obtain the following (9)Gwtpwtp,limsuptGwtplimsuptwtp=ε.

Now, if we look in the proof of Lemma 6, we can see the following (10)wt+1putpε=liminftwt+1pliminftutp.

From (8) and (10), we have (11)ε=limtutp.

By (11) and (1), one has (12)ε=limtutp=limtG1αtwt+αtGwtplimt1αtwtp+αtGwtplimt1αtwtp+limtαtGwtplimt1αtwtp+limtαtwtp=limtwtpε.

If and only if (13)ε=limt1αtwtp+αtGwtp.

One can now apply the Lemma 5, to obtain (14)limtGwtwt=0.

Conversely, we want to show that the set FG is nonempty under the assumptions that wt is bounded such that limtGwtwt=0. We may choose a point pAP,wt. If we apply Proposition 4(iv), then one can observe the following (15)rGp,wt=limsuptwtGp3+α1αlimsuptGwtwt+limsuptwtp=limsuptwtp=rp,wt.

We observed that GpA(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 of F iteration is established as follows.

Theorem 8.

Suppose J is any UCBS with Opial condition and PJ is convex nonempty and closed. If G:PP 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 Theorem 7, 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 x1P. If we apply Theorem 7 on this subsequence, we obtain limrwtrGwtr=0. Thus, by Proposition 4(v), one has x1FG. 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 x2P such that wts converges weakly to x2 and x2x1. Again by Proposition 4(v), x2FG. By Lemma 6 together with Opial property, we have (16)limtxnl1=limrwtrx1<limrwtrx2=limtwtx2=limswtsx2<limswtsx1=limtwtx1.

This is a contradiction. So, we have x1=x2. Thus, wt converges weakly to x1FG.

Now we provide some strong convergence results.

Theorem 9.

Suppose J is any UCBS and PJ is convex nonempty and compact. If G:PP 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 domain P is a compact subset of J and wtP. It follows that a subsequence wtr of wt exists such that limrwtp=0 for some pP. In the view of Theorem 7, limrPwtrwtr=0. Applying Proposition 4(iv), one has (17)wtrGp3+α1αwtrGwtr+wtrp.

Hence, if we let r, then Gp=p. The fact that p is the strong limit of wt now follows from the existence of limtwtp.

Theorem 10.

Suppose J is any UCBS and PJ is convex nonempty and closed. If G:PP is generalized α-nonexpansive operator with FG and wt is a sequence of F iterates (1) and liminftdwt,FG=0. Then, consequently, wt converges strongly to a fixed point of G.

Proof.

By using Lemma 6, one has limtwtp exists, for every fixed point of G. It follows that limtdwt,FG exists. Accordingly (18)limtdwn,FG=0.

The above limit provides us two subsequence wtr and pr of wt and FG, respectively, in the following way (19)wtrpr12rforeachr1.

By looking into the proof of Lemma 6, we see that wt is nonicreasing, therefore (20)wtr+1prwtrpr12r.

It follows that (21)pr+1prpr+1wtr+1+wtr+1pr12r+1+12r12r10,asr.

Consequently, we obtained that limrpr+1pr=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 pFG. By Lemma 6, limtwtp exists and hence p is the strong limit of wt.

Theorem 11.

Suppose J is any UCBS and PJ is convex nonempty and closed. If G:PP 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 Theorem 7 in mind, one can write (22)liminftGwtwt=0.

From the definition of condition (I), we see that (23)wtGwtfdwt,FG.

Applying (22) on (23), we have (24)liminftfdwt,FG=0.

It follows that (25)liminftdwt,FG=0.

Now applying Theorem 10, wt is strongly convergent to a fixed point of G.

3. Example

To support the main results, we provide an example of generalized α-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 set P=7,13 and set a self map on G by the following rule: (26)Gp=p+72ifp<13,7ifz=13.

We show that G is generalized α-nonexpansive having α=1/2, but not Suzuki mapping. This example thus exceeds the class of Suzuki mappings.

Case I.

When p=13=p, we have (27)12pGp+12pGp+1212pp0=GpGp.

Case II.

Choose p,p<13, we have (28)12p-Gp+12p-Gp+1-212p-p=12p-p+72+12p-p+7212p-p+72-p-p+72=122p-p-7-2p+p+72=123p-3p2=34p-p12p-p=Gp-Gp.

Case III.

When p=13 and p<13, we have (29)12p-Gp+12p-Gp+1-212p-p=12p-7+12p-p+7212p-7=p-72=Gp-Gp.

Consequently, GpGp1/2pGp+1/2pGp+121/2pp for every two points p,pG. Now if one chooses p=11.8 and p=13, we must have pp=1.2,GpGp=2.4 and 1/2pGp=1.2. It has been observed, 1/2pGppp and GpGp>pp. Thus, G exceeded the class of Suzuki mappings.

We now compare the effectiveness of the iterative scheme F  with the leading M  and Picard  and the elementry S , Ishikawa  and Mann  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  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  from the setting of contractions to the general context of generalized α-nonexpansive mappings. We have also improved the results of Ullah et al.  in the sense of better rate of convergence.

Numerical data generated by F, M, Picard-S, S, Ishikawa, and Mann iterative approximation schemes for the self map given in Example 12.

FMPicard-SSIshikawaMann
17.97.97.97.97.97.9
27.064687507.129375007.162843757.32568757.39318757.51750000
37.004649417.018597667.029464547.117858167.171773797.29756250
47.000334187.002673417.005331247.042649927.075043677.17109844
57.000024027.000384307.000964627.015433947.032784717.09838160
67.000001737.000055247.000174547.005585167.014322827.05656942
77.000000127.000007947.000031587.002021137.006257287.03252742
87.000000017.000001147.000005717.00073147.002733657.01870326
977.000000167.000001037.000264677.001194267.01075438
1077.000000027.000000197.000095787.000521747.00618377
11777.000000037.000034667.000227947.00355567
12777.000000017.000012547.000099597.00204451
137777.000004547.000043507.00117559
147777.000001647.000019017.00067597
157777.000000597.000008307.00038868
167777.000000227.000003637.00022349
177777.000000087.000001587.00012851
187777.000000037.000000697.00007389
197777.000000017.000000307.00004249
2077777.000000137.00002443
2177777.000000067.00001405
2277777.000000037.00000808
2377777.000000017.00000464
24777777.00000260

Convergence analysis view of F (cyan), M (red), Picard-S (green), S (blue), Ishikawa (magenta), and Mann (yellow) iteration process for the mapping given in Example 12.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The authors are grateful to the Basque Government for its support through grant IT1207-19.

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