# Approximation of Fixed Points for Mean Nonexpansive Mappings 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

In this paper, we establish weak and strong convergence theorems for mean nonexpansive maps in Banach spaces under the Picard–Mann hybrid iteration process. We also construct an example of mean nonexpansive mappings and show that it exceeds the class of nonexpansive mappings. To show the numerical accuracy of our main outcome, we show that Picard–Mann hybrid iteration process of this example is more effective than all of the Picard, Mann, and Ishikawa iterative processes.


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## Body

## 1. Introduction and Preliminaries

SupposeY is a Banach space and ∅≠W⊆Y. Consider a selfmap S:W⟶W. If an element e0∈W exists such that e0=Se0, then we say that e0 is a fixed point for S. In this manuscript, we essentially represent the set e0∈W:e0=Se0 by FS. The selfmap S:W⟶W is called contraction [1] if
(1)Sp−Sq≤αp−q,forallp,q∈Wandα∈0,1.The selfmapS:W⟶W is called nonexpansive if (1) holds for the value α=1. In 1965, Browder [2] and Gohde [3] proved a fixed point theorem for a nonexpansive map S:W⟶W under the restriction that Y is a uniformly convex Banach space (UCBS) and ∅≠W⊆Y is bounded as well as closed and convex.In [4], Zhang provided the following class of mappings.Definition 1.
LetW≠∅ be a subset of a Banach space. A selfmap S:W⟶W is called mean nonexpansive if for all p,q∈W there are non-negative real numbers a,b such that a+b≤1, we have
(2)Sp−Sq≤ap−q+bp−Sq.Zhang [4] provided an existence of fixed point result for mean nonexpansive mappings in Banach space setting under the normal structure assumption. After this, Wu and Zhang [5] and Zuo [6] investigated some other elementary properties and fixed point results for these mappings. In [7], Zhou and Cui used Ishikawa [8] iteration for approximating fixed points for these maps. The main aim here is to suggest some weak and strong convergence theorems for these mappings under the Picard-Mann hybrid [9] iteration and to show by a new example of mean nonexpansive maps that it converges better than the Ishikawa [8] and Mann [10] iteration processes.Remark 2.
It is easy to observe that each nonexpansive mapping is mean nonexpansive. Once again, in this research, we shall provide a new example to show that the converse is not true in general, that is, the class of mean nonexpansive maps properly includes the class of nonexpansive maps.In the following example,S is mean nonexpansive but not nonexpansive.Example 3 (see [6]).
ConsiderW=0,1 and set S:W⟶W by
(3)Sp=p5ifp<12p6ifp≥12.The Banach [1] celebrated fixed point theorem suggests the existence and uniqueness of a fixed point for a self contraction S:W⟶W under the restriction that Y is complete metric space and ∅≠W⊆Y is closed. Also this theorem essentially uses the Picard iteration [11] for finding this unique fixed point. Nevertheless, in the case of nonexpansive maps and hence for generalized nonexpansive maps, the Picard iteration fails to converge in the associated fixed point set. For some more literature on iterative schemes, please cite the work in [12–14]. Assume that W is any nonempty subset of a Banach space and S:W⟶W.The Picard [11] iterative process is stated as:
(4)p1∈W,pm+1=Spm,m≥1,Mann [10] iterative process is stated as:
(5)p1∈W,pm+1=1−αmpm+αmSpm,m≥1,where αm∈0,1.Ishikawa [8] iteration process may be viewed as a two-step Mann iteration, stated as follows:
(6)p1∈W,qm=1−βmpm+βmSpm,pm+1=1−αmpm+αmSqm,m≥1,where αm,βm∈0,1.Khan [9] introduced the Picard–Mann hybrid iteration as follows:
(7)p1∈W,qm=1−αmpm+αmSpm,pm+1=Sqm,m≥1,where αm∈0,1.Khan [9] provided the weak and strong convergence of the scheme (7) for the class of nonexpansive operators. Furthermore, he proved that the Picard–Mann hybrid iteration process is more effective than the Picard (4), Mann (5) and Ishikawa (6) iteration processes in the setting of nonexpansive maps. In this paper, we connect this scheme with the class of mean nonexpansive mappings, and in this way, we extend his results in the more general setting of mean nonexpansive mappings.Now we provide some elementary definitions and results, which will be used in sequel.Definition 4 [15].
A Banach spaceY is called UCBS if and only if for every choice of r∈0,2, one has a s>0 such that for every p,q in Y, 1/2p+q≤1−s whenever p≤1,q≤1 and p−q>r.Definition 5 [16].
A Banach spaceY is said to satisfy the Opial’s property if any weakly convergent sequence pm in Y which admits a weak limit c∈Y, one has
(8)limsupm⟶∞pm−c<limsupm⟶∞pm−c′,foreachc′∈Y−c.Definition 6 [17].
SupposeW≠∅ is any subset of a Banach space Y. A selfmap S:W⟶W is said to be endowed with the condition I if and only if a function L:0,∞⟶0,∞ exists sucht that L0=0 and Lr>0 for all r>0 and p−Sp≥Ldp,FS for each p∈W.The following lemma holds, which suggests many examples of mean nonexpansive mappings.Lemma 7.
IfS is a selfmap and nonexpansive on a subset W≠∅ of a Banach space. Then S is mean nonexpansive.From the definition of mean nonexpansive maps, we have the following facts.Lemma 8.
IfS is a selfmap and mean nonexpansive on a subset W≠∅ of a Banach space Y. Then FS is closed. Moreover, if Y is strictly convex and W is convex, then FS is also convex.Theorem 9 [6].
LetW≠∅ be a subset of a reflexive Banach space (RBS) Y having Opial property. Let S:W⟶W be a mean nonexpansive mapping. If pm⊆W be such that.
(a0) pm converges weakly to e0,
(b0) limm⟶∞Spm−pm=0,
thene0=Se0.Any UCBS can be characterized by the following way.Lemma 10 [18].
IfY is a UCBS and If sm and wm are two sequences in Y such that limsupm⟶∞sm≤l, limsupm⟶∞wm≤l and limm⟶∞δmsm+1−δmwm=l for some l≥0 and 0<q≤δm≤p<1. Then, limm⟶∞sm−wm=0.


## 2. Main Results

The following results are the main outcome of this section. Notice that all these results improve and extend some main results of Khan [9] from the case of nonexpansive maps to case of mean nonexpansive maps.Lemma 11.
LetW≠∅ be a convex closed subset of a UCBS Y and S:W⟶W be a mean nonexpansive mapping and FS≠∅. Assume that pm is a sequence of Picard–Mann hybrid iterative process (7). Consequently limm⟶∞pm−e0 exists for each e0∈FS.Proof.
Lete0∈FS. Then using (7), we have
(9)qm−e0≤1−αmpm−e0+αmpm−e0≤1−αmpm−e0+αmSpm−e0=1−αmpm−e0+αmSpm−Se0≤1−αmpm−e0+αmapm−e0+bpm−Se0=1−αmpm−e0+αmapm−e0+bpm−e0=1−αmpm−e0+αma+bpm−e0=1−αmpm−e0+αmpm−e0≤pm−e0.
This implies that(10)pm+1−e0=Sqm−e0=Sqm−Se0≤aqm−e0+bqm−Se0=aqm−e0+bqm−e0=a+bqm−e0≤qm−e0≤pm−e0.
We have showed thatpm+1−e0≤pm−e0. It follows that pm−e0 is nonincreasing and bounded. Thus limm⟶∞pm−e0 exists for each e0∈FS.☐Theorem 12.
LetW≠∅ be a convex closed subset of a UCBS Y and S:W⟶W be a mean nonexpansive mapping and FS≠∅. Assume that pm is a sequence of Picard–Mann hybrid iterative process (7). Consequently, pm is bounded in Y with the property limm⟶∞Spm−pm=0.Proof.
Since the setFS is nonempty so we may choose any e0∈FS. By Lemma 11, limm⟶∞pm−e0 exists and pm is bounded. Suppose that
(11)limm⟶∞pm−e0=s.
By looking in the proof of Lemma11, one see
(12)qm−e0≤pm−e0,⇒limsupm⟶∞qm−e0≤limsupm⟶∞pm−e0=s.
Now(13)Spm−e0=Spm−Se0≤apm−e0+bpm−Se0=a+bpm−e0≤pm−e0.
It follows that(14)limsupm⟶∞Spm−e0≤limsupm⟶∞pm−e0=s.
Again by looking in the proof of Lemma11, one see
(15)pm+1−q≤qm−q.
It follows that(16)s≤liminfm⟶∞qm−e0.
From (12) and (16), we obtain
(17)s=limm⟶∞qm−e0.
From (17), we have
(18)s=limm⟶∞qm−q=limm⟶∞1−αmpm+αmSpm−e0=limm⟶∞1−αmpm−e0+αmSpm−e0.
Hence,(19)s=limm⟶∞1−αmpm−e0+αmSpm−e0.
Now from (11), (14) and (19) together with Lemma 10, we obtain
(20)limm⟶∞Spm−pm=0.☐We now provide a weak convergence theorem under the assumption of the Opial’s condition.Theorem 13.
LetW≠∅ be a convex closed subset of a UCBS Y and S:W⟶W be a mean nonexpansive mapping and FS≠∅. Assume that pm is a sequence of Picard–Mann hybrid iterative process (7). If Y has the Opial property, then pm converges weakly to a point of of FS.Proof.
By Theorem12, the sequence pm is bounded and limn⟶∞Spm−pm=0. Since Y is UCBS, it follows that Y is RBS. Thus one has a weakly convergent subsequence pmi of pm exists with some weak limit q1∈W. By Theorem 9, q1∈FS. Next we show that pm is weakly convergent to q1. We may suppose that pm is not weakly convergent to q1, that is, one has a weakly convergent subsequence pmj of pm with a weak limit q2∈W and q2≠q1. Again applying Theorem 9, q2∈FS. By applying Opial’s condition and keeping Lemma 11 in mind, it follow that
(21)limm⟶∞pm−q1=limi⟶∞pmi−q1<limi⟶∞pmi−q2=limm⟶∞pm−q2=limj⟶∞pmj−q2<limj⟶∞pmj−q1=limm⟶∞pm−q1.
Hence we have seen a contradiction. Accordingly, we haveq1=q2. Thus, pm converges weakly to q1∈FS.☐.The strong convergence theorem under the assumption of compactness is established as follows.Theorem 14.
LetW≠∅ be a convex compact subset of a UCBS Y and S:W⟶W be a mean nonexpansive mapping and FS≠∅. Assume that pm is a sequence of Picard–Mann hybrid iterative process (7). If W is compact, then pm converges strongly to an element of FS.Proof.
SinceW is compact, and pm⊆W. One can choose a strongly convergent subsequence pmk of pm such that pmk⟶u. Now we show that Su=u. For this
(22)u−Su≤u−pmk+pmk−Spmk+Spmk−Su≤u−pmk+pmk−Spmk+apmk−u+bu−Spmk≤u−pmk+pmk−Spmk+apmk−u+bu−pmk+bpmk−Spmk=a+b+1u−pmk+b+1pmk−Spmk.☐Consequently, we obtained(23)u−Su≤a+b+1u−pmk+b+1pmk−Spmk.According to Theorem12, we have limk⟶∞pmk−Spmk=0, so applying k⟶∞, we obtain Su=u. This shows that u∈FS. By Lemma 11, limm⟶∞pm−u exists. Consequently, u is the strong limit of pm and element of FS.The strong convergence theorem without the compactness assumption is established as follows.Theorem 15.
LetW≠∅ be a convex closed subset of a UCBS Y and S:W⟶W be a mean nonexpansive mapping and FS≠∅. Assume that pm is a sequence of Picard–Mann hybrid iterative process (7). Then pm converges strongly to an element of FS if and only if liminfm⟶∞dpm,FS=0.Proof.
The necessity is obvious.
Conversely, suppose thatliminfm⟶∞dpm,FS=0 and e0∈FS. From the Lemma 11, limm⟶∞pm−e0 exists. Therefore limm⟶∞dpm,FS=0, by assumption. We prove that pm is a Cauchy sequence in W. As limm⟶∞dpm,FS=0, for a given ε>0, there exists r0∈ℕ such that for each m≥r0,
(24)dpm,FS<ε2.⇒infpm−e0:e0∈FS<ε2.
In particularinfpr0−e0:e0∈FS<ε/2. Therefore there exists e0∈FS such that
(25)pr0−e0<ε2.
Now forr,m≥k0,
(26)pm+r−pm≤pm+r−e0+pm−e0≤pr0−e0+pr0−e0=2pr0−e0<ε.
This shows thatpm is a Cauchy sequence in W. As W is closed subset of a Banach space Y, so there exists a point e0∈W such that limm⟶∞pm=e0. Now limm⟶∞dpm,FS=0 gives that de0,FS=0. Since from Lemma 8, we have the set FS a closed set in W. Hence e0∈FS.☐.The below facts are essentially due to Sentor and Dotson [17].Definition 16.
LetW≠∅ be a subset of a Banach space Y. A selfmap S:W⟶W is said to be endowed with the condition I if and only if a function L:0,∞⟶0,∞ exists such that L0=0 and Lr>0 for all r>0 and p−Sp≥Ldp,FS for each p∈W.The strong convergence theorem under the assumption of conditionI is established as follows.Theorem 17.
LetW≠∅ be a convex closed subset of a UCBS Y and S:W⟶W be a mean nonexpansive mapping and FS≠∅. Assume that pm is a sequence of Picard–Mann hybrid iterative process (7). If S is endowed with condition I, then pm converges strongly to an element of FS.Proof.
From Theorem12, we have
(27)limm⟶∞Spm−pm=0.
Condition (I) ofS provides
(28)limm⟶∞dpm,FS=0.
Now all the requirements of the Theorem15 are available, so we conclude that pm converges strongly to an element of FS.☐Remark 18.
In the view of Lemma7, our results contains the case of nonexpansive mappings.


## 3. Example

Now we want to provide a new example of mean nonexpansive maps.Example 19.
LetS:0,1⟶0,1 be defined by Sp=0 if 0≤p<1/515 and Sp=p/5 for 1/515≤p≤1. Suppose a=1/4=b. Then it is easy to show a+b≤1.
Now we may consider the following cases.(i)
Suppose that0≤p,q<1/515. Then(29)Sp−Sq=0≤ap−q+bp−Sq.(ii)
Suppose that1/515≤p,q≤1. Then(30)ap−q+bp−Sq=14p−q+bp−q5≥14p−q≥15p−q=Sp−Sq(iii)
Suppose that0≤q<1/515 and 1/515≤p≤1(31)ap−q+bp−Sq=ap−q+14p−0≥14p≥15p=Sp−Sq.(iv)
Suppose that0≤p<1/515 and 1/515≤q≤1(32)ap−q+bp−Sq=14p−q+14p−q5≥14p−q−p−q5=144q5=15q=Sp−Sq.Thus we conclude thatS is mean nonexpansive. Notice that, when p=1/600 and q=1/515, then Sp−Sq>∣p−q∣ and so S is not nonexpansive. Choose αm=0.70 and βm=0.65, then the strong convergence and effectiveness of the Picard-Mann hybrid iteration can be seen in Table 1 and Figure 1.Table 1
Strong convergence of Picard–Mann hybrid, Picard, Ishikawa and Mann iterations to the fixed pointe0=0 of S in Example 19.
mPicard-Mann hybrid (7)Picard (4)Ishikawa (6)Mann (5)10.900000000000000.900000000000000.900000000000000.9000000000000020.079200000000000.180000000000000.330480000000000.3960000000000030.006969600000000.036000000000000.121352250000000.1742400000000040.000613324800000.007200000000000.044560548403200.07666560000000500.001440000000000.016362633373650.033732864000006000.006008358974800.014842460100007000.002206269415540.006530682470008000.000661880824660.002873500286979000.000198564247390.0012643401262610000.000059569274210.0003793020378811000.000017870782260.00011379061136Figure 1
Behaviors of Picard-Mann hybrid (black), Picard (blue), Ishikawa (cyan) and Mann (red) iterates to the unique fixed pointe0=0 of the selfmap S.


## 4. Conclusions

We begun the finding of fixed points for mean nonexpansive operators in Banach spaces under the Picard–Mann hybrid iterative process. Some convergence results are established under different assumptions. It has been showen by an example that the notion of mean nonexpansive maps is properly more general than the notion of nonexpansive maps. Also, the Picard-Mann hybrid iterates of this example converge faster than the Picard, Ishikawa and Mann iterates. In particular, our results essentially improve and extended the results Khan [9] from the setting of nonexpansive operators to the larger frame work of mean nonexpansive operators.



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