# Iterative Algorithm for Mappings SatisfyingBγ,μ Condition

Authors: Thabet Abdeljawad; Kifayat Ullah; Junaid Ahmad
Journal: Journal of Function Spaces
 (2020)

---

## Abstract

In this article, we introduce a novel iterative algorithm to approximate fixed point of mappings withBγ,μ condition. We establish some strong and weak convergence results in a uniformly convex Banach space. Using a numerical example, we compare the speed of the proposed algorithm with some leading iterative algorithms.


---

## Body

## 1. Introduction and Preliminaries

Throughout the article,ℙ stands for the set of all positive numbers and ℝ for the set of all real numbers. Let V be a Banach space, K≠∅ a subset of V and f:K⟶K. An element q∈K is called a fixed point of f if q=fq. We denote the set q∈K:q=fq by ℱf. f is called quasi-nonexpansive (resp., nonexpansive) if ℱf≠∅ and fv−fq≤v−q for all v∈K and q∈ℱf (resp., fv−fw≤v−w, for all v,w∈K). It is well known that if V is uniformly convex and K is convex closed bounded then ℱf is nonempty (see [1–3]). In [4], Suzuki generalized the concept of nonexpansive mappings as follows.Definition 1.
LetK≠∅ be a subset of a Banach space V. A mapping f:K→K is said to satisfy the C condition (or Suzuki mappings) if
(1)12v−fv≤v−w⇒fv−fw≤v−w,for each v,w∈K.Remark 2.
It is easy to see that every nonexpansive mapping satisfies theC condition.
Using the following example, Suzuki [4] showed that C condition is more general than the nonexpansiveness.Example 3 (see [5]).
Define a mappingf:0,3⟶ℝ by
(2)fv=0ifv≠3,1ifv=3.
Here,f satisfies C condition, but not nonexpansive.Recently in 2018, Patir et al. [5] generalized the notion of C condition as follows.Definition 4.
LetK≠∅ be a subset of a Banach space V. A mapping f:K⟶K is said to satisfy the Bγ,μ condition if there exists γ∈0,1 and μ∈0,1/2 satisfying 2μ≤γ such that for each v,w∈K,
(3)γv−fv≤v−w+μw−fwimplies fv−fw≤1−γv−w+μv−fw+w−fv.Remark 5.
It is easy to see that a mapping withC condition satisfies the Bγ,μ condition.
Using the following example, Patir et al. [5] showed that Bγ,μ condition is more general than the C condition.Example 6 (see [5]).
Define a mappingf:0,2⟶ℝ by
(4)fv=0ifv≠2,1ifv=2.
Here,f satisfies Bγ,μ condition, but not C.The famous theorem of Banach suggests a Picard iterative algorithm, that is,vn+1=fvn to approximate the unique fixed point of f, where f is a contraction mapping. It is now well-known that for a nonexpansive mapping f the Picard iterative algorithm may fail to converge to the fixed point of f. Therefore, it is natural to investigate new algorithms to overcome such problem and to get better speed of convergence. Nowadays, the well-known algorithms are the Mann [6], Ishikawa [7], Agarwal et al. [8], Noor [9], Abbas and Nazir [10], Thakur et al. [11], and so on. For some recent literature on iterative algorithms, we refer the reader to [12–18].Mann [6] iterative algorithm is defined as
(5)v1∈K,vn+1=1−αnvn+αnfvn,n∈ℙ,where αn∈0,1.Ishikawa [7] iterative algorithm reads as follows:
(6)v1∈K,wn=1−βnvn+βnfvn,vn+1=1−αnvn+αnfwn,n∈ℙ,where αn,βn∈0,1.Agarwal et al. [8] generalized iterative algorithm (6) is as follows:
(7)v1∈K,wn=1−βnvn+βnfvn,vn+1=1−αnfvn+αnfwn,n∈ℙ,where αn,βn∈0,1.In 2016, Thakur et al. [11] used a new iterative algorithm. With the help of a numerical example, they showed that this algorithm is better than the Picard, Mann, Ishikawa, Agarwal, Noor, and Abbas iterative algorithms for the class of Suzuki mappings. The algorithm reads as follows:
(8)v1∈K,xn=1−βnvn+βnfvn,wn=f1−αnvn+αnxn,vn+1=fwn,n∈ℙ,where αn,βn∈0,1.Motivated by the above, in this article, we propose a new iterative algorithm and prove some weak and strong convergence results for mapping satisfyingBγ,μ condition in the context of uniformly convex Banach spaces. We also present a new numerical example of a mapping satisfying Bγ,μ condition and numerically compare speed of the proposed iterative algorithm with the speed of leading iterative algorithms. Our results are new and never investigated before.Following are some basic definitions and results needed in the sequel.Definition 7 (see [19]).
Recall that a Banach spaceV is said to be uniformly convex if for each η∈0,2 there is a λ>0 such that for every two elements v,w in V,
(9)v≤1w≤1v−w>ηimplies12v+w≤1−λ.Definition 8.
LetK≠∅ be a subset of a Banach space V and vn a bounded sequence in V. For each v∈V define
(i)
asymptotic radius ofvn at v by rv,vn≔limsupn→∞v−vn
(ii)
asymptotic radius ofvn relative to K by rK,vn=infrv,vn:v∈K
(iii)
asymptotic center ofvn relative to K by AK,vn=v∈K:rv,vn=rK,vn
The setAK,vn is singleton provided that V is uniformly convex. Also, AK,vn is nonempty convex provided that K is convex and weakly compact, see, e.g., [20, 21].Definition 9 (see [22]).
A Banach spaceV is said to have Opial’s property if for each sequence vn in V which weakly converges to v∈V and for every w∈V-v it follows that
(10)limsupn→∞vn−w<limsupn→∞vn−w.Definition 10 (see [23]).
LetK≠∅ be a subset of a Banach space V. A mapping f:K⟶K is said to satisfy condition I if there is a function ζ:0,∞⟶0,∞ satisfying ζ0=0 and ζr>0 for all r∈0,∞ such that v−fv≥ζdv,ℱf for all v∈K, where dv,ℱf represents distance of v from ℱf.The following lemma gives many examples of mappings withBγ,μ condition.Lemma 11.
LetK≠∅ be a subset of a Banach space V. If f satisfies C condition, then f satisfies the Bγ,μ condition.Lemma 12 (see [5]).
LetK≠∅ be a subset of a Banach space V and f:K⟶K satisfies Bγ,μ condition. If q is a fixed point of f, then for each v∈K
(11)q−fv≤q−v.Theorem 13 (see [5]).
LetK≠∅ be a subset of a Banach space V having Opial property. Let f:K⟶K satisfy Bγ,μ condition. If vn⊆K be such that
(i)
vn converges weakly to z
(ii)
limn→∞fvn−vn=0then z=fz.Proposition 14 (see [5]).
LetK≠∅ be a subset of a Banach space V. If f:K⟶K satisfies the Bγ,μ condition on K, then for all v,w∈K and s∈0,1,
(i)
fv−f2v≤v−fv
(ii)
at least one of the following ((a) and (b)) holds:(a)
s/2v−fv≤v−w
(b)
s/2fv−f2v≤fv−w
Condition (a) implies fv−fw≤1−s/2v−w+μv−fw+w−fv and condition (b) implies f2v−fw≤1−s/2fv−w+μfv−fw+w−f2v.
(iii)
v−fw≤3−sv−fv+1−s/2v−w+μ2v−fv+v−fw+w−fv+2fv−f2v
We need the following useful lemma from [24].Lemma 15.
LetV be a uniformly convex Banach space and 0<q≤δn≤p<1 for every n∈ℙ. If sn and wn are two sequences in V such that limsupn→∞sn≤l, limsupn→∞wn≤l, and limn→∞δnsn+1−δnwn=l for some l≥0, then limn→∞sn−wn=0.


## 2. Main Results

In this section, we first introduce a new iterative algorithm, namely, JA iteration, to approximate fixed point of mappings satisfying conditionBγ,μ as follows:
(12)v1∈K,xn=1−βnvn+βnfvn,wn=fxn,vn+1=f1−αnfvn+αnfwn,n∈ℙ,where αn,βn∈0,1.Now we construct a new example of mapping which satisfiesBγ,μ condition but not C. We then use this example to compare the rate of convergence of JA iterative algorithm (12) with the leading Thakur (8) and leading two-step Agarwal iterative algorithm (7). This example also shows that the converse of Lemma 11 does not hold in general.Example 16.
LetK=1,3 be endowed with absolute valued norm. Define a mapping f:K⟶K by
(13)fv=1+v2ifv≠3,1ifv=3.
Choosev=25/10 and w=3. We see that 1/2v−fv=3/8<1/2=v−w but fv−fw=3/4>1/2=v−w. Thus, f is not Suzuki mapping. Choose γ=1 and μ=1/2, we prove that f satisfies the B1,1/2 condition.Case 1.
Forv,w∈1,3, we have
(14)1−γv−w+μv−fw+w−fv=12v−fw+w−fv=12v−1+w2+w−1+v2≥123v2−3w2=34v−w≥12v−w=fv−fw.Case 2.
Forv∈1,3 and w=3, we have
(15)1−γv−w+μv−fw+w−fv=12v−fw+w−fv=12v−1+w−1+v2=12v−1+12w−1+v2≥12v−1=fv−fw.Case 3.
Forv=w=3, we have
(16)1−γv−w+μv−fw+w−fv≥0=fv−fw.Hence,f satisfies the B1,1/2 condition. Now we show that algorithm (12) converges fast to q=1 as compared to the leading Thakur (8) and Agarwal (7). Choose v1=1.5, αn=βn=1/2. Table 1 and Figure 1 show the efficiency of the proposed iterative algorithm.Table 1
Sequences generated by JA (12), Thakur et al. (8), and Agarwal (7) iterative algorithms.
JA (12)
Thakur et al. (8)
Agarwal (7)
v1
1.500000000
1.500000000
1.5000000000
v2
1.085937500
1.109375000
1.2187500000
v3
1.014770507
1.023925781
1.0957031250
v4
1.002538681
1.005233764
1.0418701171
v5
1.000436335
1.001144886
1.0183181762
v6
1.000074995
1.000250443
1.0080142021
v7
1.000012889
1.000054784
1.0035062134
v8
1.000002215
1.000011984
1.0015339683
v9
1.000000380
1.000002621
1.0006711111
v10
1.000000065
1.000000573
1.0002936111
v11
1.000000011
1.000000125
1.0001284548
v12
1.000000001
1.000000027
1.0000561990
v13
1
1.000000006
1.0000245870
v14
1
1.000000001
1.0000107568
v15
1
1
1.0000047061Figure 1
Convergence behavior of JA, Thakur et al., and Agarwal iterative algorithms for mappingf defined in Example 16.Now using Example16, we obtained the influence of initial points and parameters for the Agarwal (7), Thakur (8), and JA (12) iterations. In Table 2, items in bold show that the JA iteration (12) converges faster than others.Table 2
Influence of parameters: comparison of various iteration processes.
Iterations
Initial points
1.2
1.5
1.9
2.2
2.5
2.9
Forαn=nn+210/9,βn=1n+42/3
Agarwal
27
28
29
30
30
30
Thakur
14
15
15
15
16
16
JA
12
13
13
13
13
13
Forαn=n1/2n+51/5,βn=17n+91/3
Agarwal
22
23
23
24
24
24
Thakur
13
13
14
14
14
14
JA
8
9
9
9
9
9
Forαn=13n+51/5,βn=n7n+9
Agarwal
28
29
30
30
31
31
Thakur
14
15
16
16
16
16
JA
12
13
13
13
13
13
Forαn=n+15n+11/7,βn=12n+31/2
Agarwal
26
27
28
28
28
29
Thakur
14
14
15
15
15
15
JA
10
11
12
12
12
12Now we establish some convergence theorems for mappings withBγ,μ condition using algorithm (12). Similar results for the other algorithms can be established on the same line of proofs.Lemma 17.
LetK≠∅ be a closed convex subset of a Banach space V and f:K⟶K satisfies the Bγ,μ condition with ℱf≠∅. If vn is a sequence generated by (12), then limn→∞vn−q exists for each q∈ℱf.Proof.
Letq∈ℱf. By Lemma 12, we have
(17)xn−q=1−βnvn+βnfvn−q≤1−βnvn−q+βnfvn−q≤1−βnvn−q+βnvn−q≤vn−q,wn−q=fxn−q≤xn−q.
They imply that(18)vn+1−q=f1−αnfvn+αnfwn−q≤1−αnfvn+αnfwn−q≤1−αnfvn−q+αnfwn−q≤1−αnvn−q+αnwn−q≤1−αnvn−q+αnxn−q≤1−αnvn−q+αnvn−q=vn−q.Thus,vn−q is bounded and nonincreasing, and hence, limn→∞vn−q exists for each q∈ℱf.Theorem 18.
LetK≠∅ be a closed convex subset of a uniformly convex Banach space V and f:K⟶K a mapping satisfying the Bγ,μ condition. If vn is a sequence generated by (12), then ℱf≠∅ if and only if vn is bounded and limn→∞fvn−vn=0.Proof.
Supposeℱf≠∅ and q∈ℱf. Then, by Lemma 17, limn→∞vn−q exists and vn is bounded. Put
(19)limn→∞vn−q=l.By the proof of Lemma17 together with (19), we have
(20)limsupn→∞xn−q≤limsupn→∞vn−q=l.By Lemma12, we have
(21)limsupn→∞fvn−q≤limsupn→∞vn−q=l.Again by the proof of Lemma17, we have
(22)vn+1−q≤1−αnvn−q+αnxn−q.It follows that(23)vn+1−q−vn−q≤vn+1−q−vn−qαn≤xn−q−vn−q.So, we can getvn+1−q≤xn−q. Therefore,
(24)l≤liminfn→∞xn−q.From (20) and (24), we obtain
(25)l=limn→∞xn−q.From (25), we have
(26)l=limn→∞xn−q=limn→∞1−βnvn+βnfvn−q=limn→∞1−βnvn−q+βnfvn−q.Hence,(27)l=limn→∞1−βnvn−q+βnfvn−q.Now from (19), (21), and (27) together with Lemma 15, we obtain
(28)limn→∞fvn−vn=0.Conversely, letq∈AK,vn. By Proposition 14 (iii), for γ=s/2, s∈0,1, vn−fq≤3−svn−fvn+1−s/2vn−q+μ2vn−fvn+vn−fq+q−fvn+2fvn−f2vn≤3−svn−fvn+1−s/2vn−q+μ2vn−fvn+vn−fq+vn−q+vn−fvn+2vn−fvn (by Proposition 14 (i))
(29)⇒1−μlimsupn→∞vn−fq≤1−s2+μlimsupn→∞vn−q⇒limsupn→∞vn−fq≤1−s/2+μ1−μlimsupn→∞vn−q≤limsupn→∞vn−qas1−s/2+μ1−μ≤1,for2μ≤γ=s2⇒rfq,vn≤rq,vn.Sofq∈AK,vn. Since V is uniformly convex Banach space, so fq=q.Theorem 19.
LetK≠∅ be a closed convex subset of a uniformly convex Banach space V having Opial property. If f:K⟶K satisfies the Bγ,μ condition with ℱf≠∅, then vn generated by (12) converges weakly to an element of ℱf.Proof.
By Theorem18, the sequence vn is bounded and limn→∞fvn−vn=0. Since V is uniformly convex, we have V that is reflexive. Thus, a subsequence vni of vn exists such that vni converges weakly to some q1∈K. By Theorem 13, q1∈ℱf. It is suffice to show that vn converges weakly to q1. Indeed, if vn does not converge weakly to q1, then there exists a subsequence vnj of vn and q2∈K such that vnj converges weakly to q2 and q2≠q1. Again by Theorem 13, q2∈ℱf. By Lemma 17 together with Opial property, we have
(30)limn→∞vn−q1=limi→∞vni−q1<limi→∞vni−q2=limn→∞vn−q2=limj→∞vnj−q2<limj→∞vnj−q1=limn→∞vn−q1.This is a contradiction. So,q1=q2. Thus, vn converges weakly to q1∈ℱf.Theorem 20.
LetK≠∅ be a closed and convex subset of a uniformly convex Banach space V and f:K⟶K satisfies the Bγ,μ condition with ℱf≠∅. If vn is a sequence generated by (12), then vn converges to an element of ℱf if and only if liminfn→∞dvn,ℱf=0 or limsupn→∞dvn,ℱf=0.Proof.
The necessity is obvious.
Conversely, suppose thatliminfn→∞dvn,ℱf=0 and q∈ℱf. From Lemma 17, limn→∞vn−q exists for each q∈ℱf. Therefore, limn→∞dvn,ℱf=0, by assumption. We prove that vn is a Cauchy sequence in K. As limn→∞dvn,ℱf=0, for a given ε>0, there exists k0∈ℙ such that for each n≥k0,
(31)dvn,ℱf<ε2.⇒infvn−q:q∈ℱf<ε2.In particular,infvk0−q:q∈ℱf<ε/2. Therefore, there exists q∈ℱf such that vk0−q<ε/2.Now fork,n≥k0,
(32)vn+k−vn≤vn+k−q+vn−q≤vk0−q+vk0−q=2vk0−q<ε.This shows thatvn is a Cauchy sequence in K. As K is a closed subset of a Banach space V, so there exists a point p∈K such that limn→∞vn=p. Now limn→∞dvn,ℱf=0 gives that dp,ℱf=0. The closeness of ℱf follows from Lemma 12. Hence, p∈ℱf.We now prove the following theorem using conditionI.Theorem 21.
LetK≠∅ be a closed and convex subset of a uniformly convex Banach space V and f:K⟶K satisfies the Bγ,μ condition with ℱf≠∅. If vn is a sequence generated by (12), then vn converges strongly to an element of ℱf provided that f satisfies the condition I.Proof.
From Theorem18, we have
(33)limn→∞fvn−vn=0.
Thus, by conditionI, we obtain
(34)limn→∞dvn,Ff=0.Now all the requirements of Theorem20 are satisfied; therefore, by its conclusion vn converges strongly to a fixed point of f.Remark 22.
In the view of Lemma11, our results contain the results for mappings with C condition.


## 3. Conclusions

We have introduced a new faster iterative algorithm, namely, JA iteration, for the approximation of fixed points of the class of mapping withBγ,μ condition. Moreover, the class of mappings with Bγ,μ condition is essentially more general than the class of Suzuki mappings. Thus, our results are better in both the senses (faster iteration and general mapping). Hence, our results are the good extension of the corresponding results of Abbas and Nazir [10], Thakur et al. [11], and references therein.



---

*Source: 3492549-2020-05-05.xml*