# Fixed Point Approximation for a Class of Generalized Nonexpansive Mappings in Hadamard Spaces

Authors: Kifayat Ullah; Junaid Ahmad; Akbar Ali Khan; Manuel de la Sen
Journal: Advances in Mathematical Physics
 (2021)

---

## Abstract

In this paper, we establish strong andΔ convergence results for mappings satisfying condition Bγ,μ through a newly introduced iterative process called JA iteration process. A nonlinear Hadamard space is used the ground space for establishing our main results. A novel example is provided for the support of our main results and claims. The presented results are the good extension of the corresponding results present in the literature.


---

## Body

## 1. Introduction

Recall that a selfmapT on a subset E of a metric space is called nonexpansive if
(1)ρTa′,Ta′′≤ρa′,a′′foreverya′,a′′∈E.Once a memberw is available in the set E such that w=Tw then it is called a fixed point of T. In this research, the notation FT will throughout represent the fixed point set of T. The study of fixed points for nonexpansive operators is a crucial and busy research field now a days. One of the ealier result of Gohde [1] states that in the frame work of uniform convexity of Banach space, nonexpansive operators always admits a fixed point on closed bounded and convex subsets. Kirk [2, 3] was the first, who initiated fixed point theory of nonexpansive operators in the framework of nonlinear CAT(0) spaces. In 2008, Suzuki [4] achieved a big break-through by introducing a weak notion of nonexpansive operators. Notice that a selfmap T of a subset E of a metric space is said to satisfy Condition C (also called Suzuki map) if for any a′,a′′∈E, we have
(2)12ρa′,Ta′≤ρa′,a′′⇒ρTa′,Ta′′≤ρa′,a′′.The class of Suzuki nonexpansive mappings in linear and nonlinear setting were extensively studied by many researchers [5–12]. Very recently in 2018, Patir et al. [13] suggested a two parametric condition, which they called it Condition Bγ,μ. They proved that the Condition Bγ,μ is weaker than the corresponding condition C. Recently, Varatechakongka and Phuengrattana [14] studied Condition Bγ,μ in the setting of Hadamard spaces and proved the demiclosed principle for this class of mappings. A selfmap T of a subset E of a metric space is said to satisfy Condition Bγ,μ (or called Patir map) if there are some γ∈0,1 and μ∈0,1/2 with 2μ≤γ such that for all a′,a′′∈E,
(3)γρa′,Ta′≤ρa′,a′′+μρa′′,Ta′′⇒ρTa′,Ta′′≤1−γρa′,a′′+μρa′,Ta′′+ρa′′,Ta′.Iterative techniques for finding fixed points is very important and active research field of nonlinear analysis and has very fruitful applications in computers, applied economics, physics, and many applied sciences [15–26]. Since the Picard iteration xm+1=Txm does not always converge to a fixed point of a given nonexpansive operator, we shall present here some other well known process originally due various reseachers, which are not only converges to fixed point of a given nonexpansive operator but also have better rate of convergence as compare Picard iteration. Let we assume E be a nonempty as well as convex subset of a Banach space, αm,βm∈0,1 and T:E→E be a given operator.One of the earlier iteration process was defind by Mann [27] as follows:
(4)x1∈E,xm+1=1−αmxm+αmTxm,m≥1,The Ishikawa iteration process can be viewed as an extension of the Mann iteration, which was defined by Ishikawa in [28] as follows:
(5)x1∈E,ym=1−βmxm+βmTxm,xm+1=1−αmxm+αmTym,m≥1,Agarwal et al. [29] is the slightly modification of the Ishikawa iteration and was defined as follows:
(6)x1∈E,ym=1−βmxm+βmTxm,xm+1=1−αmTxm+αmTym,m≥1,By [29], we know that Agarwal iterative process is much better than the earlier defined process, namely, Picard, Mann and Ishikawa iterative processes.In the year 2016, Thakur et al. [30] suggested the below iterative process:
(7)x1∈E,zm=1−βmxm+βmTxm,ym=T1−αmxm+αmzm,xm+1=Tym,m≥1,Thakur et al. [30] proved that the sequence xm defined by the iterative process (7) converges (under some approperiate situations) to a fixed point of a given Suzuki map. Moreover, they constructed a new example of Suzuki mappings T and proved that the iterative process (7) converges faster to a fixed point T as compared the earlier iterative processes due to Picard, Mann [27], Ishikawa [28], S [29], Noor [31] and Abbas [15].Motivated by above, recently in 2020, Abedeljawad et al. [32] introduced a new iterative process, which they call it JA iteration process, as follows:
(8)x1∈E,zm=1−βmxm+βmTxm,ym=Tzm,xm+1=T1−αmTxm+αmTym,m≥1,Abdeljawad et al. [32] establish that the sequence xm defined by the iterative process (8) converges (under some approperiate situations) to a fixed point of a given Patir map in Banach spaces. Moreover, they constructed a new example of Patir maps T and proved that the iterative process (8) converges faster to a fixed point T as compared the leadings iterative processes due to Agarwal [29] and Thakur et al. [30]. In this paper, we improve and extend their results to the nonlinear setting of Hadamard spaces.


## 2. Preliminaries

Throughout the sequel, we will writeℙ and ℝ for natural and real numbers sets, respectively. Assume that M,ρ is a metric space. We can define a geodesic from a′ to a′′ as g:0,c→ℝ which gives g0=a′, gc=a′′ and ρgl,gl′=l−l′ for every l,l′∈0,m. In particular, the map g is an isometry and ρa′,a′′=m. The image of g is known as a geodesic (or called metric segment) joining the element a′ and a′′. If any two elements of M are connected by a geodesic then the metric space M,ρ is called a geodesic space. If one have only one geodesic joining a′ and a′′ for each a′, a′′∈M then it is called uniquely geodesic, which we often represent by a′,a′′ called the segment joining a′ to a′′.Definition 1.
Assume thata,b1,b2∈M and b0 is the midpoint of the segment b1,b2 such that,
(9)ρa,b02≤12ρa,b12+12ρa,b22−14ρb1,b22.Then (9) is known as the CN inequality of Burhat and Tits [33].A uniquely geodesic metric spaceM is called CAT0 space if and only if M is endowed with the CN inequality (cf. [34]). A CAT0 space is called Hadamard space if it is complete. For the detail study and results in Hadamard spaces, one can search [34, 35].Definition 2.
Take a bounded sequencexm in a Hadamard space M. Suppose E is closed and convex in M. Fix q∈M, then we state the following.
(10)rq,xm=limsupm→∞ρxm,q,is known as the asymptotic radius of xm at q.The asymptotic radius of the sequencexm wrt E is given by
(11)rE,xm=infrq,xm:q∈E.Moreover, the set(12)AE,xm=q∈E:rq,xm=rE,xm,is known as the asymptotic center of the sequence xm wrt to E.Remark 3.
The cardinality of the setAE,xm in any Hadamard space is always equal to one, (see e.g., [36] and others).The ([37], Proposition 2.1) tells us that in the setting of Hadamard spaces, for every bounded sequence, namely, xm⊆E, the set AE,xm is essentially the subset of E provided that E is convex and bounded. It is well-known that xm has a subsequence which Δ-converges to some point provided that the sequence is bounded.Definition 4 (see [38]).
A sequencexm in a given Hadamard space is said to be Δ-convergent to q∈M if and only if q is the unique asymptotic center of the um. Where um is any subsequence of the sequence xm. We denote by Δ−limmxm=q and call the point qtheΔ−lim of xm.Notice that a bounded sequencexm in a Hadamard space is known as regular if and only if for every subsequence, namely, um of xm one has rxm=rum. It is well-known that, in the setting of Hadamard spaces each regular sequence Δ-converges, and consequently each bounded sequence has a Δ-convergent subsequence.Definition 5 (see [37]).
LetT be a selfmap on a subset E of a given Hadamard space and f be a selfmap of 0,∞. We say that T has condition I if the following holds:
(i)
fg=0 if and only if g=0(ii)
fg>0 for every g>0(iii)
ρa′,Ta′≥fρa′,FT.
We now present some propositions and lemmas, which characterize the conditionBγ,μ.Proposition 6 (see [14]).
SupposeE is a nonempty subset of a given Hadamard space M. If T:E→E has condition Bγ,μ. Then for every fixed point w of T, one has
(13)ρw,Ta′≤ρw,a′for each a′∈E.Lemma 7 (see [14]).
SupposeE is nonempty closed convex subset of a given Hadamard space M. If T:E→E has Bγ,μ condition and the sequence xm⊆E satisfy limm→∞ρTxm,xm=0 and Δ−limmxm=w, then w=Tw.Lemma 8 (see [39]).
LetE be a nonempty subset of a given Hadamard space M. If T:E→E has the Bγ,μ condition. Then the set FT always closed.Lemma 9 (see [14], lemma 3.5).
SupposeE be nonempty subset of a given Hadamard space M. If.
T:E→E has condition Bγ,μ. Then for a′,a′′∈E and k∈0,1, the following hold:
(i)
ρTa′,T2a′≤ρa′,Ta′,(ii)
either ((h1) or (h2)) satisfy:
(h1)k/2ρa′,Ta′≤ρa′,a′′
(h2)k/2ρTa′,T2a′≤ρTa′,a′′(iii)
ρa′,Ta′′≤3−k+2μρa′,Ta′+1−k/2ρa′,a′′+μρa′,Ta′′+ρa′′,Ta′+2ρTa′,T2a′Lemma 10 (see [40]).
LetM be a Hadamard space and tm be any real sequence such that 0<a≤am≤b<1 for m≥1. Let ym and zm be any two sequences of M such that limsupm→∞ρym,x≤p,limsupm→∞ρzm,x≤p and limm→∞ρamym⊕1−amzm,x=p hold for some p≥0. Then limm→∞ρym,zm=0.


## 3. Convergence Results for Mappings SatisfyingBγ,μ Condition

This section establishes some important strong andΔ-convergence results for operators endowed with the Condition Bγ,μ.Lemma 11.
LetE≠∅ be a closed convex subset of a Hadamard space M and T:E→E satisfies the (Bγ,μ) condition with FT≠∅. Ifxm is a sequence generated by (8) (replacing + by ⊕), then limm→∞ρxm,w exists for each w∈FT.Proof.
Letw∈FT. By Proposition 6, we have
(14)ρzm,w=ρ1−βmxm⊕βmTxm,w≤1−βmρxm,w⊕βmρTxm,w≤1−βmρxm,w+βmρxm,w≤ρxm,w,ρym,w=ρTzm,w≤ρzm,w.
They imply that(15)ρxm+1,w=ρT1−αmTxm⊕αnTym,w≤ρ1−αmTxm⊕αmTym,w≤1−αmρTxm,w+αmρTym,w≤1−αmρxm,w+αmρym,w≤1−αmρxm,w+αmρzm,w≤1−αmρxm,w+αmρxm,w=ρxm,w.
Thusρxm,w is bounded below and nonincreasing and hence limm→∞ρxm,w exists for each w∈FT.Theorem 12.
SupposeE≠∅ be a closed convex subset of a Hadamard space M. Assume that T:E→E satisfies the (Bγ,μ) condition. If xm is a sequence generated by (8) (replacing + by ⊕). Then, FT≠∅ if and only if xm is bounded and limm→∞ρTxm,xm=0.Proof.
SupposeFT≠∅ and w∈FT. Then, by Lemma 11, limm→∞ρxm,w exists and xm is bounded. Put
(16)limm→∞ρxm,w=p.
By the proof of Lemma11 and (16), we have
(17)limsupm→∞ρzm,w≤limsupm→∞ρxm,w=p.
By proposition6, we have
(18)limsupm→∞ρTxm,w≤limsupm→∞ρxm,w=p.
Also by the proof of Lemma11, we have
(19)ρxm+1,w≤1−αmρxm,w+αmρzm,w
It follows that,(20)ρxm+1,w−ρxm,w≤ρxm+1,w−ρxm,wαm≤ρzm,w−ρxm,w.
Therefore(21)p≤liminfm→∞ρzm,w.p=limm→∞ρzm,w=limm→∞ρ1−βmxm⊕βmTxm,w
Applying Lemma10, we obtain
(22)limm→∞ρTxm,xm=0.
Conversely, letw∈AE,xm. By Lemma 9(iii) for γ=k/2, k∈0,1, we have
(23)ρxm,Tw≤3−k+2μρxm,Txm+1−k2ρxm,w+μρxm,Tw+ρw,Txm+2ρTxm,T2xm
So by Proposition6 and Lemma 9(i), we get
(24)ρxm,Tw≤3−k+4μρxm,Txm+1−k2+μρxm,w+μρxm,Tw
Then we have(25)ρxm,Tw≤3−k+4μ1−μρxm,Txm+1−k/2+μ1−μρxm,w
This implies that(26)rxm,Tw=limsupm→∞ρxm,Tw≤1−k/2+μ1−μlimsupm→∞ρxm,w≤limsupm→∞ρxm,w=rw,xm
SoTw∈AE,xm. By the uniqueness of asymptotic centers, one can conclude that Tw=w. This completes the proof.The below stated and proved result establishes theΔ–convergence for operators having condition Bγ,μ under JA iterations in Hadamard spaces. This improves [[14], Theorem 4.3] in the sense of better rate of convergence.Theorem 13.
SupposeE≠∅ be a closed convex subset of a Hadamard space M and T:E→E be a mapping with (Bγ,μ) condition such that FT≠∅. If xm is a sequence generated by (8) (replacing + by ⊕). Then xmΔ-converges to a fixed point of T.Proof.
By Theorem12, the sequence xm is bounded. Hence one can take Axm=p for some p∈M. We are going to prove Axmk=p for any subsequence xmk of xm. Suppose xmk be a subsequence of xm such that Axmk=q Since xmk is bounded, one can find a subsequence xmj of xmk such that xmjΔ converges to w for some w∈M. By Theorem 12, Lemma 7 one has w∈FT and hence limm→∞supρxm,w exists. If w≠q, then the singletoness of the cardinality of the asymptotic centers allows us the follwing
(27)limm→∞ρxm,w=limj→∞supρxmj,w<limj→∞supρxmj,q≤limk→∞supρxmk,q<limk→∞supρxmk,w=limm→∞supρxm,wwhich is contradiction. Therefore, q=w∈FT. Suppose that p≠q. Then
(28)limm→∞ρxm,q=limk→∞supρxmk,q≤limk→∞supρxmk,p≤limm→∞supρxm,p<limm→∞supρxm,q=limm→∞supρxm,q
xmΔ-converges to an element p∈FT.The follwing result establishes the strong–convergence for operators having conditionBγ,μ under JA iterations in Hadamad spaces. We may notice that it is that analog of ([32], Theorem 20).Theorem 14.
LetE≠∅ be a closed convex subset of a Hadamard space M and T:E→E be a map satisfying the (Bγ,μ). If FT≠∅ and limm→∞infρxm,FT=0, where xm be a sequence generated by (8) (replacing + by ⊕). Then xm converges strongly to a fixed point of T.Proof.
By Lemma11limm→∞ρxm,w exists for each w∈FT. Thus limm→∞ρxm,FT exists. Hence
(29)limm→∞ρxm,FT=0
Hence one can find a subsequencexmj of xm and wj in FT with ρxmj,wj≤1/2j, j≥1. In the view of proof of Lemma 11, one can observe that
(30)ρxm+1,wj≤ρxmj,wj≤12j.
Next it is our purpose to show that the sequencewj form a Cauchy sequence in FT. For this, we consider the following
(31)ρwj+1,wj≤ρwj+1,smj+1+ρsmj+1,wj≤12j+1+12j≤12j−1→0,asj→∞.
The above limit showes that the sequencewj is a Cauchy sequence in the set FT. By Lemma 8, the set FT is closed. Hence wj→q for some q∈FT. By Lemma 11, limm→∞ρxm,q exists. So the proof is finished.Now we establish the final result in this section, which is related to the conditionI. We may notice that it is the analog of ([32], Theorem 21).Theorem 15.
SupposeE≠∅ be a closed convex subset of a Hadamard space M. Assume that T:E→E be a map having condition Bγ,μ with FT≠∅. If xm is a sequence generated by (8) (replacing + by ⊕). Then xm converges to an element of FT provided that T has condition I.Proof.
It follows from Theorem12 the limm→∞infρxm,Txm=0. By condition I, limm→∞infρxm,FT=0. The conclusions follows from the Theorem 14.


## 4. Numerical Interpretation

In this section, we are interested in the rate of convergence. We first construct a new example of a mappingT as follows, which is Patir mapping but not Suzuki.Example 16.
Define an operatorT:1,4→1,4 as follows
(32)Ta′=a′+22ifa′≠42ifa′=4.To show thatT is not Suzuki mapping let a′=35/10,a′′=4. We see that, 1/2ρa′,Ta′=3/8<1/2=ρa′,a′′butρTa′,Ta′′=3/4>ρa′,a′′. Thus T does not satisfy condition C. Choose γ=1 and μ=1/2, we prove that T has the B1,1/2 condition.C1: If we take a′,a′′∈1,4, then
(33)1−γρa′,a′′+μρa′,Ta′′+ρa′′,Ta′=12a′−Ta′′+a′′−Ta′≥12∣3a′2−3a′′2∣=34∣a′−a′′∣≥12∣a′−a′′∣=ρTa′,Ta′′.C2: If we take a′∈1,4 and a′′=4, then
(34)1−γρa′,a′′+μρa′,Tb+ρa′′,Ta′=12∣a′−Ta′′∣+∣a′′−Ta′∣=12∣a′−2∣+a′′−a′+22=12∣a′−2∣+12a′′−a′+22≥12∣a′−2∣=ρTa′,Ta′′.C3: If we take a′=4=a′′, then we have
(35)1−γρa′,a′′+μρa′,Ta′′+ρa′′,Ta′≥0=ρTa′,Ta′′.Hence,T satisfies the B1,1/2 condition. The strong convergence of leading iterations can be seen in Table 1 and Figure 1 to the fixed point 2 of the mapping T. One can easly observe that JA iteration (8) converges faster than the leading Thakur (7) and leading Agarwal (6) iterative processes in Table 1 and Figure 1.Table 1
Strong convergence of leading iterative processes underαm=m+1/m+21/7,βm=1/2m+31/2 and x1=2.4.
JA (8)Thakur (7)Agarwal (6)x12.42.42.4x22.04164082172.07385400992.1477080198x32.00439964052.01456728492.0582691397x42.00047235162.00298129362.0238503460x52.00005136452.00062500012.0100000027x62.00000564252.00013330002.0042656021x72.00000062502.00002880432.0018434768x82.00000006972.00000628892.0008049909x92.00000000782.00000138472.0003545001x102.00000000082.00000030782.0001572218x1122.00000006852.0000701497x1222.00000001532.0000314632x1322.00000000342.0000141763x1422.00000000072.0000064132x1522.00000000012.0000029118x16222.0000013263Figure 1
Convergence behaviors of JA (8), Thakur (7) and Agarwal (6) iterations for operator T provided in the Example 16 under x1=2.4.


## 5. Conclusions

The study of fixed points in the framework of nonlinear domains gained much more attention of the mathematicians. Takahashi [41] was the first, who suggested the concept of convexity in metric spaces and proved some important fixed points results for nonexpansive operators in this setting. This convexity structure then initiated many other convexity structures in metric spaces. In this paper, we have established some strong and Δ-convergence theorems for the class of Patir et al. [13] operators in nonlinear Hadamard spaces using new up-to-date iteration process which is faster than Picard, Mann, Ishikawa, S, Noor, Abbas and Thakur iterations. Moreover the class of Patir et al. [13] operators is more general than Suzuki operators and nonexpansive operators. Hence our results extend many known results of authors [5–12, 32] whose idea was limited to the setting of Suzuki operators.



---

*Source: 6613225-2021-06-15.xml*