#### Abstract

In this paper, we establish strong and convergence results for mappings satisfying condition 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.

#### 1. Introduction

Recall that a selfmap on a subset of a metric space is called nonexpansive if

Once a member is available in the set such that then it is called a fixed point of . In this research, the notation will throughout represent the fixed point set of . 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 of a subset of a metric space is said to satisfy Condition (also called Suzuki map) if for any , we have

The class of Suzuki nonexpansive mappings in linear and nonlinear setting were extensively studied by many researchers [5â€“12]. Very recently in 2018, Patir et al. [13] suggested a two parametric condition, which they called it Condition . They proved that the Condition is weaker than the corresponding condition . Recently, Varatechakongka and Phuengrattana [14] studied Condition in the setting of Hadamard spaces and proved the demiclosed principle for this class of mappings. A selfmap of a subset of a metric space is said to satisfy Condition (or called Patir map) if there are some and with such that for all ,

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 [15â€“26]. Since the Picard iteration 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 be a nonempty as well as convex subset of a Banach space, and be a given operator.

One of the earlier iteration process was defind by Mann [27] as follows:

The Ishikawa iteration process can be viewed as an extension of the Mann iteration, which was defined by Ishikawa in [28] as follows:

Agarwal et al. [29] is the slightly modification of the Ishikawa iteration and was defined as follows:

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:

Thakur et al. [30] proved that the sequence 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 and proved that the iterative process (7) converges faster to a fixed point 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:

Abdeljawad et al. [32] establish that the sequence 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 and proved that the iterative process (8) converges faster to a fixed point 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 write and for natural and real numbers sets, respectively. Assume that is a metric space. We can define a geodesic from to as which gives , and for every . In particular, the map is an isometry and The image of is known as a geodesic (or called metric segment) joining the element and . If any two elements of are connected by a geodesic then the metric space is called a geodesic space. If one have only one geodesic joining and for each , then it is called uniquely geodesic, which we often represent by called the segment joining to .

*Definition 1. *Assume that and is the midpoint of the segment such that,

Then (9) is known as the inequality of Burhat and Tits [33].

A uniquely geodesic metric space is called space if and only if is endowed with the inequality (cf. [34]). A 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 sequence in a Hadamard space . Suppose is closed and convex in . Fix , then we state the following.
is known as the asymptotic radius of at .

The asymptotic radius of the sequence wrt is given by

Moreover, the set is known as the asymptotic center of the sequence wrt to

*Remark 3. *The cardinality of the set 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, , the set is essentially the subset of provided that is convex and bounded. It is well-known that has a subsequence which -converges to some point provided that the sequence is bounded.

*Definition 4 (see [38]). *A sequence in a given Hadamard space is said to be *-*convergent to if and only if is the unique asymptotic center of the . Where is any subsequence of the sequence We denote by and call the point *the* of

Notice that a bounded sequence in a Hadamard space is known as regular if and only if for every subsequence, namely, of one has . It is well-known that, in the setting of Hadamard spaces each regular sequence -converges, and consequently each bounded sequence has a -convergent subsequence.

*Definition 5 (see [37]). *Let be a selfmap on a subset of a given Hadamard space and be a selfmap of . We say that has condition if the following holds:
(i) if and only if (ii) for every (iii).We now present some propositions and lemmas, which characterize the condition .

Proposition 6 (see [14]). *Suppose is a nonempty subset of a given Hadamard space . If has condition . Then for every fixed point of , one has
for each *

Lemma 7 (see [14]). *Suppose is nonempty closed convex subset of a given Hadamard space . If has condition and the sequence satisfy and , then *

Lemma 8 (see [39]). *Let be a nonempty subset of a given Hadamard space . If has the condition. Then the set always closed.*

Lemma 9 (see [14], lemma 3.5). *Suppose be nonempty subset of a given Hadamard space . If.** has condition . Then for and the following hold:
*(i)*(ii)**either (() or ()) satisfy:**( h_{1})*

*(*(iii)

*h*_{2})Lemma 10 (see [40]). *Let be a Hadamard space and be any real sequence such that for . Let and be any two sequences of such that and hold for some Then .*

#### 3. Convergence Results for Mappings Satisfying Condition

This section establishes some important strong and -convergence results for operators endowed with the Condition .

Lemma 11. *Let be a closed convex subset of a Hadamard space and satisfies the () condition with . If is a sequence generated by (8) (replacing by ), then exists for each .*

*Proof. *Let . By Proposition 6, we have
They imply that
Thus is bounded below and nonincreasing and hence exists for each .

Theorem 12. *Suppose be a closed convex subset of a Hadamard space . Assume that satisfies the () condition. If is a sequence generated by (8) (replacing by ). Then, if and only if is bounded and .*

*Proof. *Suppose and . Then, by Lemma 11, exists and is bounded. Put
By the proof of Lemma 11 and (16), we have
By proposition 6, we have
Also by the proof of Lemma 11, we have
It follows that,
Therefore
Applying Lemma 10, we obtain
Conversely, let . By Lemma 9(iii) for , , we have
So by Proposition 6 and Lemma 9(i), we get
Then we have
This implies that
So . By the uniqueness of asymptotic centers, one can conclude that This completes the proof.

The below stated and proved result establishes the â€“convergence for operators having condition under iterations in Hadamard spaces. This improves [[14], Theorem 4.3] in the sense of better rate of convergence.

Theorem 13. *Suppose be a closed convex subset of a Hadamard space and be a mapping with () condition such that . If is a sequence generated by (8) (replacing by ). Then -converges to a fixed point of *

*Proof. *By Theorem 12, the sequence is bounded. Hence one can take for some . We are going to prove for any subsequence of Suppose be a subsequence of such that Since is bounded, one can find a subsequence of such that converges to for some . By Theorem 12, Lemma 7 one has and hence exists. If , then the singletoness of the cardinality of the asymptotic centers allows us the follwing
which is contradiction. Therefore, . Suppose that . Then
-converges to an element .

The follwing result establishes the strongâ€“convergence for operators having condition under iterations in Hadamad spaces. We may notice that it is that analog of ([32], Theorem 20).

Theorem 14. *Let be a closed convex subset of a Hadamard space and be a map satisfying the (). If and , where be a sequence generated by (8) (replacing by ). Then converges strongly to a fixed point of .*

*Proof. *By Lemma 11 exists for each Thus exists. Hence
Hence one can find a subsequence of and in with , In the view of proof of Lemma 11, one can observe that
Next it is our purpose to show that the sequence form a Cauchy sequence in . For this, we consider the following
The above limit showes that the sequence is a Cauchy sequence in the set . By Lemma 8, the set is closed. Hence for some . By Lemma 11, exists. So the proof is finished.

Now we establish the final result in this section, which is related to the condition . We may notice that it is the analog of ([32], Theorem 21).

Theorem 15. *Suppose be a closed convex subset of a Hadamard space . Assume that be a map having condition with . If is a sequence generated by (8) (replacing by ). Then converges to an element of provided that has condition .*

*Proof. *It follows from Theorem 12 the . By condition , 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 mapping as follows, which is Patir mapping but not Suzuki.

*Example 16. *Define an operator as follows

To show that is not Suzuki mapping let . We see that, but. Thus does not satisfy condition . Choose and , we prove that has the condition.

: If we take , then

: If we take and , then

: If we take , then we have

Hence, satisfies the condition. The strong convergence of leading iterations can be seen in Table 1 and Figure 1 to the fixed point of the mapping . 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.

#### 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 -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 [5â€“12, 32] whose idea was limited to the setting of Suzuki operators.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

We strongly declare that no one of us has conflicts of interest.

#### Authorsâ€™ Contributions

K.U, J.A, A.A.K, M.d.I.S provided equal contributions to this article.

#### Acknowledgments

The authers are grateful to the Spanish Government for Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and the Basque Government for Grant IT1207-19.