# Iterative approximation of attractive points of further generalized hybrid mappings in Hadamard spaces

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## Abstract

In this paper, we study the class of further generalized hybrid mappings due to Khan (Fixed Point Theory Appl. 2018:8, 2018) in the setting of Hadamard spaces. We prove a demiclosed principle for such mappings in Hadamard spaces. Furthermore, we also prove the Δ-convergence of the sequence generated by the S-iteration process for finding attractive points of further generalized hybrid mappings in Hadamard spaces satisfying the \((\mathbb{S})\) property and the \((\overline{Q_{4}})\) condition. Moreover, we provide a numerical example to illustrate the convergence behavior of the studied iteration and numerically compare the convergence of the studied iteration scheme with the existing schemes. Our results extend some known results which appeared in the literature.

## Keywords

Attractive point Further generalized hybrid mappings S-iteration Δ-convergence Hadamard spaces## MSC

47H09 47H10## 1 Introduction

*H*be a Hilbert space and

*C*be a nonempty subset of

*H*. Let

*T*be a mapping from

*C*into

*H*. Let \(A(T)\) denote the set of all attractive points of

*T*, i.e.,

In 2012, Takahashi et al. [3] introduced the class of normally generalized hybrid mappings in a Hilbert space.

### Definition 1.1

*normally generalized hybrid*if there exist \(\alpha ,\beta ,\gamma ,\delta \in \mathbb{R}\) such that

- (i)
\(\alpha +\beta +\gamma +\delta \geq 0\);

- (ii)
\(\alpha +\beta >0\) or \(\alpha +\gamma >0\); and

- (iii)
\(\alpha \|Tx-Ty\|^{2} +\beta \|x-Ty\|^{2} + \gamma \|Tx-y\|^{2}+ \delta \|x-y\|^{2}\leq 0\), \(\forall x,y\in C\).

*T*can be called an \((\alpha ,\beta ,\gamma ,\delta )\)-normally generalized hybrid mapping.

They also proved the weak convergence theorem of Mann type for normally generalized hybrid mappings in real Hilbert spaces without convexity assumption on the domain of mappings. To be more precise, they obtained the following result.

### Theorem 1.2

*Let*

*H*

*be a real Hilbert space and*

*C*

*be a nonempty convex subset of*

*H*.

*Let*\(T:C\to C\)

*be a normally generalized hybrid mapping*.

*Assume that*\(A(T)\ne \emptyset \).

*For*\(x_{1}\in C\),

*the sequence*\(\{x_{n}\}\)

*generated by*

*where*\(\{\alpha _{n}\}\)

*is a sequence in*\([0,1]\)

*such that*\(\liminf_{n\to \infty } \alpha _{n}(1-\alpha _{n})>0\).

*Then the sequence*\(\{x_{n}\}\)

*converges weakly to a point*\(z\in A(T)\).

*Moreover*, \(z=\lim_{n\to \infty }P_{A(T)}x_{n}\),

*where*\(P_{A(T)}\)

*is a projection of*

*H*

*onto*\(A(T)\).

In 2015, Kaewkhao et al. [4] extended the results of Takahashi et al. [3] from Hilbert spaces to Hadamard spaces.

In 2018, Khan [1] gave the concept of further generalized mappings (see Definition 1.3 below) which constitutes a generalization of normally generalized hybrid mappings due to Takahashi et al. [3] (see Definition 1.1 above).

### Definition 1.3

([1])

*further generalized hybrid*if there exist \(\alpha ,\beta ,\gamma ,\delta ,\epsilon \in \mathbb{R}\) such that

- (i)
\(\alpha +\beta +\gamma +\delta \geq 0\), \(\epsilon \geq 0\);

- (ii)
\(\alpha +\beta >0\) or \(\alpha +\gamma >0\); and

- (iii)
\(\alpha \|Tx-Ty\|^{2} +\beta \|x-Ty\|^{2} + \gamma \|Tx-y\|^{2}+ \delta \|x-y\|^{2}+ \epsilon \|x-Tx\|^{2}\leq 0\), \(\forall x,y\in C\).

Obviously, by above definitions, further generalized hybrid is a generalization of normally generalized hybrid when \(\epsilon =0\). It is noteworthy that it contains the class of generalized hybrid, quasi-nonexpansive mappings, quasi-contractive mappings and contractive mappings.

Recently, Khan [1] obtained a weak convergence theorem of Picard–Mann hybrid iterative process [5] for further generalized hybrid mappings in real Hilbert spaces without convexity assumption on the domain of mappings. The iterative process of Khan [1] is faster than Mann, Ishikawa and S-iteration process of Agarwal et al. [6] as shown by him in [5]. However, his results are in a Hilbert space and we want to have some results in Hadamard spaces. Note that no results are available at the moment for further generalized hybrid mappings even for Mann iterative process in Hadamard spaces. We further note that S-iteration process is also faster than Mann and Ishikawa iteration processes (but not Picard–Mann hybrid).

Motivated by the above works, we define a class of further generalized hybrid mappings and prove the demiclosed principle for such mapping in Hadamard spaces. Furthermore, we also obtain a Δ-convergence theorem of S-iteration process for further generalized hybrid mappings in Hadamard spaces satisfying the \((\mathbb{S})\) property and the \((\overline{Q_{4}})\) condition. Finally, we provide a numerical example to illustrate the convergence behavior of the S-iteration and numerically compare the convergence of the S-iteration schemes with the existing schemes.

## 2 Methods

The paper is organized as follows. Section 3 contains the preliminaries, including definitions and lemmas with corresponding references that will be used in the sequel. Section 4 contains the main result of the paper. In Sect. 5, we provide a numerical example to illustrate the convergence behavior of the S-iteration and numerically compare the convergence of the S-iteration schemes with the existing schemes.

## 3 Preliminaries

Let \((X,d)\) be a metric space. A geodesic from *x* to *y* is a map *γ* from the closed interval \([0,d(x,y)]\subset \mathbb{R}\) to *X* such that \(\gamma (0)=x\), \(\gamma (d(x,y))=y\) and \(d(\gamma (t_{1}), \gamma (t_{2}))=|t_{1}-t_{2}|\) for all \(t_{1},t_{2}\in [0,d(x,y)]\). The image of *γ* is called a *geodesic* (or *metric*) *segment* joining *x* and *y*. When it is unique, this geodesic segment is denoted by \([x,y]\). The space *X* is said to be a *geodesic metric space* if every two points of *X* are joined by a geodesic, and *X* is said to be *uniquely geodesic metric space* if there is exactly one geodesic joining *x* and *y* for each \(x,y\in X\). A subset *C* of *X* is said to be *convex*, if for any two points \(x,y\in C\), the geodesic joining *x* and *y* is contained in *C*.

Let *X* be a uniquely geodesic metric space. For each \(x,y\in X\) and for each \(\alpha \in [0,1]\), there exists a unique point \(z\in [x,y]\) such that \(d(x, z) = (1-\alpha ) d(x, y)\) and \(d(y, z) = \alpha d(x, y)\). We denote the unique point *z* by \(\alpha x \oplus (1-\alpha )y\).

### Lemma 3.1

([7])

*Let*

*X*

*be a uniquely geodesic metric space*.

*The following are equivalent*:

- (i)
*X**is a CAT*(0)*space*. - (ii)
*X**satisfies the*(*CN*)*inequality*:*If*\(x,y\in X\)*and*\(\frac{x\oplus y}{2}\)*is the midpoint of**x**and**y*,*then*$$ d \biggl(z,\frac{x\oplus y}{2} \biggr)^{2}\leq \frac{1}{2}d(z,x)^{2}+ \frac{1}{2}d(z,y)^{2} - \frac{1}{4}d(x,y)^{2}, \quad \textit{for all }z\in X. $$

### Lemma 3.2

*Let*

*X*

*be a CAT*(0)

*space*, \(x,y,z\in X\)

*and*\(\lambda \in [0,1]\).

*Then*

- (i)
\(d(z,\lambda x \oplus (1-\lambda )y) \leq \lambda d(z,x)+ (1- \lambda )d(z,y)\);

- (ii)
\(d(z,\lambda x \oplus (1-\lambda )y)^{2} \leq \lambda d(z,x)^{2}+ (1-\lambda )d(z,y)^{2} -\lambda (1-\lambda ) d(x,y)^{2}\).

A complete CAT(0) space is called an *Hadamard space*.

It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is an Hadamard space. Other examples include Euclidean spaces \(\mathbb{E}^{2}\), Hilbert spaces, the Hilbert ball [9], hyperbolic spaces [10], R-trees [11], and many others. The fixed point theory in Hadamard spaces was first studied by Kirk [12] in 2003. Since then many authors have published papers on the existence and convergence of fixed points for nonlinear mappings in such spaces (e.g., see [13, 14]).

*X*as follows: Let \(\{x_{n}\}\) be a bounded sequence in

*X*. For \(x\in X\), we define a mapping \(r ( \cdot ,\{x_{n}\} ):X\to [0,\infty )\) by

*asymptotic radius*of \(\{x_{n}\} \) is given by

*asymptotic center*of \(\{x_{n}\}\) is the set

We now give the definition and collect some basic properties of the Δ-convergence which will be used in the sequel.

### Definition 3.3

([16])

A sequence \(\{x_{n}\}\) in an Hadamard space *X* is said to Δ*-converge* to \(x\in X\) if *x* is the unique asymptotic center of \(\{u_{n}\}\) for every subsequence \(\{u_{n}\}\) of \(\{x_{n}\}\). In this case, we write Δ-\(\lim_{n\to \infty }x_{n}=x\) and call *x* the Δ-limit of \(\{x_{n}\}\).

### Lemma 3.5

([17])

*Let**C**be a nonempty closed convex subset of an Hadamard space**X*. *If*\(\{x_{n}\}\)*is a bounded sequence in**C*, *then the asymptotic center of*\(\{x_{n}\}\)*is in**C*.

### Lemma 3.6

([8])

*Let*\(\{x_{n}\}\)*be a sequence in an Hadamard space**X**with*\(A(\{x_{n}\}) = \{x\}\). *If*\(\{u_{n}\}\)*is a subsequence of*\(\{x_{n}\}\)*with*\(A(\{u_{n}\}) = \{u\}\)*and*\(\{d(x_{n},u)\}\)*converges*, *then*\(x = u\).

In 2008, Berg and Nikolaev [18] introduced the concept of quasilinearization in an Hadamard space *X* as follows:

*vector*. The

*quasilinearization*is a map \(\langle \cdot ,\cdot \rangle : (X\times X)\times (X\times X)\to \mathbb{R}\) defined by

*X*satisfies the

*Cauchy–Schwarz inequality*if

*X*. Then the Cauchy–Schwarz inequality implies that \(\varTheta (t,a,b)\) is the Lipschitz function with the Lipschitz seminorm \(L(\varTheta (t,a,b))=td(a,b)\), where \(t\in \mathbb{R}\), \(a,b\in X\) and \(L(\phi )=\sup \{ \frac{\phi (x)-\phi (y)}{d(x,y)}: x,y\in X, x\ne y\}\) is the Lipschitz seminorm for any function \(\phi : X\to \mathbb{R}\).

*D*on \(\mathbb{R}\times X\times X\) by

*X*, it is obtained that \(D((t,a,b),(s,u,v))=0\) if and only if \(t \langle \overrightarrow{ab},\overrightarrow{xy} \rangle = s \langle \overrightarrow{uv},\overrightarrow{xy} \rangle \) for all \(x,y\in X\). Then,

*D*can impose an equivalent relation on \(\mathbb{R}\times X\times X\), where the equivalence class of \((t, a, b)\) is

*D*, which is called the dual metric space of

*X*.

In 2013, Kakavandi [20] introduced the concept of \((\mathbb{S})\) property for an Hadamard space as follows.

### Definition 3.7

An Hadamard space *X* satisfies the \((\mathbb{S})\)*property* if for any \((x,y)\in X\times X\) there exists a point \(y_{x}\in X\) such that \([\overrightarrow{xy}]=[\overrightarrow{y_{x}x}]\).

Moreover, Kakavandi also proved the characterization of Δ-convergence for Hadamard spaces satisfying the \((\mathbb{S})\) property as follows.

### Lemma 3.8

*Let**X**be an Hadamard space*, \(\{x_{n}\}\)*be a bounded sequence in**X**and*\(x\in X\). *If**X**satisfies the*\((\mathbb{S})\)*property*, *then* Δ-\(\lim_{n\rightarrow \infty }x_{n}=x\)*if and only if*\(\lim_{n\rightarrow \infty }\langle \overrightarrow{xx_{n}}, \overrightarrow{xy}\rangle =0\)*for all*\(y\in X\).

In 2008, Kirk and Panyanak [16] introduced a geometric condition on Hadamard spaces called the \((Q_{4})\) condition as follows.

### Definition 3.9

*X*is said to satisfy the \((Q_{4})\)

*condition*if for all \(x,y,p,q \in X\),

In 2013, Kakavandi [20] modified the \((Q_{4})\) condition as follows.

### Definition 3.10

*X*is said to satisfy the \((\overline{Q_{4}})\)

*condition*if for any \(x,y,p,q\in X\),

We can see that Hilbert spaces and every Hadamard space of constant curvature satisfy the \((\overline{Q_{4}})\) condition. Anyway, since \((\overline{Q_{4}})\) implies \((Q_{4})\), there are some Hadamard spaces that do not satisfy such a condition. The following results were obtained by Kaewkhao et al. [4].

### Lemma 3.11

*Let**X**be an Hadamard space satisfying the*\((\overline{Q_{4}})\)*condition*. *Let**C**be a nonempty subset of**X*. *Then*, *for any mapping*\(T:C\rightarrow X\), \(A(T)\)*is closed and convex*.

### Lemma 3.12

*Let**X**be an Hadamard space and**C**be a closed convex subset of**X*. *Let*\(x\in X\)*and*\(y\in C\). *Then*\(y=P_{C}x\)*if and only if*\(\langle \overrightarrow{xy},\overrightarrow{yz}\rangle \geq 0\)*for all*\(z\in C\).

*μ*be a continuous linear functional on \(l^{\infty }\), the Banach space of bounded real sequences with supremum norm, and \((x_{1},x_{2},\ldots ) \in l^{\infty }\). We write \(\mu _{n}(x_{n})\) instead of \(\mu ((x_{1},x_{2},\ldots ))\). We call

*μ*a

*Banach limit*if

*μ*satisfies \(\|\mu \| = \mu (1,1,\ldots )=1\) and \(\mu _{n}(x_{n})=\mu _{n}(x_{n+1})\) for each \((x_{1},x_{2},\ldots ) \in l^{\infty }\). For a Banach limit

*μ*, we know that

We also need the following lemmas due to Kaewkhao et al. [4].

### Lemma 3.13

*Let*

*C*

*be a nonempty subset of an Hadamard space*

*X*.

*Let*\(\{x_{n}\}\)

*be a bounded sequence in*

*C*

*and*\(T:C\to C\)

*be a mapping such that*\(\lim_{n\to \infty }d(x_{n},Tx_{n})= 0\).

*Then*

- (i)
*the sequences*\(\{d(x_{n},y)\}\)*and*\(\{d(Tx_{n},y)\}\)*are bounded for all*\(y\in C\); - (ii)
\(\mu _{n} d(x_{n},y)=\mu _{n}d(Tx_{n},y)\)

*for any Banach limit*\(\mu _{n}\)*on*\(l^{\infty }\).

### Lemma 3.14

*Let**X**be an Hadamard space and**C**be a closed convex subset of**X*. *Let*\(\{x_{n}\}\)*be a bounded sequence in**X*. *If*\(d(x_{n+1},z) \leq d(x_{n},z)\)*for all*\(z\in C\), *then*\(\lim_{n\to \infty }P_{C}x _{n}= z_{0}\in C\), *where*\(P_{C}\)*is the metric projection from**X**onto**C*.

## 4 Results and discussion

*X*is an Hadamard space and

*C*is a nonempty subset of

*X*defined as

Moreover, in Hadamard spaces, a further generalized hybrid mapping is defined analogously to Definition 1.3 as follows.

### Definition 4.1

*further generalized hybrid*if there exist \(\alpha ,\beta ,\gamma ,\delta ,\epsilon \in \mathbb{R}\) such that

- (i)
\(\alpha +\beta +\gamma +\delta \geq 0\), \(\epsilon \geq 0\);

- (ii)
\(\alpha +\beta >0\) or \(\alpha +\gamma >0\); and

- (iii)
\(\alpha d(Tx,Ty)^{2}+\beta d(x,Ty)^{2}+\gamma d(Tx,y)^{2}+\delta d(x,y)^{2}+\epsilon d(x,Tx)^{2}\leq 0\), \(\forall x,y\in C\).

The following lemma is a demiclosedness principle for a further generalized hybrid mapping in an Hadamard space.

### Lemma 4.2

*Let**X**be an Hadamard space**X**satisfying the*\((\mathbb{S})\)*property*. *Let**C**be a nonempty subset of**X**and let*\(T:C\rightarrow C\)*be a further generalized hybrid mapping*. *Let*\(\{x_{n}\}\)*be a bounded sequence in**C**such that*\(\lim_{n\rightarrow \infty }d(x_{n},Tx_{n})=0\)*and* Δ-\(\lim_{n\rightarrow \infty }x_{n}=z\). *Then*\(z\in A(T)\).

### Proof

*T*is a further generalized hybrid mapping,

*μ*to both sides of this inequality. Thus, we have

*μ*to both sides of this inequality. Thus, by \(\mu _{n}d(x_{n},Tx_{n})^{2}=0\), we have

In what follows we get a Δ-convergence theorem for a further generalized hybrid mapping in an Hadamard space.

### Theorem 4.3

*Let*

*X*

*be an Hadamard space satisfying the*\((\mathbb{S})\)

*property and the*\((\overline{Q_{4}})\)

*condition*.

*Let*

*C*

*be a nonempty convex subset of*

*X*

*and*\(T:C\rightarrow C\)

*be a further generalized hybrid mapping with*\(A(T)\neq \emptyset \).

*Let*\(\{\alpha _{n}\}\), \(\{\beta _{n}\}\)

*be sequences of real numbers such that*\(0< a\leq \alpha _{n}\), \(\beta _{n} \leq b<1\)

*for all*\(n\in \mathbb{N}\)

*and for some*

*a*,

*b*.

*Suppose that*\(\{x_{n}\}\)

*is the sequence generated by the S*-

*iteration process*:

*let*\(x_{1}=x\in C\)

*and*

*Then the sequence*\(\{x_{n}\}\) Δ-

*converges to an element*\(v \in A(T)\),

*where*\(v=\lim_{n\rightarrow \infty }P_{A(T)}x_{n}\)

*and*\(P_{A(T)}\)

*is the metric projection from*

*X*

*onto*\(A(T)\).

### Proof

Step 1. We will show that \(\lim_{n\rightarrow \infty }d(u,x_{n})\) exists for all \(u\in A(T)\).

Step 2. We will show that \(\lim_{n\rightarrow \infty }d(x_{n},Tx_{n})= 0\).

*z*for some \(z\in X\). By (9) and Lemma 4.2, we have \(z\in A(T)\) and hence \(\lim_{n\rightarrow \infty }d(z,x_{n})\) exists. If \(z\neq w\), then it follows from the uniqueness of asymptotic center that

*v*, where \(v= \lim_{n\rightarrow \infty }P_{A(T)}x_{n}\). □

### Remark 4.4

Theorem 4.3 extends and improves the results of Kaewkhao et al. [4] from a normally generalized hybrid mapping to a further generalized hybrid mapping. In fact, we present the S-iteration process for solving the attractive point problem of further generalized hybrid mappings in Hadamard spaces.

It is known that a Hilbert space satisfies both the \((\mathbb{S})\) property and the \((\overline{Q_{4}})\) condition. Furthermore, Δ-convergence and weak convergence are the same in a Hilbert space. Thus, we have the following theorem.

### Theorem 4.5

*Let*

*X*

*be a Hilbert space*.

*Let*

*C*

*be a nonempty convex subset of*

*X*

*and*\(T:C\rightarrow C\)

*be a further generalized hybrid mapping with*\(A(T)\neq \emptyset \).

*Let*\(\{\alpha _{n}\}\), \(\{\beta _{n}\}\)

*be sequences of real numbers such that*\(0< a\leq \alpha _{n}\), \(\beta _{n}\leq b<1\)

*for all*\(n\in \mathbb{N}\)

*and for some*

*a*,

*b*.

*Suppose that*\(\{x_{n}\}\)

*is the sequence generated by the S*-

*iteration process*:

*let*\(x_{1}=x\in C\)

*and*

*Then the sequence*\(\{x_{n}\}\)

*converges weakly to an element*\(v\in A(T)\),

*where*\(v=\lim_{n\rightarrow \infty }P_{A(T)}x_{n}\)

*and*\(P_{A(T)}\)

*is the metric projection from*

*X*

*onto*\(A(T)\).

Moreover, the following example shows that there is an Hadamard space satisfying both the \((\mathbb{S})\) property and the \(( \overline{Q_{4}})\) condition, which is not a Hilbert space.

### Example 4.6

([4])

*d*be a metric defined by the function \(d:\mathcal{H}\times \mathcal{H}\to \mathbb{R}\) that assigns to each pair of vectors \(u=(u_{1},u_{2})\) and \(v=(v_{1},v_{2})\) the unique nonnegative number \(d(u,v)\geq 0\) such that

### Remark 4.7

Theorem 4.5 extends and improves the results of Takahashi et al. [3] from a normally generalized hybrid mapping to a further generalized hybrid mapping. In fact, we present the S-iteration process for solving the attractive point problem of further generalized hybrid mappings in Hilbert spaces.

## 5 Numerical example for the main result

In this section, we give a numerical example supporting our main results and compare the convergence of the studied method (4) with the Mann and Ishikawa iterations.

### Example 5.1

*d*, which is also an Hadamard space, and \(C =(-1,1)\). We see that

*C*is a convex subset of

*X*. Define a mapping \(T : C\to C\) by

*T*is a further generalized hybrid mapping with \(\alpha = 2\), \(\beta =\gamma =-1\), \(\delta = \epsilon =0\) and \(A(T)=[1,\infty )\). Let \(\alpha _{n}=\frac{4n}{5n+7}\) and \(\beta _{n}=\frac{n+3}{10n+1}\) for all \(n\in \mathbb{N}\). Let \(\{x_{n}\}\) be a sequence generated by S-iteration (4), \(\{u_{n}\}\) be a sequence generated by Mann iteration (12) and \(\{z_{n}\}\) be a sequence generated by Ishikawa iteration (13). The numerical experiments of all iterations for approximating the attractive point 1, where \(1=\lim_{n\rightarrow \infty }P_{A(T)}x_{n}\), and convergence of \(\{x_{n}\}\), \(\{u_{n}\}\) and \(\{z_{n}\}\) are given in Tables 1 and 2.

Iterates of S-iteration, Mann iteration, and Ishikawa iteration for \(x_{1} =u_{1}=z_{1}= -0.5\)

| S-iteration | Mann iteration | Ishikawa iteration | |||
---|---|---|---|---|---|---|

\(x_{n}\) | \(|x_{n}-x_{n-1}|\) | \(u_{n}\) | \(|u_{n}-u_{n-1}|\) | \(z_{n}\) | \(|z_{n}-z_{n-1}|\) | |

1 | −0.5000000 | – | −0.5000000 | – | −0.5000000 | – |

2 | 0.6742424 | 1.1742e + 00 | −0.0833333 | 4.1667e − 01 | −0.1590909 | 3.4091e − 01 |

3 | 0.8462461 | 1.7200e − 01 | 0.2107843 | 2.9412e − 01 | 0.1550166 | 3.1411e − 01 |

4 | 0.9271811 | 8.0935e − 02 | 0.4260250 | 2.1524e − 01 | 0.4077682 | 2.5275e − 01 |

5 | 0.9654324 | 3.8251e − 02 | 0.5960916 | 1.7007e − 01 | 0.5982240 | 1.9046e − 01 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

16 | 0.9999899 | 1.1023e − 05 | 0.9962619 | 2.1566e − 03 | 0.9974837 | 1.5930e − 03 |

17 | 0.9999952 | 5.2699e − 06 | 0.9976368 | 1.3749e − 03 | 0.9984638 | 9.8015e − 04 |

18 | 0.9999977 | 2.5201e − 06 | 0.9985102 | 8.7334e − 04 | 0.9990648 | 6.0091e − 04 |

19 | 0.9999989 | 1.2054e − 06 | 0.9990631 | 5.5292e − 04 | 0.9994320 | 3.6724e − 04 |

20 | 0.9999995 | 5.7667e − 07 | 0.9994121 | 3.4904e − 04 | 0.9996558 | 2.2380e − 04 |

Iterates of S-iteration, Mann iteration, and Ishikawa iteration for \(x_{1} =u_{1}=z_{1}= 0.4\)

| S-iteration | Mann iteration | Ishikawa iteration | |||
---|---|---|---|---|---|---|

\(x_{n}\) | \(|x_{n}-x_{n-1}|\) | \(u_{n}\) | \(|u_{n}-u_{n-1}|\) | \(z_{n}\) | \(|z_{n}-z_{n-1}|\) | |

1 | 0.4000000 | – | 0.4000000 | – | 0.4000000 | – |

2 | 0.7181818 | 3.1818e − 01 | 0.5000000 | 1.0000e − 01 | 0.5181818 | 1.1818e − 01 |

3 | 0.8669850 | 1.4880e − 01 | 0.6176471 | 1.1765e − 01 | 0.6450471 | 1.2687e − 01 |

4 | 0.9370031 | 7.0018e − 02 | 0.7219251 | 1.0428e − 01 | 0.7512207 | 1.0617e − 01 |

5 | 0.9700950 | 3.3092e − 02 | 0.8043177 | 8.2393e − 02 | 0.8312256 | 8.0005e − 02 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

16 | 0.9999913 | 9.5358e − 06 | 0.9981890 | 1.0448e − 03 | 0.9989430 | 6.6918e − 04 |

17 | 0.9999958 | 4.5591e − 06 | 0.9988551 | 6.6612e − 04 | 0.9993547 | 4.1173e − 04 |

18 | 0.9999980 | 2.1802e − 06 | 0.9992782 | 4.2311e − 04 | 0.9996071 | 2.5242e − 04 |

19 | 0.9999990 | 1.0428e − 06 | 0.9995461 | 2.6788e − 04 | 0.9997614 | 1.5427e − 04 |

20 | 0.9999995 | 4.9889e − 07 | 0.9997152 | 1.6910e − 04 | 0.9998554 | 9.4011e − 05 |

From Tables 1 and 2, we see that both \(\{x_{n}\}\), \(\{u_{n}\}\) and \(\{z_{n}\}\) converge to \(1\in A(T)\) and observe that \(|x_{n}-1|\leq |u _{n}-1|\) and \(|x_{n}-1|\leq |z_{n}-1|\), so the sequence \(\{x_{n}\}\) generated by S-iteration converges faster than both \(\{u_{n}\}\) generated by Mann iteration and \(\{z_{n}\}\) generated by Ishikawa iteration.

## 6 Conclusions

The results presented in this paper modify, extend and improve the corresponding results of Takahashi et al. [3] and Kaewkhao et al. [4], and others. The main aim of this paper is to prove the demiclosed principle for further generalized hybrid mapping and the Δ-convergence of the sequence generated by the S-iteration process for finding attractive points of such mappings in Hadamard spaces satisfying the \((\mathbb{S})\) property and the \((\overline{Q_{4}})\) condition. We also provide a numerical example to illustrate and support our results at the end.

## Notes

### Acknowledgements

The authors appreciate the support of their institutes.

### Availability of data and materials

Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.

### Authors’ contributions

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

### Funding

The first author was supported by Mahidol University Kanchanaburi Campus, Kanchanaburi, Thailand.

### Competing interests

The authors declare that they have no competing interests.

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