Computational comparative analysis of fixed point approximations of generalized $\alpha$-nonexpansive mappings in hyperbolic spaces

1.   Introduction

In a variety of general metric space contexts, the nonexpansive mappings can be defined. The $\alpha$-nonexpansive mappings was first introduced by Aoyama and Kohsaka in 2011 ^[1]. For this type of mappings, they obtained certain fixed point and convergence results in Banach spaces, which making it more general than the Suzuki's condition $(C)$ ^[2]. In 2017, Pant and Shukla introduced in ^[3] a new class of mappings, the generalized $\alpha$-nonexpansive mappings. In the setting of uniformly convex hyperbolic spaces, Mebawondu and Izuchukwu ^[4] investigated some fixed point properties and the demiclosedness principle for generalized $\alpha$-nonexpansive mappings.

Definition 1.1. Let $\mathcal{J}$ be a nonempty subset of the hyperbolic space $\mathcal{H}$. The mapping $\zeta:\mathcal{J} \rightarrow \mathcal{J}$ is called generalized $\alpha$-nonexpansive mapping or condition $(C_{\alpha})$, if

for all $s, t \in \mathcal{J}$, $0\leq \alpha < 1$ and $\hat{d}(s, \zeta(t)) = \inf\{\hat{d}(s, r): r \in \zeta(t)\}$.

In 1922, Banach ^[5] proposed the Banach Contraction Principle, which states that the Picard iterative scheme ^[6] can be used to approximate the fixed points of a contraction mapping. The Picard scheme is defined as follows

The nonexpansive mapping does not converge to a fixed point in Picard iterations. For more details, we can see ^[7].

In 1953, Mann ^[8] approximated the fixed point for nonexpansive mappings using the iterative scheme defined below as

where $\{\eta_{i}\}\in (0, 1).$

Agarwal et al. ^[9] introduced the following $S$-iterative scheme, defined as

where $\{\eta_{i}\}$, $\{\rho_{i}\}$ are the sequences of parameters in $(0, 1)$.

In 2017, Thakur et al. ^[10], introduced three step iterative scheme and the sequences of this scheme are given as

where $\{\eta_{i}\}$, $\{\rho_{i}\}$ and $\{\sigma_{i}\}$ are the sequences of parameters in $(0, 1)$.

Recently, the researchers in this area have introduced the hybrid iterative process obtained by the hybridization of Picard with Mann, Ishikawa, $S$, $S^*$ and Thakur et al., iterative schemes which results faster convergence as compared with many others schemes. Khan ^[11], introduced Picard-Mann iterative scheme which is given by the sequence $\{s_{i}\}$ as

where $\{\eta_{i}\}\in [0, 1).$

The Picard-Ishikawa hybrid iterative scheme was given by Okeke ^[12], which is given by the sequence $\{s_{i}\}$ as

where $\{\eta_{i}\}, \{\rho_{i}\} \in [0, 1).$

Similarly, Gursoy and Karakaya (see ^[13]) proposed Picard-$S$ hybrid iterative scheme given by the sequence $\{s_{i}\}$ as

where $\{\eta_{i}\}, \{\rho_{i}\} \in [0, 1).$

Also, Lamba and Panwar ^[14] gave the idea and proved the convergence results of Picard-$S^*$ iterative scheme given by the sequence $\{s_{i}\}$ as

where $\{\eta_{i}\}, \{\rho_{i}\}, \{\sigma_{i}\} \in [0, 1).$

Recently, motivated by hybridization behavior of iterative schemes Jie Jia et al.^[15] proposed the Picard-Thakur hybrid iterative scheme. They proved analytically and numerically that their scheme has better rate of convergence than all above hybrid schemes and the sequence $\{s_{i}\}$ of this scheme is defined as

where $\{\eta_{i}\}, \{\rho_{i}\}, \{\sigma_{i}\} \in [0, 1).$

For any other spaces algorithms for approximations of fixed points of $\alpha$-nonexpansive mappings, using different type of iterative schemes were recently studied in ^{[16,17,18,19]}.

Motivated by the convergence behavior of the scheme $(1.6)$, we extend these results in order to verify the $\Delta$ and strong convergence results for a generalized $\alpha$-nonexpansive mapping in hyperbolic space. Our results extend the results in Banach space to hyperbolic space. We performed a computational experiment to illustrate that the Picard-Thakur iterative scheme $(1.6)$ converges more quickly than a number of other hybrid iterative schemes. We provide an application to Fredholm integral equation.

2.   Preliminaries

Mebawondu and Izuchukwu gave the definition of generalized $\alpha$-nonexpansive mappings in the framework of uniformly convex hyperbolic spaces (UCHS) ^[4].

Definition 2.1. Let $\mathcal{J}$ be a nonempty subset of a hyperbolic space $\mathcal{H}$. A self-mapping $\zeta$ from $\mathcal{J}$ to $\mathcal{J}$ $(\zeta:\mathcal{J} \rightarrow \mathcal{J})$ is called generalized $\alpha$-nonexpansive if, for all $s, t \ \in \mathcal{J}$, there exists $\alpha\in [0, 1)$ such that

where $\hat{d}(s, \zeta(t)) = \inf\{\hat{d}(s, r): r \in \zeta(t)\}$.

It is also known in related literature as condition $(C_{\alpha})$.

In this study, we investigate some results in the framework of Kohlenbach's hyperbolic spaces ^[20]. These results concern normed linear spaces, convex subsets, and Hadamard manifolds ^[21], as well as $CAT(0)$ spaces in the Gromovian sense ^[22] and the Hilbert ball endowed with the hyperbolic metric ^[21].

Let $(\mathcal{H}, \hat{d})$ be a metric space and $\mathcal{Q}:X^{2}\times [0, 1]$. A triplet $(\mathcal{H}, \hat{d}, \mathcal{Q})$ is called a hyperbolic space, if

$(H_{1})$ $\hat{d}(s, \mathcal{Q}(t, u, \eta)) \leq (1-\eta)\hat{d}(s, t)+\eta \hat{d}(s, u)$,

$(H_{2})$ $\hat{d}(\mathcal{Q}(s, u, \eta), \mathcal{Q}(s, t, \rho)) \leq |\eta - \rho|\hat{d}(s, t)$,

$(H_{3})$ $\mathcal{Q}(s, t, \eta) = \mathcal{Q}(s, t, (1-\eta))$,

$(H_{4})$ $\hat{d}(\mathcal{Q}(s, v, \eta), \mathcal{Q}(t, u, \eta)) \leq(1-\eta)\hat{d}(s, t)+ \eta \hat{d}(v, u)$.

for all $s, t, u, v \in \mathcal{H}$ and $\eta, \rho \in [0, 1]$.

The space $(\mathcal{H}, \hat{d}, \mathcal{Q})$ is called UCHS ^[23], if for all $s, t, u \in \mathcal{H}, \xi > 0$, and $0 < \epsilon \leq 2$ there exist $\delta \in (0, 1)$ such that $\hat{d}(s, t)\leq \xi,$ $\hat{d}(s, u)\leq \xi$ & $\hat{d}(t, u)\geq \epsilon\xi.$ Then we have

A mapping $\varphi:(0, \infty)\times(0, 2] \rightarrow (0, 1]$ is called modulus of uniform convexity of $X$ if $\delta = \varphi(\xi, \epsilon)$, for a given $\epsilon\in (0, 2]$. If $\varphi$ decreases with $\xi$ (for a fixed $\epsilon$) (i.e., for a given $\epsilon > 0$ and $\xi_{2} \geq \xi_{1} > 0$, we have $\varphi(\xi_{2}, \epsilon)\leq \varphi(\xi_{1}, \epsilon)$), then $\varphi$ is monotone.

A subset $\mathcal{J}$ of a hyperbolic space $\mathcal{H}$ is convex if $\mathcal{Q}(s, t, \sigma)\in \mathcal{J}$, for any $s, t \in \mathcal{J}$ and $\sigma \in [0, 1].$

We use the notation $(1-\sigma)s\oplus \sigma t$ for $\mathcal{Q}(s, t, \sigma)$ where $s, t \in \mathcal{H}$ and $0\leq \sigma \leq 1$. Leuştean remarked in ^[23] that $\mathcal{H}$ endowed with a normed function, i.e. $(\mathcal{H}, \|.\|)$, is a hyperbolic space, with $(1-\sigma)s\oplus \sigma t = (1-\sigma)s+\sigma t$. As a result, uniformly convex Banach spaces are a generalization of the class of UCHS.

Let $\zeta:\mathcal{J} \to \mathcal{J}$ be a mapping and $s^{*}\in \mathcal{J}\subset\mathcal{H}$. If $\zeta(s^*) = s^*$ then $s^{*}$ is a fixed point of $\zeta$ and $F(\zeta)$ represents the set of all fixed points of the mapping $\zeta$.

Further, let us recall some fundamental definitions and results that will be used in our research. Dhompongsa et al. ^[24], gave the concept of $\Delta$-convergence in hyperbolic spaces and he used it as in the following.

Let $\{s_{i}\}$ be a bounded sequence in hyperbolic space $\mathcal{H}$. Consider a mapping $r_{s}(., \{s_{i}\}):\mathcal{H} \rightarrow [0, \infty)$ defined by

For each $s\in \mathcal{H}$, the value $r_{s}(s, \{s_{i}\})$ is called the asymptotic radius of $\{s_{i}\}$ at $s$.

The asymptotic radius of $\{s_{i}\}$ relative to $\mathcal{J}\subset \mathcal{H}$ is defined as

The asymptotic center of $\{s_{i}\}$ relative to $\mathcal{J}$ is the set

Definition 2.2. ^[25] The sequence $\{s_{i}\}$ in $\mathcal{H}$. $\{s_{i}\}$ is $\Delta$-convergent to $s \in \mathcal{H}$ if, for every subsequence $\{s_{i_{j}}\}$ of $\{s_{i}\}$, $s$ is the asymptotic center. So, $\Delta-\lim\limits_{i\to \infty}s_{i} = s$ and $s$ denotes the $\Delta-\lim$ of $\{s_{i}\}$.

Lemma 2.1. ^[26] If $\mathcal{J}$ is a nonempty closed convex subset of a complete UCHS $\mathcal{H}$ with a monotone modulus of uniform convexity $\varphi$, then every bounded sequence $\{s_{i}\}$ in $\mathcal{H}$ has a unique asymptotic centre with respect to $\mathcal{J}$.

Definition 2.3. ^[27] A hyperbolic space $\mathcal{H}$ is said to satisfy Opial's condition if, for any sequence $\{s_{i}\}$ with $\{s_{i}\}\rightharpoonup s \in \mathcal{H}$, the following inequality holds

for $t \in \mathcal{H}$ with $s \neq t$.

The concept of Condition (Ⅰ) was introduced by Sentor and Dotson in ^[28] and is described as follows.

Definition 2.4. Let $\Theta$ be an increasing function from $[0, \infty)$ to $[0, \infty)$ with $\Theta(0) = 0$ and $\Theta(c) > 0,$ for all $c > 0$. Let $\zeta:\mathcal{J} \rightarrow \mathcal{J}$ be a self-mapping such that

where $\hat{d}(s, F(\zeta)) = \inf\{\hat{d}(s, s^{*}): s^{*} \in F(\zeta)\}$.

Proposition 2.1. ^[3] Let $\mathcal{J}$ be a nonempty subset of the hyperbolic space $\mathcal{H}$ and $\zeta:\mathcal{J} \rightarrow \mathcal{J}$ be a generalized $\alpha$-nonexpansive mapping. Then, for all $s, t \in \mathcal{J}$, we have true

Lemma 2.2. ^[29] Let $\mathcal{H}$ be a complete UCHS with a monotone modulus of convexity $\varphi$. Let $s \in \mathcal{H}$ and $\{\eta_{i}\}$ be a sequence in $[c, d]$ for some $c, d\in (0, 1).$ If $\{s_{i}\}$ and $\{t_{i}\}$ are sequences in $\mathcal{H}$, such that $\limsup\limits_{i\to\infty}\hat{d}(s_{i}, s) \leq \omega,$ $\limsup\limits_{i\to\infty}\hat{d} (t_{i}, s) \leq \omega$ and $\limsup\limits_{i\to\infty}\hat{d}(\mathcal{Q}(s_{i}, t_{i}, \eta_{i}) = \omega$ hold for some $\omega \geq 0$, then $\lim\limits_{n\to\infty}\hat{d}(s_{i}, t_{i}) = 0.$

Lemma 2.3. ^[3] Let $\mathcal{J}$ be a nonempty subset of $\mathcal{H}$. Let $\zeta:\mathcal{J}\rightarrow \mathcal{J}$ be a generalized $\alpha$-nonexpansive mapping with $F(\zeta)\neq \emptyset$. Then $\zeta$ is quasi-nonexpansive.

Definition 2.5. A sequence $\{s_{i}\}$ is called a Fejér monotone sequence with respect to $\mathcal{J}$, subset of hyperbolic space $\mathcal{H}$, if

Proposition 2.2. ^[30] Let $\{s_{i}\}$, $\mathcal{H}$, $\mathcal{J}$ and $\zeta$ be defined as in Lemma 2.3 and $\{s_{i}\}$ is Fejér monotone w.r.t. $\mathcal{J}$. Then

$i.$ $\{s_{i}\}$ is bounded.

$ii.$ The sequence $\{\hat{d}(s_{i}, s^{*})\}$ is decreasing and converges to $s^{*}\in F(\zeta)$.

$iii.$ $\lim\limits_{i \rightarrow \infty}\hat{d}(s_{i}, F(\zeta))$ exists.

Theorem 2.1. ^[31] A monontone sequence of real numbers is convergent if and only if it is bounded. Further,

(i) If $\{\alpha_{i}\}$ is bounded decreasing then $\lim\limits_{i \rightarrow \infty}\{\alpha_{i}\} = \inf\{\alpha_{i}:i \in \mathbb{N}\}.$

(ii) If $\{\alpha_{i}\}$ is bounded increasing then $\lim\limits_{i \rightarrow \infty}\{\alpha_{i}\} = \sup\{\alpha_{i}:i \in \mathbb{N}\}.$

3.   Rate of convergence

In the following, let us construct the iterative scheme $(1.6)$ in a hyperbolic space as follows

where $\{\eta_{i}\}, \{\rho_{i}\}, \{\sigma_{i}\} \in [0, 1).$

Lemma 3.1. Let $\mathcal{J}$ be a nonempty closed convex subset of a complete UCHS $\mathcal{H}$ and $\zeta:\mathcal{J} \rightarrow \mathcal{J}$ a generalized $\alpha$-nonexpansive mapping with $F(\mathcal{J})\neq \emptyset$. Let $\{s_{i}\}$ be a sequence defined by $(3.1)$. Then $\lim\limits_{i\to\infty}\hat{d}(s_{i}, s^{*})$ exists for all $s^{*} \in F(\zeta)$.

Proof. Let $s^{*} \in F(\zeta)$. Since $\zeta$ satisfies the condition ($C_{\alpha}$) we have,

Thus, by iterative scheme $(3.1)$, we get

Using $(3.1)$ together with $(3.2)$, we get

Using $(3.1)$ together with $(3.3)$ and $(3.2)$, we get

Using $(3.1)$ together with $(3.4)$, we get

This shows that $\{\hat{d}(s_{i}, s^{*})\}$ is Fejér monotone w.r.t. $F(\zeta)$. Then, $\lim\limits_{i\to\infty}\hat{d}(s_{i}, s^{*})$ exists.

Lemma 3.2. Let $\mathcal{J}, \mathcal{H}, \zeta$ be defined as in Lemma $3.1$. For an arbitrary $s_{0} \in \mathcal{J}$, a sequence $\{s_{i}\}$ is defined by $(3.1)$. Then $\{s_{i}\}$ is bounded and $\lim\limits_{i\to\infty}\hat{d}(s_{i}, \zeta (s_{i})) = 0$ if and only if $F(\zeta)\neq \emptyset$.

Proof. Let $\{s_{i}\}$ be a sequence defined by $(3.1)$ and $s^{*} \in F(\zeta)$. From above Lemma $3.1$, $\lim\limits_{i\to\infty}\hat{d}(s_{i}, s^{*})$ exists. Let $\lim\limits_{i\to\infty}\hat{d}(s_{i}, s^{*}) = \omega > 0$. If $\omega = 0$, then $\lim\limits_{i\to\infty}\hat{d}(s_{i}, \zeta(s_{i})) = 0.$

Next, consider $\omega > 0$. By Lemma $3.1$ we obtain that

Taking $\limsup$ in $(3.6)$, we get

Since,

we get that

Then, we have

By Eqs $(3.7)$ and $(3.9)$, we get

which implies that

By Lemma 2.2, we have

On the other hand, if $\{s_{i}\}$ is bounded in $\mathcal{H},$ then by Lemma 2.1, $A_{s}\mathcal{J}, \{s_{i}\}) = \{s\}$ is a singleton (i.e, has a unique asymptotic center in $\mathcal{H}$) and $\lim\limits_{i\to\infty}\hat{d}(s_{i}, \zeta(s_{i})) = 0$. Then, $\zeta$ satisfies the condition $(C_{\alpha})$ on $\mathcal{J}$.

Using Lemma 2.1, we have

Taking $\limsup$ as $i\to \infty$ on both sides we get

By utilizing the special property of asymptotic center, we get $\zeta(s^{*}) = s^{*}$; so $s \in F(\zeta)$. Hence, $F(\zeta)$ is nonempty.

Theorem 3.1. Let $\mathcal{J}, \mathcal{H}, \zeta$ and $\{s_{i}\}$ be a sequence defined as in Lemma $3.1$ with monotone modulus of uniform convexity $\varphi$. Let $F(\zeta) \neq \emptyset$. Then $\{s_{i}\}$ $\Delta$- converges to a fixed of $\zeta$.

Proof. From Lemma $3.2$, the sequence $\{s_{i}\}$ is bounded. Therefore $\{s_{i}\}$ has a $\Delta$-convergent subsequence. We have to prove that for every $\Delta$-convergent subsequence there is a unique $\Delta$-limit, which is the fixed point of $\zeta$.

Let $\lambda$ and $\mu$ be $\Delta$-limits of the subsequences $\{\lambda_{i}\}$ & $\{\mu_{i}\}$ and respectively, of the sequence $\{s_{i}\}$. By Lemma $1$ we have $r_{s}(\mathcal{J}, \{\lambda_{i}\}) = \{\lambda\}$ and $r_{s}(\mathcal{J}, \{\mu_{i}\}) = \{\mu\}$.

By Lemma $3.2$, $\lim\limits_{i\to\infty}\hat{d} (s_{i}, \zeta(s_{i})) = 0.$ We claim that $\lambda$ and $\mu$ are two fixed points of $\zeta$. Since, $\zeta$ satisfies the condition $(C_{\alpha})$ and by Lemma $2.1$, we have

By taking $\limsup$, we get

By uniqueness of the asymptotic center, $\zeta(\lambda) = \lambda$, so $\lambda \in F(\zeta)$. Similarly, we also have $\mu \in F(\zeta).$ Further, we show that $\lambda = \mu$. Let $\lambda \neq \mu$; then by uniqueness of the asymptotic center, we have

Contradiction. Hence $\lambda = \mu$. Then the sequence $\{s_{i}\}$ $\Delta$-converges to $F(\zeta)$.

Further, we give some strong convergence theorems.

Theorem 3.2. Let $\mathcal{J}, \mathcal{H}, \zeta$ and $\{s_{i}\}$ be as in Lemma $3.1$ which satisfy condition $(C_{\alpha})$ and $F(\zeta) \neq \emptyset,$ then $\{s_{i}\}$ converges strongly to $F(\zeta)$ or ($\{s_{i}\}\rightarrow s^{*} \in F(\zeta)$) if and only if $\liminf\limits_{i\to\infty}\hat{d}(s_{i}, F(\zeta)) = 0$.

Proof. Let $\{s_{i}\}\rightarrow s^{*} \in F(\zeta).$ Then, $\lim\limits_{i\to\infty}\hat{d} (s_{i}, s^{*}) = 0.$ Because $0\leq \liminf\limits_{i\to\infty}\hat{d}(s_{i}, F(\zeta))\leq \hat{d}(s_{i}, s^{*}),$ therefore $\liminf\limits_{i\to\infty}\hat{d}(s_{i}, F(\zeta)) = 0.$

Conversely, assume that $\liminf\limits_{i\to\infty}\hat{d}(s_{i}, F(\zeta)) = 0.$ From Lemma $3.1$ we get $\hat{d}(s_{i+1}, s^{*})\leq \hat{d}(s_{i}, s^{*})$ for all $s^{*} \in F(\zeta).$

Thus $\hat{d}(s_{i+1}, F(\zeta))\leq \hat{d}(s_{i}, F(\zeta)).$ Therefore, $\lim\limits_{i\to\infty}\hat{d}(s_{i}, F(\zeta))$ exists. From the assumption of our theorem, $\liminf\limits_{i\to\infty}\hat{d}(s_{i}, F(\zeta)) = 0.$ So, we have $\lim\limits_{i\to\infty}\hat{d}(s_{i}, F(\zeta)) = 0.$

Next, we prove that $\{s_{i}\}$ is a Cauchy sequence in $\mathcal{J}$. Let $\upsilon > 0.$ Since, $\lim\limits_{i\to\infty}\hat{d}(s_{i}, F(\zeta)) = 0$, for any given $\upsilon > 0$, there is $i_{0} \in \mathbb{N}$ such that

In particular, $\inf\{\hat{d}(s_{i}, s^{*}):s^{*} \in F(\zeta)\} < (\upsilon/2)$. Then, there exists $s^{**} \in F(\zeta)$ such that $\hat{d}(s_{n_{0}}, s^{**}) < (\upsilon/2).$ For any $i, j\geq i_{0}$, we get

This implies that $\{s_{i}\}$ is a Cauchy sequence in $\mathcal{J}$. Then, $\mathcal{J}$ is a closed and complete subset of a $\mathcal{H}$. Then the sequence $\{s_{i}\}$ must to converge to a point from $\mathcal{J}$. Let $\lim\limits_{i\to\infty}s_{i} = \omega \in\mathcal{J}$.

Next, we prove $\omega\in F(\zeta)$. Since, $F(\zeta)$ satisfies the condition $(C_{\alpha})$ and using Lemma $2.1$, we have

Letting $i\to \infty$, and using Lemma $3.2$ we get $\hat{d}(s_{i}, \zeta(s_{i})) = 0$. Then, we have $\hat{d}(\omega, \zeta(\omega)) = 0$, which means $\omega$ is a fixed point of $\zeta$. Hence, $\{s_{i}\}$ converges strongly to $F(\zeta).$

Theorem 3.3. Let $\mathcal{J}, \mathcal{H}, \zeta$ and $\{s_{i}\}$ be a sequence defined as in Lemma $3.1$ with condition $(C_{\alpha})$, and $F(\zeta) \neq \emptyset$. If $\zeta$ satisfies the Sentor and Dotson condition $(I)$, then the sequence $\{s_{i}\}$ converges strongly to a fixed of $\zeta$.

Proof. From Lemma $3.2$, we have

From Condition (Ⅰ) and $(3.10)$, we get

which implies

Since $\Theta:[0, \infty) \rightarrow [0, \infty)$ is an increasing function satisfying $\Theta(0) = 0,$ $\Theta(c) > 0 \mbox{ for all } c > 0$, we have

Then, all conditions of the Theorem $3.2$ are satisfied. Hence, $\{s_{i}\}$ converges strongly to $F(\zeta).$

4.   Numerical example

This section deals with the numerical and analytical analysis of the Picard-Thakur hybrid iterative scheme $(1.6)$. We provide a numerical example for the convergence of a mapping which satisfying the condition $C_{\alpha}$) but fails to satisfy Suzuki's Condition (C).

Example 4.1. Let $\mathcal{J} = [0, \infty)$ endowed with usual norm $|.|$. Let $\zeta$ be a self map on $\mathcal{J}$ as $\zeta: [0, \infty)\rightarrow [0, \infty)$ and defined as

First, we have to show that $\zeta$ does not satisfy the Suzuki's condition $(C)$. For this, let $s = \frac {5}{2}$ and $t = \frac {7}{2}$. Since

and

which implies $\frac {1}{2}\hat{d}(s, \zeta(s)) < \hat{d}(s, t)$.

Now,

So, $\frac {1}{2}\hat{d}(s, \zeta(s)) < \hat{d}(s, t)$ but $\hat{d}(\zeta(s), \zeta(t))\geq \hat{d}(s, t)$. Hence, $\zeta$ does not satisfy the Suzuki's Condition (C).

Next, we have to show that $\zeta$ is generalized $\alpha$-nonexpansive mapping i.e.,

We take $\alpha = \frac{1}{3}$ and we consider the following cases:

Case 1: When $s\geq 3$ and $t\in [0, 3)$ then

and

Case 2: Let $s\in [0, 3)$ and $t\geq 3$, then it is obvious that

Case 3: Let $s\in[0, 3)$ and $t\in [0, 3)$; then

So, $\zeta$ is generalized $\frac{1}{3}-$nonexpansive mapping.

Further, we take different choices of parameters $\eta_{i}, \rho_{i}$ and $\sigma_{i}$ and set the stoping criterion as $\|s_{i}-s^{*}\|\leq 10^{-15}$, where $s^{*} = 5$ is fixed point of the problem. In Table 1, we obtain the convergence behavior of the iterative scheme $(1.6)$ compare with other schemes discussed in literature by choosing different initial point and fixed parameters as: $\eta_{i} = \frac{i}{(i^3+6)^2}, \ \rho_{i} = \sqrt{\frac{i}{(6i+5)^7}} \ {\rm{and}} \ \sigma_{i} = 1-\frac{i}{(i+5)^2}$. In Table 2, we compare the iterative scheme $(1.6)$ by choosing different choices of parameters.

Next, we will give a comparison of iterative schemes for different choices of parameters of the Tables 1 and 2, with initial points $s_{0} = 3.16 \ {\rm{and}}\ 6.00$.

Using the values of our tables we will generate in Matlab-version R2021a some figures proving that the iterative scheme $(1.6)$ is stable and converges faster to the fixed point, w.r.t. different choices of initial points and parameters. Then, we will obtain Figures 1–3 for parameter used in Table 1, Figures 4 and 5 for parameters B, Figures 6 and 7 for parameters C and Figures 8 and 9 for parameters D used in Table 2, respectively, for initial points $s_{0} = 3, 16$ and $6.00$.

Studying the figures and tables previous presented here we can conclude that the iterative scheme $(1.6)$ has better rate of convergence than other hybrid schemes.

5.   Application to Fredholm integral equations

This section deals with the application to Fredholm integral equations. Let us consider the following integral equation

Suppose that $\mathcal{J}$ be nonempty compact subset of a Banach space, which is special case of hyperbolic space $\mathcal{H} = C[g, h]$, where $C[g, h]$ denotes the space of all continuous real-valued functions defined on an interval $[g, h]$, with norm $\|s-t\| = \max\limits_{\bar{a}\in[g, h]}|s(\bar{a})-t(\bar{a})|$. Obviously, $(C[g, h], \|.\|)$ is a Banach space.

Consider the following conditions hold:

$C_{1}) \mathcal{F}:[g, h]\rightarrow R$ is continuous.

$C_{2}$)$f:[g, h]\times \mathcal{J}\rightarrow \mathcal{J}$ is continuous, $f(\bar{a}, s)\geq 0$ and satisfy the Lipschitzian condition w.r.t. second variable, that is, there exist $\mathcal{L}\geq 0$ such that for all $s, t\in \mathcal{J}$

$C_{3}$)$\mathcal{F}:[g, h]\times [g, h]\rightarrow R$ is continuous such that $\mathcal{W}(\bar{a}, \bar{b})\geq 0$ for all $\mathcal{W}(\bar{a}, \bar{b})\in [g, h]\times [g, h]$ and $\int_{g}^{h}\mathcal{W}(\bar{a}, \bar{b})d\bar{b}\leq \mathcal{M}$ where $\mathcal{M}\in R$ is fixed, such that $\mu = \frac{1}{\mathcal{LM}}$.

Theorem 5.1. Suppose that the conditions $(C_{1})$–$(C_{3})$ are satisfied. Then the problem $(5.1)$ has a solution $s^{*}$ in $C[g, h]$ and the iteration scheme $(1.6)$ converges to $s^{*}$.

Proof. Let $\{s_{i}\}$ be a sequence generated by $(1.6)$. Then we get

Let $s^{*}$ be fixed point of $\zeta$. We prove that $s_{i}\rightarrow s^{*}$ as $i\to \infty$. Define $v_{i} = (1-\sigma_{i})s_{i}+\sigma_{i}\zeta(s_{i})$. Then, the following estimation holds

Similarly, we have

Let $\|s_{i}-s^{*}\| = a_{i}$. Then, we have $a_{i+1}\leq a_{i}$.

Let $\{a_{i}\}$ be the sequence of positive real numbers and monotone decreasing and hence, bounded. By Theorem 2.1 we have $\lim\limits_{i\to \infty}a_{i} = 0$; so $s_{i}\rightarrow s^{*}$.

6.   Conclusions

In this paper, we investigated the fixed point approximation for generalized $\alpha$-nonexpansive mappings in hyperbolic space utilizing the Picard-Thakur hybrid iteration scheme and some strong and $\Delta$-convergence results were established. Through numerical example we have shown that the proposed the Picard-Thakur hybrid iterative scheme $(1.6)$ converges faster than (1.3)–(1.5) for different type of initial points and parameters.

We extend various results in hyperbolic space as in literature. Additionally, we also provide an application to Fredholm integral equation.

Acknowledgments

The publication of this article was partially supported by AGC grants of the Babeş-Bolyai University of Cluj-Napoca.

Conflict of interest

The authors declare that they have no conflicts of interest to report regarding the present study.