Solving an integral equation via orthogonal generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-Geraghty contractions

1.   Introduction and preliminaries

Banach ^[1] introduced one of the most essential Banach Contraction Principles. In 1993, Czerwik ^[2] initiated the notion of a $\mathfrak{b}$-metric space and proved the fixed point theorem (FPT) in this space. Aydi et al. ^[3] proved the common FPT for weak $\phi$-contractions in $\mathfrak{b}$-metric space. The existence and uniqueness of a fixed point of $\phi$-contractions was proved by Pacurar ^[4]. In 2018, Zada et al. ^[5] elaborated a FPT of a rational contraction. Geraghty ^[6] expanded the Banach contraction principle in 1973 by factoring an auxiliary function of complete metric space.

One of the interesting results was given by Samet et al. ^[7] by defining ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-contractive maps via ${\boldsymbol{\alpha}}$-admissible mappings. After that, Cho et al. ^[8] introduced the ${\boldsymbol{\alpha}}$-GC (Geraghty contraction) type maps in metric space and proved some FPT's of these functions. Popescu ^[9] developed ${\boldsymbol{\alpha}}$-GC maps and proved the fixed point theorem in complete metric space. From the above work, Karapınar ^[11] introduced generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC type maps, and see also ^[10,12]. Furthermore, several authors proved the common fixed point theorem in many metric spaces, see ^{[13,14,15,16,17,18,19]}.

An orthogonality notion in metric spaces was introduced by Gordji et al. ^[20]. The theory of an orthogonal set has general application in a number of mathematical areas, and there are many types of orthogonality, see ^{[21,22,23,24,25,26,27,28,29]}.

In this paper, we prove a FPT in an ${\bf{O}}$-complete ${\bf{O}}$-$\mathfrak{b}$-metric space (${\bf{O}}_{\mathfrak{b}}$-MS) with ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC type maps. Moreover, an example and application to an integral equation are given to strengthen our main results.

2.   Preliminaries

Throughout this paper, the standard letter $\mathbb{R}$ is the set of all real numbers, $\mathbb{R}^{+}$ represents the set of all positive real numbers, $\mathbb{N}$ denotes the set of all natural numbers, $\mathbb{N}_{0}$ denotes the set of all positive natural numbers, and $\mathbb{Z}$ denotes the set of all integers. First, we recall some standard definitions and other results that will be needed in the sequel.

In 1993, Czerwik ^[2] introduced a $\mathfrak{b}$-metric space as follows:

Definition 2.1. ^[2] Let $\mathcal{Q}$ be a non-void set. Let $\tau \colon \mathcal{Q} \times \mathcal{Q} \to \mathbb{R}^+$ be called $\mathfrak{b}$-metric on $\mathcal{Q}$ if for all $\theta, \mu, \nu \in \mathcal{Q}$ the below conditions hold:

(ⅰ) $0 \leq \tau(\theta, \mu)$ and $\tau(\theta, \mu) = 0$ iff $\theta = \mu$;

(ⅱ) $\tau (\theta, \mu) = \tau(\mu, \theta)$;

(ⅲ) $\tau(\theta, \mu) \leq \mathfrak{g}[\tau(\theta, \nu)+ \tau(\nu, \mu)]$.

Then, $(\mathcal{Q}, \tau)$ is said to be a $\mathfrak{b}$-metric space (with constant $\mathfrak{g} \geq 1$). In 2014, Karapınar ^[11] introduced the concept of ${\boldsymbol{\alpha}}$-regularity as follows:

Definition 2.2. ^[11] Let $(\mathcal{Q}, \tau)$ be a $\mathfrak{b}$-metric space and ${\boldsymbol{\alpha}} : \mathcal{Q} \times \mathcal{Q} \to \mathcal{Q}$ be a map. $\mathcal{Q}$ is called ${\boldsymbol{\alpha}}$-regular if for each sequence $\{\theta_{\eta}\}\in\mathcal{Q}$ such that ${\boldsymbol{\alpha}}(\theta_{\eta}, \theta_{\eta+1}) \geq 1, \; \; \forall\; \; \eta \in \mathbb{N}$ and $\theta_{\eta} \to \theta \in \mathcal{Q}$ as $\eta \to \infty, \; \; \exists$ a sub-sequence $\{\theta_{\eta(\gamma)}\}$ of $\{\theta_{\eta}\}$ with ${\boldsymbol{\alpha}}(\theta_{\eta(\gamma)}, \theta) \geq 1$, for all $\gamma \in \mathbb{N}$.

The notion of an orthogonal set was presented by Gordji et al. ^[21].

Definition 2.3. ^[21] Let $\mathcal{Q}$ be a non empty set and $\perp \subseteq \mathcal{Q}\times \mathcal{Q}$ be a binary relation. If $\perp$ holds with the constraint

then $(\mathcal{Q}, \perp)$ is said to be an orthogonal set.

Gordji et al. ^[21] presented the definition of an orthogonal sequence in 2017 as follows.

Definition 2.4. ^[21] Let $(\mathcal{Q}, \perp)$ be an orthogonal set. A sequence $\{\theta_\eta\}_{\eta\in \mathbb{N}}$ is called an orthogonal sequence (${\bf{O}}$-sequence) if

Definition 2.5. A tripled $(\mathcal{Q}, \perp, \tau)$ is called an ${\bf{O}}_{\mathfrak{b}}$-MS if $(\mathcal{Q}, \perp)$ is an orthogonal set and $(\mathcal{Q}, \tau)$ is a $\mathfrak{b}$-metric space.

Definition 2.6. Let $(\mathcal{Q}, \perp, \tau)$ be an ${\bf{O}}_{\mathfrak{b}}$-MS.

$(1)$ $\{\mu_{\eta}\}$, an orthogonal sequence in $\mathcal{Q}$, converges at a point $\mu$ if

$(2)$ $\{\mu_{\eta}\}, \{\mu_{\mathfrak{m}}\}$ are orthogonal sequences in $\mathcal{Q}$ and are said to be orthogonal-Cauchy sequences if

Definition 2.7. ^[20] Let $(\mathcal{Q}, \perp, \tau)$ be an ${\bf{O}}_{\mathfrak{b}}$-MS. Then, ${\mathit{E}} \colon \mathcal{Q} \to \mathcal{Q}$ is said to be orthogonally continuous in $\mu \in \mathcal{Q}$ if, for each ${\bf{O}}$-sequence $\{\mu_{\eta}\}_{\eta\in \mathbb{N}}$ in $\mathcal{Q}$ with $\mu_{\eta} \to \mu$, we have ${\mathit{E}}(\mu_{\eta}) \to {\mathit{E}}(\mu)$. Also, ${\mathit{E}}$ is said to be orthogonal continuous on $\mathcal{Q}$ if ${\mathit{E}}$ is orthogonal continuous in each $\mu \in \mathcal{Q}$.

Example 2.8. ^[20] Let $\mathcal{Q} = \mathbb{R}$ and suppose $\mu \perp \theta$ if

for some $\eta \in \mathbb{Z}$ or $\theta = 0$.

It is clear that $(\mathcal{Q}, \perp)$ is an orthogonal set. A map ${\mathit{E}} : \mathcal{Q} \to \mathcal{Q}$ is defined as ${\mathit{E}}(\theta) = [\theta]$. Then, ${\mathit{E}}$ is orthogonal-continuous on $\mathcal{Q}$, because if $\{\theta_{\mathfrak{r}}\}$ is an arbitrary ${\bf{O}}$-sequence in $\mathcal{Q}$ such that $\{\theta_{\mathfrak{r}}\}$ converges to $\theta \in \mathcal{Q}$, then the below cases hold:

Case 1: If $\theta_\mathfrak{r} = 0 \; \; \forall\; \; \mathfrak{r}$, then $\theta = 0$ and ${\mathit{E}}(\theta_\mathfrak{r}) = 0 = {\mathit{E}}(\theta)$.

Case 2: If $\theta_{\mathfrak{r}_0} \neq 0$ for some $\mathfrak{r}_0$, then there exists $m \in \mathbb{Z}$ such that $\theta_\mathfrak{r} \in (m + \frac{1}{3}, m + \frac{2}{3})$, for all $\mathfrak{r} \geq \mathfrak{r}_0$. Thus, $\theta \in [m + \frac{1}{3}, m + \frac{2}{3}]$, and ${\mathit{E}}(\theta_\mathfrak{r}) = m = {\mathit{E}}(\theta)$.

This means that ${\mathit{E}}$ is orthogonal-continuous on $\mathcal{Q}$, but it is not continuous on $\mathcal{Q}$.

The concept of orthogonal completeness in metric spaces is defined by Gordji et al. ^[21] as follows.

Definition 2.9. ^[21] Let $(\mathcal{Q}, \perp, \tau)$ be an orthogonal metric space. Then, $\mathcal{Q}$ is said to be ${\bf{O}}$-complete if every orthogonal Cauchy sequence is convergent.

Definition 2.10. ^[21] Let $(\mathcal{Q}, \perp)$ be an orthogonal set. A function ${\mathit{E}} : \mathcal{Q} \to \mathcal{Q}$ is called orthogonal-preserving if ${\mathit{E}}\theta \perp {\mathit{E}}\mu$ whenever $\theta \perp \mu$.

Ramezani ^[26] introduced the notion of being orthogonal $\alpha$-admissible as follows.

Definition 2.11. ^[26] Let ${\mathit{E}} : \mathcal{Q} \to \mathcal{Q}$ be a map and let ${\boldsymbol{\alpha}} : \mathcal{Q} \times \mathcal{Q} \to \mathbb{R}^+$ be a function. Then, ${\mathit{E}}$ is said to be ${\bf{O}}$-${\boldsymbol{\alpha}}$-admissible if $\forall\; \; \theta, \mu \in \mathcal{Q}$ with $\theta \perp \mu$

In 2021, Gnanaprakasam et al. ^[25] introduced the notion of being orthogonal triangular ${\boldsymbol{\alpha}}$-admissible, defined as below:

Definition 2.12. ^[25] A self-map ${\mathit{E}} : \mathcal{Q} \to \mathcal{Q}$ is called ${\bf{O}}$-triangular ${\boldsymbol{\alpha}}$-admissible if the following holds:

$({\mathit{E}}1)\; {\mathit{E}}$ is ${\bf{O}}$-${\boldsymbol{\alpha}}$-admissible,

$({\mathit{E}}2)\; {\boldsymbol{\alpha}}(\theta, \nu) \geq 1, \; \; {\boldsymbol{\alpha}}(\nu, \mu) \geq 1 \, \, with\, \, \theta \perp \nu\, \, and\, \, \nu \perp \mu \implies {\boldsymbol{\alpha}}(\theta, \mu) \geq 1, \forall\, \, \theta, \mu, \nu \in \mathcal{Q}.$

Recently, Popescu ^[9] has developed the notion of a triangular $\alpha$-orbital admissible mappings, and we extend it to orthogonal in ${\bf{O}}_{\mathfrak{b}}$-MS. The following definitions will be needed in the main result.

Definition 2.13. Let ${\mathit{E}} : \mathcal{Q} \to \mathcal{Q}$ be a map, and let there be a function ${\boldsymbol{\alpha}} : \mathcal{Q} \times \mathcal{Q} \to \mathbb{R}^+$. Then, ${\mathit{E}}$ is called ${\bf{O}}$-${\boldsymbol{\alpha}}$-orbital admissible if the below constraint holds:

$({\mathit{E}}3) \quad {\boldsymbol{\alpha}}(\theta, {\mathit{E}}\theta) \geq 1\, \, \, \, \implies {\boldsymbol{\alpha}}({\mathit{E}}\theta, {\mathit{E}}^{2}\theta) \geq 1, \forall\, \, \theta \in \mathcal{Q}.$

Example 2.14. Let $\mathcal{Q} = \{0, 1, 2, 3\}$, $\tau \colon \mathcal{Q} \times \mathcal{Q} \to \mathbb{R}, \tau(\theta, \mu) = |\theta - \mu|^{2}, {\mathit{E}}\colon \mathcal{Q} \to \mathcal{Q}$ such that ${\mathit{E}}0 = 0, \; {\mathit{E}}1 = 2, \; {\mathit{E}}2 = 1, \; {\mathit{E}}3 = 3$. $\alpha \colon \mathcal{Q} \times \mathcal{Q} \to \mathbb{R}, \; \alpha(\theta, \mu) = 1$ if $(\theta, \mu) \in \{(0, 1), (0, 2), (1, 1), (2, 2), (1, 2), (2, 1), (1, 3), (2, 3)\}$ with $\theta \perp \mu$, and $\alpha(\theta, \mu) = 0$ otherwise.

Since $\alpha(1, {\mathit{E}}1) = \alpha(1, 2) = 1$ and $\alpha(2, {\mathit{E}}2) = \alpha(2, 1) = 1$, we have that ${\mathit{E}}$ is orthogonal $\alpha$-orbital admissible.

Definition 2.15. Let ${\mathit{E}} : \mathcal{Q} \to \mathcal{Q}$ and ${\boldsymbol{\alpha}} : \mathcal{Q} \times \mathcal{Q} \to \mathbb{R}^+$ be maps. Then, ${\mathit{E}}$ is called ${\bf{O}}$-triangular-$\alpha$-orbital admissible if ${\mathit{E}}$ is ${\bf{O}}$-${\boldsymbol{\alpha}}$-orbital admissible and the following property holds:

$({\mathit{E}}4) \; \forall\, \, \theta, \mu\in \mathcal{Q}\; \; {\rm{with}}\, \, \theta \perp \mu, \quad {\boldsymbol{\alpha}}(\theta, \mu) \geq 1 \; \; {\rm{and}}\; \; {\boldsymbol{\alpha}}(\mu, {\mathit{E}}\mu) \geq 1\, \, \implies {\boldsymbol{\alpha}}(\theta, {\mathit{E}}\mu) \geq 1.$

Example 2.16. Let $\mathcal{Q} = \mathbb{R}, {\mathit{E}}\mu = \mu^{3}+ \sqrt[7]{\mu}$, and ${\boldsymbol{\alpha}}(\mu, \theta) = \mu^{5} - \theta^{5} +1$, for all $\mu, \theta \in \mathcal{Q}$ with $\mu \perp \theta$. Then, ${\mathit{E}}$ is an ${\bf{O}}$-triangular ${\boldsymbol{\alpha}}$-admissible mapping.

Lemma 2.17. Let ${\mathit{E}} : \mathcal{Q} \to \mathcal{Q}$ be an ${\bf{O}}$-triangular-$\alpha$-orbital admissible map. Consider that $\exists\; \theta_{0} \in \mathcal{Q}$ such that $\theta_{0}\perp {\mathit{E}}\theta_{0}$ and ${\boldsymbol{\alpha}}(\theta_{0}, {\mathit{E}}\theta_{0}) \geq 1$. An ${\bf{O}}$-sequence $\{\theta_{\eta}\}$ is defined by $\theta_{\eta+1} = {\mathit{E}}\theta_{\eta}, \; \forall\; \eta \in \mathbb{N}$ with $\theta_{\eta}\perp {\mathit{E}}\theta_{\eta}\; {\mathit{{\rm{or}}}}\; {\mathit{E}}\theta_{\eta}\perp \theta_{\eta}$. Then, we get ${\boldsymbol{\alpha}}(\theta_{\eta}, \theta_{\nu}) \geq 1, \; \forall\; \eta, \nu \in \mathbb{N}$ with $\eta < \nu$.

In this section, inspired by the concept of generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC type maps defined by Afshari et al. ^[28], we introduce a new orthogonal generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC type mapping and prove some FPT's for these contraction mappings in an ${\bf{O}}$-complete ${\bf{O}}_{\mathfrak{b}}$-MS.

3.   Main results

Let $\Lambda$ be a set of all increasing and continuous functions, and ${\boldsymbol{\psi}} \in \Lambda$ is defined as ${\boldsymbol{\psi}} : \mathbb{R}^+ \to \mathbb{R}^+$, with ${\boldsymbol{\psi}}^{-1}(\{0\}) = \{0\}$.

Let $\varGamma$ be the family of all non-decreasing functions $\lambda : [0, \infty) \to [0, \frac{1}{\mathfrak{g}})$ which satisfy the condition

First, we explain the definition of an ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC type(A) map in an ${\bf{O}}$-complete ${\bf{O}}_{\mathfrak{b}}$-MS.

Definition 3.1. Let $(\mathcal{Q}, \perp, \tau)$ be an ${\bf{O}}$-complete ${\bf{O}}_{\mathfrak{b}}$-MS, and let there be a map ${\mathit{E}} : \mathcal{Q} \to \mathcal{Q}$. ${\mathit{E}}$ is called an ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC type(A) map whenever $\exists\; {\boldsymbol{\alpha}} : \mathcal{Q} \times \mathcal{Q} \to \mathbb{R}^+$, and, for $\mathfrak{L} \geq 0$ such that

we have

for all $\theta, \mu \in \mathcal{Q}$ with $\theta \perp \mu$, where $\lambda \in \varGamma$ and ${\boldsymbol{\psi}}, \vartheta \in \Lambda.$

Now, we generalize and improve our FPT from Afshari et al. ^[28] by introducing the notion of an ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC type(A) map in ${\bf{O}}$-complete ${\bf{O}}_{\mathfrak{b}}$-MS.

Theorem 3.2. Let $(\mathcal{Q}, \perp, \tau)$ be an ${\bf{O}}$-complete ${\bf{O}}_{\mathfrak{b}}$-MS, and ${\mathit{E}}: \mathcal{Q} \to \mathcal{Q}$ satisfies the below properties:

(ⅰ) ${\mathit{E}}$ is orthogonal-preserving,

(ⅱ) ${\mathit{E}}$ is an ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC type(A) map,

(ⅲ) ${\mathit{E}}$ is ${\bf{O}}$-triangular ${\boldsymbol{\alpha}}$-orbital admissible,

(ⅳ) $\exists\; \; \theta_{0} \in \mathcal{Q}$ such that $\theta_{0} \perp {\mathit{E}}\theta_{0}$ and ${\boldsymbol{\alpha}}(\theta_{0}, {\mathit{E}}\theta_{0}) \geq 1,$

(ⅴ) ${\mathit{E}}$ is ${\bf{O}}$-continuous.

Then, ${\mathit{E}}$ has a UFP (unique fixed point).

Proof. By the condition (ⅳ), there exists $\theta_{0} \in \mathcal{Q}$ such that

Let

If $\theta_{\eta^*} = \theta_{{\eta^*}+1}$ for $\eta^* \in \mathbb{N} \cup \{0\}$, then $\theta_{\eta^*}$ is a fixed point of ${\mathit{E}}$. Therefore, the proof is complete.

So, we consider $\theta_{\eta} \neq \theta_{\eta+1}$. Thus, we have $\tau({\mathit{E}}\theta_{\eta}, {\mathit{E}}\theta_{\eta+1}) > 0$. Since ${\mathit{E}}$ is ${\bf{O}}$-preserving, we get

We construct an ${\bf{O}}$-sequence $\{\theta_{\eta}\}$. Since ${\mathit{E}}$ is an ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$ -GC type(A) map, we get

Since the map ${\mathit{E}}$ is an orthogonal triangular $\alpha$-orbital admissible, by Lemma 2.17, we get

By letting $\theta = \theta_{\eta-1}$ and $\mu = \theta_{\eta}$ in the inequality (3.1), using (3.2) and that ${\boldsymbol{\psi}}$ is an ascending map, we get

for all $\eta \in \mathbb{N}$, where

and

Since

we get

Taking (3.5) and (3.4) into account, (3.3) yields

If $\eta \in \mathbb{N}$, we get $\max\{\tau(\theta_{\eta-1}, \theta_{\eta}), \tau(\theta_{\eta}, \theta_{\eta+1})\} = \tau(\theta_{\eta}, \theta_{\eta+1}),$ and then by (3.6), we get

which is a contradiction. Thus, from (3.6) we conclude that

Hence, $\{{\boldsymbol{\psi}}(\tau(\theta_{\eta}, \theta_{\eta+1}))\}$ is a positive non-increasing sequence. Since ${\boldsymbol{\psi}}$ is ascending, the sequence $\{\tau(\theta_{\eta}, \theta_{\eta+1})\}$ is decreasing. Consequently, $\exists\; \; \epsilon \geq 0$ such that $\lim\limits_{\eta \to \infty} \tau(\theta_{\eta}, \theta_{\eta+1}) = \epsilon$. We claim that $\epsilon = 0$. Suppose, on the contrary, that

Since $\mathfrak{g} \geq 1$, (3.7) can be approximated as

With regard to (3.2), in (3.8) we get

This yields $\lim_{\eta \to \infty}\lambda({\boldsymbol{\psi}}({\bf{M}}(\theta_{\eta-1}, \theta_{\eta}))) = \dfrac{1}{ \mathfrak{g}}.$ Since $\lambda \in \varGamma,$ we get $\lim_{\eta \to \infty} {\boldsymbol{\psi}}({\bf{M}}(\theta_{\eta-1}, \theta_{\eta})) = 0$. We simplify that:

Thus, by the fact that $\tau(\theta_{\eta}, \theta_{\eta+1}) \to \epsilon$ and the continuity of ${\boldsymbol{\psi}}$, we get ${\boldsymbol{\psi}}{(\epsilon)} = 0.$ Since ${\boldsymbol{\psi}}^{-1}(\{0\}) = \{0\}$, we have $\epsilon = 0$, and this is a contradiction. Thus, we get

Now, we claim that $\lim_{\pi, \eta \to \infty} \tau(\theta_{\eta}, \theta_{\pi}) = 0.$

Consider, on the contrary, that $\exists\; \delta \succ 0$, and orthogonal subsequences $\{\theta_{\pi_{\mathfrak{j}}}\}, \; {\mathit{{\rm{and}}}} \; \{\theta_{\eta_{\mathfrak{j}}}\}$ of $\{\theta_{\eta}\}$, with $\eta_{\mathfrak{j}} \succ \pi_{\mathfrak{j}} \geq \mathfrak{j}$, such that

Additionally, for every $\pi_{\mathfrak{j}}$, we choose the smallest integer $\eta_{\mathfrak{j}}$ to fulfill (3.10), and $\eta_{\mathfrak{j}} \succ \pi_{\mathfrak{j}} \geq \mathfrak{j}$. Then, we get

From (3.10) and the condition (ⅲ) in Definition 2.1, we get

Taking $\mathfrak{j} \to \infty$ and (3.9) into account, Eq (3.12) yields

where $\lim\limits_{\mathfrak{j} \to \infty} \tau(\theta_{\eta_{\mathfrak{j}}}, \theta_{\eta_{\mathfrak{j}+1}}) = \lim\limits_{\mathfrak{j} \to \infty} \tau(\theta_{\pi_{\mathfrak{j}+1}}, \theta_{\pi_{\mathfrak{j}}}) = 0$. By Lemma 2.17, ${\boldsymbol{\alpha}}(\theta_{\pi_{\mathfrak{j}}}, \theta_{\eta_{\mathfrak{j}}}) \geq 1$. Followed by (3.1), we get

where

and

Notice that

and

Taking (3.11), (3.15) and (3.16) into account, we find that

By allowing the upper limit to be $\mathfrak{j} \to \infty$ and using constraint $({\mathit{E}}4)$, (3.13), (3.17) and (3.18), inequality (3.14) becomes

Then,

Due to the fact that $\lambda \in \varGamma$, we get

Thus, we assume that

Consequently, due to the continuity of ${\boldsymbol{\psi}}$ and ${\boldsymbol{\psi}}^{-1}{(\{0\})} = \{0\}$, we obtain

which contradicts (3.10). Therefore, $\{\theta_{\eta}\}$ is a ${\bf{O}}$-Cauchy sequence in $\mathcal{Q}$. Since $\mathcal{Q}$ is an ${\bf{O}}$-complete ${\bf{O}}_{\mathfrak{b}}$-MS, $\exists\; \; \theta^* \in \mathcal{Q}$ such that $\lim_{\eta \to \infty} \theta_{\eta} = \theta^*$. The map ${\mathit{E}}$ is ${\bf{O}}$-continuous, and it is obvious that ${\mathit{E}}\theta^* = \theta^*$. Hence, $\theta^*$ is a fixed point of ${\mathit{E}}$.

Now, we prove $\theta^*$ is a UFP of ${\mathit{E}}$. Suppose $\mu^*$ is another fixed point of ${\mathit{E}}$. If $\theta_{\eta} \to \mu^*$ as $\eta \to \infty$, we get $\theta^* = \mu^*$.

If $\lim\limits_{\eta \to \infty} \{\theta_{\eta}\} \nrightarrow \mu^*$, there is an orthogonal sub-sequence $\{\theta_{{\eta}_{\gamma}}\}$ such that ${\mathit{E}}\theta_{{\eta}_{\gamma}} \neq \mu^*, \; \; \forall\; \; \gamma \in \mathbb{N}$. By the choice of $\theta_{0}$, we get

Since ${\mathit{E}}$ is ${\bf{O}}$-preserving and ${\mathit{E}}^{\eta}\mu^* = \mu^*$, for all $\eta \in \mathbb{N}$, we get

Since ${\mathit{E}}$ is an ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC type(A) map, we get

This implies ${\boldsymbol{\psi}} (\tau({\mathit{E}}^{\eta_{\gamma}}\theta_{0}, \mu^*)) \to -\infty$ as $\gamma \to \infty$. This yields that $\theta_{\eta} \to \mu^*$ as $\eta \to \infty$, which is a contradiction. Hence, ${\mathit{E}}$ has a UFP.

We replace the continuity of map ${\mathit{E}}$ in the above theorem by a suitable condition on $\mathcal{Q}$.

Theorem 3.3. Let $(\mathcal{Q}, \perp, \tau)$ be an ${\bf{O}}$-complete ${\bf{O}}$-$\mathfrak{b}$-metric space, and ${\mathit{E}} : \mathcal{Q} \to \mathcal{Q}$ fulfills the following properties:

$(ⅰ)$ ${\mathit{E}}$ is orthogonal preserving,

$(ⅱ)$ ${\mathit{E}}$ is an ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$ -GC type(A) map,

$(ⅲ)$ ${\mathit{E}}$ is ${\bf{O}}$-triangular-$\alpha$ orbital admissible,

$(ⅳ)$ $\exists\; \; \theta_{0} \in \mathcal{Q}$ such that $\theta_{0} \perp {\mathit{E}}\theta_{0}$ and ${\boldsymbol{\alpha}}(\theta_{0}, {\mathit{E}}\theta_{0}) \geq 1,$

$(ⅴ)$ ${\mathit{E}}$ is ${\bf{O}}$-${\boldsymbol{\alpha}}$- regular.

Then, ${\mathit{E}}$ has a UFP.

Proof. From the proof of Theorem 3.2, we conclude that $\lim\limits_{\eta \to \infty} \theta_{\eta} = \theta^*$. If $\mathcal{Q}$ is ${\bf{O}}$-${\boldsymbol{\alpha}}$-regular, then ${\boldsymbol{\alpha}}(\theta_{\eta}, \theta_{\eta+1}) \geq 1$, $\exists$ a subsequence $\{\theta_{\eta_\gamma}\}$ of $\{\theta_{\eta}\}$ such that

By the (ⅲ) in Definition 2.1, we get

Letting $\gamma \to \infty$, yields

Using that ${\boldsymbol{\psi}} \in \Lambda,$ (3.19) and (3.20), we have

We have

and

Recall that

Then, by (3.9), we get

When $\gamma \to \infty$, we deduce

Since $\lambda({\boldsymbol{\psi}}({\bf{M}}(\theta_{\eta_{\gamma}}, \theta^*))) \leq \frac{1}{\mathfrak{g}}$, for all $\gamma \in \mathbb{N}$, from (3.21), we obtain

Hence, $\tau(\theta^*, {\mathit{E}}\theta^*) = 0$, that is ${\mathit{E}}\theta^* = \theta^*$. Therefore, ${\mathit{E}}$ has a fixed point.

Now, we prove $\theta^*$ is a UFP of ${\mathit{E}}$. Suppose $\mu^*$ is another fixed point of ${\mathit{E}}$. If $\theta_{\eta} \to \mu^*$ as $\eta \to \infty$, we get $\theta^* = \mu^*$.

If $\lim\limits_{\eta \to \infty}\{\theta_{\eta}\} \nrightarrow \mu^*$, there is a subsequence $\{\theta_{{\eta}_{\gamma}}\}$ such that ${\mathit{E}}\theta_{{\eta}_{\gamma}} \neq \mu^*, \; \forall\; \gamma \in \mathbb{N}$. By the choice $\theta_{0}$, we get

Since ${\mathit{E}}$ is ${\bf{O}}$-preserving and ${\mathit{E}}^{\eta}\mu^* = \mu^*, \; \forall \; \eta \in \mathbb{N}$, we get

Since ${\mathit{E}}$ is an ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$ -GC type(A) map, we get

This implies ${\boldsymbol{\psi}} (\tau({\mathit{E}}^{\eta_{\gamma}}\theta_{0}, \mu^*)) \to -\infty$ as $\gamma \to \infty$. This yields that $\theta_{\eta} \to \mu^*$ as $\eta \to \infty$, and this is contradiction. Hence, ${\mathit{E}}$ has a UFP.

We initiate the definition of an ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC type(B) map as follows:

Definition 3.4. Let $(\mathcal{Q}, \perp, \tau)$ be an ${\bf{O}}$-complete ${\bf{O}}_{\mathfrak{b}}$-MS and let ${\mathit{E}} : \mathcal{Q} \to \mathcal{Q}$ be a map. ${\mathit{E}}$ is called ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC map of type(B) whenever $\exists \; {\boldsymbol{\alpha}} : \mathcal{Q} \times \mathcal{Q} \to \mathbb{R}^+$ such that for all $\theta, \mu \in \mathcal{Q}$ with $\theta \perp \mu\; \; {\rm{or}}\; \; \mu \perp \theta$,

where

$\lambda \in \varGamma$ and ${\boldsymbol{\psi}} \in \Lambda.$

Now, we generalize and improve our fixed point theorem from Afshari et al. ^[28] by introducing the notion of an ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC type(B) map in ${\bf{O}}$-complete ${\bf{O}}_{\mathfrak{b}}$-MS.

Theorem 3.5. Let $(\mathcal{Q}, \perp, \tau)$ be an ${\bf{O}}$-complete ${\bf{O}}_{\mathfrak{b}}$-MS, and let ${\mathit{E}} : \mathcal{Q} \to \mathcal{Q}$ satisfy the below properties:

(ⅰ) ${\mathit{E}}$ is orthogonal preserving,

(ⅱ) ${\mathit{E}}$ is an ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$ -GC map of type(B),

(ⅲ) ${\mathit{E}}$ is ${\bf{O}}$-triangular-$\alpha$ orbital admissible,

(ⅳ) $\exists\; \; \theta_{0} \in \mathcal{Q}$ such that $\theta_{0} \perp {\mathit{E}}\theta_{0}$ and ${\boldsymbol{\alpha}}(\theta_{0}, {\mathit{E}}\theta_{0}) \geq 1,$

(ⅴ) ${\mathit{E}}$ is ${\bf{O}}$-continuous or $\mathcal{Q}$ is ${\bf{O}}$-${\boldsymbol{\alpha}}$-regular.

Then, ${\mathit{E}}$ has a UFP.

Now, we provide the example for Theorem 3.3.

Example 3.6. Let $\mathcal{Q}$ be a set of Lebesgue measurable functions [0, 1] such that $\int_{0}^{1} |\theta(\zeta)|\tau\zeta < 1.$ Define a relation $\perp$ on $\mathcal{Q}$ by

where $\theta(\zeta)\vee\mu(\zeta) = \theta(\zeta)\; \; {\rm{or}}\; \; \mu(\zeta)$. Define $\tau \colon \mathcal{Q} \times \mathcal{Q} \rightarrow \mathbb{R}^+$ by

Then, $(\mathcal{Q}, \tau)$ is an ${\bf{O}}$-complete ${\bf{O}}_{\mathfrak{b}}$-MS with $\mathfrak{g} = 2$. The operator ${\mathit{E}} \colon \mathcal{Q} \times \mathcal{Q} \to \mathbb{R}^{+}$ is defined by

Consider the map ${\boldsymbol{\alpha}} \colon \mathcal{Q} \times \mathcal{Q} \rightarrow \mathbb{R}^+, \; {\rm{with}}\; {\boldsymbol{\lambda}} \colon \mathbb{R}^+ \rightarrow [0, \frac{1}{2})$ and ${\boldsymbol{\psi}} \colon \mathbb{R}^+ \rightarrow \mathbb{R}^+$, defined by

Obviously, $\psi \in \Lambda$, and $\lambda \in \varGamma$. Moreover, ${\mathit{E}}$ is an ${\bf{O}}$-triangular-$\alpha$ orbital admissible map, and ${\boldsymbol{\alpha}}(1, {\mathit{E}}1) \geq 1.$

Now, we prove ${\mathit{E}}$ is an ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC type(A) map. Certainly, $\forall\; \; \zeta \in [0, 1]$, we get

and we have

Therefore,

So, we get

Hence by Theorem 3.3, we get that ${\mathit{E}}$ has a UFP.

4.   Application

As an application of Theorem 3.2, we find an existence and uniqueness result of the following type of integral equation:

Consider $\mathcal{Q} = \mathcal{C}([0, \mathfrak{a}], \mathbb{R})$ to be the real continuous functions on $[0, \mathfrak{a}]$, and a mapping $\mathcal{D}:\mathcal{Q}\to\mathcal{Q}$ is defined by

Obviously, $(\mathcal{Q}, \tau)$ is a complete $b$-metric space, and $\omega(\theta)$ is a solution of integral equation (4.1) iff $\omega(\theta)$ is a fixed point of $\mathcal{D}$.

Theorem 4.1. Suppose the following.

(1) The mappings ${\mathit{E}}: [0, \mathfrak{a}]\times\mathbb{R} \to \mathbb{R}^{+}$, $\mathcal{H}: [0, \mathfrak{a}]\times\mathbb{R} \to \mathbb{R},$ and $\lambda : [0, \mathfrak{a}] \to \mathbb{R}$ are $O$-continuous functions.

(2) There exists $\mathcal{K} > 0$, such that, for all $\theta, \mathfrak{s}\in[0, \mathfrak{a}]$ and $\omega, \mu\in\mathcal{Q}$,

(3) For all $\theta, \mathfrak{s}\in[0, \mathfrak{a}]$, we have

Then, (4.1) has a unique solution in $\mathcal{Q}$.

Proof. Define the $O$-relation $\bot$ on $\mathcal{Q}$ by

Define $\tau :\mathcal{Q}\times\mathcal{Q}\to \mathbb{R}^{+}$, given by

for all $\omega, \mu\in\mathcal{Q}$. It is easy to see that, $(\mathcal{Q}, \bot, \tau)$ is an ${\bf{O}}$-complete ${\bf{O}}_{\mathfrak{b}}$-MS. For all $\omega, \mu\in \mathcal{Q}$ with $\omega\bot \mu$ and $\theta\in [0, \mathfrak{a}]$, we have

Accordingly, $[(\mathcal{D}\omega)(\theta)][(\mathcal{D}\mu)(\theta)]\geq (\mathcal{D}\mu)(\theta)$, and so $(\mathcal{D}\omega)(\theta)\bot (\mathcal{D}\mu)(\theta)$. Then, $\mathcal{D}$ is $\bot$-preserving.

Let $\omega, \mu\in \mathcal{Q}$ with $\omega\bot \mu$. Suppose that $\mathcal{D}(\omega)\neq \mathcal{D}(\mu)$. For each $\theta\in [0, \mathfrak{a}]$, we have

Thus,

for all $\omega, \mu \in \mathcal{Q}$. Therefore, all the conditions of Theorem (3.2) are satisfied. Hence, (4.1) has a unique solution.

Example 4.2. Consider the integral equation as follows:

Clearly, the above Eq (4.5) satisfies the assumption of Theorem 4.1, that is, $\sin(\pi\theta^2) - \dfrac{\theta^2}{\pi}$ is an orthogonal continuous function on $[0, 1]$. Kernel $\mathtt{K}(\theta, \sigma)$ is orthogonal continuous on $\mathbb{R} = \{(\theta, \sigma), 0 < \theta, \sigma < 1\}$.

The solution will be determined from Eq (4.5) by the fixed point iteration method:

By choosing $\sin(\pi\theta^2) - \dfrac{\theta^2}{\pi}$ as the initial function, we can apply the fixed point iteration method to get a numerical solution:

Consider that for $|\theta| \leq 1$, an $O$-sequence $\{\mathfrak{g}_{\vartheta}(\theta)\}$ will converge to $\mathfrak{g}(\theta) = \sin(\pi\theta^2) - \dfrac{\theta^2}{\pi}$.

Error calculation for an approximate solution compared to an exact solution is given in Figure 1. Table 1 shows that the error of an approximate solution compared to an exact solution is relatively small.

5.   Conclusions

In this paper, we proved fixed point theorems for ${\bf{O}}$-generalized ${\boldsymbol{\alpha}}$-${\boldsymbol{\psi}}$-GC type contraction mappings in an ${\bf{O}}$-complete ${\bf{O}}_{\mathfrak{b}}$-MS. Furthermore, we presented some examples to strengthen our main results. Also, we provided an application to the existence of the solution of an integral equation and we have compared the approximate solution with the exact solution.

Khalehoghli et al. ^[30,31] presented a real generalization of the mentioned Banach's contraction principle by introducing $\mathbb{R}$-metric spaces, where R is an arbitrary relation on L. We note that in a special case, $\mathbb{R}$ can be considered as $\mathbb{R}$ = $\preceq$[partially ordered relation], $\mathbb{R}$ = $\perp$[orthogonal relation], etc. If one can find a suitable replacement for a Banach theorem that may determine the values of fixed points, then many problems can be solved in this $\mathbb{R}$-relation. This will provide a structural method for finding a value of a fixed point. It is an interesting open problem to study the fixed-point results on $\mathbb{R}$-complete $\mathbb{R}$-metric spaces.

Conflict of interest

The authors declare that they have no competing interests.