A Numerical Scheme and Application to the Fractional Integro-Differential Equation Using Fixed-Point Techniques

Abstract

In this paper, we introduce the notion of orthogonal $\alpha$–F–convex contraction mapping and prove some fixed-point theorems for self-mapping in orthogonal complete metric spaces. The proven results generalize and extend some of the well-known results in the literature. Following the derivation of these fixed-point results, we propose a solution for the fractional integro-differential equation, utilizing the fixed-point technique within the context of orthogonal complete metric spaces.

1. Introduction

The inception of the first fixed-point result dates back approximately a century. The technique of successive approximation for solving differential equations was initially formalized by S. Banach [1], who subsequently characterized it as a concept rooted in contraction mapping. The Banach hypothesis was not only articulated in a clear and straightforward manner but was also exemplified by illustrating the methodology to reach the desired fixed point. Fixed-point theory holds significant relevance across various qualitative sciences, capturing the fascination of researchers due to its simplicity in converting equations from diverse research domains into fixed-point problems. Numerous authors have expanded and generalized Banach’s fixed-point result in various directions, as documented in [2]. Notably, Eshaghi Gordji, Ramezani, De la Sen, and Cho [3] recently introduced the concept of an orthogonal set and put forth fixed-point results within the context of O-MS (orthogonal metric spaces).

In 2012, Wardowski [4] introduced a novel category of F-contractions, accompanied by the establishment of several fixed-point theorems. Mahendra Singh, Y., Khan, M. S., and Kang, S. M. [5] pioneered the concept of $\alpha$–F–convex contraction mappings using an admissible map and subsequently validated their proposition with the proof of various fixed-point theorems. In the year 2020, Sawangsup, Sintunavarat, and Cho [6] introduced the innovative concept of an orthogonal F–contraction map, providing proofs for fixed-point results within the orthogonal CMS (complete metric space). The exploration of orthogonal contractive-type maps has been a subject of study by various authors, yielding significant outcomes, as documented in [7,8,9,10,11,12,13].

In this paper, we adapt the notion of $\alpha$–F–convex contraction maps to orthogonal sets, presenting several fixed-point results applicable to $\alpha$–F–convex contraction mappings within the framework of orthogonal complete metric spaces. Additionally, we leverage our primary findings to determine the existence and uniqueness of solutions for fractional integro-differential equations.

2. Preliminaries

In this section, we review fundamental concepts that are employed in this paper. Throughout this paper, we denote the non-empty set, the non-negative real numbers, and the positive integer set by $\Psi$, ${\mathbb{R}}^{+}$, and $\mathtt{N}$, respectively.

First of all, we note the Wardowski [4] notion of a control function established in 2012.

Definition 1

([4]). Let $F:{\mathbb{R}}^{+}\to \mathbb{R}$ be a map with the subsequent conditions:

(F₁)

F is strictly increasing;

(F₂)

For every sequence $\left\{{\delta }_{\varrho }\right\}$ of ${\mathbb{R}}^{+}$, we have

$$\begin{array}{c}\hfill \underset{\varrho \to \infty }{lim}{\delta }_{\varrho }=0\iff \underset{\varrho \to \infty }{lim}F\left({\delta }_{\varrho }\right)=-\infty ;\end{array}$$

(F₃)

$\exists \mu \in (0,1)\ni \underset{\delta \to 0^{+}}{lim}{\delta }^{\mu }F\left(\delta \right)=0.$

ℑ denotes the family of all F functions that fulfills the conditions $\left(F_1\right)-\left(F_3\right)$.

Later, in 2018, Mahendra Singh, Khan, and Kang [5] introduced the concepts of the $\mathbb{F}$–convex contraction via admissible mapping as follows. Let $\mathcal{H}$ be a mapping on a metric space $(\Psi ,\partial )$. We denote

$$\begin{array}{c}\hfill M^k(\delta ,\xi )=max\{{\partial }^k(\delta ,\xi ),{\partial }^k(\mathcal{H}\delta ,\mathcal{H}\xi ),{\partial }^k(\delta ,\mathcal{H}\delta ),{\partial }^k(\mathcal{H}\delta ,{\mathcal{H}}^2\delta ),{\partial }^k(\xi ,\mathcal{H}\xi ),{\partial }^k(\mathcal{H}\xi ,{\mathcal{H}}^2\xi )\}\end{array}$$

Definition 2

([5]). A self mapping $\mathcal{H}$ on Ψ is said to be an α–$\mathbb{F}$–convex contraction, if there exists two functions $\alpha :\Psi \times \Psi \to [0,\infty )$, $\mathbb{F}\in \Im$ and $\tau >0$ such that

$$\begin{array}{c}\hfill {\partial }^k({\mathcal{H}}^2\delta ,{\mathcal{H}}^2\xi )>0\Rightarrow \tau +\mathbb{F}\left(\alpha (\delta ,\xi ){\partial }^k({\mathcal{H}}^2\delta ,{\mathcal{H}}^2\xi )\right)\le \mathbb{F}\left(M^k(\delta ,\xi )\right),\end{array}$$

(1)

for all $\delta ,\xi \in \Psi$, where $k\in [1,\infty )$.

Now, the definition of the O-set (or orthogonal-set), orthogonal sequence, ⊩-preserving, some premises, and examples of an orthogonal set were initiated by Gordji et al. [3], as follows:

Definition 3

([3]). Let $\Psi \ne \varphi$ and $⊩\subseteq \Psi \times \Psi$ be a binary relation. If ⊩ fulfills the consecutive condition:

$$\begin{array}{c}\hfill \exists {\delta }_0\in \Psi :\forall \delta \in \Psi ,{\delta }_0⊩\delta \left(or\right)\forall \delta \in \Psi ,\delta ⊩{\delta }_0,\end{array}$$

then $(\Psi ,⊩)$ is called an O-set.

This ${\delta }_0$ element does not have to be unique. For example:

Example 1

([3]). Let $\Psi =[0,\infty )$ and define it by $\delta ⊩\xi$ if $\delta \xi \in \{\delta ,\xi \}$. Then, by setting ${\delta }_0=0$ or ${\delta }_0=1$, $(\Psi ,⊩)$ is an O-set.

Definition 4

([3]). Let $(\Psi ,⊩)$ be an O-set. A sequence $\left\{{\delta }_{\varrho }\right\}$ is called an orthogonal sequence (briefly, O-sequence) if

$$\begin{array}{c}\hfill (\forall \varrho \in \mathtt{N},{\delta }_{\varrho }⊩{\delta }_{\varrho +1})or(\forall \varrho \in \mathtt{N},{\delta }_{\varrho +1}⊩{\delta }_{\varrho }).\end{array}$$

Definition 5

([3]). Let $(\Psi ,⊩)$ be an O-set. A function $G:\Psi \to \Psi$ is said to be ⊩-preserving if $\mathcal{H}\delta ⊩\mathcal{H}\xi$ whenever $\delta ⊩\xi$.

Now, we give the notions of O-MS, orthogonal complete (briefly, O-complete), orthogonally continuous, orthogonally $\alpha$-admissible, and orthogonally ${\alpha }^{*}$-admissible.

Definition 6

([3]). The tripled $(\Psi ,⊩,\partial )$ is said to be an O-metric space if $(\Psi ,⊩)$ is an O-set and $(\Psi ,\partial )$ is a metric space.

Definition 7

([3]). Let $(\Psi ,⊩,\partial )$ be a O-metric space. Then,

• A map $\mathcal{H}:\Psi \to \Psi$ is called orthogonal continuous (⊩-continuous) in $z\in \Psi$ if for each O-sequence $\left\{{\delta }_{\varrho }\right\}$ in Ψ with ${\delta }_{\varrho }\to \delta$ as $\varrho \to \infty$, we have $\mathcal{H}{\delta }_{\varrho }\to \mathcal{H}\delta$ as $\varrho \to \infty$. Also, $\mathcal{H}$ is called ⊩-continuous on Ψ if $\mathcal{H}$ is ⊩-continuous in each $\delta \in \Psi$.
• If every Cauchy O-sequence is convergent, then Ψ is called O-complete.

Definition 8

([14]). Let $(\Psi ,⊩)$ be an O-set and d be a metric on Ψ, $\mathcal{H}:\Psi \to \Psi$ and $\alpha :\Psi \times \Psi \to {\mathbb{R}}^{+}$ be two functions. We say that $\mathcal{H}$ is orthogonally α-admissible whenever $\delta ⊩\xi$ and $\alpha (\delta ,\xi )\ge 1\Rightarrow \alpha \left(\mathcal{H}\right(\delta ),\mathcal{H}(\xi \left)\right)\ge 1$

Example 2

([14]). Let $\Psi =(0,\infty )$. Define ⊩ on Ψ by $\delta ⊩\xi$ iff $\delta \xi =\xi$. Define $\mathcal{H}:\Psi \to \Psi$ and $\alpha :\Psi \times \Psi \to {\mathbb{R}}^{+}$ by $\mathcal{H}\left(\delta \right)=\frac{\delta +1}{2}$ if $\delta \le 1$ and $\mathcal{H}\left(\delta \right)=\frac{1}{2}$ if $\delta >1,\forall$ $\delta \in \Psi$ and $\alpha (\delta ,\xi )=\xi$. Then, $\mathcal{H}$ is orthogonally α-admissible.

Let $\mathcal{H}:\Psi \to \Psi$ be a map on $\Psi$. We denote $Fix\left(\mathcal{H}\right)=\{\delta :\mathcal{H}\delta =\delta \forall \delta \in \Psi \}$.

Definition 9.

Let $\mathcal{H}$ be an orthogonally α-admissible map. Then, $\mathcal{H}$ is called orthogonally ${\alpha }^{*}$-admissible, if for each $\delta ,\xi \in Fix\left(\mathcal{H}\right)\ne \varnothing$, we have $\delta ⊩\xi$, $\alpha (\delta ,\xi )\ge 1$. If $\delta ,\xi \in Fix\left(\mathcal{H}\right)=\varnothing$, we say that $\mathcal{H}$ is vacuously ${\alpha }^{*}$-admissible.

3. Main Results

Inspired by the $\alpha$-F-convex contraction map concepts defined by Mahendra Singh, Khan, and Kang [5], here we introduce a new orthogonal $\alpha$–F–convex contraction mapping and prove some fixed point results in an orthogonal complete metric space (CMS) for this contractive mapping.

Let $\mathcal{H}$ be a map on a O-MS $(\Psi ,\partial )$. Now

$$\begin{array}{c}\hfill M^k(\delta ,\xi )=max\{{\partial }^k(\delta ,\xi ),{\partial }^k(\mathcal{H}\delta ,\mathcal{H}\xi ),{\partial }^k(\delta ,\mathcal{H}\delta ),{\partial }^k(\mathcal{H}\delta ,{\mathcal{H}}^2\delta ),{\partial }^k(\xi ,\mathcal{H}\xi ),{\partial }^k(\mathcal{H}\xi ,{\mathcal{H}}^2\xi )\}\end{array}$$

(2)

Definition 10.

A self mapping $\mathcal{H}$ on Ψ is said to be a ${\mathcal{H}}_{⊩}$–α–F–convex contraction, if there exists two functions $\alpha :\Psi \times \Psi \to [0,\infty )$, $\mathbb{F}\in \Im$ and $\nu >0$ such that

$$\begin{array}{c}\hfill {\partial }^k({\mathcal{H}}^2\delta ,{\mathcal{H}}^2\xi )>0\Rightarrow \nu +\mathbb{F}\left(\alpha (\delta ,\xi ){\partial }^k({\mathcal{H}}^2\delta ,{\mathcal{H}}^2\xi )\right)\le \mathbb{F}\left(M^k(\delta ,\xi )\right),\end{array}$$

(3)

$\forall \delta ,\xi \in \Psi \mathrm{with}\delta ⊩\xi ,k\in [1,\infty )$.

Theorem 1.

Let $(\Psi ,⊩,\partial )$ be an O-CMS and an orthogonal element ${\delta }_0$ and $\mathcal{H}:\Psi \to \Psi$ be an ${\mathcal{H}}_{⊩}$–α-F–convex contraction mapping such that:

(i)

$\mathcal{H}$ is ⊩-preserving;

(ii)

$\mathcal{H}$ is orthogonal α-admissible;

(iii)

∃${\delta }_0\in \Psi ,\ni$${\delta }_0⊩\mathcal{H}{\delta }_0$ and $\alpha ({\delta }_0,\mathcal{H}{\delta }_0)\ge 1$.

Define a sequence $\left\{{\delta }_{\varrho }\right\}$ in Ψ by ${\delta }_{\varrho +1}=\mathcal{H}{\delta }_{\varrho }={\mathcal{H}}^{\varrho +1}{\delta }_0,\forall$ $\varrho \ge 0$, then $F\left({\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right)\le F\left(v\right)-l\nu$, whenever $\varrho =2l$ or $\varrho =2l+1$ for $l\ge 1$.

Proof. 

By condition (iii) in Theorem 1, ∃${\delta }_0\in \Psi$, $\ni {\delta }_0⊩\mathcal{H}{\delta }_0$ and $\alpha ({\delta }_0,\mathcal{H}{\delta }_0)\ge 1$. Let

$$\begin{array}{c}\hfill {\delta }_1:=\mathcal{H}{\delta }_0,{\delta }_2:=\mathcal{H}{\delta }_1={\mathcal{H}}^2{\delta }_0\dots \dots ,{\delta }_{\varrho +1}:=\mathcal{H}{\delta }_{\varrho }={\mathcal{H}}^{\varrho +1}{\delta }_0,\forall \varrho \in \mathtt{N}\cup \left\{0\right\}.\end{array}$$

Consider $\left\{{\delta }_{\varrho }\right\}$ is an O-sequence in $\Psi$.

If ${\delta }_{\varrho }={\delta }_{\varrho +1}$ for any $\varrho \in \mathtt{N}\cup \left\{0\right\}$, then ${\delta }_{\varrho }$ is a fixed point of $\mathcal{H}$.

Assume that ${\delta }_{\varrho }\ne {\delta }_{\varrho +1}$, $\forall \varrho \in \mathtt{N}\cup \left\{0\right\}$. Then,

$$\begin{array}{c}\hfill \partial ({\delta }_{\varrho },{\delta }_{\varrho +1})>0,\forall \varrho \in \mathtt{N}\cup \left\{0\right\}.\end{array}$$

Since $\mathcal{H}$ is ⊩-preserving, then $\left\{{\delta }_{\varrho }\right\}$ is an O-sequence in $\Psi$. Now, setting $v=max\{{\partial }^k({\delta }_0,{\delta }_1),{\partial }^k({\delta }_1,{\delta }_2)\}$. Condition (ii) in Theorem 1, we get

$$\begin{array}{c}\hfill \alpha ({\delta }_{\varrho },{\delta }_{\varrho +1})\ge 1,\forall \varrho \in \mathtt{N}\cup \left\{0\right\}.\end{array}$$

From (2), taking $\delta ={\delta }_0$ and $\xi ={\delta }_1$, we obtain

$$\begin{array}{cc}\hfill M^k({\delta }_0,{\delta }_1)=& max\{{\partial }^k({\delta }_0,{\delta }_1),{\partial }^k(\mathcal{H}{\delta }_0,\mathcal{H}{\delta }_1),{\partial }^k({\delta }_0,\mathcal{H}{\delta }_0),\hfill \\ \hfill & {\partial }^k(\mathcal{H}{\delta }_0,{\mathcal{H}}^2{\delta }_0),{\partial }^k({\delta }_1,\mathcal{H}{\delta }_1),{\partial }^k(\mathcal{H}{\delta }_1,{\mathcal{H}}^2{\delta }_1)\}\hfill \\ \hfill =& max\{{\partial }^k({\delta }_0,{\delta }_1),{\partial }^k({\delta }_1,{\delta }_2),{\partial }^k({\delta }_0,{\delta }_1),{\partial }^k({\delta }_1,{\delta }_2),{\partial }^k({\delta }_1,{\delta }_2),{\partial }^k({\delta }_2,{\delta }_3)\}\hfill \\ \hfill =& max\{{\partial }^k({\delta }_0,{\delta }_1),{\partial }^k({\delta }_1,{\delta }_2),{\partial }^k({\delta }_2,{\delta }_3)\}.\hfill \end{array}$$

(4)

By the Definition 1, $\left(F_1\right)$, $\alpha ({\delta }_0,{\delta }_1)\ge 1$, by (3) and (4), we have

$$\begin{array}{cc}\hfill F\left({\partial }^k({\delta }_2,{\delta }_3)\right)& =F\left({\partial }^k({\mathcal{H}}^2{\delta }_0,{\mathcal{H}}^2{\delta }_1)\right)\hfill \\ \hfill & \le F\left(\alpha ({\delta }_0,{\delta }_1){\partial }^k({\mathcal{H}}^2{\delta }_0,{\mathcal{H}}^2{\delta }_1)\right)\hfill \\ \hfill & \le F\left(M^k({\delta }_0,{\delta }_1)\right)-\nu \hfill \\ \hfill & =F(max\{{\partial }^k({\delta }_0,{\delta }_1),{\partial }^k({\delta }_1,{\delta }_2),{\partial }^k({\delta }_2,{\delta }_3)\})-\nu \hfill \\ \hfill & \le F(max\{v,{\partial }^k({\delta }_2,{\delta }_3)\})-\nu .\hfill \end{array}$$

(5)

If $max\{v,{\partial }^k({\delta }_2,{\delta }_3)\}={\partial }^k({\delta }_2,{\delta }_3)$, then (5) gives

$$\begin{array}{c}\hfill F\left({\partial }^k({\delta }_2,{\delta }_3)\right)\le F\left({\partial }^k({\delta }_2,{\delta }_3)\right)-\nu <F\left({\partial }^k({\delta }_2,{\delta }_3)\right).\end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill F\left({\partial }^k({\delta }_2,{\delta }_3)\right)\le F\left(v\right)-\nu <F\left(v\right).\end{array}$$

Since $\nu >0$ and Definition 1, $\left(F_1\right)$, we conclude that

$$\begin{array}{c}\hfill {\partial }^k({\delta }_2,{\delta }_3)<v=max\{{\partial }^k({\delta }_0,{\delta }_1),{\partial }^k({\delta }_1,{\delta }_2)\}.\end{array}$$

Again, from (2) taking with $\delta ={\delta }_1$ and $\xi ={\delta }_2$, we obtain

$$\begin{array}{cc}\hfill M^k({\delta }_1,{\delta }_2)& =max\{{\partial }^k({\delta }_1,{\delta }_2),{\partial }^k(\mathcal{H}{\delta }_1,\mathcal{H}{\delta }_2),{\partial }^k({\delta }_1,\mathcal{H}{\delta }_1),\hfill \\ \hfill & {\partial }^k(\mathcal{H}{\delta }_1,{\mathcal{H}}^2{\delta }_1),{\partial }^k({\delta }_2,\mathcal{H}{\delta }_2),{\partial }^k(\mathcal{H}{\delta }_2,{\mathcal{H}}^2{\delta }_2)\}\hfill \\ \hfill & =max\{{\partial }^k({\delta }_1,{\delta }_2),{\partial }^k({\delta }_2,{\delta }_3),{\partial }^k({\delta }_1,{\delta }_2),{\partial }^k({\delta }_2,{\delta }_3),{\partial }^k({\delta }_2,{\delta }_3),{\partial }^k({\delta }_3,{\delta }_4)\}\hfill \\ \hfill & =max\{{\partial }^k({\delta }_1,{\delta }_2),{\partial }^k({\delta }_2,{\delta }_3),{\partial }^k({\delta }_3,{\delta }_4)\}.\hfill \end{array}$$

(6)

Using (3) and (6), we get

$$\begin{array}{cc}\hfill F\left({\partial }^k({\delta }_3,{\delta }_4)\right)& =F\left({\partial }^k({\mathcal{H}}^2{\delta }_1,{\mathcal{H}}^2{\delta }_2)\right)\hfill \\ \hfill & \le F\left(\alpha ({\delta }_1,{\delta }_2){\partial }^k({\mathcal{H}}^2{\delta }_1,{\mathcal{H}}^2{\delta }_2)\right)\hfill \\ \hfill & \le F\left(M^k({\delta }_1,{\delta }_2)\right)-\nu \hfill \\ \hfill & =F(max\{{\partial }^k({\delta }_1,{\delta }_2),{\partial }^k({\delta }_2,{\delta }_3),{\partial }^k({\delta }_3,{\delta }_4)\})-\nu .\hfill \end{array}$$

If $max\{{\partial }^k({\delta }_1,{\delta }_2),{\partial }^k({\delta }_2,{\delta }_3),{\partial }^k({\delta }_3,{\delta }_4)\}={\partial }^k({\delta }_3,{\delta }_4)$, then we obtain

$$\begin{array}{c}\hfill F\left({\partial }^k({\delta }_3,{\delta }_4)\right)\le F\left({\partial }^k({\delta }_3,{\delta }_4)\right)-\nu <F\left({\partial }^k({\delta }_3,{\delta }_4)\right).\end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill max\{{\partial }^k({\delta }_1,{\delta }_2),{\partial }^k({\delta }_2,{\delta }_3)\}>{\partial }^k({\delta }_3,{\delta }_4).\end{array}$$

Therefore, $v>{\partial }^k({\delta }_2,{\delta }_3)>{\partial }^k({\delta }_3,{\delta }_4)$. Proceeding this, one can prove inductively that $\left\{{\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right\}$ is a strictly non-increasing sequence in $\Psi$. Therefore, $v>{\partial }^k({\delta }_2,{\delta }_3)$ and hence

$$\begin{array}{c}\hfill F\left({\partial }^k({\delta }_2,{\delta }_3)\right)\le F\left(v\right)-\nu .\end{array}$$

Similarly, we have

$$\begin{array}{c}\hfill F\left({\partial }^k({\delta }_3,{\delta }_4)\right)\le F\left(v\right)-\nu ;\end{array}$$

proceeding that, we get

$$\begin{array}{c}\hfill F\left({\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +\mathfrak{1}})\right)\le F\left(v\right)-l\nu ,\end{array}$$

whenever $\varrho =2l$ or $\varrho =2l+1$ for $l\ge 1$. □

Theorem 2.

Let $(\Psi ,⊩,\partial )$ be a O-CMS and an orthogonal element ${\delta }_0$, and $\mathcal{H}:\Psi \to \Psi$ be an ${\mathcal{H}}_{⊩}$–α-F–convex contraction mapping such that:

(i) 

$\mathcal{H}$ is ⊩-preserving;

(ii) 

$\mathcal{H}$ is orthogonally α-admissible;

(iii) 

∃${\delta }_0\in \Psi ,\ni$${\delta }_0⊩\mathcal{H}{\delta }_0$ and $\alpha ({\delta }_0,\mathcal{H}{\delta }_0)\ge 1$;

(iv) 

$\mathcal{H}$ is orthogonally continuous, or orthogonally orbitally continuous on Ψ.

Then, $\mathcal{H}$ has a fixed point in Ψ. Furthermore, if $\mathcal{H}$ is orthogonally ${\alpha }^{*}$-admissible, then $\mathcal{H}$ has a unique fixed point $z\in \Psi$. Moreover, for any ${\delta }_0\in \Psi$, if ${\delta }_{\varrho +1}={\mathcal{H}}^{\varrho +1}{\delta }_0\ne \mathcal{H}{\delta }_{\varrho },\forall$ $\varrho \ge 0$, then ${\lim }_{\varrho \to \infty }{\mathcal{H}}^{\varrho }{\delta }_0=z$.

Proof. 

By condition (iii) in Theorem 2, ∃${\delta }_0\in \Psi$, ∋ ${\delta }_0⊩\mathcal{H}{\delta }_0$ and $\alpha ({\delta }_0,\mathcal{H}{\delta }_0)\ge 1$. Let

$$\begin{array}{c}\hfill {\delta }_1:=\mathcal{H}{\delta }_0,{\delta }_2:=\mathcal{H}{\delta }_1={\mathcal{H}}^2{\delta }_0\dots \dots ,{\delta }_{\varrho +1}:=\mathcal{H}{\delta }_{\varrho }={\mathcal{H}}^{\varrho +1}{\delta }_0,\forall \varrho \in \mathtt{N}\cup \left\{0\right\}.\end{array}$$

Consider $\left\{{\delta }_{\varrho }\right\}$ is an O-sequence in $\Psi$.

If ${\delta }_{\varrho }={\delta }_{\varrho +1}$ for any $\varrho \in \mathtt{N}\cup \left\{0\right\}$, then ${\delta }_{\varrho }$ is a fixed point of $\mathcal{H}$. Assume that ${\delta }_{\varrho }\ne {\delta }_{\varrho +1}$, $\forall \varrho \in \mathtt{N}\cup \left\{0\right\}$. Then

$$\begin{array}{c}\hfill \partial ({\delta }_{\varrho },{\delta }_{\varrho +1})>0,\forall \varrho \in \mathtt{N}\cup \left\{0\right\}.\end{array}$$

Since $\mathcal{H}$ is ⊩-preserving, then $\left\{{\delta }_{\varrho }\right\}$ is an O-sequence in $\Psi$. Now, setting $v=max\{{\partial }^k({\delta }_0,{\delta }_1),{\partial }^k({\delta }_1,{\delta }_2)\}$. From condition (ii) in Theorem 2, we obtain

$$\begin{array}{c}\hfill \alpha ({\delta }_{\varrho },{\delta }_{\varrho +1})\ge 1,\forall \varrho \in \mathtt{N}\cup \left\{0\right\}.\end{array}$$

From (2), taking $\delta ={\delta }_{\varrho -2}$ and $\xi ={\delta }_{\varrho -1}$, where $\varrho \ge 2$, we obtain

$$\begin{array}{cc}\hfill M^k({\delta }_{\varrho -2},{\delta }_{\varrho -1})& =max\{{\partial }^k({\delta }_{\varrho -2},{\delta }_{\varrho -1}),{\partial }^k(\mathcal{H}{\delta }_{\varrho -2},\mathcal{H}{\delta }_{\varrho -1}),{\partial }^k({\delta }_{\varrho -2},\mathcal{H}{\delta }_{\varrho -2}),\hfill \\ \hfill & {\partial }^k(\mathcal{H}{\delta }_{\varrho -2},{\mathcal{H}}^2{\delta }_{\varrho -2}),{\partial }^k({\delta }_{\varrho -1},\mathcal{H}{\delta }_{\varrho -1}),{\partial }^k(\mathcal{H}{\delta }_{\varrho -1},{\mathcal{H}}^2{\delta }_{\varrho -1})\}\hfill \\ \hfill & =max\{{\partial }^k({\delta }_{\varrho -2},{\delta }_{\varrho -1}),{\partial }^k({\delta }_{\varrho -1},{\delta }_{\varrho }),{\partial }^k({\delta }_{\varrho -2},{\delta }_{\varrho -1}),\hfill \\ \hfill & {\partial }^k({\delta }_{\varrho -1},{\delta }_{\varrho }),{\partial }^k({\delta }_{\varrho -1},{\delta }_{\varrho }),{\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\}\hfill \\ \hfill & =max\{{\partial }^k({\delta }_{\varrho -2},{\delta }_{\varrho -1}),{\partial }^k({\delta }_{\varrho -1},{\delta }_{\varrho }),{\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\}.\hfill \end{array}$$

By Definition 1, $\left(F_1\right)$ and $\mathcal{H}$ are orthogonally $\alpha$-admissible, and by (3), we derive

$$\begin{array}{cc}\hfill F\left({\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right)& =F\left({\partial }^k({\mathcal{H}}^2{\delta }_{\varrho -2},{\mathcal{H}}^2{\delta }_{\varrho -1})\right)\hfill \\ \hfill & \le F\left(\alpha ({\delta }_{\varrho -2},{\delta }_{\varrho -1}){\partial }^k({\mathcal{H}}^2{\delta }_{\varrho -2},{\mathcal{H}}^2{\delta }_{\varrho -1})\right)\hfill \\ \hfill & \le F\left(M^k({\delta }_{\varrho -2},{\delta }_{\varrho -1})\right)-\nu \hfill \\ \hfill & \le F(max\{{\partial }^k({\delta }_{\varrho -2},{\delta }_{\varrho -1}),{\partial }^k({\delta }_{\varrho -1},{\delta }_{\varrho }),{\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\})-\nu .\hfill \end{array}$$

If $max\{{\partial }^k({\delta }_{\varrho -2},{\delta }_{\varrho -1}),{\partial }^k({\delta }_{\varrho -1},{\delta }_{\varrho }),{\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\}={\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})$, then we get

$$\begin{array}{c}\hfill F\left({\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right)\le F\left({\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right)-\nu <F\left({\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right).\end{array}$$

Which is a contradiction. Hence,

$$\begin{array}{c}\hfill F\left({\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right)\le F(max\{{\partial }^k({\delta }_{\varrho -2},{\delta }_{\varrho -1}),{\partial }^k({\delta }_{\varrho -1},{\delta }_{\varrho })\})-\nu .\end{array}$$

From Theorem 1, we obtain

$$\begin{array}{c}\hfill F\left({\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right)\le F\left(v\right)-l\nu ,\end{array}$$

(7)

whenever, $\varrho =2l$ or $\varrho =2l+1$ for $l\ge 1$. From (6), we obtain

$$\begin{array}{c}\hfill \underset{\varrho \to \infty }{lim}F\left({\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right)=-\infty .\end{array}$$

(8)

Using $\left(F_2\right)$, we derive

$$\begin{array}{c}\hfill \underset{\varrho \to \infty }{lim}{\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})=0.\end{array}$$

(9)

From $\left(F_3\right)$, ∃$\mu \in (0,1)$, ∋

$$\begin{array}{c}\hfill \underset{\varrho \to \infty }{lim}{\left[{\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right]}^{\mu }F\left({\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right)=0.\end{array}$$

(10)

Also, by (7), we have

$$\begin{array}{c}\hfill {\left[{\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right]}^{\mu }[F\left({\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right)-F\left(v\right)]\le -{\left[{\partial }^k({\delta }_{\varrho },{\delta }_{\varrho +1})\right]}^{\mu }]l\nu \le 0,\end{array}$$

(11)

where $\varrho =2l$ or $\varrho =2l+1$ for $l\ge 1$.

As $\varrho \to \infty$ in (11) along with (9) and (10), we derive

$$\begin{array}{c}\hfill \underset{\varrho \to \infty }{lim}l{[\partial ({\delta }_{\varrho },{\delta }_{\varrho +1})]}^{\mu }=0.\end{array}$$

(12)

Now, we have two cases arise.

Case 1: If $\varrho$ is even and $\varrho \ge 2$, then

$$\begin{array}{c}\hfill \underset{\varrho \to \infty }{lim}\varrho {[\partial ({\delta }_{\varrho },{\delta }_{\varrho +1})]}^{\mu }=0.\end{array}$$

(13)

Case 2: If $\varrho$ is odd and $\varrho \ge 3$, then

$$\begin{array}{c}\hfill \underset{\varrho \to \infty }{lim}(\varrho -1){[\partial ({\delta }_{\varrho },{\delta }_{\varrho +1})]}^{\mu }=0.\end{array}$$

(14)

Using (9) and (14) gives

$$\begin{array}{c}\hfill \underset{\varrho \to \infty }{lim}\varrho {[\partial ({\delta }_{\varrho },{\delta }_{\varrho +1})]}^{\mu }=0.\end{array}$$

(15)

Then, ∃${\varrho }_1\in \mathtt{N}$,

$$\begin{array}{c}\hfill \varrho {[\partial ({\delta }_{\varrho },{\delta }_{\varrho +1})]}^{\mu }\le 1\forall \varrho \ge {\varrho }_1.\end{array}$$

Therefore, we obtain

$$\begin{array}{c}\hfill \partial ({\delta }_{\varrho },{\delta }_{\varrho +1})\le {\displaystyle \frac{1}{{\varrho }^{{\displaystyle \frac{1}{\mu }}}}}\forall \varrho \ge {\varrho }_1.\end{array}$$

For all $p>q\ge {\varrho }_1$, we derive

$$\begin{array}{cc}\hfill \partial ({\delta }_k,{\delta }_q)& \le \partial ({\delta }_k,{\delta }_{p-1})+\partial ({\delta }_{p-1},{\delta }_{p-2})+\dots +\partial ({\delta }_{q+1},{\delta }_q)\hfill \\ \hfill & <\sum _{i=q}^{\infty }\partial ({\delta }_i,{\delta }_{i+1})\hfill \\ \hfill & \le \sum _{i=q}^{\infty }{\displaystyle \frac{1}{i^{{\displaystyle \frac{1}{\mu }}}}}.\hfill \end{array}$$

Suppose $\mathcal{H}$ is orthogonally continuous, then

$$\begin{array}{c}\hfill \partial (z,\mathcal{H}z)=\underset{\varrho \to \infty }{lim}\partial ({\delta }_{\varrho },\mathcal{H}{\delta }_{\varrho })=\underset{\varrho \to \infty }{lim}\partial ({\delta }_{\varrho },{\delta }_{\varrho +1})=0.\end{array}$$

Hence, $\mathcal{H}$ has a fixed point, that is, $z\in \Psi$.

Let $\mathcal{H}$ be orthogonally orbitally continuous on $\Psi$, then ${\delta }_{\varrho +1}=\mathcal{H}{\delta }_{\varrho }=\mathcal{H}\left({\mathcal{H}}^{\varrho }{\delta }_0\right)\to \mathcal{H}z$ as $\varrho \to \infty$. Since $(\Psi ,\partial )$ is O-complete, we have $\mathcal{H}z=z$. Therefore, $Fix\left(\mathcal{H}\right)\ne \varnothing$.

Now, let $\mathcal{H}$ be orthogonally ${\alpha }^{*}$-admissible, then $\forall z,y\in Fix\left(\mathcal{H}\right)$, we get $\varrho (z,y)\ge 1$. Using (2) and (3), we derive

$$\begin{array}{cc}\hfill F\left({\partial }^k(z,y)\right)& =F\left({\partial }^k({\mathcal{H}}^{2}z,{\mathcal{H}}^{2}y)\right)\hfill \\ \hfill & =F\left(\alpha (z,y){\partial }^k({\mathcal{H}}^{2}z,{\mathcal{H}}^{2}y)\right)\hfill \\ \hfill & \le F\left(M^k(z,y)\right)-\nu \hfill \\ \hfill & =F(max\{{\partial }^k(z,y),{\partial }^k(\mathcal{H}z,\mathcal{H}y),{\partial }^k(z,\mathcal{H}z),{\partial }^k(\mathcal{H}z,{\mathcal{H}}^{2}z),\hfill \\ & {\partial }^k(y,\mathcal{H}y),{\partial }^k(\mathcal{H}y,{\mathcal{H}}^{2}y)\left\}\right)-\nu \hfill \\ \hfill & =F\left({\partial }^k(z,y)\right)-\nu .\hfill \end{array}$$

Since $\nu >0$ and using the Definition 1, we obtain

$$\begin{array}{c}\hfill \partial (z,y)<\partial (z,y).\end{array}$$

Therefore, $\mathcal{H}$ has a unique fixed point in $\Psi$. □

Corollary 1.

Let $(\Psi ,⊩,\partial )$ be O-CMS and an orthogonal element ${\delta }_0$ and $\alpha :\Psi \times \Psi \to {\mathbb{R}}^{+}$ be a function. Suppose that $\mathcal{H}:\Psi \to \Psi$ is self mapping, such that

(i) 

for all $\delta ,\xi \in \Psi$ with $\delta ⊩\xi$ and

$$\begin{array}{cc}\hfill [& \partial ({\mathcal{H}}^2\delta ,{\mathcal{H}}^2\xi )>0\Rightarrow \alpha (\delta ,\xi )\partial ({\mathcal{H}}^2\delta ,{\mathcal{H}}^2\xi )\hfill \\ & \le kmax\{\partial (\delta ,\xi ),\partial (\mathcal{H}\delta ,\mathcal{H}\xi ),\partial (\delta ,\mathcal{H}\delta ),\partial (\mathcal{H}\delta ,{\mathcal{H}}^2\delta ),\partial (\xi ,\mathcal{H}\xi ),\partial (\mathcal{H}\xi ,{\mathcal{H}}^2\xi )\}],\hfill \end{array}$$

(16)

where $k\in [0,1)$;

(ii) 

$\mathcal{H}$ is ⊩-preserving;

(iii) 

$\mathcal{H}$ is orthogonally α-admissible;

(iv) 

∃${\delta }_0\in \Psi ,\ni$${\delta }_0⊩\mathcal{H}{\delta }_0$ and $\alpha ({\delta }_0,\mathcal{H}{\delta }_0)\ge 1$;

(v) 

$\mathcal{H}$ is orthogonally orbitally continuous or, orthogonally continuous on Ψ.

Then, $\mathcal{H}$ has a fixed point in Ψ. Furthermore, if $\mathcal{H}$ is orthogonally ${\alpha }^{*}$-admissible, then $\mathcal{H}$ has a unique fixed point $z\in \Psi$. Moreover, for every ${\delta }_0\in \Psi$ if ${\delta }_{\varrho +1}={\mathcal{H}}^{\varrho +1}{\delta }_0\ne {\mathcal{H}}^{\varrho }{\delta }_0,\forall$ $\varrho \ge 0$, then ${lim}_{\varrho \to \infty }{\mathcal{H}}^{\varrho }{\delta }_0=z$.

Proof. 

Setting $F(⋏)=ln(⋏),⋏>0$. Obviously, $F\in \Im$. From (16), we derive

$$\begin{array}{cc}\hfill -lnk+ln\alpha (\delta ,\xi )\partial ({\mathcal{H}}^2\delta ,{\mathcal{H}}^2\xi )& \le ln(max\{\partial (\delta ,\xi ),\partial (\mathcal{H}\delta ,\mathcal{H}\xi ),\partial (\delta ,\mathcal{H}\delta ),\hfill \\ & \partial (\mathcal{H}\delta ,{\mathcal{H}}^2\delta ),\partial (\xi ,\mathcal{H}\xi )\partial (\mathcal{H}\xi ,{\mathcal{H}}^2\xi )\left\}\right),\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \nu +F(\alpha (\delta ,\xi )\partial ({\mathcal{H}}^2\delta ,{\mathcal{H}}^2\xi ))\le F\left(M^1(\delta ,\xi )\right),\end{array}$$

∀$\delta ,\xi \in \Psi$ with $\delta \ne \xi$ where $\nu =-lnk$. Therefore, $\mathcal{H}$ is ${\mathcal{H}}_{⊩}$- $\alpha$-F-convex contraction mapping with $k=1$. Thus, all of the hypotheses of Theorem 2 are fulfilled. Therefore, $\mathcal{H}$ has a unique fixed point in $\Psi$. □

Example 3.

Let $\Psi =[0,\infty )$ with usual metric

$$\partial (\delta ,\xi )=\left\{\begin{array}{cc}|\delta -\xi |,\hfill & if\delta \ne \xi ;\hfill \\ 0,\hfill & if\delta =\xi .\hfill \end{array}\right.$$

Suppose $\delta ⊩\xi$ iff $\delta ,\xi \ge 0$. Clearly, $(\Psi ,⊩,\partial )$ is O-CMS. Define $\mathcal{H}:\Psi \to \Psi$ and $\alpha :\Psi \times \Psi \to {\mathbb{R}}^{+}$ by $\mathcal{H}\left(\delta \right)=\frac{{\delta }^2}{2},\forall$ $\delta \in \Psi$ and

$$\alpha (\delta ,\xi )=\left\{\begin{array}{cc}1,\hfill & if\delta ,\xi \in (0,2];\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.$$

Then, $\mathcal{H}$ is orthogonally α-admissible and orthogonally ${\alpha }^{*}$-admissible. Clearly, $\mathcal{H}$ is ⊩-preserving and orthogonally continuous. Setting $F\in \Im ,\forall$ $F(⋏)=ln(⋏),⋏>0$. Then, $\forall \delta ,\xi \in \Psi$ with $\delta \ne \xi$, we obtain

$$\begin{array}{cc}\hfill \alpha (\delta ,\xi )|{\mathcal{H}}^2\delta -{\mathcal{H}}^2\xi |& =|{\mathcal{H}}^2\delta -{\mathcal{H}}^2\xi |\hfill \\ \hfill & =|\frac{{\delta }^4}{4}-\frac{{\xi }^4}{4}|\hfill \\ \hfill & \le |\frac{{\delta }^2}{4}-\frac{{\xi }^2}{4}|+\frac{1}{2}|\delta -\xi |\hfill \\ \hfill & =\frac{1}{2}|\mathcal{H}\delta -\mathcal{H}\xi |+\frac{1}{2}|\delta -\xi |\hfill \\ \hfill & \le \frac{5}{6}max\left\{\right|\mathcal{H}\delta -\mathcal{H}\xi |,|\delta -\xi \left|\right\}\hfill \\ \hfill & \le e^{-\nu }{M}^1(\delta ,\xi ),\hfill \end{array}$$

where $-\nu =ln\frac{5}{6}$. Then

$$\begin{array}{c}\hfill \nu +F(\alpha (\delta ,\xi )\partial ({\mathcal{H}}^2\delta ,{\mathcal{H}}^2\xi ))\le F\left(M^1(\delta ,\xi )\right)\end{array}$$

Therefore, $\mathcal{H}$ is ${\mathcal{H}}_{⊩}$–α–F–convex contraction mapping with $k=1$. Hence, all the conditions of Theorem 2 are fulfilled and so $\mathcal{H}$ has a unique fixed point $\delta =2$.

4. Application to Fractional Integro-Differential Equation

We consider a Cauchy problem involving a fractional integro-differential equation with a non-local condition given by

$$\left\{\begin{array}{c}{}^k{\mathcal{D}}^{\ell }\delta \left(\mathfrak{a}\right)=\mathfrak{f}(\mathfrak{a},\delta \left(\mathfrak{a}\right))+{\int }_0^{\mathfrak{a}}\mathcal{J}(\mathfrak{a},\mathsf{ϝ},\delta \left(\mathfrak{a}\right))d\mathsf{ϝ},\mathfrak{a}\in [0,\Omega ],\Omega >0,0<\ell <1,\hfill \\ \delta \left(0\right)={\delta }_0-\mathfrak{l}\left(\delta \right),\hfill \end{array}\right.$$

(17)

where ${}^k{\mathcal{D}}^{\ell }$ denotes the Caputo fractional derivative of order ℓ, $\mathfrak{f}:[0,\Omega ]\times \mathbb{R}\to {\mathbb{R}}^{+}$, $\mathcal{J}:[0,\Omega ]\times [0,\Omega ]\times \mathbb{R}\to \mathbb{R}$ are jointly continuous, $\mathfrak{l}:K\to \mathbb{R}$ is continuous. Here, $K=K\left(\right[0,\Omega ],\mathbb{R})$ denotes the Banach space $(\Psi ,||\cdot |\left|\right)$ of all continuous maps from $[0,\Omega ]\to \mathbb{R}$ with a topology of uniform convergence with the norm denoted by $\left|\right|\cdot {\left|\right|}_K$, and $\left|\right|\delta \left|\right|={max}_{\mathfrak{a}\in [0,\Omega ]}\left|\delta \left(\mathfrak{a}\right)\right|$, see [15].

Let $K=K\left(\right[0,\Omega ],\Psi )$ be endowed with the metric $d:K\times K\to {\mathbb{R}}^{+}$ defined as $\partial (\delta ,\xi )={max}_{\mathfrak{a}\in [0,\Omega ]}|\delta \left(\mathfrak{a}\right)-\xi \left(\mathfrak{a}\right)|,\forall \delta ,\xi \in K$. Define orthogonality ⊩ on K by

$$\begin{array}{c}\hfill \delta ⊩\xi \iff \delta \left(\mathfrak{a}\right)\xi \left(\mathfrak{a}\right)\ge 0,\forall \mathfrak{a}\in [0,\Omega ].\end{array}$$

Then, $(K,⊩,\partial )$ is O-CMS. Clearly, a solution of (17) is a fixed point

$$\begin{array}{c}\hfill \delta \left(\mathfrak{a}\right)={\delta }_0-\mathfrak{l}\left(\delta \right)+\frac{1}{\mathsf{ϝ}(\ell )}{\int }_0^{\mathfrak{a}}{(\mathfrak{a}-\mathfrak{b})}^{\ell -1}[\mathfrak{f}(\mathfrak{b},\delta \left(\mathfrak{b}\right))+{\int }_{\mathfrak{b}}^{\mathfrak{a}}\mathcal{J}(\varrho ,\mathfrak{b},\delta \left(\mathfrak{b}\right))d\varrho ]d\mathfrak{b},\end{array}$$

(18)

where $\mathsf{ϝ}$ is the Gamma function.

Theorem 3.

Suppose that $(K,⊩,\partial )$ is a orthogonal CMS equipped with metric $\partial (\delta ,\xi )={max}_{\mathfrak{a}\in [0,\Omega ]}|\delta \left(\mathfrak{a}\right)-\xi \left(\mathfrak{a}\right)|,\forall \delta ,\xi \in K$ with $\delta \ne \xi$ and $\mathcal{H}:K\to K$ be an orthogonal continuous operator on K defined by

$$\begin{array}{c}\hfill \mathcal{H}\delta \left(\mathfrak{a}\right)={\delta }_0-\mathfrak{l}\left(\delta \right)+\frac{1}{\mathsf{ϝ}(\ell )}{\int }_0^{\mathfrak{a}}{(\mathfrak{a}-\mathfrak{b})}^{\ell -1}[\mathfrak{f}(\mathfrak{b},\delta \left(\mathfrak{b}\right))+{\int }_{\mathfrak{b}}^{\mathfrak{a}}\mathcal{J}(\varrho ,\mathfrak{b},\delta \left(\mathfrak{b}\right))d\varrho ]d\mathfrak{b}.\end{array}$$

(19)

∀$\delta \in K$ and $\mathfrak{b},\mathfrak{a}\in [0,\Omega ]$ satisfying the following inequality

(A1) 

$$\begin{array}{cc}\hfill \left|\mathcal{J}\right(\mathfrak{a},\mathfrak{b},\mathcal{H}\delta \left(\mathfrak{b}\right))-\mathcal{J}(\mathfrak{a},\mathfrak{b},\mathcal{H}\xi \left(\mathfrak{b}\right)\left)\right|\le & r_{1}max\left\{\right|\delta \left(\mathfrak{b}\right)-\xi \left(\mathfrak{b}\right)|,|\mathcal{H}\delta \left(\mathfrak{b}\right)-\mathcal{H}\xi \left(\mathfrak{b}\right)|,\hfill \\ \hfill & |\delta \left(\mathfrak{b}\right)-\mathcal{H}\delta \left(\mathfrak{b}\right)|,|\mathcal{H}\delta \left(\mathfrak{b}\right)-{\mathcal{H}}^2\delta \left(\mathfrak{b}\right)|,\hfill \\ \hfill & |\xi -\mathcal{H}\xi \left(\mathfrak{b}\right)|,|\mathcal{H}\xi \left(\mathfrak{b}\right)-{\mathcal{H}}^2\xi \left(\mathfrak{b}\right)\left|\right\},\hfill \end{array}$$

(20)

(A2) 

$$\begin{array}{cc}\hfill \left|\mathfrak{f}\right(\mathfrak{b},\delta \left(\mathfrak{b}\right))-\mathfrak{f}(\mathfrak{b},\xi \left(\mathfrak{b}\right)\left)\right|\le & r_{2}max\left\{\right|\delta \left(\mathfrak{b}\right)-\xi \left(\mathfrak{b}\right)|,|\mathcal{H}\delta \left(\mathfrak{b}\right)-\mathcal{H}\xi \left(\mathfrak{b}\right)|,\hfill \\ \hfill & |\delta \left(\mathfrak{b}\right)-\mathcal{H}\delta \left(\mathfrak{b}\right)|,|\mathcal{H}\delta \left(\mathfrak{b}\right)-{\mathcal{H}}^2\delta \left(\mathfrak{b}\right)|,\hfill \\ \hfill & |\xi -\mathcal{H}\xi \left(\mathfrak{b}\right)|,|\mathcal{H}\xi \left(\mathfrak{b}\right)-{\mathcal{H}}^2\xi \left(\mathfrak{b}\right)\left|\right\},\hfill \end{array}$$

(A3) 

$$\begin{array}{cc}\hfill \left|\mathfrak{l}\right(\delta )-\mathfrak{l}(\xi \left)\right|\le & r_{3}max\left\{\right|\delta \left(\mathfrak{b}\right)-\xi \left(\mathfrak{b}\right)|,|\mathcal{H}\delta \left(\mathfrak{b}\right)-\mathcal{H}\xi \left(\mathfrak{b}\right)|,\hfill \\ \hfill & |\delta \left(\mathfrak{b}\right)-\mathcal{H}\delta \left(\mathfrak{b}\right)|,|\mathcal{H}\delta \left(\mathfrak{b}\right)-{\mathcal{H}}^2\delta \left(\mathfrak{b}\right)|,\hfill \\ \hfill & |\xi -\mathcal{H}\xi \left(\mathfrak{b}\right)|,|\mathcal{H}\xi \left(\mathfrak{b}\right)-{\mathcal{H}}^2\xi \left(\mathfrak{b}\right)\left|\right\}.\hfill \end{array}$$

Then, the Cauchy problem (17) has a unique solution provided by $r_3<\frac{1}{2},r_1\le \frac{\mathsf{ϝ}(\ell +1)}{4{\Omega }^{\ell }},$$r_1\le \frac{\mathsf{ϝ}(\ell +2)}{4{\Omega }^{\ell +1}}$.

Proof. 

We define $\alpha :K\times K\to {\mathbb{R}}^{+},$ by $\alpha (\delta ,\xi )=1,\forall \delta ,\xi \in K$. Therefore, $\mathcal{H}$ is orthogonally $\alpha$-admissible and orthogonally ${\alpha }^{*}$-admissible. Now, we prove that $\mathcal{H}$ is ⊩-preserving. For every $\delta ,\xi \in K$ with $\delta ⊩\xi$ and $\mathfrak{a}\in [0,\Omega ]$, we have

$$\begin{array}{c}\hfill \mathcal{H}\delta \left(\mathfrak{a}\right)={\delta }_0-\mathfrak{l}\left(\delta \right)+\frac{1}{\mathsf{ϝ}(\ell )}{\int }_0^{\mathfrak{a}}{(\mathfrak{a}-\mathfrak{b})}^{\ell -1}[\mathfrak{f}(\mathfrak{b},\delta \left(\mathfrak{b}\right))+{\int }_{\mathfrak{b}}^{\mathfrak{a}}\mathcal{J}(\varrho ,\mathfrak{b},\delta \left(\mathfrak{b}\right))d\varrho ]d\mathfrak{b}>0.\end{array}$$

Then, $\mathcal{H}$ is ⊩-preserving. Now, we claim that $\mathcal{H}$ is a ${\mathcal{H}}_{⊩}$–$\alpha$– F–convex contraction on K. Let $\delta ,\xi \in K$ with $\delta ⊩\xi$. Suppose that $\mathcal{H}\left(\delta \right)\ne \mathcal{H}\left(\xi \right)$. Then,

$$\begin{array}{cc}\hfill |{\mathcal{H}}^2\delta \left(\mathfrak{a}\right)-{\mathcal{H}}^2\xi \left(\mathfrak{a}\right)|& \le |\mathfrak{l}\left(\delta \right)-\mathfrak{l}\left(\xi \right)|+\frac{1}{\mathsf{ϝ}(\ell )}{\int }_0^{\mathfrak{a}}{(\mathfrak{a}-\mathfrak{b})}^{\ell -1}[|\mathfrak{f}(\mathfrak{b},\delta \left(\mathfrak{b}\right))-\mathfrak{f}(\mathfrak{b},\xi \left(\mathfrak{b}\right))|\hfill \\ \hfill & +{\int }_{\mathfrak{b}}^{\mathfrak{a}}|\mathcal{J}(\varrho ,\mathfrak{b},\delta \left(\mathfrak{b}\right))-\mathcal{J}(\varrho ,\mathfrak{b},\xi \left(\mathfrak{b}\right))|d\varrho ]d\mathfrak{b}\hfill \\ \hfill & \le \left(r_3+\frac{r_1{\Omega }^{\ell }}{\mathsf{ϝ}(\ell +1)}+\frac{r_2{\Omega }^{\ell +1}}{\mathsf{ϝ}(\ell +2)}\right)max\left\{\right|\delta \left(\mathfrak{b}\right)-\xi \left(\mathfrak{b}\right)|,|\mathcal{H}\delta \left(\mathfrak{b}\right)-\mathcal{H}\xi \left(\mathfrak{b}\right)|,\hfill \\ \hfill & |\delta \left(\mathfrak{b}\right)-\mathcal{H}\delta \left(\mathfrak{b}\right)|,|\mathcal{H}\delta \left(\mathfrak{b}\right)-{\mathcal{H}}^2\delta \left(\mathfrak{b}\right)|,|\xi -\mathcal{H}\xi \left(\mathfrak{b}\right)|,|\mathcal{H}\xi \left(\mathfrak{b}\right)-{\mathcal{H}}^2\xi \left(\mathfrak{b}\right)\left|\right\}.\hfill \end{array}$$

Since $r_3<\frac{1}{2},r_1\le \frac{\mathsf{ϝ}(\ell +1)}{4{\Omega }^{\ell }},r_1\le \frac{\mathsf{ϝ}(\ell +2)}{4{\Omega }^{\ell +1}}$, therefore, $i:=r_3+\frac{r_1{\Omega }^{\ell }}{\mathsf{ϝ}(\ell +1)}+\frac{r_2{\Omega }^{\ell +1}}{\mathsf{ϝ}(\ell +2)}<1$,

$$\begin{array}{cc}\hfill |{\mathcal{H}}^2\delta \left(\mathfrak{a}\right)-{\mathcal{H}}^2\xi \left(\mathfrak{a}\right)|& \le imax\left\{\right|\delta \left(\mathfrak{b}\right)-\xi \left(\mathfrak{b}\right)|,|\mathcal{H}\delta \left(\mathfrak{b}\right)-\mathcal{H}\xi \left(\mathfrak{b}\right)|,\hfill \\ \hfill & |\delta \left(\mathfrak{b}\right)-\mathcal{H}\delta \left(\mathfrak{b}\right)|,|\mathcal{H}\delta \left(\mathfrak{b}\right)-{\mathcal{H}}^2\delta \left(\mathfrak{b}\right)|,\hfill \\ \hfill & |\xi -\mathcal{H}\xi \left(\mathfrak{b}\right)|,|\mathcal{H}\xi \left(\mathfrak{b}\right)-{\mathcal{H}}^2\xi \left(\mathfrak{b}\right)\left|\right\}.\hfill \end{array}$$

∀$\mathfrak{a}\in [0,\Omega ]$,

$$\begin{array}{cc}\hfill \partial ({\mathcal{H}}^2\delta \left(\mathfrak{a}\right),{\mathcal{H}}^2\xi \left(\mathfrak{a}\right))& =\underset{\mathfrak{a}\in [0,\Omega ]}{max}|{\mathcal{H}}^2\delta \left(\mathfrak{a}\right)-{\mathcal{H}}^2\xi \left(\mathfrak{a}\right)|\hfill \\ \hfill & \le \underset{\mathfrak{a}\in [0,\Omega ]}{max}(imax\{|\delta \left(\mathfrak{b}\right)-\xi \left(\mathfrak{b}\right)|,|\mathcal{H}\delta \left(\mathfrak{b}\right)-\mathcal{H}\xi \left(\mathfrak{b}\right)|,\hfill \\ \hfill & |\delta \left(\mathfrak{b}\right)-\mathcal{H}\delta \left(\mathfrak{b}\right)|,|\mathcal{H}\delta \left(\mathfrak{b}\right)-{\mathcal{H}}^2\delta \left(\mathfrak{b}\right)|,\hfill \\ \hfill & |\xi -\mathcal{H}\xi \left(\mathfrak{b}\right)|,|\mathcal{H}\xi \left(\mathfrak{b}\right)-{\mathcal{H}}^2\xi \left(\mathfrak{b}\right)\left|\right\})\hfill \\ \hfill & \le imax[\underset{r\in [0,\Omega ]}{max}\left\{\right|\delta \left(r\right)-\xi \left(r\right)|,|\mathcal{H}\delta \left(r\right)-\mathcal{H}\xi \left(r\right)|,\hfill \\ \hfill & |\delta \left(r\right)-\mathcal{H}\delta \left(r\right)|,|\mathcal{H}\delta \left(r\right)-{\mathcal{H}}^2\delta \left(r\right)|,\hfill \\ \hfill & |\xi -\mathcal{H}\xi \left(r\right)|,|\mathcal{H}\xi \left(r\right)-{\mathcal{H}}^2\xi \left(r\right)\left|\right\}]\hfill \\ \hfill & =imax\{\partial (\delta ,\xi ),\partial (\mathcal{H}\delta ,\mathcal{H}\xi ),\partial (\delta ,\mathcal{H}\delta ),\hfill \\ \hfill & \partial (\mathcal{H}\delta ,{\mathcal{H}}^2\delta ),\partial (\xi ,\mathcal{H}\xi ),\partial (\mathcal{H}\xi ,{\mathcal{H}}^2\xi )\}.\hfill \end{array}$$

which implies that

$$\begin{array}{cc}\hfill & \alpha (\delta ,\xi )\partial ({\mathcal{H}}^2\delta ,{\mathcal{H}}^2\xi )\le imax\{\partial (\delta ,\xi ),\partial (\mathcal{H}\delta ,\mathcal{H}\xi ),\partial (\delta ,\mathcal{H}\delta ),\hfill \\ \hfill & \partial (\mathcal{H}\delta ,{\mathcal{H}}^2\delta ),\partial (\xi ,\mathcal{H}\xi ),\partial (\mathcal{H}\xi ,{\mathcal{H}}^2\xi )\}.\hfill \end{array}$$

Hence, all the conditions of Corollary 1 are fulfilled and so $\mathcal{H}$ has a unique solution. □

Example 4.

Consider an integro-differential equation with a fractional derivative

$$\begin{array}{c}\hfill {\mathcal{D}}^{\frac{3}{4}}\left(\mathfrak{a}\right)=\left({\displaystyle \frac{-{\mathfrak{a}}^2{e}^{\mathfrak{a}}}{5}}\right)\mathfrak{g}\left(\mathfrak{a}\right)+{\displaystyle \frac{6{\mathfrak{a}}^{2.25}}{\Gamma \left(3.25\right)}}+{\int }_0^{\mathfrak{a}}{e}^{\mathfrak{a}}\mathfrak{b}\mathfrak{g}\left(\mathfrak{b}\right)d\mathfrak{b},0\le \mathfrak{a}\le 1.\end{array}$$

subject to

$$\begin{array}{c}\hfill \mathfrak{g}\left(0\right)=0.\end{array}$$

Proof. 

For each $\mathfrak{a},\mathfrak{b}\in \Psi$ with $\mathfrak{a}⊩\mathfrak{b}$ and $\mathfrak{a}\in [0,1]$, we have

$$\begin{array}{c}\hfill {\mathcal{D}}^{\frac{3}{4}}\left(\mathfrak{a}\right)=\left({\displaystyle \frac{-{\mathfrak{a}}^2{e}^{\mathfrak{a}}}{5}}\right)\mathfrak{g}\left(\mathfrak{a}\right)+{\displaystyle \frac{6{\mathfrak{a}}^{2.25}}{\Gamma \left(3.25\right)}}+{\int }_0^{\mathfrak{a}}{e}^{\mathfrak{a}}\mathfrak{b}\mathfrak{g}\left(\mathfrak{b}\right)d\mathfrak{b}.\end{array}$$

The exact solution of the above equation is

$$\begin{array}{c}\hfill \mathfrak{g}\left(\mathfrak{b}\right)={\mathfrak{b}}^3.\end{array}$$

Here, $\mathfrak{g}\left(\mathfrak{a}\right)$ is an orthogonal continuous function on $[0,\mathfrak{a}]$ where $0\le \mathfrak{a}\le 1$. We calculated value of this equation, it is called the approximate solution:

$$\begin{array}{c}\hfill {\mathcal{D}}^{\frac{3}{4}}\left(\mathfrak{a}\right)=2.3536{\mathfrak{a}}^{2.25}.\end{array}$$

Table 1. Absolute error of the present method.

$\mathfrak{a},\mathfrak{b}$	${\mathcal{D}}^{\frac{3}{4}}\left(\mathfrak{a}\right)$	$\mathfrak{g}\left(\mathfrak{b}\right)$	$|{\mathcal{D}}^{\frac{3}{4}}\left(\mathfrak{a}\right)-\mathfrak{g}\left(\mathfrak{b}\right)|$
0.0	0.0000	0.000	0.0000
0.1	0.0132	0.001	0.0122
0.2	0.0629	0.004	0.0589
0.3	0.1568	0.009	0.2835
0.4	0.2995	0.016	0.2835
0.5	0.4948	0.025	0.4698
0.6	0.7457	0.036	0.7097
0.7	1.0549	0.049	1.0059
0.8	1.4246	0.064	1.3606
0.9	1.8569	0.081	1.7759

shows a comparison of the exact and approximate solutions. For the graph of the convergence between an approximate $\left({\mathcal{D}}^{\frac{3}{4}}\left(\mathfrak{a}\right)\right)$ and the exact solution with h = 0.1, see Figure 1. □

5. Application to Nonlinear Partial Volterra Integro-Differential Equation

An application of nonlinear partial Volterra integro-differential equation for a two-dimension with desired order:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}=\mathfrak{g}(\delta ,\xi )+{\int }_0^{\xi }{\int }_0^{\delta }\mathcal{K}\left(\delta ,\xi ,\mathfrak{b},\mathfrak{a},\frac{{\partial }^{\mathfrak{b},\mathfrak{a}}\mathfrak{f}(\mathfrak{b},\mathfrak{a})}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}\right)d\mathfrak{b}d\mathfrak{a},(\delta ,\xi )\in [0,1]\times [0,1]\hfill \\ \mathrm{Appropriate}\mathrm{intial}\mathrm{conditions},\hfill \end{array}\right.\end{array}$$

(21)

where $\mathcal{K}$ is an orthogonal continuous in $\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}$ with $\delta ⊩\xi$, $\mathfrak{g}(\delta ,\xi )$ is a known function, and $\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}$ is the calculated function.

Following the approximate solution ${\displaystyle \frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}}$, for all $0\le \delta ,\xi \le 1$. Consider the mesh nodes on the square as follows:

$$\begin{array}{cc}\hfill {\delta }_{\mathfrak{m}}& ={\displaystyle \frac{\mathfrak{m}-0.5}{2\mathcal{M}}},\mathfrak{m}=1,2,\dots .2\mathcal{M}.\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\xi }_{\mathfrak{n}}& ={\displaystyle \frac{\mathfrak{n}-0.5}{2\mathcal{N}}},\mathfrak{n}=1,2,\dots .2\mathcal{N}.\hfill \end{array}$$

(23)

The two-dimensional function $\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}$ is approximated with a two-dimensional Haar wavelet on $0\le \delta ,\xi \le 1$ as follows:

$$\begin{array}{c}\hfill \frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}=\sum _{\mathfrak{i}=1}^{2\mathcal{M}}\sum _{\mathfrak{j}=1}^{2\mathcal{N}}{\mathfrak{b}}_{\mathfrak{i},\mathfrak{j}}{\mathfrak{h}}_{\mathfrak{i}}\left(\delta \right){\mathfrak{h}}_{\mathfrak{j}}\left(\xi \right).\end{array}$$

(24)

By (21), (22) and (24) in (23), we get the linear equation $4\mathcal{M}\mathcal{N}\times 4\mathcal{M}\mathcal{N}$ with ${\mathfrak{b}}_{\mathfrak{i},\mathfrak{j}}$.

$$\begin{array}{c}\hfill \frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}{|}_{\xi ={\xi }_{\mathfrak{n}}}^{\delta ={\delta }_{\mathfrak{m}}}=\sum _{\mathfrak{i}=1}^{2\mathcal{M}}\sum _{\mathfrak{j}=1}^{2\mathcal{N}}{\mathfrak{b}}_{\mathfrak{i},\mathfrak{j}}{\mathfrak{h}}_{\mathfrak{i}}\left({\delta }_{\mathfrak{m}}\right){\mathfrak{h}}_{\mathfrak{j}}\left({\xi }_{\mathfrak{n}}\right),\mathfrak{m}=1,2,\dots .,2\mathcal{M},\mathfrak{n}=1,2,\dots .,2\mathcal{N}.\end{array}$$

Unknown coefficients of ${\mathfrak{b}}_{\mathfrak{i},\mathfrak{j}}$ are achieved using Theorem 2 in [16].

Theorem 4.

Suppose a function $F(\delta ,\xi )$ of two variables δ and ξ is approximated by the Haar wavelet approximate given as

$$\begin{array}{c}\hfill F(\delta ,\xi )=\sum _{\mathfrak{i}=1}^{2\mathcal{M}}\sum _{\mathfrak{j}=1}^{2\mathcal{N}}{\mathfrak{b}}_{\mathfrak{i},\mathfrak{j}}{\mathfrak{h}}_{\mathfrak{i}}\left(\delta \right){\mathfrak{h}}_{\mathfrak{j}}\left(\xi \right),\delta ⊩\xi .\end{array}$$

Suppose that $F(\delta ,\xi )$ is known at collocation points $({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}}),\mathfrak{m}=1,2,\dots \dots ,2\mathcal{M},\mathfrak{n}=1,2,\dots \dots ,2\mathcal{N}$, and F is orthogonal continuous. Then, the approximate solution of $F(\delta ,\xi )$ at any other point is as follows:

$$\begin{array}{cc}\hfill F(\delta ,\xi )& ={\displaystyle \frac{1}{2\mathcal{M}\times 2\mathcal{N}}}\sum _{\mathfrak{k}=1}^{2\mathcal{M}}\sum _{\mathfrak{p}=1}^{2\mathcal{N}}F({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}}){\mathfrak{h}}_{\mathfrak{1}}\left(\delta \right){\mathfrak{h}}_1\left(\xi \right)\hfill \\ & +\sum _{\mathfrak{i}=1}^{2\mathcal{M}}{\displaystyle \frac{1}{{\sigma }_2\times 2\mathcal{N}}}\left(\sum _{\mathfrak{k}={\alpha }_1}^{{\beta }_1}\sum _{\mathfrak{p}=1}^{2\mathcal{N}}F({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}})-\sum _{\mathfrak{k}={\beta }_1+1}^{{\gamma }_1}\sum _{\mathfrak{p}=1}^{2\mathcal{N}}F({\delta }_{\mathfrak{m}},\xi [\mathfrak{n}])\right){\mathfrak{h}}_{\mathfrak{i}}\left(\delta \right){\mathfrak{h}}_1\hfill \\ & +\sum _{\mathfrak{j}=1}^{2\mathcal{N}}{\displaystyle \frac{1}{2\mathcal{M}\times {\sigma }_2}}\left(\sum _{\mathfrak{k}=1}^{2\mathcal{M}}\sum _{\mathfrak{p}={\alpha }_2}^{{\beta }_2}F({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}})-\sum _{\mathfrak{k}=1}^{2\mathcal{M}}\sum _{\mathfrak{p}={\beta }_2+1}^{{\gamma }_2}F({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}})\right){\mathfrak{h}}_1\left(\delta \right){\mathfrak{h}}_{\mathfrak{j}}\left(\xi \right)\hfill \\ & +\sum _{\mathfrak{i}=1}^{2\mathcal{M}}\sum _{\mathfrak{j}=1}^{2\mathcal{N}}{\displaystyle \frac{1}{{\sigma }_1\times {\sigma }_2}}(\sum _{\mathfrak{k}={\alpha }_1}^{{\beta }_1}\sum _{\mathfrak{p}={\alpha }_2}^{{\beta }_2}F({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}})-\sum _{\mathfrak{k}={\alpha }_1}^{{\beta }_1}\sum _{\mathfrak{p}={\beta }_2+1}^{{\gamma }_2}F({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}})\hfill \\ & -\sum _{\mathfrak{k}={\beta }_1+1}^{{\gamma }_1}\sum _{\mathfrak{p}={\alpha }_2}^{{\beta }_2}F({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}})+\sum _{\mathfrak{k}={\beta }_1+1}^{{\gamma }_1}\sum _{\mathfrak{p}={\beta }_2+1}^{{\gamma }_2}F({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}})){\mathfrak{h}}_{\mathfrak{i}}\left(\delta \right){\mathfrak{h}}_{\mathfrak{j}}\left(\xi \right)\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\alpha }_1& ={\alpha }_1({\mu }_1-1)+1,\hfill \\ \hfill {\beta }_1& ={\sigma }_1({\mu }_1-1)+\frac{{\sigma }_1}{2},\hfill \\ \hfill {\gamma }_1& ={\sigma }_1{\mu }_1,\hfill \\ \hfill {\sigma }_1& ={\displaystyle \frac{2\mathcal{M}}{{\nu }_1}},\hfill \\ \hfill {\mu }_1& =\mathfrak{i}-{\nu }_1,\hfill \\ \hfill {\nu }_1& =2^{\lfloor {log}_2(\mathfrak{i}-1)\rfloor }.\hfill \end{array}$$

(25)

And similarly,

$$\begin{array}{cc}\hfill {\alpha }_2& ={\sigma }_2({\mu }_2-1)+1,\hfill \\ \hfill {\beta }_2& ={\sigma }_2({\mu }_2-1)+{\displaystyle \frac{{\sigma }_2}{2}},\hfill \\ \hfill {\gamma }_2& ={\sigma }_2{\mu }_2,\hfill \\ \hfill {\sigma }_2& ={\displaystyle \frac{2\mathcal{N}}{{\nu }_2}},\hfill \\ \hfill {\mu }_2& =\mathfrak{i}-{\nu }_2,\hfill \\ \hfill {\nu }_2& =2^{\lfloor {log}_2(\mathfrak{j}-1)\rfloor }\hfill \end{array}$$

(26)

First, the Kernel of (21) is orthogonal continuous and the Haar wavelet of the two-dimensional approximate solution is as follows:

$$\begin{array}{c}\hfill \mathcal{K}\left(\delta ,\xi ,\mathfrak{b},\mathfrak{a},\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\mathfrak{b},\mathfrak{a})}{\partial {\delta }^{\mathfrak{v}\partial {\xi }^{\mathfrak{w}}}}\right)\approx \sum _{\mathfrak{i}=1}^{2\mathcal{M}}\sum _{\mathfrak{j}=1}^{2\mathcal{N}}{\mathfrak{b}}_{\mathfrak{i},\mathfrak{j}}(\delta ,\xi ){\mathfrak{h}}_{\mathfrak{i}}\left(\mathfrak{b}\right){\mathfrak{h}}_{\mathfrak{j}}\left(\mathfrak{a}\right).\end{array}$$

From the above, by (21), we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}}=\mathfrak{g}(\delta ,\xi )+{\int }_0^{\xi }{\int }_0^{\delta }\sum _{\mathfrak{i}=1}^{2\mathcal{M}}\sum _{\mathfrak{j}=1}^{2\mathcal{N}}{\mathfrak{b}}_{\mathfrak{i},\mathfrak{j}}(\delta ,\xi ){\mathfrak{h}}_{\mathfrak{i}}\left(\mathfrak{b}\right){\mathfrak{h}}_{\mathfrak{j}}\left(\mathfrak{a}\right)d\mathfrak{b}d\mathfrak{a}.\end{array}$$

By the Haar wavelet properties, in the following equation, we get

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}}=\mathfrak{g}(\delta ,\xi )+\sum _{\mathfrak{i}=1}^{2\mathcal{M}}\sum _{\mathfrak{j}=1}^{2\mathcal{N}}{\mathfrak{b}}_{\mathfrak{i},\mathfrak{j}}(\delta ,\xi ){\mathfrak{k}}_{\mathfrak{i},1}\left(\delta \right){\mathfrak{k}}_{\mathfrak{j},1}\left(\xi \right).\end{array}$$

(27)

Now, collocation points ${\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}}$ are applied in Equation (27), to get

$$\begin{array}{c}\hfill \frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}{|}_{\xi ={\xi }_{\mathfrak{n}}}^{\delta ={\delta }_{\mathfrak{m}}}=\mathfrak{g}({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}})+\sum _{\mathfrak{i}=1}^{2\mathcal{M}}\sum _{\mathfrak{j}=1}^{2\mathcal{N}}{\mathfrak{b}}_{\mathfrak{i},\mathfrak{j}}({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}}){\mathfrak{k}}_{\mathfrak{i},1}\left({\delta }_{\mathfrak{m}}\right){\mathfrak{k}}_{\mathfrak{j},1}\left({\xi }_{\mathfrak{n}}\right),\end{array}$$

(28)

$\mathfrak{m}=1,2,\dots ,2\mathcal{M},\mathfrak{n}=1,2,\dots ,2\mathcal{N}$.

The values ${\mathfrak{b}}_{\mathfrak{i},\mathfrak{j}}(\delta ,\xi )$ satisfying Theorem 1 and applied in Equation (28) reach the following equations.

$$\begin{array}{cc}\hfill \frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}{|}_{\xi ={\xi }_{\mathfrak{n}}}^{\delta ={\delta }_{\mathfrak{m}}}& =\mathfrak{g}({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}})+\frac{{\mathfrak{k}}_{1,1}\left({\delta }_{\mathfrak{m}}\right){\mathfrak{k}}_{1,1}\left({\xi }_{\mathfrak{n}}\right)}{2\mathcal{M}\times 2\mathcal{N}}\sum _{\mathfrak{k}=1}^{2\mathcal{M}}\sum _{\mathfrak{p}=1}^{2\mathcal{N}}\mathcal{K}({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}},{\mathfrak{b}}_{\mathfrak{k}},{\mathfrak{a}}_{\mathfrak{p}},\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}{|}_{\xi ={\mathfrak{a}}_{\mathfrak{p}}}^{\delta ={\mathfrak{b}}_{\mathfrak{k}}})\hfill \\ & +\sum _{\mathfrak{i}=2}^{2\mathcal{M}}\frac{{\mathfrak{k}}_{\mathfrak{i},1}\left({\delta }_{\mathfrak{m}}\right){\mathfrak{k}}_{1,1}\left({\xi }_{\mathfrak{n}}\right)}{{\sigma }_1\times 2\mathcal{N}}\left(\sum _{\mathfrak{k}={\alpha }_1}^{{\beta }_1}\sum _{\mathfrak{p}=1}^{2\mathcal{N}}\mathcal{K}\right({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}},{\mathfrak{b}}_{\mathfrak{k}},{\mathfrak{a}}_{\mathfrak{p}},\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}{|}_{\xi ={\mathfrak{a}}_{\mathfrak{p}}}^{\delta ={\mathfrak{b}}_{\mathfrak{k}}})\hfill \\ & -\sum _{\mathfrak{k}={\beta }_1+1}^{{\gamma }_1}\sum _{\mathfrak{p}=1}^{2\mathcal{N}}\mathcal{K}({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}},{\mathfrak{b}}_{\mathfrak{k}},{\mathfrak{a}}_{\mathfrak{p}},\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}{|}_{\xi ={\mathfrak{a}}_{\mathfrak{p}}}^{\delta ={\mathfrak{b}}_{\mathfrak{k}}}))\hfill \\ & +\sum _{\mathfrak{j}=2}^{2\mathcal{N}}\frac{{\mathfrak{k}}_{1,1}\left({\delta }_{\mathfrak{m}}\right){\mathfrak{k}}_{\mathfrak{j},1}\left({\xi }_{\mathfrak{n}}\right)}{2\mathcal{M}\times {\sigma }_2}\left(\sum _{\mathfrak{k}=1}^{2\mathcal{M}}\sum _{\mathfrak{p}={\alpha }_2}^{{\beta }_2}\mathcal{K}\right({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}},{\mathfrak{b}}_{\mathfrak{k}},{\mathfrak{a}}_{\mathfrak{p}},\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}{|}_{\xi ={\mathfrak{a}}_{\mathfrak{p}}}^{\delta ={\mathfrak{b}}_{\mathfrak{k}}})\hfill \\ & -\sum _{\mathfrak{k}=1}^{2\mathcal{M}}\sum _{\mathfrak{p}={\beta }_2+1}^{{\gamma }_2}\mathcal{K}({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}},{\mathfrak{b}}_{\mathfrak{k}},{\mathfrak{a}}_{\mathfrak{p}},\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}{|}_{\xi ={\mathfrak{a}}_{\mathfrak{p}}}^{\delta ={\mathfrak{b}}_{\mathfrak{k}}}))\hfill \\ & +\sum _{\mathfrak{i}=2}^{2\mathcal{M}}\sum _{\mathfrak{j}=2}^{2\mathcal{N}}\frac{{\mathfrak{k}}_{\mathfrak{i},1}\left({\delta }_{\mathfrak{m}}\right){\mathfrak{k}}_{\mathfrak{j},1}\left({\xi }_{\mathfrak{n}}\right)}{{\sigma }_1\times {\sigma }_2}\left(\sum _{\mathfrak{k}={\alpha }_1}^{{\beta }_1}\sum _{\mathfrak{p}={\alpha }_2}^{{\beta }_2}\mathcal{K}\right({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}},{\mathfrak{b}}_{\mathfrak{k}},{\mathfrak{a}}_{\mathfrak{p}},\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}{|}_{\xi ={\mathfrak{a}}_{\mathfrak{p}}}^{\delta ={\mathfrak{b}}_{\mathfrak{k}}})\hfill \\ & -\sum _{\mathfrak{k}={\alpha }_1}^{{\beta }_1}\sum _{\mathfrak{p}={\beta }_2+1}^{{\gamma }_2}\mathcal{K}({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}},{\mathfrak{b}}_{\mathfrak{k}},{\mathfrak{a}}_{\mathfrak{p}},\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}{|}_{\xi ={\mathfrak{a}}_{\mathfrak{p}}}^{\delta ={\mathfrak{b}}_{\mathfrak{k}}})\hfill \\ & -\sum _{\mathfrak{k}={\beta }_1+1}^{{\gamma }_1}\sum _{\mathfrak{p}={\alpha }_2}^{{\beta }_2}\mathcal{K}({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}},{\mathfrak{b}}_{\mathfrak{k}},{\mathfrak{a}}_{\mathfrak{q}},\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}{|}_{\xi ={\mathfrak{a}}_{\mathfrak{q}}}^{\delta ={\mathfrak{b}}_{\mathfrak{k}}})\hfill \\ & +\sum _{\mathfrak{k}={\beta }_1+1}^{{\gamma }_1}\sum _{\mathfrak{q}={\beta }_2+1}^{{\gamma }_2}\mathcal{K}({\delta }_{\mathfrak{m}},{\xi }_{\mathfrak{n}},{\mathfrak{b}}_{\mathfrak{k}},{\mathfrak{a}}_{\mathfrak{q}},\frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}{|}_{\xi ={\mathfrak{a}}_{\mathfrak{q}}}^{\delta ={\mathfrak{b}}_{\mathfrak{k}}})),\hfill \end{array}$$

where $\mathfrak{m}=1,2,\dots ,2\mathcal{M},\mathfrak{n}=1,2,\dots ,2\mathcal{N}$.

Equation (28) is a $4\mathcal{M}\mathcal{N}\times 4\mathcal{M}\mathcal{N}$ nonlinear system solved by either Broyden or Newton methods. The solution of the system gives ${\displaystyle \frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}}$ at the collocation points. Theorem 4 may be used to determine the value of ${\displaystyle \frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }^{\mathfrak{v}}\partial {\xi }^{\mathfrak{w}}}}$ at points other than collocation sites. One of the methods for partial differential equations can be used to solve the equation

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^{\mathfrak{v}+\mathfrak{w}}\mathfrak{f}(\delta ,\xi )}{\partial {\delta }_{\mathfrak{m}}^{\mathfrak{v}}\partial {\xi }_{\mathfrak{n}}^{\mathfrak{w}}}}=\mathcal{A}(\delta ,\xi ).\end{array}$$

Example 5.

Consider the following partial integro-differential equation:

$$\begin{array}{c}\hfill \frac{{\partial }^2\mathfrak{f}(\delta ,\xi )}{\partial \delta \partial \xi }=\mathfrak{g}(\delta ,\xi )+{\int }_0^{\xi }{\int }_0^{\delta }\left(\frac{{\partial }^2\mathfrak{f}(\mathfrak{b},\mathfrak{a})}{\partial \mathfrak{b}\partial \mathfrak{a}}+2\delta \xi {\left(\frac{{\partial }^2\mathfrak{f}(\mathfrak{b},\mathfrak{a})}{\partial \mathfrak{b}\partial \mathfrak{a}}\right)}^3+{\xi }^2{\left(\frac{{\partial }^2\mathfrak{f}(\mathfrak{b},\mathfrak{a})}{\partial \mathfrak{b}\partial \mathfrak{a}}\right)}^5\right)d\mathfrak{b}d\mathfrak{a},\end{array}$$

where

$$\begin{array}{c}\hfill \mathfrak{g}(\delta ,\xi )=e^{\xi }-(-1+e^{\xi })\delta -\frac{1}{5}(-1+e^{5\xi })\delta {\xi }^2-\frac{2}{3}(-1+e^{3\xi }){\delta }^2\xi .\end{array}$$

The exact solution of this problem is

$$\begin{array}{c}\hfill \mathfrak{f}(\delta ,\xi )=\delta e^{\xi }.\end{array}$$

And $\mathfrak{f}(\delta ,\xi )$ is orthogonal continuous on $[0,1]$, and supplementary conditions are

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathfrak{f}(\delta ,0)}{\partial \delta }}=1,\mathfrak{f}(0,0)=0.\end{array}$$

Graphs of the comparisons between the approximation and exact solutions with different h values are given in Figure 2, Figure 3, Figure 4 and Figure 5.

6. Conclusions

In this paper, we introduced the new orthogonal concept of $\alpha$–F–convex contraction mappings on O-CMS and proved some related fixed-point results. We have discussed a numerical experiment for approximating the fixed-point results. Finally, we obtained a simple and efficient solution for the fractional integro-differential equation by using the fixed-point technique in the setting of O-CMS.