New Extension of Darbo’s Fixed Point Theorem and Its Application to a System of Weighted-Fractional-Type Integral Equations

Abstract

In this article, we introduce several new extensions of Darbo’s fixed point theorem with newly constructed contraction functions associated with the measure of noncompactness. We apply our new extensions to prove the existence of solutions for a system of weighted fractional integral equations in Banach space $BC\left({\mathbb{R}}_{+}\right)$. At the end, we establish an example to show the applicability of our discovery.

1. Introduction

A lot of problems in real life can be described or characterized using fractional calculus. Fractional calculus has numerous and crucial applications in many fields of science and engineering. In the study of nonlinear analysis, fractional integral equations ($\mathbb{F}\mathbb{I}\mathbb{E}$) are highly helpful. Researchers can use the fixed point theory ($\mathbb{F}\mathbb{P}\mathbb{T}$) to investigate whether solutions to various kinds of fractional integral equations exist. One of the key ideas in $\mathbb{F}\mathbb{P}\mathbb{T}$ is the measure of noncompactness ($\mathbb{M}\mathbb{N}\mathbb{C}$). Kuratowski [1] originally introduced the concept of $\mathbb{M}\mathbb{N}\mathbb{C}$ in 1930, and it is highly helpful in $\mathbb{F}\mathbb{P}\mathbb{T}$. In 1955, Darbo [2] finished an expansion of Schauder’s fixed point theorem with the help of Kuratowski’s $\mathbb{M}\mathbb{N}\mathbb{C}$. Numerous authors have developed different kinds of new research papers incorporating $\mathbb{F}\mathbb{P}\mathbb{T}$ and $\mathbb{M}\mathbb{N}\mathbb{C}$, inspired by the work of Darbo (see [3,4,5]). Mathematicians have lately investigated the solvability of several $\mathbb{F}\mathbb{I}\mathbb{E}$ in different spaces using $\mathbb{M}\mathbb{N}\mathbb{C}$ (see [6,7,8,9]).

In this paper, we examine the existence of solutions of the following system of $\mathbb{F}\mathbb{I}\mathbb{E}$ (1) and (2) involving a weighted fractional integral operator with the help of generalized $\mathbb{D}\mathbb{F}\mathbb{P}$ in $BC\left({\mathbb{R}}_{+}\right)$.

$$\xi \left(\kappa \right)=\Phi (\kappa ,\xi \left(\kappa \right))+{\displaystyle \frac{w^{-1}\left(\xi \right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell$$

(1)

and

$$\zeta \left(\kappa \right)=\widehat{\Phi }(\kappa ,\zeta \left(\kappa \right))+{\displaystyle \frac{w^{-1}\left(\zeta \right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )\widehat{Ł}(\kappa ,\ell ,\zeta (\ell ))d\ell$$

(2)

where $\sigma >0$; $\kappa \in {\mathbb{R}}_{+}$; $\Phi ,\widehat{\Phi }$ are operators from ${\mathbb{R}}_{+}\times \mathbb{R}$ to $\mathbb{R}$; $Ł,\widehat{Ł}$ are operators from ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times \mathbb{R}$ to $\mathbb{R}$; $\mathcal{Z}$ is a function from ${\mathbb{R}}_{+}$ to $\mathbb{R}$; and $\Gamma (,)$ is known as Euler’s Gamma operator, which is defined as follows:

$$\Gamma \left(\sigma \right)={\int }_0^{\infty }{s}^{\sigma -1}{e}^{-s}ds.$$

Also, w is known as a weighted function, and the weighted fractional integral of an operator g with respect to another operator h of the $n^{th}$ order $(n\in \mathbb{N})$ can be written as

$$\begin{array}{cc}\hfill {(}_{c^{+}}{\mathbb{J}}_w^{n}g)\left(l\right)& =w^{-1}\left(l\right){\int }_c^l{h}^{\prime }\left({\nu }_1\right)d{\nu }_1{\int }_c^{{\nu }_1}{h}^{\prime }\left({\nu }_2\right)d{\nu }_2{\int }_c^{{\nu }_{n-1}}w\left({\nu }_n\right)g\left({\nu }_n\right)h^{\prime }\left({\nu }_n\right)d{\nu }_n\hfill \\ \hfill & ={\displaystyle \frac{w^{-1}\left(l\right)}{(n-1)!}}{\int }_c^l{(h\left(r\right)-h\left(\nu \right))}^{n-1}w\left(\nu \right)g\left(\nu \right)h^{\prime }\left(\nu \right)d\nu ,l>c\hfill \end{array}$$

(3)

where $w\left(r\right)\ne 0$ is the weighted function $w^{-1}\left(r\right)=\frac{1}{w\left(r\right)}$, and h is a strictly increasing differentiable function. Then, the corresponding derivatives are

$$\begin{array}{c}\hfill \left({\mathbb{D}}_w^{1}g\right)\left(r\right)=w^{-1}\left(r\right){\displaystyle \frac{D_r}{h^{\prime }\left(r\right)}}\left(w\left(r\right)g\left(r\right)\right);\left({\mathbb{D}}_w^{n}g\right)\left(r\right)=\left({\mathbb{D}}_w^1\left({\mathbb{D}}_w^{n-1}g\right)\right)\left(r\right),\end{array}$$

(4)

where $D_r=\frac{d}{dr}$. The fractional forms of Equations (3) and (4) are

$${(}_{c^{+}}{\mathbb{J}}_w^{b_0}g)\left(r\right)={\displaystyle \frac{w^{-1}\left(r\right)}{\Gamma \left(b_0\right)}}{\int }_c^r{(h\left(r\right)-h\left(\nu \right))}^{b_0-1}w\left(\nu \right)g\left(\nu \right)h^{\prime }\left(\nu \right)d\nu ,r>c,b_0>0$$

(5)

and

$$\begin{array}{cc}\hfill {(}_{c^{+}}{\mathbb{D}}_w^{b_0}g)\left(r\right)& =\left({\mathbb{D}}_w^n{\mathbb{J}}_w^{n-b_0}g\right)\left(r\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\Gamma (n-b_0)}}{\mathbb{D}}_w^n\left({\int }_c^r{(h\left(r\right)-h\left(\nu \right))}^{n-b_0-1}\times w\left(\nu \right)g\left(\nu \right)h^{\prime }\left(\nu \right)d\nu \right),r>c,b_0>0,\hfill \end{array}$$

(6)

respectively, where $n=\left[b_0\right]+1$ and $\left[b_0\right]$ is the integer part of $b_0$ (see [10]).

2. Auxiliary Facts and Notations

Let $\left(\mho ,\Vert .\Vert \right)$ be a real Banach space and $B(ϵ,u_0)=\left\{s\in \mho :\Vert s-ϵ\Vert \le u_0\right\}.$

Let $\Lambda (\ne Ø)\subset \mho$. Assume that

• the set of all bounded and non-empty subsets of ℧ is taken as $B_{\mho }$ and the set of all non-empty and relatively compact subsets of ℧ is taken as $R{C}_{\mho }$;
• $\overline{\Lambda }$ and $Conv\Lambda$ are the closure and the convex closure of $\Lambda$, respectively;
• $\mathbb{R}=(-\infty ,\infty )$; and
• ${\mathbb{R}}_{+}=\left[0,\infty \right).$

• Recall the following definitions as follows:

Definition 1 

([11]).  Let ${{\rm Y}}_{\gamma }$ be the family of all operators $\gamma :{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to \mathbb{R}$, which satisfy the following:

(i)

$\gamma$ is non-decreasing and continuous;

(ii)

${lim}_{t\to \infty }\gamma \left({\iota }_t\right)=0$ if and only if ${lim}_{t\to \infty }{\iota }_t=0$ for each sequence $\left\{{\iota }_t\right\}\subset (0,\infty )$.

For example, $\gamma \left({\iota }_0\right)={\iota }_0$ for all ${\iota }_0\in {\mathbb{R}}_{+}$ is an element of ${{\rm Y}}_{\gamma }$.

Definition 2. 

Let $\mathcal{O}:{\mathbb{R}}^n\to {\mathbb{R}}_{+}$ be an operator which satisfies the following:

$$\mathcal{O}({\iota }_1^{\prime },{\iota }_2^{\prime },\dots \dots ,{\iota }_n^{\prime })\le max\{{\iota }_1^{\prime },{\iota }_2^{\prime },\dots \dots ,{\iota }_n^{\prime }\}.$$

We denote this class of operators by $\overline{\mathcal{O}}.$

We can see the following examples:

(1)

$\mathcal{O}({\iota }_1^{\prime },{\iota }_2^{\prime },\dots \dots ,{\iota }_n^{\prime })=max\{{\iota }_1^{\prime },{\iota }_2^{\prime },\dots \dots ,{\iota }_n^{\prime }\},$

(2)

$\mathcal{O}({\iota }_1^{\prime },{\iota }_2^{\prime },\dots \dots ,{\iota }_n^{\prime })=\frac{1}{n}\{{\iota }_1^{\prime }+{\iota }_2^{\prime }+\dots \dots +{\iota }_n^{\prime }\};{\iota }_1^{\prime },{\iota }_2^{\prime },\cdots ,{\iota }_n^{\prime }\in \mathbb{R}.$

Recall the concept of a measure of noncompactness as follows:

Definition 3 

([12]).  $\pi :B_{\mho }\to {\mathbb{R}}_{+}$ is defined as an $\mathbb{M}\mathbb{N}\mathbb{C}$ in ℧ if it holds the axioms given below:

(i)

$\pi (\Lambda )=0$ yields that $\Lambda$ is relatively compact: $\forall \Lambda \in B_{\mho }$;

(ii)

ker $\pi =\left\{\Lambda \in B_{\mho }:\pi \left(\Lambda \right)=0\right\}\ne Ø$ and ker $\pi \subset R{C}_{\mho }$;

(iii)

$\Lambda \subseteq {\Lambda }_1\Rightarrow \pi \left(\Lambda \right)\le \pi \left({\Lambda }_1\right)$;

(iv)

$\pi \left(\overline{\Lambda }\right)=\pi \left(\Lambda \right)$;

(v)

$\pi \left(Conv\Lambda \right)=\pi \left(\Lambda \right)$;

(vi)

$\pi \left(\varrho \Lambda +\left(1-\varrho \right){\Lambda }_1\right)\le \varrho \pi \left(\Lambda \right)+\left(1-\varrho \right)\pi \left({\Lambda }_1\right)$ for all $\varrho \in \left[0,1\right]$;

(vii)

if ${\Lambda }_s\in B_{\mho },{\Lambda }_s={\overline{\Lambda }}_s,{\Lambda }_{s+1}\subset {\Lambda }_s$ for $s=1,2,3,4,\dots$ and $\underset{s\to \infty }{lim}\pi \left({\Lambda }_s\right)=0$,

then ${\bigcap }_{s=1}^{\infty }{\Lambda }_s\ne Ø.$

The family $ker\pi$ is known as the kernel of measure $\pi$. Since $\pi \left({\Lambda }_{\infty }\right)\le \pi \left({\Lambda }_s\right)$ for any $s,$ we conclude that $\pi \left({\Lambda }_{\infty }\right)=0.$ Then ${\Lambda }_{\infty }={\bigcap }_{s=1}^{\infty }{\Lambda }_s\in ker\pi .$

Definition 4 

([13]).  Let ${\Psi }_{\psi }$ be the family of all operators $\psi :{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to \mathbb{R}$, which satisfy the following:

(i)

$\psi$ is non-decreasing and continuous;

(ii)

$max\{s,{\iota }^{\prime }\}\le \psi (s,{\iota }^{\prime })$ for $s,{\iota }^{\prime }\ge 0$;

(iii)

$\psi ({\iota }_1+{\iota }_2,{\iota }_1^{\prime }+{\iota }_2^{\prime })\le \psi ({\iota }_1,{\iota }_1^{\prime })+\psi ({\iota }_2,{\iota }_2^{\prime }).$

For example, $\psi (s,{\iota }^{\prime })=s+{\iota }^{\prime }$, where $s,{\iota }^{\prime }\in {\mathbb{R}}_{+}$ is a element of ${\Psi }_{\psi }$.

We recall several fixed point theorems as follows:

Theorem 1 

([14]).  (Schauder’s fixed point theorem) Taking $\mathcal{D}$ as a non-empty, bounded, closed and convex subset ($\mathbb{NBCC}$) of a Banach space $\mho ,$ and $\mathcal{I}:\mathcal{D}\to \mathcal{D}$ as a continuous and compact mapping, then $\mathcal{I}$ has at least one fixed point.

Theorem 2 

([2]).  Assume that $\mathcal{D}$ is an $\mathbb{NBCC}$ of a Banach space ℧ and let $\mathcal{I}:\mathbb{D}\to \mathcal{D}$ be a continuous mapping. Taking a positive number $\varrho \in \left[0,1\right)$, so that

$$\pi \left(\mathbb{IE}\right)\le \varrho \pi \left(\mathbb{E}\right),\mathbb{E}\subseteq \mathcal{D},$$

then $\mathcal{I}$ has a fixed point in $\mathcal{D}$.

Definition 5 

([15]).  Let $\overline{\mathcal{U}}$ be the family of all operators $\mu :{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to \mathbb{R}$, which satisfy the following:

(i)

$\mu$ is continuous and monotone-increasing;

(ii)

$\underset{t\to \infty }{lim}{\mu }^t\left(s\right)=0,\mathrm{forall}s>0.$

For example, $\mu \left({\iota }^{\prime }\right)=\hslash ·{\iota }^{\prime }$ for all ${\iota }^{\prime }\in {\mathbb{R}}_{+}$, where $\hslash \in [0,1)$ belongs to $\overline{\mathcal{U}}$.

3. New Extensions of Darbo’s Fixed Point Theorem

In this section, we introduce several new extensions of Darbo’s fixed point theorem. We start this section with the following theorem.

Theorem 3. 

Let $\mathcal{D}$ be an $\mathbb{NBCC}$ set of ℧ and ${\mathcal{I}}_1,{\mathcal{I}}_2:\mathcal{D}\to \mathcal{D}$ be continuous operators, so that

$$\gamma \left[\psi \left(\pi \left({\mathcal{I}}_1^r\mathbb{E}\right),\alpha \left(\pi \left({\mathcal{I}}_2^r\mathbb{F}\right)\right)\right)\right]\le \mu \left[\gamma \left[\psi \left({\mathbb{T}}_{r-1}\left(\mathbb{E}\right),\alpha \left({\mathbb{T}}_{r-1}^{\prime }\left(\mathbb{F}\right)\right)\right)\right]\right],$$

(7)

where

$${\mathbb{T}}_{r-1}\left(\mathbb{E}\right)=\mathcal{O}(\pi \left(\mathbb{E}\right),\pi \left({\mathcal{I}}_1\mathbb{E}\right),\dots ,\pi \left({\mathcal{I}}_1^{r-1}\mathbb{E}\right))$$

and

$${\mathbb{T}}_{r-1}^{\prime }\left(\mathbb{F}\right)=\mathcal{O}(\pi \left(\mathbb{F}\right),\pi \left({\mathcal{I}}_2\mathbb{F}\right),\dots ,\pi \left({\mathcal{I}}_2^{r-1}\mathbb{F}\right))$$

for all $\mathbb{E},\mathbb{F}(\ne Ø)\subseteq \mathcal{D}$; π is an arbitrary $\mathbb{M}\mathbb{N}\mathbb{C}$, $\gamma \in {{\rm Y}}_{\gamma },$ $\mathcal{O}\in \overline{\mathcal{O}},$ $\psi \in {\Psi }_{\psi },$ $\mu \in \overline{\mathcal{U}}$, and $\alpha :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a non-decreasing and continuous mapping. Then, both ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$ have fixed points in $\mathcal{D}.$

Proof. 

Take ${\mathcal{J}}_0={\mathcal{L}}_0=\mathcal{D}$, ${\mathcal{J}}_1=\overline{conv\left({\mathcal{I}}_1{\mathcal{J}}_0\right)}$,..., ${\mathcal{J}}_{m-1}=\overline{conv\left({\mathcal{I}}_1^{m-1}{\mathcal{J}}_0\right)}$ and ${\mathcal{J}}_{n+m}=\overline{conv\left({\mathcal{I}}_1^m{\mathcal{J}}_n\right)}$ for all $n=0,1,2,\dots$.

Also, take ${\mathcal{J}}_0={\mathcal{L}}_0=\mathcal{D}$, ${\mathcal{L}}_1=\overline{conv\left({\mathcal{I}}_2{\mathcal{L}}_0\right)}$,..., ${\mathcal{L}}_{m-1}=\overline{conv\left({\mathcal{I}}_2^{m-1}{\mathcal{L}}_0\right)}$ and ${\mathcal{L}}_{n+m}=\overline{conv\left({\mathcal{I}}_2^m{\mathcal{L}}_n\right)}$ for all $n=0,1,2,\dots$.

Then, ${\left\{{\mathcal{J}}_g\right\}}_{g\in \mathbb{N}}$ and ${\left\{{\mathcal{L}}_g\right\}}_{g\in \mathbb{N}}$ are sequences of $\mathcal{N}.\mathbb{B}.\mathbb{C}.\mathbb{C}.S.s$, such that

$${\mathcal{J}}_0\supseteq {\mathcal{J}}_1\supseteq \cdots \supseteq {\mathcal{J}}_g\supseteq \cdots \supseteq {\mathcal{J}}_{g+r}\supseteq \cdots ,$$

and

$${\mathcal{L}}_0\supseteq {\mathcal{L}}_1\supseteq \cdots \supseteq {\mathcal{L}}_g\supseteq \cdots \supseteq {\mathcal{L}}_{g+r}\supseteq \cdots .$$

If $g_0\in \mathbb{N}$ is an integer such that $\pi \left({\mathcal{J}}_{g_0}\right)=\pi \left({\mathcal{L}}_{g_0}\right)=0$, then ${\mathcal{J}}_{g_0}$ and ${\mathcal{L}}_{g_0}$ are relatively compact. Thus, from Schauder’s theorem, it can be said that ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$ have a fixed point.

So, we can take $\pi \left({\mathcal{J}}_g\right)>0$ and $\pi \left({\mathcal{L}}_g\right)>0$ for all $g\in \mathbb{N}\cup \left\{0\right\}$.

Since $\pi$ is monotone, hence the sequences $\left\{\pi \left({\mathcal{J}}_g\right)\right\}$ and $\left\{\pi \left({\mathcal{L}}_g\right)\right\}$ are non-negative and non-increasing, and we can deduce that

$$\underset{g\to \infty }{lim}\pi \left({\mathcal{J}}_g\right)=\underset{g\to \infty }{lim}\pi \left({\mathcal{J}}_{g+r}\right)=w_1,$$

$$\underset{g\to \infty }{lim}\pi \left({\mathcal{L}}_g\right)=\underset{g\to \infty }{lim}\pi \left({\mathcal{L}}_{g+r}\right)=w_2,$$

where $w_1,w_2\ge 0$ are real numbers.

Now, from (7),

$$\begin{array}{cc}\hfill \gamma \left[\psi \left(\pi \left({\mathcal{J}}_{g+r}\right),\alpha \left(\pi \left({\mathcal{L}}_{g+r}\right)\right)\right)\right]& =\gamma \left[\psi \left(\pi \left(\overline{Conv{\mathcal{I}}_1^r{\mathcal{J}}_g}\right),\alpha (\pi \left(\overline{Conv{\mathcal{I}}_2^r{\mathcal{L}}_g)}\right)\right)\right]\hfill \\ \hfill & =\gamma \left[\psi \left(\pi \left({\mathcal{I}}_1^r{\mathcal{J}}_g\right),\alpha \left(\pi \left({\mathcal{I}}_2^r{\mathcal{L}}_g\right)\right)\right)\right]\hfill \\ \hfill & \le \mu \left[\gamma \left[\psi \left({\mathcal{T}}_{r-1}\left({\mathcal{J}}_g\right),\alpha \left({\mathcal{T}}_{r-1}^{\prime }\left({\mathcal{L}}_g\right)\right)\right)\right]\right]\hfill \\ \hfill & \le \mu \left[\gamma \left[\psi \left(\pi \left({\mathcal{J}}_g\right),\alpha \left(\pi \left({\mathcal{L}}_g\right)\right)\right)\right]\right]\hfill \end{array}$$

(8)

• for $g=0,1,2,\dots$,
• where

$$\begin{array}{cc}\hfill {\mathcal{T}}_{r-1}\left({\mathcal{J}}_g\right)& =\mathcal{O}\{\pi \left({\mathcal{J}}_g\right),\pi \left({\mathcal{J}}_{g+1}\right),\cdots ,\pi \left({\mathcal{J}}_{g+r-1}\right)\}\hfill \\ \hfill & \le max\{\pi \left({\mathcal{J}}_g\right),\pi \left({\mathcal{J}}_{g+1}\right),\cdots ,\pi \left({\mathcal{J}}_{g+r-1}\right)\}\le \pi \left({\mathcal{J}}_g\right),\hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\hfill {\mathcal{T}}_{r-1}^{\prime }\left({\mathcal{L}}_g\right)& =\mathcal{O}\{\pi \left({\mathcal{L}}_g\right),\pi \left({\mathcal{L}}_{g+1}\right),\cdots ,\pi \left({\mathcal{L}}_{g+r-1}\right)\}\hfill \\ \hfill & \le max\{\pi \left({\mathcal{L}}_g\right),\pi \left({\mathcal{L}}_{g+1}\right),\cdots ,\pi \left({\mathcal{L}}_{g+r-1}\right)\}\le \pi \left({\mathcal{L}}_g\right).\hfill \end{array}$$

(10)

Next, using Equation (8), we have

$$\begin{array}{cc}\hfill \gamma \left[\psi \left(\pi \left({\mathcal{J}}_{g+r}\right),\alpha \left(\pi \left({\mathcal{L}}_{g+r}\right)\right)\right)\right]& \le \mu \left[\gamma \left[\psi \left(\pi \left({\mathcal{J}}_g\right),\alpha \left(\pi \left({\mathcal{L}}_g\right)\right)\right)\right]\right]\hfill \\ \hfill & \le \mu \left[\gamma \left[\psi \left(\pi \left(\overline{Conv\left({\mathcal{I}}_1{\mathcal{J}}_{g-1}\right)}\right),\alpha \left(\pi \left(\overline{Conv\left({\mathcal{I}}_2{\mathcal{L}}_{g-1}\right)}\right)\right)\right)\right]\right]\hfill \\ \hfill & =\mu \left[\gamma \left[\psi \left(\pi \left({\mathcal{I}}_1{\mathcal{J}}_{g-1}\right),\alpha \left(\pi \left({\mathcal{I}}_2{\mathcal{L}}_{g-1}\right)\right)\right)\right]\right]\hfill \\ \hfill & \le {\mu }^2\left[\gamma \left[\psi \left(\pi \left({\mathcal{J}}_{g-1}\right),\alpha \left(\pi \left({\mathcal{L}}_{g-1}\right)\right)\right)\right]\right]\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le {\mu }^{g+1}\left[\gamma \left[\psi \left(\pi \left({\mathcal{J}}_0\right),\alpha \left(\pi \left({\mathcal{L}}_0\right)\right)\right)\right]\right].\hfill \end{array}$$

(11)

Taking $g\to \infty$ in Equation (11) and applying the conditions of $\mu$, we obtain

$$\begin{array}{cc}\hfill & \underset{g\to \infty }{lim}\gamma \left[\psi \left(\pi \left({\mathcal{J}}_{g+r}\right),\alpha \left(\pi \left({\mathcal{L}}_{g+r}\right)\right)\right)\right]=0\hfill \\ \hfill & \Rightarrow \underset{g\to \infty }{lim}\psi \left(\pi \left({\mathcal{J}}_{g+r}\right),\alpha \left(\pi \left({\mathcal{L}}_{g+r}\right)\right)\right)=0.\hfill \end{array}$$

Thus,

$$\underset{g\to \infty }{lim}\pi \left({\mathcal{J}}_{g+r}\right)=\underset{g\to \infty }{lim}\pi \left({\mathcal{J}}_g\right)=0$$

and

$$\underset{g\to \infty }{lim}\pi \left({\mathcal{L}}_{g+r}\right)=\underset{g\to \infty }{lim}\pi \left({\mathcal{L}}_g\right)=0.$$

Hence, we infer that $\pi \left({\mathcal{J}}_g\right)\to 0$ and $\pi \left({\mathcal{L}}_g\right)\to 0$ as $g\to \infty$. Therefore, according to Definition 3 (vii), ${\mathcal{J}}_{\infty }={\displaystyle \bigcap _{g=1}^{\infty }}{\mathcal{J}}_g$ and ${\mathcal{L}}_{\infty }={\displaystyle \bigcap _{g=1}^{\infty }}{\mathcal{L}}_g$ are convex, non-empty, and closed subsets of $\mathcal{D}$. Additionally, the sets ${\mathcal{J}}_{\infty }$ and ${\mathcal{L}}_{\infty }$ are invariant under the mappings ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$, respectively, and ${\mathcal{J}}_{\infty },{\mathcal{L}}_{\infty }\in ker\pi$. So, by applying Schauder’s theorem, ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$ have fixed points in $\mathcal{D}$. □

Corollary 1. 

Let $\mathcal{D}$ be an $\mathbb{N}\mathbb{B}\mathbb{C}\mathbb{C}$ set of ℧ and ${\mathcal{I}}_1,{\mathcal{I}}_2:\mathcal{D}\to \mathcal{D}$ be continuous operators so that

$$\gamma \left[\pi \left({\mathcal{I}}_1^r\mathbb{E}\right)+\alpha \left(\pi \left({\mathcal{I}}_2^r\mathbb{F}\right)\right)\right]\le \mu \left[\gamma \left[{\mathcal{T}}_{r-1}\left(\mathbb{E}\right)+\alpha \left({\mathcal{T}}_{r-1}^{\prime }\left(\mathbb{F}\right)\right)\right]\right],$$

(12)

where

$${\mathcal{T}}_{r-1}\left(\mathbb{E}\right)=\mathcal{O}(\pi \left(\mathbb{E}\right),\pi \left({\mathcal{I}}_1\mathbb{E}\right),\dots ,\pi \left({\mathcal{I}}_1^{r-1}\mathbb{E}\right))$$

and

$${\mathcal{T}}_{r-1}^{\prime }\left(\mathbb{F}\right)=\mathcal{O}(\pi \left(\mathbb{F}\right),\pi \left({\mathcal{I}}_2\mathbb{F}\right),\dots ,\pi \left({\mathcal{I}}_2^{r-1}\mathbb{F}\right))$$

for all $\mathbb{E},\mathbb{F}(\ne Ø)\subseteq \mathcal{D}$; π is an arbitrary $\mathbb{M}\mathbb{N}\mathbb{C}$, $\gamma \in {{\rm Y}}_{\gamma }$, $\mathcal{O}\in \overline{\mathcal{O}}$, $\mu \in \overline{\mathcal{U}}$, and $\alpha :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a non-decreasing and continuous mapping. Then, both ${\mathcal{I}}_1,{\mathcal{I}}_2$ have fixed points in $\mathcal{D}.$

Proof. 

The proof is obtained by assuming $\psi (s,{\iota }^{\prime })=s+{\iota }^{\prime }$ ($s,{\iota }^{\prime }\in {\mathbb{R}}_{+}$) in inequality (7) of Theorem 3. □

Corollary 2. 

Let $\mathcal{D}$ be an $\mathbb{N}\mathbb{B}\mathbb{C}\mathbb{C}$ of ℧ and ${\mathcal{I}}_1,{\mathcal{I}}_2:\mathcal{D}\to \mathcal{D}$ be continuous operators so that

$$\pi \left({\mathcal{I}}_1^r\mathbb{E}\right)+\alpha \left(\pi \left({\mathcal{I}}_2^r\mathbb{F}\right)\right)\le \hslash \left[{\mathcal{T}}_{r-1}\left(\mathbb{E}\right)+\alpha \left({\mathcal{T}}_{r-1}^{\prime }\left(\mathbb{F}\right)\right)\right]\hslash \in [0,1);$$

(13)

where

$${\mathcal{T}}_{r-1}\left(\mathbb{E}\right)=\mathcal{O}(\pi \left(\mathbb{E}\right),\pi \left({\mathcal{I}}_1\mathbb{E}\right),\dots ,\pi \left({\mathcal{I}}_1^{r-1}\mathbb{E}\right))$$

and

$${\mathcal{T}}_{r-1}^{\prime }\left(\mathbb{F}\right)=\mathcal{O}(\pi \left(\mathbb{F}\right),\pi \left({\mathcal{I}}_2\mathbb{F}\right),\dots ,\pi \left({\mathcal{I}}_2^{r-1}\mathbb{F}\right))$$

for all $\mathbb{E},\mathbb{F}(\ne Ø)\subseteq \mathcal{D}$; π is an arbitrary $\mathbb{M}\mathbb{N}\mathbb{C}$, $\mathcal{O}\in \overline{\mathcal{O}}$, and $\alpha :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a non-decreasing and continuous mapping. Then, both ${\mathcal{I}}_1,{\mathcal{I}}_2$ have fixed points in $\mathcal{D}.$

Proof. 

This corollary can be proved by putting $\gamma \left({\iota }^{\prime }\right)={\iota }^{\prime }$ and $\mu \left({\iota }^{\prime }\right)=\hslash ·{\iota }^{\prime },$ for all ${\iota }^{\prime }\in R_{+}$ where $\hslash \in [0,1)$, in inequality (12) of Corollary 1. □

Corollary 3. 

Let $\mathcal{D}$ be an $\mathbb{N}\mathbb{B}\mathbb{C}\mathbb{C}$ set of ℧ and ${\mathcal{I}}_1,{\mathcal{I}}_2:\mathcal{D}\to \mathcal{D}$ be continuous operators so that

$$\pi \left({\mathcal{I}}_1\mathbb{E}\right)+\pi \left({\mathcal{I}}_2\mathbb{F}\right)\le \hslash \left[\pi \left(\mathbb{E}\right)+\pi \left(\mathbb{F}\right)\right]\hslash \in [0,1)$$

(14)

for all $\mathbb{E},\mathbb{F}(\ne Ø)\subseteq \mathcal{D}$, and π is an arbitrary $\mathbb{M}\mathbb{N}\mathbb{C}$ Then, both ${\mathcal{I}}_1,{\mathcal{I}}_2$ have fixed points in $\mathcal{D}.$

Proof. 

This corollary can be proved by putting $\alpha \left({\iota }^{\prime }\right)={\iota }^{\prime }$ and $r=1$ for all ${\iota }^{\prime }\in {\mathbb{R}}_{+}$ in inequality (13) of Corollary 2. □

Remark 1. 

Let ${\mathcal{I}}_1={\mathcal{I}}_2=\mathcal{I}$, $\alpha \left({\iota }^{\prime }\right)=0,\gamma \left({\iota }^{\prime }\right)={\iota }^{\prime },\mu \left({\iota }^{\prime }\right)=\hslash {\iota }^{\prime };r=1;$ for all ${\iota }^{\prime }\in {\mathbb{R}}_{+}andforsome\hslash \in [0,1)$ in inequality (12) of corollary 1. Then, we obtain

$$\pi \left(\mathcal{I}\mathbb{E}\right)\le \hslash \pi \left(\mathbb{E}\right);\hslash \in [0,1)$$

(15)

where $\mathbb{E}(\ne Ø)\subseteq \mathcal{D}$ and π is an arbitrary $\mathbb{M}\mathbb{N}\mathbb{C}$, which is known as $\mathcal{D}.\mathcal{F}.\mathcal{P}.\mathcal{T}$.

Next, we have the following corollaries on the basis of the examples of $\mathcal{O}$.

Corollary 4. 

Let $\mathcal{D}$ be an $\mathbb{N}\mathbb{B}\mathbb{C}\mathbb{C}$ set of ℧ and ${\mathcal{I}}_1,{\mathcal{I}}_2:\mathcal{D}\to \mathcal{D}$ be continuous operators so that

$$\gamma \left[\psi \left(\pi \left({\mathcal{I}}_1^r\mathbb{E}\right),\alpha \left(\pi \left({\mathcal{I}}_2^r\mathbb{F}\right)\right)\right)\right]\le \mu \left[\gamma \left[\psi \left({\mathcal{T}}_{r-1}\left(\mathbb{E}\right),\alpha \left({\mathcal{T}}_{r-1}^{\prime }\left(\mathbb{F}\right)\right)\right)\right]\right],$$

(16)

where

$${\mathcal{T}}_{r-1}\left(\mathbb{E}\right)=max\{\pi \left(\mathbb{E}\right),\pi \left({\mathcal{I}}_1\mathbb{E}\right),\dots ,\pi \left({\mathcal{I}}_1^{r-1}\mathbb{E}\right)\}$$

and

$${\mathcal{T}}_{r-1}^{\prime }\left(\mathbb{F}\right)=max\{\pi \left(\mathbb{F}\right),\pi \left({\mathcal{I}}_2\mathbb{F}\right),\dots ,\pi \left({\mathcal{I}}_2^{r-1}\mathbb{F}\right)\}$$

for all $\mathbb{E},\mathbb{F}(\ne Ø)\subseteq \mathcal{D}$ and π is an arbitrary $\mathbb{M}\mathbb{N}\mathbb{C}$ $\gamma \in {{\rm Y}}_{\gamma },\psi \in {\Psi }_{\psi },\mu \in \overline{\mathcal{U}}$, and $\alpha :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a non-decreasing and continuous mapping. Then, both ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$ have fixed points in $\mathcal{D}.$

Proof. 

The proof is obtained by using

$$\mathcal{O}({\iota }_1^{\prime },{\iota }_2^{\prime },\dots \dots ,{\iota }_n^{\prime })=max\{{\iota }_1^{\prime },{\iota }_2^{\prime },\dots \dots ,{\iota }_n^{\prime }\};{\iota }_1^{\prime },{\iota }_2^{\prime },\dots \dots ,{\iota }_n^{\prime }\in \mathbb{R}$$

in Theorem 3. □

Corollary 5. 

Let $\mathcal{D}$ be an $\mathbb{N}\mathbb{B}\mathbb{C}\mathbb{C}$ set of ℧ and ${\mathcal{I}}_1,{\mathcal{I}}_2:\mathcal{D}\to \mathcal{D}$ be continuous operators so that

$$\gamma \left[\psi \left(\pi \left({\mathcal{I}}_1^r\mathbb{E}\right),\alpha \left(\pi \left({\mathcal{I}}_2^r\mathbb{F}\right)\right)\right)\right]\le \mu \left[\gamma \left[\psi \left({\mathcal{T}}_{r-1}\left(\mathbb{E}\right),\alpha \left({\mathcal{T}}_{r-1}^{\prime }\left(\mathbb{F}\right)\right)\right)\right]\right],$$

(17)

where

$${\mathcal{T}}_{r-1}\left(\mathbb{E}\right)=\frac{1}{r}\{\pi \left(\mathbb{E}\right)+\pi \left({\mathcal{I}}_1\mathbb{E}\right)+\dots +\pi \left({\mathcal{I}}_1^{r-1}\mathbb{E}\right)\}$$

and

$${\mathcal{T}}_{r-1}^{\prime }\left(\mathbb{F}\right)=\frac{1}{r}\{\pi \left(\mathbb{F}\right)+\pi \left({\mathcal{I}}_2\mathbb{F}\right)+\dots +\pi \left({\mathcal{I}}_2^{r-1}\mathbb{F}\right)\}$$

for all $\mathbb{E},\mathbb{F}(\ne Ø)\subseteq \mathcal{D}$ and π is an arbitrary $\mathbb{M}\mathbb{N}\mathbb{C}$, and $\gamma \in {{\rm Y}}_{\gamma },\psi \in {\Psi }_{\psi },\mu \in \overline{\mathcal{U}}$, and $\alpha :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a non-decreasing and continuous mapping. Then, both ${\mathcal{I}}_1,{\mathcal{I}}_2$ have fixed points in $\mathcal{D}.$

Proof. 

The proof is obtained by using $\mathcal{O}({\iota }_1^{\prime },{\iota }_2^{\prime },\dots \dots ,{\iota }_r^{\prime })=\frac{1}{r}\{{\iota }_1^{\prime }+{\iota }_2^{\prime }+\dots \dots +{\iota }_r^{\prime }\};{\iota }_1^{\prime },{\iota }_2^{\prime },\dots \dots ,{\iota }_r^{\prime }\in \mathbb{R}$ in Theorem 3. □

Measure of Noncompactness in $BC\left({\mathbb{R}}_{+}\right)$: Let $\mho =BC\left({\mathbb{R}}_{+}\right)$ (the family of all real operators that are bounded and continuous on ${\mathbb{R}}_{+}=[0,\infty )).$ It is easy to find that ℧ is a Banach space with the norm

$$\Vert \mathfrak{X}\Vert =sup\left\{\left|\mathfrak{X}(\hslash )\right|:\hslash \in R_{+}\right\},\mathfrak{X}\in \mho .$$

Let $\mathcal{W}(\ne Ø)$ be a non-empty, bounded subset of $BC\left({\mathbb{R}}_{+}\right)$ and $K>0$. For a given $\mathfrak{X}\in \mathcal{W}$ and an arbitrary $\varrho >0,$

$${\eta }^K(\mathfrak{X},\varrho )=sup\left\{\left|\mathfrak{X}\left({\hslash }_1\right)-\mathfrak{X}\left({\hslash }_2\right)\right|:{\hslash }_1,{\hslash }_2\in [0,K],\left|{\hslash }_2-{\hslash }_1\right|\le \varrho \right\}$$

is the modulus of the continuity of $\mathfrak{X}$ on the interval $[0,K]$. Also, we define

$$\begin{array}{cc}\hfill & {\eta }^K(\mathcal{W},\varrho )=sup\left\{{\eta }^K(\mathfrak{X},\varrho ):\mathfrak{X}\in \mathcal{W}\right\},\hfill \\ \hfill & {\eta }_0^K\left(\mathcal{W}\right)=\underset{\varrho \to 0}{lim}{\eta }^K(\mathcal{W},\varrho ),\hfill \\ \hfill & {\eta }_0\left(\mathcal{W}\right)=\underset{K\to \infty }{lim}{\eta }_0^K\left(\mathcal{W}\right).\hfill \end{array}$$

Moreover, let

$$\begin{array}{cc}\hfill & \mathcal{W}(\hslash )=\left\{\mathfrak{X}\right(\hslash ):\hslash \in \mathcal{W}\},\hfill \\ \hfill & diam\left(\mathcal{W}(\hslash )\right)=sup\left\{\right|\mathfrak{X}(\hslash )-{\mathfrak{X}}_1(\hslash )|:\mathfrak{X},{\mathfrak{X}}_1\in \mathcal{W}\},\hfill \\ \hfill & \delta \left(\mathcal{W}\right)={\eta }_0\left(\mathcal{W}\right)+\underset{\hslash \to \infty }{lim\; sup}diam\left(\mathcal{W}(\hslash )\right).\hfill \end{array}$$

Further, it can be seen that the mapping $\delta$ is an $\mathbb{M}\mathbb{N}\mathbb{C}$ in the space $BC\left({\mathbb{R}}_{+}\right)$. One can see [12] and the references therein for more details.

4. Solvability of a System of Fractional Integral Equations

In this portion, we prove the existence of solutions of the following system of $\mathbb{F}\mathbb{I}\mathbb{E}$ (18) and (19) in the Banach space $\mho =BC\left({\mathbb{R}}_{+}\right)$. We establish an example in support of our claim. We start this section with the following definition and lemma:

Lemma 1 

([16]). For a non-decreasing and upper semi-continuous operator $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, the following conditions are equivalent:

(1) 

$\underset{t\to \infty }{lim}{\phi }^t(\hslash )=0$ for any $\hslash >0$;

(2) 

$\phi (\hslash )<\hslash$ for any $\hslash >0.$

Now, consider the following $\mathbb{F}\mathbb{I}\mathbb{E}$:

$$\xi \left(\kappa \right)=\Phi (\kappa ,\xi \left(\kappa \right))+{\displaystyle \frac{w^{-1}\left(\xi \right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell$$

(18)

and

$$\zeta \left(\kappa \right)=\widehat{\Phi }(\kappa ,\zeta \left(\kappa \right))+{\displaystyle \frac{w^{-1}\left(\zeta \right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )\widehat{Ł}(\kappa ,\ell ,\zeta (\ell ))d\ell$$

(19)

where $\sigma >0,\kappa \in {\mathbb{R}}_{+}$; $\Phi ,\widehat{\Phi }$ are operators from ${\mathbb{R}}_{+}\times \mathbb{R}$ to $\mathbb{R}$; $Ł,\widehat{Ł}$ are operators from ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times \mathbb{R}$ to $\mathbb{R}$; $\mathcal{Z}$ is a function from ${\mathbb{R}}_{+}$ to $\mathbb{R}$; $\Gamma (,)$ is the Euler’s Gamma operator; and w is known as a weighted function. Let ${\mathbb{A}}_{\iota }=\left\{\xi \in \mho :\Vert \xi \Vert \le \iota \right\}$, and assumptions for the system of $\mathbb{F}\mathbb{I}\mathbb{E}.s$ (18) and (19) are as follows:

(1)

$\Phi ,\widehat{\Phi }:{\mathbb{R}}_{+}\times \mathbb{R}\to \mathbb{R}$ are continuous and ∃ an operator $\phi \in \overline{\mathcal{U}}$, such that $\left|\Phi (\kappa ,\xi \left(\overline{\kappa }\right))-\Phi ({\kappa }^{\prime },{\xi }^{\prime }\left(\overline{{\kappa }^{\prime }}\right))\right|\le \phi \left[\right|\kappa -{\kappa }^{\prime }|+|\xi \left(\overline{\kappa }\right)-{\xi }^{\prime }\left(\overline{{\kappa }^{\prime }}\right)\left|\right],$ for any $\kappa ,{\kappa }^{\prime }\in {\mathbb{R}}_{+}$ and $\forall \xi ,{\xi }^{\prime }\in BC\left({\mathbb{R}}_{+}\right)$ and $\left|\widehat{\Phi }(\kappa ,\zeta \left(\kappa \right))-\widehat{\Phi }({\kappa }^{\prime },{\zeta }^{\prime }\left({\kappa }^{\prime }\right))\right|\le \phi \left[|\kappa -{\kappa }^{\prime }|+|\zeta \left(\kappa \right)-{\zeta }^{\prime }\left({\kappa }^{\prime }\right)|\right],$ for any $\kappa ,{\kappa }^{\prime }\in {\mathbb{R}}_{+}$ and $\forall \zeta ,{\zeta }^{\prime }\in BC\left({\mathbb{R}}_{+}\right).$ Also, let $\phi$ be supper-additive, i.e, $\phi \left({\kappa }_1\right)+\phi \left({\kappa }_2\right)\le \phi ({\kappa }_1+{\kappa }_2),\forall {\kappa }_1,{\kappa }_2\in {\mathbb{R}}_{+},$ and $\kappa \to \Phi (\kappa ,0)$ and $\kappa \to \widehat{\Phi }(\kappa ,0)$ are two members of the space $BC\left({\mathbb{R}}_{+}\right)$;

(2)

w and $w^{-1}$ are continuous and bounded, i.e., $\left|w\left(\kappa \right)\right|\le {\mathcal{E}}_1\mathrm{and}\left|w^{-1}\left(\kappa \right)\right|\le {\mathcal{E}}_2$;

(3)

$Ł,\widehat{Ł}:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times \mathbb{R}\to \mathbb{R}$ are continuous and ∃ operators $\mathcal{M},\widehat{\mathcal{M}},\mathcal{N},\widehat{\mathcal{N}}:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, so that ${lim}_{\kappa \to \infty }O\left(\kappa \right)={lim}_{\kappa \to \infty }\widehat{O}\left(\kappa \right)=0$, where

$$O\left(\kappa \right)={\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2\mathcal{M}\left(\kappa \right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }\left|{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )\right|\mathcal{N}(\ell )d\ell$$

and

$$\widehat{O}\left(\kappa \right)={\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2\widehat{\mathcal{M}}\left(\kappa \right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }\left|{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )\right|\widehat{\mathcal{N}}(\ell )d\ell$$

where $Ł(\kappa ,\ell ,\xi (\ell \left)\right)\le \mathcal{M}\left(\kappa \right)\mathcal{N}(\ell )$ and $\widehat{Ł}(\kappa ,\ell ,\zeta \left(\kappa \right))\le \widehat{\mathcal{M}}\left(\kappa \right)\widehat{\mathcal{N}}(\ell )$ for all $\kappa ,\ell \in {\mathbb{R}}_{+}$ and $\forall \xi ,\zeta \in BC\left({\mathbb{R}}_{+}\right)$;

(4)

$\mathcal{Z}:{\mathbb{R}}_{+}\to \mathbb{R}$ is a continuously differentiable operator;

(5)

For some positive integer ${\iota }_0$, $\phi \left({\iota }_0\right)+\mathcal{G}\le {\iota }_0,$ and $\phi \left({\iota }_0\right)+\widehat{\mathcal{G}}\le {\iota }_0,$ where $\mathcal{G}=sup\left\{\right|\Phi (\kappa ,0)|+O(\kappa ):\kappa \ge 0\}$ and $\widehat{\mathcal{G}}=sup\{|\widehat{\Phi }(\kappa ,0)|+\widehat{O}\left(\kappa \right):\kappa \ge 0\}.$

The next theorem is as follows:

Theorem 4. 

If the axioms (1)–(5) are satisfied, then Equations (18) and (19) have solutions in $\mho =BC\left({\mathbb{R}}_{+}\right)$.

Proof. 

We consider the following mappings $\mathcal{H},\widehat{\mathcal{H}}:\mho \to \mho$, which are defined as follows:

$$\begin{array}{c}\hfill \left(\mathcal{H}\xi \right)\left(\kappa \right)=\Phi (\kappa ,\xi \left(\kappa \right))+{\displaystyle \frac{w^{-1}\left(\xi \right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell ,\end{array}$$

(20)

and

$$\begin{array}{c}\hfill \left(\widehat{\mathcal{H}}\zeta \right)\left(\kappa \right)=\widehat{\Phi }(\kappa ,\zeta \left(\kappa \right))+{\displaystyle \frac{w^{-1}\left(\zeta \right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )\widehat{Ł}(\kappa ,\ell ,\zeta (\ell ))d\ell .\end{array}$$

(21)

Step (1): Here, we show that the mappings $\mathcal{H},\widehat{\mathcal{H}}:{\mathbb{A}}_{{\iota }_0}\to {\mathbb{A}}_{{\iota }_0}$. Taking $\xi ,\zeta \in {\mathbb{A}}_{{\iota }_0}$ and $\kappa \in R_{+}$, we have

$$\begin{array}{cc}\hfill & \left|\right(\mathcal{H}\xi \left)\right(\kappa \left)\right|\hfill \\ \hfill & \le |\Phi (\kappa ,\xi \left(\kappa \right))|+{\displaystyle \frac{|w^{-1}\left(\xi \right(\kappa \left)\right)|}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }|{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )| |w(\ell )\left|\right|Ł(\kappa ,\ell ,\xi (\ell ))|d\ell \hfill \\ \hfill & \le |\Phi (\kappa ,\xi \left(\kappa \right))-\Phi (\kappa ,0)|+|\Phi (\kappa ,0)|+{\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2\mathcal{M}\left(\kappa \right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }\left|{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )\right|\mathcal{N}(\ell )d\ell \hfill \\ \hfill & \le \phi \left|\xi \right(\kappa )-0|+|\Phi (\kappa ,0\left)\right|+O\left(\kappa \right)\hfill \\ \hfill & \le \phi (\Vert \xi \Vert )+|\Phi (\kappa ,0\left)\right|+O\left(\kappa \right)\hfill \\ \hfill & \le \phi (\Vert \xi \Vert )+\mathcal{G},[where\mathcal{G}=sup\{|\Phi (\kappa ,0\left)\right|+O\left(\kappa \right):\kappa >0\}.]\hfill \end{array}$$

Hence, $\Vert \xi \Vert \le {\iota }_0$ gives

$$\begin{array}{c}\hfill \left|\right|\mathcal{H}\xi \left|\right|\le \phi \left({\iota }_0\right)+\mathcal{G}\le {\iota }_0.\end{array}$$

On account of axiom (5), $\mathcal{H}:{\mathbb{A}}_{{\iota }_0}\to {\mathbb{A}}_{{\iota }_0}$. In the same way, it can be shown that $\widehat{\mathcal{H}}:{\mathbb{A}}_{{\iota }_0}\to {\mathbb{A}}_{{\iota }_0}$.

Step (2): Here, we show the continuity of $\mathcal{H}$ and $\widehat{\mathcal{H}}$ on ${\mathbb{A}}_{{\iota }_0}.$ All functions in $\mathcal{H}$ are continuous, so, $\mathcal{H}$ is then continuous on ${\mathbb{A}}_{{\iota }_0}.$ Similarly, $\widehat{\mathcal{H}}$ is continuous on ${\mathbb{A}}_{{\iota }_0}.$

Step (3): Here, with the help of $\eta$, we show the estimation of $\mathcal{H}$ and $\widehat{\mathcal{H}}$. Take ${\Delta }_{\xi },{\Delta }_{\zeta }(\ne Ø)\subseteq {\mathbb{A}}_{{\iota }_0}$. Assume that $K>0$ and $\varrho >0$ are arbitrary. Let $\xi \in {\Delta }_{\xi }$, $\zeta \in {\Delta }_{\zeta }$ and $\kappa ,{\kappa }^{\prime }\in [0,K]$, such that $\left|{\kappa }^{\prime }-\kappa \right|\le \varrho$ with ${\kappa }^{\prime }\ge \kappa$. Next, we have

$$\begin{array}{cc}\hfill & \left|\left(\mathcal{H}\xi \right)\left({\kappa }^{\prime }\right)-\left(\mathcal{H}\xi \right)\left(\kappa \right)\right|\hfill \\ \hfill & \le \left|\Phi ({\kappa }^{\prime },\xi \left({\kappa }^{\prime }\right))-\Phi (\kappa ,\xi \left(\kappa \right))\right|\hfill \\ \hfill & +|{\displaystyle \frac{w^{-1}\left(\xi \left({\kappa }^{\prime }\right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{{\kappa }^{\prime }}{(\mathcal{Z}\left({\kappa }^{\prime }\right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł({\kappa }^{\prime },\ell ,\xi (\ell ))d\ell \hfill \\ \hfill & -{\displaystyle \frac{w^{-1}\left(\xi \left(\kappa \right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell |\hfill \\ \hfill & \le \left|\Phi ({\kappa }^{\prime },\xi \left({\kappa }^{\prime }\right))-\Phi (\kappa ,\xi \left({\kappa }^{\prime }\right))\right|+\left|\Phi (\kappa ,\xi \left({\kappa }^{\prime }\right))-\Phi (\kappa ,\xi \left(\kappa \right))\right|\hfill \\ \hfill & +|{\displaystyle \frac{w^{-1}\left(\xi \left({\kappa }^{\prime }\right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{{\kappa }^{\prime }}{(\mathcal{Z}\left({\kappa }^{\prime }\right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł({\kappa }^{\prime },\ell ,\xi (\ell ))d\ell \hfill \\ \hfill & -{\displaystyle \frac{w^{-1}\left(\xi \left({\kappa }^{\prime }\right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{{\kappa }^{\prime }}{(\mathcal{Z}\left({\kappa }^{\prime }\right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell |\hfill \\ \hfill & +|{\displaystyle \frac{w^{-1}\left(\xi \left({\kappa }^{\prime }\right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{{\kappa }^{\prime }}{(\mathcal{Z}\left({\kappa }^{\prime }\right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell \hfill \\ \hfill & -{\displaystyle \frac{w^{-1}\left(\xi \left(\kappa \right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell |\hfill \\ \hfill & \le \phi \left(\right|{\kappa }^{\prime }-\kappa \left|\right)+\phi \left(\right|\xi \left({\kappa }^{\prime }\right)-\xi \left(\kappa \right)\left|\right)\hfill \\ \hfill & +|{\displaystyle \frac{w^{-1}\left(\xi \left({\kappa }^{\prime }\right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{{\kappa }^{\prime }}{(\mathcal{Z}\left({\kappa }^{\prime }\right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )\left(Ł({\kappa }^{\prime },\ell ,\xi (\ell ))-Ł(\kappa ,\ell ,\xi (\ell ))\right)d\ell |\hfill \\ \hfill & +\Omega ({\kappa }^{\prime },\kappa )\hfill \\ \hfill & \le \phi \left(\right|{\kappa }^{\prime }-\kappa \left|\right)+\phi \left({\eta }^K(\xi ,\varrho )\right)+{\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2{\eta }^K(Ł,\varrho )}{\Gamma \left(\sigma \right)}}{\displaystyle \frac{{\left[-{(\mathcal{Z}\left({\kappa }^{\prime }\right)-\mathcal{Z}(\ell ))}^{\sigma }\right]}_0^{{\kappa }^{\prime }}}{\sigma }}+\Omega ({\kappa }^{\prime },\kappa )\hfill \\ \hfill & \le \phi \left(\right|{\kappa }^{\prime }-\kappa \left|\right)+\phi \left({\eta }^K(\xi ,\varrho )\right)+{\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2{\eta }^K(Ł,\varrho )}{\Gamma (\sigma +1)}}{\left(\mathcal{Z}\left({\kappa }^{\prime }\right)\right)}^{\sigma }+\Omega ({\kappa }^{\prime },\kappa )[since,\Gamma \left(\sigma \right).\sigma =\Gamma (\sigma +1)]\hfill \\ \hfill & \le \phi \left(\right|{\kappa }^{\prime }-\kappa \left|\right)+\phi \left({\eta }^K(\xi ,\varrho )\right)+{\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2{\eta }^K(Ł,\varrho )}{\Gamma (\sigma +1)}}{\left(\mathcal{Z}\left(K\right)\right)}^{\sigma }+\Omega ({\kappa }^{\prime },\kappa )[since,\kappa \in [0,K]]\hfill \end{array}$$

(22)

whenever

$${\eta }^K(Ł,\varrho )=sup\left\{\right|Ł({\kappa }^{\prime },\ell ,\xi )-Ł(\kappa ,\ell ,\xi )|:{\kappa }^{\prime },\kappa ,\ell \in [0,K],\xi \in [-\iota ,\iota ],|{\kappa }^{\prime }-\kappa |\le \varrho \},$$

$${\eta }^K(\xi ,\varrho )=sup\left\{\left|\xi \left({\kappa }^{\prime }\right)-\xi \left(\kappa \right)\right|:{\kappa }^{\prime },\kappa \in [0,K],\left|{\kappa }^{\prime }-\kappa \right|\le \varrho \right\}$$

and

$$\begin{array}{cc}\hfill \Omega ({\kappa }^{\prime },\kappa )& =|{\displaystyle \frac{w^{-1}\left(\xi \left({\kappa }^{\prime }\right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{{\kappa }^{\prime }}{(\mathcal{Z}\left({\kappa }^{\prime }\right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell \hfill \\ \hfill & -{\displaystyle \frac{w^{-1}\left(\xi \left(\kappa \right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell |\hfill \\ \hfill & \le |{\displaystyle \frac{w^{-1}\left(\xi \left({\kappa }^{\prime }\right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{{\kappa }^{\prime }}{(\mathcal{Z}\left({\kappa }^{\prime }\right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell \hfill \\ \hfill & -{\displaystyle \frac{w^{-1}\left(\xi \left({\kappa }^{\prime }\right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{{\kappa }^{\prime }}{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell |\hfill \\ \hfill & +|{\displaystyle \frac{w^{-1}\left(\xi \left(\kappa \right)\right)}{\Gamma \left(\sigma \right)}}{\int }_{\kappa }^{{\kappa }^{\prime }}{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell |\hfill \\ \hfill & \le |{\displaystyle \frac{w^{-1}\left(\xi \left({\kappa }^{\prime }\right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{{\kappa }^{\prime }}\left({(\mathcal{Z}\left({\kappa }^{\prime }\right)-\mathcal{Z}(\ell ))}^{\sigma -1}-{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}\right){\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell |\hfill \\ \hfill & +|{\displaystyle \frac{w^{-1}\left(\xi \left(\kappa \right)\right)}{\Gamma \left(\sigma \right)}}{\int }_{\kappa }^{{\kappa }^{\prime }}{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell |\hfill \\ \hfill & \le |{\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2\mathcal{M}\left(\kappa \right)}{\Gamma \left(\sigma \right)}}{\int }_0^{{\kappa }^{\prime }}\left({(\mathcal{Z}\left({\kappa }^{\prime }\right)-\mathcal{Z}(\ell ))}^{\sigma -1}-{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}\right){\mathcal{Z}}^{\prime }(\ell )\mathcal{N}(\ell )d\ell |\hfill \\ \hfill & +|{\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2\mathcal{M}\left(\kappa \right)}{\Gamma \left(\sigma \right)}}{\int }_{\kappa }^{{\kappa }^{\prime }}{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )\mathcal{N}(\ell )d\ell |\hfill \\ \hfill & \le {\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2\mathcal{M}\left(\kappa \right){\mathcal{N}}_0\left(K\right)}{\Gamma \left(\sigma \right)\sigma }}{\left[-{(\mathcal{Z}\left({\kappa }^{\prime }\right)-\mathcal{Z}(\ell ))}^{\sigma }+{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma }\right]}_0^{{\kappa }^{\prime }}\hfill \\ \hfill & +{\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2\mathcal{M}\left(\kappa \right){\mathcal{N}}_0\left(K\right)}{\Gamma \left(\sigma \right)\sigma }}{\left[-{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma }\right]}_{\kappa }^{{\kappa }^{\prime }}[where,{\mathcal{N}}_0\left(K\right)=sup\{\mathcal{N}(\ell ):\ell \in [0,K]\}]\hfill \\ \hfill & \le {\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2\mathcal{M}\left(\kappa \right){\mathcal{N}}_0\left(K\right)}{\Gamma (\sigma +1)}}\left({\left(\mathcal{Z}\left({\kappa }^{\prime }\right)\right)}^{\sigma }+{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}\left({\kappa }^{\prime }\right))}^{\sigma }-{\left(\mathcal{Z}\left(\kappa \right)\right)}^{\sigma }\right)\hfill \\ \hfill & +{\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2\mathcal{M}\left(\kappa \right){\mathcal{N}}_0\left(K\right)}{\Gamma (\sigma +1)}}\left({(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}\left({\kappa }^{\prime }\right))}^{\sigma }\right)\hfill \\ \hfill & \le {\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2\mathcal{M}\left(\kappa \right){\mathcal{N}}_0\left(K\right)}{\Gamma (\sigma +1)}}\left(({\left(\mathcal{Z}\left({\kappa }^{\prime }\right)\right)}^{\sigma }-{\left(\mathcal{Z}\left(\kappa \right)\right)}^{\sigma })+2{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}\left({\kappa }^{\prime }\right))}^{\sigma }\right).\hfill \end{array}$$

(23)

We know that $\Phi$ and Ł are uniformly continuous on $[0,K]\times [-\iota ,\iota ]$ and ${[0,K]}^2\times [-\iota ,\iota ]$, respectively. Also, if $\varrho \to 0$, then ${\kappa }^{\prime }\to \kappa$, ${\eta }^K(Ł,\varrho )\to 0$ and $\Omega ({\kappa }^{\prime },\kappa )\to 0$. Thus, from Equation (22),

$${\eta }_0^K\left(\mathcal{H}{\Delta }_{\xi }\right)\le \underset{\varrho \to 0}{lim}\phi \left({\eta }^K(\xi ,\varrho )\right).$$

(24)

Again, from the continuity of $\phi$,

$${\eta }_0^K\left(\mathcal{H}{\Delta }_{\xi }\right)\le \phi \left({\eta }_0^K\left({\Delta }_{\xi }\right)\right)$$

(25)

and, eventually,

$${\eta }_0\left(\mathcal{H}{\Delta }_{\xi }\right)\le \phi \left({\eta }_0\left({\Delta }_{\xi }\right)\right).$$

(26)

Now, we take arbitrary operators $\xi ,{\xi }_1\in {\Delta }_{\xi }$. An analogous calculation to (22) gives

$$\begin{array}{cc}\hfill & |\left(\mathcal{H}\xi \right)\left(\kappa \right)-\left(\mathcal{H}{\xi }_1\right)\left(\kappa \right)|\hfill \\ \hfill & \le |\Phi (\kappa ,\xi \left(\kappa \right))-\Phi (\kappa ,{\xi }_1\left(\kappa \right))|\hfill \\ \hfill & +|{\displaystyle \frac{w^{-1}\left(\xi \left(\kappa \right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell \hfill \\ \hfill & -{\displaystyle \frac{w^{-1}\left({\xi }_1\left(\kappa \right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,{\xi }_1(\ell ))d\ell |\hfill \\ \hfill & \le \phi \left(\right|\xi \left(\kappa \right)-{\xi }_1\left(\kappa \right)\left|\right)\hfill \\ \hfill & +|{\displaystyle \frac{w^{-1}\left(\xi \left(\kappa \right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,\xi (\ell ))d\ell |\hfill \\ \hfill & +|{\displaystyle \frac{w^{-1}\left({\xi }_1\left(\kappa \right)\right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}{\mathcal{Z}}^{\prime }(\ell )w(\ell )Ł(\kappa ,\ell ,{\xi }_1(\ell ))d\ell |\hfill \\ \hfill & \le \phi \left(\right|\xi \left(\kappa \right)-{\xi }_1\left(\kappa \right)\left|\right)+2O\left(\kappa \right).\hfill \end{array}$$

(27)

Now, from the above inequality, we have

$$diam\left(\mathcal{H}{\Delta }_{\xi }\right)\left(\kappa \right)\le \phi \left(diam{\Delta }_{\xi }\left(\kappa \right)\right)+2O\left(\kappa \right).$$

(28)

In conclusion, from the definition of $\phi$, we calculate

$$\underset{\kappa \to \infty }{lim\; sup}diam\left(\mathcal{H}{\Delta }_{\xi }\right)\left(\kappa \right)\le \phi \left(\underset{\kappa \to \infty }{lim\; sup}diam{\Delta }_{\xi }\left(\kappa \right)\right).$$

(29)

In addition, by combining Equations (26) and (29) and considering the properties of the operator $\phi$, we obtain that

$$\begin{array}{cc}\hfill & {\eta }_0\left(\mathcal{H}{\Delta }_{\xi }\right)+\underset{\kappa \to \infty }{lim\; sup}diam\left(\mathcal{H}{\Delta }_{\xi }\right)\left(\kappa \right)\le \phi \left({\eta }_0\left({\Delta }_{\xi }\right)\right)+\phi \left(\underset{\kappa \to \infty }{lim\; sup}diam{\Delta }_{\xi }\left(\kappa \right)\right)\hfill \\ \hfill & \Rightarrow \delta \left(\mathcal{H}{\Delta }_{\xi }\right)\le \phi \left({\eta }_0\left({\Delta }_{\xi }\right)+\underset{\kappa \to \infty }{lim\; sup}diam{\Delta }_{\xi }\left(\kappa \right)\right)\hfill \\ \hfill & \Rightarrow \delta \left(\mathcal{H}{\Delta }_{\xi }\right)\le \phi \left(\delta \left({\Delta }_{\xi }\right)\right).\hfill \end{array}$$

(30)

Similarly, we obtain

$$\delta \left(\widehat{\mathcal{H}}{\Delta }_{\zeta }\right)\le \phi \left(\delta \left({\Delta }_{\zeta }\right)\right).$$

(31)

Adding the above obtained Equations (30) and (31), we can see that

$$\delta \left(\mathcal{H}{\Delta }_{\xi }\right)+\delta \left(\widehat{\mathcal{H}}{\Delta }_{\zeta }\right)\le \phi [\delta \left({\Delta }_{\xi }\right)+\delta \left({\Delta }_{\zeta }\right)].$$

(32)

From Corollary 1, we can see that $\mathcal{H}$ and $\widehat{\mathcal{H}}$ have solutions in ${\mathbb{A}}_{{\iota }_0}$. Hence, Equations (18) and (19) have solutions such as $(\xi ,\zeta )$ in ${\mho }^2=BC{\left({\mathbb{R}}_{+}\right)}^2.$ □

Remark 2. 

The presented application has been constructed for the case $r=1$. For iterated $(r\ge 1)$ integral equation systems, the reader can refer to [9].

Next, we shall establish an example to find the validation of Theorem 4.

Example 1. 

Let us define a system of $\mathbb{F}.\mathbb{I}.\mathbb{E}$ which is a special mode of Equations (18) and (19) as follows:

$$\xi \left(\kappa \right)={\displaystyle \frac{\xi \left(\kappa \right)+1}{17+{\kappa }^2}}+{\displaystyle \frac{1}{\Gamma \left(\frac{1}{5}\right)}}{\int }_0^{\kappa }{(\kappa -\ell )}^{\frac{1}{5}-1}{\displaystyle \frac{e^{-\kappa }{\kappa }^{-\frac{1}{5}}cos\left(\xi (\ell )\right)}{1+|sin\xi (\ell \left)\right|}}d\ell$$

(33)

and

$$\zeta \left(\kappa \right)={\displaystyle \frac{\zeta \left(\kappa \right)+1}{17+{\kappa }^4}}+{\displaystyle \frac{1}{\Gamma \left(\frac{1}{5}\right)}}{\int }_0^{\kappa }{(\kappa -\ell )}^{\frac{1}{5}-1}{\displaystyle \frac{e^{-\kappa }{\kappa }^{-\frac{1}{5}}{\zeta }^2(\ell )}{1+{\zeta }^2(\ell )}}d\ell .$$

(34)

Let us assume that

$$\begin{array}{cc}\hfill \Phi (\kappa ,\xi (\kappa \left)\right)& =\frac{\xi \left(\kappa \right)+1}{17+{\kappa }^2},\hfill \\ \hfill \widehat{\Phi }(\kappa ,\zeta \left(\kappa \right))& =\frac{\zeta \left(\kappa \right)+1}{17+{\kappa }^4},\hfill \\ \hfill Ł(\kappa ,\ell ,\xi (\ell \left)\right)& ={\displaystyle \frac{e^{-\kappa }{\kappa }^{-\frac{1}{5}}cos\left(\xi (\ell )\right)}{1+|sin\xi (\ell \left)\right|}},\hfill \\ \hfill \widehat{Ł}(\kappa ,\ell ,\xi (\ell ))& ={\displaystyle \frac{e^{-\kappa }{\kappa }^{-\frac{1}{5}}{\zeta }^2(\ell )}{1+{\zeta }^2(\ell )}},\hfill \\ \hfill \sigma & =\frac{1}{5},\hfill \\ \hfill \mathcal{Z}\left({\iota }^{\prime }\right)& ={\iota }^{\prime },\hfill \\ \hfill w\left(\kappa \right)=w^{-1}\left(\kappa \right)& =1.\hfill \end{array}$$

Now, we shall examine the conditions of Theorem 4.

(1)

$\Phi (\kappa ,{\xi }_1\left({\kappa }^{\prime }\right))=\frac{\xi \left({\kappa }^{\prime }\right)+1}{17+{\kappa }^2},\widehat{\Phi }(\widehat{\kappa },{\xi }_2\left({\widehat{\kappa }}^{\prime }\right))=\frac{\xi \left(\widehat{{\kappa }^{\prime }}\right)+1}{17+{\widehat{\kappa }}^4}$ are continuous operators so that

$$|\Phi (\kappa ,{\xi }_1\left({\kappa }^{\prime }\right))-\Phi (\widehat{\kappa },{\xi }_2\left({\widehat{\kappa }}^{\prime }\right))|=\left|\frac{\xi \left({\kappa }^{\prime }\right)+1}{17+{\kappa }^2}-\frac{\xi \left({\widehat{\kappa }}^{\prime }\right)+1}{17+{\widehat{\kappa }}^2}\right|\le \frac{|\xi \left({\kappa }^{\prime }\right)-\xi \left({\widehat{\kappa }}^{\prime }\right)|}{17}\le |\xi \left({\kappa }^{\prime }\right)-\xi \left({\widehat{\kappa }}^{\prime }\right)|,$$

and

$$|\widehat{\Phi }(\kappa ,{\zeta }_1\left({\kappa }^{\prime }\right))-\widehat{\Phi }(\widehat{\kappa },{\zeta }_2\left({\widehat{\kappa }}^{\prime }\right))|=\left|\frac{\zeta \left({\kappa }^{\prime }\right)+1}{17+{\kappa }^4}-\frac{\zeta \left({\widehat{\kappa }}^{\prime }\right)+1}{17+{\widehat{\kappa }}^4}\right|\le \frac{|\zeta \left({\kappa }^{\prime }\right)-\zeta \left({\widehat{\kappa }}^{\prime }\right)|}{17}\le |\zeta \left({\kappa }^{\prime }\right)-\zeta \left({\widehat{\kappa }}^{\prime }\right)|.$$

Thus, $\phi (\hslash )=\frac{\hslash }{17}$. Furthermore, $\Phi (\kappa ,0)=\frac{1}{17+{\kappa }^2},\widehat{\Phi }(\kappa ,0)=\frac{1}{17+{\kappa }^4}$ is continuous, and $\left|\Phi (\kappa ,0)\right|\le \frac{1}{17}$ and $\left|\widehat{\Phi }(\kappa ,0)\right|\le \frac{1}{17}.$ So, the functions $\kappa \to \Phi (\kappa ,0)$ and $\kappa \to \widehat{\Phi }(\kappa ,0)$ are members of $BC\left({\mathbb{R}}_{+}\right)$.

(2)

$\left|w\left(\kappa \right)\right|\le 1={\mathcal{E}}_1$ and $|w^{-1}\left(\kappa \right)|\le 1={\mathcal{E}}_2$.

(3)

It is not hard to see that $Ł(\kappa ,\ell ,\xi (\ell ))={\displaystyle \frac{e^{-\kappa }{\kappa }^{-\frac{1}{5}}cos\left(\xi (\ell )\right)}{1+|sin\xi (\ell \left)\right|}},\widehat{Ł}(\kappa ,\ell ,\xi (\ell ))={\displaystyle \frac{e^{-\kappa }{\kappa }^{-\frac{1}{5}}{\zeta }^2(\ell )}{1+{\zeta }^2(\ell )}}$ are continuous operators and there are continuous operators $\mathcal{M},\mathcal{N},\widehat{\mathcal{M}},\widehat{\mathcal{N}}:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ with $\mathcal{M}\left(\kappa \right)={\kappa }^{-\frac{1}{5}}{e}^{-\kappa }$, $\widehat{\mathcal{M}}\left(\kappa \right)={\kappa }^{-\frac{1}{5}}{e}^{-\kappa }$, $\mathcal{N}(\ell )=1$ and $\widehat{\mathcal{N}}(\ell )=1$, such that $|Ł(\kappa ,\ell ,\xi (\ell )\left)\right|\le \mathcal{M}\left(\kappa \right)\mathcal{N}(\ell )$ and $|\widehat{Ł}(\kappa ,\ell ,\zeta (\ell ))|\le \widehat{\mathcal{M}}\left(\kappa \right)\widehat{\mathcal{N}}(\ell )$ for all $\kappa ,\ell \in {\mathbb{R}}_{+}$ such that $\ell \le \kappa$, and for every $\xi ,\zeta \in BC\left({\mathbb{R}}_{+}\right)$.

• Moreover,

$$\begin{array}{cc}\hfill \underset{\kappa \to \infty }{lim}O\left(\kappa \right)& ={\displaystyle \frac{{\mathcal{E}}_1{\mathcal{E}}_2\mathcal{M}\left(\kappa \right)}{\Gamma \left(\sigma \right)}}{\int }_0^{\kappa }{(\mathcal{Z}\left(\kappa \right)-\mathcal{Z}(\ell ))}^{\sigma -1}\mathcal{N}(\ell )d\ell \hfill \\ \hfill & =\underset{\kappa \to \infty }{lim}{\kappa }^{-\sigma }{e}^{-\kappa }{\int }_0^{\kappa }\frac{{(\kappa -\ell )}^{\sigma -1}}{\Gamma \left(\sigma \right)}d\ell \hfill \\ \hfill & =\underset{\kappa \to \infty }{lim}\frac{{\kappa }^{-\sigma }{e}^{-\kappa }}{\Gamma (\sigma +1)}{\kappa }^{\sigma }\hfill \\ \hfill & =0.\hfill \end{array}$$

Similarly, $\underset{\kappa \to \infty }{lim}\widehat{O}\left(\kappa \right)=0$ can be obtained.

(4)

Here, $\mathcal{Z}\left({\iota }^{\prime }\right)={\iota }^{\prime },{\iota }^{\prime }\in {\mathbb{R}}_{+}$. Thus, we can see that $\mathcal{Z}:{\mathbb{R}}_{+}\to \mathbb{R}$ is a continuously differentiable operator.

(5)

The estimations of the constants $\mathcal{G},\widehat{\mathcal{G}}$ are as follows:

$$\begin{array}{ccc}\hfill \mathcal{G}& =& sup\left\{\right|\xi (\kappa ,0)|+O(\kappa ):\kappa \ge 0\}\hfill \\ & =& sup\{\frac{1}{17+{\kappa }^2}+\frac{{\kappa }^{-\sigma }{e}^{-\kappa }}{\Gamma (\sigma +1)}{\kappa }^{\sigma }:\kappa \ge 0\}\hfill \\ & =& sup\{\frac{1}{17+{\kappa }^2}+\frac{e^{-\kappa }}{\Gamma (\frac{1}{5}+1)}:\kappa \ge 0\}\hfill \\ & =& \frac{1}{17}+{\displaystyle \frac{e^0}{\Gamma \left(\frac{6}{5}\right)}}\hfill \\ & =& \frac{1}{17}+{\displaystyle \frac{1}{\Gamma \left(\frac{6}{5}\right)}}.\hfill \end{array}$$

Similarly, $\widehat{\mathcal{G}}=\frac{1}{17}+{\displaystyle \frac{1}{\Gamma \left(\frac{6}{5}\right)}}.$ Now, if

$$\begin{array}{cc}\hfill & \phi \left(\iota \right)+\mathcal{G}\le \iota \hfill \\ \hfill & \Rightarrow {\displaystyle \frac{\iota }{17}}+{\displaystyle \frac{1}{17}}+{\displaystyle \frac{1}{\Gamma \left(\frac{6}{5}\right)}}\le \iota \hfill \\ \hfill & \Rightarrow \iota \ge \frac{17}{16}\frac{(\Gamma (\frac{6}{5}+17)}{17\Gamma \left(\frac{6}{5}\right)}.\hfill \end{array}$$

Hence, all axioms $\left(1\right)$ to $\left(5\right)$ of Theorem 4 are satisfied. According to Theorem 4, Equations (33) and (34) have solutions in $\mho =BC\left({\mathbb{R}}_{+}\right).$