Application of Fixed Point Theorem to Solvability for Non-Linear Fractional Hadamard Functional Integral Equations

Abstract

In the present paper, our main work aims to discover the existence result of the fractional order non-linear Hadamard functional integral equations on $[1,a]$ by employing the theory of measure of non-compactness together with the fixed point theory in Banach space. An example is presented to see the validity and practicability of our existence result.

1. Introduction

Fractional calculus is the branch of mathematical analysis that plays a pivotal role in non-linear analysis, which generalizes derivatives, together with integrals of integer orders to non-integer orders, by applying the Euler Gamma function. It is a well-known mathematical tool for physical investigation, together with the description of non-local and anomalous diffusion integral operators. Abel’s study of the tautochrone problem is considered to be the first application of fractional calculus to an engineering problem. The subject of fractional calculus allows us to describe various phenomena and effects, in the diverse and widespread fields of engineering and sciences, such as fractional viscoelasticity, fractional electromagnetics, fractional electrochemistry, spatial dispersion of power type, fading memory (forgetting), frequency dispersion of power type, intrinsic dissipation, the openness of systems (interaction with the environment), fractional relaxation–oscillation, fractional biological population models, fractional diffusion-waves, the long-range interactions of power-law type, optical computing, communication, signal processing, etc.

The theory of fractional functional integral Equations (FFIE) plays a significant role in different fields, which includes various implications in the scaling system theory, the theory of algorithms, etc. The practical importance of non-linear fractional integral equations is increasingly evident from studies that incorporate the same in distinct areas of knowledge that include biology, traffic theory, the theory of optimal control, economics, acoustic scattering, etc. Precisely, extensive studies on these equations are focused on their solutions by employing the technique of measure of non-compactness. In such studies, the existence of solutions is proven with the help of the fixed point theory. Verifying the existence of solutions, their behavioral properties are also extensively studied. In recent times, the fixed point theory (FPT) is applicable in various scientific fields suggested by Stephen Banach. Furthermore, FPT can be applied to seeking solutions for FFIE. FFIE, in a variety of forms, is an extraordinary and prestigious branch of non-linear analysis and seeks various invocations in demonstrating numerous real-life scenarios, together with real-world problems (cf. [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]).

In our work, we consider the following functional non-linear integral equation with the Hadamard fractional operator:

$$\begin{array}{cc}\hfill \mathrm{x}\left(\varrho \right)& =F\left(\varrho ,\mathrm{x}\left(\varrho \right),\frac{g_1(\varrho ,\mathrm{x}\left(\chi \left(\varrho \right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{u(\varrho ,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left(\varrho \right)\right)\right)\hfill \\ \hfill & \times G\left(\varrho ,\mathrm{x}\left(\varrho \right),g_2(\varrho ,\mathrm{x}\left(\eta \left(\varrho \right)\right))\underset{1}{\overset{a}{\int }}v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right))ds,\mathrm{x}\left(d\left(\varrho \right)\right)\right),\hfill \end{array}$$

(1)

for $\varrho \in \mathcal{I}=[1,a]$, where $\mathsf{\Gamma }$ is the gamma function, $g_i:\mathcal{I}\times \mathrm{I}\mathrm{R}\to \mathrm{I}\mathrm{R}$ are the maps to be determined for values of $i=1,2,$ and $u,v:\mathcal{I}\times \mathcal{I}\times \mathrm{I}\mathrm{R}\to \mathrm{I}\mathrm{R}$ are Lebesgue integral maps.

The principal construction of our study is as follows: Section 2 explores some notations, definitions, and preliminary results. In Section 3, we adopt the technique established by the applications of measure of non-compactness (MNC) and FPT in Banach algebra to find the existence result. In Section 4, we present an example to illustrate our existence result. Our conclusion is presented in Section 5.

2. Preliminaries

In this section, we organize some notations, definitions, and auxiliary facts which we need throughout the paper.

Let us suppose $(\mathcal{E},\parallel .\parallel )$ is a real Banach space together with the zero element $\Theta$. The notations $\overline{\mathcal{X}}$, Conv $\mathcal{X}$ stand for the closure and convex closure of a subset $\mathcal{X}$ of $\mathcal{E}$, respectively. The notation $\mathcal{B}(\mathrm{x},{\mathrm{r}}_0)$ stands for the closed ball centered at $\mathrm{x}$ together with radius ${\mathrm{r}}_0$. The notation ${\mathcal{B}}_{{\mathrm{r}}_0}$ stands for the ball $\mathcal{B}(\Theta ,{\mathrm{r}}_0)$. Denoted by $\mathrm{I}\mathrm{R}$ is the set of real numbers and ${\mathrm{I}\mathrm{R}}^{+}=[0,+\infty )$. Furthermore, suppose that ${\mathcal{Z}}_{\mathcal{E}}$ represents the class of any non-empty bounded subsets of $\mathcal{E}$, and ${\mathcal{N}}_{\mathcal{E}}$ represents its subclass consist of any non-empty subsets, whose closure is compact.

The space of $\mathcal{E}$-valued continuous maps on $\mathcal{I}$ is represented by $\mathcal{C}(\mathcal{I},\mathcal{E})$ together with the usual norm:

$$\parallel \mathrm{z}\left(\varrho \right)\parallel =sup\left\{\right|\mathrm{z}\left(\varrho \right)|:\varrho \in [1,a\left]\right\},forsome\mathrm{z}\in \mathcal{C}(\mathcal{I},\mathcal{E}).$$

Definition 1.

A measurable function $\mathrm{z}:\mathcal{I}\to \mathcal{E}$ is known as the Bochner integral if $\parallel \mathrm{z}\parallel$ is Lebesgue integral. Suppose $L^1(\mathcal{I},\mathcal{E})$ represent the Banach space of measurable maps $\mathrm{z}:\mathcal{I}\to \mathcal{E}$ which are Bochner integral given by norm

$${\parallel \mathrm{z}\parallel }_{L^1}={\int }_1^a\parallel \mathrm{z}\left(\varrho \right)\parallel d\varrho .$$

Definition 2

([39,40]). The Hadamard integral of fractional order $\alpha \in {\mathrm{I}\mathrm{R}}^{+}$ for a map $f\in L^1([1,a],\mathcal{E})$ is defined by

$$I^{\alpha }f\left(\varrho \right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}\underset{1}{\overset{a}{\int }}\frac{f\left(s\right)}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds,$$

where the Euler Gamma function, Γ, is defined as

$$\mathsf{\Gamma }\left(\alpha \right)={\int }_0^{\infty }{\varrho }^{\alpha -1}{e}^{-\varrho }d\varrho ,\alpha >0.$$

Definition 3

([41]). A mapping $\nu :{\mathcal{Z}}_{\mathcal{E}}\to {\mathrm{I}\mathrm{R}}^{+}$ is known as an MNC in $\mathcal{E}$ if the following axioms hold:

 (i)

$\nu \left(\mathcal{X}\right)=0$ iff $\mathcal{X}\in {\mathcal{N}}_{\mathcal{E}}$;

 (ii)

$\mathcal{X}\subseteq \mathcal{Y}\mathrm{implies}\nu \left(\mathcal{X}\right)\le \nu \left(\mathcal{Y}\right)$;

 (iii)

$\nu \left(\overline{\mathcal{X}}\right)=\nu \left(\mathcal{X}\right)$;

 (iv)

$\nu \left(Conv\mathcal{X}\right)=\nu \left(\mathcal{X}\right)$;

 (v)

$\nu (\lambda \mathcal{X}+(1-\lambda \left)\mathcal{Y}\right)\le \lambda \nu \left(\mathcal{X}\right)+(1-\lambda )\nu \left(\mathcal{Y}\right),\mathrm{for}\mathrm{all}\lambda \in [0,1]$;

 (vi)

Suppose that a sequence $\left\{{\mathcal{X}}_{\rho }\right\}$ of closed sets obtained by ${\mathcal{Z}}_{\mathcal{E}}$ in such a way ${\mathcal{X}}_{\rho }$ contains ${\mathcal{X}}_{\rho +1}$, where $\rho =1,2,...$ and that ${lim}_{\rho \to +\infty }\nu \left({\mathcal{X}}_{\rho }\right)=0$, then, we are able to conclude that ${\mathcal{X}}_{\infty }={\bigcap }_{\rho =1}^{+\infty }{\mathcal{X}}_{\rho }$ is non-empty.

Definition 4

([41]). Assume that $\mathcal{N}$ is a non-empty subset of the Banach space $\mathcal{E}$ and let $\mathcal{S}:\mathcal{N}\to \mathcal{E}$ be a continuous operator that transforms bounded subsets of $\mathcal{N}$ to bounded ones. Furthermore, let ν be the MNC in $\mathcal{E}$. Then, we are able to deduce that $\mathcal{S}$ fulfills the Darbo condition on the set $\mathcal{N}$ together with the constant k concerning the measure ν, provided that

$$\nu \left(\mathcal{S}\mathcal{X}\right)\le k\nu \left(\mathcal{X}\right),\mathrm{for}\mathrm{all}\mathcal{X}\in {\mathcal{Z}}_{\mathcal{E}}\mathrm{such}\mathrm{that}\mathcal{X}\subset \mathcal{N}.$$

Further, when $k<1$, we are able to conclude that $\mathcal{S}$ is the contraction concerning the measure ν.

Theorem 1

([41]). Let $\mathcal{Q}$ be a non-empty, bounded, closed, and convex subset of the space $\mathcal{E}$ and let $\mathcal{H}:\mathcal{Q}\to \mathcal{Q}$ be a continuous transformation which is a contraction concerning the measure ν, i.e., there is a constant $0\le k<1$, such that $\nu \left(\mathcal{H}\mathcal{W}\right)\le k\nu \left(\mathcal{W}\right),$ for any non-empty subset $\mathcal{W}$ of $\mathcal{Q}$. Then, $\mathcal{H}$ has a fixed point in the set $\mathcal{Q}.$

Theorem 2

([42,43]). Consider $\mathcal{N}$ to be a non-empty, closed, convex, and bounded subset of $\mathcal{C}[1,a]$, together with the operators $\mathcal{P}$ and $\mathcal{T}$, which transform continuously on the set $\mathcal{N}$ into $\mathcal{C}[1,a]$ in such a manner that $\mathcal{P}\left(\mathcal{N}\right)$ and $\mathcal{T}\left(\mathcal{N}\right)$ are bounded. In addition, suppose that the operator $\mathcal{S}=\mathcal{P}·\mathcal{T}$ transforms $\mathcal{N}$ into itself. Further, each of the operators $\mathcal{P}$ and $\mathcal{T}$ fulfill the Darbo condition on the set $\mathcal{N}$, together with the constants $K_1$ and $K_2$, correspondingly. Then, we are able to conclude that the operator S fulfills the Darbo condition on $\mathcal{N}$ along with the constant

$$\parallel \mathcal{P}\left(\mathcal{N}\right)\parallel K_2+\parallel \mathcal{T}\left(\mathcal{N}\right)\parallel K_1.$$

Remark 1.

In particular, suppose that $\parallel \mathcal{P}\left(\mathcal{N}\right)\parallel K_2+\parallel \mathcal{T}\left(\mathcal{N}\right)\parallel K_1<1$, then we are able to conclude that $\mathcal{S}$ is a contraction regarding the measure ν and possesses at least one fixed point in the set $\mathcal{N}$.

The space $\mathcal{C}\left(\mathcal{I}\right)=\mathcal{C}[1,a]$, which has all real valued continuous maps defined on $\mathcal{I}=[1,a]$, together with the standard norm $\parallel \mathrm{x}\parallel =max\left\{\right|\mathrm{x}\left(\varrho \right)|:\varrho \in [1,a\left]\right\},$, is viewed as Banach space. To be precise, the space $\mathcal{C}[1,a]$ also has the composition of Banach algebra.

Definition 5

([43]). Let us suppose $\mathcal{W}$ is a non-empty, bounded, fixed subset of $\mathcal{C}\left(\mathcal{I}\right),$$\rho >0$, and $\mathrm{x}\in \mathcal{W},$; we define $w(\mathrm{x},\rho ),$ the modulus of continuity of $\mathrm{x}$ as

$$w(\mathrm{x},\rho )=sup\left\{\right|\mathrm{x}\left({\varrho }_2\right)-\mathrm{x}\left({\varrho }_1\right)|:{\varrho }_1,{\varrho }_2\in [1,a];|{\varrho }_2-{\varrho }_1|\le \rho \}.$$

Further, we define

$$w(\mathcal{W},\rho )=sup\left\{w\right(\mathrm{x},\rho ):\mathrm{x}\in \mathcal{W}\},$$

and

$$w_0\left(\mathcal{W}\right)=\underset{\rho \to 0}{lim}w(\mathcal{W},\rho ),$$

where the map $w_0\left(\mathcal{W}\right)$ is an MNC in the space $\mathcal{C}\left(\mathcal{I}\right)$ in such a way that the Hausdorff MNC is given by $\nu \left(\mathcal{W}\right)=\frac{1}{2}{w}_0(\mathcal{W}$) (see [41]).

3. New Results

In this section, we investigate the existence of a non-linear FFIE Hadamard-type Equation (1) given the following hypotheses:

$\left(H_1\right)$

The maps $g_i(\varrho ,\mathrm{x}\left(\varrho \right))=g_i:\mathcal{I}\times \mathrm{I}\mathrm{R}\to \mathrm{I}\mathrm{R}$ are continuous. Furthermore, there are continuous maps $b_i\left(\varrho \right):[1,a]\to \mathrm{I}\mathrm{R}$, such that

$$\begin{array}{cc}\hfill |g_i(\varrho ,\mathrm{x})-g_i(\varrho ,\mathrm{z})|& \le b_i\left(\varrho \right)|\mathrm{x}-\mathrm{z}|,\hfill \end{array}$$

for $i=1,2.$ Furthermore, there are positive constants M and $M^{\prime }$, so that

$${\displaystyle M=\underset{i}{max}\left\{\right|b_i\left(\varrho \right)|:\varrho \in [1,a]\},}$$

together with

$${\displaystyle M^{\prime }=\underset{i}{max}\left\{\right|g_i(\varrho ,0)|:\varrho \in [1,a]\},}$$

respectively, for $i=1,2.$

$\left(H_2\right)$

There are the continuous maps $a_j:[1,a]\to [1,a]$, for $j=5,6,7,8,9,10$, in such a way that

$$\begin{array}{cc}\hfill |F(\varrho ,{\mathrm{x}}_1,{\mathrm{z}}_1,{\mathrm{x}}_2)-F(\varrho ,{\mathrm{x}}_3,{\mathrm{z}}_2,{\mathrm{x}}_4)|& \le a_5\left(\varrho \right)|{\mathrm{x}}_1-{\mathrm{x}}_3|+a_6\left(\varrho \right)|{\mathrm{z}}_1-{\mathrm{z}}_2|+a_7\left(\varrho \right)|{\mathrm{x}}_2-{\mathrm{x}}_4|,\hfill \\ \hfill |G(\varrho ,{\mathrm{x}}_1,{\mathrm{z}}_1,{\mathrm{x}}_2)-G(\varrho ,{\mathrm{x}}_3,{\mathrm{z}}_2,{\mathrm{x}}_4)|& \le a_8\left(\varrho \right)|{\mathrm{x}}_1-{\mathrm{x}}_3|+a_9\left(\varrho \right)|{\mathrm{z}}_1-{\mathrm{z}}_2|+a_{10}\left(\varrho \right)|{\mathrm{x}}_2-{\mathrm{x}}_4|,\hfill \end{array}$$

for all $\varrho \in [1,a]and{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4,{\mathrm{z}}_1,{\mathrm{z}}_2\in \mathrm{I}\mathrm{R}.$ Furthermore, there is a non-negative constant, K, so that

$${\displaystyle K=\underset{i}{max}\{a_i\left(\varrho \right):\varrho \in [1,a]\}.}$$

$\left(H_3\right)$

The maps $u=u(\varrho ,s,\mathrm{x}\left(\acute{a}\left(s\right)\right)),togetherwithv=v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right))$, act continuously from the set $[1,a]\times [1,a]\times \mathrm{I}\mathrm{R}into\mathrm{I}\mathrm{R}$. Furthermore, there are non-decreasing maps $\mathsf{\Phi },\mathsf{\Psi }:\mathrm{I}{\mathrm{R}}^{+}\to \mathrm{I}{\mathrm{R}}^{+}$ such that $|u(\varrho ,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))|\le \mathsf{\Phi }(\left|\mathrm{x}\right|),togetherwith|v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right))|\le \mathsf{\Psi }(\left|\mathrm{x}\right|).$

$\left(H_4\right)$

There is a solution for ${\mathrm{r}}_0>0$ in the following inequality

$$\begin{array}{cc}\hfill & \left[2K{\mathrm{r}}_0+p+K(M{\mathrm{r}}_0+M^{\prime })\frac{\mathsf{\Phi }\left({\mathrm{r}}_0\right)}{\mathsf{\Gamma }(\alpha +1)}{(loga)}^{\alpha }\right]\times [2K{\mathrm{r}}_0+p\hfill \\ \hfill & +K(M{\mathrm{r}}_0+M^{\prime })\mathsf{\Psi }\left({\mathrm{r}}_0\right)(a-1)]\le {\mathrm{r}}_0.\hfill \end{array}$$

(2)

Moreover,

$$\begin{array}{cc}\hfill & \left[2K{\mathrm{r}}_0+p+K(M{\mathrm{r}}_0+M^{\prime })\frac{\mathsf{\Phi }\left({\mathrm{r}}_0\right)}{\mathsf{\Gamma }(\alpha +1)}{(loga)}^{\alpha }\right](K\left((M\mathsf{\Psi }\left({\mathrm{r}}_0\right)(a-1)+2)\right)\hfill \\ \hfill & +[2K{\mathrm{r}}_0+p+K(M{\mathrm{r}}_0+M^{\prime })\mathsf{\Psi }\left({\mathrm{r}}_0\right)(a-1)]\left(K\left(\left(\frac{M\mathsf{\Phi }\left({\mathrm{r}}_0\right){(loga)}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right)+2\right)\right)<1\hfill \end{array}$$

(3)

also holds.

Remark 2.

As a consequence of the hypothesis $\left(H_2\right)$, there is a non-negative constant, p, so that

$$\begin{array}{cc}\hfill \left|F\right(\varrho ,0,0,0\left)\right|& \le p,\hfill \\ \hfill \left|G\right(\varrho ,0,0,0\left)\right|& \le p,\hfill \end{array}$$

$\mathrm{for}\mathrm{all}\varrho \in [1,a]$.

Furthermore, the new result of the our study is as follows:

Theorem 3.

Under the hypotheses $\left(H_1\right)$–$\left(H_4\right)$, together with Remark 2, Equation (1) has at least one solution in $\mathcal{C}\left(\mathcal{I}\right).$

 Proof.

Let us propose the operator $\mathcal{T}$, for $\mathrm{x}\in \mathcal{C}\left(\mathcal{I}\right)$, as

$$\begin{array}{cc}\hfill \left(\mathcal{T}\mathrm{x}\right)\left(\varrho \right)& =\left(\mathcal{F}\mathrm{x}\right)\left(\varrho \right)\times \left(\mathcal{G}\mathrm{x}\right)\left(\varrho \right),\varrho \in [1,a],\hfill \\ \hfill where\\ \hfill \left(\mathcal{F}\mathrm{x}\right)\left(\varrho \right)& =F\left(\varrho ,\mathrm{x}\left(\varrho \right),\frac{g_1(\varrho ,\mathrm{x}\left(\chi \left(\varrho \right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{u(\varrho ,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left(\varrho \right)\right)\right),\hfill \\ \hfill and\\ \hfill \left(\mathcal{G}\mathrm{x}\right)\left(\varrho \right)& =G\left(\varrho ,\mathrm{x}\left(\varrho \right),g_2(\varrho ,\mathrm{x}\left(\eta \left(\varrho \right)\right))\underset{1}{\overset{a}{\int }}v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right))ds,\mathrm{x}\left(d\left(\varrho \right)\right)\right).\hfill \end{array}$$

We split the proof of main theorem into following steps:

• Step (1):
  We claim that $\mathcal{F},\mathcal{G}\in \mathcal{C}\left(\mathcal{I}\right),$ where $\mathcal{I}=[1,a]$.

For this, it is sufficient to verify that $k,k^{\prime }\in \mathcal{C}\left(\mathcal{I}\right),$ wherein

$$\begin{array}{cc}\hfill k\left(\varrho \right)& =\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{u(\varrho ,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{(log\frac{\varrho }{s})}^{\alpha -1}ds,\hfill \\ \hfill \mathrm{and}{k}^{\prime }\left(\varrho \right)& =\underset{1}{\overset{a}{\int }}v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right))ds.\hfill \end{array}$$

To prove this, let us fix $\rho >0$ and ${\varrho }_1,{\varrho }_2\in [1,a]$ so that $|{\varrho }_1-{\varrho }_2|\le \rho ,$. We estimate

$$\begin{array}{cc}\hfill & |k\left({\varrho }_2\right)-k\left({\varrho }_1\right)|\hfill \\ \hfill & =|\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_2}{\int }}\frac{u({\varrho }_2,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_1}{\int }}\frac{u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_1}{s}\right)}^{\alpha -1}ds|\hfill \\ \hfill & \le |\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_2}{\int }}\frac{u({\varrho }_2,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_2}{\int }}\frac{u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds|\hfill \\ \hfill & +|\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_2}{\int }}\frac{u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_1}{\int }}\frac{u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds|\hfill \\ \hfill & +|\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_1}{\int }}\frac{u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_1}{\int }}\frac{u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_1}{s}\right)}^{\alpha -1}ds|\hfill \\ \hfill & \le \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_2}{\int }}\frac{|u({\varrho }_2,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))-u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))|}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{{\varrho }_1}{\overset{{\varrho }_2}{\int }}\frac{\left|u\right({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right)\left)\right|}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_1}{\int }}\frac{\left|u\right({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right)\left)\right|}{s}|{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds-{\left(log\frac{{\varrho }_1}{s}\right)}^{\alpha -1}|ds\hfill \\ \hfill & \le \frac{w(u,\rho )}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_2}{\int }}\frac{1}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds+\frac{\mathsf{\Phi }(\parallel \mathrm{x}\parallel )}{\mathsf{\Gamma }\left(\alpha \right)}\underset{{\varrho }_1}{\overset{{\varrho }_2}{\int }}\frac{1}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\frac{\mathsf{\Phi }(\parallel \mathrm{x})\parallel }{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_1}{\int }}\frac{1}{s}|{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds-{\left(log\frac{{\varrho }_1}{s}\right)}^{\alpha -1}|ds\hfill \\ \hfill & \le \frac{w(u,\rho )}{\mathsf{\Gamma }(\alpha +1)}{(log{\varrho }_2)}^{\alpha }+\frac{\mathsf{\Phi }(\parallel \mathrm{x}\parallel )}{\mathsf{\Gamma }(\alpha +1)}{\left(log\frac{{\varrho }_2}{t_1}\right)}^{\alpha }+\frac{\mathsf{\Phi }(\parallel \mathrm{x}\parallel )}{\mathsf{\Gamma }(\alpha +1)}\left[{(log{\varrho }_2)}^{\alpha }-{(log{\varrho }_1)}^{\alpha }+{\left(log\frac{{\varrho }_2}{{\varrho }_1}\right)}^{\alpha }\right],\hfill \end{array}$$

where $w(u,\rho )=sup\left\{\right|u({\varrho }_2,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))-u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))|:{\varrho }_1,{\varrho }_2\in [1,a],|{\varrho }_2-{\varrho }_1|\le \rho \}.$

We know that u is uniformly continuous on $\mathcal{I}$; therefore, $w(u,\rho )\to 0$ when $\rho \to 0$. In this way, ${\varrho }_1\to {\varrho }_2$,

$$|k\left({\varrho }_2\right)-k\left({\varrho }_1\right)|\le 0.$$

Thus, we summarize that $k\in \mathcal{C}\left(\mathcal{I}\right)$.

Now, let us fix $\varrho \in \mathcal{I}$; ${\varrho }_m$ is a sequence in $\mathcal{I}$, such that ${\varrho }_m\to \varrho$ when $m\to +\infty ,s\in \mathcal{I}.$

As a result of the continuous map v, we write $v({\varrho }_m,s,\mathrm{x}\left(c\left(s\right)\right))\to v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right)),$ as $m\to +\infty ,s\in \mathcal{I}$. Furthermore, from hypothesis $\left(H_3\right)$, we write

$$|v({\varrho }_m,s,\mathrm{x}\left(c\left(s\right)\right))|\le \mathsf{\Psi }(\left|\mathrm{x}\right|),\mathrm{for}\mathrm{all}m.$$

Now, by applying dominated convergence theorem, we have

$$k^{\prime }\left({\varrho }_m\right)\to k^{\prime }\left(\varrho \right)asm\to +\infty .$$

Thus, $k^{\prime }\in \mathcal{C}\left(\mathcal{I}\right).$ Hence, $\mathcal{T}:\mathcal{C}\left(\mathcal{I}\right)\to \mathcal{C}\left(\mathcal{I}\right)$ is clearly established.

• Step (2):
  We claim that $\mathcal{T}{\mathcal{B}}_{{\mathrm{r}}_0}\subset {\mathcal{B}}_{{\mathrm{r}}_0}.$

For this, suppose that $\mathrm{x}\in \mathcal{C}\left(\mathcal{I}\right)$ be such that $\left|\mathrm{x}\right|\le {\mathrm{r}}_0.$ Using hypothesis $\left(H_1\right)and\left(H_3\right)$, for all $\varrho \in \mathcal{I}$, we estimate that

$$\begin{array}{cc}\hfill & \left|\right(\mathcal{F}\mathrm{x}\left)\right(\varrho \left)\right|\hfill \\ & =|F\left(\varrho ,\mathrm{x}\left(\varrho \right),\frac{g_1(\varrho ,\mathrm{x}\left(\chi \left(\varrho \right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{u(\varrho ,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left(\varrho \right)\right)\right)|\hfill \\ \hfill & \le |F\left(\varrho ,\mathrm{x}\left(\varrho \right),\frac{g_1(\varrho ,\mathrm{x}\left(\chi \left(\varrho \right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{u(\varrho ,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left(\varrho \right)\right)\right)\hfill \\ & -F(\varrho ,0,0,0)|+|F(\varrho ,0,0,0)|\hfill \\ \hfill & \le a_5\left(\varrho \right)\left|\mathrm{x}\left(\varrho \right)\right|+a_6\left(\varrho \right)\frac{|g_1(\varrho ,\mathrm{x}\left(\chi \left(\varrho \right)\right))|}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{\left|u\right(\varrho ,s,\mathrm{x}\left(\acute{a}\left(s\right)\right)\left)\right|}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds+a_7\left(\varrho \right)\left|\mathrm{x}\left(b\left(\varrho \right)\right)\right|\hfill \\ & +p\hfill \\ \hfill & \le a_5\left(\varrho \right)\left|\mathrm{x}\left(\varrho \right)\right|+a_7\left(\varrho \right)\left|\mathrm{x}\left(b\left(\varrho \right)\right)\right|+p\hfill \\ & +a_6\left(\varrho \right)\left(\left[\right|g_1(\varrho ,\mathrm{x}\left(\chi \left(\varrho \right)\right))-g_1(\varrho ,0)|+|g_1(\varrho ,0)\left|\right]\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{\mathsf{\Phi }\left(\right|\mathrm{x}\left(s\right)\left|\right)}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds\right)\hfill \\ \hfill & \le 2K\parallel \mathrm{x}\parallel +p+K(M\parallel \mathrm{x}\parallel +M^{\prime })\frac{\mathsf{\Phi }(\parallel \mathrm{x}\parallel )}{\mathsf{\Gamma }(\alpha +1)}{(loga)}^{\alpha }.\hfill \end{array}$$

Now,

$$\begin{array}{cc}\hfill \parallel {\mathcal{FB}}_{\mathrm{r}0}\parallel & \le 2K{\mathrm{r}}_0+p+K(M{\mathrm{r}}_0+M^{\prime })\frac{\mathsf{\Phi }\left({\mathrm{r}}_0\right)}{\mathsf{\Gamma }(\alpha +1)}{(loga)}^{\alpha }.\hfill \end{array}$$

(4)

Similarly, for all $\varrho \in \mathcal{I}$, we obtain

$$\begin{array}{cc}\hfill & \left|\right(\mathcal{G}\mathrm{x}\left)\right(\varrho \left)\right|\hfill \\ & =\left|G\left(\varrho ,\mathrm{x}\left(\varrho \right),g_2(\varrho ,\mathrm{x}\left(\eta \left(\varrho \right)\right))\underset{1}{\overset{a}{\int }}v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right))ds,\mathrm{x}\left(d\left(\varrho \right)\right)\right)\right|\hfill \\ \hfill & \le |G\left(\varrho ,\mathrm{x}\left(\varrho \right),g_2(\varrho ,\mathrm{x}\left(\eta \left(\varrho \right)\right))\underset{1}{\overset{a}{\int }}v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right))ds,\mathrm{x}\left(d\left(\varrho \right)\right)\right)-G(\varrho ,0,0,0)|+|G(\varrho ,0,0,0)|\hfill \\ \hfill & \le a_8\left(\varrho \right)\left|\mathrm{x}\left(\varrho \right)\right|+a_9\left(\varrho \right)\left(\right|g_2(\varrho ,\mathrm{x}\left(\eta \left(\varrho \right)\right))|\underset{1}{\overset{a}{\int }}\left|v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right))\right|ds)+a_{10}\left(\varrho \right)\left|\mathrm{x}\left(d\left(\varrho \right)\right)\right|+p\hfill \\ \hfill & \le a_8\left(\varrho \right)\left|\mathrm{x}\left(\varrho \right)\right|+a_9\left(\varrho \right)\left(\left[\right|g_2(\varrho ,\mathrm{x}\left(\eta \left(\varrho \right)\right))-g_2(\varrho ,0)|+|g_2(\varrho ,0)\left|\right]\underset{1}{\overset{a}{\int }}\mathsf{\Psi }\left(\right|\mathrm{x}\left(s\right)\left|\right)ds\right)\hfill \\ & +a_{10}\left(\varrho \right)\left|\mathrm{x}\left(d\left(\varrho \right)\right)\right|+p\hfill \\ \hfill & \le 2K\parallel \mathrm{x}\parallel +p+K(M\parallel \mathrm{x}\parallel +M^{\prime }\left)\mathsf{\Psi }\right(\parallel \mathrm{x}\parallel )(a-1).\hfill \end{array}$$

Furthermore,

$$\begin{array}{cc}\hfill \parallel {\mathcal{GB}}_{{\mathrm{r}}_0}\parallel & \le 2K{\mathrm{r}}_0+p+K(M{\mathrm{r}}_0+M^{\prime })\mathsf{\Psi }\left({\mathrm{r}}_0\right)(a-1).\end{array}$$

(5)

The above inequalities, (4) and (5), prove that $\mathcal{F}\left({\mathcal{B}}_{{\mathrm{r}}_0}\right),\mathcal{G}\left({\mathcal{B}}_{{\mathrm{r}}_0}\right)$ are bound in $\mathcal{C}\left(\mathcal{I}\right)$. Furthermore, we find $\left(\mathcal{T}\mathrm{x}\right)\left(\varrho \right)=\left(\mathcal{F}\mathrm{x}\right)\left(\varrho \right)\times \left(\mathcal{G}\mathrm{x}\right)\left(\varrho \right)$ on the Banach algebra $\mathcal{C}\left(\mathcal{I}\right)$. Therefore, by considering the inequalities (4) and (5) together with the hypothesis $\left(H_4\right)$, we obtain

$$\begin{array}{cc}\hfill \left|\right(\mathcal{T}\mathrm{x}\left)\right(\varrho \left)\right|& =\left|\right(\mathcal{F}\mathrm{x}\left)\right(\varrho )\times (\mathcal{G}\mathrm{x}\left)\right(\varrho \left)\right|\hfill \\ \hfill & \le [2K{\mathrm{r}}_0+p+K(M{\mathrm{r}}_0+M^{\prime })\frac{\mathsf{\Phi }\left({\mathrm{r}}_0\right)}{\mathsf{\Gamma }(\alpha +1)}{(loga)}^{\alpha }]\hfill \\ & \times [2K{\mathrm{r}}_0+p+K(M{\mathrm{r}}_0+M^{\prime })\mathsf{\Psi }\left({\mathrm{r}}_0\right)(a-1)]\hfill \\ \hfill & \le {\mathrm{r}}_0,\hfill \end{array}$$

for any $\mathrm{x}\in {\mathcal{B}}_{{\mathrm{r}}_0}$ and $\varrho \in \left(\mathcal{I}\right)$, which implies that $\mathcal{T}{\mathcal{B}}_{{\mathrm{r}}_0}\subset {\mathcal{B}}_{{\mathrm{r}}_0}.$

• Step (3):
  We show that $\mathcal{F}and\mathcal{G}$ are continuous in ${\mathcal{B}}_{{\mathrm{r}}_0}.$

For this, suppose that ${\mathrm{x}}_m$ is a sequence so that ${\mathrm{x}}_m\to \mathrm{x}$ is together with $\mathrm{x}\in {\mathcal{B}}_{{\mathrm{r}}_0}.$ Then, according to hypotheses $\left(H_1\right)and\left(H_3\right)$, we estimate that

$$\begin{array}{cc}\hfill & |\mathcal{F}\left({\mathrm{x}}_m\right)\left(\varrho \right)-\mathcal{F}\left(\mathrm{x}\right)\left(\varrho \right)|\hfill \\ & \le |F\left(t,{\mathrm{x}}_m\left(\varrho \right),\frac{g_1(\varrho ,{\mathrm{x}}_m\left(\chi \left(\varrho \right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{u(\varrho ,s,{\mathrm{x}}_m\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds,{\mathrm{x}}_m\left(b\left(\varrho \right)\right)\right)\hfill \\ \hfill & -F\left(\varrho ,\mathrm{x}\left(\varrho \right),\frac{g_1(\varrho ,\mathrm{x}\left(\chi \left(\varrho \right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{u(\varrho ,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left(\varrho \right)\right)\right)|\hfill \\ \hfill & \le |F\left(\varrho ,{\mathrm{x}}_m\left(\varrho \right),\frac{g_1(\varrho ,{\mathrm{x}}_m\left(\chi \left(\varrho \right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{u(\varrho ,s,{\mathrm{x}}_m\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds,{\mathrm{x}}_m\left(b\left(\varrho \right)\right)\right)\hfill \\ \hfill & -F\left(\varrho ,\mathrm{x}\left(\varrho \right),\frac{g_1(\varrho ,{\mathrm{x}}_m\left(\chi \left(\varrho \right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{u(\varrho ,s,{\mathrm{x}}_m\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left(\varrho \right)\right)\right)|\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +|F\left(\varrho ,\mathrm{x}\left(\varrho \right),\frac{g_1(\varrho ,{\mathrm{x}}_m\left(\chi \left(\varrho \right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{u(\varrho ,s,{\mathrm{x}}_m\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left(\varrho \right)\right)\right)\hfill \\ \hfill & -F\left(\varrho ,\mathrm{x}\left(\varrho \right),\frac{g_1(\varrho ,\mathrm{x}\left(\chi \left(\varrho \right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{u(\varrho ,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left(\varrho \right)\right)\right)|\hfill \\ \hfill & \le a_5\left(\varrho \right)|{\mathrm{x}}_m\left(\varrho \right)-\mathrm{x}\left(\varrho \right)|+a_7\left(\varrho \right)|{\mathrm{x}}_m\left(b\left(\varrho \right)\right)-\mathrm{x}\left(b\left(\varrho \right)\right)|\hfill \\ \hfill & +a_6\left(\varrho \right)|\frac{g_1(\varrho ,{\mathrm{x}}_m\left(\chi \left(\varrho \right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{u(\varrho ,s,{\mathrm{x}}_m\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds\hfill \\ \hfill & -\frac{g_1(\varrho ,\mathrm{x}\left(\chi \left(\varrho \right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{u(\varrho ,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds|\hfill \\ \hfill & \le a_5\left(\varrho \right)|{\mathrm{x}}_m\left(\varrho \right)-\mathrm{x}\left(\varrho \right)|+a_7\left(\varrho \right)|{\mathrm{x}}_m\left(b\left(\varrho \right)\right)-\mathrm{x}\left(b\left(\varrho \right)\right)|\hfill \\ \hfill & +a_6\left(\varrho \right)(\frac{|g_1(\varrho ,{\mathrm{x}}_m\left(\chi \left(\varrho \right)\right))|}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{\varrho }{\int }}\frac{|u(\varrho ,s,{\mathrm{x}}_m\left(\acute{a}\left(s\right)\right))-u(\varrho ,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))|}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds\hfill \\ \hfill & +\frac{|g_1(\varrho ,{\mathrm{x}}_m\left(\chi \left(\varrho \right)\right))-g_1(\varrho ,\mathrm{x}\left(\chi \left(\varrho \right)\right))}{\mathsf{\Gamma }\left(\alpha \right)|}\underset{1}{\overset{\varrho }{\int }}\frac{\left|u\right(\varrho ,s,\mathrm{x}\left(\acute{a}\left(s\right)\right)\left)\right|}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds)\hfill \\ \hfill & \le a_5\left(\varrho \right)|{\mathrm{x}}_m\left(\varrho \right)-\mathrm{x}\left(\varrho \right)|+a_7\left(\varrho \right)|{\mathrm{x}}_m\left(b\left(\varrho \right)\right)-\mathrm{x}\left(b\left(\varrho \right)\right)|\hfill \\ \hfill & +a_6\left(\varrho \right)(\frac{|g_1(\varrho ,{\mathrm{x}}_m\left(\chi \left(\varrho \right)\right))|}{\mathsf{\Gamma }\left(\alpha \right)}{\delta }_u\left(\rho \right)\underset{1}{\overset{\varrho }{\int }}\frac{1}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds\hfill \\ & +b_1\left(\varrho \right)\parallel {\mathrm{x}}_m\left(\chi \left(\varrho \right)\right)-\mathrm{x}\left(\chi \left(\varrho \right)\right)\parallel \mathsf{\Phi }(\parallel \mathrm{x}\parallel )\underset{1}{\overset{\varrho }{\int }}\frac{1}{s}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}ds),\hfill \\ \hfill & \le 2K\parallel {\mathrm{x}}_m-\mathrm{x}\parallel +K\left((M\parallel \mathrm{x}\parallel +M^{\prime })\frac{{\delta }_u\left(\rho \right)}{\mathsf{\Gamma }(\alpha +1)}{(loga)}^{\alpha }+M\parallel {\mathrm{x}}_m-\mathrm{x}\parallel \frac{\mathsf{\Phi }(\parallel \mathrm{x}\parallel )}{\mathsf{\Gamma }(\alpha +1)}{(loga)}^{\alpha }\right),\hfill \end{array}$$

where ${\delta }_u\left(\rho \right)=sup\left\{\right|u(\varrho ,s,{\mathrm{x}}_m)-u(t,s,\mathrm{x})|:\varrho ,s\in [1,a];{\mathrm{x}}_m,\mathrm{x}\in [-{\mathfrak{r}}_{\mathfrak{0}},{\mathrm{r}}_0];∥{\mathrm{x}}_m-\mathrm{x}∥\le \rho \}.$

Now, ${\mathrm{x}}_m\to \mathrm{x}asm\to +\infty$. Hence,

$$∥\mathcal{F}\left({\mathrm{x}}_m\right)\left(\varrho \right)-\mathcal{F}\left(\mathrm{x}\right)\left(\varrho \right)∥\to 0,asm\to +\infty .$$

Thus, the operator $\mathcal{F}:{\mathcal{B}}_{{\mathrm{r}}_0}\to \mathcal{C}\left(\mathcal{I}\right)$ is continuous.

Similarly,

$$\begin{array}{cc}\hfill & |\mathcal{G}\left({\mathrm{x}}_m\right)\left(\varrho \right)-\mathcal{G}\left(\mathrm{x}\right)\left(\varrho \right)|\hfill \\ & \le |G\left(\varrho ,{\mathrm{x}}_m\left(\varrho \right),g_2(\varrho ,{\mathrm{x}}_m\left(\eta \left(\varrho \right)\right))\underset{1}{\overset{a}{\int }}v(\varrho ,s,{\mathrm{x}}_m\left(c\left(s\right)\right))ds,{\mathrm{x}}_m\left(d\left(\varrho \right)\right)\right)\hfill \\ \hfill & -G\left(\varrho ,\mathrm{x}\left(\varrho \right),g_2(\varrho ,\mathrm{x}\left(\eta \left(\varrho \right)\right))\underset{1}{\overset{a}{\int }}v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right))ds,\mathrm{x}\left(d\left(\varrho \right)\right)\right)|\hfill \\ \hfill & \le |G\left(\varrho ,{\mathrm{x}}_m\left(\varrho \right),g_2(\varrho ,{\mathrm{x}}_m\left(\eta \left(\varrho \right)\right))\underset{1}{\overset{a}{\int }}v(\varrho ,s,{\mathrm{x}}_m\left(c\left(s\right)\right))ds,{\mathrm{x}}_m\left(d\left(\varrho \right)\right)\right)\hfill \\ \hfill & -G\left(\varrho ,\mathrm{x}\left(\varrho \right),g_2(\varrho ,{\mathrm{x}}_m\left(\eta \left(\varrho \right)\right))\underset{1}{\overset{a}{\int }}v(\varrho ,s,{\mathrm{x}}_m\left(c\left(s\right)\right))ds,\mathrm{x}\left(d\left(\varrho \right)\right)\right)|\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +|G\left(\varrho ,\mathrm{x}\left(\varrho \right),g_2(\varrho ,{\mathrm{x}}_m\left(\eta \left(\varrho \right)\right))\underset{1}{\overset{a}{\int }}v(\varrho ,s,{\mathrm{x}}_m\left(c\left(s\right)\right))ds,\mathrm{x}\left(d\left(\varrho \right)\right)\right)\hfill \\ \hfill & -G\left(\varrho ,\mathrm{x}\left(\varrho \right),g_2(\varrho ,\mathrm{x}\left(\eta \left(\varrho \right)\right))\underset{1}{\overset{a}{\int }}v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right))ds,\mathrm{x}\left(d\left(\varrho \right)\right)\right)|\hfill \\ \hfill & \le a_8\left(\varrho \right)|{\mathrm{x}}_m\left(\varrho \right)-\mathrm{x}\left(\varrho \right)|+a_{10}\left(\varrho \right)|{\mathrm{x}}_m\left(d\left(\varrho \right)\right)-\mathrm{x}\left(d\left(\varrho \right)\right)|\hfill \\ \hfill & +a_9\left(\varrho \right)|g_2(\varrho ,{\mathrm{x}}_m\left(\eta \left(\varrho \right)\right))\underset{1}{\overset{a}{\int }}v(\varrho ,s,{\mathrm{x}}_m\left(c\left(s\right)\right))ds-g_2(\varrho ,\mathrm{x}\left(\eta \left(\varrho \right)\right))\underset{1}{\overset{a}{\int }}v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right))ds|\hfill \\ \hfill & \le a_8\left(\varrho \right)|{\mathrm{x}}_m\left(\varrho \right)-\mathrm{x}\left(\varrho \right)|+a_{10}\left(\varrho \right)|{\mathrm{x}}_m\left(d\left(\varrho \right)\right)-\mathrm{x}\left(d\left(\varrho \right)\right)|\hfill \\ \hfill & +a_9\left(\varrho \right)\left(\right|g_2(\varrho ,{\mathrm{x}}_m\left(\eta \left(\varrho \right)\right))|\underset{1}{\overset{a}{\int }}\left[\right|v(\varrho ,s,{\mathrm{x}}_m\left(c\left(s\right)\right))-v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right))\left|\right]ds\hfill \\ \hfill & +|g_2(\varrho ,{\mathrm{x}}_m\left(\eta \left(\varrho \right)\right))-g_2(\varrho ,\mathrm{x}\left(\eta \left(\varrho \right)\right))|\underset{1}{\overset{a}{\int }}\left|v(\varrho ,s,\mathrm{x}\left(c\left(s\right)\right))\right|ds)\hfill \\ \hfill & \le 2K\parallel {\mathrm{x}}_m-\mathrm{x}\parallel +K\left[\right(M\parallel \mathrm{x}\parallel +M^{\prime }){\delta }_v\left(\rho \right)(a-1)+M\parallel {\mathrm{x}}_m-\mathrm{x}\parallel \mathsf{\Psi }(\parallel \mathrm{x}\parallel )(a-1)],\hfill \end{array}$$

where ${\delta }_v\left(\rho \right)=sup\left\{\right|v(\varrho ,s,{\mathrm{x}}_m)-u(\varrho ,s,\mathrm{x})|:\varrho ,s\in [1,a];{\mathrm{x}}_m,\mathrm{x}\in [-{\mathfrak{r}}_{\mathfrak{0}},{\mathrm{r}}_0];∥{\mathrm{x}}_m-\mathrm{x}∥\le \rho \}.$

Furthermore, ${\mathrm{x}}_m\to \mathrm{x}asm\to +\infty$. Hence, we are enabled to find that

$$∥\mathcal{G}\left({\mathrm{x}}_m\right)\left(\varrho \right)-\mathcal{G}\left(\mathrm{x}\right)\left(\varrho \right)∥\to 0,asm\to +\infty .$$

Thus, the operator $\mathcal{G}:{\mathcal{B}}_{{\mathrm{r}}_0}\to \mathcal{C}\left(\mathcal{I}\right)$ is continuous.

• Step (4):
  We prove that $\mathcal{F}and\mathcal{G}$ fulfill Darbo’s condition on ${\mathcal{B}}_{{\mathrm{r}}_0}.$

For this, suppose an arbitrary, fixed $\rho >0$ and $\mathcal{W}$ is a non empty subset of ${\mathcal{B}}_{{\mathrm{r}}_0}.$ Furthermore, we take $\mathrm{x}\in \mathcal{W}and{\varrho }_1,{\varrho }_2\in [1,a]togetherwith{\varrho }_1\le {\varrho }_{2}sothat|{\varrho }_2-{\varrho }_1|\le \rho .$ Then, we estimate that

$$\begin{array}{cc}\hfill & |\mathcal{F}\left(\mathrm{x}\right)\left({\varrho }_2\right)-\mathcal{F}\left(\mathrm{x}\right)\left({\varrho }_1\right)|\hfill \\ & \le |F\left({\varrho }_2,\mathrm{x}\left({\varrho }_2\right),\frac{g_1({\varrho }_2,\mathrm{x}\left(\chi \left({\varrho }_2\right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_2}{\int }}\frac{u({\varrho }_2,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left({\varrho }_2\right)\right)\right)\hfill \\ \hfill & -F\left({\varrho }_1,\mathrm{x}\left({\varrho }_1\right),\frac{g_1({\varrho }_1,\mathrm{x}\left(\chi \left({\varrho }_1\right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_1}{\int }}\frac{u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_1}{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left({\varrho }_1\right)\right)\right)|\hfill \\ \hfill & \le |F\left({\varrho }_2,\mathrm{x}\left({\varrho }_2\right),\frac{g_1({\varrho }_2,\mathrm{x}\left(\chi \left({\varrho }_2\right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_2}{\int }}\frac{u({\varrho }_2,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left({\varrho }_2\right)\right)\right)\hfill \\ \hfill & -F\left({\varrho }_2,\mathrm{x}\left({\varrho }_1\right),\frac{g_1({\varrho }_1,\mathrm{x}\left(\chi \left({\varrho }_1\right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_1}{\int }}\frac{u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_1}{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left({\varrho }_1\right)\right)\right)|\hfill \\ \hfill & +|F\left({\varrho }_2,\mathrm{x}\left({\varrho }_1\right),\frac{g_1({\varrho }_1,\mathrm{x}\left(\chi \left({\varrho }_1\right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_1}{\int }}\frac{u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_1}{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left({\varrho }_1\right)\right)\right)\hfill \\ \hfill & -F\left({\varrho }_1,\mathrm{x}\left({\varrho }_1\right),\frac{g_1({\varrho }_1,\mathrm{x}\left(\chi \left({\varrho }_1\right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_1}{\int }}\frac{u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_1}{s}\right)}^{\alpha -1}ds,\mathrm{x}\left(b\left({\varrho }_1\right)\right)\right)|\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le a_5\left({\varrho }_2\right)|\mathrm{x}\left({\varrho }_2\right)-\mathrm{x}\left({\varrho }_1\right)|+a_6\left({\varrho }_2\right)|\frac{g_1({\varrho }_2,\mathrm{x}\left(\chi \left({\varrho }_2\right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_2}{\int }}\frac{u({\varrho }_2,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds\hfill \\ \hfill & -\frac{g_1({\varrho }_1,\mathrm{x}\left(\chi \left({\varrho }_1\right)\right))}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_1}{\int }}\frac{u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))}{s}{\left(log\frac{{\varrho }_1}{s}\right)}^{\alpha -1}ds|+a_7\left({\varrho }_2\right)|\mathrm{x}\left(b\left({\varrho }_2\right)\right)-\mathrm{x}\left(b\left({\varrho }_1\right)\right)|\hfill \\ \hfill & +w_1(F,\rho )\hfill \\ \hfill & \le a_5\left({\varrho }_2\right)|\mathrm{x}\left({\varrho }_2\right)-\mathrm{x}\left({\varrho }_1\right)|\hfill \\ \hfill & +a_6\left({\varrho }_2\right)(\frac{|g_1({\varrho }_2,\mathrm{x}\left(\chi \left({\varrho }_2\right)\right))|}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_2}{\int }}\frac{|u({\varrho }_2,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))-u({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right))|}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds\hfill \\ \hfill & +\frac{|g_1({\varrho }_2,\mathrm{x}\left(\chi \left({\varrho }_2\right)\right))-g_1({\varrho }_1,\mathrm{x}\left(\chi \left({\varrho }_1\right)\right))|}{\mathsf{\Gamma }\left(\alpha \right)|}\underset{1}{\overset{{\varrho }_2}{\int }}\frac{\left|u\right({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right)\left)\right|}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds\hfill \\ \hfill & +\frac{|g_1({\varrho }_1,\mathrm{x}\left(\chi \left({\varrho }_1\right)\right))|}{\mathsf{\Gamma }\left(\alpha \right)}\{\underset{1}{\overset{{\varrho }_2}{\int }}\frac{\left|u\right({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right)\left)\right|}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds\hfill \\ \hfill & -\underset{1}{\overset{{\varrho }_1}{\int }}\frac{\left|u\right({\varrho }_1,s,\mathrm{x}\left(\acute{a}\left(s\right)\right)\left)\right|}{s}{\left(log\frac{{\varrho }_1}{s}\right)}^{\alpha -1}ds\left\}\right)+a_7\left({\varrho }_2\right)|\mathrm{x}\left(b\left({\varrho }_2\right)\right)-\mathrm{x}\left(b\left({\varrho }_1\right)\right)|+w_1(F,\rho ),\hfill \end{array}$$

i.e.,

$$\begin{array}{cc}\hfill & |\mathcal{F}\left(\mathrm{x}\right)\left({\varrho }_2\right)-\mathcal{F}\left(\mathrm{x}\right)\left({\varrho }_1\right)|\hfill \\ & \le Kw(\mathrm{x},\rho )+K(\left[\frac{b_1\left({\varrho }_2\right)|\mathrm{x}\left(\chi \left({\varrho }_2\right)\right)|+|g_1({\varrho }_2,0)|w(u,\rho )}{\mathsf{\Gamma }\left(\alpha \right)}\right]\underset{1}{\overset{{\varrho }_2}{\int }}\frac{1}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds\hfill \\ \hfill & +\left[\right|g_1({\varrho }_2,\mathrm{x}\left(\chi \left({\varrho }_2\right)\right))-g_1({\varrho }_2,\mathrm{x}\left(\chi \left({\varrho }_1\right)\right))|\hfill \\ \hfill & +|g_1({\varrho }_2,\mathrm{x}\left(\chi \left({\varrho }_1\right)\right))-g_1({\varrho }_1,\mathrm{x}\left(\chi \left({\varrho }_1\right)\right))\left|\right]\frac{\mathsf{\Phi }(\parallel \mathrm{x}\parallel )}{\mathsf{\Gamma }\left(\alpha \right)}\underset{1}{\overset{{\varrho }_2}{\int }}\frac{1}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds\hfill \\ \hfill & +\frac{[b_1\left({\varrho }_1\right)|\mathrm{x}\left(\chi \left({\varrho }_1\right)\right)|+|g_1({\varrho }_1,0)\left|\right]\mathsf{\Phi }(\parallel \mathrm{x}\parallel )}{\mathsf{\Gamma }\left(\alpha \right)}\left\{\underset{1}{\overset{{\varrho }_2}{\int }}\frac{1}{s}{\left(log\frac{{\varrho }_2}{s}\right)}^{\alpha -1}ds-\underset{1}{\overset{{\varrho }_1}{\int }}\frac{1}{s}{\left(log\frac{{\varrho }_1}{s}\right)}^{\alpha -1}ds\right\})\hfill \\ \hfill & +Kw(\mathrm{x},w(b,\rho ))+w_1(F,\rho )\hfill \\ \hfill & \le Kw(\mathrm{x},\rho )+K(\frac{(M\parallel \mathrm{x}\parallel +M^{\prime })w(u,\rho )}{\mathsf{\Gamma }(\alpha +1)}{(log{\varrho }_2)}^{\alpha }\hfill \\ \hfill & +\frac{\left[b_1\left({\varrho }_2\right)\right|\mathrm{x}\left(\chi \left({\varrho }_2\right)\right)|-\mathrm{x}\left(\chi \left({\varrho }_1\right)\right)|+w(g_1,\rho )\mathsf{\Phi }(\parallel \mathrm{x})]}{\mathsf{\Gamma }(\alpha +1)}{(log{\varrho }_2)}^{\alpha }\hfill \\ \hfill & +\frac{(M\parallel \mathrm{x}\parallel +M^{\prime }\left)\mathsf{\Phi }\right(\parallel \mathrm{x}\parallel )}{\mathsf{\Gamma }(\alpha +1)}{[{(log{\varrho }_2)}^{\alpha }-log{\varrho }_1)}^{\alpha }\left]\right)+Kw(\mathrm{x},w(b,\rho ))+w_1(F,\rho )\hfill \\ \hfill & \le Kw(\mathrm{x},\rho )+K(\frac{(M\parallel \mathrm{x}\parallel +M^{\prime })w(u,\rho )}{\mathsf{\Gamma }(\alpha +1)}{(log{\varrho }_2)}^{\alpha }\hfill \\ & +\frac{\left[Mw(\mathrm{x},w(\chi ,\rho ))\right|+w(g_1,\rho )]\mathsf{\Phi }(\parallel \mathrm{x}\parallel )}{\mathsf{\Gamma }(\alpha +1)}{(log{\varrho }_2)}^{\alpha }\hfill \\ & +\frac{(M\parallel \mathrm{x}\parallel +M^{\prime }\left)\mathsf{\Phi }\right(\parallel \mathrm{x}\parallel )}{\mathsf{\Gamma }(\alpha +1)}{[{(log{\varrho }_2)}^{\alpha }-log{\varrho }_1)}^{\alpha }\left]\right)\hfill \\ \hfill & +Kw(\mathrm{x},w(b,\rho ))+w_1(F,\rho )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le Kw(\mathrm{x},\rho )+K(\frac{(M{\mathrm{r}}_0+M^{\prime })w(u,\rho )}{\mathsf{\Gamma }(\alpha +1)}{(log{\varrho }_2)}^{\alpha }\hfill \\ \hfill & +\frac{\left[Mw(\mathrm{x},w(\chi ,\rho ))\right|+w(g_1,\rho )]\mathsf{\Phi }\left({\mathrm{r}}_0\right)}{\mathsf{\Gamma }(\alpha +1)}{(log{\varrho }_2)}^{\alpha }+\frac{(M{\mathrm{r}}_0+M^{\prime })\mathsf{\Phi }\left({\mathrm{r}}_0\right)}{\mathsf{\Gamma }(\alpha +1)}{[{(log{\varrho }_2)}^{\alpha }-log{\varrho }_1)}^{\alpha }\left]\right)\hfill \\ \hfill & +Kw(\mathrm{x},w(b,\rho ))+w_1(F,\rho ),\hfill \end{array}$$

wherein

$$\begin{array}{cc}\hfill w(u,\rho )& =sup\left\{\right|u(\varrho ,s,\mathrm{x})-u({\varrho }^{\prime },s,\mathrm{x})|:\varrho ,{\varrho }^{\prime },s\in [1,a];|\varrho -{\varrho }^{\prime }|\le \rho ;\mathrm{x}\in [-{\mathrm{r}}_0,{\mathrm{r}}_0]\},\hfill \\ \hfill w_1(F,\rho )& =sup\left\{\right|F(\varrho ,{\mathrm{x}}_1,{\mathrm{z}}_1,{\mathrm{x}}_2)-F({\varrho }^{\prime },{\mathrm{x}}_1,{\mathrm{z}}_1,{\mathrm{x}}_2)|:\varrho ,{\varrho }^{\prime }\in [1,a];|\varrho -{\varrho }^{\prime }|\le \rho ;\hfill \\ \hfill & {\mathrm{x}}_1,{\mathrm{x}}_2\in [-{\mathrm{r}}_0,{\mathrm{r}}_0];{\mathrm{z}}_1\in [-K^{\prime }a,K^{\prime }a]\},K^{\prime }=sup\left\{\right|u(\varrho ,s,\mathrm{x})|:\varrho ,s\in [1,a];\mathrm{x}\in [-{\mathrm{r}}_0,{\mathrm{r}}_0]\},\hfill \\ \hfill w(g_1,\rho )& =sup\left\{\right|g_1(\varrho ,\mathrm{x})-g_1({\varrho }^{\prime },\mathrm{x})|:\varrho ,{\varrho }^{\prime }\in [1,a];|\varrho -{\varrho }^{\prime }|\le \rho ;\mathrm{x}\in [-{\mathrm{r}}_0,{\mathrm{r}}_0]\},\hfill \\ \hfill w(\chi ,\rho )& =sup\left\{\right|\chi \left(\varrho \right)-\chi \left({\varrho }^{\prime }\right)|:\varrho ,{\varrho }^{\prime }\in [1,a];|\varrho -{\varrho }^{\prime }|\le \rho \}.\hfill \end{array}$$

By applying the uniform continuity of the maps $F(\varrho ,{\mathrm{x}}_1,{\mathrm{z}}_1,{\mathrm{x}}_2)$ on the set $\mathcal{I}\times \mathrm{I}\mathrm{R}\times \mathrm{I}\mathrm{R}\times \mathrm{I}\mathrm{R}$, together with the map $u=u(\varrho ,s,\mathrm{x})$ on the set $\mathcal{I}\times \mathcal{I}\times \mathrm{I}\mathrm{R}$, we obtain $w(u,\rho )\to 0,w(g_1,\rho )\to 0,w(\chi ,\rho )\to 0,$ and $w_1(F,\rho )\to 0,$ when $\rho \to 0$. Thus, we find

$$w_0\left(\mathcal{F}\mathcal{W}\right)\le 2K{w}_0\left(\mathcal{W}\right)+\frac{M\mathsf{\Phi }\left({\mathrm{r}}_0\right){(loga)}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}{w}_0\left(\mathcal{W}\right).$$

(6)

Similarly,

$$\begin{array}{cc}\hfill & |\mathcal{G}\left(\mathrm{x}\right)\left({\varrho }_2\right)-\mathcal{G}\left(\mathrm{x}\right)\left({\varrho }_1\right)|\hfill \\ & \le |G\left({\varrho }_2,\mathrm{x}\left({\varrho }_2\right),g_2({\varrho }_2,\mathrm{x}\left(\eta \left({\varrho }_2\right)\right))\underset{1}{\overset{a}{\int }}\frac{v({\varrho }_2,s,\mathrm{x}\left(c\left(s\right)\right))}{s},\mathrm{x}\left(d\left({\varrho }_2\right)\right)\right)\hfill \\ \hfill & -G\left({\varrho }_1,\mathrm{x}\left({\varrho }_1\right),g_2({\varrho }_1,\mathrm{x}\left(\eta \left({\varrho }_1\right)\right))\underset{1}{\overset{a}{\int }}\frac{v({\varrho }_1,s,\mathrm{x}\left(c\left(s\right)\right))}{s},\mathrm{x}\left(d\left({\varrho }_1\right)\right)\right)|\hfill \\ \hfill & \le |G\left({\varrho }_2,\mathrm{x}\left({\varrho }_2\right),g_2({\varrho }_2,\mathrm{x}\left(\eta \left({\varrho }_2\right)\right))\underset{1}{\overset{a}{\int }}v({\varrho }_2,s,\mathrm{x}\left(c\left(s\right)\right))ds,\mathrm{x}\left(d\left({\varrho }_2\right)\right)\right)\hfill \\ \hfill & -G\left({\varrho }_2,\mathrm{x}\left({\varrho }_1\right),g_2({\varrho }_1,\mathrm{x}\left(\eta \left({\varrho }_1\right)\right))\underset{1}{\overset{a}{\int }}v({\varrho }_1,s,\mathrm{x}\left(c\left(s\right)\right))ds,\mathrm{x}\left(d\left({\varrho }_1\right)\right)\right)|\hfill \\ \hfill & +|G\left({\varrho }_2,\mathrm{x}\left({\varrho }_1\right),g_2({\varrho }_1,\mathrm{x}\left(\eta \left({\varrho }_1\right)\right))\underset{1}{\overset{a}{\int }}v({\varrho }_1,s,\mathrm{x}\left(c\left(s\right)\right))ds,\mathrm{x}\left(d\left({\varrho }_1\right)\right)\right)\hfill \\ \hfill & -G\left({\varrho }_1,\mathrm{x}\left({\varrho }_1\right),g_2({\varrho }_1,\mathrm{x}\left(\eta \left({\varrho }_1\right)\right))\underset{1}{\overset{a}{\int }}v({\varrho }_1,s,\mathrm{x}\left(c\left(s\right)\right))ds,\mathrm{x}\left(d\left({\varrho }_1\right)\right)\right)|\hfill \\ \hfill & \le a_8\left({\varrho }_2\right)|\mathrm{x}\left({\varrho }_2\right)-\mathrm{x}\left({\varrho }_1\right)|+a_9\left({\varrho }_2\right)|g_2({\varrho }_2,\mathrm{x}\left(\eta \left({\varrho }_2\right)\right))\underset{1}{\overset{a}{\int }}v({\varrho }_2,s,\mathrm{x}\left(c\left(s\right)\right))ds\hfill \\ \hfill & -g_2({\varrho }_1,\mathrm{x}\left(\eta \left({\varrho }_1\right)\right))\underset{1}{\overset{a}{\int }}v({\varrho }_1,s,\mathrm{x}\left(c\left(s\right)\right))ds|+a_{10}\left({\varrho }_2\right)|\mathrm{x}\left(d\left({\varrho }_2\right)\right)-\mathrm{x}\left(d\left({\varrho }_1\right)\right)|+w_1(G,\rho )\hfill \\ \hfill & \le Kw(\mathrm{x},\rho )+K\left(\right|g_2({\varrho }_2,\mathrm{x}\left(\eta \left({\varrho }_2\right)\right))|\underset{1}{\overset{a}{\int }}|v({\varrho }_2,s,\mathrm{x}\left(c\left(s\right)\right))-v({\varrho }_1,s,\mathrm{x}\left(c\left(s\right)\right))|ds\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +|g_2({\varrho }_2,\mathrm{x}\left(\eta \left({\varrho }_2\right)\right))\underset{1}{\overset{a}{\int }}v({\varrho }_1,s,\mathrm{x}\left(c\left(s\right)\right))ds\hfill \\ \hfill & -g_2({\varrho }_1,\mathrm{x}\left(\eta \left({\varrho }_1\right)\right))\underset{1}{\overset{a}{\int }}v({\varrho }_1,s,\mathrm{x}\left(c\left(s\right)\right))ds\left|\right)+Kw(\mathrm{x},w(d,\rho ))+w_1(G,\rho )\hfill \\ \hfill & \le Kw(\mathrm{x},\rho )+K\left(\right[b_2\left({\varrho }_2\right)|\mathrm{x}\left(\eta \left({\varrho }_2\right)\right)|+|g_2({\varrho }_2,0)\left|\right]w(v,\rho )(a-1)\hfill \\ \hfill & +|g_2({\varrho }_2,\mathrm{x}\left(\eta \left({\varrho }_2\right)\right))-g_2({\varrho }_2,\mathrm{x}\left(\eta \left({\varrho }_1\right)\right))|\underset{1}{\overset{a}{\int }}\left|v({\varrho }_1,s,\mathrm{x}\left(c\left(s\right)\right))\right|ds\hfill \\ \hfill & +|g_2({\varrho }_2,\mathrm{x}\left(\eta \left({\varrho }_1\right)\right))-g_2({\varrho }_1,\mathrm{x}\left(\eta \left({\varrho }_1\right)\right))|\underset{1}{\overset{a}{\int }}\left|v({\varrho }_1,s,\mathrm{x}\left(c\left(s\right)\right))\right|ds)\hfill \\ \hfill & +Kw(\mathrm{x},w(d,\rho ))+w_1(G,\rho )\hfill \\ \hfill & \le Kw(\mathrm{x},\rho )+Kw(\mathrm{x},w(d,\rho ))+w_1(G,\rho )\hfill \\ \hfill & +K\left(\right(M\parallel \mathrm{x}\parallel +M^{\prime })w(v,ϵ)(a-1)+M|\mathrm{x}\left(\eta \left({\varrho }_2\right)\right)-\mathrm{x}\left(\eta \left({\varrho }_1\right)\right)\left|\mathsf{\Psi }\right(\parallel \mathrm{x}\parallel )(a-1)\hfill \\ & +w(g_2,ϵ)\mathsf{\Psi }(\parallel \mathrm{x}\parallel )(a-1))\hfill \\ \hfill & \le Kw(\mathrm{x},\rho )+Kw(\mathrm{x},w(d,\rho ))+w_1(G,\rho )\hfill \\ \hfill & +K((M{\mathrm{r}}_0+M^{\prime })w(v,\rho )(a-1)+MKw(\mathrm{x},w(\eta ,\rho ))\mathsf{\Psi }\left({\mathrm{r}}_0\right)(a-1)+w(g_2,\rho )\mathsf{\Psi }\left({\mathrm{r}}_0\right)(a-1)),\hfill \end{array}$$

wherein

$$\begin{array}{cc}\hfill w(v,\rho )& =sup\left\{\right|v(\varrho ,s,\mathrm{x})-u({\varrho }^{\prime },s,\mathrm{x})|:\varrho ,{\varrho }^{\prime },s\in [1,a];|\varrho -{\varrho }^{\prime }|\le \rho ;\mathrm{x}\in [-{\mathrm{r}}_0,{\mathrm{r}}_0]\},\hfill \\ \hfill w_1(G,\rho )& =sup\left\{\right|G(\varrho ,{\mathrm{x}}_1,{\mathrm{z}}_1,{\mathrm{x}}_2)-G({\varrho }^{\prime },{\mathrm{x}}_1,{\mathrm{z}}_1,{\mathrm{x}}_2)|:\varrho ,{\varrho }^{\prime }\in [1,a];|\varrho -{\varrho }^{\prime }|\le \rho ;\hfill \\ & {\mathrm{x}}_1,{\mathrm{x}}_2\in [-{\mathrm{r}}_0,{\mathrm{r}}_0];{\mathrm{z}}_1\in [-K^{\prime }a,K^{\prime }a]\},K^{\prime }=sup\left\{\right|v(\varrho ,s,\mathrm{x})|:\varrho ,s\in [1,a];\mathrm{x}\in [-{\mathrm{r}}_0,{\mathrm{r}}_0]\},\hfill \\ \hfill w(g_2,\rho )& =sup\left\{\right|g_2(\varrho ,\mathrm{x})-g_1({\varrho }^{\prime },\mathrm{x})|:\varrho ,{\varrho }^{\prime }\in [1,a];|\varrho -{\varrho }^{\prime }|\le \rho ;\mathrm{x}\in [-{\mathrm{r}}_0,{\mathrm{r}}_0]\},\hfill \\ \hfill w(\eta ,\rho )& =sup\left\{\right|\eta \left(\varrho \right)-\eta \left({\varrho }^{\prime }\right)|:\varrho ,{\varrho }^{\prime }\in [1,a];|\varrho -{\varrho }^{\prime }|\le \rho ;\mathrm{x}\in [-{\mathrm{r}}_0,{\mathrm{r}}_0]\}.\hfill \end{array}$$

By applying the uniform continuity of the maps $G(\varrho ,{\mathrm{x}}_1,{\mathrm{z}}_1,{\mathrm{x}}_2)$ on the set $\mathcal{I}\times \mathrm{I}\mathrm{R}\times \mathrm{I}\mathrm{R}\times \mathrm{I}\mathrm{R}$, together with the map $v=v(\varrho ,s,\mathrm{x})$ on the set $\mathcal{I}\times \mathcal{I}\times \mathrm{I}\mathrm{R}$, $w(v,\rho )\to 0,w(g_2,\rho )\to 0,w(\eta ,\rho )\to 0,$ and $w_1(G,\rho )\to 0,$ when $\rho \to 0$. Thus,

$$w_0\left(\mathcal{G}\mathcal{W}\right)\le 2K{w}_0\left(\mathcal{W}\right)+KM\mathsf{\Phi }\left({\mathrm{r}}_0\right)(a-1)w_0\left(\mathcal{W}\right).$$

(7)

Finally, from the estimate (4)–(7), together with Theorem 2, the operator $\mathcal{T}$ fulfills the Darbo’s condition on ${\mathcal{B}}_{{\mathrm{r}}_0}$ regarding the measure $w_0$, together with the constant $∥\mathcal{F}\left({\mathcal{B}}_{{\mathrm{r}}_0}\right)∥\left(K(M\mathsf{\Phi }\left({\mathrm{r}}_0\right)(a-1)+2)\right)+∥\mathcal{G}\left({\mathcal{B}}_{{\mathrm{r}}_0}\right)∥\left(K\left(\frac{M\mathsf{\Phi }\left({\mathrm{r}}_0\right){(loga)}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+2\right)\right).$ Now, taking into account (4) and (5) and hypothesis $\left(H_6\right)$, we infer that

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left({\mathcal{B}}_{{\mathrm{r}}_0}\right)∥\left(K(M\mathsf{\Phi }\left({\mathrm{r}}_0\right)(a-1)+2)\right)+∥\mathcal{G}\left({\mathcal{B}}_{{\mathrm{r}}_0}\right)∥\left(K\left(\frac{M\mathsf{\Phi }\left({\mathrm{r}}_0\right){(loga)}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+2\right)\right)& <1.\hfill \end{array}$$

(8)

Therefore, the operator $\mathcal{T}$ is a contraction on ${\mathcal{B}}_{{\mathrm{r}}_0}$ regarding measure $w_0$. Hence, by utilizing Theorem 2, together with Remark 1, $\mathcal{T}$ possesses at least one fixed point in ${\mathcal{B}}_{{\mathrm{r}}_0}.$ As a result, the non-linear functional fractional integral equation, Equation (1), possesses at least one solution in ${\mathcal{B}}_{{\mathrm{r}}_0}.$ □

4. Example

In this section, we examine the following example to validate our result.

Let us suppose the following non-linear Hadamard fractional functional integral equation:

$$\begin{array}{cc}\hfill \mathrm{x}\left(\varrho \right)& =\left[{\displaystyle \frac{\frac{{\varrho }^{1/2}{e}^{-\varrho /2}}{4}+\frac{\varrho e^{-{\varrho }^2/2}\mathrm{x}}{6}}{7\mathsf{\Gamma }\left(\alpha \right)}}\underset{1}{\overset{\varrho }{\int }}\left\{{\displaystyle \frac{\frac{\varrho +s}{2}+sin\mathrm{x}\left(\sqrt{s}\right)}{s}}{\left(log\frac{\varrho }{s}\right)}^{\alpha -1}\right\}ds\right]\hfill \\ \hfill & \times \left[{\displaystyle \frac{1}{14}}\left({\displaystyle \frac{\varrho }{3}}+{\displaystyle \frac{e^{-{(\varrho -1)}^2}\mathrm{x}}{5}}\right)\underset{1}{\overset{2}{\int }}\left\{{\displaystyle \frac{cos\left(\mathrm{x}\right(1-s\left)\right)}{2}}+3{\varrho }^{2}arctan\left({\displaystyle \frac{\left|\mathrm{x}\right(1-s\left)\right|}{1+\left|\mathrm{x}\right(1-s\left)\right|}}\right)\right\}ds\right],\hfill \end{array}$$

(9)

where $\varrho \in [1,2]$.

Now, we equate Equation (9) with Equation (1). We obtain

$$g_1(\varrho ,\mathrm{x})=\frac{{\varrho }^{1/2}{e}^{-\varrho /2}}{4}+\frac{\varrho e^{-{\varrho }^2/2}\mathrm{x}}{6},$$

$$u(\varrho ,s,\mathrm{x})=\frac{\varrho +s}{2}+sin\mathrm{x}\left(\sqrt{s}\right),$$

$$g_2(\varrho ,\mathrm{x})={\displaystyle \frac{\varrho }{3}}+{\displaystyle \frac{e^{-{(\varrho -1)}^2}\mathrm{x}}{5}},$$

$$v(\varrho ,s,\mathrm{x})={\displaystyle \frac{cos\mathrm{x}}{2}}+3{\varrho }^{2}arctan\left({\displaystyle \frac{\left|\mathrm{x}\right|}{1+\left|\mathrm{x}\right|}}\right),$$

$$F(\varrho ,{\mathrm{x}}_1,{\mathrm{z}}_1,{\mathrm{x}}_2)={\displaystyle \frac{1}{7}}{\mathrm{z}}_1,G(\varrho ,{\mathrm{x}}_1,{\mathrm{z}}_1,{\mathrm{x}}_2)={\displaystyle \frac{1}{14}}{\mathrm{z}}_1.$$

Observe that the maps $g_i:\mathcal{I}\times \mathrm{I}\mathrm{R}\to \mathrm{I}\mathrm{R}$ are continuous, where $i=1,2$. In addition, $|g_1(\varrho ,\mathrm{x})-g_1(\varrho ,\mathrm{z})|=|\left(\frac{{\varrho }^{1/2}{e}^{-\varrho /2}}{4}+\frac{\varrho e^{-{\varrho }^2/2}\mathrm{x}}{6}\right)-\left(\frac{{\varrho }^{1/2}{e}^{-\varrho /2}}{4}+\frac{\varrho e^{-{\varrho }^2/2}\mathrm{z}}{6}\right)|=\frac{\varrho e^{-{\varrho }^2/2}}{6}|\mathrm{x}-\mathrm{z}|$ and $|g_2(\varrho ,\mathrm{x})-g_2(\varrho ,\mathrm{z})|=|\left({\displaystyle \frac{\varrho }{3}}+{\displaystyle \frac{e^{-{(\varrho -1)}^2}\mathrm{x}}{5}}\right)-\left({\displaystyle \frac{\varrho }{3}}+{\displaystyle \frac{e^{-{(\varrho -1)}^2}\mathrm{z}}{5}}\right)|=\frac{e^{-{(\varrho -1)}^2}}{5}|\mathrm{x}-\mathrm{z}|.$

Thus, we have $b_1\left(\varrho \right)=\frac{\varrho e^{-{\varrho }^2/2}}{6}and{b}_2\left(\varrho \right)=\frac{e^{-{(\varrho -1)}^2}}{5}.$ Obviously, each of the maps achieves their largest value at $\varrho =1.$ As a result, we obtain

$$M=max\left\{{\displaystyle \frac{e^{-1/2}}{6}},{\displaystyle \frac{1}{5}}\right\}=0.2.$$

In addition,

$$M^{\prime }=max\left\{{\displaystyle \frac{e^{-1/2}}{4}},{\displaystyle \frac{1}{7}}\right\}=0.1516326649281.$$

Furthermore, it has been witnessed that the maps $F,G:[1,2]\times \mathrm{I}\mathrm{R}\times \mathrm{I}\mathrm{R}\times \mathrm{I}\mathrm{R}\to \mathrm{I}\mathrm{R}$ and $u,v:\mathcal{I}\times \mathcal{I}\times \mathrm{I}\mathrm{R}\to \mathrm{I}\mathrm{R}$ are continuous and fulfill the hypothesis $\left(H_2\right)$ with $a_5=a_7=a_8=a_{10}=0,a_6={\displaystyle \frac{1}{7}},a_9={\displaystyle \frac{1}{14}},p=0.$

In this case, we have

$$K=max\left\{{\displaystyle \frac{1}{7}},{\displaystyle \frac{1}{14}}\right\}={\displaystyle \frac{1}{7}}.$$

Furthermore, the first inequality, arising in Hypothesis $\left(H_4\right)$, becomes

$$\begin{array}{cc}\hfill & \left[0.2857{\mathrm{r}}_0+0.1428(0.2{\mathrm{r}}_0+0.1516)\frac{{\mathrm{r}}_0}{\mathsf{\Gamma }(\alpha +1)}{(log2)}^{\alpha }\right]\hfill \\ & \times [0.2857{\mathrm{r}}_0+0.1428(0.2{\mathrm{r}}_0+0.1516){\mathrm{r}}_0]\le {\mathrm{r}}_0,\hfill \end{array}$$

(10)

and the second inequality, arising in Hypothesis $\left(H_4\right)$, becomes

$$\begin{array}{cc}\hfill & \left[0.2857{\mathrm{r}}_0+0.1428(0.2{\mathrm{r}}_0+0.1516)\frac{{\mathrm{r}}_0}{\mathsf{\Gamma }(\alpha +1)}{(log2)}^{\alpha }\right]\left(0.1428(\left(0.2{\mathrm{r}}_0\right)+2)\right)\hfill \\ \hfill & +[0.2857{\mathrm{r}}_0+0.1428(0.2{\mathrm{r}}_0+0.1516){\mathrm{r}}_0]\left(0.1428\left(\left(\frac{0.2{\mathrm{r}}_0{(log2)}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right)+2\right)\right)<1.\hfill \end{array}$$

(11)

Thus, each of the above inequalities are fulfilled for ${\mathrm{r}}_0=1$ as well as for each $\alpha >0$. Hence, ${\mathrm{r}}_0=1$ is a solution of the inequalities which have occurred in Hypothesis $\left(H_4\right)$. Furthermore, all the hypotheses from $(H_1-H_4)$ are fulfilled. Finally, in accordance with Theorem 3, Equation (1) possesses at least one solution, $\mathrm{x}\left(\varrho \right)\in {\mathcal{B}}_1\subset \mathcal{C}[1,2]$.

5. Conclusions

In this work, we have thus investigated, verified, and proved the existence of the solution for the non-linear fractional Hadamard functional integral equations in the Banach algebra $\mathcal{C}[1,a]$. The result was established by applying MNC and FPT in Banach algebra. A numerical example is provided to support the practicability of our existence result.