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\fancyhead[CE]{A. Devi and A. Kumar} 
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{\noindent Journal of Mathematical Extension \\
Vol. XX, No. XX, (), pp-pp (Will be inserted by layout editor)}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
‎\vspace*{9mm}
‎
\begin{center}

{\Large \bf 
Existence result for integro fractional differential equations in the frame of Atangana-Baleanu fractional derivative \\}



\let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)}

{\bf Amita Devi$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\


\vspace{2mm} {\small    
	Central University of Punjab
} \vspace{2mm}

{\bf  Anoop Kumar}\vspace*{-2mm}\\
\vspace{2mm} {\small     
	Central University of Punjab} \vspace{2mm}

\end{center}

\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} 	In this article, we present the existence and uniqueness(EU) results for fractional differential equations(FDEs) with a new direction in Atangana-Baleanu-Caputo (ABC) fractional derivative approach.  The studied problem is considered with non-local integral initial condition.  The existence of solution is investigated by the implementation of Krasnoselskii's fixed point theorem for proposed equations. The uniqueness of the result is derived with the help of the Banach contraction mapping principle. In the end, an example is presented to smooth the understanding of the derived results..
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:}  26A33; 34A08; 34A12

\noindent{\bf Keywords and Phrases:}  Atangana-Baleanu-Caputo fractional derivative, Banach contraction mapping principle, Fractional differential equation, Fixed point theorems, initial condition.
\end{quotation}}





%\end{frontmatter}
%
%\linenumbers

\section{Introduction}
Fractional calculus is an expeditiously outpacing branch of mathematical analysis that unifies the integer-order derivatives and integrals to random order. Many FDEs involve the fractional in- integral and derivative because the fractional order model can describe several real-world phenomena more realistically. FDEs has many applications in various field of real-world problems. Recently, it has been noticed that the cardiac tissues, ultra waves propagation, speech signals and tautochrone problems are studied in the form of FDEs \cite{1}. Many researchers also stated that the fractional integrals and derivatives are convenient for modelling some disorder regions and hereditary properties of several complex phenomena. For more applications of FDEs, the reader can see \cite{2,3,4,5}.  \\
Many mathematicians define different type fractional derivatives(FD). This task facilitates the researchers to take the most appropriate FD to obtain the batter description of results to model various fields' additional problem. Leibnitz proposed the fractional order derivative \cite{6,7}.  After that various type of FD are established by the many authors \cite{8,9,10,11,12}. The more generally used derivatives are Riemann-Liouville and Caputo derivatives. But Riemann-Liouville and Caputo derivatives have the singular kernels i.e. the kernels used in these derivatives contains the singularities. The singular kernels creates many difficulties in applying these fractional operators. To overcome this problem Caputo and Fabrizio innovate the Caputo-Fabrizio(CF) fractional derivative. This FD contains the non-singular kernels, but this fractional derivative still conserves the non-locality property. However, CF fractional derivative gives a better description as compared to the other derivatives with the singular kernels. The associated integral with the CF derivative is in the term of classical order. To reduce this problem, Atangana and Baleanu \cite{13} proposed a derivative with generalised Mittag Leffler(ML) function. This so-called is known as Atangana-Baleanu(AB-derivative) fractional derivative.
Many researchers provided their contribution in the evolution of FDEs related to ABC-derivative see \cite{14,15,16,17}. The rubella disease is analysed by Koca with FDEs involving AB-derivative \cite{18}. Alka- htani and Atangana established a model by using ML function for a mixture of groundwater and chemical waste with decay objects \cite{19}. A general model with ML \& the exponential laws was proposed by Atangana and Gomez-Aguilar\cite{20}. Koca and Atangana established the results for the elastic heat conduction equation with ML kernels \cite{21}. Analytical solutions for the fractional diffusion equation with fractional derivative help are
discussed by Morales-Delgado et al.\cite{22}. Abdel- jawad  \cite{23,24} proposed the fractional difference operators in both senses ABR and ABC derivatives with generalised ML kernels and established the fractional integrals of general order with the help of the infinite binomial theorem. Khan et al.  \cite{25} demonstrated the HU stability and existence of FDEs involving AB-derivative with the p-Laplacian operator. Alqahtani et al. \cite{26}. derived the EU of the solutions for non-linear F-contractions involving AB-derivative in the structure of b-metric spaces. Chua’s circuit model is proposed by Alkahtani  \cite{27}. with the help of the new derivative. The HU stability and existence are analysed by Devi et al. for general FDEs via the fixed point technique  \cite{28}. Prakasha et al.  \cite{29} analysed the hepatitis E virus model by using AB-derivative. AB-derivative is used by Ullah et al.\cite{30} to establish the fractional HBV model. For more about the FDEs reader can see \cite{31,32,33,34,35}. Recently, Ravichandran et al. \cite{36} derived the HU and existence stability for a integro FDEs by using explored AB-fractional derivative. Logeswari and Ravichandarn established the EU results for netural integro-differential equations via AB-fractional derivative  \cite{37}. Abdo et al. \cite{38} analysed the fractional boundary problem involving AB-derivative with non-linear integral conditions.
Motivated by the afore-mentioned work, in this manuscript, we discuss the EU results for integro fractional differential equations involving ABC-fractional derivatives with the non-local initial condition:

\begin{equation}{\label{r1}}
\begin{cases} \mathcal{~^{ABC}_{0}D}^{\omega}\bigg[{\mathfrak{u}(\varkappa)+\psi^*(\varkappa,\mathfrak{u}(\varkappa))}\bigg] =\mathfrak{f}\bigg(\varkappa, \mathfrak{u}(\varkappa), \int_{0}^{\varkappa}\mathcal{g}(\varkappa,\varsigma,\mathfrak{u}(\varsigma))\mathcal{d}\varsigma,\int_{0}^{T}\varphi(\varkappa,\varsigma)\mathcal{d}\varsigma\bigg),   \  \ \ \  \ \      \varkappa\in[0,1],\\
\mathcal{\mathfrak{u}}(0)=\int_{0}^{1}\mathcal{h(\varsigma,\mathfrak{u}(\varsigma))d\varsigma},   
\end{cases} 
\end{equation}
where   $  \mathcal{^{ABC}_{0}D}^{\mathcal{\omega}}  $ be the left Caputo AB-derivative of fractional order $ \omega,~0< \mathcal{\omega}\le1, ~ \mathcal{\varkappa,\varsigma},T\in [0,1]$. $\psi^{*},\mathcal{h}: [0,1]\times\mathbb{R}\longrightarrow\mathbb{R}$ and
${\mathfrak{f}: [0,1]}\times\mathbb{R}^{3}\longrightarrow\mathbb{R}$ are continuous functions.\\ 
Consider $I_1\mathfrak{u}(\varkappa)=\int_{0}^{\varkappa}\mathcal{g}(\varkappa,\varsigma,\mathfrak{u}(\varsigma))\mathcal{d}\varsigma~~~ \text{and}~~~I_2\mathfrak{u}(\varkappa)=\int_{0}^{T}\varphi(\varkappa,\varsigma)\mathcal{d}\varsigma$. Then (\ref{r1}) becomes,
\begin{equation}{\label{r1.1}}
\begin{cases} \mathcal{~^{ABC}_{0}D}^{\mathcal{\omega}}\bigg[\mathcal{\mathfrak{u}(\varkappa)+\psi^*(\varkappa,\mathfrak{u}(\varkappa))}\bigg] =\mathfrak{f}\bigg(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa)\bigg),   \  \ \ \  \ \      \varkappa\in[0,1],\\
\mathcal{\mathfrak{u}}(0)=\int_{0}^{1}\mathcal{h}(\varsigma,\mathfrak{u}(\varsigma))\mathcal{d}\varsigma,   
\end{cases} 
\end{equation}
This manuscript is outlined as: in the $2^{nd}$ section, we serve some definitions, theorems, and lemmas necessary for problem evolution. We discuss the EU of solution in the  $3^{rd}$ section. An example discussed numerically in the next part to illustrate the derived results. The last section includes the conclusion of the solved problem.

\section{ Basic results and Preliminaries}
Here, we contemplate some definition, lemmas and actual results.\\
%We define the term $\mathcal{AC}^{m}_{\gamma}\mathcal{[a,b]}$ which contains all absolutely-continuous functions $\mathcal{\varphi^{\ast}}$ and having $\gamma^{m-1}$-derivative absolutely continuous on $\mathcal{[a,b]}$ $\big(\gamma^{m-1}\mathcal{\varphi^{\ast}}\in \mathcal{AC([a,b]},\mathbb{R})\big)$. \\
{\bf Definition 2.1.}{\cite{3}}
For $\omega>0,$ Riemann-Liouville (R-L) fractional integral of order $ \omega \in\mathbb{R}$  is defined as
\begin{equation}{\label{r2.1}}
I^{\omega}\mathfrak{u}(\varkappa) =\dfrac{1}{\Gamma(\omega)}\int_{0}^{\varkappa}(\mathcal{t-x})^{\omega-1}\mathcal{\mathfrak{u}(x)dx}.
\end{equation}
\vspace{0.2cm}\\
{\bf Definition 2.2.}{\cite{3}}
For $0<\omega\le1$, the R-L fractional derivative and Caputo fractional derivative are defined as
\begin{equation}{\label{r2.2}}
\mathcal{D^{\omega}\mathfrak{u}(t)}=\frac{1}{\Gamma(1-\omega)}\mathcal{\frac{d}{dt}\bigg(\int_{0} ^{t}(t-x)^{-\omega}\mathfrak{u}(x)dx}\bigg).
\end{equation}
% part of the real number $\delta$.\\

\begin{equation}{\label{r2.3}}
\mathcal{~^{c}D}^{\omega}\mathfrak{u}(\varkappa) =\dfrac{1}{\Gamma(\mathcal{1}-\omega)}\int_{0}^{\varkappa}(\mathcal{t-x})^{-\omega}\mathcal{\mathfrak{u}^{'}(x)dx}
\end{equation}
respectively.
\vspace{0.2cm}\\
{\bf Definition 2.3.}{\cite{13}} 
Let $0<\omega\le 1 $ and $\mathcal{\mathfrak{u}}\in C^1[a,b], \mathcal{\mathfrak{u}}^{'}\in L^1[a,b]$, where $0\le a<b$, the Caputo AB-fractional derivative and the R-L AB-fractional derivative of order $\omega$ are defined by
\begin{equation}{\label{r2.4}}
\mathcal{~^{ABC}D}^{\omega}\mathfrak{u}(\varkappa) =\dfrac{B(\omega)}{\mathcal{1}-\omega}\int_{0}^{\varkappa}\mathcal{\mathfrak{u}^{'}(x)} E_{\omega}\bigg[-\omega\dfrac{(\mathcal{t-x})^{\omega}}{1-\omega}\bigg]\mathcal{dx},
\end{equation}
%\vspace{0.2cm}\\
%{\bf Definition 2.5.} Let $0<\omega\le 1 $ and $\mathcal{\mathfrak{u}}\in C^1[a,b], \mathcal{\mathfrak{u}}^{'}\in L^1[a,b]$, where $0\le a<b$, the R-L AB-fractional derivative is 
and
\begin{equation}{\label{r2.5}}
\mathcal{~^{ABR}D}^{\omega}\mathfrak{u}(\varkappa) =\dfrac{B(\omega)}{\mathcal{1}-\omega}\mathcal{\dfrac{d}{dt}}\bigg(\int_{0}^{\varkappa}\mathcal{\mathfrak{u}(x)} E_{\omega}\bigg[-\omega\dfrac{(\mathcal{t-x})^{\omega}}{1-\omega}\bigg]\mathcal{dx}\bigg).
\end{equation}
respectively, where $E_{\omega}$ is called the Mittag-Leffler function and given by 
\begin{equation*}
E_{\omega}(\mathcal{\mathfrak{u}})=\sum_{k=0}^{\infty}\dfrac{\mathcal{\mathfrak{u}}^k}{\Gamma(k\omega+1)},
\end{equation*}
and $B(\omega)$ is a normalizing positive function satisfying $B(0)=B(1)=1$.
\vspace{0.2cm}\\
{\bf Definition 2.4.} {\cite{13}} Let $0<\omega\le 1 $ and $\mathcal{\mathfrak{u}}\in C^1[a,b]$,  where $0 \le a<b$, the associated  AB-fractional integral is 
\begin{equation}{\label{r2.6}}
\mathcal{~^{AB}I}^{\omega}\mathfrak{u}(\varkappa) =\dfrac{(1-\omega)}{B(\omega)}\mathfrak{u}(\varkappa)+\dfrac{\omega}{B(\omega)}I^{\omega}\mathfrak{u}(\varkappa),
\end{equation}
where $I^{\omega}$ is the R-L fractional integral defined in (\ref{r2.1}).\\
The following results are based on the fixed point technique for equ. (\ref{r1.1}). The following assumptions are needed for establish the EU results.\\
Let $ \mathcal{Y}=\mathcal{ C}\big([0,1],\mathbb{R}\big)$ be the Banach space of continuous  functions $\mathcal{\mathfrak{u}}:[0,1]\longrightarrow\mathbb{R}$, with the norm $ \|{\mathcal{\mathfrak{u}}}\|= \underset{\varkappa\in[0,1]}{\text{sup}}{|\mathfrak{u}(\varkappa)|}.$
\begin{itemize}
\item $(\mathcal{R_1})~~$ Suppose that $\mathcal{\mathfrak{f}}\in\big([0,1]\times{\mathbb{R}^{3}},\mathbb{R}\big)~~ \exists$ positive constants $\mathcal{L_{1}}$ and $\mathcal{L_{2}}$ such that
\begin{equation*}
| \mathcal{\mathfrak{f}}( \varkappa,\mathfrak{u}_1,\mathfrak{v}_1,\mathfrak{z}_1)-\mathfrak{f}( \varkappa,\mathfrak{u}_2,\mathfrak{v}_2,\mathfrak{z}_2)|\le \mathcal{L_{1}}\big( \|\mathcal{x}_1-\mathcal{x}_2\|+\|\mathcal{y}_1-\mathcal{y}_2\|+\|\mathcal{z}_1-\mathcal{z}_2\|\big) 
\end{equation*}
for all $\mathfrak{u}_1,\mathfrak{v}_1,\mathfrak{z}_1,\mathfrak{u}_2,\mathfrak{v}_2,\mathfrak{z}_2\in \mathcal{Y},\varkappa\in [0,1]$ and $\mathcal{L_{2}}=  
\underset{\varkappa\in[0,1]}{\text{max}}\|\mathcal{f}(\varkappa,0,0,0)\|$.
\vspace*{0.3 cm}\\
\item 	$(\mathcal{R}_2) ~~ $	 Let $\varkappa\in [0,1]$ and   $\psi^*\in\big([0,1]\times{\mathbb{R}},\mathbb{R}\big)$, there exist positive constants $\mu_{1}$ and $\mu_{2}$ such that
\begin{equation*}
| \psi^*( \varkappa,\mathfrak{u}_1)-\psi^*( \varkappa,\mathfrak{u}_2|\le \mu_{1}\big( \|\mathfrak{u}_1-\mathfrak{u}_2)\|\big) 
\end{equation*}
for all $\mathfrak{u}_1,\mathfrak{u}_2\in \mathcal{Y}$ and $\mu_{2}=  
\underset{\varkappa\in[0,1]}{\text{max}}\|{\psi^*}(\varkappa,0)\|$.

\item 	$(\mathcal{R}_3) ~~ $	 If $\mathcal{\varkappa,\varsigma}\in [0,1]$ then  $\mathcal{g}\in\big([0,1]\times[0,1]\times{\mathbb{R}},\mathbb{R}\big)$, $\exists$ positive constants $\lambda_{1}$ and $\lambda_{2}$ such that
\begin{equation*}
| \mathcal{g}( \varkappa,\varsigma,\mathfrak{u}_1)-\mathcal{g}( \varkappa, \varsigma,\mathfrak{u}_2|\le \lambda_{1}\big( \|\mathfrak{u}_1-\mathfrak{u}_2)\|\big) 
\end{equation*}
for all $\mathfrak{u}_1,\mathfrak{u}_2\in \mathcal{Y}$ and $\lambda_{2}=  
\underset{{\varkappa,\varsigma}\in[0,1]}{\text{max}}\|\mathcal{g}({\varkappa,\varsigma},0)\|$.
\item $(\mathcal{R}_4) ~~ $	 There exist positive constants $\mathcal{M}_{1}$  and $\mathcal{M}_{2}$ for  $\varphi\in\big([0,1]\times[0,1]\times{\mathbb{R}},\mathbb{R}\big)$, such that
\begin{equation*}
| \varphi( \varkappa,\varsigma,\mathfrak{u}_1)-\varphi( \varkappa, \varsigma,\mathfrak{u}_2|\le \mathcal{M}_{1}\big( \|\mathfrak{u}_1-\mathfrak{u}_2)\|\big) 
\end{equation*}
for all  ${\varkappa,\varsigma}\in [0,1], \mathfrak{u}_1,\mathfrak{u}_2\in \mathcal{Y}$ and $\mathcal{M}_{2}=  
\underset{{\varkappa,\varsigma}\in[0,1]}{\text{max}}\|\varphi({\varkappa,\varsigma},0)\|$.
\item 	$(\mathcal{R}_5) ~~ $	Let $\mathcal{h}\in\big([0,1]\times{\mathbb{R}},\mathbb{R}\big)$, $\exists$ positive constants $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ such that
\begin{equation*}
| \mathcal{h}( \varkappa,\mathfrak{u}_1)-\mathcal{h}( \varkappa,\mathfrak{u}_2)|\le \mathcal{N}_{1}\big( \|\mathfrak{u}_1-\mathfrak{u}_2)\|\big) 
\end{equation*}
for all $\mathfrak{u}_1,\mathfrak{u}_2\in \mathcal{Y}$ and  $\mathcal{N}_{2}=  
\underset{\varkappa\in[0,1]}{\text{max}}\|\mathcal{h}(\varkappa,0)\|$.
\item 	$(\mathcal{R}_6) ~~ $ For any positive ${\bar{r}}$, we take $\mathcal{B}_{\bar{r}}=\{{ \mathfrak{u}}\in \mathcal{Y} : \mathcal{\|\mathfrak{u}\|}\le {\bar{r}}\}\subset \mathcal{Y},$
where ${\bar{r}}\ge \dfrac{\mathcal{Q}}{(1-\mathcal{P})}$,  where $\mathcal{P}=\mu_1 +\mathcal{N}_1+\bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{\Gamma(\omega) B(\omega)}\bigg)\mathcal{L}_1\big(1+\lambda_1+\mathcal{T}~\mathcal{M}_1\big)$ and   $\mathcal{Q}=\mu_2 +\mathcal{N}_2+\bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{\Gamma(\omega) B(\omega)}\bigg)\mathcal{L}_2\big(1+\lambda_2+\mathcal{T}~\mathcal{M}_2\big)$ then $\mathcal{B}_{\bar{r}}$  is bounded, closed and convex subset in $\mathcal{C}([0,1],\mathbb{R}).$ \\

\end{itemize}

{\bf Lemma 2.5.} 
If $(\mathcal{R}_3) $	and $(\mathcal{R}_4) $	 are satisfied, then the estimate
\begin{equation*}
\| I_{1}\mathfrak{u}(\varkappa)\|\le \varkappa\big(\lambda_{1}\|\mathcal{\mathfrak{u}}\|+\lambda_{2}\big), 
\end{equation*}
and
\begin{equation*}
\| I_{2}\mathfrak{u}(\varkappa)\|\le \mathcal{T}\big(\mathcal{M}_{1}\|\mathcal{\mathfrak{u}}\|+\mathcal{M}_{2}\big), 
\end{equation*}
are hold true for any $\varkappa\in [0,1]$ and $\mathcal{\mathfrak{u}}\in \mathcal{Y}$.\\
{\bf Proposition 2.6}{\cite{5,15}  If $0<\omega\le1$, then
\begin{equation}
\begin{aligned} 
\big(\mathcal{~^{AB}_{0}I}^{\omega}\big(\mathcal{~^{ABC}_{0}D}^{\omega}\big)\mathcal{\mathfrak{u}\big)}(\varkappa)&=\mathfrak{u}(\varkappa)-\mathcal{\mathfrak{u}}(0)E_{\omega}(\lambda\varkappa^{\omega})-\dfrac{\omega}{1-\omega}\mathcal{\mathfrak{u}}(0)E_{\omega,\omega+1}(\lambda\varkappa^{\omega})\\&
=\mathfrak{u}(\varkappa)-\mathcal{\mathfrak{u}}(0)
\end{aligned}
\end{equation} 
%{\bf Theorem 2.7.} (Ascoli-Arzela Theorem)
%Let $\mathcal{A}$ be a compact Hausdroff metric spaces. Then $\mathcal{S} \subset \mathcal{C(A)}$ is relatively compact iff $\mathcal{S}$ is uniformly bounded and uniformly equicontinuous.
%\vspace*{0.5cm}\\
{\bf Theorem 2.7.} (Krasnoselkii Fixed Point Theorem){\cite{39}}
Let $ \mathcal{S} $ is a nonempty, closed, bounded and convex subset of a Banach space $ \mathsf{E}$. Let $A_1, A_2$ be the operators from $\Omega$ to $ \mathsf{E}$ such that:\\(i) $A_{1}\mathcal{x}+A_{2}\mathcal{y}\in \Omega $ whenever $\mathcal{x ,y}\in \Omega;$\\ (ii)  $A_1$ is continuous and compact; \\(iii) $A_2$ is a contraction map. \\ Then there exists $\mathcal{z}\in \Omega$ such that $\mathcal{z}=A_1\mathcal{z}+A_2\mathcal{z}.$ \\
\section{ Derived Results}
First, we observe  the existence of the problem (\ref{r1.1}) by using the fixed point technique.\\
{\bf Theorem 3.1.} Let $0<\omega\le1$ and $\exists~~ \mathfrak{f}\in \big([0,1]\times\mathbb{R}^{3},\mathbb{R}\big)$ with $\mathcal{f}^{*}(0,\mathcal{\mathfrak{u}}(0),0,\int_{0}^{T}\varphi(0,\varsigma,\mathcal{\mathfrak{u}(\varsigma)d\varsigma}))=\psi^*(0,\mathcal{\mathfrak{u}}(0))=0$. A function $\mathcal{\mathfrak{u}}\in \mathcal{C}[0,1]$ be a solution of  the integral equation 
\begin{equation}{\label{r3.1}}
\mathfrak{u}(\varkappa)=\psi^{*}(\varkappa,\mathfrak{u}(\varkappa))-\psi^{*}(0,\mathcal{\mathfrak{u}}(0))+\int_{0}^{1}\mathcal{h(\varsigma,x(\varsigma))d\varsigma}+~^{AB}_{0}I^{\omega}\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa)),
\end{equation}
iff $\mathfrak{u}(\varkappa)$ is a solution of the ABC-problem (\ref{r1.1}).\\
{\bf Proof.} Let $\mathfrak{u}(\varkappa)$  satisfy  (\ref{r1.1}). Applying the AB-fractional integarl of (\ref{r1.1}),\\
we get
\begin{equation}
\big(\mathcal{~^{AB}_{0}I}^{\mathcal{\omega}}(\mathcal{~^{ABC}_{0}D}^{\mathcal{\omega}})[\mathcal{\mathfrak{u}(t)-\psi^*(\varkappa,\mathfrak{u}(\varkappa))}]\big) = \mathcal{~^{AB}_{0}I}^{\mathcal{\omega}}\mathfrak{f}\bigg(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa)\bigg).
\end{equation}
By using {\bf Proposition 2.6}, we obtain
\begin{equation}
\mathfrak{u}(\varkappa)-\psi^*(\varkappa,\mathfrak{u}(\varkappa))-(\mathcal{\mathfrak{u}}(0)-\psi^*(0,\mathcal{\mathfrak{u}}(0))) = \mathcal{~^{AB}_{0}I}^{\mathcal{\omega}}\mathfrak{f}\bigg(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa)\bigg).
\end{equation}
Since $ \mathcal{\mathfrak{u}}(0)=\int_{0}^{1}\mathcal{h}(\varsigma,\mathfrak{u}(\varsigma))\mathcal{d}\varsigma$,  then (\ref{r3.1}) is satisfied.\
Now, consider $\mathfrak{u}(\varkappa)$  satisfies the (\ref{r3.1}), then by $\mathcal{f}^{*}(0,\mathcal{\mathfrak{u}}(0),0,\int_{0}^{T}\varphi(0,\varsigma,\mathfrak{u}(\varsigma)\mathcal{d}\varsigma))=\psi^*(0,\mathcal{\mathfrak{u}}(0))=0$ it is visible that $ \mathcal{\mathfrak{u}}(0)=\int_{0}^{1}\mathcal{h(\varsigma},\mathfrak{u}(\varsigma))\mathcal{d}\varsigma$.
Applying AB-derivative in R-L sense of (\ref{r3.1}) and by using $\big(\mathcal{~^{AB}_{0}D}^{\omega}\big(\mathcal{~^{AB}_{0}I}^{\omega}\big)\mathcal{\mathfrak{u}\big)}(\varkappa)=\mathfrak{u}(\varkappa)$, we get
\begin{equation}
\begin{aligned}
\big(\mathcal{~^{ABR}_{0}D}^{\omega}\mathfrak{u}\big)(\varkappa)&=\bigg(\int_{0}^{1}\mathcal{h}(\varsigma,\mathfrak{u}(\varsigma))\mathcal{d}\varsigma\bigg)\big(\mathcal{~^{ABR}_{0}D}^{\omega}1\big)(\varkappa)+\big(\mathcal{~^{ABR}_{0}D}^{\omega}\big)\psi^*(\varkappa,\mathfrak{u}(\varkappa))\\&+\big(\mathcal{~^{ABR}_{0}D}^{\omega}\mathcal{~^{AB}_{0}I}^{\mathcal{\omega}}\big)\mathfrak{f}\bigg(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa)\bigg).
\end{aligned}
\end{equation}
Thus, 
\begin{equation}
\big(\mathcal{~^{ABR}_{0}D}^{\omega}\big)\big(\mathfrak{u}(\varkappa)-\psi^*(\varkappa,\mathfrak{u}(\varkappa))\big)=\bigg(\int_{0}^{1}\mathcal{h}(\varsigma,\mathfrak{u}(\varsigma))\mathcal{d}\varsigma \bigg)E_{\omega}(\dfrac{-\omega}{1-\omega}\varkappa^{\omega})+\mathfrak{f}\bigg(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa)\bigg).
\end{equation}
Hence, the equation (\ref{r1}) can be obtained by the Theroem 1 in \cite{13}.\\
Now, let us define the operator $\mathcal{F}$ on $\mathcal{B_{\bar{r}}}$ as follows
\begin{equation}{\label{r3.2}}
\mathcal{F}\mathfrak{u}(\varkappa)=\psi^{*}(\varkappa,\mathfrak{u}(\varkappa))+\int_{0}^{1}\mathcal{h}(\varsigma,x(\varsigma))\mathcal{d}\varsigma+~^{AB}_{0}I^{\omega}\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa)).
\end{equation}
It is obeserved that $\mathfrak{u}(\varkappa)$ is the solution of (\ref{r1}) iff the operator $\mathcal{F}$ has a fixed point.\\
{\bf Theorem 3.2.} Assume that $\mathcal{R_1}-\mathcal{R_6}$ are satisfied and $q  (\varkappa_2-\varkappa_1)=\mathcal{L_{1}}\big[\|(\varkappa, \upsilon(\varkappa), I_1\upsilon(\varkappa),I_2\upsilon(\varkappa))\|+\varkappa(\lambda_{1}\|(\varkappa, \upsilon(\varkappa), I_1\upsilon(\varkappa),I_2\upsilon(\varkappa))\|)+\mathcal{T}(\mathcal{M}_{1}\|(\varkappa, \upsilon(\varkappa), I_1\upsilon(\varkappa),I_2\upsilon(\varkappa))\|)\big]$. If $\Lambda=\mu_1+\mathcal{N}\le1$, then problem (\ref{r1.1}) has a solution.\\
{\bf Proof.} Let us define the operators $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ on $\mathcal{B_{\bar{r}}}$ such that $\mathcal{F}=\mathcal{F}_{1}+\mathcal{F}_{2}$
\begin{equation}{\label{r3.3}}
\mathcal{F}_{1}\mathfrak{u}(\varkappa)=\psi^{*}(\varkappa,\mathfrak{u}(\varkappa))+\int_{0}^{1}\mathcal{h}(\varsigma,x(\varsigma))\mathcal{d}\varsigma,
\end{equation}
\begin{equation}{\label{r3.4}}
\mathcal{F}_{2}\mathfrak{u}(\varkappa)=~^{AB}_{0}I^{\omega}\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa)).
\end{equation}
The following three steps are required to apply Theorem $2.1$.
\begin{enumerate}
	\item   $\|\mathcal{F_{1}\mathfrak{u}}+\mathcal{F_{2}\mathfrak{z}\|\le ~\bar{r}~~~~~ \text{where}~~\bar{r}\in B}_{\bar{r}},$ \\
	For any $\mathcal{\mathfrak{u},\mathfrak{z} \in B}_{\bar{r}}$ \\
	\begin{equation*}
	\begin{aligned}
	\|	\mathcal{F_{1}}&{\mathfrak{u}}+\mathcal{F_{2}\mathfrak{z}}\|\\&
	= ~\underset{\varkappa\in[0,1]}{\text{sup}}\biggl\{\bigg|\psi^{*}(\varkappa,\mathfrak{u}(\varkappa))+\int_{0}^{1}\mathcal{h}(\varsigma,\mathfrak{u}(\varsigma))\mathcal{d}\varsigma+\dfrac{(1-\omega)}{B(\omega)}\mathfrak{f}(\varkappa, \mathfrak{z}(\varkappa), I_1\mathfrak{z}(\varkappa),I_2\mathfrak{z}(\varkappa))\\&~~~+\dfrac{\omega}{B(\omega)}~_{0}I^{\omega}\mathfrak{f}(\varkappa, \mathfrak{z}(\varkappa), I_1\mathfrak{z}(\varkappa),I_2\mathfrak{z}(\varkappa))\bigg|\biggr\}\\&
	\le ~ \underset{\varkappa\in[0,1]}{\text{sup}}\biggl\{\big|\psi^{*}(\varkappa,\mathfrak{u}(\varkappa))\big|+\bigg|\int_{0}^{1}\mathcal{h}(\varsigma,\mathfrak{z}(\varsigma))\mathcal{d}\varsigma\bigg|+\dfrac{(1-\omega)}{B(\omega)}\big|\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathcal{\mathfrak{z}(t)},I_2\mathcal{\mathfrak{z}(\varkappa)})\big|\\&~~~+\dfrac{\omega}{B(\omega)}~_{0}I^{\omega}\big|\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{z}(\varkappa),I_2\mathfrak{z}(\varkappa))\big|\biggr\}\\&
	\le\underset{\varkappa\in[0,1]}{\text{sup}}\biggl\{\big|\psi^{*}(\varkappa,\mathfrak{u}(\varkappa))-\psi^{*}(\varkappa,\mathcal{\mathfrak{u}}(0))+\psi^{*}(\varkappa,\mathcal{\mathfrak{u}}(0))\big|\\&~~~+\int_{0}^{1}\big|\mathcal{h(\varsigma,\mathcal{\mathfrak{u}}(\varsigma))d\varsigma}-\mathcal{h}(\varsigma,\mathcal{\mathfrak{u}}(0))+\mathcal{h}(\varsigma,\mathcal{\mathfrak{u}}(0))\big|\mathcal{d}\varsigma\\&~~~+\dfrac{(1-\omega)}{B(\omega)}\big|\mathfrak{f}(\varkappa, \mathfrak{z}(\varkappa), I_1\mathfrak{z}(\varkappa),I_2\mathfrak{z}(\varkappa))-\mathfrak{f}(\varkappa, \mathfrak{z}(0), I_1\mathfrak{z}(0),I_2\mathfrak{z}(0))+\mathfrak{f}(\varkappa, \mathfrak{z}(0), I_1\mathfrak{z}(0),I_2\mathfrak{z}(0))\big|\\&~~~+\dfrac{\omega}{B(\omega)}~_{0}I^{\omega}\big|\mathfrak{f}(\varkappa, \mathfrak{z}(\varkappa), I_1\mathfrak{z}(\varkappa),I_2\mathfrak{z}(\varkappa))-\mathfrak{f}(\varkappa, \mathfrak{z}(0), I_1\mathfrak{z}(0),I_2\mathfrak{z}(0))+\mathfrak{f}(\varkappa, \mathfrak{z}(0), I_1\mathfrak{z}(0),I_2\mathfrak{z}(0))\big|\biggr\}\\&
	\le\mu_{1}\|\mathcal{\mathfrak{u}}\|+\mu_{2}+\mathcal{N}_{1}\|\mathcal{\mathfrak{u}}\|+\mathcal{N}_{2}+\dfrac{(1-\omega)}{B(\omega)}\bigg(\mathcal{L_{1}}\big[\|\mathcal{\mathfrak{u}}\|+\varkappa(\lambda_{1}\|\mathcal{\mathfrak{u}}\|+\lambda_{2})+\mathcal{T}(\mathcal{M}_{1}\|\mathfrak{z}\|+\mathcal{M}_{2})\big]\bigg)\\&~~~+\dfrac{(1-\omega)}{B(\omega)}\mathcal{L_{2}}
	+\dfrac{\omega}{B(\omega)}\bigg(\mathcal{L_{1}}\big[\|\mathfrak{z}\|+\varkappa(\lambda_{1}\|\mathcal{\mathfrak{u}}\|+\lambda_{2})+\mathcal{T}(\mathcal{M}_{1}\|\mathfrak{z}\|+\mathcal{M}_{2})\big]\bigg)\dfrac{(1)^{\omega}}{\omega\Gamma(\omega)}\\&
	~~~+\bigg(\dfrac{\omega}{B(\omega)}\mathcal{L_{2}}\bigg)\dfrac{(1)^{\omega}}{\omega\Gamma(\omega)}\\&
	\le \big(\mu_1 +\mathcal{N}_1\big)\|\mathcal{\mathfrak{u}}\|+\bigg[\bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{\Gamma(\omega) B(\omega)}\bigg)\mathcal{L}_1\big[1+\lambda_1+\mathcal{T}~\mathcal{M}_1\big]\bigg]\|\mathfrak{z}\|\\&~~~~+ \bigg[\mu_2 +\mathcal{N}_2+\bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{\Gamma(\omega) B(\omega)}\bigg)\mathcal{L}_2\big[1+\lambda_2+\mathcal{T}~\mathcal{M}_2\big]\bigg]\\&
	=\mathcal{P}\|\mathcal{\mathfrak{u}}\|+\mathcal{Q}\\&
	\le\mathcal{P}\bar{r}+\mathcal{Q}\le \bar{r}\\& 
	\end{aligned}
	\end{equation*}
	\item 	$\mathcal{F}_{1}$ is the contraction on $\mathcal{B_{\bar{r}}}$.\\
	For each $\mathcal{\mathfrak{u},\mathfrak{z} \in B}_{\bar{r}}$, by using $\mathcal{R_2}$ and $\mathcal{R_5}$.
	\begin{equation*}
	\begin{aligned}
	\|\mathcal{F}_{1}\mathfrak{u}(\varkappa)-\mathcal{F}_{1}\mathfrak{z}(\varkappa)\|&=\underset{\varkappa\in[0,1]}{\text{sup}}\bigg|\biggl\{\psi^{*}(\varkappa,\mathfrak{u}(\varkappa))+\int_{0}^{1}\mathcal{h}(\varsigma,\mathfrak{u}(\varsigma))\mathcal{d}\varsigma-\psi^{*}(\varkappa,\mathcal{\mathfrak{z}(\varkappa)})-\int_{0}^{1}\mathcal{h(\varsigma,\mathfrak{z}(\varsigma))d\varsigma}\bigg|\biggr\}
	\\&
	\le\underset{\varkappa\in[0,1]}{\text{sup}}\biggl\{\big|\psi^{*}(\varkappa,\mathfrak{u}(\varkappa))-\psi^{*}(\varkappa,\mathfrak{z}(\varkappa))\big|+\int_{0}^{1}\big|\mathcal{h}(\varsigma,\mathfrak{u}(\varsigma))-\mathcal{h}(\varsigma,\mathfrak{z}(\varsigma))\big|\mathcal{d}\varsigma\biggr\}\\&
	\le \mu_{1}\|{\mathfrak{u}}-\mathfrak{z}\|+ \mathcal{N}_{1}\|\mathfrak{u}-\mathfrak{z}\|\\&
	\le \Lambda\|{\mathfrak{u}}-\mathfrak{z}\|,
	\end{aligned}
	\end{equation*}
	$\implies~~~~\|\mathcal{F}_{1}\mathfrak{u}(\varkappa)-\mathcal{F}_{1}\mathfrak{z}(\varkappa)\|\le\Lambda\|\mathcal{\mathfrak{u}}-\mathfrak{z}\|,~~$
	where $\Lambda=\mu_1+\mathcal{N}_1$. \\ As $\Lambda<1$ . Thus $\mathcal{F}_1$ is a contraction operator. 
	\item We prove that $\mathcal{F}_{2}$ is completely continuous operator.\\
	For completeness of  $\mathcal{F}_{2}$, firstly we prove that $\mathcal{F}_{2}$ is continuous.\\
	With $\lim\limits_{n \rightarrow \infty} \|\mathcal{\mathfrak{u}_{n}}-\mathcal{\mathfrak{u}}\|=0$ for any $\mathcal{\mathfrak{u}_{n}}, \mathcal{\mathfrak{u}} \in \mathcal{B_{\bar{r}}}, \mathcal{n}= 1,2,...$.\\ Then $\lim\limits_{n \rightarrow \infty} \mathfrak{u}_{n}(\varkappa)=\mathfrak{u}(\varkappa)~~~\forall \varkappa\in[0,1]$\\
	Therefore $\lim\limits_{n \rightarrow \infty} \mathfrak{f}\bigg(\varkappa, \mathfrak{u}_{n}(\varkappa), I_1\mathfrak{u}_{n}(\varkappa),I_2\mathfrak{u}_{n}(\varkappa)\bigg)=\mathfrak{f}\bigg(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa)\bigg)~~~\forall \varkappa\in[0,1]$
	Now, for $\varkappa\in[0,1]$

	\begin{equation*}
	\begin{aligned}
	\|\mathcal{F}_{2}\mathfrak{u}_{n}(\varkappa)-&\mathcal{F}_{2}{\mathfrak{u}{(\varkappa)}}\|\\&=\underset{\varkappa\in[0,1]}{\text{sup}}\biggl\{\bigg|\dfrac{(1-\omega)}{B(\omega)}
	\big(\mathfrak{f}\varkappa, \mathfrak{u}_{n}(\varkappa), I_1\mathfrak{u}_{n}(\varkappa),I_2\mathfrak{u}_{n}(\varkappa))-\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa))\big)\\&
	~~~+\dfrac{\omega}{B(\omega)}~_{0}I^{\omega}\big(\mathfrak{f}\varkappa, \mathfrak{u}_{n}(\varkappa), I_1\mathfrak{u}_{n}(\varkappa),I_2\mathfrak{u}_{n}(\varkappa))-\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa))\big)\bigg|\biggr\}\\&
	\le \bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{B(\omega)\Gamma(\omega)}\bigg)\\&~~~\times\underset{\varkappa\in[0,1]}{\text{sup}}\|\mathfrak{f}\varkappa, \mathfrak{u}_{n}(\varkappa), I_1\mathfrak{u}_{n}(\varkappa),I_2\mathfrak{u}_{n}(\varkappa))-\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa))\|
	\end{aligned}
	\end{equation*} 
	Thus $\|\mathcal{F}_{2}\mathfrak{u}_{n}(\varkappa)-\mathcal{F}_{2}\mathcal{\mathfrak{u}{(t)}}\|\longrightarrow 0$ as $\mathcal{n}\longrightarrow\infty$.\\
	Hence $\mathcal{F}_{2}$ is continuous.\\
	Now, we prove that   $\mathcal{F}_{2}$ is compact.
	\begin{equation*}
	\begin{aligned}
	\|\mathcal{F}_{2}\mathfrak{u}(\varkappa)\|&=\underset{\varkappa\in[0,1]}{\text{sup}}\biggl\{\bigg|\dfrac{(1-\omega)}{B(\omega)}\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa))
	+\dfrac{\omega}{B(\omega)}~_{0}I^{\omega}\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa))\bigg|\biggr\}\\&
	\le \bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{B(\omega)\Gamma(\omega)}\bigg)\underset{\varkappa\in[0,1]}{\text{sup}}\|\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa))\|\\&
	\le\bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{B(\omega)\Gamma(\omega)}\bigg)
	\bigg(\mathcal{L_{1}}\big[\|\mathcal{\mathfrak{u}}\|+\varkappa(\lambda_{1}\|\mathcal{\mathfrak{u}}\|+\lambda_{2})+\mathcal{T}(\mathcal{M}_{1}\|\mathcal{\mathfrak{u}}\|+\mathcal{M}_{2})\big]+\mathcal{L_{2}}\bigg)\\&
	\le\bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{B(\omega)\Gamma(\omega)}\bigg)\bigg[ \mathcal{L}_1\big[1+\lambda_1+\mathcal{T}~\mathcal{M}_1\big]\|\mathcal{\mathfrak{u}}\|+\mathcal{L}_2\big[1+\lambda_2+\mathcal{T}~\mathcal{M}_2\big]\bigg]\\&
	\le \bigg[\big(\mathcal{P}-\mu_1 -\mathcal{N}_1\big)\bar{r}+\big(\mathcal{Q}-\mu_2 -\mathcal{N}_2\big)\bigg]<\infty.
	\end{aligned}
	\end{equation*} 
	which shows that $\mathcal{F}_{2}$ is bounded on $\mathcal{B_{\bar{r}}}$.\\
	%	Next we prove that $\mathcal{F}_{2}$ is equicontinuous.\\
	For any  $0<{\varkappa_1}<\mathcal{\varkappa_2}<\varkappa<1,$ we have \\
	\begin{equation*}
	\begin{aligned}
	\|\mathcal{F}_{2}{\mathfrak{u}(\varkappa_{2})}&-\mathcal{F}_{2}\mathfrak{u}(\varkappa_{1})\|\\&=\underset{\varkappa\in[0,1]}{\text{sup}}\biggl\{\bigg|\dfrac{(1-\omega)}{B(\omega)}\mathfrak{f}(\varkappa_2, \mathfrak{u}(\varkappa_2), I_1\mathfrak{u}(\varkappa_2),I_2\mathfrak{u}(\varkappa_2))\\&~~~+\dfrac{\omega}{B(\omega)}~_{0}I^{\omega}\mathfrak{f}(\varkappa_2, \mathfrak{u}(\varkappa_2), I_1\mathfrak{u}(\varkappa_2),I_2\mathfrak{u}(\varkappa_2))\\&~~~-\dfrac{(1-\omega)}{B(\omega)}\mathfrak{f}(\varkappa_1, \mathfrak{u}(\varkappa_1), I_1\mathfrak{u}(\varkappa_1),I_2\mathfrak{u}(\varkappa_1))-\dfrac{\omega}{B(\omega)}~_{0}I^{\omega}\mathfrak{f}(\varkappa_1, \mathfrak{u}(\varkappa_1), I_1\mathfrak{u}(\varkappa_1),I_2\mathfrak{u}(\varkappa_1))\bigg|\biggr\}\\&
\le \dfrac{(1-\omega)}{B(\omega)}\|\mathfrak{f}(\varkappa_2, \mathfrak{u}(\varkappa_2), I_1\mathfrak{u}(\varkappa_2),I_2\mathfrak{u}(\varkappa_2))-\mathfrak{f}(\varkappa_1, \mathfrak{u}(\varkappa_1), I_1\mathfrak{u}(\varkappa_1),I_2\mathfrak{u}(\varkappa_1))\|\\&~~~~+\dfrac{\omega}{B(\omega)}~_{0}I^{\omega}\|\mathfrak{f}(\varkappa_2, \mathfrak{u}(\varkappa_2), I_1\mathfrak{u}(\varkappa_2),I_2\mathfrak{u}(\varkappa_2))-\mathfrak{f}(\varkappa_1, \mathfrak{u}(\varkappa_1), I_1\mathfrak{u}(\varkappa_1),I_2\mathfrak{u}(\varkappa_1))\|\\&
		\end{aligned}
	\end{equation*}
	\begin{equation*}
	\begin{aligned}
	&
	\le\dfrac{(1-\omega)}{B(\omega)}\bigg(\mathcal{L_{1}}\big[\|(\varkappa, \mathfrak{z}(\varkappa), I_1\mathfrak{z}(\varkappa),I_2\mathfrak{z}(\varkappa))\|+\varkappa(\lambda_{1}\|(\varkappa, \mathfrak{z}(\varkappa), I_1\mathfrak{z}(\varkappa),I_2\mathfrak{z}(\varkappa))\|)\\&~~~+\mathcal{T}(\mathcal{M}_{1}\|(\varkappa, \mathfrak{z}(\varkappa), I_1\mathfrak{z}(\varkappa),I_2\mathfrak{z}(\varkappa))\|)\big]\bigg)\\&~~~
	+\dfrac{\omega}{B(\omega)}\bigg(\mathcal{L_{1}}\big[\|(\varkappa, \mathfrak{z}(\varkappa), I_1\mathfrak{z}(\varkappa),I_2\mathfrak{z}(\varkappa))\|+\varkappa(\lambda_{1}\|(\varkappa, \mathfrak{z}(\varkappa), I_1\mathfrak{z}(\varkappa),I_2\mathfrak{z}(\varkappa))\|)\\&~~~+\mathcal{T}(\mathcal{M}_{1}\|(\varkappa, \mathfrak{z}(\varkappa), I_1\mathfrak{z}(\varkappa),I_2\mathfrak{z}(\varkappa))\|)\big]\bigg)\dfrac{  (\varkappa_2-\varkappa_1)^{\omega}}{\omega\Gamma(\omega)}\\&
	\le\dfrac{(1-\omega)}{B(\omega)}q  (\varkappa_2-\varkappa_1)+\dfrac{\omega}{B(\omega)}q  (\varkappa_2-\varkappa_1)\dfrac{  (\varkappa_2-\varkappa_1)^{\omega}}{\omega\Gamma(\omega)}\\&
	\le ~q\bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{  (\varkappa_2-\varkappa_1)^{\omega}}{B(\omega)\Gamma(\omega)}\bigg)
	(\varkappa_2-\varkappa_1),
	\end{aligned}
	\end{equation*} 
		$	\|\mathcal{F}_{2}{\mathfrak{u}(\varkappa_{2})}-\mathcal{F}_{2}{\mathfrak{u}(\varkappa_{1})}\|\longrightarrow0$ as ${\varkappa_2}\longrightarrow{\varkappa_1}$. Consequently, $\mathcal{F}_{2}$ is equicontinuous operator on $\mathcal{B_{\bar{r}}}$. Therefore by the Arzela-Ascoli theorem $\mathcal{F}_{2}$  is relatively compact on  $\mathcal{B_{\bar{r}}}$.
	Hence by theorem (2.8) $\mathcal{F}$ has at least one fixed point.  Thus ${\mathfrak{u}}$ is that fixed point of $\mathcal{F}$. Consequently, ${\mathfrak{u}}$ is solution of equ. (\ref{r1}).
\end{enumerate}
\section*{Uniqueness Result}
{\bf Theorem 3.3.} Assume that $\mathcal{R_1}-\mathcal{R_6}$ are satisfied. If  $\mathcal{f}^{*}(0,\mathcal{\mathfrak{u}}(0),0,\int_{0}^{T}\varphi(0,\varsigma,\mathcal{\mathfrak{u}(\varsigma)d\varsigma}))=\psi^{*}(0,\mathcal{\mathfrak{u}}(0))=0$ and $\mu_{1}+\mathcal{N}_1+\bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{\Gamma(\omega) B(\omega)}\bigg)\mathcal{L_{1}}\big(1+\lambda_{1}+\mathcal{T}~\mathcal{M}_{1}\big)\le 1$. Then problem (\ref{r1}) has unique solution on $[0,1]$.\\
{\bf Proof.} 	For any $\mathcal{\mathfrak{u} \in B}_{\bar{r}}.$ 
\begin{equation*}
\begin{aligned}
\|\mathcal{F\mathfrak{u}}\|\\& 
= ~\underset{\varkappa\in[0,1]}{\text{sup}}\biggl\{\bigg|\psi^{*}(\varkappa,\mathfrak{u}(\varkappa))+\int_{0}^{1}\mathcal{h(\varsigma,\mathcal{\mathfrak{u}}(\varsigma))d\varsigma}+\dfrac{(1-\omega)}{B(\omega)}\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa))\\&~~~+\dfrac{\omega}{B(\omega)}~_{0}I^{\omega}\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa))\bigg|\biggr\}\\&
\le ~ \underset{\varkappa\in[0,1]}{\text{sup}}\biggl\{|\psi^{*}(\varkappa,\mathfrak{u}(\varkappa))|+\bigg|\int_{0}^{1}\mathcal{h(\varsigma,\mathcal{\mathfrak{u}}(\varsigma))d\varsigma}\bigg|+\dfrac{(1-\omega)}{B(\omega)}|\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa))|\\&~~~+\dfrac{\omega}{B(\omega)}~_{0}I^{\omega}|\mathfrak{f}(\varkappa, \mathfrak{u}(\varkappa), I_1\mathfrak{u}(\varkappa),I_2\mathfrak{u}(\varkappa))|\biggr\}\\&
\le\underset{\varkappa\in[0,1]}{\text{sup}}\biggl\{|\psi^{*}(\varkappa,\mathfrak{u}(\varkappa))-\psi^{*}(\varkappa,\mathcal{\mathfrak{u}}(0))+\psi^{*}(\varkappa,\mathcal{\mathfrak{u}}(0))|\\&+\int_{0}^{1}|\mathcal{h(\varsigma,\mathfrak{u}(\varsigma))d\varsigma}-\mathcal{h}(\varsigma,\mathcal{\mathfrak{u}}(0))+\mathcal{h}(\varsigma,\mathcal{\mathfrak{u}}(0))|\mathcal{d\varsigma}\\&~~~+\dfrac{(1-\omega)}{B(\omega)}|\mathfrak{f}(\mathcal{\varkappa, {\mathfrak{u}}(\varkappa)}, I_1{\mathfrak{u}}(\varkappa),I_2\mathfrak{u}(\varkappa))-\mathfrak{f}(\varkappa, {\mathfrak{u}}(0), I_1{\mathfrak{u}}(0),I_2{\mathfrak{u}}(0))+\mathfrak{f}(\varkappa,{\mathfrak{u}}(0), I_1{\mathfrak{u}}(0),I_2{\mathfrak{u}}(0))|\\&~~~+\dfrac{\omega}{B(\omega)}~_{0}I^{\omega}|\mathfrak{f}({\varkappa, {\mathfrak{u}}(\varkappa)}, I_1\mathfrak{u}(\varkappa),I_2{\mathfrak{u}(\varkappa)})-\mathfrak{f}(\varkappa, {\mathfrak{u}}(0), I_1{\mathfrak{u}}(0),I_2{\mathfrak{u}}(0))+\mathfrak{f}(\varkappa, \mathcal{\mathfrak{u}}(0), I_1{\mathfrak{u}}(0),I_2{\mathfrak{u}}(0))|\biggr\}\\&
\le\mu_{1}\|{\mathfrak{u}}\|+\mu_{2}+\mathcal{N}_{1}\|{\mathfrak{u}}\|+\mathcal{N}_{2}+\dfrac{(1-\omega)}{B(\omega)}\bigg(\mathcal{L_{1}}\big[\|{\mathfrak{u}}\|+\varkappa\|(\lambda_{1}\|{\mathfrak{u}}\|+\lambda_{2})+\mathcal{T}(\mathcal{M}_{1}\|{\mathfrak{u}}\|+\mathcal{M}_{2})\big]\bigg)\\&~~~+\dfrac{(1-\omega)}{B(\omega)}\mathcal{L_{2}}
+\dfrac{\omega}{B(\omega)}\bigg(\mathcal{L_{1}}\big[\|{\mathfrak{u}}\|+\varkappa\|(\lambda_{1}\|{\mathfrak{u}}\|+\lambda_{2})+\mathcal{T}(\mathcal{M}_{1}\|{\mathfrak{u}}\|+\mathcal{M}_{2})\big]\bigg)\dfrac{(1)^{\omega}}{\omega\Gamma(\omega)}\\&
~~~+\bigg(\dfrac{\omega}{B(\omega)}\mathcal{L_{2}}\bigg)\dfrac{(1)^{\omega}}{\omega\Gamma(\omega)}\\&
\le \bigg[\mu +\mathcal{N}+\bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{\Gamma(\omega) B(\omega)}\bigg)\mathcal{L}\big[1+\lambda+\mathcal{T}~\mathcal{M}\big]\bigg]\|{\mathfrak{u}}\|\\&~~~~+ \bigg[\mu +\mathcal{N}+\bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{\Gamma(\omega) B(\omega)}\bigg)\mathcal{L}\big[1+\lambda+\mathcal{T}~\mathcal{M}\big]\bigg]\\&
=\mathcal{Q}\|\mathcal{\mathfrak{u}}\|+\mathcal{Q}\\&
\le\mathcal{Q}\bar{r}+\mathcal{Q}\le \bar{r}
\end{aligned}
\end{equation*}
Now to prove uniqueness
\begin{equation*}
\begin{aligned}
\|	\mathcal{F}\mathcal{\mathfrak{u}}_{1}(\varkappa)-&\mathcal{F}\mathcal{\mathfrak{u}}_{2}(\varkappa)\|\\&=\underset{\varkappa\in[0,1]}{\text{sup}}\bigg|\biggl\{\psi^{*}(\varkappa,\mathcal{\mathfrak{u}}_{1}(\varkappa))+\int_{0}^{1}\mathcal{h}(\varsigma,{\mathfrak{u}}_{1}(\varsigma))\mathcal{d}\varsigma+\dfrac{(1-\omega)}{B(\omega)}\mathfrak{f}(\varkappa, \mathfrak{u}_{1}(\varkappa), I_1\mathfrak{u}_{1}(\varkappa),I_2\mathfrak{u}_{1}(\varkappa))\\&~~~+\dfrac{\omega}{B(\omega)}~_{0}I^{\omega}\mathfrak{f}(\varkappa, \mathfrak{u}_{1}(\varkappa), I_1\mathfrak{u}_{1}(\varkappa),I_2\mathfrak{u}_{1}(\varkappa))-\psi^{*}(\varkappa,\mathcal{\mathfrak{u}}_{2}(\varkappa))-\int_{0}^{1}\mathcal{h}(\varsigma,{\mathfrak{u}}_{2}(\varsigma))\mathcal{d}\varsigma\\&~~~-\dfrac{(1-\omega)}{B(\omega)}\mathfrak{f}(\varkappa, \mathfrak{u}_{2}(\varkappa), I_1\mathfrak{u}_{2}(\varkappa),I_2\mathfrak{u}_{2}(\varkappa))-\dfrac{\omega}{B(\omega)}~_{0}I^{\omega}\mathfrak{f}(\varkappa, \mathfrak{u}_{2}(\varkappa), I_1\mathfrak{u}_{2}(\varkappa),I_2\mathfrak{u}_{2}(\varkappa))\bigg|\biggr\}\\&
\le \mu_{1}\|\mathcal{\mathfrak{u}}_{1}-\mathcal{\mathfrak{u}}_{2}\|+ \mathcal{N}_{1}\|\mathcal{\mathfrak{u}}_{1}-\mathcal{\mathfrak{u}}_{2}\|\\&~~~+\dfrac{(1-\omega)}{B(\omega)}\bigg(\mathcal{L_{1}}\big[\|\mathfrak{u}_{1}(\varkappa)-\mathfrak{u}_{2}(\varkappa)\|+\varkappa(\lambda_{1}\|\mathfrak{u}_{1}(\varkappa)-\mathfrak{u}_{2}(\varkappa)\|)+\mathcal{T}(\mathcal{M}_{1}\|\mathfrak{u}_{1}(\varkappa)-\mathfrak{u}_{2}(\varkappa)\|)\big]\bigg)\\&~~~+\dfrac{(\omega)}{B(\omega)}\bigg(\mathcal{L_{1}}\big[\|\mathfrak{u}_{1}(\varkappa)-\mathfrak{u}_{2}(\varkappa)\|+\varkappa(\lambda_{1}\|\mathfrak{u}_{1}(\varkappa)-\mathfrak{u}_{2}(\varkappa)\|)\\&~~~+\mathcal{T}\big(\mathcal{M}_{1}\|\mathfrak{u}_{1}(\varkappa)-\mathfrak{u}_{2}(\varkappa)\|\big)\big]\bigg)\dfrac{(1)^{\omega}}{\omega\Gamma(\omega)}\\&
\le\bigg[\mu_{1}+\mathcal{N}_1+\bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{\Gamma(\omega) B(\omega)}\bigg)\mathcal{L_{1}}\big(1+\lambda_{1}+\mathcal{T}~\mathcal{M}_{1}\big)\bigg]\|{\mathfrak{u}_{1}}-{\mathfrak{u}_{2}}\|
\end{aligned}
\end{equation*}

Since $$\mu_{1}+\mathcal{N}_1+\bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{\Gamma(\omega) B(\omega)}\bigg)\mathcal{L_{1}}\big(1+\lambda_{1}+\mathcal{T}~\mathcal{M}_{1}\big)\le1.$$\\ Consequently, $\mathcal{F}$ is a contraction mapping. Therefore by the Banach contraction principle, the operator $\mathcal{F}$ has a unique fixed point. Hence  equ. (\ref{r1}) has a unique solution.
\section{Example}
This section of the article produce an example related to EU of solutions of the discussed problem. Let us analyse the given below  FDEs:
\begin{equation}{\label{r4.1}}
\begin{cases} \mathcal{~^{ABC}_{0}D}^{\mathcal{\omega}}\bigg[\mathcal{\mathfrak{u}(\varkappa)+\psi^*(\varkappa,\mathfrak{u}(\varkappa))}\bigg] =\mathfrak{f}\bigg(\varkappa, \mathfrak{u}(\varkappa), \int_{0}^{\varkappa}\mathcal{g}(\varkappa,\varsigma,\mathfrak{u}(\varsigma))\mathcal{d}\varsigma,\int_{0}^{T}\varphi(\varkappa,\varsigma)\mathcal{d}\varsigma\bigg),   \  \ \ \  \ \      \varkappa\in[0,1],\\
\mathcal{\mathfrak{u}}(0)=\int_{0}^{1}\mathcal{h(\varsigma,\mathfrak{u}(\varsigma))d\varsigma},   
\end{cases} 
\end{equation}
\vspace{0.3 cm}
where $~~~~\mathcal{\omega}=\dfrac{1}{2},~~~\mathcal{T}=\dfrac{\pi}{4}$,\\ 
$$\psi^*(\varkappa, \mathfrak{u}(\varkappa))=\dfrac{1}{100+\varkappa^{2}}\mathfrak{u}(\varkappa) $$\\ \begin{equation*}
\begin{aligned}
\mathfrak{f}\bigg(\varkappa, \mathfrak{u}(\varkappa), \int_{0}^{\varkappa}&\mathcal{g}(\varkappa,\varsigma,\mathfrak{u}(\varsigma))\mathcal{d}\varsigma,\int_{0}^{T}\varphi(\varkappa,\varsigma)\mathcal{d}\varsigma\bigg)\\&
=\dfrac{1}{\varkappa^2+20}\bigg(\dfrac{1}{50}\int_{0}^{\varkappa}(\varkappa^{2}+\varsigma^{2})\mathcal{\mathfrak{u}(\varsigma)d\varsigma}+\dfrac{1}{10}\int_{0}^{\mathcal{T}}(\varkappa^{2}\sin \varsigma)~{\mathfrak{u}(\varsigma)d\varsigma}\bigg),
\end{aligned}
\end{equation*} 
$$g(\varkappa,\varsigma,\mathfrak{u}(\varsigma))=(\varkappa^{2}+\varsigma^{2})\mathcal{\mathfrak{u}(\varsigma)},$$\\
$$\varphi(\varkappa,\varsigma)=(\varkappa^{2}\sin \varsigma)\mathfrak{u}(\varsigma),$$\\$$\mathcal{h}(\varsigma,\mathfrak{u}(\varsigma))=\varsigma^{2}\mathcal{\mathfrak{u}(\varsigma)}.$$\\
Now,
$$|\psi^*(\varkappa, \mathfrak{u}(\varkappa))-\psi^*(\varkappa, \mathfrak{u}(\varkappa))|\le\dfrac{1}{100}\|{\mathfrak{u}}-{\mathfrak{u}_1}\|, $$\\
$$| \mathcal{g}( \varkappa,\varsigma,\mathcal{\mathfrak{u}})-\mathcal{g}( \varkappa, \varsigma,{\mathfrak{u}}_1|\le \dfrac{1}{25}\big( \|{\mathfrak{u}}-{\mathfrak{u}}_1)\|\big) 
$$\\
$$| \psi( \varkappa,\varsigma,{\mathfrak{u}})-\psi( \varkappa, \varsigma,{\mathfrak{u}}_1|\le \dfrac{1}{10}\big( \|{\mathfrak{u}}-{\mathfrak{u}}_1)\|\big) 
$$\\
\begin{equation*}
\begin{aligned}
\bigg|\mathfrak{f}\bigg(\varkappa, \mathfrak{u}(\varkappa), \int_{0}^{\varkappa}\mathcal{g}(\varkappa,\varsigma,\mathfrak{u}(\varsigma))\mathcal{d}\varsigma,\int_{0}^{T}\varphi(\varkappa,\varsigma)\mathcal{d}\varsigma\bigg)-&\mathfrak{f}\bigg({\varkappa, \mathfrak{u}_1(\varkappa)}, \int_{0}^{\varkappa}\mathcal{g}(\varkappa,\varsigma,\mathfrak{u}_1(\varsigma))\mathcal{d}\varsigma,\int_{0}^{T}\varphi(\varkappa,\varsigma,\mathfrak{u}_1(\varsigma))\mathcal{d}\varsigma\bigg)\bigg|\\&\le\dfrac{1}{20}\bigg(\dfrac{2}{75}+\dfrac{1-\cos T}{10}\bigg)\|\mathfrak{u}-\mathfrak{u}_1)\|,
\end{aligned}
\end{equation*}
thus assumptions $(\mathcal{R_1}), (\mathcal{R_2}), (\mathcal{R_3}) ~  \text{and } ~(\mathcal{R_4}) $ are hold true. 
Hence,   $\Lambda=\mu_{1}+\mathcal{N}_1\approx0.021111<1.$ Consequently, the theorem $3.1$ implies that equ.(\ref{r4.1}) has a  solution.\\
In addition, $\mu_{1}+\mathcal{N}_1+\bigg(\dfrac{(1-\omega)}{B(\omega)}+\dfrac{1}{\Gamma(\omega) B(\omega)}\bigg)\mathcal{L_{1}}\big(1+\lambda_{1}+\varkappa~\mathcal{M}_{1}\big) \approx0.024441
<1,$ hence using  theorem $3.2$  the equ. (\ref{r4.1}) has a unique. 
\section{Conclusion}
In this paper, we examined the EU of solutions for integro FDEs involving ABC-derivative and non-local initial condition. Recently, AB-derivative gained much attention due to the non-singular property of the kernels. An example is produced to prove the usefulness of the proposed result. Although EU results for different types of FDEs in terms of ABC derivative have been investigated with non-local conditions, but this type of problem is not studied yet.
\vspace{4mm}\\
\noindent{\bf Acknowledgements}\\
The first author would like to thanks for funding provided by the  Council of Scientific and Industrial Research (CSIR)- New Delhi, India under grant no. 09/1051(0031)/2019-EMR-1  for this research work.


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\end{center}
{\small

\noindent{\bf Amita Devi}

\noindent Research Scholar

\noindent Department of Mathematics \& Statistics, 
School of Basic and Applied Sciences,   
Central University of Punjab Bathinda 


\noindent Bathinda, Punjab, India 

\noindent E-mail: amitabeniwal86@gmail.com}\\

{\small
\noindent{\bf  Dr. Anoop Kumar  }
\noindent Assistant Professor of Mathematics

\noindent Department of Mathematics \& Statistics, 
School of Basic and Applied Sciences,   
Central University of Punjab Bathinda 


\noindent Bathinda, Punjab, India 

\noindent E-mail: anoop.kumar@cup.edu.in}\\



\end{document}