Introduction

The fractional differential equations (FDEs) have been widely deployed to diverse areas of aeronautical engineering, biology, physics, mechanics, and chemistry. Mainly, FDEs have a lot of application in the modeling of numerous phenomena in different fields of engineering and science such as heat conduction, signals, fluid mechanics, and hydrodynamics[1, 2]. FDEs that globalize the usual notion of derivative and integral operators to derivatives and integrals of arbitrarily (fractional) order. In the last few years, the notion of FDEs received more attention due to their considerable interest in various fields [3,4,5]. Recently, FDEs with multiple types of derivatives like Caputo derivative, Riemann-Liouville (R-L) derivative, Hilfer fractional derivative, Hadamard derivative have been studied by many researchers. Recently, Caputo and Fabrizio [6] developed a derivative based on the exponential decay law, which helped to explore the area of FDEs. The influence of this derivative is that this derivative does not have singularity since the exponential kernel function is used. The FDEs using this new derivative have been established with significant results [7, 8]. But, the associated integral with the Caputo and Fabrizio derivative is in the term of classical order. Atangana and Baleanu established the new derivative with Mittag Leffler (ML) function to fix this problem. Atangana-Baleanu derivative and its considerable results in diverse fields we suggest the reader refer [9].

There has been a growing interest in the evolution of sequential FDEs with the p-Laplacian operator in many fields of research. The turbulent flow is a fundamental mechanics problem in a porous medium. For solving this type of issue, Leibenson [10] developed the p-Laplacian operator as follows:

$$\begin{aligned} {\phi _{{\mathcal {p}}}\big (\mathcal {v}^{'}(\mathcal {t})\big )=-{\mathcal {f}}(\mathcal {t}, \mathcal {v}(\mathcal {t}),\mathcal {v}^{'}(\mathcal {t})) \quad {\mathcal {t}}\in (0,1).} \end{aligned}$$
(1)

Later on, many important results have been established related to (1) with various type of boundary conditions see the reader [11,12,13,14,15]. Many researchers have described the EU results for FDEs with the p-Laplacian operator; for example, Li and He [16] established the positive solutions for FDEs involving the p-Laplacian operator with four-point boundary conditions. The existence of upper and lower solutions is proved using the monotone iterative method:

where \(1< \chi ,\varsigma ^*\le 2\) and \( {\mathcal {D}}^{\varsigma ^*} ,{\mathcal {D}}^{\chi } \) are fractional derivative in the form of Riemann-Liouville differentiations. The following is the integral boundary value problem involving a fractional p-Laplacian equation in the form of mixed fractional derivatives[17]:

where \(1< \chi ,\varsigma ^*\le 2\) and \( {\mathcal {D}}^{\varsigma ^*} ,{\mathcal {D}}^{\chi } \) are fractional derivative in the form of Riemann-Liouville differentiations, denotes the -Laplacian operator and satisfies the . The following is the integral boundary value problem involving a fractional p-Laplacian equation in the form of mixed fractional derivatives[17]:

where \(f \in C([0, 1] \times {\mathbb {R}}^{2},{\mathbb {R}})\) is a nonlinear function, \( ~^{\mathcal {C}}_{0}\mathcal {D}^{{\chi }}, ~^{\mathcal {C}}_{0}\mathcal {D}^{\varsigma ^*} \) be the derivatives of fractional order \({ \chi }\) and \(\varsigma ^*\) are the Caputo fractional such that \(0<\varsigma ^*, \chi \le 1\), \(\nu , \epsilon _{1}, j \in {\mathbb {R}}, \xi , \alpha \in [0, 1].\) The HU stability and existence of positive solution for a class of singular FDEs in Caputo sense with nonlinear p-Laplacian operator is discussed by Khan et al. [18]. Aslam et al. explored the EU and HU stability results for singular delay FDEs with p-Laplacian fractional boundary conditions[19]. Matar et al. studied the existence criterion for nonperiodic FDEs in the form of generalized Caputo sense with the p-Laplacian[20]. The EU result for Langevin FDEs by using the Bielecki norm with the modified argument is investigated by Faruk Develi [21]. A particular form of the equations is derived, and the uniqueness of the solutions is studied by utilizing the Burton’s method. Seda Igret Araz [22] presented a new general condition for the EU of solutions for the integro FDEs by using the notion of fractal-fractional derivative and AB- integral under contraction mapping. Devi et al. [23] analyzed the stability and EU of solutions for a general FDEs with the help of green function and nonlocal conditions by utilizing the fixed point technique. Khan et al.[24] and Xiaoyan Li [25] investigated the existence and HU stability for a nonlinear singular FDEs with ML-kernel involving local conditions. However, several FDEs are yet to be investigated and enlarged to real space problems. The theory of FDEs with various boundary conditions is still in the beginning phase, and many characteristics of this theory need to be investigate. A boundary condition is a direction, some sequence of values of the unknown solution. Recently, the notion of boundary value problem for FDEs with the positive solution is studied very little in Banach space until now. Also, there are very few papers on FDEs with the p-Laplacian operator involving AB-fractional derivative in Caputo’s sense.

Motivated by the above discussions, our primary objective is to investigate the EU of solutions for singular FDEs in Banach space. The main contribution of this article is the singular FDEs is discussed with nonlocal integral boundary conditions instead of local conditions. Moreover, the accomplished solution in this manuscript are generalized and improve the existing literature works.

$$\begin{aligned} {} {\left\{ \begin{array}{ll} ~^{\mathcal {ABC}}_{0}\mathcal {D}^{{\chi }}\phi ^{*}_{{\mathcal {p}}}\big [~^{\mathcal {ABC}}_{0}\mathcal {D}^{\varsigma ^*}\big (\mathcal {v(t)}\big )\big ] =-{\mathcal {f}}\mathcal {(t, v(t))}, \ \ \ \mathcal {t}\in [0,1],\\ \phi ^{*}_{\mathcal {p}}\big [~^{\mathcal {ABC}}_{0}\mathcal {D}^{\varsigma ^*}\mathcal {v(t)}\big ]\big |_{t=0} =0, \ \ \ {\mathcal {v}}^{'}({{0}})=0, \ \ \ \ {\mathcal {v}}(1)=\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta } ~~\phi ^{*}_q[^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\chi }\big ({\mathcal {f}}(\kappa , \mathcal{sv}\mathcal{}(\kappa ))\big )]{\mathcal {d}}\kappa , \\ \end{array}\right. }\nonumber \\ \end{aligned}$$
(2)

where \( ~^{\mathcal {ABC}}_{0}\mathcal {D}^{{\chi }}, ~^{\mathcal {ABC}}_{0}\mathcal {D}^{\varsigma ^*} \) be the derivatives of fractional order \({ \chi }\) and \(\varsigma ^*\) in the left AB-derivative Caputo’s sense respectively and is continuous function. The orders \(0< \chi \le 1, 1<\varsigma ^*\le 2 , \delta >0,<\epsilon _{1}<1 \) and be the -Laplacian operator and satisfies the .

The scheme of the remaining article is as planned out as follows: we recall some notations and basic definitions which are desired in the discussion in the next section. \( 2^{nd}\) section would introduce certain properties of the green function to find the desired results. \( 3^{rd}\) is devoted to deriving the EU results through the fixed point technique. Further, we also establish the HU stability in \( 4^{th}\) part. At the end, we discuss an application to analyze our outcome more apparently.

Basic results and Preliminaries

Definition 2.1

[2] For \(\chi >0,\) The R-L fractional integral of order \( \chi \in {\mathbb {R}}\) for a continuous function is defined as

$$\begin{aligned} {} I^{\chi }\mathcal {f}({\mathcal {t}}) =\dfrac{1}{\varGamma (\chi )}\int _{0}^{{\mathcal {t}}}(\mathcal {t-x})^{\chi -1}\mathcal {f(x)dx}. \end{aligned}$$
(3)

provided that such integral exists.

Definition 2.2

[1] For \(\chi \ge 0\), is a integrable and continuous function, the R-L fractional derivative of order \(\chi \) is defined as

$$\begin{aligned} {} {\mathcal {D}^{\chi }_{\mathcal {t}}{\mathcal {f}}(\mathcal {t}) =\frac{1}{\varGamma ({\mathcal {n}}-\chi )}\bigg (\frac{\mathcal {d}}{\mathcal{dt}\mathcal{}} \bigg )^{\mathcal {n}}\int _{0}^{\mathcal {t}}(\mathcal {t-x})^{\mathcal {n}-\chi -1}\mathcal {f(x)dx}, \ \ \ \ \mathcal {n} =[\chi ]+1,}\qquad \end{aligned}$$
(4)

where \([\chi ]\) represent the gratest-integer.

Definition 2.3

[1] For n-times continuously differentiable function , Caputo fractional derivative of order \( \chi \in {\mathbb {R}} ~(\chi >0)\) is defined as

$$\begin{aligned} {} {~^{\mathcal {c}}\mathcal {D}^{\chi }{\mathcal {f(t)}} =\dfrac{1}{\varGamma ({\mathcal {n}}-\chi )}\int _{0}^{{\mathcal {t}}}(\mathcal {t-x})^{{\mathcal {n}}-\chi -1} (\mathcal {f})^{\mathcal {n}}(\mathcal {x})\mathcal{dx}\mathcal{}, \ \ \ \ \ \mathcal {n-1}<\chi <{\mathcal {n}},\ \ \ \ {\mathcal {n}}=[\chi ]+1,} \end{aligned}$$
(5)

where \([\chi ]\) represent the gratest-integer.

Definition 2.4

[27] Let \(0<\chi \le 1 \) and , where \(0\le a<b\), the Caputo AB-fractional derivative and the R-L AB-fractional derivative of order \(\chi \) are defined as

$$\begin{aligned} {} {~^{\mathcal {ABC}}\mathcal {D}^{\chi }\mathcal {v(t)} =\dfrac{B(\chi )}{{{1}}-\chi }\int _{0}^{{\mathcal {t}}}\mathcal {v}^{'}(\mathcal {x}) E_{\chi }\bigg [-\chi \dfrac{(\mathcal {t-x})^{\chi }}{1-\chi }\bigg ]\mathcal{dx}\mathcal{},} \end{aligned}$$
(6)

and

$$\begin{aligned} {} {~^{\mathcal {ABR}}\mathcal {D}^{\chi }\mathcal {v(t)} =\dfrac{B(\chi )}{{{1}}-\chi }\dfrac{\mathcal {d}}{\mathcal{dt}\mathcal{}} \bigg (\int _{0}^{{\mathcal {t}}}\mathcal {v(x)} E_{\chi }\bigg [-\chi \dfrac{(\mathcal {t-x})^{\chi }}{1-\chi }\bigg ]\mathcal{dx}\mathcal{}\bigg ).} \end{aligned}$$
(7)

respectively, where \(E_{\chi }\) is called the Mittag-Leffler function and given by

and \(B(\chi )\) is a normalizing positive function satisfying \(B(0)=B(1)=1\).

Definition 2.5

[27] Let \(0<\chi \le 1 \) and , where \(0 \le a<b\), the associated AB-fractional integral is

$$\begin{aligned} {} {~{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\mathcal {v(t)} =\dfrac{(1-\chi )}{B(\chi )}\mathcal {v(t)}+\dfrac{\chi }{B(\chi )}I^{\chi }\mathcal {v(t)},} \end{aligned}$$
(8)

where \(I^{\chi }\) is the R-L fractional integral defined in (3).

Lemma 2.1

[26] The fractional derivative AB-integral of the fuction is given by

$$\begin{aligned}{} {{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\varsigma *}\mathcal {(v(t))}=\dfrac{1-\chi ^*+n}{B(\chi ^*-1)}I^{n}_{a}(v(t))+\dfrac{\chi ^*-n}{B(\chi ^*-n)}I^{\chi ^*}_{a}(v(t)),} \end{aligned}$$

where \(I^{\chi ^*}\) is the R-L fractional integral defined in (3).

Lemma 2.2

[2] Let and , then

$$\begin{aligned}{} {I^{ \chi }\mathcal {D}^{ \chi }\mathcal {{f}(v)}=\mathcal {{f}(v)}+a_0+a_1v+a_2v^2+a_3v^3+...+a_{k-1}v^{k-1},} \end{aligned}$$

for the \(a_{j}\in {\mathbb {R}}\) for \(j=0,1,2,\ldots ,k-1.\)

Theorem 2.3

[29, 30](Guo-Krasnosel’skii Theorem ) Consider \({{\mathcal {Y}}}^{*}\) be a Banach space and let a cone \({\mathfrak {P}}^{*}\in {\mathcal {Y}}^{*} \). Assume that \(\mathcal {A}^{*}_1,\mathcal {A}^{*}_2\) are two bounded subset of \({\mathcal {Y}}^{*}\) such that \(0\in \mathcal {A}^{*}_{1},~ \overline{\mathcal {A}^{*}_1}\subset \mathcal {A}^{*}_{2}\). Then, a operator \(\mathcal {G}^{*}:{\mathfrak {P}}^{*}\cap (\overline{\mathcal {A}^{*}_2} \setminus \mathcal {A}^{*}_{1})\longrightarrow {\mathfrak {P}}^{*},\) which is completely continuous and satisfying the following

Then \( \mathcal {G}^{*}\) has a fixed point in \({\mathfrak {P}}^{*}\cap (\overline{\mathcal {A}^{*}_{2}} \setminus \mathcal {A}^{*}_{1}).\)

Lemma 2.4

[23, 24] For the -Laplacian operator , the following conditions are hold true \((1)\, \text { If }\, |\gamma _{1}|, |\gamma _{2}|\ge \mu>0,~1<{\mathcal {p}}\le 2,~\gamma _{1},\gamma _{2}>0\), then

\((2)\,\text { If }\, {\mathcal {p}}> 2,|\gamma _{1}|, |\gamma _{2}|\le \mu ^{*}>0\), then

Green function

Theorem 3.1

Let and \( \varsigma ^* \in (1,2)\) such that then FDEs (2) with \(\phi ^{*}_p\) operator has the following integral form

$$\begin{aligned} {} {\begin{aligned} \mathcal {v(t)}=&\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{t}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa +\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}, {\kappa })\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\&+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa , \end{aligned}} \end{aligned}$$
(9)

where the Green’s function is definied by

$$\begin{aligned} {} {\begin{aligned}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}, {\kappa })= {\left\{ \begin{array}{ll} \dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}-\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{({\mathcal {t}}- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}, \ \ \ \ 0< \kappa \le t<1,\\ \dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma ( \varsigma ^*)}, \ \ \ \ \ \ \ \ \ ~~~~~~~~~~~ \ \ \ \ 0<t\le \kappa <1. \end{array}\right. } \end{aligned}} \end{aligned}$$
(10)

Proof

Taking the integral operator \(^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\chi }\) to both sides (2) and using Lemma (2.1) the given equation (2) becomes

$$\begin{aligned} {} {\phi ^{*}_{\mathcal {p}}\big (~^{\mathcal {ABC}}_{0}\mathcal {D}^{\varsigma ^*}[\mathcal {v(t)}]\big ) =-^{\mathcal{AB}\mathcal{}}I^{\chi }[{\mathcal {f}}\mathcal {(t, v(t))}]+b_0.} \end{aligned}$$
(11)

As given condition , put the value of \(b_0=0\), then (11) convert in

$$\begin{aligned} {} {\phi ^{*}_{\mathcal {p}}\big (~^{\mathcal {ABC}}_{0}\mathcal {D}^{\varsigma ^*}[\mathcal {v(t)}]\big ) =-^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\chi }[{\mathcal {f}}\mathcal {(t, v(t))}].} \end{aligned}$$
(12)

Applying q-Laplacian opertaor further (12) get the form

$$\begin{aligned} {} {~^{\mathcal {ABC}}_{0}\mathcal {D}^{\varsigma ^*}[\mathcal {v(t)}]=-\phi ^{*}_{q} \big (^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\chi }[{\mathcal {f}}\mathcal {(t, v(t))}]\big ).} \end{aligned}$$
(13)

Again using Lemma (2.1), then (13) becomes

$$\begin{aligned} {} {\mathcal {v(t)}=-^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^{*}}\Big (\phi ^{*}_{q} \big (^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\chi }[{\mathcal {f}}\mathcal {(t, v(t))}]\big )\Big )+a_0+a_1t,} \end{aligned}$$
(14)

where \( a_{j}\in {\mathbb {R}}\) for \(j=0,1.\)

Using boundary conditions in (14), \( \implies a_1=0 \) and , implies that .

Putting value of constants \(a_0\) and \(a_1\) in (14), we get

$$\begin{aligned} {} {\begin{aligned} \mathcal {v(t)}&= - ^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]+ ^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]\big |_{t=1}\\ {}&~~+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}} \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]\big ){\mathcal {d}}\kappa \\ {}&=\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{t}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\&~~+\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}, {\kappa })\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\ {}&~~+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa , \end{aligned}} \end{aligned}$$
(15)

where is defined in (10). \(\square \)

Lemma 3.2

be the Green’s function specified in (10) satisfies the following properties:

;

\((\mathcal {B}_{2})~~\)The function is decreasing multivalued function and ;

.

Proof

To assess \((\mathcal {B}_{1})\), we will discuss the following cases

Case 1. For \( \kappa \le t\), then

$$\begin{aligned} {} {\begin{aligned} {\mathcal {H}}^{\varsigma ^*}(\mathcal {t}, {\kappa })&= \dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}-\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{({\mathcal {t}}- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}\\ {}&=\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}-{\mathcal {t}}^{\varsigma ^*-1}\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \frac{\kappa }{{\mathcal {t}}})^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}\\ {}&\ge \dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}-{\mathcal {t}}^{\varsigma ^*-1}\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}>0. \end{aligned}} \end{aligned}$$
(16)

Case 2. When \(t\le \kappa \), we evaluate

$$\begin{aligned} {} \begin{aligned} \dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma ( \varsigma ^*)}>0. \end{aligned} \end{aligned}$$
(17)

From (16) and (17), verified that .

Now, to evaluate \((\mathcal {B}_{2})\), we suppose that

Case 1. For \( \kappa \le t\), then

$$\begin{aligned} {} {\begin{aligned} \dfrac{\partial }{\partial {\mathcal {t}}} {\mathcal {H}}^{\varsigma ^*}(\mathcal {t}, {\kappa })&= \dfrac{-(\varsigma ^*-1)({\mathcal {t}}- \kappa )^{\varsigma ^*-2}}{B(\varsigma ^*)\varGamma (\varsigma ^*-1)} <0. \end{aligned}} \end{aligned}$$
(18)

Case 2. When \(t\le \kappa \), we find that

$$\begin{aligned} {} {\begin{aligned} \dfrac{\partial }{\partial {\mathcal {t}}}{\mathcal {H}}^{\varsigma ^*}(\mathcal {t}, {\kappa })=\dfrac{\partial }{\partial {\mathcal {t}}}\bigg (\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma ( \varsigma ^*)}\bigg )=0. \end{aligned}} \end{aligned}$$
(19)

From equation (18) and (19), we observe , accordingly, is decreasing function. Therefore, we have for

$$\begin{aligned} {} {\begin{aligned} \underset{t\in [0,1]}{\text {max}}{\mathcal {H}}^{\varsigma ^*}(\mathcal {t}, {\kappa })&=\lim \limits _{{\mathcal {t}}\rightarrow 0}\bigg \{\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}-\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{({\mathcal {t}}- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}\bigg \}\\ {}&=\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}\\ {}&={\mathcal {H}}^{\varsigma ^*}({0, \kappa }) \end{aligned}} \end{aligned}$$
(20)

and for

$$\begin{aligned} {} {\underset{t\in [0,1]}{\text {max}}{\mathcal {H}}^{\varsigma ^*}(\mathcal {t}, {\kappa })=\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}={\mathcal {H}}^{\varsigma ^*}({0, \kappa }).} \end{aligned}$$
(21)

For \((\mathcal {B}_{3})\), we assume that

Case 1. For , then with the condition , we have

$$\begin{aligned} {} {\begin{aligned} {\mathcal {H}}^{\varsigma ^*}(\mathcal {t}, {\kappa })&= \dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}-\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{({\mathcal {t}}- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}\\ {}&=\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}-{\mathcal {t}}^{\varsigma ^*-1}\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \frac{\kappa }{{\mathcal {t}}})^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}\\ {}&\ge {\mathcal {t}}^{\varsigma ^*-1}\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}={\mathcal {t}}^{\varsigma ^*-1}{\mathcal {H}}^{\varsigma ^*}({0, \kappa }) \end{aligned}} \end{aligned}$$
(22)

Case 2. For , then

$$\begin{aligned} {} {\begin{aligned} {\mathcal {H}}^{\varsigma ^*}(\mathcal {t}, {\kappa })&=\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}\\ {}&\ge {\mathcal {t}}^{\varsigma ^*-1}\dfrac{\varsigma ^*-1}{B(\varsigma ^*-1)}\dfrac{(1- \kappa )^{ \varsigma ^* -1}}{\varGamma (\varsigma ^* )}={\mathcal {t}}^{\varsigma ^*-1}{\mathcal {H}}^{\varsigma ^*}({0, \kappa }). \end{aligned}} \end{aligned}$$
(23)

Consequently, from equations (22) and (23), proof of condition \(\mathcal {B}_{3}\) is completed.

Existence result

Here, we establish the existence criterion for FDEs (2) by inserting the below mentioned conditions.

Consider \({\mathcal {Y}}^{*}=\mathcal { C[0,1]}\) be the Banach space endowed with the . Suppose that \({\mathfrak {P}}^{*}\in {\mathcal {Y}}^{*} \) be a cone having non-negative functions, . Let , the given equation (2), using Theorem (3.1) has alternative form

$$\begin{aligned} {} {\begin{aligned} \mathcal {v(t)}=&\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{t}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\ {}&+\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}, {\kappa })\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\&\quad +\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa . \end{aligned}} \end{aligned}$$
(24)

Let us define a operator \({\mathfrak {F}}^{*}:{\mathfrak {P}}^{*}\setminus \{0\}\rightarrow {\mathcal {Y}}^{*}\) assosiated with problem (2), such that

$$\begin{aligned} {} {\begin{aligned} {\mathfrak {F}}^{*}\mathcal {v(t)}=&\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{t}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\&\quad +\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}, {\kappa })\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\ {}&+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa . \end{aligned}} \end{aligned}$$
(25)

By using Theorem 3.1, be a fixed point of \( {\mathfrak {F}}^{*}\) i.e.,

$$\begin{aligned} {\mathcal {v(t)}={\mathfrak {F}}^{*}\mathcal {v(t)}.} \end{aligned}$$
(26)

To find the desired result we define the some assumptions:

  • and .

  • .

Theorem 4.1

Let us consider that conditions \( (\mathcal {R}_{1}) -(\mathcal {R}_{2})\) are satisfied and . Then \({\mathfrak {F}}^{*}\) is completely continuous.

Proof

For any , using Lemma 3.2 and (25), we have

$$\begin{aligned} {} {\begin{aligned} {\mathfrak {F}}^{*}\mathcal {v(t)}\le&\bigg \{\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{0}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\&\quad +\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {0}, {\kappa })\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\ {}&+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \bigg \}. \end{aligned}} \end{aligned}$$
(27)

Now using \(\mathcal {B}_{3}\) of Lemma 3.2 we have

$$\begin{aligned} {} {\begin{aligned} {\mathfrak {F}}^{*}\mathcal {v(t)}\ge&\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{t}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa +{\mathcal {t}}^{\varsigma ^* -1}\\&\quad \int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {0}, {\kappa })\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\ {}&+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q [{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}((\kappa ,\mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa . \end{aligned}} \end{aligned}$$
(28)

For continuity of \({\mathfrak {F}}^{*},\) we shows that as , let us contrive

$$\begin{aligned} {} {\begin{aligned} \Vert {\mathfrak {F}}^{*}\mathcal {v}_{\mathcal {n}}(\mathcal {t})-{\mathfrak {F}}^{*}\mathcal {v(t)}\Vert&= \underset{{\mathcal {t}}\in [0,1]}{\text {max}}\bigg \{\bigg |\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{t}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa ,\mathcal {v}_{\mathcal {n}}(\kappa ))\big )]{\mathcal {d}}\kappa \\&\quad +\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}, {\kappa })\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa ,\mathcal {v}_{\mathcal {n}}(\kappa ))\big )]{\mathcal {d}}\kappa \\&+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa ,\mathcal {v}_{\mathcal {n}}(\kappa )) \big )]{\mathcal {d}}\kappa -\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\\&\quad \int _{t}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\ {}&-\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}, {\kappa })\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa -\epsilon _{1}\\&\quad \int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa ,\mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \bigg |\bigg \}\\ {}&\le \dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{t}^{1}\big |\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa ,\mathcal {v}_{\mathcal {n}}(\kappa ))\big )]\\&\quad -\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]\big |{\mathcal {d}}\kappa \\ {}&+\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}, {\kappa })\big |\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa ,\mathcal {v}_{\mathcal {n}}(\kappa ))\big )]\\&\quad -\phi ^{*}_q [{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]\big |{\mathcal {d}}\kappa \\ {}&+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta } \big |\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa ,\mathcal {v}_{\mathcal {n}}(\kappa ))\big )]\\&\quad -\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]\big |{\mathcal {d}}\kappa \end{aligned}} \end{aligned}$$
(29)

By continuity of function , we have . This implies that \({\mathfrak {F}}^{*}\) is a continuous operator. \(\square \)

Now, we discuss the uniform boundedness of \({\mathfrak {F}}^{*} \) on \((\overline{\mathcal {A}^{{*}}_2(r)}) \setminus \mathcal {A}^{{*}}_1(r).\)

By (25) and using \((\mathcal {R}_{1})-(\mathcal {R}_{2})\), for any , we get

$$\begin{aligned} {} \Vert {\mathfrak {F}}^{*} {\mathcal {v}}\Vert= & {} \underset{{\mathcal {t}}\in [0,1]}{\text {max}}\bigg \{\bigg |\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)} \int _{t}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \nonumber \\&+\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}, {\kappa })\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \nonumber \\&+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta } \phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \bigg |\bigg \}\nonumber \\\le & {} \dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{0}^{1}\phi ^{*}_q\Bigg [\dfrac{1-\chi }{B(\chi )}\Vert \mathcal {f}(\kappa , \mathcal {v}(\kappa ))\Vert \nonumber \\&+\dfrac{\chi }{B(\chi )\varGamma (\chi )}\int _{0}^\kappa (\kappa -\xi )^{\chi -1}\Vert {\mathcal {f}}(\xi ,{\mathcal {v}}(\xi )) \Vert {\mathcal {d}} \xi \Bigg ]{\mathcal {d}}\kappa \nonumber \\&+\,\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}, {\kappa })\phi ^{*}_q\Bigg [\dfrac{1-\chi }{B(\chi )}\Vert \mathcal {f}(\kappa , \mathcal {v}(\kappa ))\Vert \nonumber \\&+\dfrac{\chi }{B(\chi )\varGamma (\chi )}\int _{0}^\kappa (\kappa -\xi )^{\chi -1}\Vert {\mathcal {f}}(\xi ,{\mathcal {v}}(\xi )) \Vert {\mathcal {d}} \xi \Bigg ]{\mathcal {d}}\kappa \nonumber \\&+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q\Bigg [\dfrac{1-\chi }{B(\chi )}\Vert \mathcal {f}(\kappa , \mathcal {v}(\kappa ))\Vert \nonumber \\&+\dfrac{\chi }{B(\chi )\varGamma (\chi )}\int _{0}^\kappa (\kappa -\xi )^{\chi -1}\Vert {\mathcal {f}}(\xi ,{\mathcal {v}}(\xi )) \Vert {\mathcal {d}} \xi \Bigg ]{\mathcal {d}}\kappa \nonumber \\\le & {} \dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{0}^{1}\phi ^{*}_q\Big [\dfrac{1-\chi }{B(\chi )}\varTheta +\dfrac{\varTheta }{B(\chi )\varGamma (\chi )}\Big ]{\mathcal {d}}\kappa \nonumber \\&+\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }({0, \kappa })\phi ^{*}_q\Big [\dfrac{1-\chi }{B(\chi )}\varTheta +\dfrac{\varTheta }{B(\chi )\varGamma (\chi )}\Big ]{\mathcal {d}}\kappa \nonumber \\&+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q\Big [\dfrac{1-\chi }{B(\chi )}\varTheta +\dfrac{\varTheta }{B(\chi )\varGamma (\chi )}\Big ] {\mathcal {d}}\kappa \nonumber \\< & {} \infty , \end{aligned}$$
(30)

Consequently, \( {\mathfrak {F}}^{*} \) is uniformly bounded.

Next, we verify that compactness of operator \({\mathfrak {F}}^{*} \), for this firstly we discuss the equicontinuity of \({\mathfrak {F}}^{*} \).

For , we have

$$\begin{aligned} |{\mathfrak {F}}^{*} {\mathcal {v}}(\mathcal {t}_{2})- & {} {\mathfrak {F}}^{*} {\mathcal {v}}(\mathcal {t}_{1})|\nonumber \\&=\underset{{\mathcal {t}}\in [0,1]}{\text {max}}\bigg \{\bigg |\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{t_2}^{1}\phi ^{*}_q [{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \nonumber \\&\quad -\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{t_1}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \nonumber \\&+\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}_2, \kappa )\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \nonumber \\&\quad -\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}_1,{\kappa })\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \nonumber \\&+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q [{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \nonumber \\&\quad -\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q [{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \bigg |\bigg \}\nonumber \\\le & {} \bigg [\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\bigg ]\bigg |\int _{t_2}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \nonumber \\&\quad -\int _{t_1}^{t_2}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \nonumber \\&-\int _{t_2}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \bigg |\nonumber \\&\quad +\int _{0}^{1}\big |{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}_2, \kappa )-{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}_1,{\kappa })\big |\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (|\mathcal {f}(\kappa , \mathcal {v}(\kappa ))|\big )]{\mathcal {d}}\kappa \nonumber \\&\le ~\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\bigg |-\int _{t_1`}^{t_2}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \bigg |\nonumber \\&\quad +\int _{0}^{1}\big |{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}_2, \kappa )-{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}_1,{\kappa })\big |\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (|\mathcal {f}(\kappa , \mathcal {v}(\kappa ))|\big )]{\mathcal {d}}\kappa \nonumber \\&\le ~\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{t_1`}^{t_2}\phi ^{*}_q\Big [\dfrac{1-\chi }{B(\chi )} \varTheta +\dfrac{\varTheta }{B(\chi )\varGamma (\chi )}\Big ]{\mathcal {d}}\kappa \nonumber \\&+\bigg |\dfrac{\big (\mathcal {t}_{2}-\kappa \big )^{\varsigma ^*}}{\varGamma (\varsigma ^*+1)} -\dfrac{\big (\mathcal {t}_{1}-\kappa \big )^{\varsigma ^*}}{\varGamma (\varsigma ^*+1)}\bigg |\phi ^{*}_q \Big [\dfrac{1-\chi }{B(\chi )}\varTheta +\dfrac{\varTheta }{B(\chi )\varGamma (\chi )}\Big ]\nonumber \\&|{\mathfrak {F}}{\mathcal {v}}(\mathcal {t}_{2})-{\mathfrak {F}}{\mathcal {v}}(\mathcal {t}_{1})|\longrightarrow {0} ~~~~~ as~~~~~ (\mathcal {t}_{2}-\mathcal {t}_{1})\longrightarrow {0}. \end{aligned}$$
(31)

Thus, \({\mathfrak {F}}^{*}\) is an equicontinuous operator on \((\overline{\mathcal {A}^{{*}}(r_2)}) \setminus \mathcal {A}^{{*}}(r_1)\) and by Arzela Ascoli theorem \({\mathfrak {F}}^{*}\) is compact on \((\overline{\mathcal {A}^{{*}}(r_2)}) \setminus \mathcal {A}^{{*}}(r_1).\) Thus \({\mathfrak {F}}^{*}\) is completely continuous.

Let us calculate the height functions for for

$$\begin{aligned} {} \begin{aligned}{\left\{ \begin{array}{ll} \phi ^{*}_{\text {min}}({\mathcal {t}},r)=\underset{{\mathcal {t}}\in [0,1]}{\text {min}}\{{\mathcal {f}}\mathcal {(t, v(t))}:{\mathcal {t}}^{\varsigma ^*-1}r\le {\mathcal {v}}\le r \}>-\infty ,\\ \phi ^{*}_{\text {max}}({\mathcal {t}},r)=\underset{{\mathcal {t}}\in [0,1]}{\text {max}}\{{\mathcal {f}}\mathcal {(t, v(t))}:{\mathcal {t}}^{\varsigma ^*-1}r\le {\mathcal {v}}\le r \}<+\infty .\\ \end{array}\right. } \end{aligned} \end{aligned}$$
(32)

Theorem 4.2

Suppose that assumptions \((\mathcal {R}_{1})-(\mathcal {R}_{2})\), are satisfied and \(\exists ~~ {\mathfrak {K}}_1,{\mathfrak {K}}_2\in \mathbb {R}^{{+}}\) such that any of the below mentioned condition hold:

$$\begin{aligned} {} \begin{aligned} (\mathcal {S}_{1})~~&\text {if} ~~\mathcal {v(t)}\in {\mathfrak {P}}^{*}\cap \partial {{\mathcal {A}}^{*}_1} ({\mathfrak {K}}_1),~~~ \int _{0}^{1}\bigg [{\mathcal {H}}^{\varsigma ^*}(\mathcal {0}, {\kappa })+\epsilon _{1}(1-\kappa )^{\delta }\bigg ]\\&\quad \phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\phi ^{*}_{\text {min}}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \ge {\mathfrak {K}}_1 \\&\text {and}\\&~~~\mathcal {v(t)}\in {\mathfrak {P}}^{*}\cap \partial {{\mathcal {A}}^{*}_1} ({\mathfrak {K}}_2),~~ \int _{0}^{1}\bigg [\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}+{\mathcal {H}}^{\varsigma ^*}(\mathcal {0}, {\kappa })+\epsilon _{1}(1-\kappa )^{\delta }\bigg ]\\&\quad \phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\phi ^{*}_{\text {max}}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \le {\mathfrak {K}}_2 , \end{aligned} \end{aligned}$$
(33)

or

$$\begin{aligned} {} \begin{aligned} (\mathcal {S}_{2})~~&\text {if}~~~ \mathcal {v(t)}\in {\mathfrak {P}}^{*}\cap \partial {{\mathcal {A}}^{*}_1} ({\mathfrak {K}}_1),~~ \int _{0}^{1}\bigg [\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}+{\mathcal {H}}^{\varsigma ^*}(\mathcal {0}, {\kappa })+\epsilon _{1}(1-\kappa )^{\delta }\bigg ]\\&\quad \phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\phi ^{*}_{\text {max}}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \le {\mathfrak {K}}_1\\ {}&\text {and} \\ {}&\text {if} ~~\mathcal {v(t)}\in {\mathfrak {P}}^{*}\cap \partial {{\mathcal {A}}^{*}_1} ({\mathfrak {K}}_2),~~~ \int _{0}^{1}\bigg [{\mathcal {H}}^{\varsigma ^*}(\mathcal {0}, {\kappa })+\epsilon _{1}(1-\kappa )^{\delta }\bigg ]\\&\quad \phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\phi ^{*}_{\text {min}}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \ge {\mathfrak {K}}_2. \end{aligned} \end{aligned}$$
(34)

Then the given equation (2) has a positive solution .

Proof

Let us consider the case . If then and . Using (32), , we devise

$$\begin{aligned} {} \begin{aligned} \Vert {\mathfrak {F}}^{*} {\mathcal {v}}\Vert&=\underset{{\mathcal {t}}\in [0,1]}{\text {max}}\bigg \{\bigg |\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\int _{t}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\&\quad +\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}, {\kappa })\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\ {}&+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \bigg |\bigg \}\\ {}&\le ~ \int _{0}^{1}\bigg [\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}+{\mathcal {H}}^{\varsigma ^*}({0, \kappa })+\epsilon _{1}(1-\kappa )^{\delta }\bigg ]\phi ^{*}_q\Big [\dfrac{1-\chi }{B(\chi )}\Vert \mathcal {f}(\kappa , \mathcal {v}(\kappa ))\Vert \\ {}&+\dfrac{\chi }{B(\chi )\varGamma (\chi )}\int _{0}^\kappa (\kappa -\xi )^{\chi -1}\Vert {\mathcal {f}}(\xi ,{\mathcal {v}}(\xi ))\Vert {\mathcal {d}} \xi \Big ]{\mathcal {d}}\kappa \\ {}&\le \int _{0}^{1}\bigg [\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}+{\mathcal {H}}^{\varsigma ^*}({0, \kappa })+\epsilon _{1}(1-\kappa )^{\delta }\bigg ]\phi ^{*}_q\Big [\dfrac{1-\chi }{B(\chi )}\Vert \phi ^{*}_{\text {max}}(\xi ,{\mathfrak {K}}_2)\Vert \\&+\dfrac{\chi }{B(\chi )\varGamma (\chi )}\int _{0}^\kappa (\kappa -\xi )^{\chi -1}\Vert \phi ^{*}_{\text {max}}(\xi ,{\mathfrak {K}}_2)\Vert {\mathcal {d}} \xi \Big ]{\mathcal {d}}\kappa \\ {}&\le {\mathfrak {K}}_2 \\ {}&=\Vert {\mathcal {v}}\Vert . \end{aligned} \end{aligned}$$
(35)

If then . Using (32) and . From the Green function property with , we have

$$\begin{aligned} {} \begin{aligned} \Vert {\mathfrak {F}}^{*} {\mathcal {v}}\Vert&=\underset{{\mathcal {t}}\in [0,1]}{\text {max}}\bigg \{\bigg |\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)} \int _{t}^{1}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\&\quad +\int _{0}^{1}{\mathcal {H}}^{ \varsigma ^* }(\mathcal {t}, {\kappa })\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \\ {}&+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi } \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )]{\mathcal {d}}\kappa \bigg |\bigg \}\\ {}&\ge ~ \int _{0}^{1}\bigg [{\mathcal {t}}^{\varsigma ^*-1}{\mathcal {H}}^{\varsigma ^*}({0, \kappa })+\epsilon _{1}(1-\kappa )^{\delta }\bigg ]\phi ^{*}_q\Big [\dfrac{1-\chi }{B(\chi )}\Vert \mathcal {f}(\kappa , \mathcal {v}(\kappa ))\Vert \\&\quad +\dfrac{\chi }{B(\chi )\varGamma (\chi )}\int _{0}^\kappa (\kappa -\xi )^{\chi -1}\Vert {\mathcal {f}}(\xi ,{\mathcal {v}}(\xi )) \Vert {\mathcal {d}} \xi \Big ]{\mathcal {d}}\kappa \\ {}&\ge \int _{0}^{1}\bigg [{\mathcal {H}}^{\varsigma ^*}({0, \kappa })+\epsilon _{1}(1-\kappa )^{\delta }\bigg ]\phi ^{*}_q\Big [\dfrac{1-\chi }{B(\chi )}\Vert \phi ^{*}_{\text {min}}(\kappa ,{\mathfrak {K}}_1) \Vert \\&\quad +\dfrac{\chi }{B(\chi )\varGamma (\chi )}\int _{0}^\kappa (\kappa -\xi )^{\chi -1}\Vert \phi ^{*}_{\text {min}}(\xi ,{\mathfrak {K}}_1)\Vert {\mathcal {d}} \xi \Big ]{\mathcal {d}}\kappa \\ {}&\ge {\mathfrak {K}}_1\\ {}&=\Vert {\mathcal {v}}\Vert . \end{aligned} \end{aligned}$$
(36)

Thus is a fixed point of \({\mathfrak {F}}^{*}\). By using Lemma 3.2 and Theorem 2.3, for and , we have . Therefore is positive solution. \(\square \)

Uniqueness result

Theorem 4.3

Let us consider assumptions \((\mathcal {R}_1)~\text {and}~ (\mathcal {R}_2)\) are satisfied. If

$$\begin{aligned} {} \varDelta ^{*}<1, \end{aligned}$$
(37)

where

$$\begin{aligned} {} \begin{aligned} \varDelta ^{*}=&{\mathcal {L}}(\mathcal {q-1})\varsigma ^{\mathcal {q-2}}\bigg [\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}\bigg (~\dfrac{1-\chi }{B(\chi )}+\dfrac{1}{B(\chi )\varGamma (\chi +1)}\bigg )\\ {}&+\dfrac{\varsigma ^*-1}{\varGamma (\varsigma ^*+1)}\bigg (\dfrac{\varGamma (\varsigma ^*+1)\varGamma (\chi +1)-\varGamma (\chi +1)\varGamma (\varsigma ^*+\chi +1)+\varGamma (\chi )\varGamma (\varsigma ^*+\chi +1)}{B(\chi )B(\varsigma ^*-1)\varGamma (\chi )\varGamma (\varsigma ^*+\chi +1)}\bigg )\\ {}&+\epsilon _{1}\bigg (\dfrac{\delta \varGamma (\delta +1)\varGamma (\chi +1)+\varGamma (\chi +1)\varGamma (\delta +1)-\varGamma (\chi +1)\varGamma (\chi +\delta +2)+\varGamma (\chi )\varGamma (\chi +\delta +2)}{B(\chi )(\delta +1)\varGamma (\chi )\varGamma (\chi +\delta +2)}\bigg )\bigg ], \\ {}&\text {and}\\ {}&\varsigma \ge |{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))\big )d\kappa |. \end{aligned}\nonumber \\ \end{aligned}$$
(38)

Then there exist unique solution for the given equation (2) on (0, 1).

Proof

For \(q\ge 2 \).

For each , we have

but in (37) assumed that \(\varDelta ^{*}<1.\) Therefore \({\mathfrak {F}}^{*}\) is contraction map. Accordingly, by Banach contraction mapping principle that there exist a unique fixed point for operator \({\mathfrak {F}}^{*}\). Hence we conclude that, unique solution exist for (2). \(\square \)

Hyers-Ulam stability

Here, we analysis the HU stability of (2). We define the HU stability as following:

Definition 5.1[28] The integral equation (9) is said to be HU stable if there exists non negative constant \(\varLambda \), for for every \(\gamma ^{*}>0\) satisfying the following:

If,

$$\begin{aligned} {} \begin{aligned}&\big |\mathcal {v(t)}+ ^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]- ^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]\big |_{t=1}\\ {}&-\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}} \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]\big ){\mathcal {d}}\kappa \big |\le \gamma ^{*},\\ {}&\end{aligned} \end{aligned}$$
(39)

then there exist a function \(\upsilon (t)\), which is continuous and satisfying the given below equation:

$$\begin{aligned} {} \begin{aligned} \upsilon ({\mathcal {t}})&= ^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\mathcal {f}(\kappa ,\upsilon (\kappa ))]- ^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]\big |_{t=1}\\ {}&~~~-\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}} \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]\big ){\mathcal {d}}\kappa \end{aligned} \end{aligned}$$
(40)

implies

$$\begin{aligned} {|\mathcal {v(t)}-\upsilon ({\mathcal {t}})|\le \varLambda \gamma ^{*}.} \end{aligned}$$
(41)

Theorem 5.1

The FDE (2) with operator is HU stable for \(q\ge 2\) provided that \((\mathcal {R}_1)\text {and}~ (\mathcal {R}_2)\) are satisfied. satisfies

$$\begin{aligned} \begin{aligned} \big |\mathcal {v(t)}+ ^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]&- ^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]\big |_{t=1}\\ {}&-\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}} \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]\big ){\mathcal {d}}\kappa \big |\le \gamma ^{*}, \end{aligned} \end{aligned}$$
(42)

under the condition , where be the approximate solution (42).

Proof

Let be the solution of (2) and \(\upsilon (t)\) be the approximate solution and satisfying (42). Consider that

$$\begin{aligned} \begin{aligned} \mathcal {v(t)}=- ^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]&+ ^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}}^{\chi }\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]\big |_{t=1}\\&+\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }\phi ^{*}_q[{^{\mathcal{AB}\mathcal{}}\mathcal {I}^{\varsigma ^*}} \big (\mathcal {f}(\kappa , \mathcal {v}(\kappa ))]\big ){\mathcal {d}}\kappa +\mathcal {g(t)} \end{aligned} \end{aligned}$$
(43)

and . We prove that the integral equation (9) is HU stable, with assumptions \((\mathcal {R}_1) \text {and}~ (\mathcal {R}_2).\) we have

(44)

Let

$$\begin{aligned} \begin{aligned} \mathcal {c(t)}={\mathcal {L}}(\mathcal {q-1}){\varsigma }^{q-2}\int _{t}^{1} \bigg [\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}+(t-\kappa )^{\varsigma ^*-1}\bigg ]\Big [\dfrac{1-\chi }{B(\chi )} +\dfrac{\kappa ^\chi }{B(\chi )\varGamma (\chi )}\Big ]{\mathcal {d}}\kappa . \end{aligned} \end{aligned}$$
(45)

By using Gronwall inequality, we get

$$\begin{aligned} {\Vert \mathcal {v}-\upsilon \Vert \le \Vert \mathcal {g(t)}\Vert e^{\mathcal {c(t)}}\le \gamma ^{*} \varLambda .} \end{aligned}$$
(46)

Hence the equation (9) is HU stable. As a result, the FDE (2) is HU stable. \(\square \)

Example

Here, we present some example to illustrate our results.

Example 6.1

Assume that a singular FDEs with laplacian operator:

$$\begin{aligned} {} {\begin{aligned}{\left\{ \begin{array}{ll} ~^{\mathcal {ABC}}_{0}\mathcal {D}^{\chi }\phi ^{*}_{{\mathcal {p}}}\big [~^{\mathcal {ABC}}_{0}\mathcal {D}^{\varsigma ^*}\big (\mathcal {v(t)}\big ] =-{\mathcal {f}}\mathcal {(t, v(t))}, \ \ \ \mathcal {t}\in (0,1),\\ \phi ^{*}_{\mathcal {p}}\big [~^{\mathcal {ABC}}_{0}\mathcal {D}^{\varsigma ^*}\mathcal {v(t)}\big ]\big |_{t=0} =0, \ \ \ {\mathcal {v}}^{'}({{0}})=0, \ \ \ \ {\mathcal {v}}(1)=\epsilon _{1}\int _{0}^{1}(1-\kappa )^{\delta }I^{\chi }\big ({\mathcal {f}}(\kappa , \mathcal {v}(\kappa ))\big ){\mathcal {d}}\kappa , \ \ \end{array}\right. } \end{aligned}} \end{aligned}$$
(47)

where .

, then we have

Assume \({\mathfrak {K}}_1=0.0001\), \({\mathfrak {K}}_2=1\) and and using Theorem 4.2. Then,

$$\begin{aligned} \begin{aligned}&\int _{0}^{1}\bigg [{\mathcal {H}}^{\varsigma ^*}({0, \kappa })+\epsilon _{1}(1-\kappa )^{\delta }\bigg ]\phi ^{*}_q\Big [\dfrac{1-\chi }{B(\chi )}\Vert \phi ^{*}_{\text {min}}(\kappa ,{\mathfrak {K}}_1)\Vert \\ {}&~~+\dfrac{\chi }{B(\chi )\varGamma (\chi )}\int _{0}^\kappa (\kappa -\xi )^{\chi -1}\Vert \phi ^{*}_{\text {min}}(\xi ,{\mathfrak {K}}_1) \Vert {\mathcal {d}} \xi \Big ]{\mathcal {d}}\kappa \\ {}&\ge \int _{0}^{1}\bigg [\dfrac{(1-\kappa )^{0.5}}{\varGamma (1.5)}+\dfrac{1}{20}(1-\kappa )\bigg ]\\&\quad \phi ^{*}_q\Big [\dfrac{0.5}{3}\sqrt{({\mathfrak {K}}_1)}\kappa ^{\frac{1}{4}}+\dfrac{0.5}{3\varGamma (0.5)}\int _{0}^\kappa (\kappa -\xi )^{-0.5} \sqrt{({\mathfrak {K}}_1)}\xi ^{\frac{1}{4}}{\mathcal {d}} \xi \Big ]{\mathcal {d}}\kappa \\ {}&>0.00085982>{\mathfrak {K}}_1. \end{aligned} \end{aligned}$$
(48)

Now, , we have

$$\begin{aligned} \begin{aligned}&\int _{0}^{1}\bigg [\dfrac{2-\varsigma ^*}{B(\varsigma ^*-1)}+{\mathcal {H}}^{\varsigma ^*}({0, \kappa })+\epsilon _{1}(1-\kappa )^{\delta }\bigg ]\phi ^{*}_q\Big [\dfrac{1-\chi }{B(\chi )}\Vert \phi ^{*}_{\text {max}}(\kappa ,{\mathfrak {K}}_2)\Vert \\ {}&+\dfrac{\chi }{B(\chi )\varGamma (\chi )}\int _{0}^\kappa (\kappa -\xi )^{\chi -1}\Vert \phi ^{*}_{\text {max}}(\xi ,{\mathfrak {K}}_2) \Vert {\mathcal {d}} \xi \Big ]{\mathcal {d}}\kappa \\ {}&\le \int _{0}^{1}\bigg [0.5+\dfrac{(1-\kappa )^{0.5}}{\varGamma (1.5)}+\dfrac{1}{20}(1-\kappa )\bigg ]\\&\quad \phi ^{*}_q\Big [\dfrac{0.5}{2}\sqrt{({\mathfrak {K}}_2)}+\dfrac{0.5}{2\varGamma (0.5)}\int _{0}^\kappa (\kappa -\xi )^{-0.5} \sqrt{({\mathfrak {K}}_2)}{\mathcal {d}} \xi \Big ]{\mathcal {d}}\kappa \\ {}&<0.7648988<{\mathfrak {K}}_2=1. \end{aligned} \end{aligned}$$
(49)

Using Theorem 4.2, given equation (47) has a solution which satisfies .

Conclusion

The main goal of this article was to illustrate extensive applications of AB-fractional derivative in various problem. This derivative opened the scope of exploration of the analytical, theoretical, and quantitative behaviour for various problem due to nonlocal kernel. The EU of FDEs is investigated by employed point theorem. The HU stability is also estimated. An example is also proposed to demonstrate the accomplished solution.