On a strong-singular fractional differential equation

Abstract

It is important we try to solve complicate differential equations specially strong singular ones. We investigate the existence of solutions for a strong-singular fractional boundary value problem under some conditions. In this way, we provide a new technique for our study. We provide an example to illustrate our main result.

1 Introduction

Fractional arithmetic theory has gained a special place in various sciences. In recent years, numerous works have been published in the field of fractional integro-differential equations such as q-differences [1–6], positive solutions [7, 8], fractional integro-differential equations [9–13], approximate solutions [14–16], hybrid problems [17, 18], and applied modelings [19–23]. It has been showed that one of the best methods for mathematical describing of complicate phenomena is modeling of the problems as singular fractional integro-differential equations (see [24–26]) which have been studied by some researchers (see, for example, [27–30]). Note that most published works on singular fractional equations have studied weak singularities, while it is important we try to review strong singular fractional integro-differential equations. There are a few works on strong singularities [31–33].

In 2014, Jleli et al. studied the existence of a positive solution for the singular fractional boundary value problem $D^{\alpha } u(t)+ f(t, u(t))=0$ with boundary value conditions $u(0) = u'(0) =0$ and $u'(1) = \sum_{i=1}^{m-2} \beta _{i} u'(\xi _{i})$, where $0 < t <1$, $2 < \alpha \leq 3, 0 <\xi _{1} < \cdots< \xi _{m-2} < 1$, $f: (0,1] \times \mathbb{R} \to \mathbb{R}$ is a continuous function, $f(t, x)$ is singular at $t=0$, and $D^{\alpha }$ is the Caputo derivative [8]. In 2016, Shabibi et al. reviewed the multi-singular pointwise defined fractional integro-differential equation $D^{\mu } x(t)+ f(t, x(t), x'(t), D^{\beta }x(t), I^{p}x(t)) =0$ under different boundary conditions, where $\mu \in [2,3)$ or $\mu \in [3,\infty )$, $0\leq t\leq 1$, $x \in C^{1}[0,1]$, $\beta , \xi , \eta \in (0,1)$, $p>1$, $D^{\mu }$ is the Caputo fractional derivative of order μ and $f:[0,1] \times \mathbb{R}^{5} \to \mathbb{R}$ is a function such that $f(t,\cdot,\cdot,\cdot,\cdot)$ is singular at some points $t\in [0,1]$ [28].

In 2018, Baleanu et al. investigated the existence of solutions for the pointwise defined problem $D^{\alpha } x(t)+ f(t , x(t), x'(t), D^{\beta }x(t), \int _{0}^{t} h( \xi ) x(\xi ) \,d\xi , \phi (x(t)))=0$ with boundary value conditions $x(1)=x(0)=x''(0)=x^{n}(0)=0$, where $\alpha \geq 2$, $\lambda , \mu , \beta \in (0,1)$, $\phi: X \rightarrow X$ is a mapping such that $\| \phi (x) - \phi (y)\| \leq \theta _{0} \|x-y\| + \theta _{1} \|x'-y' \|$ for some nonnegative real numbers $\theta _{0}$ and $\theta _{1} \in [0,\infty )$ and all $x,y \in X$, $D^{\alpha }$ is the Caputo fractional derivative of order α, $f(t,x_{1}(t),\ldots, x_{5}(t))=f_{1}(t,x_{1}(t),\ldots, x_{5}(t))$ for all $t\in [0,\lambda )$, $f(t,x_{1}(t),\ldots, x_{5}(t))=f_{2}(t,x_{1}(t),\ldots, x_{5}(t))$ for all $t\in [\lambda ,\mu ]$ and $f(t,x_{1}(t),\ldots, x_{5}(t))=f(t,x_{1}(t),\ldots, x_{5}(t))$ for all $t\in (\mu ,1]$, $f_{1}(t,\cdot,\cdot,\cdot,\cdot)$ and $f_{3}(t,\cdot,\cdot,\cdot,\cdot)$ are continuous on $[0,\lambda )$ and $(\mu ,1]$, and $f_{2}(t,\cdot,\cdot,\cdot,\cdot)$ is multi-singular [25]. They published another work on a three-step crisis integro-differential equation [26]. In 2020, Talaee et al. reviewed the existence of solutions for the fractional differential pointwise defined problem $D^{\alpha } x(t) = f(t, x(t), x'(t), D^{\beta }x(t), \int _{0}^{t} g( \xi ) x(\xi ) \,d\xi )$ with boundary value conditions $x(\mu )=\int _{0}^{1} h(z) x(z) \,dz$ and $x(0)= x^{(j)} (0) = 0$ for $2 \leq j\leq n-1$, where $\alpha \geq 2$, $n = [\alpha ] + 1$, $\mu , \beta \in (0,1)$, $g,h:[0,1] \to \mathbb{R}$ are mappings such that $g, zh \in L^{1}[0,1]$ and $f\in L^{1}$ is singular at some points $[0,1]$ [30].

By using the main idea of these works, we investigate the existence of solutions for the strong singular fractional differential equation

$$\begin{aligned} D^{\alpha } x(t) = f\bigl(t, x(t), I^{p_{1}}x(t), \ldots, I^{p_{m}}x(t) \bigr), \end{aligned}$$

(1)

with some boundary value conditions, where $\alpha \geq 1$, $p_{1},\dots , p_{m}>0$, $m\geq 1$, $D^{\alpha }$ is the fractional Caputo derivative of order α and $f(t,\cdot, \ldots,\cdot)$ is strong singular at some points $[0,1]$.

The Riemann–Liouville integral of order p with the lower limit $a\geq 0$ for a function $f:(a,\infty )\to \mathbb{R}$ is defined by $I^{p}_{a^{+}}f(t)=\frac{1}{\varGamma (p)} \int _{a}^{t} (t-s)^{p-1} f(s)\,ds$ provided that the right-hand side is pointwise define on $(a,\infty )$ [34]. We denote $I^{p}_{0^{+}}f(t)$ by $I^{p}f(t)$. The Caputo fractional derivative of order $\alpha >0$ is defined by ${}^{c}D^{\alpha }f(t)=\frac{1}{\varGamma (n-\alpha )} \int _{0}^{t} \frac{f^{n}(s)}{(t-s)^{\alpha +1-n}}\,ds$, where $n=[\alpha ]+1$ and $f:(a,\infty )\to \mathbb{R}$ is a function [34].

Let Ψ be the family of nondecreasing functions $\psi:[0,\infty ) \to [0,\infty )$ such that $\sum_{n=1}^{\infty } \psi ^{n}(t)<\infty$ for all $t> 0$ [35]. One can check that $\psi (t)< t$ for all $t>0$ [35]. Let $T:X \to X$ and $\alpha:X \times X \to [0,\infty )$ be two maps. Then T is called an α-admissible map whenever $\alpha (x,y) \geq 1$ implies $\alpha (Tx,Ty) \geq 1$ [35]. Let $(X,d)$ be a metric space, $\psi \in \varPsi$, and $\alpha:X \times X \to [0,\infty )$ be a map. A self-map $T:X \to X$ is called an α-ψ-contraction whenever $\alpha (x,y) \,d(Tx,Ty) \leq \psi (d(x,y))$ for all $x,y \in X$ [35]. We need the next results.

Lemma 1

([35])

Let$(X,d)$be a complete metric space, $\psi \in \varPsi$, $\alpha:X \times X \to [0,\infty )$be a map, and$T:X \to X$be anα-admissibleα-ψ-contraction. IfTis continuous and there exists$x_{0} \in X$such that$\alpha (x_{0}, Tx_{0}) \geq 1$, thenThas a fixed point.

Lemma 2

([34])

Let$n-1\leq \alpha < n$and$x\in C(0,1)$. Then$I^{\alpha } D^{\alpha }x(t)=x(t)+ \sum_{i=0}^{n-1} c_{i}t^{i}$for some real constants$c_{0},\dots ,c_{n-1}$.

Lemma 3

([36])

For all$z > 0$and$\omega >-1$, we have$\int ^{t}_{0} (t-s)^{\omega - 1} s^{z} \,ds = B(z + 1, \omega ) t^{ \omega + z}$, where$B(z, \omega ) = \frac{\varGamma (\omega ) \varGamma (z)}{\varGamma (\omega +z)}$.

2 Main results

Now, we are ready for preparing our main results. For the next key result, we use the main idea of [25] to conclude that it is valid on $L^{1}[0,1]$.

Lemma 4

Let$\alpha \geq 1$, $[\alpha ] =n-1$, kbe a natural number, $\mu \in (0,1)$, $\gamma _{1},\dots ,\gamma _{k} \in (0,1)$, $\lambda _{1},\dots ,\lambda _{k} \geq 0$and$q_{1},\dots ,q_{k} >0$be such that$\sum_{i=1}^{k} \frac{\lambda _{i} \gamma _{i}^{q_{i}}}{\varGamma (q_{i} + 1)} <1$and$f\in L^{1} [0,1]$. Then the solution of the problem$D^{\alpha } x(t)=f(t)$with boundary conditions$x^{(2)}(0)=\cdots =x^{(n-1)}(0)=0$, $x(0) = \int _{0}^{1} x(\xi ) \,d\xi$, and$x(\mu ) =\sum_{i=1}^{k} \lambda _{i} I^{q_{i}}x(\gamma _{i})$is$x(t) = \int _{0}^{1} G(t,s) f(s) \,ds$, where the Green function$G(t,s)$is defined by

$$\begin{aligned} G(t,s)= {}& \frac{ (t-s)^{\alpha - 1} }{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j} (\gamma _{j} -s)^{\alpha + q_{j} - 1}}{\theta _{q} \varGamma (\alpha + q_{j})} \\ &{}+ \frac{2( \mu - \theta _{q+1} -t \theta _{q} -1)(1-s)^{\alpha }}{\theta _{q} \varGamma (\alpha +1)} - \frac{(\mu -s)^{\alpha - 1}}{\theta _{q}\varGamma (\alpha )}, \end{aligned}$$

whenever$s \leq \mu$, $s \leq t$, $s \leq \gamma _{1} < \cdots < \gamma _{k} <1$,

$$\begin{aligned} G(t,s)= {}& \frac{ (t-s)^{\alpha - 1} }{\varGamma (\alpha )} + \sum_{j=j_{0}}^{k} \frac{\lambda _{j} (\gamma _{j} -s)^{\alpha + q_{j} - 1}}{\theta _{q} \varGamma (\alpha + q_{j})} \\ &{}+ \frac{2( \mu - \theta _{q+1} -t \theta _{q} -1)(1-s)^{\alpha }}{\theta _{q} \varGamma (\alpha +1)} - \frac{(\mu -s)^{\alpha - 1}}{\theta _{q}\varGamma (\alpha )}, \end{aligned}$$

whenever$s \leq \mu$, $s \leq t$, $\gamma _{1} < \gamma _{2} < \cdots \gamma _{j_{0}-1} \leq s \leq \gamma _{j_{0}} < \cdots < \gamma _{k} <1$,

$$\begin{aligned} G(t,s)= {}& \frac{ (t-s)^{\alpha - 1} }{\varGamma (\alpha )} + \sum_{j=j_{0}}^{k} \frac{\lambda _{j} (\gamma _{j} -s)^{\alpha + q_{j} - 1}}{\theta _{q} \varGamma (\alpha + q_{j})} \\ &{}+ \frac{2( \mu - \theta _{q+1} -t \theta _{q} -1)(1-s)^{\alpha }}{\theta _{q} \varGamma (\alpha +1)}, \end{aligned}$$

whenever$s \geq \mu$, $s \leq t$, $\gamma _{1} < \gamma _{2} < \cdots <\gamma _{j_{0}-1} \leq s \leq \gamma _{j_{0}}< \cdots < \gamma _{k} <1$,

$$\begin{aligned} G(t,s)= \frac{ (t-s)^{\alpha - 1} }{\varGamma (\alpha )} + \frac{2( \mu - \theta _{q+1} -t \theta _{q} -1)(1-s)^{\alpha }}{\theta _{q} \varGamma (\alpha +1)}, \end{aligned}$$

whenever$s \geq \mu$, $s \leq t$, $\gamma _{1} < \gamma _{2} < \cdots \gamma _{k} <s<1$,

$$\begin{aligned} G(t,s)= \frac{ (t-s)^{\alpha - 1} }{\varGamma (\alpha )} + \frac{2( \mu - \theta _{q+1} -t \theta _{q} -1)(1-s)^{\alpha }}{\theta _{q} \varGamma (\alpha +1)}, \end{aligned}$$

whenever$s \geq \mu$, $s \leq t$, $\gamma _{1} < \gamma _{2} < \cdots <\gamma _{j_{0}-1} \leq s \leq \gamma _{j_{0}}< \cdots < \gamma _{k} <1$,

$$\begin{aligned} G(t,s)= \sum_{j=j_{0}}^{k} \frac{\lambda _{j} (\gamma _{j} -s)^{\alpha + q_{j} - 1}}{\theta _{q} \varGamma (\alpha + q_{j})} \frac{2( \mu - \theta _{q+1} -t \theta _{q} -1)(1-s)^{\alpha }}{\theta _{q} \varGamma (\alpha +1)}, \end{aligned}$$

whenever$s \geq \mu$, $s \geq t$, $\gamma _{1} < \gamma _{2} < \cdots <\gamma _{j_{0}-1} \leq s \leq \gamma _{j_{0}}< \cdots < \gamma _{k} <1$, and

$$\begin{aligned} G(t,s)= \frac{2( \mu - \theta _{q+1} -t \theta _{q} -1)(1-s)^{\alpha }}{\theta _{q} \varGamma (\alpha +1)}, \end{aligned}$$

whenever$s \geq \mu$, $s \geq t$, $\gamma _{1} < \gamma _{2} < \cdots < \gamma _{k} < s <1$. Here, $\theta _{q+1}:= 1- \sum_{i=1}^{k} \frac{\lambda _{i} \gamma _{i}^{q_{i}+ 1}}{\varGamma (q_{i} + 2)}$and$\theta _{q}:= 1- \sum_{i=1}^{k} \frac{\lambda _{i} \gamma _{i}^{q_{i}}}{\varGamma (q_{i} + 1)}$.

Proof

Let x be a solution for the problem. By using Lemma 2, we have

$$\begin{aligned} x(t)= \frac{1}{\varGamma (\alpha )} \int ^{t}_{0} (t-s)^{\alpha - 1} f(s) \,ds +c_{0} + c_{1} t + \cdots+ c_{n} t^{n} \end{aligned}$$

for some real constants $c_{0},\dots ,c_{n}$. Since $x^{(2)}(0)=\cdots =x^{(n-1)}(0)=0$, we get $c_{2}= \cdots= c_{n} =0$, and so $x(t)= \frac{1}{\varGamma (\alpha )} \int ^{t}_{0} (t-s)^{\alpha - 1} f(s) \,ds + c_{0} +c_{1} t$. Thus, $x(0)=c_{0}$ and

$$\begin{aligned} \int _{0}^{1} x(t)&= \frac{1}{\varGamma (\alpha )} \int _{0}^{1} \int ^{t}_{0} (t-s)^{\alpha - 1} f(s) \,ds + c_{0} +\frac{c_{1}}{2} \\ &= \frac{1}{\varGamma (\alpha +1)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds + c_{0} +\frac{c_{1}}{2}. \end{aligned}$$

Now, by using the condition $x(0)= \int _{0}^{1} x(\xi ) \,d\xi$, we obtain $\frac{1}{\varGamma (\alpha +1)} \int _{0}^{1} (t-s)^{\alpha } f(s) \,ds + c_{0} +\frac{c_{1}}{2}= c_{0}$, and so $c_{1}= \frac{-2}{\varGamma (\alpha +1)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds$. Hence,

$$\begin{aligned} x(t)= \frac{1}{\varGamma (\alpha )} \int ^{t}_{0} (t-s)^{\alpha - 1} f(s) \,ds +c_{0} -\frac{2t}{\varGamma (\alpha +1)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds, \end{aligned}$$

(2)

and so $x(\mu )= \frac{1}{\varGamma (\alpha )} \int ^{\mu }_{0} (\mu -s)^{ \alpha - 1} f(s) \,ds +c_{0} -\frac{2 \mu }{\varGamma (\alpha +1)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds$. On the other hand, we have $I^{q_{i}}(t)= \frac{1}{\varGamma (q_{i})} \int _{0}^{t} (t-s)^{q_{i} - 1} \,ds = \frac{t^{q_{i}}}{\varGamma (q_{i} + 1)}$ for all $1 \leq i \leq k$. By using Lemma 3, we get $I^{q_{i}}(t)=\frac{1}{\varGamma (q_{i})} \int _{0}^{t} t(t-s)^{q_{i} - 1} \,ds = \frac{1}{\varGamma (q_{i} )} B(2, q_{i}) t^{2+ q_{i} -1} = \frac{1}{\varGamma (q_{i} )}. \frac{\varGamma (2) \varGamma (q_{i})}{\varGamma (2 + q_{i})} t^{ q_{i} +1} = \frac{t^{ q_{i} +1}}{\varGamma (2 + q_{i} )}$. Since $I^{q_{i}} I^{\alpha } f(t) = I^{q_{i} + \alpha } f(t)$, by using (2) we obtain

$$\begin{aligned} I^{q_{i}}x(t) ={}& \frac{1}{\varGamma (\alpha + q_{i})} \int _{0}^{t} (t-s)^{ \alpha + q_{i} - 1} f(s)\,ds + c_{0} \frac{t^{q_{i}}}{\varGamma (q_{i} + 1)} \\ & {} - \frac{2 t^{q_{i} + 1}}{\varGamma (\alpha +1) \varGamma (q_{i} +2)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds. \end{aligned}$$

Hence,

$$\begin{aligned} \lambda _{i} I^{q_{i}}x(\gamma _{i})={}& \frac{\lambda _{i}}{\varGamma (\alpha + q_{i})} \int _{0}^{\gamma _{i} }( \gamma _{i} -s)^{\alpha + q_{i} - 1} f(s)\,ds + \frac{\lambda _{i} c_{0} \gamma _{i}^{q_{i}}}{\varGamma (q_{i} + 1)} \\ &{}- \frac{2 \lambda _{i} \gamma _{i}^{q_{i} + 1}}{\varGamma (\alpha +1) \varGamma (q_{i} +2)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds \end{aligned}$$

for all $1 \leq i \leq k$, and so

$$\begin{aligned} \sum_{i=1}^{k} \lambda _{i} I^{q_{i}}x(\gamma _{i}) ={}&\sum_{i=1}^{k} \frac{\lambda _{i}}{\varGamma (\alpha + q_{i})} \int _{0}^{\gamma _{i} }( \gamma _{i} -s)^{\alpha + q_{i} - 1} f(s)\,ds \\ &{} + c_{0} \sum_{i=1}^{k} \frac{\lambda _{i} \gamma _{i}^{q_{i}}}{\varGamma (q_{i} + 1)}- \frac{2}{\varGamma (\alpha +1)}\sum_{i=1}^{k} \frac{ \lambda _{i} \gamma _{i}^{q_{i} + 1}}{ \varGamma (q_{i} +2)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds. \end{aligned}$$

Since $x(\mu )=\sum_{i=1}^{k} \lambda _{i} I^{q_{i}}x(\gamma _{i})$, we get

$$\begin{aligned} &\frac{1}{\varGamma (\alpha )} \int ^{\mu }_{0} (\mu -s)^{\alpha - 1} f(s) \,ds +c_{0} -\frac{2 \mu }{\varGamma (\alpha +1)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds \\ &\quad =\sum_{i=1}^{k} \frac{\lambda _{i}}{\varGamma (\alpha + q_{i})} \int _{0}^{ \gamma _{i} }(\gamma _{i} -s)^{\alpha + q_{i} - 1} f(s)\,ds + c_{0} \sum_{i=1}^{k} \frac{\lambda _{i} \gamma _{i}^{q_{i}}}{\varGamma (q_{i} + 1)} \\ &\qquad{}- \frac{2}{\varGamma (\alpha +1)}\sum_{i=1}^{k} \frac{ \lambda _{i} \gamma _{i}^{q_{i} + 1}}{ \varGamma (q_{i} +2)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds, \end{aligned}$$

and so

$$\begin{aligned} c_{0} ={}&\sum_{i=1}^{k} \frac{\lambda _{i}}{\theta _{q} \varGamma (\alpha + q_{i})} \int _{0}^{ \gamma _{i} }(\gamma _{i} -s)^{\alpha + q_{i} - 1} f(s)\,ds + \frac{2 \mu }{\varGamma (\alpha +1)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds \\ &{}-\frac{1}{\theta _{q}\varGamma (\alpha )} \int ^{\mu }_{0} (\mu -s)^{ \alpha - 1} f(s) \,ds - \frac{2}{\theta _{q} \varGamma (\alpha +1)}\sum_{i=1}^{k} \frac{ \lambda _{i} \gamma _{i}^{q_{i} + 1}}{ \varGamma (q_{i} +2)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds, \end{aligned}$$

where $\theta _{q}:= 1- \sum_{i=1}^{k} \frac{\lambda _{i} \gamma _{i}^{q_{i}}}{\varGamma (q_{i} + 1)}$ and $\theta _{q+1}:= 1- \sum_{i=1}^{k} \frac{\lambda _{i} \gamma _{i}^{q_{i}+ 1}}{\varGamma (q_{i} + 2)}$. Thus,

$$\begin{aligned} x(t)={} & \frac{1}{\varGamma (\alpha )} \int ^{t}_{0} (t-s)^{\alpha - 1} f(s) \,ds \\ &{}+\sum_{i=1}^{k} \frac{\lambda _{i}}{\theta _{q} \varGamma (\alpha + q_{i})} \int _{0}^{ \gamma _{i} }(\gamma _{i} -s)^{\alpha + q_{i} - 1} f(s)\,ds \\ &{}+ \frac{2 \mu }{\varGamma (\alpha +1)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds \\ &{}-\frac{1}{\theta _{q}\varGamma (\alpha )} \int ^{\mu }_{0} (\mu -s)^{ \alpha - 1} f(s) \,ds \\ &{} - \frac{2}{\theta _{q} \varGamma (\alpha +1)}\sum_{i=1}^{k} \frac{ \lambda _{i} \gamma _{i}^{q_{i} + 1}}{ \varGamma (q_{i} +2)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds \\ &{}- \frac{2t}{\varGamma (\alpha +1)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds \end{aligned}$$

or

$$\begin{aligned} x(t)= {}& \frac{1}{\varGamma (\alpha )} \int ^{t}_{0} (t-s)^{\alpha - 1} f(s) \,ds \\ &{} + \frac{\lambda _{1}}{\theta _{q} \varGamma (\alpha + q_{1})} \int _{0}^{ \gamma _{1} }(\gamma _{1} -s)^{\alpha + q_{1} - 1} f(s)\,ds \\ &{}+ \frac{\lambda _{2}}{\theta _{q} \varGamma (\alpha + q_{2})} \int _{0}^{ \gamma _{2} }(\gamma _{2} -s)^{\alpha + q_{2} - 1} f(s)\,ds \\ &{}+\dots + \frac{\lambda _{k}}{\theta _{q} \varGamma (\alpha + q_{k})} \int _{0}^{\gamma _{k} }(\gamma _{k} -s)^{\alpha + q_{k} - 1} f(s)\,ds \\ &{}+ \frac{2( \mu - \theta _{q+1} -t \theta _{q} -1)}{\theta _{q} \varGamma (\alpha +1)} \int _{0}^{1} (1-s)^{\alpha } f(s) \,ds \\ &{}-\frac{1}{\theta _{q}\varGamma (\alpha )} \int ^{\mu }_{0} (\mu -s)^{ \alpha - 1} f(s) \,ds = \int _{0}^{1} G(t,s) f(s), \end{aligned}$$

where $G(t,s)$ is the Green function. This completes the proof. □

Note that in the last result, it remains only the boundary value conditions $x(0) = \int _{0}^{1} x(\xi ) \,d\xi$ and $x(\mu ) =\sum_{i=1}^{k} \lambda _{i} I^{q_{i}}x(\gamma _{i})$ whenever $1\leq \alpha <2$. It is easy to see

$$\begin{aligned} \bigl\vert G(t,s) \bigr\vert \leq{} & \frac{ (t-s)^{\alpha - 1} }{\varGamma (\alpha )} + \sum _{j=1}^{k} \frac{\lambda _{j} (\gamma _{j} -s)^{\alpha + q_{j} - 1}}{\theta _{q} \varGamma (\alpha + q_{j})} \\ &{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert (1-s)^{\alpha }}{\theta _{q} \varGamma (\alpha +1)} + \frac{(\mu -s)^{\alpha - 1}}{\theta _{q}\varGamma (\alpha )} \end{aligned}$$

and G is continuous with respect to t. Consider the Banach space $X= C[0,1]$ with the sup norm. Let $g:[0,1]\times X^{m+1}\to \mathbb{R}$ be singular at the points $\gamma _{1}< \gamma _{2} <\cdots< \gamma _{k}$ in $[0,1]$. Define the map $F:X \to X$ by

$$\begin{aligned} Fx(t) ={}& \int _{0}^{1} G(t,s) g\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds \\ ={}& \frac{1}{\varGamma (\alpha )} \int _{0}^{t} (t-s)^{\alpha -1} g\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds \\ &{} + \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \int _{0}^{ \gamma _{j}} (\gamma _{j} -s)^{\alpha + q_{j} - 1} g\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds \\ &{}+ \frac{2( \mu - \theta _{q+1} -t \theta _{q} -1)}{\theta _{q} \varGamma (\alpha +1)} \int _{0}^{1} (1-s)^{\alpha } g\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds \\ &{} -\frac{1}{\theta _{q}\varGamma (\alpha )} \int _{0}^{\mu } (\mu -s)^{ \alpha - 1} f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds. \end{aligned}$$

Let $0 = t_{0} < t_{1} < \cdots < t_{r-1} < t_{r} = 1$ and $f(s,\cdot, \ldots,\cdot)$ is singular at each $t_{i}$ for $1 \leq i \leq r$. Put $n_{0} = [ \frac{2}{\min_{0 \leq i \leq r} (t_{i+1} - t_{i})} ] + 1$. For $n \geq n_{0}$, define $F^{n}: X \to X$ by

$$\begin{aligned} F^{n}x(t) ={}& \sum_{i=0}^{r-1} \int _{t_{i} + \frac{1}{n}}^{t_{i+1} - \frac{1}{n}} G(t,s) f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds \\ ={}& \frac{1}{\varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[0,t] \cap [t_{i} + \frac{1}{n}, t_{i+1} - \frac{1}{n}]} (t-s)^{\alpha -1} f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds \\ &{} + \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \\ & {}\times \Biggl( \sum_{i=0}^{r-1} \int _{[0,\gamma _{j} ] \cap [t_{i} + \frac{1}{n}, t_{i+1} - \frac{1}{n}]} (\gamma _{j} -s)^{\alpha + q_{j} - 1} f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds \Biggr) \\ &{}+ \frac{2( \mu - \theta _{q+1} -t \theta _{q} -1)}{\theta _{q} \varGamma (\alpha +1)} \\ & {}\times\sum_{i=0}^{r-1} \int _{[t_{i} + \frac{1}{n}, t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha } f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds \\ & {}- \frac{1}{\theta _{q}\varGamma (\alpha )} \\ &{}\times \sum_{i=0}^{r-1} \int _{[0, \mu ] \cap [t_{i} + \frac{1}{n}, t_{i+1} - \frac{1}{n}]} (\mu -s)^{\alpha - 1} f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds. \end{aligned}$$

(3)

Note that each fixed point of F is a solution for problem (1).

Theorem 5

Assume that$\alpha \geq 1$, $[\alpha ] =n-1$, $r,k,m\geq 1$, $\mu \in (0,1)$, $\gamma _{1},\dots ,\gamma _{k} \in (0,1)$, $\lambda _{1},\dots ,\lambda _{k} \geq 0$, $q_{1},\dots ,q_{k} >0$, $p_{1},\dots ,p_{m} >0$, $a_{1},\dots.a_{m+1}$, and$\varLambda _{1},\dots , \varLambda _{m+1}: \mathbb{R} \to [0, \infty )$are some functions such that$\hat{a}_{i}(t)=(1-t)^{\alpha -1} a_{i}(t) \in L^{1}(K_{j})$for every compact subset$K_{j} \subseteq (t_{j}, t_{j+1})$for$i=1,\dots ,m+1$and$j=1,\dots ,r-1$, $\lim_{z \to 0^{+}} \frac{\varLambda _{i}(z)}{z}=B_{i} \geq 0$, and we have$|f(t, x_{1}, \ldots, x_{m+1}) - f(t, y_{1}, \ldots, y_{m+1})| \leq \sum_{i=1}^{m+1} a_{i}(t) \varLambda _{i}( |x_{i} - y_{i}|)$for all$(x_{1}, \ldots, x_{m+1})$and$(y_{1}, \ldots, y_{m+1})$in$X^{m+1}$and almost all$t \in [0,1]$. Suppose that

$$\begin{aligned} \Delta \sum_{i=0}^{r-1} \Biggl( \Biggl[ \frac{ 1}{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j} }{\theta _{q} \varGamma (\alpha + q_{j})} + \frac{2( \theta _{q+1} - \mu +1)}{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q}\varGamma (\alpha )} \Biggr] \sum _{j^{\prime }=1}^{m+1} B_{j'} \Vert \hat{a}_{j',i,n} \Vert \Biggr) < 1, \end{aligned}$$

where$\|\hat{a}_{j',i,n} \|:= \int _{t_{i} + \frac{1}{n}}^{t_{i+1} - \frac{1}{n}} (1-s)^{\alpha - 1} a_{j'}(s) \,ds$and$\Delta:= \max \{ 1, \frac{1}{\varGamma ( p_{1} +1)}, \ldots , \frac{1}{\varGamma ( p_{m} +1)} \}$. Assume that there are two mapsband$N: X^{m+1} \to [0, \infty )$such that$(1-t)^{\alpha -1} b(t) \in L^{1}(K_{j})$for every compact subset$K_{j} \subseteq (t_{j}, t_{j+1})$for$j=1,\dots ,r-1$andNis nondecreasing with respect to all its components and$\lim_{z \to 0^{+}} \frac{N(z, \ldots, z)}{z}=\eta \geq 0$. Suppose that$|f(t, x_{1}, \ldots, x_{m+1}) | \leq b(t) N( x_{1}, \ldots, x_{m+1})$for all$(x_{1}, \ldots, x_{m+1}) \in X^{m+1}$and almost all$t \in [0,1]$. If

$$\begin{aligned} \eta \sum_{i=0}^{r-1} \Vert \hat{b}_{i,n} \Vert \Biggl[ \frac{ 1}{\varGamma (\alpha )} + \sum _{j=1}^{k} \frac{\lambda _{j} }{\theta _{q} \varGamma (\alpha + q_{j})} + \frac{2( \theta _{q+1} - \mu +1)}{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q}\varGamma (\alpha )} \Biggr] \in \biggl[0, \frac{1}{\Delta }\biggr), \end{aligned}$$

then the singular problem (1) has a solution.

Proof

Let $x, y \in X$ and $t \in [0, 1]$. Then we have

$$\begin{aligned} & \bigl\vert F^{n}x(t)- F^{n}y(t) \bigr\vert \\ &\quad \leq \sum_{i=0}^{r-1} \int _{t_{i} + \frac{1}{n}}^{t_{i+1} - \frac{1}{n}} \bigl\vert G(t,s) \bigr\vert \bigl\vert f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \\ & \qquad{} - f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \bigr\vert \,ds \\ & \quad\leq \Biggl\vert \sum_{i=0}^{r-1} \int _{t_{i} + \frac{1}{n}}^{t_{i+1} - \frac{1}{n}} \Biggr\vert G(t,s)| \bigl[ a_{1}(s) \varLambda _{1}\bigl( \bigl\vert x(s) - y(s) \bigr\vert \bigr) + a_{2}(s) \varLambda _{2}\bigl( \bigl\vert I^{p_{1}}\bigl(x(s) - y(s)\bigr) \bigr\vert \bigr) \\ &\qquad{} + \cdots + a_{m+1}(s) \varLambda _{m+1}\bigl( \bigl\vert I^{p_{m}}\bigl(x(s) - y(s)\bigr) \bigr\vert \bigr) \bigr] \,ds. \end{aligned}$$

For $1 \leq i \leq m$ and $t \in [0,1]$, we obtain

$$\begin{aligned} \bigl\vert I^{p_{i}}x(t) \bigr\vert \leq \frac{1}{p_{i}} \int _{0}^{t} (t-s)^{p_{i} -1} \bigl\vert x(s) \bigr\vert \,ds \leq \frac{ \Vert x \Vert }{p_{i}} \int _{0}^{t} (t-s)^{p_{i} -1} \,ds = \frac{ \Vert x \Vert }{\varGamma ( p_{i} +1)} t^{p_{i}}, \end{aligned}$$

and so $|I^{p_{i}}x(t) | \leq \frac{\|x\|}{p_{i} +1}$. Hence,

$$\begin{aligned} & \bigl\vert F^{n}x(t)- F^{n}y(t) \bigr\vert \\ &\quad \leq \sum_{i=0}^{r-1} \int _{t_{i} + \frac{1}{n}}^{t_{i+1} - \frac{1}{n}} \bigl\vert G(t,s) \bigr\vert \biggl[ a_{1}(s) \varLambda _{1}\bigl( \Vert x - y \Vert \bigr) + a_{2}(s) \varLambda _{2}\biggl(\frac{ \Vert x - y \Vert }{\varGamma ( p_{1} +1)} \biggr) \\ &\qquad{} + \cdots + a_{m+1}(s) \varLambda _{m+1}\biggl( \frac{ \Vert x - y \Vert }{\varGamma ( p_{m} +1)} \biggr) \biggr] \,ds. \end{aligned}$$

Since $\Delta:= \max \{ 1, \frac{1}{\varGamma ( p_{1} +1)}, \ldots , \frac{1}{\varGamma ( p_{m} +1)} \}$, we get

$$\begin{aligned} \bigl\vert F^{n}x(t)- F^{n}y(t) \bigr\vert \leq \sum _{i=0}^{r-1} \Biggl( \int _{t_{i} + \frac{1}{n}}^{t_{i+1} - \frac{1}{n}} \bigl\vert G(t,s) \bigr\vert \Biggl[\sum_{j=1}^{m+1} a_{j}(s) \varLambda _{j}\bigl(\Delta \Vert x - y \Vert \bigr) \Biggr] \,ds \Biggr). \end{aligned}$$

(4)

Let $\epsilon >0$ be given. Since $\lim_{z \to 0^{+}} \frac{\varLambda _{j}(z)}{z}=B_{j}$ for all $1 \leq j \leq m+1$, there exists $\delta (\epsilon ) >0$ such that $|z| \leq \delta '$ implies $|\frac{\varLambda _{j}(z)}{z} - B_{j} | \leq \epsilon$, where $\delta ' \leq \delta (\epsilon )$. Thus, $\varLambda _{j}(z) \leq (\epsilon + B_{j}) z$ for all $|z| \leq \delta '$. Let $\delta '_{0}:= \min \{ \epsilon , \delta (\epsilon ) \}$ and $|z| \leq \delta '_{0}$. Then we have $\varLambda _{j}(z) \leq (\epsilon + B_{j}) z$ for all $1 \leq j \leq m+1$. If $\Delta \|x - y \| \leq \delta '_{0}$, then $\varLambda _{j}(\Delta \|x - y \| ) \leq (\epsilon + B_{j}) \Delta \|x - y \| \leq (\epsilon + B_{j}) \delta '_{0} \leq (\epsilon + B_{j}) \epsilon$ for all $1 \leq j \leq m+1$. Also, $\Delta \|x - y \| \leq \delta '_{0}$ implies $\|x - y \| \leq \frac{\epsilon }{\Delta }$. Let $t \in [0,1]$. By using (4), we conclude that

$$\begin{aligned} &\bigl\vert F^{n}x(t)- F^{n}y(t) \bigr\vert \\ &\quad \leq \sum_{i=0}^{r-1} \Biggl( \int _{t_{i} + \frac{1}{n}}^{t_{i+1} - \frac{1}{n}} \Biggl[ \frac{ (1-s)^{\alpha - 1} }{\varGamma (\alpha )} + \sum _{j=1}^{k} \frac{\lambda _{j} (1 -s)^{\alpha + q_{j} - 1}}{\theta _{q} \varGamma (\alpha + q_{j})} \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert (1-s)^{\alpha }}{\theta _{q} \varGamma (\alpha +1)} + \frac{(1 -s)^{\alpha - 1}}{\theta _{q}\varGamma (\alpha )} \Biggr] \times \Biggl[ \sum _{j^{\prime }=1}^{m+1} a_{j'}(s) (\epsilon + B_{j'}) \Biggr] \,ds\Biggr) \epsilon \\ &\quad \leq \epsilon \sum_{j^{\prime }=1}^{m+1} \epsilon + B_{j'} \Biggl( \sum_{i=0}^{r-1} \Biggl( \Biggl[ \frac{ 1}{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j} }{\theta _{q} \varGamma (\alpha + q_{j})} \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q}\varGamma (\alpha )} \Biggr] \times \biggl[ \int _{t_{i} + \frac{1}{n}}^{t_{i+1} - \frac{1}{n}} (1-s)^{\alpha - 1} a_{j'}(s) \biggr] \,ds\Biggr) \Biggr). \end{aligned}$$

If $\|\hat{a}_{j',i,n} \|:= \int _{t_{i} + \frac{1}{n}}^{t_{i+1} - \frac{1}{n}} (1-s)^{\alpha - 1} a_{j'}(s) \,ds$, then

$$\begin{aligned} \bigl\vert F^{n}x(t)- F^{n}y(t) \bigr\vert \leq{}& \epsilon \sum_{i=0}^{r-1} \Biggl( \Biggl[ \frac{ 1}{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j} }{\theta _{q} \varGamma (\alpha + q_{j})} \\ &{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q}\varGamma (\alpha )} \Biggr] \sum _{j^{\prime }=1}^{m+1} ( \epsilon + B_{j'}) \Vert \hat{a}_{j',i,n} \Vert \Biggr). \end{aligned}$$

Since $1 - \sum_{i=1}^{k} \frac{\lambda _{i} \gamma _{i}^{q_{i} +1}}{ \varGamma ( q_{i} + 2)} > 1 - \sum_{i=1}^{k} \frac{\lambda _{i} \gamma _{i}^{q_{i} }}{ \varGamma ( q_{i} +1)}$, $\theta _{q+1} > \theta _{q} \geq t \theta _{q} >0$ for all $t \in [0,1]$, and so

$$ \sup_{t \in [0,1]} \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert \leq \sup_{t \in [0,1]} \bigl( \vert \mu - 1 \vert + \vert \theta _{q+1} -t \theta _{q} \vert \bigr) = 1 - \mu + \theta _{q+1}. $$

Thus, we find

$$\begin{aligned} \bigl\Vert F^{n}x- F^{n}y \bigr\Vert \leq{}& \epsilon \sum_{i=0}^{r-1} \Biggl( \Biggl[ \frac{ 1}{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j} }{\theta _{q} \varGamma (\alpha + q_{j})} \\ &{}+ \frac{2( \theta _{q+1} - \mu +1)}{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q}\varGamma (\alpha )} \Biggr] \sum _{j^{\prime }=1}^{m+1} ( \epsilon + B_{j'}) \Vert \hat{a}_{j',i,n} \Vert \Biggr), \end{aligned}$$

and so $\| F^{n}x- F^{n}y \| \leq \epsilon M_{n}$, where

$$\begin{aligned} M_{n} ={}& \sum_{i=0}^{r-1} \Biggl( \Biggl[ \frac{ 1}{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j} }{\theta _{q} \varGamma (\alpha + q_{j})} \\ &{}+ \frac{2( \theta _{q+1} - \mu +1)}{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q}\varGamma (\alpha )} \Biggr] \sum _{j^{\prime }=1}^{m+1} ( \epsilon + B_{j'}) \Vert \hat{a}_{j',i,n} \Vert \Biggr). \end{aligned}$$

Since $\epsilon >0$ was arbitrary, $F^{n}x \to F^{n}y$ as $x \to y$ for all $n \geq n_{0}$. Thus, $F^{n}$ is continuous. Since $\lim_{z \to 0^{+}} \frac{N(\Delta z,\ldots,\Delta z)}{\Delta z}=\eta$, there exists $r(\epsilon ) >0$ such that $\frac{N(\Delta z,\ldots,\Delta z)}{\Delta z} \leq \eta + \epsilon$ for all $z \in (0, r(\epsilon )]$, and so $N(\Delta z,\ldots,\Delta z) \leq (\eta + \epsilon ) \Delta z$. Since

$$\begin{aligned} \eta \sum_{i=0}^{r-1} \Vert \hat{b}_{i,n} \Vert \Biggl[ \frac{ 1}{\varGamma (\alpha )} + \sum _{j=1}^{k} \frac{\lambda _{j} }{\theta _{q} \varGamma (\alpha + q_{j})} + \frac{2( \theta _{q+1} - \mu +1)}{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q}\varGamma (\alpha )} \Biggr] \in \biggl[0,\frac{1}{\Delta }\biggr), \end{aligned}$$

there is $\epsilon _{0} >0$ such that

$$\begin{aligned} (\eta + \epsilon _{0}) \sum_{i=0}^{r-1} \Vert \hat{b}_{i,n} \Vert \Biggl[ \frac{ 1}{\varGamma (\alpha )} + \sum _{j=1}^{k} \frac{\lambda _{j} }{\theta _{q} \varGamma (\alpha + q_{j})} + \frac{2( \theta _{q+1} - \mu +1)}{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q}\varGamma (\alpha )} \Biggr] \in \biggl[0,\frac{1}{\Delta }\biggr). \end{aligned}$$

On the other hand, we have $\lim_{z \to 0^{+}} \frac{\varLambda _{j'}(\Delta z)}{\Delta z}= B_{j'} \geq 0$ for all $1 \leq j' \leq m+1$. Let $\epsilon >0$ be given. Choose $\delta (\epsilon ) >0$ such that $\frac{\varLambda _{j'}(\Delta z)}{\Delta z} < B_{j'} + \epsilon$ for all $0 \leq z \leq \delta (\epsilon )$. Hence, $\varLambda _{j'}(\Delta z) < (B_{j'} + \epsilon ) \Delta z$ for $1 \leq j' \leq m+1$. Since

$$\begin{aligned} \Delta \sum_{i=0}^{r-1} \Biggl( \Biggl[ \frac{ 1}{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j} }{\theta _{q} \varGamma (\alpha + q_{j})} + \frac{2( \theta _{q+1} - \mu +1)}{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q}\varGamma (\alpha )} \Biggr] \sum _{j^{\prime }=1}^{m+1} B_{j'} \Vert \hat{a}_{j',i,n} \Vert \Biggr) < 1, \end{aligned}$$

there is $\epsilon _{1}> 0$ such that

$$\begin{aligned} &\Delta \sum_{i=0}^{r-1} \Biggl( \Biggl[ \frac{ 1}{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j} }{\theta _{q} \varGamma (\alpha + q_{j})} + \frac{2( \theta _{q+1} - \mu +1)}{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q}\varGamma (\alpha )} \Biggr] \sum _{j^{\prime }=1}^{m+1} (B_{j'} + \epsilon _{1}) \Vert \hat{a}_{j',i,n} \Vert \Biggr) \\ &\quad < 1. \end{aligned}$$

Let $\delta _{1} = \delta (\epsilon _{1})$, $z \in (0, \delta _{1}]$ and $1 \leq j' \leq m+1$. Then we have

$$\begin{aligned} \varLambda _{j'}(\Delta z) \leq (B_{j'} + \epsilon _{1}) \Delta z. \end{aligned}$$

(5)

If $r_{0} = \min \{ r(\epsilon _{0}), \frac{\delta _{1}}{2} \}$, then $N(\Delta z, \ldots, \Delta z) \leq (\eta + \epsilon _{0}) \Delta z$ for all $z \in (o, r_{0}]$. Specially for $z = r_{0}$, we have $N(\Delta r_{0}, \ldots, \Delta r_{0}) \leq (\eta + \epsilon _{0}) \Delta r_{0}$. Put $C= \{ x \in X: \|x\| \leq r_{0} \}$. Define the map $\alpha: X^{2} \to [0, \infty )$ by $\alpha (x,y) = 1$ whenever $x,y \in C$ and $\alpha (x,y)=0$ elsewhere. If $\alpha (x,y) \geq 1$, then $\|x\| \leq r_{0}$ and $\|y\| \leq r_{0}$, and so

$$\begin{aligned} & \bigl\vert F^{n}x(t) \bigr\vert \\ &\quad \leq \Biggl\vert \sum _{i=0}^{r-1} \int _{t_{i} + \frac{1}{n}}^{t_{i+1} - \frac{1}{n}} G(t,s) f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds \Biggr\vert \\ &\quad \leq \frac{1}{\varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[0,t] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (t-s)^{\alpha -1} \bigl\vert f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \bigr\vert \,ds \\ &\qquad{}+ \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \Biggl( \sum_{i=0}^{r-1} \biggl( \int _{[0,\gamma _{j}] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (\gamma _{j} -s)^{\alpha + q_{j} - 1} \\ &\qquad{} \times \bigl\vert f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \bigr\vert \,ds\biggr) \Biggr) \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha } \\ & \qquad{} \times\bigl\vert f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \bigr\vert \,ds \\ &\qquad{}+ \frac{1}{\theta _{q} \varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[0, \mu ] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (\mu -s)^{ \alpha } \\ & {}\qquad \times\bigl\vert f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \bigr\vert \,ds \\ &\quad \leq \frac{1}{\varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[0,t] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (t-s)^{\alpha -1}b(s) N\bigl(x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds \\ &\qquad{}+ \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \Biggl( \sum_{i=0}^{r-1} \biggl( \int _{[0,\gamma _{j}] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (\gamma _{j} -s)^{\alpha + q_{j} - 1} \\ & \qquad{}\times b(s) N\bigl(x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds\biggr) \Biggr) \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha } \\ & \qquad{}\times b(s) N\bigl(x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds \\ &\qquad{}+ \frac{1}{\theta _{q} \varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[0, \mu ] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (\mu -s)^{ \alpha } \\ & \qquad{}\times b(s) N\bigl(x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \,ds \\ &\quad \leq \frac{1}{\varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[0,t] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (t-s)^{\alpha -1}b(s) N\biggl( \Vert x \Vert , \frac{ \Vert x \Vert }{ \varGamma (p_{1} +1)}, \ldots, \frac{ \Vert x \Vert }{ \varGamma (p_{m} +1)} \biggr) \,ds \\ &\qquad{}+ \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} ( \sum_{i=0}^{r-1} \biggl( \int _{[0,\gamma _{j}] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (\gamma _{j} -s)^{\alpha + q_{j} - 1} \\ & \qquad{}\times b(s) N\biggl( \Vert x \Vert , \frac{ \Vert x \Vert }{ \varGamma (p_{1} +1)}, \ldots, \frac{ \Vert x \Vert }{ \varGamma (p_{m} +1)} \biggr) \,ds\biggr) \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha } \\ &\qquad{}\times b(s) N\biggl( \Vert x \Vert , \frac{ \Vert x \Vert }{ \varGamma (p_{1} +1)}, \ldots, \frac{ \Vert x \Vert }{ \varGamma (p_{m} +1)} \biggr) \,ds \\ &\qquad{}+ \frac{1}{\theta _{q} \varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[0, \mu ] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (\mu -s)^{ \alpha } \\ & \qquad{} b(s) N\biggl( \Vert x \Vert , \frac{ \Vert x \Vert }{ \varGamma (p_{1} +1)}, \ldots, \frac{ \ll\times Vert x \Vert }{ \varGamma (p_{m} +1)} \biggr) \,ds \\ &\quad \leq \frac{1}{\varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[0,t] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (t-s)^{\alpha -1}b(s) N\bigl(\Delta \Vert x \Vert , \ldots, \Delta \Vert x \Vert \bigr) \,ds \\ &\qquad{}+ \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \\ &\qquad{}\times \Biggl( \sum_{i=0}^{r-1} \biggl( \int _{[0,\gamma _{j}] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (\gamma _{j} -s)^{\alpha + q_{j} - 1} b(s) N \bigl(\Delta \Vert x \Vert , \ldots, \Delta \Vert x \Vert \bigr) \,ds \biggr) \Biggr) \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha } b(s) N\bigl(\Delta \Vert x \Vert , \ldots, \Delta \Vert x \Vert \bigr) \,ds \\ &\qquad{}+ \frac{1}{\theta _{q} \varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[0, \mu ] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (\mu -s)^{ \alpha } b(s) N\bigl(\Delta \Vert x \Vert , \ldots, \Delta \Vert x \Vert \bigr) \,ds \\ & \quad\leq \frac{1}{\varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1 -s)^{\alpha -1}b(s) N\bigl( \Delta \Vert x \Vert , \ldots, \Delta \Vert x \Vert \bigr) \,ds \\ &\qquad{}+ \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \Biggl( \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1 -s)^{ \alpha + q_{j} - 1} b(s) N\bigl(\Delta \Vert x \Vert , \ldots, \Delta \Vert x \Vert \bigr) \,ds \Biggr) \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha } b(s) N\bigl(\Delta \Vert x \Vert , \ldots, \Delta \Vert x \Vert \bigr) \,ds \\ &\qquad{}+ \frac{1}{\theta _{q} \varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1 -s)^{\alpha } b(s) N\bigl( \Delta \Vert x \Vert , \ldots, \Delta \Vert x \Vert \bigr) \,ds \\ & \quad \leq \frac{N(\Delta r_{0}, \ldots, \Delta r_{0})}{\varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1 -s)^{\alpha -1}b(s) \,ds \\ &\qquad{}+ N(\Delta r_{0}, \ldots, \Delta r_{0}) \sum _{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \sum _{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1 -s)^{ \alpha - 1} b(s) \,ds \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert N(\Delta r_{0}, \ldots, \Delta r_{0})}{\theta _{q} \varGamma (\alpha +1)} \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha -1} b(s) \,ds \\ &\qquad{}+ \frac{N(\Delta r_{0}, \ldots, \Delta r_{0})}{\theta _{q} \varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1 -s)^{\alpha -1} b(s) \,ds \\ &\quad \leq \frac{(\eta +\epsilon _{0}) \Delta r_{0}}{\varGamma (\alpha )} \sum_{i=0}^{r-1} \Vert \hat{b}_{i,n} \Vert + (\eta +\epsilon _{0}) \Delta r_{0} \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \sum_{i=0}^{r-1} \Vert \hat{b}_{i,n} \Vert \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} (\eta +\epsilon _{0}) \Delta r_{0} \sum_{i=0}^{r-1} \Vert \hat{b}_{i,n} \Vert + \frac{(\eta +\epsilon _{0}) \Delta r_{0}}{\theta _{q} \varGamma (\alpha )} \sum_{i=0}^{r-1} \Vert \hat{b}_{i,n} \Vert \\ &\quad = (\eta +\epsilon _{0}) \Delta \Vert \hat{b}_{i,n} \Vert \Biggl( \frac{1}{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \\ & \qquad{} + \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q} \varGamma (\alpha )} \Biggr) r_{0}\\ & \quad \leq (\eta +\epsilon _{0}) \Delta \Vert \hat{b}_{i,n} \Vert \Biggl( \frac{1}{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \\ &\qquad{} + \frac{2( \vert \mu -1 \vert + \vert \theta _{q+1} -t \theta _{q} \vert ) }{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q} \varGamma (\alpha )} \Biggr) r_{0}. \end{aligned}$$

Let $t \in [0,1]$ and $n \geq n_{0}$. Then

$$\begin{aligned} \bigl\Vert F^{n}x \bigr\Vert \leq{} & (\eta +\epsilon _{0}) \Delta \Vert \hat{b}_{i,n} \Vert \Biggl[ \frac{1}{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \\ & {} + \frac{2( 1- \mu + \theta _{q+1} ) }{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q} \varGamma (\alpha )} \Biggr] r_{0} \leq r_{0}, \end{aligned}$$

and so $F^{n}x \in C$ for $n \geq n_{0}$. By using the same reasons, one can conclude that $F^{n}y \in C$ for $y \in C$. Thus, $\alpha (F^{n}x, F^{n}y) \geq 1$. Since $F^{n}x_{0} \in C$ for $x_{0} \in C$, $\alpha (x_{0}, F^{n}x_{0}) \geq 1$ for all $n \geq n_{0}$. Now, let $x,y \in X$, $t \in [0,1]$ and $n \geq n_{0}$. Then we have

$$\begin{aligned} & \bigl\vert F^{n}x(t) - F^{n}y(t) \bigr\vert \\ &\quad \leq \frac{1}{\varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[0,t] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (t-s)^{\alpha -1} \bigl\vert f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) \\ &\qquad{} - f\bigl(s, y(s), I^{p_{1}}y(s), \ldots, I^{p_{m}}y(s) \bigr) \bigr\vert \,ds \\ &\qquad{}+ \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \sum _{i=0}^{r-1} \biggl( \int _{[0,\gamma _{j}] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (\gamma _{j} -s)^{\alpha + q_{j} - 1} \\ & \qquad{} \times\bigl\vert f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) - f\bigl(s, y(s), I^{p_{1}}y(s), \ldots, I^{p_{m}}y(s) \bigr) \bigr\vert \,ds\biggr) \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha } \\ &\qquad{} \times\bigl\vert f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) - f\bigl(s, y(s), I^{p_{1}}y(s), \ldots, I^{p_{m}}y(s) \bigr) \bigr\vert \,ds \\ &\qquad{}+ \frac{1}{\theta _{q} \varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[0, \mu ] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (\mu -s)^{ \alpha } \\ & \qquad{} \times\bigl\vert f\bigl(s, x(s), I^{p_{1}}x(s), \ldots, I^{p_{m}}x(s) \bigr) - f\bigl(s, y(s), I^{p_{1}}y(s), \ldots, I^{p_{m}}y(s) \bigr) \bigr\vert \,ds \\ & \quad\leq \frac{1}{\varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[0,t] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (t-s)^{\alpha -1} \bigl[a_{1}(s) \varLambda _{1}\bigl( \bigl\vert x(s) - y(s) \bigr\vert \bigr) \\ & \qquad{}+ a_{2}(s) \varLambda _{2}\bigl( \bigl\vert I^{p_{1}}\bigl(x(s) - y(s) \bigr) \bigr\vert \bigr) + \cdots+ a_{m+1}(s) \varLambda _{m+1}\bigl( \bigl\vert I^{p_{m}}\bigl(x(s) - y(s) \bigr) \bigr\vert \bigr) \bigr] \,ds \\ &\qquad{}+ \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \sum _{i=0}^{r-1} \int _{[0,\gamma _{j}] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (\gamma _{j} -s)^{\alpha + q_{j} - 1} \bigl[a_{1}(s) \varLambda _{1}\bigl( \bigl\vert x(s) - y(s) \bigr\vert \bigr) \\ & \qquad{}+ a_{2}(s) \varLambda _{2}\bigl( \bigl\vert I^{p_{1}}\bigl(x(s) - y(s) \bigr) \bigr\vert \bigr) + \cdots+ a_{m+1}(s) \varLambda _{m+1}\bigl( \bigl\vert I^{p_{m}}\bigl(x(s) - y(s) \bigr) \bigr\vert \bigr) \bigr] \,ds \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha } \bigl[a_{1}(s) \varLambda _{1}\bigl( \bigl\vert x(s) - y(s) \bigr\vert \bigr) \\ & \qquad{}+ a_{2}(s) \varLambda _{2}\bigl( \bigl\vert I^{p_{1}}\bigl(x(s) - y(s) \bigr) \bigr\vert \bigr) + \cdots+ a_{m+1}(s) \varLambda _{m+1}\bigl( \bigl\vert I^{p_{m}}\bigl(x(s) - y(s) \bigr) \bigr\vert \bigr) \bigr] \,ds \\ &\qquad{}+ \frac{1}{\theta _{q} \varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{[0, \mu ] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (\mu -s)^{ \alpha } \bigl[a_{1}(s) \varLambda _{1}\bigl( \bigl\vert x(s) - y(s) \bigr\vert \bigr) \\ & \qquad{}+ a_{2}(s) \varLambda _{2}\bigl( \bigl\vert I^{p_{1}}\bigl(x(s) - y(s) \bigr) \bigr\vert \bigr) + \cdots+ a_{m+1}(s) \varLambda _{m+1}\bigl( \bigl\vert I^{p_{m}}\bigl(x(s) - y(s) \bigr) \bigr\vert \bigr) \bigr] \,ds \\ & \quad\leq \frac{1}{\varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha -1} \biggl[a_{1}(s) \varLambda _{1}\bigl( \Vert x - y \Vert \bigr) \\ & \qquad{}+ a_{2}(s) \varLambda _{2}\biggl( \frac{ \Vert x - y \Vert }{\varGamma (p_{1} +1)} \biggr) + \cdots+ a_{m+1}(s) \varLambda _{2}\biggl( \frac{ \Vert x - y \Vert }{\varGamma (p_{m} +1)}\biggr) \biggr] \,ds \\ &\qquad{}+ \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \sum _{i=0}^{r-1} \int _{[0,\gamma _{j}] \cap [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1 -s)^{\alpha + q_{j} - 1} \biggl[a_{1}(s) \varLambda _{1}\bigl( \Vert x - y \Vert \bigr) \\ & \qquad{}+ a_{2}(s) \varLambda _{2}\biggl( \frac{ \Vert x - y \Vert }{\varGamma (p_{1} +1)} \biggr) + \cdots+ a_{m+1}(s) \varLambda _{2}\biggl( \frac{ \Vert x - y \Vert }{\varGamma (p_{m} +1)}\biggr) \biggr] \,ds \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha } \biggl[a_{1}(s) \varLambda _{1}\bigl( \Vert x - y \Vert \bigr) \\ &\qquad{} + a_{2}(s) \varLambda _{2}\biggl( \frac{ \Vert x - y \Vert }{\varGamma (p_{1} +1)} \biggr) + \cdots+ a_{m+1}(s) \varLambda _{2}\biggl( \frac{ \Vert x - y \Vert }{\varGamma (p_{m} +1)}\biggr) \biggr] \,ds \\ &\qquad{}+ \frac{1}{\theta _{q} \varGamma (\alpha )} \sum_{i=0}^{r-1} \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1 -s)^{\alpha } \biggl[a_{1}(s) \varLambda _{1}\bigl( \Vert x - y \Vert \bigr) \\ & \qquad{}+ a_{2}(s) \varLambda _{2}\biggl( \frac{ \Vert x - y \Vert }{\varGamma (p_{1} +1)} \biggr) + \cdots+ a_{m+1}(s) \varLambda _{2}\biggl( \frac{ \Vert x - y \Vert }{\varGamma (p_{m} +1)}\biggr) \biggr] \,ds \\ &\quad \leq \frac{1}{\varGamma (\alpha )} \sum_{i=0}^{r-1} \Biggl[\sum_{j=1}^{m+1} \varLambda _{j}\bigl(\Delta \Vert x - y \Vert \bigr) \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha -1} a_{j}(s) \,ds \Biggr] \\ &\qquad{}+ \Biggl(\sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \Biggr) \sum_{i=0}^{r-1} \Biggl[\sum_{j=1}^{m+1} \varLambda _{j}\bigl(\Delta \Vert x - y \Vert \bigr) \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha -1} a_{j}(s) \,ds \Biggr] \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} \sum_{i=0}^{r-1} \Biggl[\sum_{j=1}^{m+1} \varLambda _{j}\bigl(\Delta \Vert x - y \Vert \bigr) \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha -1} a_{j}(s) \,ds \Biggr] \\ &\qquad{}+ \frac{1}{\theta _{q} \varGamma (\alpha )} \sum_{i=0}^{r-1} \Biggl[\sum_{j=1}^{m+1} \varLambda _{j}\bigl(\Delta \Vert x - y \Vert \bigr) \int _{ [t_{i} + \frac{1}{n} , t_{i+1} - \frac{1}{n}]} (1-s)^{\alpha -1} a_{j}(s) \,ds \Biggr]. \end{aligned}$$

If $x, y \notin C$, then $\alpha (x, y) = 0$, and so $\alpha (x, y) \,d(F^{n}x, F^{n}y) =0 \leq d(x,y)$ for $x, y \notin C$. Hence, $\|x- y\| \leq 2r_{0} \leq 2 \frac{\delta _{1}}{2} = \delta _{1}$. Now, by using (5), $\varLambda _{j} (\Delta \|x - y \|) \leq (B_{j} + \epsilon _{1}) \Delta \|x - y \|$. Thus,

$$\begin{aligned} & \bigl\vert F^{n}x(t) - F^{n}y(t) \bigr\vert \\ &\quad \leq \frac{1}{\varGamma (\alpha )} \sum_{i=0}^{r-1} \Biggl[ \sum_{j=1}^{m+1} (B_{j} + \epsilon _{1}) \Delta \Vert x - y \Vert \Vert \hat{a}_{j,i,n} \Vert \Biggr] \\ &\qquad{}+ \Biggl(\sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} \Biggr) \sum_{i=0}^{r-1} \Biggl[\sum_{j=1}^{m+1} (B_{j} + \epsilon _{1}) \Delta \Vert x - y \Vert \Vert \hat{a}_{j,i,n} \Vert \Biggr] \\ &\qquad{}+ \frac{2 \vert \mu - \theta _{q+1} -t \theta _{q} -1 \vert }{\theta _{q} \varGamma (\alpha +1)} \sum_{i=0}^{r-1} \Biggl[\sum_{j=1}^{m+1} (B_{j} + \epsilon _{1}) \Delta \Vert x - y \Vert \Vert \hat{a}_{j,i,n} \Vert \Biggr] \\ &\qquad{}+ \frac{1}{\theta _{q} \varGamma (\alpha )} \sum_{i=0}^{r-1} \Biggl[\sum_{j=1}^{m+1} (B_{j} + \epsilon _{1}) \Delta \Vert x - y \Vert \Vert \hat{a}_{j,i,n} \Vert \Biggr], \end{aligned}$$

and so

$$\begin{aligned} & \bigl\Vert F^{n}x - F^{n}y \bigr\Vert \\ &\quad \leq \Delta \Biggl( \frac{1}{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} + \frac{2(1- \mu + \theta _{q+1}) }{\theta _{q} \varGamma (\alpha +1)} \frac{1}{\theta _{q} \varGamma (\alpha )} \Biggr) \\ & \qquad{}\times \Biggl( \sum_{j=1}^{m+1} \Biggl[ (B_{j} + \epsilon _{1}) \sum_{i=0}^{r-1} \Vert \hat{a}_{j,i,n} \Vert \Biggr] \Biggr) \Vert x - y \Vert = \lambda \Vert x - y \Vert , \end{aligned}$$

where $\lambda:= \Delta ( \frac{1}{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} + \frac{2(1- \mu + \theta _{q+1}) }{\theta _{q} \varGamma (\alpha +1)} \frac{1}{\theta _{q} \varGamma (\alpha )} ) \times ( \sum_{j=1}^{m+1} [ (B_{j} + \epsilon _{1}) \sum_{i=0}^{r-1} \| \hat{a}_{j,i,n}\| ] )$. Hence, $\| F^{n}x - F^{n}y \| \leq \lambda \|x - y \|$ for all $x,y \in X$. Now consider the map $\psi: [0, \infty ) \to [0, \infty )$ defined by $\psi (t) = \lambda t$. Then we have $\sum_{i=1}^{\infty } \psi ^{i} (t) =\lambda t + \lambda ^{2} t + \cdots = \frac{\lambda }{1 - \lambda } t < \infty$ for all $t \in [0, \infty )$. Thus, $\alpha (x,y) \,d(F^{n}x, F^{n}y) \leq \psi ( d(x,y) )$ for all $x,y \in X$. Now, by using Lemma 1, we conclude that $F^{n}$ has a fixed point $x_{n}$ for each $n \geq n_{0}$, that is, $x_{n}(t) = F^{n}{x_{n}}(t)$ for all $t \in [0,1]$. Here, the map $F^{n}$ is defined by (3). Let $\{x_{n} \}$ be a sequence of the fixed points. By using the proof, $\{x_{n} \} \subset C$ and so $\{x_{n} \}$ is bounded in X. In fact, we have

$$ x_{n}(t) = \int _{ [0,1]\backslash \{ \bigcup _{i=0}^{r-1} [t_{i} + \frac{1}{n}, t_{i+1} -\frac{1}{n} ] \} } G(t,s) f\bigl(s, x_{n}(s), I^{p_{1}}x_{n}(s), \dots , I^{p_{m}}x_{n}(s) \bigr) \,ds $$

for all $t \in [0,1]$. Note that G is continuous with respect to t on $[0,1]\backslash \{ \bigcup_{i=0}^{r-1} [t_{i} +\frac{1}{n}, t_{i+1} - \frac{1}{n} ] \}$ as well as the maps $\frac{\partial G}{\partial t}, \ldots, \frac{\partial ^{[\alpha ]+1} G}{\partial t^{[\alpha ] +1}}$. Hence,

$$\begin{aligned} &\lim_{t_{k} \to t} \frac{\partial ^{m} x_{n}(t_{k})}{\partial t_{k}^{m}} \\ &\quad = \lim_{t_{k} \to t} \int _{ [0,1]\backslash \{ \bigcup _{i=0}^{r-1} [t_{i} +\frac{1}{n}, t_{i+1} -\frac{1}{n} ] \} } \frac{\partial ^{m} G}{\partial t_{k}^{m}}(t_{k},s) f\bigl(s, x_{n}(s), I^{p_{1}}x_{n}(s), \ldots, I^{p_{m}}x_{n}(s) \bigr) \,ds \\ &\quad = \int _{ [0,1]\backslash \{ \bigcup _{i=0}^{r-1} [t_{i} +\frac{1}{n}, t_{i+1} -\frac{1}{n} ] \} } \frac{\partial ^{m} G}{\partial t^{m}}(t,s) f\bigl(s, x_{n}(s), I^{p_{1}}x_{n}(s), \ldots, I^{p_{m}}x_{n}(s) \bigr) \,ds = \frac{\partial ^{m} x_{n}(t)}{\partial t^{m}} \end{aligned}$$

for all $1 \leq m \leq [\alpha ]+1$ and $t \in [0,1]\backslash \{ \bigcup_{i=0}^{r-1} [t_{i} +\frac{1}{n}, t_{i+1} -\frac{1}{n} ] \}$. Thus, the fixed points $x_{n}$ belong to the space $X^{\alpha }= \{ x: D^{\alpha }x \in C[0,1] \}$. This implies that the sequence $\{x'_{n} \}$ is equi-continuous, and so $\{x_{n} \}$ is relatively compact in X. Now, by using the Arzela–Ascoli theorem, there exists $x_{0} \in X$ such that $\lim_{n \to \infty } x_{n} = x_{0}$. One can check that $x_{0}$ satisfies the boundary value conditions of problem (1). Since $x_{n} \in C$ for all n, we have

$$\begin{aligned} &\chi _{ [0,1]\backslash \{ \bigcup _{i=0}^{r-1} [t_{i} +\frac{1}{n}, t_{i+1} -\frac{1}{n} ] \}}(s) \bigl\vert G(t,s) f\bigl(s, x_{n}(s), I^{p_{1}}x_{n}(s), \ldots, I^{p_{m}}x_{n}(s) \bigr) \bigr\vert \\ &\quad \leq (\eta +\epsilon _{0}) \Delta \Biggl[ \frac{1}{\varGamma (\alpha )} + \sum _{j=1}^{k} \frac{\lambda _{j}}{\theta _{q} \varGamma (\alpha + q_{j})} + \frac{2( 1- \mu + \theta _{q+1} ) }{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q} \varGamma (\alpha )} \Biggr] r_{0} \\ &\qquad{}\times\chi _{ [0,1]\backslash \{ \bigcup _{i=0}^{r-1} [t_{i} +\frac{1}{n}, t_{i+1} -\frac{1}{n} ] \}}(s) (1-s)^{\alpha - 1} b(s), \end{aligned}$$

where $\chi _{E}(s) = 1$ whenever $s \in E$ and $\chi _{E}(s) = 0$ whenever $s \notin E$. Note that the map $(1-s)^{\alpha - 1} b(s)$ belongs to $L^{1}(K_{j})$ for every compact subset $K_{j} \subseteq (t_{j}, t_{j+1})$ for $j=1,\dots ,r-1$, and so $\chi _{ [0,1]\backslash \{ \bigcup _{i=0}^{r-1} [t_{i} +\frac{1}{n}, t_{i+1} -\frac{1}{n} ] \}}(s)(1-s)^{\alpha - 1} b(s) \in L^{1}[0,1]$. Now, by using the Lebesgue dominated theorem, we conclude that

$$\begin{aligned} x_{0}(t) ={}& \lim_{n \to \infty } x_{n}(t) \\ ={}& \lim_{n \to \infty } \int _{0}^{1} \chi _{ [0,1]\backslash \{ \bigcup _{i=0}^{r-1} [t_{i} +\frac{1}{n}, t_{i+1} -\frac{1}{n} ] \}}(s) G(t,s) f \bigl(s, x_{n}(s), I^{p_{1}}x_{n}(s),\dots , I^{p_{m}}x_{n}(s) \bigr) \,ds \\ ={}& \int _{0}^{1} G(t,s) f\bigl(s, x_{0}(s), I^{p_{1}}x_{0}(s),\dots , I^{p_{m}}x_{0}(s) \bigr) \,ds = Fx_{0}(t). \end{aligned}$$

In fact, by using a similar method in (4), we have

$$\begin{aligned} &\bigl\vert f\bigl(s, x_{n}(s), I^{p_{1}}x_{n}(s), \dots , I^{p_{m}}x_{n}(s) \bigr)- f\bigl(s, x_{0}(s), I^{p_{1}}x_{0}(s),\dots , I^{p_{m}}x_{0}(s) \bigr) \bigr\vert \\ &\quad \leq \sum_{j=1}^{m+1} a_{j}(s) \varLambda _{j}\bigl(\Delta \Vert x_{n} - x \_0 \Vert \bigr). \end{aligned}$$

Let $\epsilon >0$ be given. Choose $\delta (\epsilon ) >0$ such that $\varLambda _{j}(\Delta \|x_{n} - x_{0}\|) \leq (\epsilon + B_{j}) \epsilon$ for all $n\geq n_{0}$ with $\|x_{n} - x_{0} \| <\delta (\epsilon )$. Hence,

$$\begin{aligned} &\bigl\vert f\bigl(s, x_{n}(s), I^{p_{1}}x_{n}(s), \dots , I^{p_{m}}x_{n}(s) \bigr)- f\bigl(s, x_{0}(s), I^{p_{1}}x_{0}(s),\dots , I^{p_{m}}x_{0}(s) \bigr) \bigr\vert \\ &\quad \leq \epsilon \sum_{j=1}^{m+1} (\epsilon + B_{j}) a_{j}(s), \end{aligned}$$

and so $f(s, x_{n}(s), I^{p_{1}}x_{n}(s),\dots , I^{p_{m}}x_{n}(s) ) \to f(s, x_{0}(s), I^{p_{1}}x_{0}(s),\dots , I^{p_{m}}x_{0}(s) )$ as $x_{n} \to x_{0}$. This implies that F has the fixed point $x_{0}$ which is a solution for problem (1). □

Now, we provide an example to illustrate our main result.

Example 1

Consider the strong singular problem

$$\begin{aligned} D^{\frac{3}{2}} x(t) = f\bigl(t, x(t), I^{\frac{1}{2}}x(t) \bigr), \end{aligned}$$

(6)

with boundary conditions $x(0)=\int _{0}^{1} x(\xi ) \,d\xi$ and $x(\frac{1}{3} ) = I^{\frac{5}{2} }x(\frac{1}{2})$, where

$$ f(t, x_{1}, x_{2}) = \frac{0.1}{(1-t)} \bigl( \vert x_{1} \vert + \vert x_{2} \vert \bigr). $$

Put $m=1$, $k=1$, $t_{0}= 0$, $t_{1}= 1$, $\mu = \frac{1}{3}$, $\lambda _{1}= 1$, $q_{1}= \frac{5}{2}$, $\gamma _{1} = \frac{1}{2}$, $\varLambda _{1}(x)=\varLambda _{2}(x)=x$, $a_{1}(t)=a_{2}(t)=b(t)= \frac{0.1}{1-t}$, and $N(x_{1}, x_{2})= |x_{1}| +|x_{2}|$. Then $B_{1}=B_{2}=1$, where $B_{i}= \lim_{z \to 0^{+}} \frac{\varLambda _{i}(z)}{z}$. Note that $(1-t)^{\alpha -1} a_{i}(t) \in L^{1}(K_{j})$ for all compact subsets $K_{j} \in (t_{j}, t_{j+1})$ ($j=0,1$), ${\theta }_q:=1-{\sum }_{i=1}^k\frac{{\lambda }_i{\gamma }_i^{q_i}}{\Gamma (q_i+1)}=1-\frac{{(\frac{1}{2})}^{\frac{5}{2}}}{\Gamma (\frac{7}{2})}=1-\frac{2}{15\sqrt{2\pi }}$, ${\theta }_{q+1}:=1-{\sum }_{i=1}^k\frac{{\lambda }_i{\gamma }_i^{q_i+1}}{\Gamma (q_i+2)}=1-\frac{{(\frac{1}{2})}^{\frac{7}{2}}}{\Gamma (\frac{9}{2})}=1-\frac{2}{105\sqrt{2\pi }}$,

$$\mathrm{\Delta }:=max\{1,\frac{1}{\Gamma (p_1+1)},\dots ,\frac{1}{\Gamma (p_m+1)}\}=max\{1,\frac{1}{\Gamma (\frac{3}{2})}\}=\frac{2}{\sqrt{\pi }},$$

$\|\hat{b}_{i,n} \|= \|\hat{a}_{j',i,n} \| \leq \int _{ \frac{1}{n}}^{1 - \frac{1}{n}} (1-s)^{\frac{1}{2}} \frac{0.1}{1-s} \,ds =0. 2$,

$$ \bigl\vert f(t, x_{1}, \ldots, x_{m+1}) \bigr\vert \leq b(t) N( x_{1}, \ldots, x_{m+1}), $$

$N(x_{1}, x_{2}) = |x_{1}|+ |x_{2}|$, $\eta:= \lim_{z \to 0^{+}} \frac{N(z, z)}{z}=2 \in [0,\infty )$,

$$\begin{array}{rl}& \mathrm{\Delta }\sum _{i=0}^{r-1}\left(\right[\frac{1}{\Gamma (\alpha )}+\sum _{j=1}^k\frac{{\lambda }_j}{{\theta }_q\Gamma (\alpha +q_j)}+\frac{2({\theta }_{q+1}-\mu +1)}{{\theta }_q\Gamma (\alpha +1)}\\ & +\frac{1}{{\theta }_q\Gamma (\alpha )}]\sum _{j^{\prime }=1}^{m+1}{B}_{j^{\prime }}\parallel {\hat{a}}_{j^{\prime },i,n}\parallel )\frac{2}{\sqrt{\pi }}\left(\right[\frac{1}{\Gamma (\frac{3}{2})}+\frac{1}{{\theta }_q\Gamma (\frac{7}{2})}\\ & +\frac{2({\theta }_{q+1}-\frac{1}{3}+1)}{{\theta }_q\Gamma (\alpha +1)}+\frac{1}{{\theta }_q\Gamma (\alpha )}]\times 0.2)<1,\end{array}$$

and $\eta \sum_{i=0}^{r-1} \| \hat{b}_{i,n} \| [ \frac{ 1}{\varGamma (\alpha )} + \sum_{j=1}^{k} \frac{\lambda _{j} }{\theta _{q} \varGamma (\alpha + q_{j})} + \frac{2( \theta _{q+1} - \mu +1)}{\theta _{q} \varGamma (\alpha +1)} + \frac{1}{\theta _{q}\varGamma (\alpha )} ] \in [0,\frac{1}{\Delta })$. Now, by using Theorem (5), we conclude that problem (6) has a solution.

3 Conclusion

There are some phenomena that can be modeled by fractional differential equations. But most singular fractional differential equations studied by researchers have simple singularity. In this work, by providing a new technique we review a strong singular fractional differential equation under some boundary value conditions.