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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.

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 [16], positive solutions [7, 8], fractional integro-differential equations [913], approximate solutions [1416], hybrid problems [17, 18], and applied modelings [1923]. 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 [2426]) which have been studied by some researchers (see, for example, [2730]). 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 [3133].

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\(B(z, \omega ) = \frac{\varGamma (\omega ) \varGamma (z)}{\varGamma (\omega +z)}\).

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 \(K_{j} \in (t_{j}, t_{j+1})\) (\(j=0,1\)), θ q :=1 i = 1 k λ i γ i q i Γ ( q i + 1 ) =1 ( 1 2 ) 5 2 Γ ( 7 2 ) =1 2 15 2 π , θ q + 1 :=1 i = 1 k λ i γ i q i + 1 Γ ( q i + 2 ) =1 ( 1 2 ) 7 2 Γ ( 9 2 ) =1 2 105 2 π ,

Δ:=max { 1 , 1 Γ ( p 1 + 1 ) , , 1 Γ ( p m + 1 ) } =max { 1 , 1 Γ ( 3 2 ) } = 2 π ,

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

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

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

Δ i = 0 r 1 ( [ 1 Γ ( α ) + j = 1 k λ j θ q Γ ( α + q j ) + 2 ( θ q + 1 μ + 1 ) θ q Γ ( α + 1 ) + 1 θ q Γ ( α ) ] j = 1 m + 1 B j a ˆ j , i , n ) 2 π ( [ 1 Γ ( 3 2 ) + 1 θ q Γ ( 7 2 ) + 2 ( θ q + 1 1 3 + 1 ) θ q Γ ( α + 1 ) + 1 θ q Γ ( α ) ] × 0.2 ) < 1 ,

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.

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.

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Acknowledgements

The second and third authors were supported by Azarbaijan Shahid Madani University. Also, the fourth author was supported by Islamic Azad University, Mehran Branch. The authors express their gratitude dear unknown referees for their helpful suggestions which improved the final version of this paper.

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Baleanu, D., Ghafarnezhad, K., Rezapour, S. et al. On a strong-singular fractional differential equation. Adv Differ Equ 2020, 350 (2020). https://doi.org/10.1186/s13662-020-02813-8

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MSC

  • 34A08
  • 37C25

Keywords

  • Boundary value conditions
  • Integro-differential equation
  • Strong-singularity
  • The fractional Caputo derivative