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.
Similar content being viewed by others
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
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
whenever\(s \leq \mu \), \(s \leq t\), \(s \leq \gamma _{1} < \cdots < \gamma _{k} <1\),
whenever\(s \leq \mu \), \(s \leq t\), \(\gamma _{1} < \gamma _{2} < \cdots \gamma _{j_{0}-1} \leq s \leq \gamma _{j_{0}} < \cdots < \gamma _{k} <1\),
whenever\(s \geq \mu \), \(s \leq t\), \(\gamma _{1} < \gamma _{2} < \cdots <\gamma _{j_{0}-1} \leq s \leq \gamma _{j_{0}}< \cdots < \gamma _{k} <1\),
whenever\(s \geq \mu \), \(s \leq t\), \(\gamma _{1} < \gamma _{2} < \cdots \gamma _{k} <s<1\),
whenever\(s \geq \mu \), \(s \leq t\), \(\gamma _{1} < \gamma _{2} < \cdots <\gamma _{j_{0}-1} \leq s \leq \gamma _{j_{0}}< \cdots < \gamma _{k} <1\),
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
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
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
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,
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
Hence,
for all \(1 \leq i \leq k\), and so
Since \(x(\mu )=\sum_{i=1}^{k} \lambda _{i} I^{q_{i}}x(\gamma _{i})\), we get
and so
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,
or
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
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
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
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
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
then the singular problem (1) has a solution.
Proof
Let \(x, y \in X\) and \(t \in [0, 1]\). Then we have
For \(1 \leq i \leq m\) and \(t \in [0,1]\), we obtain
and so \(|I^{p_{i}}x(t) | \leq \frac{\|x\|}{p_{i} +1}\). Hence,
Since \(\Delta:= \max \{ 1, \frac{1}{\varGamma ( p_{1} +1)}, \ldots , \frac{1}{\varGamma ( p_{m} +1)} \} \), we get
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
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
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
Thus, we find
and so \(\| F^{n}x- F^{n}y \| \leq \epsilon M_{n}\), where
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
there is \(\epsilon _{0} >0\) such that
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
there is \(\epsilon _{1}> 0\) such that
Let \(\delta _{1} = \delta (\epsilon _{1})\), \(z \in (0, \delta _{1}]\) and \(1 \leq j' \leq m+1\). Then we have
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
Let \(t \in [0,1]\) and \(n \geq n_{0}\). Then
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
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,
and so
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
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,
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
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
In fact, by using a similar method in (4), we have
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,
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
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
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 \(K_{j} \in (t_{j}, t_{j+1})\) (\(j=0,1\)), , ,
\(\|\hat{b}_{i,n} \|= \|\hat{a}_{j',i,n} \| \leq \int _{ \frac{1}{n}}^{1 - \frac{1}{n}} (1-s)^{\frac{1}{2}} \frac{0.1}{1-s} \,ds =0. 2\),
\(N(x_{1}, x_{2}) = |x_{1}|+ |x_{2}|\), \(\eta:= \lim_{z \to 0^{+}} \frac{N(z, z)}{z}=2 \in [0,\infty )\),
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.
References
Ntouyas, S.K., Samei, M.E.: Existence and uniqueness of solutions for multi-term fractional q-integro-differential equations via quantum calculus. Adv. Differ. Equ. 2019, 475 (2019). https://doi.org/10.1186/s13662-019-2414-8
Liang, S., Samei, M.E.: New approach to solutions of a class of singular fractional q-differential problem via quantum calculus. Adv. Differ. Equ. 2020, 14 (2020). https://doi.org/10.1186/s13662-019-2489-2
Samei, M.E.: Existence of solutions for a system of singular sum fractional q-differential equations via quantum calculus. Adv. Differ. Equ. 2020, 23 (2020). https://doi.org/10.1186/s13662-019-2480-y
Samei, M.E., Yang, W.: Existence of solutions for k-dimensional system of multi-term fractional q-integro-differential equations under anti-periodic boundary conditions via quantum calculus. Math. Methods Appl. Sci. 43(7), 4360–4382 (2020). https://doi.org/10.1002/mma.6198
Etemad, S., Rezapour, S., Samei, M.E.: α-ψ-contractions and solutions of a q-fractional differential inclusion with three-point boundary value conditions via computational results. Adv. Differ. Equ. 2020, 218 (2020). https://doi.org/10.1186/s13662-020-02679-w
Rezapour, S., Samei, M.E.: On the existence of solutions for a multi-singular pointwise defined fractional q-integro-differential equation. Bound. Value Probl. 2020, 38 (2020). https://doi.org/10.1186/s13661-020-01342-3
Hedayati, V., Samei, M.E.: Positive solutions of fractional differential equation with two pieces in chain interval and simultaneous Dirichlet boundary conditions. Bound. Value Probl. 2019, 141 (2019). https://doi.org/10.1186/s13661-019-1251-8
Jleli, M., Karapinar, E., Samet, B.: Positive solutions for multi-points boundary value problems for singular fractional differential equations. J. Appl. Math. 2014, Article ID 596123 (2014). https://doi.org/10.1186/s13662-020-02614-z
Aydogan, M.S., Baleanu, D., Mousalou, A., Rezapour, S.: On high order fractional integro-differential equations including the Caputo–Fabrizio derivative. Bound. Value Probl. 2018, 90 (2018). https://doi.org/10.1186/s13661-018-1008-9
Baleanu, D., Rezapour, S., Mohammadi, H.: Some existence results on nonlinear fractional differential equations. Philos. Trans. A Math. Phys. Eng. Sci. 371, 20120144 (2013). https://doi.org/10.1098/rsta.2012.0144
Baleanu, D., Mousalou, A., Rezapour, S.: The extended fractional Caputo–Fabrizio derivative of order \(0 \leq \sigma <1\) on \(c_{\mathbb{r}}[0,1]\) and the existence of solutions for two higher-order series-type differential equations. Adv. Differ. Equ. 2018, 255 (2018). https://doi.org/10.1186/s13662-018-1696-6
Baleanu, D., Mousalou, A., Rezapour, S.: On the existence of solutions for some infinite coefficient-symmetric Caputo–Fabrizio fractional integro-differential equations. Bound. Value Probl. 2017, 145 (2017). https://doi.org/10.1186/s13661-017-0867-9
Baleanu, D., Rezapour, S., Saberpour, Z.: On fractional integro-differential inclusions via the extended fractional Caputo–Fabrizio derivation. Bound. Value Probl. 2019, 79 (2019). https://doi.org/10.1186/s13661-019-1194-0
Kojabad, E.A., Rezapour, S.: Approximate solutions of a sum-type fractional integro-differential equation by using Chebyshev and Legendre polynomials. Adv. Differ. Equ. 2017, 351 (2017). https://doi.org/10.1186/s13662-017-1404-y
Aydogan, S.M., Baleanu, D., Mousalou, A., Rezapour, S.: On approximate solutions for two higher-order Caputo–Fabrizio fractional integro-differential equations. Adv. Differ. Equ. 2017, 221 (2017). https://doi.org/10.1186/s13662-017-1258-3
Baleanu, D., Mousalou, A., Rezapour, S.: A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo–Fabrizio derivative. Adv. Differ. Equ. 2017, 51 (2017). https://doi.org/10.1186/s13662-017-1088-3
Etemad, S., Rezapour, S., Samei, M.E.: On fractional hybrid and non-hybrid multi-term integro-differential inclusions with three-point integral hybrid boundary conditions. Adv. Differ. Equ. 2020, 161 (2020). https://doi.org/10.1186/s13662-020-02627-8
Baleanu, D., Etemad, S., Rezapour, S.: On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators. Alex. Eng. J. (2020). https://doi.org/10.1016/j.aej.2020.04.053
Alizadeh, S., Baleanu, D., Rezapour, S.: Analyzing transient response of the parallel RCL circuit by using the Caputo–Fabrizio fractional derivative. Adv. Differ. Equ. 2020, 55 (2020). https://doi.org/10.1186/s13662-020-2527-0
Baleanu, D., Jajarmi, A., Mohammadi, H., Rezapour, S.: A new study on the mathematical modeling of human liver with Caputo–Fabrizio fractional derivative. Chaos Solitons Fractals 134, 109705 (2020). https://doi.org/10.1016/j.chaos.2020.109705
Baleanu, D., Mohammadi, H., Rezapour, S.: Analysis of the model of HIV-1 infection of CD4+ T-cell with a new approach of fractional derivative. Adv. Differ. Equ. 2020, 71 (2020). https://doi.org/10.1186/s13662-020-02544-w
Baleanu, D., Etemad, S., Rezapour, S.: A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions. Bound. Value Probl. 2020, 64 (2020). https://doi.org/10.1186/s13661-020-01361-0
Baleanu, D., Mohammadi, H., Rezapour, S.: Mathematical theoretical study of a particular system of Caputo–Fabrizio fractional differential equations for the rubella disease model. Adv. Differ. Equ. 2020, 184 (2020). https://doi.org/10.1186/s13662-020-02614-z
Almeida, R., Bastos, B.R.O., Monteiro, M.T.T.: Modeling some real phenomena by fractional differential equations. Math. Methods Appl. Sci. 39(16), 4846–4855 (2016). https://doi.org/10.1002/mma.3818
Baleanu, D., Ghafarnezhad, K., Rezapour, S., Shabibi, M.: On the existence of solutions of a three steps crisis integro-differential equation. Adv. Differ. Equ. 2018, 135 (2018). https://doi.org/10.1186/s13662-018-1583-1
Baleanu, D., Ghafarnezhad, K., Rezapour, S.: On a three steps crisis integro-differential equation. Adv. Differ. Equ. 2019, 153 (2019). https://doi.org/10.1186/s13662-019-2088-2
Shabibi, M., Rezapour, S., Vaezpour, S.M.: A singular fractional integro- differential equation. UPB Sci. Bull. Ser. A 79(1), 109–118 (2017)
Shabibi, M., Postolache, M., Rezapour, S., Vaezpour, S.M.: Investigation of a multi-singular pointwise defined fractional integro-differential equation. J. Math. Anal. 7(5), 61–77 (2016)
Shabibi, M., Postolache, M., Rezapour, S.: Positive solutions for a singular sum fractional differential system. Int. J. Anal. Appl. 13(1), 108–118 (2016)
Talaee, M., Shabibi, M., Gilani, A., Rezapour, S.: On the existence of solutions for a pointwise defined multi-singular integro-differential equation with integral boundary condition. Adv. Differ. Equ. 2020, 41 (2020). https://doi.org/10.1186/s13662-020-2517-2
Gu, Y., Gao, H., Wang, H., Zhang, G.: A general algorithm for evaluating nearly strong-singular (and beyond) integrals in three-dimensional boundary element analysis. Comput. Mech. 59, 779–793 (2017)
Wang, L., Cheng, K., Zhang, B.: A uniqueness result for strong singular Kirchhoff-type fractional Laplacian problems. Appl. Math. Optim. (2019). https://doi.org/10.1007/s00245-019-09612-y
Wei, L., Du, Y.: Positive solutions of elliptic equations with a strong singular potential. Bull. Lond. Math. Soc. 51(2), 251–266 (2019)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Anal. 75, 2154–2165 (2012)
Tatar, N.: An impulsive nonlinear singular version of the Gronwall–Bihari inequality. J. Inequal. Appl. 2006, Article ID 84561 (2006).
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.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-020-02813-8